0198529716 Accurate Clock Pendulums

Accurate Clock Pendulums

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Accurate Clock Pendulums

Robert James Matthys

1

3Great Clarendon Street, Oxford OX2 6DP

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CONTENTS

Preface viiIntroduction ix

Part I: General 11. Better accuracy from simple pendulums 32. A short history of temperature compensation 73. Scaling the size of a pendulum 134. Finding a pendulum’s axis of rotation 195. Does a pendulum’s axis of rotation shift

with amplitude? 216. Some practical properties of quartz 237. Putting Q in perspective 278. The Allan variance and the rms time error 379. Transient temperature effects in a pendulum 43

10. Transient response of a pendulum to temperature change 47

11. Dimensional stability of pendulum materials 5712. Variations on a Riefler bob shape 7913. Bob shape 8714. Rate adjustment mechanisms 91

Part II: Suspension spring 9515. Spring suspensions for accurate pendulums 9716. James’ suspension spring equations 12117. Barometric compensation with a crossed

spring suspension? 12718. Solid one-piece suspension springs 13119. Stable connections to a pendulum’s suspension spring 13920. Stability of suspension spring materials 143

Part III: Pendulum rod 15321. Pendulum rod materials 15522. The heat treatment of invar 15923. The instability of invar 16324. Position sensitivity along the pendulum rod 16725. Fasteners for quartz pendulum rods 17126. Effect of the pendulum rod on Q 179

Part IV: Air and clock case effects 18327. Correcting the pendulum’s air pressure error 18528. Pendulum air movement: A failed experiment 19129. Pendulum air movement: A second try 19530. Time error due to air pressure variations 20531. Effect of the clock case walls on a pendulum 211

Part V: Electronics 22132. An electronically driven pendulum 22333. Sinusoidal drive of a pendulum 22734. Photoelectronics for pendulums 24135. Check your clock against WWV 24936. Electronic correction for air pressure variations 255

Conversion Table 261Index 263

Contents

vi

Preface

Almost all of the accuracy of a pendulum clock resides in its pendulum.Hence this book is about pendulums and about how to design anaccurate one.

This book is aimed at those people who want to make their pendu-lum clock run more accurately, and at those who want to make anaccurate one. In simple terms and with very little math, the bookdescribes many scientific aspects of pendulum design and operation,backed up by experimental data. One chapter covers improving theaccuracy of the ordinary run-of-the-mill pendulum clock, which is con-structed differently and more economically than a pendulum clockintended to be more accurate.

Making a pendulum run accurately requires paying attention to itsmany different parts, and the book is written in that fashion, with aseparate chapter on each part or subject. Subjects covered include thedimensional stability of different pendulum materials, good and poorsuspension spring designs (the suspension spring has more effect onpendulum performance than almost anything else—only temperaturecompensation is more important), the design of mechanical joints andclamps, effect of Q on accuracy, temperature compensation, air drag ofdifferent bob shapes, making a sinusoidal electromagnetic drive, andmany other topics related to pendulums and pendulum accuracy.

RJM

Introduction

The Shortt clock, made in the 1920s, is the most famous accuratependulum clock ever known, with an accuracy of 1 s/year when kept ata nearly constant temperature. In its day, it was the world’s standard fortime. Since then, people have been trying to make pendulum clockswith the same accuracy.

A pendulum clock’s accuracy is in its pendulum. If the pendulum isaccurate, the clock is accurate. Now, a pendulum’s timing is proportionalto the square root of its length. And there are about 30 million secondsin a year. To attain an accuracy of 1 s/year means that the pendulum’slength must be constant to 1 part in 15 million for a whole year. Anyonewho has any experience in designing and building accurate mechanicalmechanisms will tell you that such an extreme stability tolerance iseither impossible or next to impossible. Trying to approach or meetthis stability tolerance is the central problem in making an accuratependulum.

This book describes some of what I have learned about the makingof accurate pendulums. About three-fourths of the material has beenpublished before in magazine and newsletter articles. This book collectsthem all together in one place.

As a general guide, for most users, the single biggest improvement inpendulum stability will most likely come from replacing a suspensionspring having screwed-on chops with a suspension spring havingsoldered-on chops, or better yet, with a solid one-piece suspensionspring. The second biggest improvement will most likely come fromchanging over from a metal pendulum rod to a quartz rod. A in.quartz rod diameter is recommended for a 2 s period pendulum length.

I hope this book is useful to you the reader in making or improvingyour own pendulum.

58

part I

General

3

chapter 1

Better accuracy from simplependulums

Some simple things to improve the accuracy of simple pendulums aredescribed.

While most of this book deals with accurate pendulums, this chapterdeals with simple pendulums and with several things that can be doneto improve their accuracy. Most of the items have only a minor effecton accuracy, but they add up.

1. If possible, enclose the pendulum in a case. A case protects thependulum from the air currents of an open room, which will pushthe pendulum around and give erratic timing. Also, if possible, mountthe pendulum in a separate case from the hour strike or chimes. Thisprotects the pendulum from strike vibrations and from the stray air cur-rents caused by the swinging of chime tubes. The case will also smoothout any short-term variations in room temperature, for the benefit ofthe pendulum inside the case.

2. Use a metal pendulum rod rather than a wooden one. A woodenpendulum rod expands less with temperature (along the grain, but notacross the grain) than a metal rod. But the expansion of wood withhumidity is 10 times greater than that of metal with temperature, and willcause 10 times greater error in the clock rate (see Chapter 21). Varnishingthe wood, shellacking it, or soaking it in oil or wax does not help much[1]. It just takes longer to absorb or release moisture from or to the air.

3. If the pendulum is not temperature compensated, pick a low ther-mal expansion metal like iron for the pendulum rod. Invar would beeven better but it costs more. Temperature change is the biggest error,or one of the biggest, in a pendulum clock. Using a low expansion metalwill minimize the thermal expansion of the pendulum rod, and therebyminimize any changes in clock rate due to temperature.

4. If the pendulum is not temperature compensated, support the bobat its bottom edge rather than at its middle or top edge. The upwardthermal expansion of the bob, from its bottom edge up to its middle,will partially compensate for the downward thermal expansion of thependulum rod, and thereby provide a little better timekeeping. The

biggest correction will occur with a low expansion rod material likeiron or invar, and a high expansion bob material like brass. And using atall (or large diameter) bob rather than a short (or small diameter) onewill improve the temperature compensation, because the longer dis-tance between the bob’s middle and its bottom-supported edge willgive more upward thermal expansion, and hence more temperaturecompensation.

5. Use a low drag bob shape. Over 90% of the drive energy put into apendulum is dissipated in air drag losses as the pendulum swingsthrough the air. Less drag means that less drive energy is needed, whichin turn means that a smaller drive spring can be used, or a longer clockrunning time obtained. A pendulum that requires less drive energy isalso more accurate, because the less you disturb a pendulum, the moreaccurate it is (see Chapter 7). The Riefler bob, shown in Figure 1.1, has alow drag shape [2] and is a good one to use, particularly when the front-to-back distance available inside the clock case is limited. The Riefler bobshape requires about 3–28% less drive energy than the lens-shaped bobthat is frequently used. The percent energy saved can provide about thesame increase in timing accuracy.

6. When possible, keep the walls of the clock case at least 2–5 in.(depending on the bob’s shape) away from the pendulum, that is, fromthe pendulum’s closest point of approach. This is because all four wallsof the clock case both slow the pendulum down a little and increase theenergy needed to drive the pendulum (see Chapter 31). The walls dothis by increasing the air drag on the pendulum. Any closer than 2–5 in.and the walls’ effects on the pendulum increase considerably.

7. Walls close to the pendulum also cause a problem with relativehumidity. Relative humidity expands and shrinks the wooden clockwalls, changing the wall-to-pendulum spacings and thereby also chang-ing the clock rate. The effect is small but not negligible (see Chapter 30).At spacings closer than 2–5 in., a 24% change in relative humidity (from40% to 64%, say) can move the walls enough to change the clock rate by0.04–0.16 s/day (total of all four walls). The change in the pendulum’senergy drive needs is quite small and can be ignored (0–0.5%). Thesenumbers assume that the clock case walls are non-plywood, with thewalls expanding across the wood’s grain. These numbers can be cutapproximately in half by making the walls out of plywood (the 90

cross-plying reduces the expansion/contraction). The numbers can bereduced even further by making the walls out of glass, plastic, or metal.If glass or plastic, the walls need their inside surfaces coated with anelectrically conductive (anti-static) coating to avoid electrostatic chargeeffects, which can be as large as 100 s/day.

8. A common rate adjustment scheme for simple pendulums is toslide the top end of the suspension spring up and down through anarrow slot. This varies the pendulum’s length and thereby varies the

Accurate clock pendulums

4

Figure 1.1. Riefler bob shape.

T

D

T = toD2

D4

= 50° to 75°

clock rate. Figures 1.2a and b show two similar ways of doing this, butthe second way in Figure 1.2b gives a more constant clock rate. InFigure 1.2a, the slot itself is moved up and down by a slide mechanism,and the looseness necessary for movement in the slide adds to thelooseness of the suspension spring in the slot, increasing the variabilityof the pendulum’s timing. In Figure 1.2b, the slot is fixed solidly to theclock’s frame, and the slide moves the suspension spring itself up anddown through a fixed slot. Fixing the slot solidly to the clock frame asshown in Figure 1.2b gives a much more solid pendulum mounting,thereby giving a more constant clock rate. The difference between the twoapproaches is small, but the difference in timing accuracy is considerable.

9. If possible, use a weight-driven pendulum rather than a spring-driven one. The constant force provided by a weight drive allows thependulum to swing at a constant amplitude, making a big improvementin its accuracy.

10. The drive weight and its support chain should be kept eitherlonger or shorter than the pendulum, to avoid the energy exchangeeffect that occurs between two adjacent pendulums having the same oralmost the same frequency. The weight and its support chain form acrude pendulum. When two pendulums exchange energy, the amplitudeof one increases and the other decreases to zero. Then they reverse, withthe first pendulum’s amplitude decreasing to zero and the second one’samplitude increasing back up from zero. The energy exchange continuesindefinitely. The problem is that when the real pendulum’s amplitudegoes to zero, the clock stops. The two-pendulum problem can beavoided by using a short support chain and an automatic weight rewindevery few hours, or by doubling the weight and using a pulley to halvethe weight’s travel, so that the weight and its chain never get as long asthe pendulum. Alternatively, the two-pendulum problem can be

chapter i | Better accuracy from simple pendulums

5

Figure 1.2. Clock rate adjustment: (a) movable slots,(b) movable anchor.

Worm gear

(b)(a)

Rod bearing

Threaded rod

Movable slide

Fixed anchor

Movable anchor

Movable slot

Fixed slot

Suspension spring

Pendulum rod

Rate adjustmentshaft

minimized by putting a lot of damping and/or resistive losses in thehanging drive weight and its chain, so that it does not want to swing as apendulum and requires a lot of energy to force it to swing as a pendulum.

11. Keep the number of piece parts and mechanical joints in apendulum to a minimum. The more pieces and mechanical joints thereare, the more opportunity there is for movement and uncertainty in theclock rate. The most stable pendulum with the most accuracy has theleast number of parts and joints.

References1. A. Heldman. “Wooden pendulum rods and change of weight with change

in humidity,” Hor. Sci. Newslett. (April 2000), Available from NAWCClibrary, Columbia, PA.

2. D. Bateman. “Is your bob in better shape?” Clocks, ( June 1988), pp. 34–37.


6

7

chapter 2

A short history of temperaturecompensation

This chapter covers the temperature compensation of pendulum clocks only. It is based largely on two references: Rees [1], and Roberts [2].

Temperature compensators can be divided into four groups:

Mercury for medium expansion pendulum rods (iron) Gridiron for medium expansion pendulum rods (iron) Sleeve for low expansion pendulum rods (invar, quartz) Miscellaneous low accuracy schemes.

The story starts in the early 1700s, when people noticed that differ-ent metals expanded at different rates. Several clockmakers (Berthoud,Harrison, and Graham) independently measured the relative thermalexpansions of different metals, with each using his own arbitrary scaleof expansion.

Mercury

In 1721, Graham successfully tested a mercury compensated pendulumin which mercury was put in a glass jar and used as the bob. The pen-dulum is shown in Figure 2.1(a). The mercury expanded faster withtemperature than the glass jar did, and the rise in the mercury’s centerof mass speeded up the pendulum, compensating for the pendulum’sslowdown due to temperature, increasing the length of the pendulumrod. It takes about a 7 in. depth of mercury to compensate for an ironpendulum rod.

The mercury took a long time (up to 3 days) to change temperature,because of its large mass and also because the glass jar acted as a ther-mal insulator. This meant that the temperature correction was wrongevery time the temperature changed, until the mercury could stabilizeat the new temperature. Then it would be correct again. In the moreaccurate mercury pendulums, thin-walled cast iron jars were usedinstead of thick-walled glass ones to reduce the mercury’s temperature

Figure 2.1. Temperature compensation usingmercury: (a) Graham (1721) and (b) Riefler(1891).

Tube two-thirdsfilled withmercury

Bob

Glass jarfilled withmercury

Iron

(a) (b)

stabilization time. The cast iron jars provided less insulation and a fastertemperature response.

In 1891, Riefler [2], [3] (copyright D. Roberts, with permission) madea direct attack on the stabilization time problem in his mercury com-pensated pendulum, shown in Figure 2.1(b). Riefler used a hollow tubefor the pendulum rod, and filled it two-thirds full of mercury. The tubeis thus tightly coupled thermally to the mercury, so that both the tubeand the compensating mercury have nearly the same temperature evenwhen the temperature changes. This drastically reduced the clock’stime error due to the mercury’s long stabilization time.

Gridiron

The gridiron temperature compensator consists of a parallel array ofrods having alternating high and low coefficients of thermal expansion,as shown in Figure 2.2 (a and b). The low coefficient rods expand down-ward and the high coefficient rods expand upward. When the amountof upward expansion equals the amount of downward expansion, theoverall length of the pendulum remains fixed, and the clock runs at aconstant rate independent of the temperature.


8

Figure 2.2. Four gridiron temperaturecompensators: (a) five iron and four brass rods(Harrison 1726), (b) three iron and two zincrods, (c) three bars—two iron and one zinc(Ward 1806), and (d) three iron rods and onezinc tube.

Ironrod

(1 of 3)

Zinctubewithironrodinside

ZincBrass

Iron

IronCross-piece

Anti-bendstraps

Loosescrewin slot(1 of 4)

Bob Bob Bob Bob

(a) (b) (c) (d)

In 1726, Harrison [1] made the first gridiron, using five iron rods andfour brass rods (Figure 2.2(a) ). The key to making a gridiron is to havea high ratio between the thermal expansion coefficients of the twotypes of rods used. If the ratio is greater than 2 to 1, the grid can bemade with two rods expanding downward and one rod expandingupward. If the ratio is between 1.5 to 1 and 2 to 1 the gridiron is madewith three rods expanding downward and two rods expanding upward.The thermal expansion ratio for Harrison’s iron and brass rods was 1.7to 1, so his gridiron was made with two sets of five rods. Two sets wereused to provide symmetry on the pendulum. One rod in each set can becombined into one common rod, making a total of nine rods.

The dots in the cross-pieces in the figures mean that a mechanicalconnection is made to the rod underneath. With no dot the rod passesthrough a slightly larger hole in the cross-piece. There are usually twoanti-bend straps in the middle of the grid to keep the rods that are incompression from bowing and collapsing.

If iron and zinc rods are used, the thermal expansion ratio is higherat 2.8 : 1, and the gridiron can then be made with only five rods, asshown in Figure 2.2(b), instead of with nine rods as in Figure 2.2(a).

The gridiron has a couple of advantages. First, it looks beautiful ona pendulum, and that helps in the sale of clocks. It is so beautiful thateven today in the year 2003 many pendulums carry fake gridirons, thatis, a parallel array of unconnected rods. Second, the materials arecheaper for a gridiron than for an invar rod, which can be significant fora large pendulum. An invar rod (1.5 in. diameter 15 ft long) for a pen-dulum with a 4-s period costs US$1770 as of 2003 from Fry Steel Co.,Santa Fe Springs, CA, USA.

The gridiron also has a couple of disadvantages. First, the rods tendto rub against the anti-bend straps, creating friction and jerky motion.The through-rods also tend to bind up in the holes through the cross-pieces, due to rust or to lacquer put on the rods. Second, the gridiron inits parallel rod form does not meet the KISS rule (Keep It Simple,Stupid). It has far too many piece parts and joints, both of which leadto instability. The tubular version in Figure 2.2(d) is not as bad, but isnothing to brag about either. Invar and quartz were not available yet, sothe only real alternative at the time was mercury compensation.

In 1806, Ward proposed a three-bar gridiron arrangement using ironand zinc bars, as shown in Figure 2.2(c). There are four loose-fittingmachine screws through slots in the bars to keep the bars togetherwhile still allowing them to independently expand thermally. To be real-istic, I think this configuration needs interlocking grooves on the twointerfaces between the three bars.

The most practical version of the gridiron is shown in Figure 2.2(d).Figure 2.2(d) is the same as Figure 2.2(b) except that the two rods incompression in the latter have been replaced with one common tube.

chapter 2 | History of temperature compensation

9

The rods are iron and the tube is zinc. The tube’s larger diametereliminates any chance of buckling in compression, so there is no needfor any anti-bend straps. Large air holes (not shown) are cut in the tubeto allow air to reach the rod inside, and speed up temperature changein the rod inside. Over time, the two outer rods in Figure 2.2(d) werealso replaced with a common tube in most pendulums. In 1905Goodrich [4] described how to make a gridiron with a central rod ofsteel and two concentric tubes: one of aluminum and one of iron.

Also, by 1905, clockmakers knew [5] that zinc was an unstable metaland should not be used in accurate pendulums. Fortunately, in 1901, thevery low expansion metal called invar was invented by Guillaume inFrance, but it would take awhile to come into wide use. The modernview of zinc is “because pure zinc will creep under load at room temper-ature, alloying additions are necessary in … structural applications.” [6]Even zinc with alloying additions is not very stable dimensionally, so zincis not a desirable material for pendulums.

If one were to make a gridiron today, one would use the concentrictube approach, but with different materials. Stainless steel type 410 or416 would be used in place of iron. These two stainless types rust lessand have lower thermal expansion rates than iron. Also, aluminum type6061, hard and preferably stretched to reduce internal stress (T651 tem-per), would be used in place of zinc. Aluminum 6061 is not the moststable of metals, but it is better than zinc. The thermal expansion ratioof aluminum 6061 to stainless steel 410 is 2.3 : 1.

Sleeve

In 1809, Adam Reid [2] (copyright D. Roberts, with permission)invented a thermal compensation scheme for low expansion rod mate-rials such as invar and quartz. His scheme is shown in Figure 2.3. It issimple and effective, and is the preferred method of temperature com-pensation for invar and quartz pendulum rods today.

Referring to Figure 2.3, the pendulum rod extends down throughand below the bob for some distance. A nut called the rating nut isthreaded onto the bottom end of the rod. A loose-fitting sleeve of highthermal expansion material is placed around the bottom end of the rod,and rests on the rating nut. The sleeve is the temperature compensator,and the bob rests on top of the sleeve. When temperature expands therod downward, the compensator sleeve expands upward by the sameamount, holding the bob at the same distance from its axis of rotation.The pendulum’s time of swing then holds constant and is independentof temperature. Adam Reid’s idea meets the KISS rule, and has no bigdisadvantages.


10

Figure 2.3. Temperature compensation for alow thermal expansion pendulum rod(Adam Reid 1809).

Invarorquartzrod

Bob

Ratingnut

Temperaturecompensatingsleeve

Miscellaneous low accuracy schemes

Bimetal

A bimetal sandwich consists of two pieces of metal with widely differ-ent thermal expansion coefficients that are soldered or brazed together.When the temperature changes, one metal expands or contractsmore than the other, forcing the bimetal sandwich to bend in a curvealong the soldered (or brazed) surface joint. If the bimetal sandwich isthin, that is, about 0.05 in. thick or less, the stresses in the piece parts arereasonable and the bimetal concept works. If the bimetal sandwich isthick, the stresses generated in the piece parts and in the joint betweenthem become very large and exceed the elastic limit. Then somethinghas to deform itself and give way. Also, the large temperature drop backto room temperature after soldering or brazing the two pieces togethercan introduce more stress and curvature than will ever occur over thebimetal’s entire operating temperature range. And the mediocre springproperties (high relaxation under stress over time) of suitable bimetaland soldering/brazing materials limit the accuracy of all bimetaldevices.

As a pendulum’s temperature compensator, a horizontal bimetalstrip is attached to the pendulum rod and a weight is hung out at theend of the bimetal strip. When the temperature changes, the bimetalmoves the weight up or down to correct the pendulum’s time of swing.But you can only hang a small weight on a thin bimetal, which restrictsthe concept to small lightweight pendulums. The problems of limitedaccuracy and low compensation weight prevented the bimetal frombeing widely used on pendulums.

Length of the suspension spring

There are a number of schemes that use the thermal expansion of ametal component to pull a flat suspension spring up through a slot in thependulum’s support cock, or to let the spring down through the slot,correcting for temperature effects by decreasing or increasing the lengthof the whole pendulum. Le Roy (1739), Deparcieux (1739), and Fordyce(1794) [1] all proposed schemes of this type. But the uncertainties of theslot on the suspension spring limit the idea to lower accuracy clocks.

Barometer on pendulum

In 1895, Riefler in Germany experimented with putting a barometer ona pendulum for temperature correction and Robinson in England puttwo barometers on a pendulum, one on each side of the pendulum rod

chapter 2 | History of temperature compensation

11

for symmetry. This idea was too complicated for practical use, andnever developed into anything.

Gears and levers

There is a wide assortment of these schemes, all of them worthless.Rees [1] shows sketches of several of them. Friction in the joints and con-tact points made any movement jerky, and they tended to bind up.Troughton made a temperature-compensated multi-scissored pendulumrod structure that looks like an adjustable width gate hung vertically [1].Ellicott made some rod and lever arrangements (with high friction levels) that lifted up the bob on the pendulum rod [1]. Both Ritchie andDoughty used bimetal-operated levers to change the overall length ofthe pendulum rod [1].

References1. Abraham Rees. Rees’s clocks, watches and chronometers, Charles Tuttle Co.,

Rutland, Vermont, 1819, reprinted 1970.2. Derek Roberts. Precision pendulum clocks, privately published, 1986.3. D. Riefler. “Riefler-Präzisionspendeluhren,” Verlag George D. W. Callwey,

München, Germany, 1981, p. 36.4. W. L. Goodrich. The modern clock, privately published, 1905, reprinted 1950,

pp. 48–52.5. W. L. Goodrich. The modern clock, privately published, 1905, reprinted 1950,

pp. 54–5.6. Amer. Soc. Metals. A.S.M. metals handbook, Desk reference version, Amer.

Soc. Metals, Metals Park, Ohio, 1985, pp. 11–15.


12

chapter 3

Scaling the size of a pendulum

Making a pendulum bigger or smaller involves more than just a linear scalingup or down of the pendulum’s dimensions.

If you want to make a larger or smaller pendulum than the one youhave now, how should the dimensions change? Should they change lin-early with size, that is, with pendulum length? It turns out that not allof the dimensions should change linearly with pendulum length. Butsuppose they did. Suppose you had a 1 s beat (2 s period) pendulum witha 15 lb bob, which is a common bob size for a 1 s pendulum. Supposefurther that you wanted to make a larger version of it, a 2 s beatpendulum, which is four times longer. If you scale up the bob diameterproportionately by four times, the bob weight will go up by 64 times(four cubed), giving a 2 s bob weight of 960 lb! A wee bit heavy, you say?

Or suppose you wanted to make a smaller one-third second beat pen-dulum, which is one-ninth the size of the 1 s pendulum. If the bobdiameter is scaled down linearly, then the bob weight scales down by729 (9 cubed) to 0.020 lb (0.33 oz)! A wee bit light, you say? Obviously,making the bob weight proportional to the cube of the pendulumlength is making the weight change too fast.

Suppose we slow down the rate of change by making the bob weightproportional to just the square of the pendulum’s length, instead of tothe cube. The bob weights for different pendulum lengths are listed inTable 3.1. The bob weight is 0.25 lb for the little one-third second beatpendulum, and 240 lb for the large 2 s beat pendulum. The 0.25 lb is tooheavy, going by the standard practice in mantel clocks, which is0.06–0.18 lb (1–3 oz). And 240 lb is a little light for the big 2 s pendulum,if we go by the 506 lb bob weight used in Big Ben [1], which has a 2 s beatpendulum. From this, it is apparent that the bob weight changes tooslowly when it is proportional to the square of the pendulum length.

Well then, let us pick something in the middle—let us arbitrarilymake the bob weight proportional to the 2.5 power of pendulumlength. As Table 3.1 shows, we then get bob weights close to “currentpractice” in the clock trade; 0.11 lb (2 oz) for the small one-third secondpendulum, and 480 lb for the large 2 s pendulum. For a lens-shaped bob,selecting the 2.5 power of pendulum length is equivalent to scaling the

13

bob’s height and width directly proportional to the pendulum’s length,but making the bob’s thickness proportional to the square root of thependulum’s length.

How about the diameter of the pendulum rod? The purpose of thependulum rod’s diameter is to make the pendulum rigid, so that thependulum acts as a solid one-piece object instead of like a springy pieceof rubber. To this end, the rod’s diameter should be scaled for a constantstiffness in the pendulum rod. A constant stiffness means a constant bendangle () per unit of transverse force (F) applied, regardless of the pen-dulum rod’s length. This constant stiffness, /F, is defined geometricallyin Figure 3.1. The standard beam deflection equations show that a con-stant stiffness is obtained when the beam diameter is proportional to thesquare root of its length. So let us scale the pendulum rod’s diameter asproportional to the square root of the pendulum’s length.

The suspension spring obviously should be scaled for the weight ithas to carry (bob pendulum rod), but also for two other less import-ant items: (1) minimizing the spring’s bending torque on the pendu-lum, which minimizes the temperature compensation problem, and(2) minimizing the horizontal oscillation of the top of the pendulumrod without the bob following along. Using the minimum spring thick-ness will give the minimum bending torque on the pendulum, since thebending torque is proportional to the cube of the spring’s thickness.And using the shortest practical spring length will minimize the unde-sired horizontal oscillation of the pendulum rod. Contrary-wise, thebending stresses in the spring increase as its length is reduced, so a com-promise on length is needed. A good compromise is to make the lengthtwice the distance that the pendulum’s axis of rotation is below the topend of the suspension spring. This puts the pendulum’s axis of rotationat the center of the spring’s length.

The free length of the suspension spring should be scaled directlyproportional to the pendulum’s length. James [2] has shown that thevertical distance from the pendulum’s axis of rotation up to the free top


14

Table 3.1. Bob weight proportional to 2nd, 2.5, and 3rd powers of the pendulum’s length

Pendulum length (in.) Beat time (s) Bob weight (lb) proportional to

(Pendulum length)2 (Pendulum length)2.5 (Pendulum length)3

5.5 0.33 0.25 0.11 0.03

10.0 0.5 1.0 0.5 0.25

22.0 0.75 4.0 3.6 2.0

39.0 1.0 15.0 15.0 15.0

88.0 1.5 60.0 115.0 120.0

156.0 2.0 240.0 480.0 960.0

Figure 3.1. Constant stiffness, /F, inpendulum rod.

2

2

L

F

2L2

S

R R

edge of the suspension spring is directly proportional to the spring’sthickness. So the spring’s thickness is then scaled directly proportionalto the suspension spring’s length (and incidentally, directly proportionalto the pendulum’s length also).

The suspension spring’s width is adjusted to keep the static and bend-ing stresses approximately constant for all pendulum lengths. A muchsimplified version of James’ complicated equation for bending stress is,using James’ notation,

where fb bending stress in psi, pendulum half angle in radians,E modulus of elasticity in psi, W weight of bob (pendulum rod)in pounds, b width of suspension spring in inches, t thickness ofsuspension spring in inches.

James gives a lot of spring stress data [2] for a 1 s beat pendulum witha 16 lb bob. In particular, he gives a total stress level of 49,000 psi for a16 lb bob at 3 off vertical with a steel suspension spring of dimensions0.25 0.5 0.004 in.3 (L W T). The weight of the pendulum rod isapparently included in the bob weight in his analysis. Subtracting thestatic stress of the bob weight, converting to a smaller more modernpendulum angle of 1.50 half arc, and changing to a bob ( pendulumrod) weight of 16.5 lb gives a net bending stress of 28,800 psi for the1 s beat pendulum in Table 3.2. James’ bending stresses and those inTable 3.2 are based on the suspension spring being made of steel.

A few things should be mentioned about the baseline dimensionsgiven in Table 3.2 for the 1 s beat pendulum. They are important,

fb 3EWbt

,

chapter 3 | Scaling the size of a pendulum

15

Table 3.2. Scaled pendulum dimensions, scaled up and down from the 1 s beat pendulum

Pendulum Beat Pendulum rod (invar) Bob Suspension spring (steel)

# timeLength Diameter Weight

weightFree Width Thickness Static Bending Total

(s)(in.) (in.) (lb)

(lb)length (in.) (in.) stress stress at stress

(in.) (psi) 1.5 at 1.5

half arc half

(psi) arc (psi)

1 0.33 5.5 0.12 0.02 0.11 1/16a 1/16a 0.001a 2100 10,400 13,000

2 0.5 10.0 0.18 0.10 0.50 1/16 0.08 0.001 7500 19,800 27,000

3 0.75 22.0 0.25 0.38 3.6 1/8 1/4 0.002 8000 20,500 29,000

4 1.0 39.0 0.37 1.5 15.0 1/4 1/2 0.004 8300 20,800 29,000

5 1.5 88.0 0.56 7.8 115.0 9/16 2.0 0.009 7700 19,900 28,000

6 2.0 156.0 0.75 24.0 480.0 1.0 4.0 0.016 7900 21,300 29,000

Notea Limiting practical value.

as the other pendulum sizes are to be ratioed from these baselinedimensions. As mentioned earlier, 15 lb seems to be about the averagebob weight for a 1 s pendulum. As for the pendulum rod diameter, a in. diameter is about the minimum practical size, in the writer’sopinion. Anything smaller and the pendulum feels too rubbery or“twangy.” The spring dimensions given happen to be those of theauthor’s own 1 s beat pendulum. The one-quarter inch spring length isa very practical length for a 1 s pendulum. And the pendulum’s axis ofrotation is located in. down from the top of the spring, in the middleof its length. By actual test, the axis of rotation was put in the middleof the spring’s length with a 0.004 in. spring thickness (berylliumcopper), which is a little different from the 0.006 in. thickness (steel)indicated analytically by James for putting the axis of rotation in thislocation. The half inch width consists of two one-quarter inch widesprings in parallel, spaced apart with a 1.5 in. gap between them. Thesprings “handle” very well. The original pair of springs were still in use,unbent and unwrinkled, after having been inserted and removed fromthe clock about 100 times in various tests over a 1-year interval.

Table 3.2 gives the scaled dimensions of pendulums of various sizes.Using the scaling rules given herein, the pendulum dimensions inTable 3.2 have been scaled up and down from those of the 1 s beatpendulum, which has the assumed baseline dimensions. The static,bending, and total stresses in the suspension springs are also given inTable 3.2. The bending stresses are ratioed up and down from the 1 sbeat pendulum values, using the simplified James equation for bendingstress that was given above.

The total spring stress of every pendulum listed in Table 3.2 is withinthe 100 year stress limits of both steel (55,000 psi) [3] and berylliumcopper (30,000 psi) [4]. For beryllium copper, the bending stresses inTable 3.2 should be reduced to 79% of the values given, as the bendingstresses are proportional to the square root of the modulus of elasticity(18.5 106 psi for beryllium copper, 30 106 psi for steel).

For the convenience of the reader, all of the different scaling factorsused are listed in Table 3.3. Note that two of the five pendulum dimen-sions given in Table 3.3 (bob weight and diameter of the pendulum rod)do not scale directly proportional to the pendulum’s length. Note alsothat the width of the suspension spring, although scaled in principle fora roughly constant stress level, in practice, does scale directly propor-tional to the pendulum’s length, as can be seen in Table 3.2.

In summary, it makes more sense to scale the pendulum’s dimen-sions using scaling factors like those given here, rather than to just scaleall its dimensions linearly with the pendulum’s length.

18

38


16

References1. D. Bateman and K. James. “The pendulum of Big Ben,” Hor. J. (February

1977), pp. 3–9.2. K. James. “Design of suspension springs for pendulum clocks,” Timecraft-

clocks and watches, ( June 1983), pp. 9–11; ( July 1983), pp. 14–15; (August 1983),pp. 10–15; (November 1983), p. 27.

3. C. MacGregor. “Strength of materials” in L. S. Marks Mechanical engineers’handbook, 6th edn, McGraw Hill, New York, 1958, p. 11. Based on SAE 1050steel, quenched and drawn. Fatigue life extrapolated from 2 107 cycledata.

4. Beryllium copper data booklet, NGK Metals Corp., 1987. Fatigue life extrap-olated from 2 107 cycle data.

chapter 3 | Scaling the size of a pendulum

17

Table 3.3. Pendulum scaling factors

The scaling of Is proportional to

diameter of pendulum rod (pendulum length)1/2

bob weight (pendulum length)2.5

suspension spring

free length pendulum length

thickness pendulum length

widtha adjust for constant level of bending stress

Notea In practice, spring width turns out to be directly proportional to pendulum

length.

19

chapter 4

Finding a pendulum’s axis ofrotation

It is relatively easy to find a pendulum’s axis of rotation. Temporarilymount a small piece of paper on the front of the pendulum rod at the rod’stop end, so that the paper extends up an inch or two past the suspensionspring, as shown in Figure 4.1(a and b). With the pendulum stopped, marktwo small dots (A and B) on the paper about 1 in. directly above and 1 in.directly below the top end of the free unclamped part of the suspensionspring. The 1-in. dimensions are not critical, but accurately measure theactual distance L between the two dots (A and B). Set the pendulum swing-ing at its normal swing amplitude, and using an accurate ruler (a 6-in.machinist’s scale calibrated in decimal inches is ideal), measure the hori-zontal motion of each dot (A1 and B1 in Figure 4.1(c) ). The location x ofthe axis of rotation in Figure 4.1 is given by:

and then

This locates the axis of rotation on the paper. To locate it on the sus-pension spring, push a straight pin horizontally through the paper at thedistance x below dot A, and measure the distance y from the straight pinup to the top of the free unclamped part of the suspension spring. The

x A1

A1 B1L.x

L x A1

B1

Figure 4.1. Axis of rotation: (a) frontview, stopped, (b) side view, stopped, and(c) front view, swinging.

Axis of

Suspensionspring

A1

A

B

A

BTemporarysheet ofpaper

Papertaped topendulum rod

rotation Paper

B1

A

B

Ly

x

L

xy

(a) (b) (c)

Pendulum rod

axis of rotation is then y inches below the top end of the freeunclamped part of the suspension spring. This method has beendescribed before in the clock literature. There are other ways of doingthis, but the basic approach should be evident.

The location of the axis of rotation changes slightly with the pendu-lum’s swing amplitude. If desired, one can determine how much the axisof rotation moves by measuring y at different swing amplitudes, andthen plotting y vs the swing amplitude on a graph (see Chapter 5).


20

21

chapter 5

Does a pendulum’s axis of rotationshift with amplitude?

Does a pendulum’s axis of rotation move when the swing amplitude isincreased? Alan Heldman raised this question when we met at theNAWCC 2001 National Convention in New Orleans. The question pre-supposes that the pendulum has a flat spring type of suspension. A flatspring bends all along a section of its length, so there is no obvious rea-son to indicate that a pendulum’s axis of rotation would remain fixedwhen the swing amplitude is increased.

To find out whether the axis of rotation does or does not move, I ranan experiment, with the results shown in Figure 5.1. The test pendulumhas a in. diameter invar rod, a 19 lb brass bob, and a 2 s period. Thesuspension uses two flat beryllium copper springs, spaced 1.75 in.apart. The springs are silver soldered into very heavy stainless steel endpieces. The free unclamped size of each spring is 0.004 in. (L

W T ).The axis of rotation is found by measuring the horizontal travel of

two points on the pendulum rod, one point a little above the axis ofrotation and the other a little below it (see Chapter 4). The measure-ment accuracy is limited by the 0.01 in. markings on the 6-in. machin-ist’s ruler used to measure the horizontal travels of the two points onthe pendulum rod. The accuracy limitation shows up as scatter in thedata points in Figure 5.1, with the scatter inherently getting larger as theswing amplitude gets smaller.

The line in Figure 5.1 is visually drawn through the center of datascatter, and shows that the axis of rotation does move downwardslightly (down the suspension spring), as the pendulum’s swingamplitude increases. This should make the pendulum speed up as the

38

14

38

Figure 5.1. Distance of axis of rotationbelow top end of suspension spring vsswing amplitude.

0

0

0.1

0.22 4 6 8 10

Dis

tan

ce (

in.)

Swing amplitude (deg) (half angle)

amplitude increases, but in reality a pendulum actually slows downwith increasing amplitude. Apparently there are other bigger factorspresent.

The downward movement amounts to 0.0033 in./deg. of swing (halfangle), or 3.3 s/day/deg. of swing (half angle).


22

23

chapter 6

Some practical properties ofquartz

The big attraction of quartz as a pendulum material is its good dimen-sional stability over time. Stability over time is the biggest and mostneeded characteristic in an accurate pendulum. In contrast to invar,which was known to be unstable almost from its beginning, quartz hasa long history of being a stable material. For over 50 years, quartz hasbeen the preferred material for optical flats and telescope mirrors, bothof which are highly demanding in terms of stability. Its thermal expan-sion coefficient is 0.55 106 /C, which is lower than that of invar(0.7–3.0 106 /C, depending on heat treatment and coldworking).And quartz is much much cheaper and much more available than themore modern ultra-low thermal expansion glasses, such as CER-VIT(Owens Illinois), ULE (Corning), and ZERODUR (Schott).

Dimensional stability is not the same as low thermal expansion. If apendulum is temperature compensated, as all accurate pendulums are,then it does not matter much what the thermal expansion coefficient is,so long as the compensation has been done accurately. And in thewriter’s experience, the accuracy of compensation is limited by factorsother than the thermal expansion coefficient.

Because of their low density, quartz pendulum rods do have onedrawback. They have a much higher sensitivity to barometric pressurechanges than invar—2.7 times higher in my case. My pendulum has apressure sensitivity of 0.71 s/day/in. of Hg with a quartz rod, and only0.26 s/day/in. of Hg with an invar rod.

The next two paragraphs contain material paraphrased from theGlass engineering handbook [1] (G. McLellan and E. Shand., copyright1984, McGraw Hill, with permission).

In the trade, quartz is a type of glass that is more properly called fusedsilica. It is 99.9% silicon dioxide. Its tensile strength is usually given as6000–8000 psi, and its compressive strength as 160,000–280,000 psi.These are short-term (a few seconds) strengths. Longer term (1 h ormore) strengths are usually only 35–45% of the short-term values. Thesestress levels are not basic characteristics of the material, but are almostwholly determined by how the glass is fabricated (pressed, blown,

drawn, floated) and by the size of the micro-cracks in the surface of theglass. Each fabrication process produces its own characteristic surfaceflaws and size range of micro-cracks. When the glass surface is severelysandblasted, the tensile strength (1 h) is a low 2000 psi. With the “asreceived” surface, the tensile strength (1 h) increases to 6500 psi. Butwith an acid etched surface (the acid removes most or all of the micro-cracked material), and coated with a surface protecting lacquer, the ten-sile strength (1 h) skyrockets to 250,000 psi. The Handbook’s authorsrecommend maximum working stress levels of 500–1500 psi in tensionand 5000–10,000 psi in compression. These stress levels apply to all ofthe common types of glass, including quartz.

The Handbook also shows two remarkable photographs of three in.diameter glass rods, which are laid across two parallel edges that are 8 in.apart. The rods are bowed down in the middle by hooks fastened to 100 lbweights. The rods are sodalime (window) glass, and are bowed down 0.67 in. in the middle by the 100 lb weights, producing a 150,000 psi stressin the rods. The rods were undamaged and protected with a coat oflacquer. They were left in an unheated building for 26 years, where thetemperature ranged from 30 F to 100 F. When the loads wereremoved from the rods (after 26 years), they recovered their initial straightshape. Recovery to the straight shape was complete in 48 h. The twophotographs show (1) the bowed rods under their 100 lb loads, and(2) the straight shape they returned to when the loads were removed.

A significant point for the clockmaker is that the rods did not imme-diately recover their initial shape, but required 48 h to do so. There was a delay in the elastic response of the glass rods. This delay in theelastic response of glass is discussed further (below) by Murgatroyd andSykes [2].

In a torsion experiment, they measured the strain response to anapplied stress over a 1-month interval. Three different types of glasseswere tested: fused silica (quartz), a borosilicate, and a sodalime (window)glass. They found that whatever the amount of initial strain that occurredduring the first minute after the application of stress, during the follow-ing month, the strain would increase by a small amount directly propor-tional to the initial strain. This small increase in strain over a month’s timeamounted to 0.12% of the initial strain for quartz, 0.55% for the borosil-icate glass and 1.2% for the window glass. The increases in straindecreased exponentially with time over the 1-month interval.

Moreover, when the applied stress was removed, the strain wentmost of the way back to zero in the first minute, but stopped slightlyshort of zero. During the following month, the small remaining straindecayed toward the initial zero starting position, and decreased at essen-tially the same exponential rate it had increased when stress was initiallyapplied. In other words, the strain increased and decreased at essentiallythe same exponential rate and amount for a month after both the

14


24

application and removal of stress. This delayed response characteristicis not limited to just quartz, and shows up in most metals as well [3, 4].

What the above means to the clockmaker is that each time a pendu-lum with a quartz or metal rod is hung, you can expect its timing rateto exponentially slow down over the first month or so, graduallyapproaching a constant timing rate as the rod stretches to its finalweighted length. How much timing error does the stretching amountto? By calculation, it is very little. By my tests, it is much more. Thestretching is calculated as

where L increase in the pendulum length, L length of the pendu-lum rod, W bob weight, A cross-sectional area of pendulum rod,E Young’s modulus 10.6 106 psi for quartz.

For my quartz pendulum rod, which is 0.641 in. in diameter and 50.5 in.long (effective), and with a bob weight of 18.4 lb, the rod stretches272 in. Using Murgatroyd and Sykes’ delayed stretching percentage of0.12%, the delayed pendulum stretch is (272 106) (0.0012)

0.33 in. A 0.001 in. change in the length of a 1 s beat pendulum willchange the clock rate by approximately 1 s/day. The 0.33 in. delayedstretch of the pendulum rod thus corresponds to a slowing down timingerror of 330 s/day, or a total time loss over a one month interval of about0.01 s. This is pretty small. Experimentally, I find the effect is much bigger,amounting to a total accumulated time error of about 0.27 s over 5.5 daysfor a quartz pendulum rod, and about 0.75 s over 5.5 days for an invar rod.About 90–95% of the total accumulated time error occurs in the first 5-dayinterval after the pendulum is hung. The pendulums continue to stretchbeyond this time period at an exponentially decreasing rate.

The accumulated time error is much larger for the invar rod than forthe quartz rod. Note that the whole pendulum is involved in the experi-mental numbers, not just the pendulum rod by itself. I have no explana-tion for the big difference between the calculated and measured timeerrors. May be it is due to the stretching of the suspension springs (beryl-lium copper), or to a difference between torsional stress and linear stress.

References1. G. McLeIIan and E. Shand. Glass engineering handbook, 3rd edn, McGraw-Hill,

New York, 1984.2. J. Murgatroyd and R. Sykes. “Delayed elastic effect in silicate glasses at

room temperature,” J. Soc. Glass Technol. 31 (1947), 17–35.3. C. Zener. Elasticity and anelasticity of metals, University of Chicago Press,

New York, 1948.4. J. Woirgard et al. “Apparatus for the measurement of internal friction as a

function of frequency between 105 and 10 Hz,” Rev. Sci. Inst. 48(10)(October 1977), 1322–5.

L LWAE ,

chapter 6 | Practical properties of quartz

25

27

chapter 7

Putting Q in perspective

Many articles have been written about the relative importance of Q in aclock. Q is a clock oscillator’s quality factor, a measure of how low itsenergy losses are with respect to the total energy stored in the motion ofthe oscillator. Bateman [1, 2] and Woodward [3, 4] have been the prin-cipal ones writing for Q. Boucheron [5], Matthys [6], and Cain [7] havewritten against Q. This chapter provides some new information on Q,and tries to provide some perspective. Arguments both for and against Qwill be presented. And discussions along the way will try to make senseout of the various arguments. First, the arguments against Q.

Arguments against Q

First. This argument starts with the pendulum’s equivalent electricalcircuit, shown in Figure 7.1. When the L, R, and C components inFigure 7.1 are used as the frequency controlling elements in an oscil-lator, the oscillator’s center frequency is controlled by the reactive com-ponents L and C only, and is independent of the series resistance R.Q specifies the frequency bandwidth of the circuit, and the resistancedoes affect this, as Q 2 f L/R, where f is the frequency of oscillation,L is the inductance (corresponding to the mass of a bob), and R is theseries resistance (corresponding to the energy losses in a pendulum).

Now the oscillation frequency will wander up and down slightlyfrom the center frequency, depending on the value of Q. Increasing theresistance R will decrease the Q, and will allow the oscillation frequencyto wander up and down a bit more from the center frequency. Thepoint here is that, over time, the oscillation frequency will still averageout at the center frequency, regardless of whether Q is a small or largenumber. Thus, in the short term, Q will have an effect on the frequency.But in the long term, the oscillation frequency will average out at thecenter frequency regardless of the value of Q.

Second. It is pretty much a given fact that the less you disturb a pen-dulum, the more accurate it will be. So the less hard or less often youhave to push a pendulum to keep it going, the more accurate it will be.Yes, you can say that this is an argument for making a pendulum as

Figure 7.1. Equivalent electrical circuit for anoscillating pendulum.

L R C

Part of the information on atomic frequencystandards came from physicist T. Parker atNIST in Boulder, Colorado, USA.

efficient as possible. And yes, you can make the efficiency number partof a pendulum’s Q. But I think that detracts from the primary idea,which is to disturb the pendulum as little as possible.

Third. Without periodic re-enforcement, the energy stored in theswinging of a pendulum will die out in approximately Q cycles of oscil-lation, that is, the amplitude will go to zero in a time span of approx-imately Q cycles of oscillation. With a Q of 10,000, it will die out in 10,000cycles of oscillation. Having died out, the stored energy can have no effectbeyond that point in time, which is about 5.6 h for a pendulum with a 2 speriod. So how can Q, which inherently involves the concept of storedenergy, have an effect on longer time intervals, such as a month or a year?It cannot, at least not in any way that I can see. But as in the first argu-ment, Q can have an effect in short time intervals of 5.6 h or less.

Now you can increase Q by going to a shorter pendulum, but thatdoes not help matters [7]. Suppose you were able to double the Q to20,000 by going to a shorter pendulum with a 1 s period. The shorter(and higher Q) pendulum has twice as many oscillations per unit oftime, and hence its stored energy will die out in the same span of timeas the longer (but lower Q) pendulum: 5.6 h. The increased Q of theshorter pendulum is of no help at all, as it has not increased the timespan of the pendulum’s stored energy.

Fourth. What is noticeable in most graphs of clock time error vs timeis that the clock will run at a reasonably constant rate for 3–6 months,and then will suddenly jump to a new rate and run reasonably constantat the new rate. The effect of the sudden jump in clock rate at 3–6 monthsusually far exceeds that of any other error in a good clock, with theexception of inaccurate or no temperature compensation.

I think that this error, the big jump in the clock rate at long timeintervals, is the biggest problem in designing a good clock, because itseffect is so large and it is so hard to find and fix the cause of it. Now ifQ is important, it has to be connected with the biggest problems inclock design. But I do not see it connected to this “biggest” problem.What could Q possibly do that would have no effect on a clock untilafter 3–6 months had passed by? This is a time span more than 300 timeslonger than the time needed for a pendulum’s stored energy to die out(assuming the same pendulum with a Q of 10,000 and a 2 s period). ButI do see this “biggest” problem being caused by mechanical design prob-lems, such as materials instability, or micro-slippage in a mechanicaljoint like (1) the clamp on the lower end of a suspension spring, or(2) the thread of a rating nut.

Fifth. This one is not an argument but two independent opinions,given with reasons why:

The long term stability of a [pendulum] clock is dictated not by system Qbut by its inevitable involvement with the adverse environment in whichit is operated. (L. Leeds [8])


28

For the medium- or long-term [timekeeping errors], it is not the Q thatis the crucial factor but the stability of the oscillator parameters in theface of temperature and barometric changes and aging phenomena.(H. Wallman [9])

Sixth. This is an argument plus three opinions, all on a side issue. Theargument is the same as in argument three against Q, but is applied toa quartz crystal instead of to a pendulum. With a Q of 1 million and afrequency of 1 MHz, a crystal’s oscillation will die out in 1 s withoutperiodic reinforcement. So how can Q have any effect on longer timeintervals such as a month or a year? The answer is the same as before:it cannot, not in any way that I can see.

Q does correlate with a crystal’s short-term stability (0.01–1 s,depending on frequency) but not at all with its long-term stability(months). “There is no correlation whatsoever between Q and a crystal’slong term stability,” says Virgil Bottom [10], an acknowledged expert oncrystals. My own opinion, based on many years of designing crystaloscillators [11], is in agreement with Bottom’s.

The good long-term stability of a quartz crystal comes from theextreme simplicity of its design (a small piece of quartz with two plated-on electrodes, inside a sealed container), plus the use of very stablematerials (quartz is one of the most stable materials known to man).This is just my opinion, but it is based on a lifetime of designing andbuilding precision mechanical and electronic apparatus. In practice, oneof the primary limitations to a crystal’s long-term stability is the move-ment of molecular-sized “dirt,” with the “dirt” moving back and forthfrom the container walls to the crystal surface.

Seventh. This is an argument on a side issue. An atomic frequencystandard has two oscillators inside it. One is a 5 MHz crystal oscillator,with a Q of about 1 million. The other is an atomic oscillator whosegigahertz frequency is the frequency difference between two atomicenergy levels. The gigahertz oscillator does not have a real Q, as its fre-quency is the frequency difference between the two energy levels. Butan artificial or imputed Q is sometimes used, obtained by dividingthe gigahertz oscillator’s frequency by the small variation detected inthe gigahertz oscillator’s frequency [12]. The small frequency variationcan be reduced (and the imputed Q increased) by keeping the atomslonger at the higher energy level before they fall back to their lowerenergy level.

The imputed Q affects the short-term (hours, a few days) accuracy ofthe gigahertz oscillator, and has no effect on its long-term (months, years)accuracy. The gigahertz oscillator’s long-term accuracy is controlled bysystematic errors: magnetic shielding, cavity dimensions, atomic inter-actions with cavity walls, ageing of detection electronics, etc.

Only the 5 MHz oscillator signal is brought out for external use bythe user. The signal from the atomic gigahertz oscillator is not brought

chapter 7 | Putting Q in perspective

29

out but is used internally, its only purpose being to correct the long-termdrift of the crystal oscillator, via a varicap diode in the crystal’s voltage-controlled oscillator circuit.

A crystal oscillator has good short-term stability and good long-termstability. The gigahertz oscillator has poor short-term stability but out-standingly good long-term stability (1 s in 1000 to 1 million years,depending on the atom used). The two oscillators are combined intoone to gain a superior performance that neither could provide on itsown. The point here is that the atomic frequency standard should beconsidered for what it really is, a 5 MHz crystal oscillator with a Q of1 million (not 107–1010), the good short-term (0.2 s) stability of a crystal,a good medium term (hours, a few days) stability that is dependent onthe gigahertz oscillator’s imputed Q, and with essentially zero long-term drift (1 s in 1000 to 1 million years).

For Q comparisons to other clocks, one would use the imputedQ value of the gigahertz oscillator, because of the longer time intervalof interest. The imputed Q value varies from 107 to 1010, depending onthe atom used. But as mentioned above, the imputed Q controls onlythe medium term (hours, a few days) accuracy of the 5 MHz crystaloscillator, but not its long-term (months, years) accuracy.

Discussion

The first argument against Q, that over time the oscillation frequencywill average out at the center frequency, does not mention the resultingtime error, which is the accumulation (i.e. the mathematical integ-ration) of the frequency error over time. Now a lower Q with its inher-ently larger frequency wander up and down from the center frequencywill generate a larger plus and minus time error than a higher Q wouldwith its smaller frequency wander. So at first glance one might say thatthis first argument against Q is really an argument for (high) Q. But thetime error vs time plots of real clocks show very little or no randomtime wander of this type, particularly on the short time scale (minutes,hours) that would be expected. One, therefore, concludes that the oscil-lation frequency averaging out at the center frequency is effective inreducing any time error effect to insignificance. Thus, the first argu-ment is a valid one against Q.

Five of the seven arguments against Q show that Q is not connectedwith the long-term accuracy of pendulums, crystals, and atomic fre-quency standards, although Q is connected to their short-term accuracy.And although it was not shown, applying argument three—that thestored energy dies out in a short time interval, so how can Q have anyeffect on long time intervals?—to the other clock types would undoubt-edly show that Q is not connected to their long-term accuracy either,


30

although it is connected to their short-term accuracy. Thus, it is probablytrue (but not proven here) that Q is not tied to the long-term accuracyof any clock type, although it is tied to the short-term accuracy of all ofthem.

Another argument says that Q cannot be too important—it cannothelp solve the problem of the big rate jumps that occur in most goodclocks every 3–6 months. And one argument implies that Q is just asmaller version of a more general rule “for maximum accuracy, disturbthe pendulum as little as possible.”

Comment on a side issue: because of the big jumps in clock rate at3–6 month intervals, I think clocks should be rated on their perform-ance over a long time interval such as 1 year, so as to include the ratejumps and give a better measure of their long-term accuracy. If ashorter 2–4-month interval is used, it is too easy to pick an intervalbetween rate jumps and get a clock accuracy number that is better thanthe long-term performance actually is.

Arguments for Q

First. Starting from a general equation of motion and assuming a sinu-soidal driving force, Bateman [1] derived an equation showing that thefrequency error in the frequency of a clock is proportional to theangle variation in the basic drive angle , and that it also variesinversely with both Q and the sine squared of the basic drive angle :

This equation (Bateman’s eq. 12) says that a clock’s performance can beimproved just by increasing its Q, without making any modifications tothe escapement. The equation applies to clock drive errors. The deriva-tion further mentions that the drive angle variation becomes the lim-iting factor on clock accuracy after temperature and external errors havebeen reduced to a minimum. This implies that the temperature and exter-nal errors are bigger errors, and only after they have been made smalldoes the drive angle variation error become the dominant error.

Second. Bateman [1] plotted the Q vs the accuracy of a wide varietyof clocks on a graph, reprinted here as Figure 7.2. The wide clock vari-ety covered everything: wristwatches, tuning forks, pendulums, crystaloscillators, and atomic frequency standards. Figure 7.2 shows that theQ vs accuracy points of all of the various types of clocks fall in a broadstraight line (Bateman used three parallel lines) across the graph, indic-ating a linear connection between Q and clock accuracy. The highera clock’s Q, the higher its accuracy is, in broad proportion to its Q.

2Q sin2.


31

At a fixed accuracy level, the Q width of the broad line connectingall the clock points in Figure 7.2 is about 3.3 orders of magnitude, or2000 to 1. At a fixed Q level, the accuracy width of the broad line isabout 3 orders of magnitude, or 1000 to 1. Even with a fixed Q, a clockaccuracy variation of 1000 to 1 certainly leaves plenty of room for anindividual clockmaker to express his own ideas and taste in clock design.

Discussion

I was not much impressed with the first argument for Q, as all of thecomponents in its error equation were previously known. That is, tomake an accurate pendulum clock, impulse it at the center of swing, asconstantly and repeatedly as possible; and make the pendulum as effi-cient as possible, so that the impulse is as small as possible and disturbsthe pendulum as little as possible.

But the second argument for Q, that the accuracy of a clock isroughly proportional to its Q, is one that I found to be very convincing.That a plot of Q vs the accuracy of actual working clocks should fallalong a (broad) straight line, indicating a linear relationship between


32

Figure 7.2. Q vs accuracy for a wide variety ofclocks. (After Bateman, courtesy HorologicalJournal.)

Balance wheels(f, g chronometers)

Torsional pendulum(atmos clock)Tuning forks (i Dye & Essen,i clocks & wristwatches)Pendulums(p shortt clock, r Big ben)Quartz crystals(v Essen ring)Electromagnetic

Atomic (Cs–2 caesium, NPL3)

10

10–2

10–4

10–6

10–8

10–10

10–12

10–14

10–16

1

0.1

0.01

0.001

0.0001

1 s in a1000 years

1 s in amillion years

second per day

100 102 104

Quality factor (Q)106 108 1010 1012

Q Proportionalto accuracy

Fe57

Rb–1

Rb–2

Rb–3

CH3

H2

Cs–2

Cs–1

Cs–3Cs–3

xw

v

q

t

p

u

sn

a

d bce

hg

f lA

kr

C

B E

NH3

Acc

ura

cy

Q and accuracy, is something I find hard to argue with. And it is long-term accuracy, not short term, that is plotted in Figure 7.2, sinceit is the actual accuracy (estimated in some cases) of real clocks thathas been plotted here. And how can this be squared with the informa-tion presented earlier in the arguments against Q—that Q is connectedonly to the short-term accuracy of pendulums, crystal oscillators, andatomic frequency standards—and is not connected to their long-termaccuracy?

The short answer is that both statements are right, but by differentpaths. That Q is related to short-term accuracy but not to long-termaccuracy is scientifically ascertainable in the laboratory. That Q isrelated to long-term accuracy is historically ascertainable from actualclock records.

Others have noticed the dichotomy that Q is not related to long-termaccuracy but is related to it anyway. Workers at the National Institute ofStandards and Technology (NIST) in Boulder, Colorado, USA, in theirever-continuing search for ever more accurate oscillators and time stand-ards, have noticed that the standards with good long-term stabilityalways seem to have high Q, even though the high Q is not needed forlong-term stability. There must be a connection there somewhere, buteven they do not know what it is. One NIST physicist thought it mightbe due to the high Q reducing the effect of other errors in an oscillator.He said high Q seems to be a necessary but not a sufficient condition forgood long-term stability. And even though Q is not scientifically relatedto it, they do use Q as an indicator to direct their oscillator work.

Figure 7.2 shows that on a global scale (from wristwatches to atomicfrequency standards) Q is broadly proportional to clock accuracy. Andalso on a global scale, Bateman’s statement that the type of oscillator ismore important than the type of escapement, is true. After all, a crys-tal oscillator (with a Q of 1 million) is unquestionably more accuratethan a mechanical wristwatch (with its lower Q of 100–1000).

But suppose one decides to make one of these clock types, say apendulum type. Then the picture changes to a smaller scale, a pendulum-sized scale. And Bateman’s statement about the type of oscillatorbecomes immaterial, and the choice of escapement becomes relevant.To make the picture clearer, the pendulum data in Figure 7.2 has beentransferred to Figure 7.3, leaving the rest of the clock data behind.Figure 7.3 then is the same as Figure 7.2 except that only the pendulumdata is shown in Figure 7.3. The accuracy range of just the pendulumdata, as shown in Figure 7.3, is 27,000 to 1 (8 s/day to 0.0003 s/day), andthe Q range is 227 to 1 (500,000 to 2,200).

Two horizontal lines have been added to Figure 7.3. The firsthorizontal line is at 1.1 s/day, representing the uncompensated tem-perature error of a clock operating 4 F (2.22 C) away from its calibra-tion temperature. An iron pendulum rod is assumed, which has a


33

temperature coefficient of 11.7 106 in./in.C. The time error perday is then

The high location of the uncompensated temperature line in Figure 7.3says that you are not going to get much of a pendulum’s accuracy cap-ability unless you compensate it for temperature.

The second horizontal line in Figure 7.3 is really a horizontal band,with upper and lower limits of 0.62 and 0.008 s/day. This band repres-ents the range of rate jumps found in the time error vs time records ofan assortment of pendulum clocks. A Shortt clock had the lowest ratejump (0.008 s/day), and a Pulsynetic clock with additional secondscontacts had the highest (0.62 s/day). Most of the pendulum recordsexamined showed rate jumps in the middle of the range (0.1–0.3 s/day),and the pendulums usually ran smoothly at their new rate for2–6 months before some of them exhibited another rate jump. Verylittle of the clock error data available extended beyond 1 year, so it wasnot possible to determine when or if the other clocks exhibited morerate jumps.

11.7 106(2.22) 86,4002 1.1 s/day.


34

Figure 7.3. Q vs accuracy for pendulumclocks only. (After Bateman, courtesyHorological Journal.)

Uncompensatedtemperature

m

Ratejumps

l r

n

p

q

Q Proportionalto accuracy

10

1

0.1

0.01

0.001

0.0001

1 s in a1000 years

1 s in amillion years

10–2

10–4

10–6

10–8

10–10

10–12

10–14

10–16

100 102 104 106 108 1010 1012

Quality factor (Q)

Acc

ura

cy

second per day

The maximum Q for a pendulum at atmospheric pressure is about25,000. Placing this in the overall 2200–500,000 range of pendulum Qmeans that about two-thirds of the total accuracy improvement of 227to 1 that is available from increasing a pendulum’s Q over its minimumvalue cannot be obtained unless the pendulum is put in a vacuumenclosure.

Conclusions

The value of Q rests on two facts: (1) that increasing a pendulum’s Qmeans lower drive forces, which means less pendulum disturbance,which means a more accurate clock; and (2) that historically (but notscientifically), a higher Q broadly means a higher long-term accuracy.

On a global scale of wristwatches to atomic frequency standards,Q is historically broadly proportional to long-term clock accuracy. Thismakes Q significant as an indicator of clock accuracy on a global clockscale. But on the more limited scale of pendulum clocks by themselves,both temperature compensation and aging are more important issuesthan Q, as their errors are bigger than the increased accuracy availablefrom an increased Q.

Q is scientifically connected to a clock’s short-term accuracy only.This is true whether the clock is a pendulum, a crystal oscillator, or anatomic frequency standard. And it is probably true of all clock types,but that has not been proven here. Q is historically broadly proportionalto a clock’s long-term accuracy. The reason for the historical connectionof Q to long-term accuracy is not known.

Q may be a smaller version of the more general rule that the less youdisturb a pendulum, the more accurate it will be. And about two-thirdsof the total accuracy increase (227 to 1) available from increasing a pen-dulum’s Q is not available unless the pendulum is put in a vacuumenclosure.

In summary, on a global scale covering all types of clocks, I agree withBateman and Woodward—long-term clock accuracy is broadly propor-tional to Q. But on a smaller scale, a pendulum clock scale, I agree withLeeds and Wallman: temperature and aging effects are more importantthan Q. Q has its place in the sun, but from a pendulum’s point of view,I believe it is a limited place, exceeded by more important issues such astemperature compensation (or control) and the dimensional stability ofpendulum materials and joints. The pendulum’s temperature and agingproblems, being bigger, must be solved first before the accuracy can beraised very much by improving Q.

And finally, in spite of disagreeing with some of his conclusions,I still consider Bateman’s article on vibration theory [1] to be one of thebest clock articles I have ever read.


35

References1. D. Bateman. “Vibration theory and clocks,” Hor. J., in 7 parts, ( July 1977

through January 1978).2. D. Bateman. “Quality factor—the practical approach,” Hor. Sci. Newsletter

of chapter #161, NAWCC ( July 1994).3. P. Woodward. “A note on Q values,” Hor. J. (May 1975), 3–4.4. P. Woodward. “Q is really a simple concept,” Hor. J. (December 1996),

414–15.5. P. Boucheron. “The facts about pendulum Q,” Hor. Sci. Newslett. chapter

#161, NAWCC (April 1993).6. R. Matthys. “Q, bob shape, scaling, and air currents,” Hor. Sci. Newsletter of

chapter #161, NAWCC (December 1994).7. D. Cain. “The ultimate pendulum: Higher Q or better drive?”, Hor. Sci.

Newsletter of chapter #161, NAWCC (February 1995).8. L. Leeds. “Clock pendulums and the Q parameter,” NAWCC Library,

February 1970. Also Hor. Sci. Newsletter of chapter #161, NAWCC(September 1995).

9. H. Wallman. “Comments on Bateman’s vibration theory,” Hor. J. (August1978), 48–50.

10. V. Bottom. Private communication, and “Introduction to quartz crystal unitdesign,” Van Nostrand Reinhold, published simultaneously in New Yorkand Canada, 1982.

11. R. Matthys. Crystal oscillator circuits, Wiley & Sons, New York, 1981.12. W. Itano and N. Ramsey. “Accurate measurement of time,” Sci. Amer. ( July

1993), 56–65.


36

37

chapter 8

The Allan variance and the rmstime error

Introduction

There is no question in my mind that the traditional plot of a clock’stime error vs time is far superior to the Allan variance for showing apendulum clock’s performance. The Allan variance is admittedly a uni-versal and statistically more accurate measure of a clock or oscillator’srandom frequency and time variations (i.e. variance), because it is aver-aged over multiples of each time interval. But it is a whole curve on agraph instead of being just a single memorable number, and its value isdrastically reduced by the short time span it is able to cover. The vari-ance, however, can be used to generate an oscillator’s “root meansquare (rms) time error vs time” curve, a curve that is much easier tounderstand. But the rms time error’s equally short time span drasticallylimits its value to the clockmaker also.

Traditional “time error vs time” plots

If one measures the time difference between his clock and the radiotime standard WWV, measures it every day or so1 over a year’s time,and then plots it up on a graph as time error vs time, one has an almostperfect picture of his clock’s time performance. It shows what theclock’s time error was over a reasonably long period of time, and showsat a glance how accurate the clock was. Which is exactly what you wantto know about a clock, and it is presented in an easy, simple, and directmanner.

Figure 8.1 shows what a pendulum’s time error vs time plot mightlook like. And when someone asks “how accurate is the clock, with noexcuses?”—the right answer is that the clock gave the correct timewithin 30 s over a 1-year period. It is impossible to set a clock to exacttime with no drift, so I am willing to subtract out the average drift froma clock’s actual performance over the year. And once a year I wouldpermit re-setting the clock’s hands back to zero error, and re-trimmingthe clock’s drift rate closer to zero.

Author’s note: The Allan variance is acomplicated subject. This chapter representsthe author’s understanding of it, and ispresented here to promote discussion of itsrelative worth to the pendulum clockmaker,as distinct from the needs of those workingwith atomic frequency standards.

1 If you are going to calculate the Allanvariance or the rms time error, you shouldmeasure the clock regularly at uniformintervals, and not erratically.

If allowed to subtract out the average slope over the 1-year interval,then the right answer to the “how is it performing?” question is that theclock gave the correct time within 15 s over a 1-year period. To reducethe 15 s maximum error to a standard deviation of (say) 5 s is to mejust “smoke and mirrors,” to make the error appear smaller than itreally is. When the average person asks how accurate a clock is, hewants to know the maximum error, not the standard deviation (“Youmean the error can be bigger than 5 s? How much bigger?”).

Figure 8.1 also shows another characteristic typical of pendulumclocks—that the clock will run relatively smoothly at one rate, and thenafter 3–6-months it will suddenly jump to a new rate and run relativelysmoothly at the new rate, as shown in Figure 8.1. This characteristic iscalled a random walk by the oscillator noise people. These suddenchanges in clock rate are usually the biggest error source in a goodpendulum clock. Because of their large effect, these rate changes at 3–6-month intervals should be included in a clock’s performance rating,and are the primary reason for specifying a clock’s performance over thelonger 1-year interval. Specifying a clock’s performance over a shorterinterval allows exclusion of these big 3–6-month rate errors, and willgive a better performance rating than a clock actually has long term.

Allan variance

The Allan variance is a statistical measure of the normalized frequency(f/f ) and time (t/t) variations in an oscillator, over various intervalsof time. The variance is usually presented as a line on a graph, since thevariance for a given oscillator changes with the length of the time inter-val over which the oscillator is allowed to drift. And since a pendulumclock is a mechanical oscillator, the Allan variance can be applied to apendulum clock, and might look like that shown in Figure 8.2.

An oscillator’s variance is a dimensionless number. Multiplying thesquare root of the variance by the oscillator’s frequency will give theoscillator’s frequency variation, in Hertz, over each time interval. Andmultiplying the square root of the oscillator’s variance by its associatedtime intervals will give the oscillator’s time variation, in seconds, overeach time interval. The industry has agreed on a common set of defi-nitions, rules, and equations for the Allan variance,2 so that the random


38

Figure 8.1. Time error vs time for a typicalpendulum clock.

1 12

+30

20

10

0

–10

Slopechange

Tim

e er

ror

(s)

14

34

Year

2 The Allan variance is officially defined asa squared error function. What is actuallywanted, and is plotted on the graphs in thischapter (and in many other published articlesas well), is the square root of the Allanvariance. Unless it makes a difference, thelonger more correct terminology is frequentlyshortened to the generic name “Allanvariance.”

frequency and time variations of any oscillator, and in particular anyatomic frequency standard, can be statistically specified by its Allanvariance.

The square root of the variance is usually plotted on log–log graphpaper, whereupon the slope of the plotted line corresponds to the dif-ferent types of modulation listed in Table 8.1. The slopes are importantbecause they describe different statistical noise processes. Two modula-tion types are listed for a slope of 1.0. There is a modified Allan vari-ance available which will give separate slopes to each (1.0 and 1.5).

For pendulum clocks, the zero (0) slope at 1-day time intervals inFigure 8.2 usually changes to (0.5) at the 3–6-month interval, becauseof the frequent appearance of a step change in clock rate (frequency) atabout that interval of time. It does not show in Figure 8.4 because theplot ends at 3 months.

What does the Allan variance give us? It gives us two things: (1) a nor-malized curve representing a clock’s random variation in frequency andtime over a range of time intervals, and (2) the slope of the curvedescribes what type of modulation is represented by the clock’s vari-ance over different time intervals. But knowing what modulation typethat a clock’s variance represents is useless information to the clock-maker. Granted, a variance slope of 0.5 indicates that a step change inclock rate occurred, but the traditional time error vs time plot will also

chapter 8 | Allan variance and rms time error

39

Figure 8.2. Allan variance for a typicalpendulum clock.

101 100 1000

Time interval (days)

10–5

10–6

10–7

Var

ian

ce,

∆T T Slope = 0

= +0.5

Table 8.1. Modulation types for various line slopes on

log–log paper [1]

Slope Modulation

1.0 Frequency steadily changing same direction

0.5 Random walk FM (step change in frequency)

0 Flicker FM (1/f noise)

0.5 White FM

1.0 Flicker PM or white PM

Note: FM frequency modulation; PM phase

modulation.

show the step change, and in addition will show when it happened, sothat you can dig into the data at that point in time and hopefully findwhat caused it.

If you are interested in how your oscillator (clock) stands on a globalperformance scale representing all types of oscillators from waterclocks to atomic frequency standards, then the Allan variance (actuallythe square root of the Allan variance), which is that global scale, is foryou. It is a complicated scale, as the variance is not a single number fora given oscillator, but varies with the length of the time interval overwhich the oscillator may vary. And the variance numbers all carry neg-ative exponents: about 106 for a pendulum clock, to 1011–1015 forthe atomic oscillators. Remember, though, that this is a random errorscale, and not a total error scale.

But if you are interested in something that is less complicated butmore useful, such as what can the variance tell you about your clock’stime errors—then the rms time error is for you.

Rms time error

As Woodward [2] pointed out, a clock’s (rms) time error can beobtained by multiplying the square root of its Allan variance by itsassociated time intervals. Allan discusses the rms time error, which is arandom error, in two articles [3, 4].

The published data on the Bateman pendulum clock will be used asan example of the rms time error. In Figure 8.3, curve A shows the


40

Figure 8.3. Performance of the Bateman clock:(A) actual time error (after Bateman [4]),(B) leveled time error, (C) rms time error.(Curve A copyright Horological Journal withpermission.)

J F M A M J J A S O N D J F M A M J

C

30

20

10

0

–10

–20

–30

–40

1974 1975Time

Tim

e er

ror

(s)

A

B

actual measured time error of the clock over an 18-month period [5].Curve B shows the leveled time error with the average slope of curve Aremoved. The average slope is defined here by the beginning and end-ing points of curve A.3 The square root of the Allan variance on thedata has been calculated by Woodward [2], and is shown in Figure 8.4.And multiplying the square root of the variance by its associated timeintervals gives the rms time error in seconds, which is shown as curveC in Figure 8.3. The rms time error is both plus and minus, and isshown as such in Figure 8.3.

All three measures of the clock’s time error are plotted together inFigure 8.3, all to the same scale, for easy comparison. The actual andleveled time error curves (A and B in Figure 8.3) both cover 18 months.The rms time error (C in Figure 8.3) and the variance (in Figure 8.4)extend only one-fifth as far, 3.6 months, because of the statistical needfor a minimum of five samples to obtain a reasonably accurate Allanvariance. The short length of the rms time error curve severely limitsits usefulness as an indicator of clock performance, and raises the ques-tion of whether it is worth the effort of calculating it. For atomic timestandards it certainly would be, but for pendulum clock purposes,I do not think it is.

Conclusions

I think the traditional graphs of a clock’s actual time error and leveledtime error vs time are still the best way to show a clock’s time per-formance. A clock’s Allan variance is essentially just a number on aglobal performance scale containing all types of oscillators, and pro-vides no really useful information to the pendulum clockmaker.However, the variance can be used to calculate the rms time error,which is useful to him. Unfortunately, both the variance and the rmstime error have inherently short time scales, amounting at most to one-fifth of the length of the actual and leveled time error curves. Theirshort time spans drastically reduce their usefulness as indicators of aclock’s time performance.

chapter 8 | Allan variance and rms time error

41

Figure 8.4. Allan variance for the Batemanclock. After Woodward [1]. (Copyright OxfordUniversity Press, with permission.)

Var

ian

ce,

(ppm

)∆T T

1.00

0.50

0.20

0.10

0.05

0.02

0.011 2 5 10 20 50 100 200 500

Time interval (days)

3 Allan recommends this method forremoving the slope in [3, p. 652] and thenhedges by saying the optimum methoddepends on the oscillator’s noise characteristics.

Look at Figure 8.3, where all three time error curves are shown: theactual, leveled, and rms time error curves. Then decide for yourselfwhether the effort to calculate the variance and the rms time error isworthwhile. A few people should still calculate them, just to see ifsomething more useful might come out of it.

And finally, a few words on perspective. The Allan variance is ameasure of an oscillator’s random errors only, with all of the systematicerrors carefully removed. It is not a measure of the total oscillator error,which has to include the systematic errors as well. And as anyone whohas done experimental work knows, the effects of systematic errorsare usually 10–100 times larger than those of random ones. Even withthe atomic frequency standards, the systematic errors are bigger thanthe random ones [6, p. TN-23]. With pendulum clocks, the biggest timeerrors are also systematic: temperature, and setting the clock at exactlythe right rate. A pendulum’s random errors, although interesting, are offar less importance than the systematic ones.

References1. D. Allan et al. “Standard terminology for fundamental frequency and time

metrology,” 42nd Annu. Freq. Control Symp. (1988), pp. 419–25. Also in [6,TN–139].

2. P. Woodward. My own right time, Oxford University Press, 1995, pp. 124–5.3. D. Allan and H. Hellwig. “Time deviation and time prediction error for

clock specification, characterization, and application,” Proc. Position Locat.Navig. Symp. (PLANS), 1978, p. 29.

4. D. Allan. “Time and frequency (time-domain) characterization, estimation,and prediction of precision clocks and oscillators,” IEEE Trans. Ultrason.Ferroelectr. Freq. Control UFFC-34(6) (November 1987), 647–54.

5. D. Bateman. “Electronically maintained precision pendulum clock—longer term performance,” Hor. J. (October 1975), 3–11.

6. D. Sullivan et al. (eds). “Characterization of clocks and oscillators,” NISTTechnical Note 1337 (1990). Available (free!) from NIST, Boulder, Colorado,Phone 303-497-3212. This is an updated collection of papers on the Allanvariance. Recommended.


42

chapter 9

Transient temperature effects ina pendulum

This chapter describes some transient temperature measurements made on apendulum with a quartz pendulum rod.

I was having difficulty trying to correct the time offset error causedby transient temperature changes in a pendulum, one with a quartzrod, when George Feinstein suggested going back to fundamentals andactually measuring the temperature and how it varied at multiplepoints across the pendulum. This would be done with the pendulumoperating normally (i.e. swinging), and during a step change in ambienttemperature. It would give some idea of the heat flow between thependulum’s various parts, and might provide some insight into the timeoffset error.

The time offset error occurs because different parts of the pendulumchange temperature at different rates. Before and after a temperaturechange, the pendulum is the right length (hopefully) and runs at theright rate. But during the temperature change, the pendulum is thewrong length, due to its different parts changing temperature at differ-ent rates, and it runs at the wrong rate during the temperature changeinterval. This chapter is intended to be about the transient temperaturechanges only, so their effect on the offset time error will not be covered.

Small thermistors, about 0.1 0.1 0.05 in. size, were used to meas-ure the pendulum temperatures. These were connected to an externalmultipoint switch and a digital ohmmeter via #35 fine copper wire, withthe wires passing up the pendulum rod and forming a loose half inch cir-cular loop (i.e. a low friction “hinge”) as they went past the suspensionspring. The thermistors were calibrated for resistance vs temperatureusing a small oil bath and a mercury-in-glass thermometer with a NIST-traceable calibration. Each thermistor was placed on the pendulum sur-face using a dab of heat-conducting grease, and covered over and heldthere with a in. rectangle of Scotch tape. Figure 9.1( b) shows thelocations of the 15 temperature sensors on the pendulum.

The pendulum’s mechanical design is shown in Figure 9.1(a). Thependulum has a 2 s period. The quartz rod is 0.641 in. in diameter and

58

12

43

51.5 in. long. The spherical brass bob is 4.9 in. in diameter, weighs 18.4 lb, and is supported at its center by a pyrex sleeve. This sleeve isusually quartz, but was changed to pyrex in this case. The pyrex sleeveprovides about one-third of the temperature compensation.

The other two-thirds of the temperature compensation is provided bytwo thin-walled pyrex tubes 7.8 in. long, that are located parallel to andon opposite sides of the quartz pendulum rod. These temperature com-pensating tubes are located in the plane of swing, and are supported topand bottom by stiff brass end caps. The end caps provide no temperaturecompensation, other than the unavoidable change in the brass’ springconstant with temperature (beam bending mode). The bottom end capis supported on the quartz rod by an invar sleeve, with an invar dowelpin that passes horizontally through both the sleeve and the rod.Although not shown in Figure 9.1, the outer surface of the invar sleeveis threaded for a movable invar nut, to adjust the clock rate. The invarsleeve adds a very small amount of temperature compensation.

The clock case is wrapped in electric blankets and heated for 2.5 daysto stabilize the internal temperature. A 17 C step down in tem-perature is provided by removing the blankets and the front door ofthe clock case. The stabilized temperatures are shown at 8 AM in


44

Figure 9.1. (a) Pendulum’s mechanical design and (b) locations of temperaturesensors.

BeCu suspension spring

(a) (b)

Quartz rod

Brass bob

Pyrex sleeve

Brass topend cap

Pyrex temperaturecompensatingtube (1 of 2)

Brass bottomend cap

Invar sleeve

Invar dowel pin

1

2

3

4

5

6

7

8

9

11 10

12 14 13

15

Figures 9.2 and 9.3. Three minutes later, the blankets and the front doorof the clock case were removed, and Figures 9.2 and 9.3 show the ensu-ing changes in temperature over the next 15 h.

Ambient temperature is read on a glass-mercury thermometer in theroom at a height of about 50 in., not at the clock. Temperatures takennear the bottom of the clock can stabilize at values less than ambient.

The two suspension spring temperatures (1 and 2) stayed togetherand are plotted as one (suspension spring). The two bob temperatures(6 and 7) stayed together and are plotted as one (bob). The two pyrexsleeve temperatures (8 and 9) stayed together and are plotted as one(pyrex sleeve). The two top end-cap temperatures (10 and 11) stayedtogether and are plotted as one (top end cap). And the two pyrex com-pensating tube temperatures (12 and 13) stayed together and are plottedas one (pyrex compensating tubes).

chapter 9 | Transient temperature effects

45

Figure 9.2. Temperature vs time of variouspendulum parts after a 17 C step down inambient temperature.

34 in. above bob18 in. above bob

7 in. below bob

Quartz rod,in. above bob

12

Pyrex sleevebelow bob

Pyrex tubescompensating

38

34

30

26

22

8 10 12 2 4 6 8 10Time (h)

AM PMTe

mpe

ratu

re (

°C)

Figure 9.3. More temperature vs time ofvarious pendulum parts after a 17 C stepdown in ambient temperature.

38

34

30

26

22

8 AM 10 12 2 4 6 8 10 PM

Time (h)

Tem

pera

ture

(°C

)

Room ambientSuspension springTop end cap

Bottom end cap

Bob

Results

1. The ambient room temperature rose 0.2 C at 3 PM and fell 0.6 C at6 PM in Figures 9.2 and 9.3, and all the recorded temperatures roseand fell along with that. Because of this, the 0.2 C rise and 0.6 Cfall in the recorded temperatures at 3 and 6 PM should be ignored.

2. Both the quartz rod and the pyrex temperature compensatingtubes change temperature quickly, within less than 1 h. The topand bottom end caps also change temperature fairly quickly, butnot as fast as the quartz rod and temperature compensating tubes.

3. The slow temperature drop in the pyrex sleeve below the bob andparticularly in the quartz rod in. above the bob indicate thatsome heat is flowing downward out of the bob, and that a lot ofheat is flowing upward out of the bob and into the quartz rod.These heat flows would reverse direction, of course, with a step upin ambient temperature instead of a step down.

4. The suspension spring assembly changes temperature relativelyslowly.

5. The bob, with its large thermal mass, changes temperature theslowest of any part of the pendulum. This is to be expected.

In summary, the temperature data in Figures 9.2 and 9.3 provide aninteresting look into the thermodynamics of a pendulum.

12


46

chapter 10

Transient response of a pendulumto temperature change

This chapter describes the transient response of a pendulum to a step changein temperature. It also describes how to eliminate the resulting permanenttime error.

Permanent time error

What happens when a temperature compensated pendulum encoun-ters a change in temperature, other than the obvious change in clockrate if the pendulum is not perfectly compensated? This questionstarted from several mentions [1, 2] in old issues of the HorologicalJournal concerning mercury-type pendulums (mercury bobs, actually)that took 2–3 days to stabilize and run at a constant rate after a changein temperature. The delay is due to the large thermal mass of the mer-cury, which is enclosed in a glass or metal container. How big is thisdelay effect with the 19 lb metal bobs that I am currently using? A heattest in which the pendulum is given a step change in temperature wouldshow what was happening.

A heat test requires making an oven for the clock. To make theoven, the clock case is wrapped with three electric blankets andheated to 17 C (30 F) above the room’s ambient temperature for2.5 days to make sure the pendulum is stabilized at the highertemperature. Removing the electric blankets and the front side ofthe clock case then provides a 17 C step down in temperature tothe pendulum inside which has a 2 s period. The pendulum’s timeerror response to the step input of temperature is monitored byrecording over a 12–16 h interval the time difference between thependulum’s ends-of-swing and WWV, the radio time standard in theUnited States. A moving magnet on the pendulum and a fixed coil onthe clock case give a polarity-reversing voltage null at each end of thependulum swing that accurately defines (electrically) the two ends of thependulum’s swing.

47

Figure 10.1(d) illustrates the time response of a typical temperaturecompensated pendulum to a step (down) change in temperature.Figure 10.1(d) shows a permanent T time error or time change of upto 0.2 s in the pendulum’s timing, due to a 17 C step change in tem-perature. If the pendulum’s temperature compensation is not perfect,then the time response will also show a rising or falling slope as well asthe permanent T increase, as shown in Figure 10.1( b or c).

What causes the permanent T time error? In general terms, thepermanent T time error is caused by mismatches in the thermalmasses and thermal conductances of the different parts of the pendu-lum. A temperature compensated pendulum runs correctly at a giventemperature. Then the temperature changes. The different parts of thependulum change temperature (and length) at different rates. Until allparts of the pendulum have stabilized at the new temperature, the over-all pendulum length will be wrong and the pendulum will run at thewrong rate. But after all the parts have stabilized at the new tempera-ture, the pendulum will again be at the right length, and will again runat the right rate. The rate error integrates over a period of time into apermanent time error, as shown in Figure 10.1(d).

In more specific terms, the error starts with a bob that has a largethermal mass and changes temperature slowly. In a typical temperaturecompensated pendulum, the temperature compensating sleeve islocated inside the bob, as shown in Figure 10.2, so the sleeve cannotchange its temperature (or its length) any faster than the bob does. Incontrast, the pendulum rod has a low thermal mass and can change itstemperature (and its length) rather quickly. The result is a temporarychange in the pendulum’s length and in its clock rate, which integratesover a short time interval into a permanent time error, as shown inFigure 10.1(d).

When the temperature falls, a positive permanent time error occurs.When the temperature rises, a negative permanent time error occurs,so the two errors would appear to cancel, at least partially. The crit-ical issue is whether the permanent time errors cancel completelywhen the temperature changes in opposite directions at different rates(the usual situation) but with the same total rise and fall in temperature.I do not know yet. My test data is conflicting on this issue, some ofit saying yes and some saying no. But whether or not the plus andminus time errors do cancel each other out completely, the clock’stime is almost always in error by a small amount (up to 0.2 s for a17 C temperature change), and that small amount changes everytime the temperature changes. And when I compare my clock’s timeagainst WWV with 0.001 s precision (see Chapter 35), the error looksas big as a house, and it shows up as an error in all of my meas-urements. But there is a way to make the error go to zero, and that iscovered next.


48

Figure 10.1. Clock time error vs time: (a) heat input, (b) temperature compensatortoo short, (c) too long, (d) right length,(e) 2.5 days: startup transient decay, (f ) 2 h: slope calibration, and (g) 12 h: recordtest data.

1

0

0+

–

0+

–

0+

(e) (g)

Time

–

Hea

t in

put

Clo

ck t

ime—

WW

V (

s)

∆T

∆T

∆T = timeoffset

∆ = changein

slope

(a)

(b)

(c)

(d)

(f)

Figure 10.2. A “typical” temperaturecompensated pendulum design, showingthe temperature compensator up insidethe bob, and the bob being supported at itscenter.

3.8

6.0

in.

3.0

3.1

Bob, brass,19 lb

Temperature compensatorsleeve, brass

Threaded nut,invar

38

ø rod,invar

“Zeroing out” the permanent time error

The permanent time error is more sensitive to change in some partsof the pendulum than in others. To show this, the dimensions of a“typical” temperature compensated pendulum are given in Figure 10.2.This “typical” pendulum has a measured permanent time error of0.12 s. One at a time, various changes were made in different parts ofthis pendulum, to see which parts had the most effect on the permanenttime error. These changes and their effect on the permanent time errorare listed in Table 10.1. All of the testing described in this chapter is onpendulums with a 2 s period.

The effect of changing the bob’s height is shown in Figure 10.3. Theweight of each bob in Figure 10.3 was kept at 19 lb. When the bobheight was changed, the bob’s diameter was also changed, so that eachbob would have the same 19 lb weight. Figure 10.3 shows that the per-manent time error did not change much with bob height except for thetallest (12 in. high) bob, where the permanent time error increased to0.25 s, compared to 0.12 s for the “typical” bob with a 6 in. height.

The vertical location of the bob’s internal support point also affectsthe permanent time error. It does so by determining whether the bob’scenter of mass moves upward or downward (with respect to the bob’ssupport point) with increasing temperature, and with what amplitude.This movement is delayed in time because of the bob’s large thermalmass. Figure 10.4 shows how much the permanent time error changeswhen the bob’s support point is moved from the bottom of a bob to upnear its top. Figure 10.4 also shows the effects of changing the bob’sshape, weight, and material, and also of changing the pendulum rod’smaterial. The data in Figure 10.4 were taken with the temperature com-pensator located approximately 1.3 in.1 below the bob, so the perman-ent time errors shown in Figure 10.4 are about 0.07 s less than if thecompensator were up inside the bob, as shown in Figure 10.2. Thelinotype metal bob in Figure 10.4 is a 19 lb cylinder 3.1 in. in diameterand 6.9 in. long.

Table 10.1 shows that the two biggest sources of the permanent timeerror are (1) the closeness of the temperature compensator to the bob,

chapter 10 | Transient response to temperature change

49

Figure 10.3. Permanent time error vs bobheight. All bobs 19 lb brass cylinders.

0.1

02 4 6 8 10 120

0.2

0.3

Bob height (in.)

Per

man

ent

tim

e er

ror

(s)

1 1.3 in. between the bob’s bottom endand the compensator’s top end.

Table 10.1. Changes in the permanent time error caused by various changes in a “typical” temperature compensated

pendulum. The “typical” pendulum’s dimensions are given in Figure 10.2. The step change in temperature is 17 C (30 F)

Item Detail Change in permanent

time error

Increase (s) Decrease (s)

Increase bob weight From 5.3–19 lb, both brass cylinders 0.01

Change bob shape From cylinder to sphere, both 19 lb brass 0.06

Change bob height From 6.0 in. to 2.25, 3.25, 9.25, and 12 in. See Figure 10.3

heights. Bob diameters changed to maintain

constant 19 lb weight; all brass cylinders

Raise bob support point From center of bob to 2.5 in. above center 0.05 (0.10)

invar rod (quartz rod)

Lower bob support point From center of bob to 2.0 in. below center 0.04 (0.08)

invar rod (quartz rod)

Increase bob’s thermal From linotype metala to unleaded brass,a 0.05

mass and thermal both 19 lb cylinders

conductivity

Increase pendulum rod From to in. diameter, both invar 0.01

diameter

Change pendulum rod material From invar to quartz, approximately same 0.025

in. diameter

Increase temperature From to in., both brass 0.03

compensator wall

thickness

Move temperature compensator From inside bob to below bobb

0.5 in. below bob: invar rod (quartz rod) 0.04 (0.065)



Add thermal isolation At top and bottom ends of temperature 0.02

washers compensator. Compensator located 0.17 in.b

below 13 lb ellipsoidal bob. Washers are

window glass, O.D. I.D. in.

thick; invar rod

Notesa Thermal mass of 19 lb pure lead is 0.72 BTU/F; 19 lb of unleaded brass is 1.7 BTU/F. Linotype metal is 84% lead; thermal

conductivity of pure lead is 12 W/M K, unleaded brass 123 W/M K. Linotype metal is 84% lead.b Measured from bottom end of bob to top end of temperature compensator.

18

38

34

316

116

58

58

14

and (2) the vertical location of the bob’s internal support point. Thesetwo effects together can be made negative enough to “zero out” or evenreverse the polarity of the permanent time error. If the bob issupported at a point 2.5 in. above its center and the temperaturecompensator is moved to a location below the bob, then using thedimensions shown in Figure 10.5(a) (invar rod) or (b) (quartz rod) thepermanent time error will be zeroed out, as is shown by the time errorgraphs in Figure 10.6.

The actual permanent time error shown in Figure 10.6(b) for theinvar rod is 0.02 s rather than 0.00 s. If the pendulum with the invarrod were taken apart and then reassembled, the transient time errorwould vary about 0.02 s, which includes 0.00 s. That is close enoughto zero (on the far side of it, in fact) to prove the point that the error canbe zeroed out. The quartz rod pendulum has less variability than theinvar rod pendulum, and is easier to set to the zero position.


51

Figure 10.4. Permanent time error vslocation of bob support point, for (a) 19 lb spherical brass bob invar rod, (b) 19 lb spherical brass bob quartz rod, (c) 5.3 lb spherical brass bob quartz rod, (d) 19 lb cylindrical brass bob invar rod, and(e) 19 lb cylindrical linotype metal bob invar rod.

–4

–0.1

+0.2

0–2 +2

Below bob center (in.) Above bob center (in.)

(a)

(b)(c)

(d)

(e)

Per

man

ent

tim

eer

ror

(s)

Figure 10.5. The “typical” pendulum ofFigure 10.2 re-arranged for zero permanent time error for: (a) invar rod and(b) quartz rod.

Bob, brass,19 lb

Temperaturecompensator,brass

End cap,invar

Pin,invar

Threaded nut,invar

Spacer,invar

Spacer,quartz

4.7

0.5

0.5

0.5

5.6

2.1

6.0

in.

6.0

in.

0.64 ø Rod,quartz

38

18

Rod,invar

(a) (b)

ø

ø

Conclusions

The permanent time error can be zeroed out by a combination of mov-ing the bob’s support point to up near the top of the bob and movingthe temperature compensator to a position down below the bob.Using the pendulum dimensions given in Figure 10.5(a) (invar rod) or(b) (quartz rod), the permanent time error will either be zero or verysmall, even for a one-directional step in temperature.

The correction technique is not perfect. Although Figure 10.6 showsthat there is little or no permanent time error, a short-term transienttime error of 0.01–0.025 s amplitude remains. The short-term transienttime error in Figure 10.6 lasts for 7–8 h. And the pendulum rod has tohave a little extra length to permit mounting the temperature compen-sator down below the bob.

Figure 10.4 shows that supporting the bob at a point above the bob’scenter reduces the permanent time error. And supporting the bob at apoint below the bob’s center increases the permanent time error.

Suppose that one cannot move an existing bob’s internal supportpoint higher in the bob. Then on a smaller and simpler scale, Table 10.1shows that you can still eliminate two-thirds of the permanent timeerror just by moving the temperature compensator down to a position2–3 in. below the bob. The spacer used to fill the compensator’s formerspace inside the bob should be of the same material as the pendulumrod, so as not to disturb the pendulum’s temperature compensation.

Appendix: Comments on side issues

1. Supporting the bob at a point away from the bob’s center affectsthe length of the temperature compensator. When the bob issupported at its center, the correct temperature compensator


52

Figure 10.6. Pendulum’s transient response toa 17 C step down in temperature, with thependulum adjusted to zero or minimumpermanent time error: (a) heat input, (b) withinvar pendulum rod, (c) with quartzpendulum rod, and (d) pretest slopecalibration interval.

100

0

0.2

0.1

0.2

0.1

–2 0 4 8 12 16 +20

Time (h)

(a)

(b) –0.02 s

0.00 s

Permanenttime error

(c)

Hea

tC

lock

tim

e er

ror

(s)

(d)

length (by actual test) for the pendulum used is 2.0 in., as shown inFigure 10.7(b). If the bob is supported at 2.5 in. above its center,then the compensator must be 2.5 in. longer, or 4.5 in. total, asshown in Figure 10.7(c). If the bob is supported at 2.0 in. below itscenter, then the temperature compensator must be 2.0 in. shorter,in which case there is no apparent temperature compensator at all,the actual temperature compensator being the 2.0 in. of bobmaterial between the bob’s center and its support point 2.0 in.lower down, as shown in Figure 10.7(a). This assumes that the boband the compensator are both of the same type of metal—brass inthis case. If they are of different types of metal, then the amountadded to or subtracted from the compensator’s length must bemultiplied by the ratio of their thermal expansion coefficients. Thependulum dimensions given in Figure 10.7(a–c) are the dimensionsused to obtain the curve in Figure 10.4(d).

2. Both Leeds [3] and Feinstein [4] state that the correct supportpoint for a bob is not at the geometric center but “slightly below”it, so as to correct for the bob’s radius of gyration about its centerof mass. “Slightly below” means 0.10–0.15 in. below, dependingon the bob’s shape. Then the pendulum’s timing is independentof any thermal expansion effects in the bob itself. Theoreticallythis is true, but practically it is irrelevant. The pendulum must bethermally compensated by experiment, which means testing andcompensating it as a whole, as one piece, as there are parts ofthe pendulum whose thermal expansion cannot be calculated(i.e. suspension spring elasticity and end clamping effects, theuncertainty in the expansion coefficient of this particular pendulumrod, etc.).

I have made many month-long pendulum tests with the bobsupported at a variety of points above and below the bob’s center,


53

Figure 10.7. Variation in temperaturecompensator length with the bob’s supportpoint. Bob supported at (a) 2.0 in. belowcenter, (b) at center, (c) at 2.5 in. above center.

Bob, brass,19 lb

Bob, brass,19 lb

Spacer,invar

Spacer,invar

Threaded nut,invar

Temperature compensator,brass

Threaded nut,invar

6.0

in.

5.0

6.0

in.

6.0

in.3.

0 0.5

1.3

1.3

4.5

2.0

2.5

5.8

38 ø

(a) (b) (c)

Rod,invar 3

8 ø Rod,invar

with no observable bad effects and one good effect—the correctionof the pendulum’s thermal transient response.

3. The support point within a bob is easily moved. The bob is initiallybored out putting the support point 0.5 in. from the top of thebob. Then loose fitting sleeve bushings of varying lengths and ofthe same material as the bob are inserted in the bored-out hole tolower the support point to whatever level is desired. But this wasnot done with the linotype metal bob. With the linotype metal bob,no sleeve bushings were used as the softness of the metal mightintroduce undesired compression effects in the sleeve bushings.The testing of the linotype metal bob was done starting with thesupport point at the bottom of the bob, then repeatedly boring outthe bob a little more each time to move the bob’s support pointa little higher.

4. Figure 10.4 contains additional information on the variation in thepermanent time error with the location of the bob’s internalsupport point:(a) The variation is linear with the vertical location of the bob’s

support point, even with spherical bobs.(b) The permanent time error’s magnitude is directly proportional

to the bob’s thermal mass and the bob’s thermal expansioncoefficient. It is inversely proportional to the pendulum rod’sexpansion coefficient.

(c) Bob shape also affects the permanent time error. The perman-ent time error is much larger with a spherical bob than it iswith a cylindrical bob of equal weight.

5. The temperature compensators used for the data in Figure 10.4 areshorter than those used for the data in the rest of this chapter.Compare the compensator length in Figure 10.7(c) (used for part ofthe Figure 10.4(d) data) with that in Figure 10.5(a), and that inFigure 10.7(b) with that in Figure 10.2. The Figure 10.4 data wastaken 2 years earlier using hardened brass parts. The rest of thedata was taken using annealed brass parts and a slightly differentsuspension spring. This indicates that the thermal expansioncoefficient, and possibly Young’s modulus, change significantly withthe temper of the material.

6. The minimum wall thickness of the temperature compensatorsleeve is about 0.06 in. If made thinner, say 0.03 in., the requiredcompensator length for correct temperature compensationchanges considerably. Apparently, the material’s elastic modulusbecomes more of a factor in thermal expansion under axial loadwith the thinner wall. Compensator walls thicker than 0.06 in. donot affect the compensator’s length.


54

References1. L. Waldo. “Mercurial vs. zinc and steel pendulums,” Hor. J. ( July 1886),

161–5.2. T. Buckley. “On the superiority of zinc and steel pendulums,” Hor.

J. (October), 17–21; (November), 46; 62–3; (December 1886), ( June 1889),156–7.

3. L. M. Leeds. “The pendulum; Six scientific papers,” Pend. Temp.Compensation. No. 5, MB150LEE. Available from NAWCC Library,Columbia, PA, USA.

4. G. Feinstein. “Pendulum bob configuration and thermal compensation,”Hor. Sci. Newslett. of NAWCC chapter 161 (May 1994). Available fromNAWCC Library, Columbia, PA, USA.


55

chapter 11

Dimensional stability ofpendulum materials

Silicon, quartz, and type 642 aluminum silicon bronze were the most stablependulum materials tested. Free machining invar was the least stablematerial, partly because of its long length as a pendulum rod. Two differenttypes of heat treatment were needed to get good dimensional stability in thependulum metals.

The dimensional stability of a pendulum directly affects its accuracy.If the bob sags downward or warps upward, the clock slows down orspeeds up. If the temperature compensator shrinks or expands in length,the same thing happens. If the pendulum rod gets longer or shorter,again the clock slows down or speeds up. And it does not take much todisturb an accurate clock. A 27 in. change in the length of a pendulumwith a 2 s period will change the clock’s timing by 0.027 s/day, or10 s/year. And the Shortt clock’s total error of 1 s/year is 10 timessmaller than even these small numbers.

No one wants to use unstable materials on a pendulum. Are somematerials more stable than others? Very definitely yes! For instance,silicon and quartz are both very stable materials, by actual test. Andmetal alloys containing tin or nickel are relatively unstable metals, againby actual test. Some of the instability is caused by internal stressesrelieving themselves in a material. The situation is further complicatedby heat treatment, which can make a 2–8 times improvement in thedimensional stability of many metals. The internal stresses must beremoved before any accurate usage. If the internal stresses are notremoved, they will cause the parts to slowly relax themselves and slowlychange their shape (slightly) over many years’ time, causing the clockrate to continuously change over the same many years’ time.

Now, what is or is not dimensionally stable depends on where youstand—the builders of bridges and skyscrapers do not care if a beam movesa few thousandths of an inch. Only a very small group of users (whichincludes clockmakers) cares about stabilities of millionths of an inch.

Thermal hysteresis, which is the cycling of a material (or a pendu-lum) over a small temperature range to see if it will come back to its

57

original dimensions (or clock rate) at the original temperature, makes agood dimensional stability test. Thermal hysteresis generates a muchbigger dimensional change in a part than just letting the part age qui-etly, sitting in a drawer for 6 months (at room temperature (RT) ). Mostpendulum metals, as purchased, do not come back to the same clockrate after cycling over a small temperature range. The clock rate after asmall temperature cycle (28 F) can differ from the clock rate before thetemperature cycle by up to 0.2 s/day (at the same original tempera-ture). Errors this large will destroy any claim to clock accuracy.

I am interested in stable materials for pendulum rods, bobs, temper-ature compensators, and suspension springs. Nonmagnetic materialsare desired, such as the copper alloys, stainless steel, silicon, aluminum,and quartz. There is a lot of metallurgy in this chapter, as metallurgy isthe subject that deals with the stability of metals. Suspension springmaterials are covered separately in Chapter 20.

General metal errors

Most of the fundamental research in metals was done back in the1910–50 era—too old to be found in computer databases. There are fivegeneral errors in metals: creep, relaxation, fatigue, thermal hysteresis,and internal stresses. When a spring is deflected by a weight, over aperiod of time the deflection will increase a little. The increase in deflec-tion over time is called creep. After the spring has been deflected forsome time, suppose the deflecting weight is removed. The spring willimmediately move back toward but will not quite reach its original posi-tion. Over a period of time, the spring will move back even closer to itsoriginal position, but still will not reach it. This later movement overtime back closer to the original position is called relaxation error. Andwhen a spring is bent back and forth enough times, it will break fromthe repeated bending stresses. This is called a fatigue break. The num-ber of bend cycles until the spring breaks varies with the stress level inthe spring. A metal’s fatigue life is usually shown as a graph of the num-ber of cycles to failure vs the level of stress in the spring.

A material is slowly increased (or decreased) in temperature, andthen slowly brought back to its original temperature. The materialexpands (or contracts) with the temperature change, and then contracts(or expands) back toward but does not quite go back to its originaldimension. The difference between the original dimension and thedimension after the temperature cycle is called thermal hysteresis. Andunless a clock is kept in a temperature-controlled room, its pendulumwill cycle over a small temperature range every day.

To get a dimensionally stable material, all of these errors must bedealt with. One of the most effective ways is heat treatment.


58

Effects of manufacture

The manufacturing process has a big effect [13] on metals. Pure metals(i.e. not alloyed with other metals) are generally weak metals. Duringmanufacture, a small amount of the right alloying element(s) is added,making many metals 3–10 times stronger. The pure base metal is madeharder even if the alloying element is softer than the base metal. It usu-ally takes only a few percent of an alloying element to make a very bigdifference in the base metal. And heat treating steel and aluminumalloys can make them 2–10 times stronger yet. Copper alloys (with afew exceptions) are different in that they can only be softened by heat-ing. Copper alloys can only be hardened by coldworking—rolling thicksheets into thinner sheets, or extruding large diameter rods into smallerdiameter rods. Unfortunately, the rolling and extruding put lots ofstress into the metal. The copper alloys get stronger, but at the sametime they are also picking up a lot of internal stress.

One interesting property of metals is that magnetic metals becomenonmagnetic when they are mixed with and go into solution with othernonmagnetic metals in the making of alloys. As an example, 304 stain-less steel is 68% iron, which is obviously magnetic. Yet 304 stainlessitself is nonmagnetic.

Metals can be obtained in all tempers, but in general the metals pur-chased in the “retail” market have the following characteristics:

Steels—soft, with medium internal stress Stainless steels—annealed, with low internal stress Aluminums—hard (relatively speaking), with lots of internal stress Copper alloys—hard, with lots of internal stress.

For applications requiring a high level of dimensional stability, thestress level in most purchased materials must be reduced before theycan be used.

The naming of alloys varies. Steel alloys are just called steel, and alu-minum alloys are just called aluminum. Copper alloys are called brass ifzinc is the only significant additive (any lead content is ignored). But thenames of other copper alloys are a mishmash, depending on history,color, and content.

In describing an alloy’s content, the largest component is listed firstand the smallest last, such as for type 642 aluminum silicon bronze,which is 91% copper, 7% aluminum, and 2% silicon. A common short-hand listing of the content of this metal would be 91Cu7AL2Si, usingthe chemical name abbreviations.

The type metals

Type metals are used in the printing industry for making inked impres-sions on paper. The type themselves are made up as individual letter

chapter 11 | Dimensional stability of pendulum materials

59

and word “slugs” of type metal in linotype machines, which are nowmostly obsolete. The type metals are also used for large printing platesin huge printing presses. Our horological interest is in the type metals’high density, a useful property for making low-volume high-weight low-cost pendulum bobs. Such bobs have less air drag, and consequentlyrequire less driving force for a given bob weight. The type metals’ den-sities vary, from about 60 to 90% of that of pure lead, or from about 0.8to 1.2 times the density of brass. (The density of the type metals isapproximately the same as that of brass, but somehow this never getsmentioned.)

Pure lead is too soft to use for a bob, as pure lead sags with time (atRT). The type metals are about 10 times stronger than pure lead, byactual test (mine). The type metals are mostly lead, but contain varyingamounts of antimony and tin for strengthening and hardening. Thereare five different type metals. Table 11.1 lists some data on them fromseveral sources and from some personally conducted tests.

The clockmaker can make his own type metal bob. First, buy a 25 lbingot of (say) linotype metal at about US$1.50 per lb (in 2003) fromyour local lead supplier, and melt it down in a tin can of appropriatesize, using multiple propane torches or one large propane torch forheat. After melting, I let my tin can of linotype metal “air cool” sittingon a concrete driveway, which is a simple but poor way of doing it. Theslow air cooling does anneal the bob, however. The type metal bob is


60

Table 11.1. Properties [1, 2] of the type metals and pure lead

Electrotype Linotype Stereotype Monotype Foundry Pure

type lead

UNS No. 52,830 53,425 53,530 53,570 53,750 51,120

Density, lb/in.3 0.37a 0.28a 0.409

Thermal expansion, 106/C 24a 29.3

Modulus of elasticity, psi 106 2

Yield strength, psi ≈6000 ≈8000 ≈7000 ≈10,000 800

Creep strength, %/Yr @ psi 3 @ 300

Fatigue strength, psi for 107 cycles 620

before failure

Thermal conductivity, W/M K 35

Specific heat, Cal/g C ≈0.036 0.0309

% Antimony 3 11 14 15 24 0

% Tin 3 5 6 7 12 0

% Lead 94 84 80 78 64 99.9

Notea Measured by author.

machined to size afterward in a lathe. Pure lead and pure antimonydiffer in density and solidification temperature, and this causes a ten-dency for them to separate on cooling. In addition, a sizable air hole(approximately 0.8 in. in diameter in my case) appears up the center ofthe casting in the tin can. Both problems can be avoided by using a fastcool, that is, dunking the can of hot type metal in RT water. Thenanneal the cast type metal to remove the internal stresses you put intoit previously with the fast cool.

Heat treatment

There are at least two useful heat treatments for pendulum materials:annealing and temperature cycling. Annealing reduces the internalstresses in a material. Temperature cycling improves the dimensionalstability of metal alloys by a different mechanism. There is a third heattreatment called cryogenic cooling, but metallurgists are divided on itsbenefits. A 6-month literature search found three reports on dimen-sional stability efforts in the past, each trying to understand andimprove the stability of metals. Two involved hardened steel (gageblocks) [4, 5, 26] and one involved hardened non-ferrous metals [3, 25].But as discussed later, pendulum metals can be soft (annealed) and donot need to be hard. This is a big advantage, heat treatment-wise andstability-wise.

The effect of annealing, and the high stress level existing in coldworkedcopper alloys was shown in four 4 in. long type 360 brass temperaturecompensators that grew 0.1–0.2% (0.004–0.008 in.) in length when I hadthem annealed. For a pendulum with a 2 s period, a length change of0.004–0.008 in. corresponds to a clock rate change of 4–8 s/day. Fora clock to be accurate to 10 s/year, the maximum daily error rate is0.027 s/day, which is equal to a length change of 27 in. in a pendulumwith a 2 s period. The dimensional change from annealing exceeded thiserror by more than 100 to 1. That whopping change is why pendulummetals need to be annealed for long-term stability.

The heat treatment of metals has been around so long that many ofthe procedures are listed as “recipes” in a book. Heat treating deals mostlywith steel and aluminum—making them harder or softer as needed.Copper alloys are only a small part of the heat treating business, partlybecause heat cannot strengthen (harden) copper alloys. Heat can onlysoften copper alloys, and that is not nearly as useful as strengthening.(There are a couple of exceptions to the copper “heat softening only”rule. For instance, beryllium copper can be made 2.5 times strongerby heat.)

To relieve internal stresses, a part is heated in a furnace until the metalbecomes soft, but not a liquid. The softer metal allows the internal


61

stresses to relieve themselves much faster at the higher temperature,allowing the part to pull itself into a slightly distorted but (almost)stress-free shape. Even at the high annealing temperature, a metal is stillnot very viscous and it takes time for the stresses to move the metal intoa stress-free shape. The longer the part soaks at high temperature, themore the internal stresses are relieved, and the more dimensionallystable the part becomes. The usual rule is a 1 h soak at high tempera-ture per inch of thickness of the part.

The part is slowly brought back down to RT, to prevent temperaturegradients across the part from putting more stresses into the metal.About a 1 h cool down to RT with the furnace fan shut off, or a “fur-nace cool” (shut off the heat and let the furnace cool down to RTby itself, with the parts inside) is appropriate for copper alloys. The cool-down interval can be shorter for copper alloys than for steel, because the2–3 times greater thermal conductivity of copper (with respect to steel)reduces any thermal gradients across the part in the same ratio.

Heat treatment inherently softens copper alloys (with a few excep-tions), and reduces their yield strength. By unspoken convention, a“stress relief ” heat treatment will remove as much stress as possible, butspecifically must not reduce a metal’s yield strength by more than asmall amount, say 10%. One of my tests showed that only about one-third of the internal stress is actually removed with a “stress-relief ” heattreatment. Others say that about three-fourths of the internal stress isremoved. Whatever the actual value, it is not enough for accurate pen-dulum work.

To remove more stress, you soak the part longer and/or at a highertemperature. The process is the same as before, but because the highertemperature soak will cause a copper alloy’s yield strength at RT todrop to about one-third of its initial (coldworked) value, the process’name is changed from “stress relief ” to “anneal.” For accurate pendu-lum work, a “full anneal” heat treatment is desired, that is, the removalof the maximum amount of internal stress. The big 3 to 1 reduction ina metal’s yield strength that comes with the annealing heat treatmentdoes not matter for pendulum work, because except for the suspensionspring, a pendulum is all low stress. The highest stress in my pendulumis a mere 220 psi (suspension spring excepted), and occurs at the in.diameter crosspin that fastens the pendulum rod to the suspensionspring.

As to cost, heat treating a few pendulum parts will cost US$60 (airfurnace) or $200 (vacuum furnace). These are the minimum furnacecharges. Price goes according to weight and furnace time. Annealing50 lb of brass for 1–3 h costs the $60 minimum charge (in the year 2000)at the last heat treat company I used. If the processing temperature isunder 1000 F, an open air furnace is used. Over 1000 F, a vacuumfurnace is used, to minimize surface scale on the parts.

18


62

Warpage of a long skinny object like an invar pendulum rod is aproblem in heat treating. To minimize this, put a cross-hole through therod at one end, and put a wire loop through it so that the rod can behung straight vertical in the furnace. If the rod is laid horizontal in thefurnace, it will bend and conform to whatever it is laying on. My experi-ence is an invar rod 4 ft long and hung vertical in the furnace will warp(bow) about 0.02 in. out of true. That is an acceptable amount. But onlyone of the local heat treating companies has a furnace that can handlea 4 ft vertical rod and also quickly lower it into cold water as is neededafter heating it.

Temperature cycling

Temperature cycling was recommended to me by a local metallurgist,as a way of getting better dimensional stability than what an annealingheat treatment will give just by itself. Temperature cycling is doneafter the annealing heat treatment, and consists of multiple cycles ofraising the temperature of a part to or a little above its maximum oper-ating temperature, and then down to or a little below its lowest operat-ing temperature. Five or so cycles are considered enough, with the partbeing allowed to soak for awhile (several hours to a day) at each tem-perature. For operation at RT, I picked high and low cycle temperaturesof 200 F and 40 F because they are easily done in the homekitchen, moving the part back and forth between the oven and therefrigerator. To reduce thermal shock, the part is allowed to warm up(or cool down) for 15–30 min at room ambient on the kitchen tablewhen being moved from the refrigerator to the oven or vice versa. Soaktimes at these low temperatures have to be long because the metal isless viscous here than at the high annealing temperatures.

Temperature cycling by itself produces only a small reduction in ther-mal hysteresis. But temperature cycling after annealing was found tobe very effective, reducing the thermal hysteresis of annealed 642aluminum silicon bronze by 6.7 times, and the hysteresis of annealed304 stainless steel by 2.5 times. Epprecht [24] gives a description of howthermal cycling works. It is based on the differences in the thermalexpansions of the different metal components in an alloy. What temper-ature cycling does to a pure metal is left unexplained.

All of the bobs and temperature compensators were temperaturecycled 5–7 times between 200 F and 40 F. A few of the early pieceswere soaked for only 2 h/cycle at each temperature, but the rest weresoaked all night at 200 F in the kitchen oven, and all day at 40 F inthe kitchen refrigerator (one cycle per day). Some pieces were tested forthermal hysteresis both before and after being temperature cycled, tosee how much improvement the temperature cycling would provide.


63

Cryogenic cooling

Cryogenic cooling involves cooling a part down to liquid nitrogentemperature—about 300 F. In aluminum, it reduces or eliminates theresidual stresses [27]. In steel, it makes hardened steel even harder. As tobrass, very little is known about the effects of cryogenic cooling.Cryogenic cooling has even been used to improve the performance ofsome plastics.

A cryogenic cooling cycle (cooling to 300 F, soaking awhile, thenwarming back up to RT) is very long time-wise—about 1–5 days. Thatis because of the low “viscosity” of metal at such a low temperature.Two and three to one improvements in metal performance in many realworld applications show that the process works. The fly in the ointmentis that no one understands how it works. Hence the controversy as toits value among metallurgists. Research on it is currently being done inseveral places. Cryogenic cooling is not too expensive. Locally, it costsabout US$60 minimum for up to 50 lb of material. The CryogenicSociety of America [31] can provide more information on cryogeniccooling.

None of the pendulum parts described in this chapter were cryo-genically cooled. But if cryogenic cooling can reduce or eliminate resid-ual stresses in aluminum, would it not do the same for brass pendulumparts? Would it reduce thermal hysteresis? This is worth investigating inthe future.

Selection of test materials

For pendulum use we would like materials with zero thermal hystere-sis. However, there is almost no information on thermal hysteresis inthe literature. Strong’s remark that quartz’s thermal hysteresis is only5% of invar’s was one of only two references [6, 7] found. The materialsselected for testing were nonmagnetic, mostly medium density, and ifthe data was available, with a low relaxation error. If a metal’s relaxationerror is low, then hopefully its thermal hysteresis might also be low. Mostof the materials selected were copper alloys, with one stainless steel andthree aluminums picked as well.

Both beryllium copper and phosphor bronze are known to have goodspring properties, and were included. Brass was included because of itslong horological history and common availability—with and withoutlead and silicon additives. One cast brass (875 silicon brass) was pickedbecause it contained a lot (4%) of silicon. A local foundry provided thematerial and cast a bob-sized cylinder of it, which I then machined intoa bob.

As to stainless, a local metallurgist recommended type 304, sayingthat type 303 has too many additives in it for good stability. The stability


64

of invar is inversely proportional to the amount of carbon it contains.Because of this, the low carbon version (304L) of 304 stainless steel wasselected. Some metals were picked on the basis of isolated remarksfound in the literature: that a particular alloy had a low relaxation error,that adding a little silver to copper reduced the copper’s relaxationerror, that 729 copper nickel tin had a long fatigue life and low relax-ation, etc.

Aluminum alloys are known to be unstable. The American Societyfor Metals (ASM) Metals handbook says that 2024 aluminum is morestable than types 6061 or 7075. Samples of all three aluminums wereincluded to see how bad they were. A linotype metal bob was includedfor comparison purposes. A tungsten bob was not included becauseof its high cost. To make it machinable, it would have been one ofabout six tungsten alloys (2% Mo, 15% Mo, 2–11% NiCu or NiFe)rather than pure tungsten, with a density of about 90% of that of puretungsten.

Zinc has the highest thermal expansion of any practical metal (40 106 /C). As such, in the old days (around 1900), zinc rods wereused together with iron rods, which have a low thermal expansion, tomake gridiron pendulums. Unfortunately, zinc is dimensionally unsta-ble [32]. Pure zinc is soft and will creep under load at RT. Zinc alloys


65

Table 11.2. Pendulum rod materials

Invar Quartz Sapphire Silicon

(free machining) (fused silica) (single crystal)a

Polycrystallineb Single crystalc

Thermal expansion, 106 /C 0.8–3.0 0.5 5.4–8.8 2.6 2.33

Density, lb/in.3 0.291 0.135 0.143 0.084 0.083

Magnetic? Yes No No No No

Modulus of elasticity, 106 psi 20.5 10.6 50.–63d 22.6 27.6

Yield strength, 103 psi 40 8.5 38.–150d 174 1010

Tensile strength, 103 psi 65 8.5 38.–150d 174 1010

Design strength, 103 psi — 0.5–1.5 50 — —

Thermal conductivity, W/M K 10.5 1.4 41.9 141 157

Specific heat, Cal/g C 0.12 0.17 0.18 0.17 —

Thermal mass, Cal/in.3 C 15.8 10.4 11.7 6.5 —

Thermal hysteresis, s/day (as 0.097 0.01 — 0.003 —

pendulum, 1 s beat, T ≈ 28 F)

Notesa From [28].b From [27] except for thermal hysteresis.c From [8].d Varies with orientation or manufacturer.

are stronger but are still dimensionally unstable. Typical zinc alloys AG40Aand AC41A increased their dimensions by an average of80 106 in./in. over a 6-year time span in an indoor environment at 70 F(and presumably at zero load) [33]. Because of zinc’s instability, no metalscontaining zinc were selected, other than brass and 7075 aluminum.

Three pendulum rod materials were tested: quartz (fused), invar(free machining type), and silicon (polycrystalline). A 4 ft long pieceof single crystal silicon was too expensive (US$10,000), so I settledfor the much cheaper (US$100 in 1998) polycrystalline silicon [28].Sapphire [29] is available in a 2 foot maximum length, with about a0.2 in. maximum diameter. The properties of these materials are listedin Table 11.2.

Single crystal silicon is a unique material in that it has perfect springproperties—zero creep, zero relaxation error, and zero thermal hys-teresis. Silicon is brittle, like glass. And fatigue strength has no meaning


66

Table 11.3. Materials selected for thermal hysteresis testing

Material Composition, % Costa Density Thermal Young’s

US$/lb lb/in.3 expansion modulus

106 /C 106 psi

— Brass, unleaded (old) 65Cu35Zn (est.) 3.70 — — —

360 Brass, leaded 61Cu36Zn3Pb 3.– to 5.50 0.307 20.5 14.6

464 Brass, naval 60Cu39Zn0.75Sn 4.25 to 5.– 0.304 21 15

875 Brass, silicon 82Cu14Zn4Si 2.– to 12.– 0.299 19.6 15.4

107 Silver copper 99.9Cu0.085Ag 10.40 0.323 17 17

172 Beryllium copper 98Cu2Be 18.– to 21.– 0.298 17.5 18.5

510 Phosphor bronze 95Cu5Sn0.1P 5.– to 20.– 0.320 17.8 16

630 Aluminum nickel bronze 80Cu10AL5Ni3Fe 4.– 0.274 16 17.5

642 Aluminum silicon bronze 91Cu7AL2Si 4.– to 7.– 0.278 18 16

655 Silicon bronze 97Cu3Si 7.50 to 9. – 0.308 18 15

715 Copper nickel 70Cu30Ni 9.– 0.323 16 22

729 Copper nickel tin 77Cu15Ni8Sn 11.– to 18.– 0.323 16.4 18.5

304L Stainless steel 68Fe18Cr10Ni2Mn 3.– to 4.50 0.29 17.2 28

2024 Aluminum 93.5AL4.4Cu1.5Mg0.6Mn 5.80 0.101 23.2 10

6061 Aluminum 97.9AL1.0Mg0.6Si0.3Cu 2.90 0.098 23.6 10

7075 Aluminum 90.0AL5.6Zn2.5Mg1.6Cu 5.80 0.101 23.6 10

53425 Linotype metal 84Pb11Sb5Sn 1.– 0.37 24 —

— Quartz rod 100SiO2 (fused) 28.– 0.135 0.55 10.6

— Invar (free machining) 64Fe35Ni0.2Se 13.– 0.291 0.8–3.2 20.5

— Silicon rod 100Si (polycrystalline) 62.– 0.084 2.6 16.4

Notea In 1998. Price varies with rod diameter and vendor.

with silicon. Polycrystalline silicon is almost but not quite as good. Ittoo is brittle. I am told that single crystal sapphire also has perfect springproperties. The existence of the perfect spring properties of silicon andsapphire seems to be well known within the silicon and sapphire indus-tries, but is almost totally unknown outside them.

Some properties of the materials selected for testing are given inTable 11.3. Table 11.4 lists sources for the pendulum materials, some ofwhich have only one or two sources.


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Table 11.4. Sources for pendulum materials

Material Sources (all in USA)

— Copper alloys (general) Standard Metals

Hartford, Connecticut

— Copper alloys (general) Copper & Brass Sales

Chicago, Illinois

107 Silver copper U.S. Brass & Copper

Downers Grove, Illinois

172 Beryllium copper (strip) Mead Metals

St. Paul, Minnesota

510 Phosphor bronze (sheet) Lewis Brass & Copper

Middle Village, New York

729 Copper nickel tin Brush Wellman

Cleveland, Ohio

875 Silicon brass (18 lb ingots) St. Paul Brass & Aluminum

St. Paul, Minnesota

304L Stainless steel (Process 70) Carpenter Technology Corp.

Reading, Pennsylvania

902 Ni Span C (bar) Special Metals Corp.

Huntington, West Virginia

902 Ni Span C (strip) Hamilton Precision Metals

Lancaster, Pennsylvania

Quartz rod GM Associates

Oakland, California

Quartz rod (avail. annealed) Quartz Scientific

Fairport Harbor, Ohio

Invar rod (free machining) Fry Steel

Santa Fe Springs, California

Invar (all three types) Scientific Alloys

Westerly, Rhode Island

Silicon rod (polycrystalline) Mitsubishi Polysilicon

Theodore, Alabama

Test program

An oven is needed for temperature testing. The clock case is made intoan oven by wrapping three electric heating blankets around the case,and varying the electrical power into the heating blankets. The clockcase temperature is cycled three times over a small temperature range.The first two cycles are not measured, and represent an attempt to givea common thermal history to each metal tested. The first cycle movesthe clock case from RT to RT16 C, and then back to RT. The secondcycle moves the clock case up halfway to RT8 C, and then back toRT. On the third cycle, test data on clock rate vs temperature is taken:initially at RT, then at RT8 C and at RT16 C, both with increas-ing temperature, then at RT8 C and at RT, both of these withdecreasing temperature, and finally at RT8 C again with increasingtemperature.

Figure 11.1 is a typical thermal hysteresis curve—a graph of clockrate vs temperature. It does not always form a closed loop, dependingon the relative amounts of hysteresis of the different pendulum parts.In Figure 11.1, the pendulum is continuously running faster, as if thetemperature compensator was growing longer. The data in Figure 11.1is for a pendulum with a temperature compensator of 7075 aluminum(hard and temperature cycled), a bob of 642 aluminum silicon bronze(annealed and temperature cycled), and a quartz pendulum rod. Fromother tests, both the bob and the pendulum rod were known to bequite stable, so the hysteresis shown in Figure 11.1 is due almostentirely to the aluminum temperature compensator. Thermal hys-teresis is defined here as the maximum variation in clock rate at themiddle temperature, that is, at about 30 C, and measures 0.061 s/dayin Figure 11.1.

Figure 11.1 shows two extra factors that need explaining but do notaffect the ranking of the hysteresis data. First, if the pendulum had zerohysteresis and perfect temperature compensation, all of the data inFigure 11.1 would lie on a straight true vertical line. Second, if the pen-dulum had zero hysteresis but the temperature compensation was a littleoff, the data in Figure 11.1 would still lie on a straight line but the linewould have a slope to it—it would no longer be true vertical. And third,if the thermal expansion is non linear, the line would be curved (twobent segments) as shown in Figure 11.1. But as long as the nonlineartemperature conditions are the same for every test, they do not affect therelative rankings of the thermal hysteresis measured on each material.

Figure 11.2 shows the effect of continuously thermally cycling a pen-dulum over a 6-week interval. The pendulum has a quartz rod and bothits bob and its temperature compensator are made of type 464 navalbrass, annealed and temperature cycled. The dotted lines show the firsttwo temperature cycles that normally were not recorded. It is a little


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Figure 11.1. Thermal hysteresis—thevariation in clock rate after a temperaturecycle.

40

35

30

25

+0.2 +0.1 0 –0.1 –0.220

Clock rate (s/day)

Tem

pera

ture

(°C

)

End

Start

Hysteresis =0.061 (s/day)

hard to see in Figure 11.2, but the clock rate is slowly increasingthroughout the 6-week run. The thermal hysteresis is more easily seenif the clock rates at the temperature midpoint (29 C) in Figure 11.2 arereplotted as a function of time, as in Figure 11.3. The thermal hystere-sis is the difference in the clock rate with the temperature going up andgoing down. The scatter in the data is larger than what one would like,but Figure 11.3 clearly shows that the thermal hysteresis remainsroughly constant over time and multiple thermal cycles. It does notgradually grow smaller or bigger.

The three pendulum arrangements used for hysteresis testing areshown in Figure 11.4. Because of its low hysteresis, the quartz rod inFigure 11.4(c) was used for all of the testing except for two of the pen-dulum rod tests at the bottom of Table 11.5. The arrangementsin Figure 11.4 are the traditional one for pendulum rods with a low


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Figure 11.2. Effect of repeated temperaturecycling on clock rate.

35

30

25

+0.1 0 –0.2–0.1 –0.320

Clock rate (s/day)

Tem

pera

ture

(°C

)

End Start

Figure 11.3. Clock rate at mid-temperature ofrepeated temperature cycles.

+0.2

+0.1

0

–0.1

–0.20 10 20 30 40 50

Clo

ck r

ate

(s/d

ay)

Time (days)

Temperature going up

Temperature going down

Thermalhysteresis

Figure 11.4. Pendulum arrangement for thermal hysteresis testing, with (a) invar rod, ( b) silicon rod, and (c) quartz rod.

0.37 øInvar rod

0.75 øSilicon rod

0.64 øQuartz rod

Bob, brass19 lb

(Test) bob19 lb

(Test) temperaturecompensator

Spacers,quartz

Endcap + pin,invar

Spacer,invar

Spacer,pyrex


Endcap + pin,304 stainlesssteel

6.0

2.1

5.6

in.

Nut,invar

0.5 0.5

4.5

2.0

4.1

in.

0.5

6.0

0.5

0–2

3–5

in.

(a) (b) (c)

thermal expansion, but with two changes. The bobs are internallysupported a half inch (1.7 in. for the linotype bob) below the top of thebob instead of at the bob center, and the temperature compensatingsleeve is located below the bob instead of inside the bob. These twochanges make a big improvement in a pendulum’s transient response totemperature change [30]. All of the bobs tested were 19 lb cylinders.And all were 6.0 in. tall as shown in Figure 11.4, except for the linotypebob which was 6.8 in. tall.

All of the test pendulums have a 2 s period. Mechanical escapementswere not used. The pendulums were driven electromagnetically by ashort electrical pulse at the center of swing.

In Figure 11.4(b), the spacer material for the polycrystalline silicon rodis pyrex. The thermal expansion coefficient of pyrex (3.5 106 /C) isthe closest I could find to that of polycrystalline silicon (2.6

106 /C). The silicon rod’s temperature compensator is brass and israther long, 8.6 in., because of silicon’s relatively high thermal expan-sion coefficient. The 48 in. long silicon rod was not long enough to putall of the compensator below the bob. So about half of the temperaturecompensator was inside the bob and about half was below the bob, asshown in Figure 11.4(b).

The temperature compensation in each of the thermal hysteresis testsis only approximately correct. Small errors in the length of the tempera-ture compensator (0.3 in. in a typical 4–5 in. length) have little effect onthe thermal hysteresis. Thus, a large amount of time need not be spentgetting the temperature compensation exactly correct for each test.

Thermal hysteresis tests are very time consuming. A hysteresis test onany given material takes 3 weeks, because of the long 2–3-day tempera-ture stabilization time needed at each temperature before accurate clockdata can be taken. Four to ten days additional are needed at the begin-ning of many of the tests to determine the barometric correction factorfor that particular pendulum, so that clock rate changes due to air pres-sure changes can be subtracted out of the data and not be consideredpart of the hysteresis error. The correction factor changes with the den-sities and amounts of the different materials used in a pendulum.

All of the hysteresis testing was done with one or the other of two iden-tical spring suspensions. Each has dual type 510 phosphor bronze springs,with each spring and its thick ends of solid one-piece construction. Thecentral free part of each individual spring is 0.006 in. (L W T ). The two suspensions differ only in the hole size providedfor the pendulum rod.

Test philosophy

I got into thermal hysteresis when my pendulum would not repeat itsprevious clock rate after having been cycled over a small temperature

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12


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range and then being returned to its original RT. The non-repeatabilitywas quite marked, up to 0.2 s/day with some metals. This was after cor-rection for any changes in barometric pressure. Obviously, one or moreparts of the pendulum were not returning to their original length. Andif a pendulum will not repeat its previous clock rate at the same tem-perature and pressure, it is useless as an accurate time source.

How do you measure the thermal hysteresis of the individual partsof a pendulum? What you can measure is the pendulum’s total thermalhysteresis, which is the sum of the hysteresis of all its parts—pendulumrod, bob, temperature compensator, compensator spacers, and suspen-sion spring. The solution is to make up a “test pendulum” of reasonablylow hysteresis parts, and then one at a time substitute different materi-als for each of its parts, noting the change in the pendulum’s totalhysteresis as the material of each part is changed. The “test pendulum”cannot have high hysteresis, as any variations in its high hysteresiswould swamp out and ruin the measurement of any low hysteresismaterial. To get this low hysteresis test pendulum required 3 years ofpreliminary testing of all the pendulum materials, incorporating anynew material into the pendulum that gave a lower total hysteresis thanthe last material tested. Finally, after many tests and substitutions oflower hysteresis materials into the pendulum, a test pendulum of relat-ively low hysteresis was obtained that could be used for testing thevarious parts of the pendulum.

Some materials are suitable for both the bob and the temperaturecompensator. The temperature compensator is the most convenientplace to test them, the compensator being small, low cost, and requiringlittle effort to make a test part. When only the compensator’s materialwas being tested, the bob normally used was a low hysteresis material(642 aluminum silicon bronze). If the material was not too expensiveand was available in a large enough size, a bob was also made of thesame material, and was included in the same test with the compensator.

Making a single hysteresis test with both the bob and compensatormade of the same material had two advantages and one disadvantage.The biggest advantage was the 3–4 weeks of time saved by not testinga bob and compensator of the same material separately. The overallmaterials stability testing took a long 7 years to do, so anything thatwould hurry things up was important. Second, a material (if suitable)would most likely be used in both places, both as a bob and as a com-pensator. And in the final analysis, a pendulum’s total thermal hysteresisis more important than that of its parts, some of which may cancel thatfrom other parts.

The disadvantage of the combined hysteresis test is the difficulty ofcomparing the thermal hysteresis obtained by testing a material in twopendulum parts at the same time with the hysteresis obtained by testing the material in only one pendulum part. The difficulty is caused


71

by differences between the two locations. The compensator is incompression with the same stress throughout its length. The bob is intension, as it is supported at or near its top, with the stress varying fromzero at the bob’s bottom edge up to a maximum near the bob’s top atthe bob’s support plane. Both tension and compression stresses willproduce creep effects.

A small length change at the bob’s bottom end will move very littlemass, and will have little effect on the clock rate. A small length changeup near the bob’s top end just below the bob’s support plane will movenearly the whole bob, with maximum effect on the clock rate. At anaverage position, halfway up the bob, a small length change will movehalf the bob’s mass, with half the maximum effect on the clock rate.

In contrast, in the compensator, a small length change anywherealong its length will move the whole bob. So on average the pendulumis twice as sensitive to length changes in the compensator, which movesthe whole bob, than to length changes in the bob, which on averagemove only half the bob’s mass.

If the bob were supported at its center, the pendulum’s 2 : 1 ratio ofthe effect (on clock rate) of length change in the compensator to that inthe bob would double to 4 : 1. That is because a centrally supported bobcuts the effect of a bob length change in half. With central support, asmall length change at the bob’s top and bottom edges would move verylittle bob mass, with little effect on the clock rate. A small length changenear the bob’s center just above or below the bob’s support plane wouldmove half the bob’s mass, the bob’s maximum effect on clock rate. At anaverage location, halfway between the bob’s center and the bob’s top (orbottom) edge, a small length change would move only one-fourth of thebob’s mass. Thus, if the bob were centrally supported, the clock rate (onaverage) would be only one fourth as sensitive to length change in thebob as it is to length change in the compensator. With central support, auniform length change throughout the bob would approximately cancelitself out, with equal and opposite movements of the bob’s top and bot-tom halves. The cancelation is only approximate because of the smalldifference in the distances of the bob’s top and bottom halves to thependulum’s axis of rotation. Conversely, with the bob supported at ornear its top edge, a uniform length change throughout both the bob andthe compensator will cancel out half to all of the length change in boththe bob and the compensator, with the actual amount canceled outdepending on the relative lengths of the bob and the compensator.

Now let us come back to the question of how to compare the hys-teresis values obtained by changing just the temperature compensatorwith those obtained by changing both the compensator and the bob.On the basis of the pendulum being (on average) twice as sensitive tosmall length changes in the compensator as to small length changes inthe bob, one might reduce by one-third those thermal hysteresis values


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obtained with a combined material change in both the compensatorand the bob. This was not done. The hysteresis values listed in Table 11.5are the actual test values measured for the compensators and bobslisted in Table 11.5. The one-third correction was not done for two rea-sons. First, the actual 2 : 1 ratio of the pendulum’s sensitivity to lengthchange in the compensator and the bob is quite variable, depending asit does on where the length change occurs in the bob. And second, itwould not change which materials were best, that is, which had thelowest hysteresis. The best materials are the ones we are most inter-ested in and would want to use in a pendulum.

Results

Table 11.5 lists the thermal hysteresis of all the materials tested. Thematerials in each section of Table 11.5 are listed in the order of increas-ing hysteresis. About half the tests are tests of the temperature com-pensator’s material only, using the pendulum rod and bob with thelowest hysteresis (quartz rod and 642 aluminum silicon bronze bob). Inthe other half of the tests, the temperature compensator and the bobare of the same material and are tested together.

Looking first at the heat treatments, it is apparent that annealing andtemperature cycling each by itself helps a little in lowering the thermalhysteresis. But using both of them in sequence (annealing first) helpsmuch more than the individual amounts of each in reducing hysteresis.Temperature cycling by itself reduced the hysteresis of a 360 brass tem-perature compensator from 0.098 s/day in the “as received” conditiondown to 0.087 s/day after temperature cycling. And annealing by itselfreduced the hysteresis of a 642 aluminum silicon bronze temperaturecompensator from 0.084 s/day in the “as received” condition down to0.067 s/day after annealing. But both annealing and then temperaturecycling reduced the hysteresis of a 360 brass temperature compensatorfrom 0.098 s/day in the “as received” state down to 0.020 s/day. Thesecomparisons are a little in error because of the use of two differentbrass bobs (see Table 11.5), one being leaded and the other unleaded.But the overall trend in Table 11.5 is clear. Except for the aluminumalloys, the lowest hysteresis is obtained when materials are bothannealed and temperature cycled.

Looking next at the pendulum rod materials, the bottom section ofTable 11.5 shows that the thermal hysteresis of the quartz rod is5.8 times less than that of the invar (free machining) rod. And thehysteresis of the silicon rod is slightly more than that of the quartz rod(0.023 s/day for silicon vs 0.020 s/day for quartz).

The silicon rod was expected to show less hysteresis than what thequartz rod had. But silicon’s higher temperature coefficient required atemperature compensator length of 8.4 in. (brass), about twice what


73


74

Table 11.5. Thermal hysteresis of pendulum materials

Material Thermal

Temperature Hard/ Bob Temperature Hard/ Pendulumhysteresisb

compensator Annealed cycleda Annealed rod(s/day)

Yes/No

As received6061 Aluminum H 642 Alum. silicon bronze Y A Quartz 0.0252024 Aluminum H 642 Alum. silicon bronze Y A Quartz 0.042642 Alum. silicon bronze H — Old brass, unleaded N H Quartz 0.084360 Brass, leaded H — Old brass, unleaded Y A Quartz 0.098

Temp. cycleda

7075 Aluminum H 642 Alum. silicon bronze Y A Quartz 0.0626061 Aluminum H 510 Phosphor bronze N A Quartz 0.067510 Phosphor bronze H 510 Phosphor bronze N H Quartz 0.073360 Brass, leaded H — Old brass, unleaded N H Quartz 0.087

Annealed630 Alum. nickel bronze A 630 Alum. nickel bronze N A Quartz 0.063642 Alum. silicon bronze A 642 Alum. silicon bronze Y A Quartz 0.067464 Naval brass A 464 Naval brass N A Quartz 0.079304 Stainless steel A 304L Stainless steel N A Quartz 0.093

Annealed Temperature cycleda

642 Alum. silicon bronze A 642 Alum. silicon bronze Y A Quartz 0.010360 Brass, leaded A — Old brass, leaded Y A Quartz 0.0202024 Aluminum A 642 Alum. silicon bronze Y A Quartz 0.023642 Alum. silicon bronze A 53425 Linotype Y A Quartz 0.034172 Beryllium copper A 642 Alum. silicon bronze Y A Quartz 0.035304 Stainless steel A 304L Stainless steel Y A Quartz 0.037107 Silver copper A 642 Alum. silicon bronze Y A Quartz 0.037875 Silicon brass A 875 Silicon brass Y A Quartz 0.041655 Silicon bronze A 655 Silicon bronze Y A Quartz 0.041630 Alum. nickel bronze A 630 Alum. nickel bronze Y A Quartz 0.046464 Naval brass A 464 Naval brass Y A Quartz 0.047510 Phosphor bronze A 510 Phosphor bronze Y A Quartz 0.050304L Stainless steel A 304L Stainless steel Y A Quartz 0.052715 Copper nickel A 642 Alum. silicon bronze Y A Quartz 0.0857075 Aluminum A 642 Alum. silicon bronze Y A Quartz 0.0856061 Aluminum A 642 Alum. silicon bronze Y A Quartz 0.094729 Copper nickel tin A 642 Alum. silicon bronze Y A Quartz 0.096

Pendulum rod tests360 Brass, leaded A — Old brass, unleaded Y A Quartz 0.020360 Brass, leaded A — Old brass, unleaded Y A Silicon 0.023360 Brass, leaded A — Old brass, unleaded Y A Invarc 0.117

Notesa Between 200 F and 40 F, 5–7 times.b For a temperature variation of approximately 28 F.c Free machining type.

was needed for the quartz rod. The double length of compensatorneeded for the silicon rod introduced extra hysteresis into the siliconrod assembly, making it appear that the silicon rod had more hysteresisthan the quartz rod.

The residual stress levels in the silicon and quartz rods are unknown—both were tested in their “as received” state, and were not heat treated.The invar rod was given the first two steps of Lement’s [26] three-stepheat treatment at MIT: 1500 F for 30 min, water quench. Reheat to600 F for 1 h, air cool. The third step—reheat to 200 F for 24 h, aircool—was skipped to save money, figuring that several years at RTwould produce the same result. About 4 years elapsed between the two-step heat treatment and this hysteresis test. In an ideal world, all threerods would be heat treated to reduce their internal stresses, and thequartz rod would probably look even better then than it does now.

As to the bob and temperature compensator materials, Table 11.5shows that 642 aluminum silicon bronze has the lowest hysteresis(0.010 s/day) of all the metals tested. 360 brass has the second lowest hys-teresis, 0.020 s/day. This is after both metals have been annealed and tem-perature cycled. Table 11.5 also shows that metals containing tin or nickel(invar is 36% nickel) have medium to high hysteresis, with values rangingfrom 0.037 to 0.096 s/day, again after annealing and temperature cycling.

The aluminum alloys 2024, 6061, and 7075 are a mostly poor bag—only two specific cases had low hysteresis. All of the temperature com-pensators of all the materials were machined from rod stock, which inthe case of 6061 aluminum turned out to be significant. 6061 aluminumin its “as received” condition had low hysteresis (0.025 s/day), mostlikely because the manufacturers stretch the rod form of 6061 to give ita permanent set of 1–3%. The purpose of the stretching is to reducethe residual stresses in the rods, and the stretching obviously does that.The 2024 and 7075 aluminums are not stretched. Type 2024 aluminum,when annealed and temperature cycled, is the second aluminum alloywith low hysteresis (0.023 s/day). The effect of pure aluminum as anadditive to other metal alloys is quite variable, giving low hysteresis insome alloys and high hysteresis in others, as shown in Table 11.5.

Aluminum alloys 2024, 6061, and 7075 have little use as pendulummaterials, primarily because of aluminum’s low density. The low density(1) increases the air drag for a given bob weight, and (2) increases the pen-dulum’s sensitivity to changes in air pressure. In addition, two differentmetal handbooks say that the aluminum alloys are dimensionally unstable,some of them over a short term (days) but others over periods of years.

Conclusions

Unless a pendulum is kept in a constant temperature room, it isapparent that heat treatment will improve the pendulum’s stability by


75

2–8 times, the exact amount of improvement depending on thematerial, as shown in Table 11.5. The combination of annealing plustemperature cycling is very effective in reducing the thermal hysteresisof pendulum materials, making a pendulum clock more stable andmore accurate in the typical home environment where the temperatureis not perfectly constant.

For the pendulum rod, quartz is the best material. It has the lowesthysteresis of the three materials tested: quartz, silicon, and invar (freemachining).

For the bob and the temperature compensator, the metal with thelowest thermal hysteresis is 642 aluminum silicon bronze, annealedand temperature cycled. The metal with the second lowest hysteresisis 360 brass, annealed and temperature cycled. 360 brass is relativelycheap and is the most widely available of all the brasses. 642 alu-minum silicon bronze costs about 30% more than 360 brass (seeTable 11.3).

Surprisingly, 304 stainless has less hysteresis than 304L stainless.Although it is a little lower in performance, 304 stainless would make agood low-cost pendulum bob. It comes already annealed from themanufacturer, and thus avoids the cost of annealing it yourself. Thetemperature cycling needed can be done at home, using the oven andrefrigerator in the kitchen. Type 304 stainless costs about 10% less than360 brass (see Table 11.3).

In summary, this search for dimensionally stable materials, that is,low thermal hysteresis materials, has been in the general directionof annealed, nonmagnetic, nonrusting materials. A few relativelystable materials have been found. Most likely, there are a few moreout there.

Further reading

Material characteristics1. Properties of lead and lead alloys, Lead Industries Assoc., 292 Madison Ave.,

New York, NY, USA, December 1983.2. J. Thompson. “Properties of lead–bismuth, lead–tin, type metal, and

fusible alloys,” Res. Paper RP248, J. Res. Natl. Bur. Stds. 5 ( July–December1930), 1085–102.

3. L. Schetky. “Properties of metals and alloys of particular interest in preci-sion instrument construction,” Report R-137. Instrumentation Lab., Mass.Inst. Tech. Library, Cambridge, Mass., USA, January 1957. Measureddimensional stabilities. Available MIT library archives.

4. M. Meyerson and M. Sola. “Gage blocks of superior stability III: Theattainment of ultrastability,” Trans. Amer. Soc. Metals 57 (1964), 164–85.

5. B. Lement, B. Averbach, and M. Cohen. “The dimensional stability ofsteel part IV—tool steels,” Trans. Amer. Soc. Metals 41 (1949), 1061–92.


76

6. J. Strong. Procedures in experimental physics, Prentice Hall, 1938. p. 190. One of only two references found on thermal hysteresis.

7. C. Marschall and R. Maringer. Dimensional instability, Pergamon Press,1977. The second of only two references found on thermal hysteresis.

8. K. Petersen. “Silicon as a mechanical material,” Proc. IEEE 70(5) (May1982), 420–57. Tutorial on silicon.

9. J. Berthold, S. Jacobs, and M. Norton. “Dimensional stability of fusedsilica, invar, and several ultralow thermal expansion materials,” Appl. Optics15(8) (August 1976), 1898–9.

10. S. Jacobs. “Dimensional stability of materials useful in optical engineer-ing,” Optica Acta 33(11) (November 1986), 1377–88.

11. S. Jacobs. “Variable invariables—dimensional instability with time andtemperature,” SPIE Crit. Rev CR43 ( July 1992), 181–204.

12. C. Barrett. Structure of metals, 2nd edn, McGraw-Hill, 1952. On metal-lurgy. Excellent.

13. K. Van Horn. “Residual stresses introduced during metal fabrication,”Trans. Amer. Inst. Mining Engrs. 5(1) (March 1953), 405–22. Manufacturingrealities.

Aluminum Bronze14. F. Wilson. “The copper-rich corner of the copper–aluminum–

silicon diagram,” Trans. Amer. Inst. Mining Engrs. 175 (1948), 262–82.15. M. Cook, W. Fentiman, and E. Davis. “Observations on the structure and

properties of wrought copper–aluminium–nickel–iron alloys,” J. Inst.Metals 80 (1951–2), 419–29, photo plates 63–6.

16. J. McKeown, D. Mends, E. Bale, and A. Michael. “The creep and fatigueproperties of some wrought complex aluminium bronzes,” J. Inst. Metals83 (1954–5), 69–79, photo plates 13, 14 facing p. 116.

17. P. Macken and A. A. Smith. The aluminium bronzes, Publication No. 31.Copper Dev. Assoc., London, 1938, 2nd edn, 1966.

Creep, relaxation, and anelasticity18. A. Scully. Metallic creep and creep resistant alloys, Butterworth’s Scientific

Publications, London, 1949. Much experimental data.19. F. Garofalo. Fundamentals of creep and creep-rupture in metals, Macmillan

Co., New York, 1965. General reference in understandable English.20. J. Gittus. Creep, viscoelasticity and creep fracture in solids, Applied

Science Publishers, London, 1975. Good on basic sources of anti-creepstrength.

21. D. Uhlmann and N. Kreidl (eds). “Viscosity and relaxation,” Glass: Scienceand Technology, Vol. 3, Academic Press, 1986. Relaxation properties ofquartz.

22. C. Zener. Elasticity and anelasticity of metals, University of ChicagoPress, Chicago, Illinois, USA, 1948. Good chapter on interpretation ofanelasticity.

23. J. Woirgard, Y. Sarrazin, and H. Chaumet. “Apparatus for the measurementof internal friction as a function of frequency between 105 and 10 Hz,”Rev. Sci. Inst. 48(10) (October 1977), 1322–5.


77

Heat treatment24. W. Epprecht. “Behavior of complex alloys under thermal cycling,” (in

German). Zeit. Metallkd. 59(1) (1968), 1–12. English translation availablefrom Copper Dev. Assoc. via American Fulfillment LLC, Oxford,Connecticut, USA, Accession no. 4787. Only reference found on thermalcycling.

25. B. Lement and B. Averbach. “Measurement and control of the dimensionalbehavior of metals,” Summary Report #1. Metals Processing Div., Dept. ofMetallurgy, Mass. Inst. Tech., Cambridge, Massachusetts, USA, (December1955). Available MIT library archives. Heat treatments for dimensional sta-bility of hard non-ferrous metals.

26. B. Lement. “Distortion in tool steels,” Amer. Soc. Metals (1959), Novelty,Ohio, USA. Using heat to improve dimensional stability. Excellent.

27. L. Leonard. “Enhancing metals properties with supercold: Fact or fancy?”Mater. Eng. 102 ( July 1985), 29–32.

Miscellaneous28. Mitsubishi Polysilicon, Theodore, Alabama, USA.29. Saphikon, Milford, New Hampshire, USA.30. R. Matthys. “Transient response of a pendulum to temperature change,”

Hor. J. (December 2000), 417–19, 424.31. Cryogenic Society of America, Oak Park, Illinois, USA.

www.cryogenicsociety.org32. W. L. Goodrich. The modern clock, privately published 1905, reprinted 1950.33. Metals handbook, desk reference version. Amer. Soc. Metals (1985). (Original

source: New Jersey Zinc Co.)34. H. Gough. “Crystalline structure in relation to failure of metals—

Especially by fatigue,” Marburg lecture, Proc. ASTM 33 (part II) (1933),3–114.


78

www.cryogenicsociety.org

79

chapter 12

Variations on a Riefler bob shape

For the lowest air drag, Riefler bobs should have an edge angle of 50–75 anda diameter-to-thickness ratio of 2. Such bobs need 2% less drive force than asphere of equal volume. Information on bob aerodynamics is included.

Riefler’s bob shape, shown in Figure 12.1(a), is well known. It consistsof two truncated cones, back-to-back. It is a shape with low air drag,and works well inside the clock case, where front-to-back space isfrequently limited. In 1988, Bateman [1] did a study on the air draglosses of various bob shapes, and found that the football and oblatespheroid shapes had even less drag than Riefler’s two truncated cones.But they require more front-to-back space, assuming equal volume, andare harder to make.

Reducing a bob’s air drag is worthwhile for two reasons. First, itreduces the weight needed to drive the escapement’s gear train, whichin turn reduces the wear on the gearing and the escapement’s pallets.Second, the less you disturb a pendulum, the more accurate it becomes(see Chapter 7).

Riefler’s bob shape has three advantages: (1) it is easy to make on thelathe, as its surface curves in only one dimension, (2) it has low draglosses, lower than the bi-convex lens shape [1], and (3) it can be madethin (front-to-back) with only a small increase in air drag. I had twoquestions: (1) does the Riefler bob shape have an optimum diameter-to-thickness ratio, and (2) is there an optimum angle for the bob’s edge? Itturns out that the answer is yes to both questions. This gets into theaerodynamics of bob shape, but more on that later.

To test these two variables, 17 bobs were made, with diameter-to-thickness ratios of 0.63–4.0 and edge angles () of 28–180. The bobswere painted a dark matte red color and are shown in Figure 12.2. Fourof the bobs have a round edge as shown in Figure 12.1(b) instead of asharp edge, just to see how round-edge bobs perform. A sphere is around-edge bob with a diameter-to-thickness ratio of 1. And four of thebobs, cylinders, have flat bob edges. All of these bobs are mounted withtheir axis perpendicular to the plane of swing. There is one 6-in. tallcylinder with a diameter-to-thickness ratio of 0.63 that is mounted withits cylindrical axis parallel and concentric to the pendulum rod.

Thanks to Wayne Bohannon for educatingme about bob aerodynamics, and to BillWerner for providing the maple and oak forthe bobs.

Fourteen more bobs were made later to fill in gaps in the data, fora total of 31 bobs. To reduce cost, the bobs were made of maple, exceptfor the sphere and the 6-in. tall cylinder which were made of oak. Thebobs are all of equal volume, 70 in.3, and would weigh 21 lb eachif made of brass. The bobs are 4–10 in. in diameter, and 2–6 in. thick.The bobs are mounted one by one on a quarter inch diameter woodenpendulum rod, and oscillate with a 2 s period. With a swing angleof 2.4 (half angle), the bobs have Reynolds numbers between 320 and750 at the halfway point across the bobs.

The bobs’ edges were made neither exceptionally round nor excep-tionally sharp. They were sanded a little with fine sand paper (150 grit)to smooth out the rough spots, but otherwise they were left as machinedwith edge radii of 0.01–0.04 in.

The sinusoidal electromagnetic servo described in Chapter 33 wasused to drive the pendulum. Figure 12.3 shows the two moving perman-ent magnets and a bob mounted on the pendulum rod. The electricalwires visible in Figure 12.3 go to the two fixed coils. One magnet

Figure 12.2. The first 17 bob shapes, all ofequal volume.


80

Figure 12.1. Bob shape: (a) Riefler’s back-to-back truncated cones and (b) round-edgeversion.

T

T

D

D

r = T2

(a)

(b)

+

+

+

+

generates a velocity signal in one fixed coil. The second magnet withthe second fixed coil generates the pendulum’s drive force. The peakvalue of the sinusoidal drive current is used as a measure of the driveforce needed to keep each bob swinging at a constant 2.4 half angle.The atmospheric air pressure, at 29.0–29.2 in. of mercury, was relativelyconstant during the measurements.

The pendulum has no escapement, so the drive current in the fixeddrive coil is a good measure of the pendulum’s air drag, as about99% of the pendulum’s energy input then goes into air drag losses.This is known to be true [2], because if a pendulum without anescapement is operated in a vacuum, its Q, which is an inversemeasure of loss, increases by a factor of about 100. To do that, the non-air-drag losses have to be 1% or less of the total loss at atmosphericair pressure.

Figure 12.3. The two moving magnets andbob on the pendulum.

chapter 12 | Variations on a Riefler bob shape

81

Air drag results

Figure 12.4 gives the test results, with the pendulum’s peak drive cur-rent shown for each bob as a function of its diameter-to-thickness ratioand edge angle . The peak drive currents for the round-edged bobs arealso shown along the right hand edge of Figure 12.4. Figure 12.4 showsthat the most efficient bob shape, the one requiring the least drive cur-rent, has an edge angle between 50 and 75, and a diameter-to-thickness ratio of 2. With very large edge angles, 140 to 180, the bestdiameter-to-thickness ratio increases to 4. The round-edge bobs withdiameter-to-thickness ratios of 1 (sphere), 2, and 3 are almost as effi-cient as the best sharp-edged bobs.

The peak drive current for the pendulum with no bob attached is0.44 mA, which is mostly for the pendulum rod’s air drag. With no bobpresent, there is an extra 4–10 in. of rod exposed, increasing the rod’s airdrag by an estimated 25%. The drive currents given in Figure 12.4 are thetotal pendulum drive currents. And the percent efficiency ratios given forthe bobs are based on the pendulum’s total drive currents. If one wantedto compare the drive currents on the basis of just the bobs’ air drag alonewithout including the rod’s air drag, then 75% of 0.44 mA 0.33 mAshould be subtracted from the drive current values given in Figure 12.4,and the bob efficiency ratios recalculated using the new drive currentvalues.

Bob aerodynamics

Wayne Bohannon, a retired aeronautical engineer, gave me an hour’slecture on aerodynamics as applied to pendulum bobs. After a long

Figure 12.4. Pendulum drive current vs the bob’s edge angle and diameter-to-thickness ratio. The drive current for the round-edge bobs is shown along the right margin.


82

1.2

1.0

0.8

Pen

dulu

m d

rive

cu

rren

t, (p

eak

mA

)

=1.3

= 0.63= 1

=1=2

=3

=4

DT

Sphere

Rou

nd

edge

Flat

edg

e (c

ylin

ders

)

0 20 40 60 80 100 120 140 160 180Edge angle, (deg)

=2=3

=4

career designing aircraft and missiles at speeds of Mach 0.1 to aboutMach 10, he was surprised that anyone was interested in lift and drageffects at the ridiculously low speed of 2 in./s, which is the averagevelocity of the bob on my clock. This velocity, together with a bobdiameter of about 3.6 in., gives a Reynolds number of 150 at a pointhalfway across the bob. There is very little aerodynamic data availablefor guidance at such a low number. The next four paragraphs summar-ize Bohannon’s advice.

In general, surfaces curved in two dimensions (sphere, ellipsoid, etc.)have less drag than surfaces curved in only one dimension (cylinder).For the lowest drag, you want turbulent non-laminar air flow within theboundary layer on the bob’s leading and trailing sides. Well-roundededges will reduce drag by making it easier for the boundary layer to stayin contact with the bob’s surface when the layer goes around an edge.The boundary layer has zero thickness at the very front edge of thebob’s leading surfaces, and increases linearly in thickness as it extendsback over the bob. (The boundary layer is about 0.08 in. thick at thehalfway point around a 3.6 in. diameter cylinder, from visual observa-tion in an earlier smoke test.)

If laminar flow does occur, use a “tripwire” to break it up. A tripwireis located on the bob’s surface and wrapped vertically about the bob ina plane perpendicular to the direction of swing. A tripwire is a thin wall,wire, or break in the bob’s surface that extends outward (or inward)from the bob’s surface. The wall (or whatever) is oriented perpendic-ular to the airflow across the surface, and must protrude up through theboundary layer’s thickness at that point. The tripwire itself will intro-duce some drag. If the tripwire is too high, it will introduce more dragthan it eliminates in redirecting the airflow.

An effect called the Von Karman vortex has a high drag coefficient andshould be eliminated if present. The Von Karman vortex causes theoscillatory flapping of a flag in the wind, and will alternately push thebob frontward and rearward (most likely at an odd frequency) as the boboscillates sideways in its normal left-right and right-left motion. Wrappinga tripwire vertically around the center of the bob and orienting itperpendicular to the airflow should get rid of a Von Karman vortex.

The drag coefficient for a leading surface is 0.105 for laminar flowand 0.002 for turbulent flow. The drag coefficient for a trailing surfaceis 0.30 for separated flow (away from the surface) and 0.06 for attachedflow. Thus, there is at least a 5 to 1 possible advantage in lower dragin having turbulent flow in the boundary layer and in maintainingattached flow on the bob’s trailing surfaces, where it tends to separate.

But after all of the above, remember that aerodynamics is still verymuch an art, requiring a lot of experience and scientific wild-assedguesses (SWAGs). This is at least partly due to the nasty and very diffi-cult Navier–Stokes equations at the center of aerodynamic design.


83

Aerodynamic results

In photographs of smoke from a previous airflow test (see Chapter 29),Bohannon could see that there was some separation of the boundarylayer on the trailing surfaces of the large sphere. And if the photographof the 6 in. tall large vertical cylinder were better, he was sure thatboundary layer separation would also be seen on the cylinder’s trailingsurfaces. There was no sign of a Von Karman vortex in any of thesmoke photographs. The aerodynamic effects observed there wouldalso occur with the sphere and the 6 in. tall vertical cylinder in the pre-sent airflow tests.

Tripwires were tried on an assortment of bob shapes with disappoint-ing results. A vertical “wall” (see Figure 12.5(a) ) 0.16 in. high and madefrom 0.03 in. thick aluminum sheet was placed around the center of avertical cylinder 6 in. tall and 3.6 in. in diameter. This increased the airdrag by 20% instead of decreasing it. A vertical wall (see Figure 12.5(b) )0.16 in. high placed around the center of a 4.9 in. diameter sphereincreased the air drag by 60%. A sloped vertical wall (see Figure 12.6)0.15 in. high placed around the periphery of both side walls on eightdifferent Riefler bob shapes did decrease the air drag on all eight bobs,but only by 3–6%. The walls’ height was not varied. A different wallheight might produce better results.

Bohannon said that because the tripwires’ drag reduction was sosmall, it indicated that the bobs’ trailing edge drag is much smallerthan the leading edge drag, that is, that most of the drag is comingfrom the bobs’ leading surfaces and not from their trailing surfaces.This is reversed from the usual aircraft situation where if the flowseparated (detached), most of the drag would come from the trailingsurfaces.

Conclusions

The Riefler bob shape with the lowest drag has an edge angle between50 and 75, and a diameter-to-thickness ratio of 2. This shape requiresa trifle less (2% less) drive force than a sphere of equal volume. Theedge angle has six times greater effect on the air drag than thediameter-to-thickness ratio. Figure 12.4 shows that, if needed, the bobcan easily be made thinner—cutting the thickness in half to get a biggerdiameter-to-thickness ratio of 4 would increase the pendulum’s driveforce by only 4%.

The round-edged bobs also did well. Round-edged bobs with adiameter-to-thickness ratio of 2 or 3 required only 3% more drive forcethan the best sharp-edged bobs. The cylindrical bobs were the worstperformers, requiring 24–39% more drive force than the spherical bob.

Figure 12.5. Tripwires on (a) a verticalcylinder and (b) a sphere. For clarity, thetripwire height is exaggerated.

Figure 12.6. Sloping tripwire edge aroundperiphery of bob’s side walls: (a) bob cross-section and (b) isometric view. For clarity, thetripwire’s depth into the side walls isexaggerated.


84

(a) (b)

Tripwire edge (1 of 2)

Plane

of swing

0.15 (2)

(a)

(b)

The aerodynamic tripwires tried were a disappointment, in that atbest they only decreased the pendulum’s drive force by a small 3–6%.A different tripwire design might be more helpful.

In sum, the best sharp-edged and round-edged Riefler bobs both hadthe same drive force as a spherical bob within 2%, with all bobs hav-ing the same volume.

As a final note, I think that the sphere is better than Riefler’s bobshape for an accurate pendulum, because then you do not have toworry about aligning the bob’s plane of symmetry into parallelism withthe plane of swing. This alignment affects the pendulum’s clock rate(see Figure 13.3). It also assumes that the additional front-to-back spaceneeded for a spherical bob is available.

References1. D. Bateman. “Is your bob in better shape?” Clocks 2 ( June 1988), 34–7.2. E. T. Hall. “The Littlemore clock,” Hor. Sci. Newslett. NAWCC chapter 161

(August 1996). Available in NAWCC library, Columbia, Pennsylvania,17512, USA.


85

chapter 1 3

Bob shape

The sphere and the vertically oriented cylinder are more repeatable andpredictable bob shapes than the more efficient, lower-drag prolate spheroidand football shapes.

One of my pendulum bobs has the shape of a prolate spheroid. Asa bob shape, prolate spheroids have low air drag for their volume, andgive a high Q pendulum [1]. Only football-shaped bobs, that is, bobswith pointed ends (120 included angles) in the direction of travel givea higher Q. A prolate spheroid shape is obtained by rotating an ellipse360 around its long axis:

The prolate spheroid bob was machined out of solid brass. Thebob is 7.0 in. long, 3.5 in. in diameter, and weighs 13.3 lb. It has alow amplitude Q of 22,000 (1 half angle) and a high amplitude Q of18,000 (1 half angle). Figure 13.1 shows the partially machinedbob in the lathe. Figure 13.2 shows the completed bob mounted on its pendulum rod. There is a in. diameter 3 in. long magnet rodburied flush beneath the surface at each end of the bob’s long axis topermit driving the pendulum electromagnetically.

The bob sits atop the rating nut and temperature compensator, andcan be rotated around the pendulum rod. And this is where the prolatespheroid shape gets into trouble. If the bob’s long axis is rotated out ofparallel with the plane of swing, the clock speeds up. And the fartherthe bob is out of parallel, the more the clock speeds up. Figure 13.3shows the measured clock speedup vs misalignment angle. If the bob’slong axis is set at 90 to the plane of swing, the clock speeds up by65 s/day. That is not a misprint—that is a whopping 1.08 min/day! Thefootball shape has the same rotational alignment problem.

The increased air drag (the viscous portion) from a misaligned bobwas expected to slow the clock down. It took a long time to figure outwhy bob misalignment made the clock run faster rather than slower.The rationale is as follows. The bob’s long axis is a straight line, andis perpendicular to the pendulum rod. When the bob’s long axis is

18

xa

2 y

b2 1.

87

parallel to the plane of swing, the parts of the bob at the ends of its longaxis are slightly farther from the pendulum’s axis of rotation than is thebob’s center portion. When the bob’s long axis is perpendicular to theplane of swing, the parts of the bob at the ends of its long axis are at


88

Figure 13.1. Brass bob being machined in the lathe.

Figure 13.2. Finished bob mounted onpendulum rod, between the two magneticdrive coils. Two optical amplitude sensors are located below the bob.

the same distance from the pendulum’s axis of rotation as the bob’scenter portion. So the average distance of the bob from the pendulum’saxis of rotation is slightly shorter when the bob’s long axis is perpendi-cular to the plane of swing, and the clock speeds up in response to theshorter pendulum length.

The pendulum’s Q is lower at the bob’s 90 orientation than it iswhen the bob’s long axis is parallel to the plane of swing. At the prolatespheroid bob’s 90 orientation, its low amplitude Q is 12% lower (1

half angle) and its high amplitude Q is 29% lower (1 half angle),compared to the values obtained with the bob’s long axis parallel to theplane of swing.

The bob can be visually aligned to the plane of swing within about1 accuracy. From Figure 13.3, the uncertainty in clock rate for the pro-late spheroid is then about 0.05 s/day. Both bob misalignment and itsassociated timing uncertainty can be eliminated by changing to a bobwith a spherical or cylindrical shape (cylinder axis vertical), both ofwhich are immune to rotational effects. But the sphere has a 14% largerair-drag loss, and the cylinder has a 52–78% [1] larger air-drag lossdepending on its length-to-diameter ratio. These loss numbers are basedon bob shape alone. The pendulum rod has significant air-drag losses ofits own (see Chapter 24), so the net percentage increase in the pendu-lum’s total air-drag loss will be roughly half of what is indicated here.

In summary, bobs shaped like a sphere or a vertically oriented cylin-der have somewhat higher air-drag losses, but have better repeatabilityand predictability than the prolate spheroid or football bob shapes. Theimproved repeatability and predictability is due to their not having therotational alignment uncertainty that is present in the prolate spheroidand football bob shapes.

Reference1. D. Bateman. “Is your bob in better shape?,” Clocks ( June 1988), 34–7.

Figure 13.3. Clock rate vs bob misalignment.

chapter 1 3 | Bob shape

89

–16 –12 –8 –4 0 +4 +8 +12 +16Angle between bob’s long axis and plane of swing (deg)

Clo

ck

rat

e ch

ange

(s/

day)

+6

+4

+2

–2

chapter 14

Rate adjustment mechanisms

Using a sleeve bushing or a thick washer of pre-determined length underneaththe bob as a coarse rate adjustment, and a small threaded nut at the top of thependulum rod as a fine rate adjustment provide a better way to trim apendulum’s clock rate.

Coarse rate adjustment

The usual coarse adjustment for trimming a pendulum’s clock rate isa threaded nut beneath the bob, which moves the whole bob up anddown the pendulum rod. The surface of the thread is somewhat rough,particularly if the material is invar, which machines poorly. The thread’sroughness prevents any sort of smooth adjustment. Axially, 0.001 in.equals 1 s/day on a pendulum with a 2 s period. To get an adjustmentsensitivity of 1 s/day, the thread must be lapped, or more correctly,rubbed smooth. And if the thread is not smooth, the whole bob weightwill rest on the small raised points of the rough thread’s two facingsurfaces, creating high stress points and a potentially unstable joint(at the micro-inch level).

A better coarse adjustment, recommended by Doug Bateman, is toput a dowel pin crosswise through the pendulum rod and insert a stackof washers (as many as needed) between the dowel pin and the bob’scenter point of suspension. It would be even better to go further andreplace the stack of washers with a single sleeve bushing whose lengthis equal to the washers’ stack height. This gets rid of all the jointsbetween the washers and reduces the number of pendulum parts, bothof which would improve the pendulum’s stability.

Fine rate adjustment

The usual fine rate adjustment is to put small weights (grams andmilligrams) on a small weight pan attached to the pendulum rod.The weight pan can be located anywhere on the rod except at the

91

rod’s top and bottom ends (see Chapter 24). The pan is usually locatedhalf to two-thirds of the way up the rod. The sensitivity depends onthe ratio of the added weight to the bob’s weight, and also on theweight pan’s location along the pendulum rod (see Figure 24.1).With a pendulum period of 2 s and a 19 lb bob, a 1 g weight addedto the pan located two-thirds of the way up the rod will change theclock rate by about 1 s/day. Weights less than 1 g are usually small bitsof aluminum foil with one edge bent up at 90 for easier grabbing withtweezers.

The weight pan is relatively small and moves back and forth with thependulum, so it is not easy to add or remove the small bits of aluminumfoil without knocking other bits off the pan. If your hands are a littleshaky, adding or removing weights can be difficult.

A better fine rate adjustment is a threaded nut, moving vertically,located near the top of the pendulum. The nut can be located anywhereon the pendulum rod except at the rod’s midpoint (see Chapter 24), butthe top location works best as it moves very little when the pendulumswings, and the nut can be easily adjusted with your fingers with littleor no disturbance of the pendulum’s timing.

The sensitivity of the threaded nut can be varied over a wide range,as it depends on (1) the fineness of the thread, (2) the nut’s locationon the pendulum rod (see Figure 24.2), and (3) the ratio of the nut’sweight to the bob’s weight. With a pendulum period of 2 s, 32 threadsper inch, and a 19 lb bob, a 0.1 lb nut will change the clock rate by about0.2 s/day/revolution of the nut, when located immediately below thesuspension spring, that is, at the top of the pendulum rod. If 20 equallyspaced vertical lines are engraved around the nut’s periphery, a nut rota-tion of one line (18) will change the clock rate by 0.01 s/day, an easyfactor to remember. A 2 in. length of thread will give a total fine adjust-ment range of about 10 s/day (the nut takes up part of the thread’slength).

Why use a threaded nut for the fine adjustment but not for the coarseadjustment? The reason is that the effect of the thread’s rough-ness depends on the weight the thread is carrying. As a coarse adjust-ment, the thread carries the whole weight of the bob, 19 lb in theexample given. But as a fine adjustment, the thread carries only theweight of the nut, 0.1 lb, a 190 to 1 reduction in the effect of threadroughness.

A non-trivial advantage of putting the threaded nut at the top of thependulum rod is that the thread can be cut in the brass (or stainless?)piece connecting the dual suspension springs to the top of the pendu-lum rod, as shown in Figure 14.1, instead of cutting the thread in invar.As mentioned before, threads cut in invar are not of good qualitybecause of invar’s poor machining properties.


92

Appendix: Smoothing the threads

A female threaded nut cannot be lapped directly together with its malethread, as it will bind up. An extra thread set of steel or cast iron (notinvar) is made up with an oversize (about 0.010 in. diameter) femalethread and an undersize (about 0.010 in. diameter) male thread, tomake room for the lapping compound between the threads. A lappingcompound contains many small diamond particles in a grease base.A medium grit (30 m particle size) works well. A small tube of dia-mond lapping compound, more than enough for the job, costs US$30.The undersized male thread is rubbed against the normal-sized femalethread with a layer of the lapping compound in between the thread sur-faces. In a similar fashion, the oversized female thread is rubbed againstthe normal-sized male thread.

The male part is put in a lathe or an electric drill for rotation, and thefemale part, held with the fingers, is run back and forth 20–50 timesover the male threads. While running, pressure is axially applied to thefemale part, pushing it sometimes up the thread and sometimes downthe thread, and once in a while radially, with the intention of puttingpressure on the thread faces being smoothed. The process continuouslyknocks off the high points on each thread face, so that the thread facescontinuously become smoother as the rubbing continues. Eventually asmoothing limit is reached, and to go smoother, the lapping compoundand the steel and invar (or whatever) particles imbedded in it must be

chapter 14 | Rate adjustment mechanisms

93

Figure 14.1. Fine rate adjustment at top ofpendulum rod. A is 0.1 lb threaded nut madeof type 304 stainless steel. B is the connectionpiece (304 stainless steel) between the twosuspension springs and the pendulum rod,and has the male thread for the nut.

Pin(1 of 4)

Suspensionspring(1 of 2)

B

Rod pin

A

32 Threads per inch

Pendulumrod

2.0

in.

washed out with solvent, and replaced with a finer grit of lappingcompound. The rubbing process is then repeated. But I have found thatusing the medium grit lapping compound only once makes the threadsurfaces smooth enough without using any of the finer grits.

The smoothing process can be done without the diamond lappingcompound. Instead, the rubbing is done with a light grease or heavy oilbetween the threads, and the steel and invar (or whatever) particles thatare knocked off the thread surfaces take the place of the diamondparticles. However, the rubbing process takes longer and the threadsurfaces are not quite as smooth.


94

part i i

Suspension spring

chapter 15

Spring suspensions for accuratependulums

This chapter covers the design and performance of various types of springsuspensions for an accurate clock pendulum. It also covers the three basicoscillation modes of a pendulum, the interactions between them, and theeffects they have on the clock’s timing accuracy. These oscillation modes affect the design of the suspension.

To simplify the following discussion, left-right pendulum motion willrefer to the normal left-right motion of the pendulum bob, as viewedfrom in front of the clock case. And front-to-back pendulum motionwill refer to the front-to-back motion of the pendulum bob, again asviewed from in front of the clock case. The pendulum used for thevarious tests weighs 15 lb, has a 2 s period, and is driven by a shortelectromagnetic pulse at the center of its swing. The in. diameter pen-dulum rod is made of invar. The lengths given for suspension springsare the free flexure lengths, and do not include the clamped portions atthe ends of the springs.

The springs in this chapter are made of thin flat stock, with their endsthickened afterward by clamping or soldering “chops” (thicker endpieces) onto the spring’s two ends.

General spring characteristics

A big problem in designing a spring suspension is that the accuracystriven for in a good pendulum clock is so great that very small errorsbecome large and significant. A good pendulum will have an accuracyof about 1 s/month (0.03 s/day), which requires a constant pendulumlength of one part in 1.3 million, or 31 in. in a 40-in. pendulum length.Because there are many small errors, all of which add up, an arbitraryrule of thumb that individual errors should be no more than one-thirdof the total is used. To meet a total error budget of 1 s/month, theindividual errors, or more correctly the changes in the individual errors,should not exceed 0.01 s/day or 10 in. in a 40-in. pendulum length.

38

The author wishes to thank Bill Volna forhelpful suggestions on some mechanicalproblems, Dick Porter for making alignmentpins, and John Shallcross at the TimeMuseum for making the measurements ontheir Shortt clock.

97

This 0.01 s/day provides a useful scale of importance in the discussionsof errors and error tolerances that follow.

Now most timing errors can be compensated for if they remainfixed. It is mostly changes in the errors over time that cause a pendulumclock to deviate from a constant time rate. Initial tests indicated that thebiggest source of timing uncertainties in a suspension is in the variableclamping pressures on the ends of the suspension springs.

How long should the suspension spring be? How thick? How wide?What material? James [1, 2] has dealt with the theoretical aspects ofthese issues, and Boucheron [3, 4] with some of the practical. Practicalanswers are:

Thickness—0.002 to 0.010 in., in general the thinner the better.Width— to 1.5 in, the smaller the better.Length— to in, the shorter the better.Material—spring brass, beryllium copper, phosphor bronze, spring

steel, stainless steel, Ni Span C.

The easiest way is to pick the material first and the thickness second. Fordiscussion purposes here, however, material selection is covered last.

The thickness should be relatively thin, so as to have the least effect onthe clock’s timing rate and the pendulum’s temperature compensation.The problem is that thin springs are easily bent, causing a compromiseon thickness. What is “too thin” depends on both the suspension design,mostly its handle-ability without creasing or wrinkling the suspensionspring, and the mechanical aptitude of the user. Thus, the thicknessshould be made as thin as the suspension’s handle-ability and the user’smechanical aptitude will permit.

The width is used to adjust the spring’s cross-sectional area, so that ithas enough strength to carry both the pendulum’s weight and thespring’s bending stresses. The bending stresses are usually 3–4 timesbigger than the static stress from the bob’s weight [1]. Attention mustbe paid to the fatigue life vs stress level of the material. Maximum bend-ing stress occurs at the top end of the spring, as can be seen in Figure 15.1.The fatigue life of several materials is listed in Table 20.3. The best wayof calculating the thickness and width is by means of James’ equationsgiven in Chapter 16.

The spring’s length is made short so as to minimize an undesiredpendulum vibration: the horizontal vibration of the top of the pendu-lum rod without the bob following along. To complicate matters, thespring has to be made a little longer as its thickness is increased, to keepthe bending stress within a reasonable limit. Conversely, the spring canbe shortened as the thickness is reduced.

Well, how short is short? To answer that, the axis of rotation for thenormal left-right motion of the pendulum is located below the top edgeof the suspension spring. With a 15 lb pendulum and a flat beryllium

12

18

18


98

copper spring of dimensions 0.75 0.5 0.004 in. (L W T ), theaxis of rotation is 0.12 in. below the spring’s top edge. If the spring’slength is shortened up to 0.25 in., the axis of rotation is located 0.11 in.below the spring’s top edge. In other words, the axis of rotation essen-tially did not move when the length was shortened from in. to in.,indicating that the lower half inch of the spring length was superfluous.This leads to a second rule of thumb, which is to put the axis of left-right motion in the middle of the spring, that is, make the spring’slength approximately twice the distance that the left-right axis of rota-tion is below the top edge of the spring.

The location of the axis of rotation for left-right motion is found bytemporarily extending the pendulum rod about an inch or two up pastthe suspension spring with a lightweight pointer. The pendulum is setswinging at a reasonably large angle, and the horizontal amplitude isthen measured at two points: at the top of the pointer, and at thebottom of the pendulum. The ratio of these two horizontal amplitudesis equal to the ratio of the vertical distances each measuring point isabove and below the axis of rotation (see Chapter 4).

Spring steel is the most common material used for suspension springs.Spring steel has the disadvantage of rusting, which changes the springconstant and introduces an error in the clock’s timing rate. However,industry in general uses several other materials such as berylliumcopper, phosphor bronze, nickel iron, spring brass, and stainless steelfor springs. Beryllium copper and phosphor bronze are of particular

14

34

chapter i 5 | Spring suspensions for accurate pendulums

99

Figure 15.1. Calculated shape of a steelsuspension spring, with a 16 lb pendulum at 5 off vertical. (From K. James [1].)

Horizontal spring deflection (in.)

Spring thickness0.001 in.

0.002

0.004

0.006

0.008

0.010

0.012

0

0

0.1

0.2

0.3

0.4

0.5

0.6 Tops of pendulum rods

Dis

tan

ce b

elow

top

of s

prin

g (i

n.)

0.5 wide, in.W = 16 lb1 s pendulumIp = 1000 lb in.2

0.01 0.02 0.03 0.04 0.05

interest as both have lower flexure losses than spring steel, and thuseither one will give a higher Q pendulum than steel. The mechanicalhysteresis in beryllium copper is at least an order of magnitude less thanthat of steel, while that of phosphor bronze falls in between that ofberyllium copper and steel. Table 20.3 lists some properties of thesemetals. Beryllium copper’s modulus of elasticity is only 60% thatof steel, giving a 40% softer spring for equal thickness. Berylliumcopper does have one disadvantage. It contains 2% beryllium, whichin small-particle or powder form can cause berylliosis, a chronicirreversible lung disease. A filter mask should be worn in any sanding orsmall-particle-generating operations on beryllium copper, and anyresidue should be disposed of promptly and not left lying around. Bothberyllium copper and phosphor bronze are available from Mead Metals,St. Paul, Minnesota, USA.

Elinvar is an old metal that had the useful property that its springconstant did not change with temperature. Its thermoelastic coefficientof zero was hard to control, however, so Elinvar was replaced withtype 902 nickel iron (trade name is Ni Span C), whose spring constantalso does not change with temperature. Elinvar and Ni Span C areoccasionally used for pendulum suspension springs, but usually the cor-rection for thermoelastic changes in the suspension spring is included inthe thermal expansion correction for the pendulum rod.

Ni Span C is magnetic, rusts very little, and has low mechanical hys-teresis (0.03%). It is machined with carbide cutting tools and thenheated to approximately 1100 F for 3–5 h so that its spring constantdoes not change with temperature.

Ni Span C has some disadvantages. First, its availability and size selec-tion are limited, because it is a narrow-use material produced only insmall quantities. And second, making suspension springs out of it isexpensive in small quantities, because it needs heat treatment and theminimum charge for heat treating anything in a vacuum furnace isabout $200. Ni Span C is available from Special Metals Corp., (formerlyINCO Alloys International), Huntington, West Virginia, and fromHamilton Precision Metals, Lancaster, Pennsylvania, as Precision C.

The spring material used should be flat without any wrinkles or dim-ples, which might pop in and out (“oil can” effect) as the spring flexes.One way to make springs without edge burrs is to put several (2–6) in astack, and put the stack between two -in. thick aluminum strips like asandwich. The long sides of the sandwich are machined in a vertical mill,and the end holes are drilled in the same setup. This gives a burr-free setof springs of equal width and length, with extra springs available in casethe ones in use get bent.

The mounting angle to the wall at the top end of the suspensionspring is important. If the spring’s top end is tipped forward or back-ward, that is, rotated about a horizontal left-right axis, the clock’s

14


100

timing rate is affected, as shown in Figures 15.12 and 15.13. A tip of 0.3

forward or backward will change the clock’s timing rate by 1.2 s/day.This is discussed in more detail later on in this chapter under the sectionFront-to-Back Suspensions. And if the spring’s top end is tipped side-ways, that is, rotated about a horizontal front-to-back axis, the clock’stiming is affected by 0 to 1 s/day for 1.5 of rotation. The magnitudeof this timing error varies from spring to spring. Figure 15.2 shows theouter limits of the error as measured on 10 different suspension springsand suspension designs. No specific cause was found for this sidewaystipping error. Much testing showed it repeated on any given spring,but independent of length, thickness, material, suspension design, oruneven location of the end stiffeners (chops). Some springs would havea low slope to their curve of timing error vs sideways mounting angle,while others would have a high slope. Even with two apparently ident-ical suspensions, one would have a low-slope curve and the other a high-slope curve. This lack of an identifiable cause applies only to sidewaystipping of the top mounting of the suspension spring. The cause of thetiming error due to front-to-back tipping of the top mounting of thesuspension spring is known and discussed later in the section Front-to-Back Suspensions.

Left-right suspensions

Clock pendulums normally have two suspensions: a low loss one forleft-right motion, and a high loss one to allow some front-to-backmovement. This section covers suspension designs for left-right motion.

The first suspension design is shown in Figure 15.3(a and b). In thisdesign, the pendulum’s temperature compensator is located at the topof the pendulum rod. The rectangular structure at the top end of thesuspension spring is designed to go around the pendulum rod andthe temperature compensator, with the intent of shortening up thevertical height needed for the pendulum. There are two suspensionsprings in parallel, with four in. diameter steel dowel pins through1

8

Figure 15.2. Clock rate vs mounting angle oftop end of suspension spring, arounda horizontal front-to-back axis.


101

+6

+4

+2

–2–4 +4

Lower limit

Upper limit

0

Top suspension angle off vertical (deg)

–4

–6 Ch

ange

in c

lock

rate

, s/d

ay

+2

Figure 15.3. Double spring suspension: (a) bottom and (b) top structures.

2 in.

Spacer

Clampbolt

Dowelpin

Dowelpin

A

Suspensionspring

(a)

(b)

the four in. diameter reamed holes in the ends of the springs. Forclarity, the two suspension springs are shown in both Figure 15.3(aand b). The two halves of the top structure are clamped together bytwo long bolts. The bottom structure contains the pendulum’s temper-ature compensation and is held together by four screws. The suspen-sion springs are made of beryllium copper with dimensions of

0.004 in.3 (L W T ) each.This suspension design did not perform well, as the clock’s timing

rate changed a large 9 s/day when the two clamp bolts in the top struc-ture were changed from snug to very tight. Changing the tightness ofthe clamping bolts turned out to be a good test for measuring the sen-sitivity of the different suspension designs to clamping pressure. Thistest was given to all of the suspension designs.

There is a second problem in this design. Even with the temperaturecompensation in the lower structure disconnected (by moving thedowel pin through the pendulum rod down to the position marked A inFigure 15.3(a) ), the suspension grossly overcompensates the thermalexpansion of the invar pendulum rod by 10 times. This is believed to bedue to the thermal change in spring clamping pressure, caused by thedifference in the thermal expansion coefficients of the steel clampingbolts and the aluminum in the aluminum top structure. The defect inthe design is that the weight of the pendulum tends to pull the clampsapart at the top ends of the two suspension springs, making them moresensitive to clamping pressures.

An improved double spring suspension is shown in Figure 15.4(a and b).Here the length of the top structure clamped by the through-bolt hasbeen shortened to in., to reduce any differential thermal expansioneffects, and the weight of the pendulum does not pull the suspensionspring’s top clamp open. The clamping of the spring ends was moveddirectly to the dowel pins going through the spring ends, by usingshoulder screws here instead of ordinary dowel pins. Shoulder screwsare dowel pins with a screw head at one end and a short length ofthread (for adding a nut) at the other end. The length of the shoulderscrews was shortened to in., again to reduce any differential thermalexpansion effects between the bolts and the support parts beingclamped together.

The suspension design in Figure 15.4(a and b) also has the pendulumrod’s temperature compensator at the top of the pendulum. Thesuspension springs are beryllium copper, with dimensions of

0.004 in. (L W T) each. The in. diameter holes in the ends of thesprings are reamed to size for a close fit onto the in. diameter shoulderscrews.

This suspension design performed better than the one in Figure 15.3,but it still was not good enough. The clock’s timing rate changed6 s/day when the springs’ top clamp screws were moved from snug to

18

18

14

14

58

14

14

14

18


102

-20 Threaded rod

Shoulderscrew

Dowelpin

Shoulderscrew

Spacer

3.6

in.

2.7

in.

Tem

pera

ture

com

pen

sato

r

(a)

(b)

A

12

Figure 15.4. (a) Top and (b) bottom ofimproved double spring suspension.

very tight, and 3 s/day when the bottom spring screws were movedfrom snug to very tight. This suspension behaved properly over tem-perature, and did not exhibit the large thermal over-compensation thatthe suspension in Figure 15.3(a and b) did.

The suspension in Figure 15.4(a and b) has two design defects. First,the top of the suspension requires two long fingers coming downaround the temperature compensator to pick up the top ends of thesuspension springs, as shown in Figure 15.4(a). These long fingerscontribute an undesirable springiness to an otherwise stiff wall supportfor the pendulum. These fingers have to be very thick to combat thespringiness they introduce. And second, the added weight and bulk ofthe stiffened fingers makes it very easy to bend the delicate suspensionsprings while handling the pendulum.

How about putting end stiffeners, sometimes called “chops,” on theends of the suspension spring—do they help? The answer is yes they dohelp, but to get much benefit, the end stiffeners have to be soldered tothe suspension springs, and not be mechanically clamped to them withscrews. On a micro-inch scale, soldering provides a more positive stopor “end” to the free length of the spring than can be obtained withoutgrinding and optically polishing the spring ends and their associatedclamping surfaces to micro-inch tolerances.

End stiffeners have three advantages. First, they thicken the springends where the dowel pins go through, so that if necessary you can sub-stitute a threaded screw in place of the more preferred dowel pin withits smooth surface. Second, an end stiffener provides a square stop tothe free length of the spring where it attaches to the normally roundedtop end of the pendulum rod. And third, it is much easier to solidly grabhold of a large thick relatively rough-surfaced end stiffener with a relat-ively rough and not perfectly flat clamping surface, than it is to solidlygrab hold of a limp suspension spring where the actual points of clamp-ing contact depend critically on the parallelism and micro-inch surfacefinish of the mating surfaces involved. Figure 15.5 shows a suspensionspring with end stiffeners attached.

Since a positive length stop to the flexing of the suspension spring isdesired, the end stiffeners should be stiff, which means they should bethick. There are two end stiffeners located at each end of the spring, fora total of four end stiffeners for each suspension spring. Minimumthickness for an end stiffener is about in, with in. being much better,and in. being the maximum useful thickness. Steel end stiffeners aretwice as good as brass ones because steel’s modulus of elasticity isabout twice that of brass.

To find out how much effect end stiffeners have on the variability ofspring clamping pressures, brass end stiffeners in. thick were silver sol-dered onto the ends of two beryllium copper suspension springs withdimensions of 0.004 in.3 (L W T ) each. The two slots for3

838

116

14

18

116

Figure 15.5. Suspension spring with endstiffeners.


103

in.to116–

14–

the springs in the suspension of Figure 15.4(a and b) were widened toaccept the increased width of the end stiffeners on the suspensionspring. The clock’s timing rate changed 0.7 s/day when the springs’ topshoulder screws were moved from snug to very tight, and 0.4 s/daywhen the springs’ bottom shoulder screws were moved from snug tovery tight.

This is an 8–9 times reduction in sensitivity to clamping pressure.Other tests with other suspension designs showed that silver solderedend stiffeners typically reduced the clock’s timing sensitivity to clamp-ing pressures on the ends of the suspension spring by 5–10 times. Anda change in clamping pressure on the top end of a suspension springtypically produced 2–3 times the change in clock rate that an equalchange in clamping pressure produced on the bottom end of the spring.This difference is caused by more of the spring’s bending occurringnear the top of the spring. These tests also showed that silver solderingto the ends of the suspension spring gave a more stable joint thanmechanically clamping them.

Because of the importance of the soldering process, additional infor-mation on it is given in the Appendix.

The next left-right suspension design to be considered is shown inFigure 15.6. The temperature compensator has been left out, and endstiffeners are used on the suspension springs. Shoulder screws are usedin the end locating holes of the suspension springs, so as to locate theclamping pressures directly over the spring ends. The pendulum rodis inserted in a hole in the bottom piece, and its position is fixed witha dowel pin.

This design was not built, as it could be improved even further, asshown in Figure 15.7. First, the end stiffeners and shoulder screws wereeliminated, with the ends of the suspension springs being soldereddirectly to the top and bottom pieces of the suspension. And second,the hole at the bottom for the pendulum rod was lengthened, to reducethe angular uncertainty of the pendulum rod in the hole. This hole isreamed to size for a good fit to the pendulum rod.

Figure 15.7 shows five small alignment holes in the top and bottompieces. Short lengths of #14 small diameter copper wire are placed inthese holes, to hold the components in place during the oven solderingprocess.

A possible objection to the soldered joint construction in Figure 15.7is that it is hard to keep the suspension springs from bending or twisting“out of flat” during the soldering process. The answer to that is to notuse a hand soldering iron, but instead to pre-tin the top and bottompieces and then use an oven for soldering them to the suspension springs.An ordinary kitchen oven works fine for this. In the oven, all of the com-ponent parts are at the same temperature. And a slow even cooldownafterward to room temperature, without any thermal distortions, is

Figure 15.6. Double spring suspension withend stiffeners.

Figure 15.7. Double spring suspension withsoldered joints.


104

End stiffeners

Threaded rod

Dowelpin

B

B

A

A

A–A B–B

–24 Threaded rod–38

Solderingalignment hole (5)

2.1 in.

Dowelpin Soldered

jointSolderedjoint

Spacer

.12

easily obtained by shutting off the oven heat but leaving the oven doorclosed for several more hours. Oven soldering, rather than hand solder-ing, is ideal for obtaining a minimum stress assembly. Additional infor-mation on oven soldering is given in the Appendix.

Three spacers, the same thickness as the suspension springs, areshown in Figure 15.7. With hindsight, these could have been omitted bymachining a shallow clearance slot for each suspension spring in the topand bottom pieces of the suspension.

The dimensions of the beryllium copper springs in Figure 15.7 are0.30 0.25 0.004 (L W T). The 0.30 in. length of the suspensionsprings in Figure 15.7 is about as short as it was practical to make it. The0.30 in. distance is needed for clearing chips and the end of the drill bitwhen drilling the hole for the pendulum rod in the bottom piece of thesuspension.

This suspension design, shown in Figure 15.7, is a stable one. Thereare of course no clamping screws on the springs to cause variations inthe clock’s timing rate. Changing the tightness of the two nuts on theshaft going through the top piece from snug to very tight changedthe clock’s timing rate by 0.9 s/day. The thermal time correction witha thermally uncorrected pendulum measures 0.11 s/day/C, which isreasonably close to what it should be for an invar pendulum rod.

What about single spring suspensions? Single spring suspensionspretty much require the use of end stiffeners, because the connection ofthe suspension spring to the pendulum rod is frequently loose, deliber-ately so, so as to permit the easy removal of the pendulum. This is nota good idea from a stability standpoint, but is done anyway for the con-venience of pendulum removal. A mitigating factor is that the clock’stiming rate is 2–3 times less sensitive to variations in clamping pressureat the bottom end of the suspension spring than at the top.

Figure 15.8 shows a single spring suspension with end stiffeners.The bottom end stiffeners are in. thick stainless steel, and the top onesare in. thick steel. To reduce the volume of metal stressed by the tight-ening of the nuts, the nuts are recessed in. into the top end stiffeners.In this design, the pendulum rod does not need an open hook to fastenonto the suspension spring, because the top structure, including the hor-izontal shaft, is small and light, and easily removed with the pendulum.The size of the beryllium copper suspension spring is 0.004 in.3

(L W T ).The clock’s timing rate changed 1 s/day when the top clamping nuts

were moved from snug to very tight. The loose dowel pin at the bottomof the suspension spring was then replaced with a shoulder screw,so that clamping pressure could be applied to the lower end of thesuspension spring. The clock’s timing rate changed 0.4 s/day when theclamping pressure on the spring’s lower end was changed from snug tovery tight.

58

38

18

14

116

Figure 15.8. Single spring suspension.


105

Dowelpin

Threaded rod

Endstiffeners

0.25 in.

The single spring suspension shown in Figure 15.8 has a designdefect—it has an unnecessary axis of rotation in the dowel pin at thebottom end of the suspension spring. The weight of the pendulumtends to pull the suspension spring into a vertical orientation, but theinherent friction in the front-to-back suspension bearing at the top ofthe suspension spring, and in the dowel pin pivot at the bottom of thesuspension spring, can keep the suspension spring slightly off from atrue vertical orientation. This is shown in an exaggerated fashion inFigure 15.9. Measurements of the friction offset angle show it can beas large as 1.4. This is with a in. diameter steel dowel pin, steel endstiffeners on a beryllium copper suspension spring of size

0.004 in.3 (L W T), and a front-to-back suspension consisting of a in. diameter steel shaft lying in two 120 steel Vees. The distance from

the front-to-back axis of rotation to the dowel pin is 1.06 in. Moving thespring’s orientation from vertical to 1 off vertical changed the clock’stiming rate by 1.5 s/day.

This rotational design defect is eliminated in the design shown inFigure 15.10. Here the pendulum rod is inserted in a hole bored in thebottom piece, and cannot rotate. The pendulum is still separable fromthe clock by removing the dowel pin shown in Figure 15.10.

Energy coupling between modes of oscillation

A pendulum has three primary modes of oscillation: (1) the left-rightmotion of the bob using the suspension spring as a pivot, (2) the front-to-back motion of the bob using the horizontal left-right shaft that islocated just above the suspension spring as a pivot, and (3) rotationabout the long axis of the pendulum rod.

There are two secondary modes of oscillation as well, both of whichare undesirable. First is the pendulum rod’s vibrating sideways by itself,like a guitar string, at the fundamental and harmonic frequencies of thependulum’s length. This mode of oscillation can be quite prominentfor small diameter pendulum rods, of in. diameter or less. It isminimized by using larger diameter (i.e. stiffer) pendulum rods, of and

in. diameter. Second is the horizontal vibration of the top of the pen-dulum rod without the bob at the bottom following along. This vibra-tion is minimized by making the suspension spring’s length as short aspractical.

All modes of oscillation except the normal left-right motion of thebob should be suppressed, as they introduce undesirable errors in theclock’s time and timing rate. Front-to-back oscillation will be discussedfirst. Front-to-back oscillation is excited either by a small portion ofthe energy stored in the left-right motion of the pendulum, or by asmall portion of the external maintaining force that drives the normal

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38

516

18

58

38

18

Figure 15.9. Offset angle (exaggerated) inthe suspension spring.

Figure 15.10. Single spring suspensionwithout rotation about the dowel pin.


106

Front-to-backsuspension axis

Dowelpin

Ver

tica

l

Endstiffeners

Solderedjoints

Dowelpin

left-right mode of oscillation. These two drive forces couple into thefront-to-back mode via non-orthogonality of the front-to-back’s axis ofrotation to the left-right’s axis of rotation, or via the non-parallelismof the front-to-back’s axis of rotation to the direction of the externalforce driving the left-right oscillation. The energy stored in the left-rightoscillation is usually the biggest driver of front-to-back oscillation, asthe gravitational drive force at the ends of a left-right swing is normallymuch bigger than the external left-right drive force.

What happens when the pendulum oscillates front-to-back at thesame time as it is oscillating left-to-right? To find out, the left-right’s axisof rotation was moved 4 away from being perpendicular to the front-to-back’s axis of rotation. This provided a sizable drive force for thefront-to-back oscillation. A misalignment of 1 would be more typical,but 4 makes the effects of misalignment much larger and easier to see.The front-to-back suspension was a very low friction type—a roundshaft rolling on a flat surface, so that any front-to-back oscillationswould not die out right away.

At the start, friction is added to the front-to-back suspension, so thatthere is 2 in. of straight line left-right bob motion, and no front-to-backmotion. Periodic external drive pulses from an external bang-bangservo keep the left-right motion at a 2 in. minimum amplitude. Thefriction is then removed from the front-to-back suspension, andthe straight line bob motion expands out into an oval. The swing ampli-tude at the bob is now 2 in. left-right and 0.10 in. front-to-back. At thesame time, the clock’s timing rate slows down by 0.68 s/day.

Now the axes of rotation for left-right and front-to-back motionsare at slightly different vertical locations, so their natural resonantfrequencies are slightly different. But the periodicity of the left-rightdriving force, that is coupling over into the front-to-back plane, forces(i.e. synchronizes) the front-to-back oscillation to be at the same fre-quency as the left-right oscillation.

Every 6 min, the amplitude of the front-to-back oscillation slowlydecreases from 0.10 in. down to zero and then back up to 0.10 in. again,indicating an energy interchange between the front-to-back and left-right oscillation modes. Time comparisons with WWV show that theclock’s timing rate is constant and a little slow (0.68 s/day), and doesnot change over the 6-min cycle. Although the left-right amplitudealways has to slightly exceed the 2 in. minimum amplitude establishedby the external bang-bang servo, it is affected by the 6-min cycle. Whenthe front-to-back suspension has high friction and thereby preventsfront-to-back oscillation, the external drive pulses for the left-rightmotion come at a fairly steady rate of one pulse every 74 s. But whenthe pendulum is oscillating in the front-to-back mode because of verylow front-to-back suspension friction, the left-right drive pulse ratevaries from one pulse every 20 s to one pulse every 140 s, and does this


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in a steady repetitive pattern in synchronism with the 6-min front-to-back amplitude cycle.

The path traversed by the pendulum bob varies over the 6-min cycle.At zero front-to-back amplitude, the bob moves in a straight line 2 in.long in a left-right plane. As the front-to-back amplitude builds up, thebob motion expands into an oval, 2 in. long by 0.1 in. wide, with theoval’s long axis oriented left-right. The oval then slowly rotates 4 untilits long axis is parallel to the actual misaligned rotational axis of front-to-back motion. The bob’s oval motion then slowly collapses back intoa straight line, still parallel to the misaligned rotational axis of front-to-back motion. This straight line motion slowly changes back into a longoval, 2 in. long by 0.1 in. wide, with the oval’s long axis still parallelto the actual misaligned rotational axis of front-to-back motion. Theoval then slowly rotates back 4 to where its long axis is again oriented left-right. The oval slowly collapses back into a straight line motion 2 in.long in a left-right plane, and the front-to-back amplitude is again backdown to zero. This cycle repeats itself every 6 min.

All through the 6-min cycle, the clock runs at a constant time rate,0.68 s/day slower than when front-to-back motion is prevented. Frictionin the front-to-back suspension pivots will reduce or prevent any front-to-back oscillation. A friction torque of 0.07 in. lb was found to reducefront-to-back oscillation amplitude by a factor of 5 in 2 min. A frictionof 0.07 in. lb is also enough to prevent any new front-to-back oscilla-tions from starting up. A method of measuring these friction torques isdescribed in the Appendix.

The third primary mode of pendulum oscillation is rotation about thelong axis of the pendulum rod. This rotation is excited either by a smallportion of the energy stored in the suspension spring, or by a smallportion of the external drive force that maintains left-right motion ofthe pendulum. The left-right’s external drive force will couple into therotational mode if the external drive force is not centered in the planeof left-right swing containing the pendulum’s center of percussion. Andenergy stored in the suspension spring will couple into the rotationalmode if the suspension spring is not centered about the same plane.

The oscillation frequency of the rotational mode falls within a rangeof 0.3–3.5 Hz, with 0.3 Hz corresponding to a single long-spring sus-pension ( 0.004 in., L W T ), and 3.5 Hz corresponding toa double short-spring suspension ( 0.004 in. each, L W T )with a 1.5 in. gap between the springs. The rotational oscillations dieout rather quickly (30 s) with a double short-spring suspension. It takesmuch longer, up to 20 min, for the oscillations to die out with a singlelong-spring suspension.

What sort of timing error does the rotational mode introduce? To findout, two suspensions were tested by manually starting a rotational oscil-lation about the long axis of their pendulum rods. The first suspension

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was a medium length single spring design ( 0.004 in.3, L W T,beryllium copper), with a 6.5 rotational amplitude and a 0.87 Hz rota-tional frequency. The pendulum is simultaneously swinging left-rightin its normal timing mode (0.5 Hz). Figure 15.11(a) shows the timedifference between the pendulum and WWV, measured at 1 s intervalsover a one minute period at the start of the rotational oscillation. Therotational oscillation shown in Figure 15.11(a) continues on with decreas-ing amplitude for another four minutes before dying out.

The second suspension was a short length double spring design

0.004 in.3 each, (L W T, beryllium copper) with a 1.5 in. gapbetween the springs. With a double spring suspension, the rotationalamplitude about the long axis of the pendulum rod is physically limitedto a small value. The pendulum had a 0.8 rotational amplitude and a3.5 Hz rotational frequency, and was simultaneously swinging left-rightin its normal timing mode (0.5 Hz). Figure 15.11( b) shows the timedifference between the pendulum and WWV, measured at 1 s intervalsover a 36 s period.

Figure 15.11(a and b) show that the timing error introduced bypendulum oscillation about the long axis of the pendulum rod is small(0.016 s max). And it is not a time rate error, but a time error. That is, itis an error in pendulum angle, the integrated total of which goes to zerowhen the rotational oscillation goes to zero. In other words, there isa small plus or minus short-term time error while the pendulum is oscil-lating rotationally about the long axis of the pendulum rod, but after therotational oscillation has stopped, there is no longer a time error.

Front-to-back suspensions

The primary purpose of the front-to-back suspension is to allow thependulum to hang and swing in a vertical plane that is as close as pos-sible to true vertical. If the left-right pendulum motion is not in a trulyvertical plane, the clock’s timing rate will slow down, as shown in

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Figure 15.11. Clock time error fromrotational oscillation about the long axis ofthe pendulum rod for a: (a) single springsuspension, and (b) double spring suspension.


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0.12

0.11

0.10

0.11

0.10

0 20 40Time (s)

60

Clo

ck a

hea

d of

WW

V (

s)

(a)

(b)

Figure 15.12. The data in Figure 15.12 was taken by moving thependulum’s left-right plane of swing away from true vertical by attach-ing a horizontal rod, oriented front-to-back, onto the top suspensionpiece in Figure 15.3(a), and hanging a weight at various positions alongthe horizontal rod.

The component of gravity that couples into the plane of swingvaries with the cosine of the angle between true vertical and the planeof swing. So if the plane of swing is not vertical, the error in the clock’stiming rate in s/day is:

Error (1 cos )(86,400 s/day)

This calculated error is plotted as a solid line in Figure 15.12. The exper-imental data are plotted as individual data points in Figure 15.12, and theyagree with the calculated solid line curve. The experimental data wastaken with three different lengths of suspension spring, and Figure 15.12shows that the slowdown in timing rate is essentially independent ofspring length. If the left-right plane of swing is 0.28 off true vertical,Figure 15.12 shows that the pendulum will slow down by 1 s/day.

The sensitivity or slope of the curve is rather high at 1 s/day. Toreduce the sensitivity and also to get a more reasonable error rate of say0.01 s/day, the plane of swing must be within 0.026 of true vertical.This is shown in Figure 15.13, which is the same as Figure 15.12 exceptthat the angle scale has been expanded 10 times for better resolutionat small angles near true vertical. One can conclude from Figure 15.13that it is desirable to have as low a friction as possible in the front-to-back suspension, so as to get the left-right plane of swing closer totrue vertical, reducing the timing rate error and also the timing rate’ssensitivity to changes in the off vertical angle .

A contrary need for adding more friction to the front-to-back sus-pension comes from the desire to damp out and prevent any front-to-back oscillations, which can couple over into the left-right motion andcause timing rate errors. A friction test back in the previous sectionshowed that 0.07 in. lb of friction torque was enough to stop front-to-back oscillation. So what is needed in a front-to-back suspension is nofriction or as little friction as possible during the initial setup of theclock, so as to get the pendulum hanging as close as possible to truevertical. And then during the actual running of the clock, the friction inthe front-to-back suspension should equal or exceed 0.07 in. lb, so as toprevent front-to-back oscillation.

Two suspension designs that will do this are shown in Figures 15.14and 15.15. The suspension in Figure 15.14 is a horizontal shaft withsmall diameter ( in.) ends resting in two 120 Vees. The pendulum andits left-right suspension are attached to the horizontal shaft in the man-ner shown in Figure 15.7(a), and hang between the two Vees. The shaftis a -24 threaded rod over most of its short length, with only the two3

8

18

Figure 15.12. Effect of left-right plane ofswing being tilted away from true vertical. Spring length: · 0.25 in., · 0.37 in., · 0.75 in.

Figure 15.13. Effect of left-right plane ofswing being tilted away from true vertical, atsmall angles.


110

–0.6 –0.3

–2

–4

–6

–886,400 (1–cos )

+0.3 +0.60Front-to-back angle off vertical (deg)

Ch

ange

in c

lock

rat

e (s

/day

)

–0.06 –0.03

–0.02

–0.04

–0.06

86,400 (1–cos )

0Front-to-back angle off vertical (deg)

Ch

ange

in c

lock

rat

e (s

/day

)

+0.03 +0.06

ends of the shaft being reduced to a small in. diameter where they restin the two Vees. The friction level of the in. dia. steel shaft in two 120

steel Vees measures 0.16 in. lb, which is more than enough to damp outany front-to-back oscillations. And the friction angle measures 0.018,which from Figure 15.13 gives a 0.004 s/day maximum timing rateerror.

The friction level of a round shaft lying in a Vee is an inverse functionof the Vee’s included angle, as described in the Appendix. Using a 120

Vee angle reduces the friction by 31%, compared to that of the morecommonly used 90 Vee. An even larger 135 Vee angle was tried and hadto be rejected, as the in. diameter shaft then rolled up the sides of theVee instead of staying at the bottom of the Vee and rotating there.

The front-to-back suspension design shown in Figure 15.14 has twoadvantages: (1) a positive angle alignment defined by the locations ofthe two Vees, and (2) the pendulum’s left-right plane of swing willalways be within 0.018 of true vertical, even if the clock case tipsforward or backward by more than that amount.

The second front-to-back suspension design is shown in Figure 15.15.This suspension design uses a large diameter (0.3 in.) shaft rolling ontwo flat surfaces. The two flat surfaces are separated a short distance,and the large diameter shaft is horizontal and rests on the two flat bear-ing surfaces. The shaft is a -24 threaded rod, with its two ends turneddown to a smooth 0.30 in. diameter for very low friction contact on theflat bearing surfaces. The pendulum and its left-right suspension arefastened to the shaft in between the two flat surfaces, in the mannershown in Figure 15.7(a).

Two flat springs (0.6 0.3 0.031 in.3 each, L W T) add frictionto the suspension, by rubbing on the top of the shaft. Each spring islocated directly over one of the flat bearing surfaces. This friction pre-vents front-to-back oscillation. During startup of the clock, one screw isloosened on each spring, lifting the springs off the shaft and removingthe friction, which allows the pendulum’s left-right plane of swing tomove closer to true vertical. To align the shaft, a spacer is temporarilyinserted between the shaft and a fixed shoulder at the side of each flatbearing surface, as shown in Figure 15.15.

The advantage of this design over that in Figure 15.14 is its lower bear-ing friction when the friction springs are lifted. This allows the pendu-lum’s left-right plane of swing to get closer to true vertical, and reducesthe timing error caused by the left-right plane of swing being slightly offvertical. The clamping of the friction spring has to be carefully arrangedso that the shaft does not rotate when the friction spring touches it.

Almost any friction level can be obtained by changing the thicknessof the clamping spring. If a low friction level such as 0.07 in. lb is picked,the left-right plane of swing will always be close to vertical even if theclock case gets tipped backward or forward by a large amount. But to

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18

18

18

Figure 15.14. Front-to-back suspension: Smalldiameter shaft in Vee.

Figure 15.15. Front-to-back suspension: Largediameter shaft rolling on a flat surface.


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Small diametershaft

Vee supportarm (1 of 2)

120° Vee

Temporary alignment spacer

Friction spring

Shaft supportarm (1 of 2)

Large diameter shaft

keep things in perspective, the size of the advantage that this design hasover that in Figure 15.14 is only 0.004 s/day.

Shortt and Riefler suspensions

For comparison purposes, it may be useful to look at the suspensiondesigns of two acknowledged clock experts, William Shortt and SigmundRiefler. In doing this, one should remember that their designs were done80–100 years ago when less was known about clock design.

In Shortt’s clock, a single short suspension spring (0.25 0.39

0.0062 in.3, L W T ) is used for the left-right suspension. It is integralwith its end stiffeners, that is, both the spring and the end stiffenerswere made together out of a single piece of Elinvar. Making thesuspension spring integral with its end stiffeners gives a lower loss sus-pension, helping to give the Shortt pendulum its high Q of 25,000 atatmospheric pressure, and 110,000 at 25 mm of mercury.

For the front-to-back suspension, Shortt used a “large diameter shaft”rolling on a “flat surface.” During the initial setup of the clock, a in.diameter shaft rolls freely on the flat surface, to get the pendulumhanging close to true vertical. The shaft is then locked rigidly in posi-tion by four horizontal screws. One consequence of locking the shaft inposition is that the wall mount for the clock case must be very stableangle-wise, as any front-to-back tipping of the clock case that mightoccur over time will introduce a change in the clock’s timing rate, asdescribed earlier in the last section.

The “large diameter shaft” actually consists of two short lengths ofin. diameter rod, axially placed, with one end of the suspension spring

sandwiched in between them as shown in Figure 15.16. The two shortlengths of in. diameter rod and the suspension spring are clampedtogether by a in. diameter bolt down through the center. The reasonfor using such a large in. diameter shaft is to decrease the angle3

4

316

34

34

34

Figure 15.16. Front-to-back suspension in theShortt clock.


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1.8 in.0.060

A – AA

Lockscrew A

Round shaft onflat surface

sensitivity of the shaft’s horizontal locking screws, by giving them alonger moment arm.

The “flat surface” actually consists of two long flat ledges, in. wideand spaced 1.12 in. apart. The pendulum hangs between the two ledges.With this construction, the weight of the pendulum tends to pull theclamp apart at the top end of the suspension spring. The testing in thesection Left-Right Suspensions showed that this type of constructionintroduces temperature variations in the clock’s timing rate. It is knownthat some Shortt clock owners complained about the clock’s tempera-ture compensation not being as good as it should be. The design of itsfront-to-back suspension may be the reason why.

Reifler made quite a few different clock designs, each with a differentlevel of accuracy. The following applies to his most accurate types(A, D, and E).

Riefler’s most accurate clocks used two left-right pendulum suspen-sions in series. One was a knife edge attached to the clock frame, andthe other was a flat spring (actually two flat springs in parallel) attachedto the top of the pendulum rod and connected to the knife edgethrough the front-to-back suspension. The three-dimensional arrange-ment of this is difficult to convey in a two-dimensional sketch, so thereader is referred to the photographs in [5]. The flat spring suspensionis the main left-right suspension for the pendulum, with the extra knifeedge suspension being used to allow the escapement and its gear trainto drive the pendulum through “over-flexing” of the suspension springs,instead of through the traditional crutch which is not used in thesedesigns. The two connecting pieces between the knife edge suspensionand the top ends of the suspension springs “flip” back and forth at thecenter of the pendulum’s swing, giving drive energy to the pendulumthrough the “over-flexing” of the springs. The idea is that better time-keeping will result from driving the pendulum softly through the topends of the consistent and low-loss suspension springs, rather than withhard strikes from a crutch with its variable friction losses.

The two flat springs in the suspension are quite short (0.10 in.), andare spaced 0.24 in. apart. The springs’ thickness (i.e. stiffness) is scaledto meet the needs of the escapement drive, rather than just the needsof the pendulum alone. The ends of the springs are mechanicallyclamped with screws.

The type of front-to-back suspension used in Riefler’s clocksdepended on the accuracy level. His highest accuracy clocks used two60 cone points, spaced 0.6 in. apart and resting in the ends of two small0.03 in. diameter holes, to obtain a low but “sufficient frictional resist-ance against possible transverse oscillations [5].” This statement,together with the inherently low friction in his front-to-back suspensiondesign, indicates that Riefler knew about front-to-back pendulum oscil-lations, and about how much friction it takes to damp them out.

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In Riefler’s medium accuracy clocks, the front-to-back suspension isa short 0.10 in. diameter steel pin lying across two 90 Vees. Riefler’slower accuracy clocks had no front-to-back suspension, other than thependulum’s swiveling on the pin at the bottom end of the suspensionspring. In all of Riefler’s pendulums, even in his most accurate clocks,the slotted attachment hole in the top of the pendulum rod is a roundhole and not an inverted Vee. The diameter of the hole is nominally thesame as that of the pin in the hole.

Table 15.1 lists some suspension and pendulum characteristics of boththe Shortt and Riefler pendulum clocks. Some of the numbers in thissection and in Table 15.1 are only approximate, having been scaled fromdrawings and photographs. Most of the Shortt data is from measure-ments made on Shortt clock No. 6 at the Time Museum in Rockford,Illinois (now closed). Most of the Riefler data is from Ref. 5.


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Table 15.1. Some characteristics of the Shortt and Riefler clocks

Item Shortt clock Riefler clocks

Most accurate Medium accurate

Suspension, left-right Flat spring Flat spring plus knife edgea Flat spring plus knife edgea

No. of suspension springs 1 2 2

Spring material Elinvar NAb NAb

Spring size L W T, in., each 0.25 0.39 0.0062 0.1 0.11 0.0045 0.15 0.15 NAb

Gap spacing between springs, in. — 0.24 0.18

Connections to ends of springs Integralc Screw clamped Screw clamped

End stiffeners used? Yes Yes Yes

End stiffener thickness, 0.030d 0.075 0.075

each side, in.

Suspension, front-to-Back in. diameter shaft Two 60 cone points 0.10 in. diameter shaft lying

rolling on flat surface, in two 0.03 in. diameter across two 90 Vees

with position clamp holes, 0.6 in. apart

Bob weight, lb 14 16 NAb

Bob material Type metal NAb NAb

Pendulum attachment to 0.078 in. diameter pin 0.09 in. diameter pin 0.09 in. diameter pin

suspension spring

Pendulum rod diameter, in. 0.31 0.55 0.39

Beat time, s 1 1 1

Operating air pressure, mm Hg 20 50e 760

Notesa 60 or 90 steel knife edge on flat agate surface, knife edge radius 80 in. or less.b Not available.c Suspension spring and end stiffeners are all one piece.d 0.060 in. total thickness of end stiffener.e Type D and E clocks only.

34

Summary

The best left-right suspension design tested is the one shown in Figure 15.7.It is a double spring arrangement, using relatively short and thin sus-pension springs. It is a minimum parts count design, with the springends silver soldered directly to the top and bottom suspension pieces.Beryllium copper is the best material for the suspension springs,because of its low flexure losses. Phosphor bronze is the second bestmaterial. These two conclusions do not include the torque effects ofspring bending, which couple over into the pendulum’s timing. Thetorque effects are discussed in Chapters 16 and 20.

Attachment by soldering to the ends of the suspension springs makesa more stable joint than mechanical clamping. Silver soldering gives amore rigid and more stable joint than lead soldering. An appropriatelow temperature silver soldering process is described in the Appendix.

The characteristics needed in a front-to-back pendulum suspensionare: (1) zero or very low friction when the clock is started up, so as toget the pendulum swinging in a plane as close as possible to true verti-cal, and (2) a low friction of 0.07 in. lb during normal running of theclock, which is sufficient friction to prevent or damp out any front-to-back oscillations.

Two suitable front-to-back suspension designs are shown inFigures 15.14 and 15.15. Both have been tested. The design in Figure 15.14is the simplest, and consists of a in. diameter shaft lying across two120 Vees. To reduce friction, the shaft diameter is reduced to in. in theVees. Its friction level of 0.16 in. lb is a little higher than the 0.07 in. lbmentioned previously, but it is still suitable for all but the most accuratependulum clocks.

The second front-to-back suspension design is shown in Figure 15.15,and uses the concept of a large diameter shaft rolling on a flat surfaceto get a very low friction level. This in turn gets the pendulum’s left-right plane of swing closer to true vertical, resulting in a lower error inthe clock’s timing rate. The friction level is adjustable in this design.

Energy in the left-right motion of the pendulum can couple over intothe other two oscillation modes of the pendulum, and cause both front-to-back oscillation of the pendulum and rotary oscillation about the longaxis of the pendulum rod. Both of these modes of oscillation coupleback into the pendulum’s normal left-right oscillation mode, and unde-sirably affect the clock’s timing. When the mode coupling was deliber-ately made extra large, the energy exchange between the left-right andfront-to-back oscillations was plainly visible in the motion of the pendu-lum bob. Front-to-back oscillation slows down the clock’s timing rate,which is derived of course from the pendulum’s left-right motion.

Any rotational oscillations about the long axis of the pendulumrod couple back into the left-right motion of the pendulum as small

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115

amplitude variations in the pendulum’s left-right position, and show upin the clock’s timing as small variations in the time readout. The sumtotal of these time variations goes to zero when the rotational oscilla-tions go to zero.

Front-to-back oscillation can be eliminated or damped out in 2 minby putting a small amount of friction (0.07 in. lb) in the front-to-backsuspension. Rotational oscillation can be eliminated or quickly dampedout by using a double spring suspension with short springs and a widespacing between the two springs.

Bush and Jackson [6] have previously given some translationalstability requirements for the wall or base that a pendulum clock ismounted on. They showed that a horizontal left-right springiness inthe suspension spring’s top mounting piece of 285 in./lb changes theclock’s timing rate by 6.42 s/day. This was with a 14 lb pendulum boband a swing (half ) amplitude of 0.75. They also showed that verticalspringiness in the suspension spring’s top mounting piece has no effecton the clock’s timing.

This chapter gives some angular stability requirements for the wallor base that the clock is mounted on. Rotation of the spring suspen-sion’s top mounting piece in the left-right plane of swing affects theclock’s timing rate by anywhere from 0 to 0.5 s/day per degree ofrotation, as is shown in Figure 15.2. No cause was found for this effect.

Rotation of the spring suspension’s top mounting piece in the planeof front-to-back motion also affects the clock’s timing rate, amount-ing to 1.2 s/day for 0.3 of rotation, or 0.01 s/day for 0.027 of rotation.This error is a cosine effect and is due to the gravity vector being out ofthe plane of left-right pendulum motion. The error rate applies up toan angle equal to the maximum friction angle in the front-to-back sus-pension. This error can be reduced by using the smallest possible fric-tion level in the front-to-back suspension that will stop or damp out anyfront-to-back oscillation. That minimum friction level is 0.07 in. lb, asdescribed in the section Front-to-Back Suspensions.

Appendix

Silver soldering process

Soldering a suspension spring to its mating parts is a critical process, andmuch effort has been expended in finding a good way to do it. The firstrecommendation is use a low temperature silver solder rather than alead solder. Silver solder is more rigid and more stable, and gives astronger bond. It also gives a much smaller solder fillet, because of itslower surface tension. The solder composition to use is 2% silver, 98%tin. This composition is normally used on jewelry and melts at 450 F,


116

a little higher than lead solder (40% lead, 60% tin melts at 375 F). Thesilver solder is manufactured by Alpha Metals, Jersey City, New Jersey,and is commonly available in hardware stores. The flux that comes withit is a liquid rather than a paste, making it easier to accurately stackparts together before soldering.

The overall soldering concept is to individually pre-tin all of the mat-ing pieces except the suspension springs, and then solder the pre-tinnedpieces to the suspension springs in an oven. The suspension springs arenot pre-tinned because of the difficulty in determining where the bound-ary is between the spring’s clamping and free flexure areas, that is,between the solder and non-solder areas. Ease of soldering is an advant-age to making the suspension springs out of brass, beryllium copper, orphosphor bronze, rather than steel, stainless steel, or Ni Span C.

Oven soldering offers many advantages. It gives the lowest jointstress of any soldering method, because of its uniform temperatureduring cooldown after soldering. Because of fixturing, it gives betteraligned springs and a more constant spring length. A stack of partscomes out uniform in height and uniform in thickness of the solderjoint. A uniform joint thickness is a big help when making end stiffen-ers that have to go in a slot of fixed width at the top of a pendulum rod.And finally, the slow cooldown from 500 F to room temperature (aftersoldering) is conveniently the proper heat treat needed to convert softberyllium copper springs to their hardened spring temper condition.Phosphor bronze springs do not need heat treating, and the 500 F oventemperature does not affect them. Steel spring material can be pur-chased in a hardened spring temper as feeler gage stock, and is also notaffected by the 500 F oven temperature.

Pre-tinning of the parts to be soldered is important as it drasticallyreduces the occurrence of a bad solder joint. The materials used alsohave a strong effect on soldering as some materials are harder to “wet”or “tin” than others. Brass parts can usually be tinned in one try. Steelparts usually take three tries, and stainless steel ones can take up to tentries. The edges of the parts immediately adjacent to the free flexureportion of the suspension spring is the most critical area, and should beinspected with a magnifying glass for a good pre-tinning.

The amount of solder put on a part during pre-tinning is important.Too much solder and some of it will squeeze out of the joint and forma big fillet on the suspension spring. Too little and an un-soldered gapwill form in the solder joint. The right amount of solder is just slightlymore than the bare minimum that will “wet” the surface. Normaltinning leaves too much solder on the surface. To remove it, tin thesurface normally and then tip the tinned surface to vertical with onecorner pointing down, and shake off the excess solder. Scrape off abouthalf of the one remaining drop of solder at the bottom corner. Movethe tinned surface back to horizontal, and let the remainder of the


117

solder drop that is still present flow back uniformly over the surface.This will give the right amount of solder for a good joint with either nofillet or just a small fillet extending out on the suspension spring.

Pre-tinning is most conveniently done with a hot plate, as shown inFigure 15.17. The hot plate itself is covered with a small piece of sheetaluminum, to keep the solder and corrosive flux out of the heatingcoils. The Variac adjusts the hot plate temperature to a small amountabove the melting point of the solder.

When soldering end stiffeners (chops) onto a suspension spring, asquarer and more accurate job can be obtained by soldering the springsin joined pairs in the oven, as shown in Figure 15.18. The end stiffenersat each end are left joined together during the soldering process, and arecut apart afterwards. Small locating holes are drilled in both the endstiffeners and the suspension springs, as shown in Figure 15.18. Shortloose-fitting copper pins are inserted in these holes for alignment dur-ing soldering. The end stiffeners are then pulled apart slightly, toremove the looseness from the copper alignment pins.

A small squaring tool keeps the end stiffeners square with the sus-pension springs. A good machinist’s square (Starrett, etc.) should not beused here, as the corrosive soldering flux will stain and corrode thesquare.

Soldering in an oven requires a minimum amount of fixturing.Figure 15.19 shows the setup used. The parts to be soldered are stackedvertically, so that a weight placed on top of the stack will press thesoldered surfaces together. A firebrick is placed above and below thestack, to provide non-solderable surfaces that can withstand the 500

oven temperature. Suitable firebricks are readily available from ceramicsupply companies.

After soldering, all of the visible flux is removed by soft brushing inwarm water. The flux contains hydrochloric acid and is corrosive. Thesmall amount of flux remaining in the small cracks can be ignoredunless one of the materials being soldered is steel, in which case ultra-sonic cleaning is needed to get the flux out of the cracks. Otherwise, therust occurring in these small cracks will come out over time and coverthe whole steel part.

An even better procedure might be to tin the individual parts usingthe acidic flux, but to solder the tinned parts together in the oven using

Figure 15.17. Hot plate setup for pre-tinningsuspension parts.

Figure 15.18. Soldering suspension springs inpairs gives better alignment of the endstiffeners.


118

120Vac

Variac

Aluminumplate

Part to be tinned

Electricalhot plate

Cut apartafter soldering

Alignment pin

End stiffeners

Squaring tool

Figure 15.19. Soldering setup in 500 F oven.

Weight

Firebrick

Firebrick

Oven

Suspensionassembly

Baseplate

an electronic grade of solder flux, which has a neutral pH. It is easyto clean the acid flux off the individual parts after tinning, and anyneutral pH flux left after washing the oven soldered parts stack will notcause rust.

Friction torque vs Vee angle

The friction torque Tf of a round shaft weighing W and lying in a Veeof angle is proportional to the coefficient of friction Cf, the radius Rof the shaft, and the force F normal to the Vee’s surfaces, as shown inFigure 15.20. From Figure 15.20,

Tf 2FtR,Ft Cf F,

then;

Thus, the Vee’s frictional torque Tf increases inversely with the sineof half the Vee angle.

Measuring a suspension’s pivot friction

A suspension’s pivot friction can be measured using the setup inFigure 15.21, which shows the friction being measured in a front-to-back suspension. A stiff board (1.5 3.5 in.) is mounted across the front of the clock case, in front of the pendulum bob. This provides areference surface for the depth micrometer, so it can measure the hori-zontal distance from the front of the clock case to the pendulum bob.An electrical buzzer is connected between the micrometer and the topof the pendulum, so as to give an accurate aural indication of when themicrometer touches the pendulum bob.

The idea is to pull the bob about 2 in. toward the front of the clockcase, and then very slowly let it move back toward the pendulum’svertical position. It will stop short of true vertical by an angle , due topivot friction. The bob is then pushed about 2 in. toward the back of theclock case, and very slowly allowed to return toward the pendulum’svertical position. It will again stop short of true vertical by an angle ,due to pivot friction. The micrometer measures the horizontal distanceto both the front (D1) and back (D2) positions of the bob, giving thevalue of D. The length of the pendulum is L.

,

sin1(D/L).

D D2 D1

2

Tf Cf RW

sin(/2)

W 2F sin(/2),


119

Figure 15.20. Round shaft lying in a Vee.

Ft F t

F FW

R

The friction torque Tf in the front-to-back suspension is then thebob’s weight times the horizontal distance D to the pendulum’s pivotpoint:

Tf WD WL sin .

References1. K. James. “Design of suspension springs for pendulum clocks,”

Timecraft—Clocks and Watches, ( June 1983), pp. 9–11; ( July 1983) pp. 14–15;(August 1983), pp. 10–15; (November 1983), p. 27.

2. K. James. “Precision pendulum clocks—circular error and the suspensionspring,” Antiquarian Hor. (September 1974), pp. 868–83.

3. P. Boucheron. “Pendulum suspensions,” NAWCC Bulletin (April 1987),pp. 98–104.

4. P. Boucheron, private communication, 1993.5. D. Riefler. Riefler-Präzisionspendeluhren (in German). Verlag George D. W.

Callwey, München, Germany, 1981.6. V. Bush and J. Jackson. “Amateur scientist,” Sci. Amer. ( July 1960),

pp. 165–76; (August 1960), pp. 158–68.

Figure 15.21. Measuring pendulum pivotfriction.


120

Electrical buzzer Si

de w

all o

fcl

ock

case

Depthmicrometer

Stiffboard

2D∆L

2

L

D1 D2

W

chapter 16

James’ suspension spring equations

James’ two equations are used to design and evaluate a pendulum’ssuspension spring.

The two equations

In 1983, James [1] published two useful but rather complicatedequations, whose purpose was to help design the suspension spring.The equations show the effect that different spring lengths, widths, andthicknesses will have on the pendulum. The two equations are quitehelpful, as the suspension spring is without doubt the most complicatedpart of a pendulum, despite the spring’s apparent physical simplicity.The first equation calculates the maximum stress in the spring, whichoccurs at the spring’s top end at the maximum angle of swing. Thesecond calculates how much the pendulum will speed up due tothe inherent torque the suspension spring exerts on the pendulumrod. And as James suggested, the second equation is used here to showthat the suspension spring exerts a temperature effect on the pen-dulum’s timing that is roughly as big as the thermal expansion of thependulum rod.

James’ article was apparently not checked for mistakes before beingprinted, as his two published equations are full of errors. In 1995,Bigelow [2, 3], corrected James’ equations, compared the first one withLeeds’ equations [4], and found they agreed on the maximum stressin the suspension spring. In 1999, also checking James’ equations,Woodward [5] derived his own equations for stress in the spring, andcould find no fault in James’ work. In addition, James’ second equation(pendulum speedup) agrees with the experimental data given inChapter 20. And using James’ equations, my calculated stress andspeedup values using James’ equations agree with Bigelow’s at the sixpoints listed in Bigelow’s paper [3]. As a result of all this cross-checking,I consider both of James’ equations to be correct.

121

James’ equations, as corrected by Bigelow, are reproduced here asEqs (16.1) and (16.2):

(16.1)

(16.2)

(16.3)

V K sin W cos2, lb, (16.4)

(16.5)

(16.6)

Q ekL, (16.7)

(16.8)

IpML2, in. lb s2, (16.9)

(16.10)

(16.11)

(16.12)

(16.13)

where

ft spring’s total stress, psifb spring’s bending stress, psifw spring’s static weight stress, bob rod, psiT/Td speedup change in clock rate, s/dayE spring’s Young’s modulus of elasticity, psi

MWg , lb s2/in.

EIEbt3

12 , lb in.2,

fwWbt

, psi,

Mo

Lp

2VQ tan [K(1 Q)2cos ]

kLp(1 Q2),

Lp 0.5LsLs24Ip

M, in.,

kVEI , in.1,

KIpW sin

M(Lp)2 Ip

, lb,

ft fb fw, psi,

TTd

86,400

2 Mo

Lp 1

W sin Mo

Lp, s/day,

fbEkt t an

2 1Q 2

1Q 2 Ip

MLpIp, psi ,


122

b spring width, in.t spring thickness, in.L spring’s free unclamped length, in. swing (half ) angle, deg.g gravity 386.4 in./s2

Ls pendulum length 39.1505 in.Tp pendulum period, sTd 1 day 86,400 sW weight of bob rod, lb.

Equations (16.1) and (16.2) are plotted in Figures 16.1–16.7. Thefigures show the maximum stress in the spring and the speedup in theclock rate as functions of (1) the swing angle, (2) the bob-plus-the-rod’sweight, and (3) the spring’s length, width, and thickness. The curves aredrawn for the author’s pendulum dimensions. The following values

chapter 16 | James’ suspension spring equations

123

0

30 K

20 K

10 KTot

al s

tres

s (p

si)

01 2 3

b=0.5

0 in.

=0.75

=1.50

2Swing angle, (deg)0

30 K

20 K

10 K

0

Tot

al s

tres

s (p

si)

1 2 3

t=0.0

03 in

.=0.0

20

=0.006

=0.015

=0.010

2Swing angle, (deg)

30 K

20 K

10 K

0

Tot

al s

tres

s (p

si)

Thickness, t (in.)0 0.005 0.010 0.015 0.020

W=

30 lb=20

=15

=10

=5

30 K

20 K

10 K

0

Tot

al s

tres

s (p

si)

Thickness, t (in.)0 0.005 0.010 0.015 0.020

L = 0.25 in.

=0.50=1.0

Figure 16.1. Spring stress vs swing angle for three springwidths.

Figure 16.2. Spring stress vs swing angle for several springthicknesses.

Figure 16.3. Spring stress vs spring thickness for severalpendulum weights.

Figure 16.4. Spring stress vs spring thickness for three springlengths.


124

0 0.005

1000

100

108

4

2

10.010

Thickness, t (in.)

Pen

dulu

m s

peed

up,

∆T

/Td

(s/

day)

0.015 0.020

L = 0.25 in.

=0.50

=1.0

30 K

20 K

10 K

0

Tot

al s

tres

s (p

si)

Spring length, L (in.)0 0.5 1.0 1.5

t = 0.003 in.

=0.006

=0.010

=0.020

Figure 16.5. Spring stress vs length for four springthicknesses.

0 0.005

1,000

10,000

100

108

4

2

10.010

Thickness, t (in.)

Pen

dulu

m s

peed

up,

∆T

/Td

(s/d

ay)

0.015 0.020

W= 5 lb

=10

=20

=30

Figure 16.7. Speedup vs spring thickness for fourpendulum weights.

0.0010 0.005 0.010 0.015 0.020

0.01

0.1

0.2

0.4

0.81.0

Invar rod

Quartz rod

alone

alone

Spri

ng’

s th

erm

al s

low

dow

n (

s/da

y °C

)

Thickness, t (in.)

L = 0.25 in.

= 0.50

= 1.0

Figure 16.8. Spring’s thermal slowdown vs springthickness for three spring lengths.

Figure 16.6. Speedup vs spring thickness for three springlengths.

were used unless otherwise marked in the figures:

b 0.75 in.t 0.006 in.L 0.25 in.E 16 106 psi (type 510 phosphor bronze) 1 (half angle)

Tp 2 sW 21 lbIp 2.59 in. lb s2

Comment

The curves show that for reasonable dimensions, the spring’s maximumstress is very roughly proportional to:

(16.14)

And the pendulum’s speedup is very roughly proportional to:

(16.15)

The curves show that for reasonable parameter values, the totalspring stress is in the range of 10,000–20,000 psi, which is within therange of copper alloy springs. Compared to steel springs, copper alloyones have two advantages: not rusting, and a Young’s modulus onlyabout half that of steel. Because of its higher modulus, a steel springwould have twice the speedup effect shown in Figure 16.7.

The Young’s moduli of metals are slightly temperature sensitive(4.2104 psi/psi C for 510 phosphor bronze [6]), which means thatthe pendulum’s speedup effect from the suspension spring is also tem-perature sensitive. Figure 16.8 shows the change in speedup with tem-perature. The speedup becomes less (pendulum slows down) as thetemperature increases. Figure 16.8 also shows that the spring’s thermaleffect is roughly of the same magnitude as that produced by the ther-mal expansion of the whole pendulum rod.

With a spring material, thickness, and suspended weight of 510 phos-phor bronze, 0.006 in., and 20 lb, respectively, the spring introduces atemperature coefficient of 0.0096 s/day C, about one-sixth of that ofan invar rod alone (0.06 s/day C). In contrast, with a spring material,thickness, and suspended weight of 17–4 stainless steel, 0.010 in., and 10 lb, respectively, the spring introduces a temperature coefficient of0.34 s/day C, which is a little over five times bigger than that of thewhole invar rod. Both of these examples used the same spring length(0.25 in.) and width (0.75 in.), and the same (half ) swing angle (1).

The end result is that anywhere from 16% to 84% of a pendulum’stotal temperature sensitivity is due to the suspension spring, with theactual amount depending on the spring’s dimensions, modulus of elasti-city, and suspended weight. Information such as this makes James’equations helpful and worthwhile in designing a pendulum’s suspen-sion spring.

TTd

t3E

L2 (swing angle)(spring’s thickness)3(Young’s modulus)

(spring’s length)2

ft E

bt

(swing angle) Young’s modulus(spring’s crosssectional area)

chapter 16 | James’ suspension spring equations

125

References1. K. James. “The design of suspension springs for pendulum clocks,”

Timecraft ( June–August 1983).2. N. Bigelow. “On suspension springs,” Hor. Sci. Newslett. NAWCC chapter 161

(February 1995). Available NAWCC Library, Columbia, PA 17512, USA.3. N. Bigelow. “A comparison of James and Leeds on suspension spring

stress,” Hor. Sci. Newslett. NAWCC chapter 161 (December 1995).4. L. Leeds. “The design of suspension springs for heavy bob pendulums,”

MB150 LEE. Available from NAWCC Library, Columbia, PA 17512, USA.5. P. Woodward. “Circular error of a pendulum on a suspension spring,”

Hor. Sci. Newslett. NAWCC chapter 161 (December 2001).6. H. Imai and K. Iitzuka. “Temperature dependence of Young’s modulus of

materials used for elastic transducers,” Proc. 10th Conf. Imeko TC-3 onMeasurement of Force and Mass, Kobe, Japan, (September 1984), pp. 29–31.


126

chapter 17

Barometric compensation with a crossed spring suspension?

There is a question mark in the title because although this articleproposes an idea for barometric compensation of a pendulum using acrossed spring suspension, I have not built a model to see how well theidea works.

Suspension characteristics

A crossed spring suspension provides an axis of rotation characterized byextremely low friction.1 That is something that a pendulum can use.Figure 17.1 shows a crossed spring suspension applied to a pendulum.The suspension normally consists of four flat strips of spring metal(two sets of two, with the two sets oriented at 90 to each other) that areclamped at the ends, as shown in Figure 17.1. For small rotation angles,the axis of rotation is nominally located at the springs’ crossover point inthe middle of the springs. The suspension has three important character-istics. First, it has extremely low friction. Second, the rotational stiffnessof the suspension springs (in units of torque per unit angle about the axisof rotation) varies with the total weight suspended from the springs, thatis, with the weight of the pendulum. And third, the axis of rotationmoves horizontally (and upward, to a lesser degree, as I remember it from45 years ago) as the pendulum swings away from its vertical orientationin the center of swing. These effects are analyzed and their magnitudesgiven in [1–4]. Bateman found a 52-page bibliography by P. J. Geary [5]on flexure devices that lists more articles on crossed spring suspensions.

As the crossed springs carry more weight, that is, a heavier pendulum,the crossed springs’ rotational stiffness about the axis of rotationdecreases. Adding still more weight, the rotational stiffness will decreaseto zero—or even further, to a negative rotational stiffness which pushesthe pendulum away from the center position, with the negative forceincreasing with the angle away from center. This change in rotationalstiffness with the spring supported weight is the characteristic to beexploited for barometric pressure compensation.

1 The single vertical spring suspension alsohas extremely low friction.

127

Forty-five years ago I designed and built an instrument [6] usinga crossed spring suspension, in which the weight supported by thecrossed springs was adjusted to give zero rotational spring stiffness.The instrument worked exceptionally well, primarily due to its crossedspring suspension. The suspension used four beryllium copper springs,each 3.0 in. long (free unclamped length) by 0.25 in. wide by 0.015 in.thick. The weight used to bring the crossed springs to zero rotationalstiffness was 2.05 lb. Most pendulums with a 2 s period weigh more than2 lb, so the point of zero rotational stiffness is easily reached. And I wouldguess that many of the crossed spring pendulums I have seen are operat-ing beyond that point, in the negative rotational stiffness region.

Barometric compensation

“The [suspension] spring is at all times trying to drive the pendulumtowards the vertical or zero position. It therefore adds to the force ofgravity, causing the clock to gain.” [7] With this quote, K. James was refer-ring to the effect of a single vertical spring suspension, but it also appliesto crossed spring suspensions supporting only a small weight. Now apendulum “floats” in a sea of air. When the air pressure increases, the bobbecomes lighter, and the clock (with any kind of suspension) runs slower.This is called the barometric effect. But with a crossed spring suspension,


128

Figure 17.1. A crossed spring suspension.

Axis of rotation (at null)

Axis movementwith rotation

End clamp(1 of 4)

Pendulum

Flat springs(2 sets of 2)

Bob

the reduction in bob weight with increased air pressure also increases thesprings’ rotational stiffness, making the clock run faster. By optimizingthe design of the suspension springs, it may be possible to equalize thesetwo opposing effects, and make a pendulum that is free and independentof barometric pressure effects. At a minimum, the pendulum’s barometriceffect will be smaller with a crossed spring suspension than with a singlevertical spring suspension. Note that the barometric correction occurseven with a heavy pendulum and the crossed spring suspension operatingin the negative rotational stiffness region.

Movement of the axis of rotation

When the pendulum swings away from center, the crossed springs’ axis ofrotation moves horizontally in the direction of bob motion. This meansthat the clock slows down as the pendulum’s swing angle increases, andimplies that the circular error effect will be larger with a crossed springsuspension than with a single vertical spring suspension. But this is onlypartially true, as the slowdown effect with larger amplitude will be modi-fied (or even reversed?) by the unknown effect of the vertical rise of therotational axis that also occurs as the pendulum rotates away from center.The effect would be known if the rise occurred at 90 to what actuallyoccurs with a crossed reed suspension (the effect is zero [8] if the verticalmotion is sinusoidal at 90 phasing).

Miscellaneous

One of the reasons for proposing the barometric compensation idea isthat it provides a platform for describing the basic characteristics ofa crossed spring suspension, of which many clockmakers seem to beunaware. Crossed spring suspensions are useful and have their place inthe sun. But I have no interest in building a model of the barometriccompensation idea, because I think a crossed spring suspension is toocomplicated for pendulum use. However, there are people who do likeusing the crossed spring suspension on a pendulum, and they may beinterested in the barometric compensation idea.

References1. W. H. Wittrick. “Theory of symmetrical crossed flexure pivots,” Austral.

J. Sci. Res. A 1(2) (1948), 121–34.2. W. H. Wittrick. “The properties of crossed flexure pivots,” Aeronautical Q.

2 (February 1951), 272–92.

chapter 17 | Barometric compensation

129

3. L. W. Nichols and W. L. Wunsch. “Design characteristics of cross springpivots,” Machinery 79 (October 11, 1951), 645–51.

4. W. E. Young. “An investigation of the cross spring pivot,” J. Appl. Mech.11(2) ( June 1944), 113–20.

5. P. J. Geary. “Flexure devices, pivots movements suspensions,” Available atBritish Scientific Instrument Research Association (SIRA, South Hill,Chislehurst, Kent, England), 1954, reprinted 1957, 1961.

6. R. Matthys. “Precision torque balance for the measurement of smalltorques,” AIEE Commun. Elect. 20 (September 1955), 485–90.

7. K. James. “Design of suspension springs for pendulum clocks,” Timecraft( July 1983), 14.

8. V. Bush and J. Jackson. “Amateur scientist,” Sci. Amer. ( July 1960), 165–76(August 1960), 158–68.


130

chapter 18

Solid one-piece suspension springs

This chapter is mostly about how to make the solid one-piece type ofpendulum suspension spring. It also contains some information about the three different types of suspension springs.

There are three types of pendulum suspension springs that I amaware of. There is the mechanically clamped type shown in Figure 18.1(a),the soldered assembly type shown in Figure 18.1(b), and the solid one-piece type shown in Figure 18.1(c). The mechanically clamped type is theeasiest to make, but I have found its clock rate to be variable and depend-ent on the clamping forces and clamping surfaces. A 10 in. change in thelength of a 1 s beat pendulum amounts to an error of 3.7 s/year. To meetthe microinch length tolerance needed for an accurate clock, a mechanic-ally bolted assembly must have microinch tolerances on the clamped sur-faces, that is, an optical finish. Mechanically machined surfaces are notgood enough, and introduce uncertainty as to where the spring ends andthe end clamp actually begins. The uncertainty causes the pendulum’slength and its timing to vary with temperature and the suspensionspring’s clamping pressure.

A soldered spring assembly gets around the mechanical clampingproblem, but the soft solder can introduce some relaxation and move-ment over time in the soldered joint. A low temperature hard soldersuch as silver solder (2% silver, 98% tin) will relax less than a soft leadsolder, and thereby provide a more stable joint. Pendulums with sol-dered suspension springs normally gradually slow down when initiallyhung, asymptotically approaching a constant clock rate. One can saythat such pendulums are slowly stretching to their final length. I havealso had pendulums with soldered spring assemblies speed up when ini-tially hung, asymptotically approaching a constant clock rate. These are alittle harder to explain (“you mean the pendulum is actually gettingshorter over time?”). In an interesting article in 1938, Atkinson [1] blamedthe speedup effect on hardening of the spring due to coldworking.

The solid one-piece construction gets rid of the problems of both themechanically clamped and the soldered approaches. The length of the

131

solid one-piece suspension spring is determined by the location of itsthick ends, and not by the mechanical clamps or clamping pressures onthe ends. The spring ends must be much thicker than the center section(25–50 times thicker) if the spring’s length is to be well isolated from theend clamps and clamping pressures. If the ends are not thick enough,they will bend when the center bends, and thereby become part ofthe length of the spring. The timing of a solid one-piece spring whoseends are not thick enough will still be sensitive to the end clamps andclamping pressures.

The Shortt clock uses a single one-piece suspension spring. I hadalways thought that such springs had to be made on a grinding machine(which I do not have), but it turns out they can also be made on avertical milling machine (which I do have).

Because suspension springs are so thin, there are some crucial stepsin the making of them that I learned the hard way. My first 11 tries,improving each time, were failures. The twelfth try was a success. Thecrucial steps in making the springs are described in the Appendix. I havealso put all the machining information in the Appendix, so that thosereaders who are not interested can skip it.

What material should the suspension spring be made from? Type 172beryllium copper has the best spring properties of any practical material.It has the least mechanical hysteresis (lowest energy loss), and it does notrust. Beryllium copper is 2% beryllium, so it does have the drawback of being a minor health hazard, berylliosis, during machining opera-tions. Berylliosis is incurable, and is caused by getting fine (about 1 m)beryllium particles, mostly from grinding operations, in your lungs.Machining beryllium copper is discussed further in the Appendix.Phosphor bronze (type 510) has the second best spring properties, and itdoes not rust. Stainless steel is in third place—it is nonmagnetic and doesnot rust. Its mechanical hysteresis and “taking a set” are the highest of thethree materials, however. I chose to use phosphor bronze.

When phosphor bronze is rolled into sheets, it acquires a grainorientation. This is not true of all metals, but in phosphor bronze, the


132

Figure 18.1. Three suspension spring designs:(a) mechanically coupled, (b) soldered, and(c) solid one-piece.

Screws Endstiffeners

Suspensionspring

Solder

1.25

0.35

0.82

0.005

38

316

316

18 R(4)

(2)

1 4

2.00

in.

(a) (b) (c)

grain orientation gives a 15% difference in yield strength in specificdirections within the metal. Pendulum suspension springs are a highstress application, and can take advantage of this. To take advantage ofit, you bend the material across the direction of rolling, which is indi-cated by the direction of the material’s surface marks (from the rollingprocess). Figure 18.2 illustrates the right and wrong directions of bend.

After I had made the one-piece suspension springs as described in theAppendix, how did they work? Good. Better than the mechanicallyclamped or soldered types. Pendulums with soldered suspensionsprings, when hung, take a week to exponentially arrive at a constantclock rate. New springs take even longer. In contrast, the solid one-piece springs immediately start out at and continue at a constant clockrate, as shown in Figure 18.3. Figure 18.3 also shows what happens ifyou skip the stress relieving heat treatment, which consists of 1 h at 375 F in your kitchen oven, with a slow cooldown afterward. The slowcooldown consists of shutting off the oven’s heat source, and with theoven door kept closed, letting the oven cool down by itself to roomtemperature.

The data in Figure 18.3 is not a fluke. The constant clock rate char-acteristic at startup repeated itself on three different pairs of one-piecesuspension springs. Atkinson also remarked on the same characteristicof constant clock rate at startup, in [1]. Figure 18.3 only shows short-term performance, but it augurs well for the long term.

Phosphor bronze cannot be hardened by heat treatment to improveits spring properties. In fact, heat treatment anneals it. The only way toharden phosphor bronze is by coldworking. The only heat treatmentsfor phosphor bronze are for stress relief and annealing. Annealing doesnot change the yield strength very much, but it does equalize the yieldstrengths across and parallel to the grain orientation. According to theMiller Co., Meriden, Connecticut, USA, annealing a piece of hard phos-phor bronze drops its yield strength from 95,000 psi (hard) to about88,000 psi (annealed).

chapter 18 | Solid one-piece suspension springs

133

Figure 18.2. Taking advantage of the higherbend strength caused by grain orientation: (a) right and (b) wrong.

Figure 18.3. Initial clock rate after pendulumis hung.

Surface marks indicatedirection of rolling

(a) (b)

Without stress relief

With stress relief

0.7

0.6

0.5

0.4

0 1 2 3Time (days)

Clo

ck e

rror

(s)

4 5

Appendix: Machining solid one-piece suspension springs

The solid one-piece suspension springs are cut to size and the mountingholes drilled before machining the thin section, which is the only diffi-cult part. The tolerances on thickness, parallelism, and uniformity inthe thin section are at the limit of what can be done with a mill, and thisdrives much of the following procedure.

Referring to Figure 18.4, the springs are mounted on an aluminumsubplate, which is in turn mounted in a vise on the mill table. The frontand back faces of the subplate are machined flat and parallel, so that thesubplate can be removed and replaced accurately in the vise withoutlosing its angular orientation. The bottom edge of the subplate in thevise is also machined flat for the same reason.

Before mounting the springs on the subplate, the specific end mill tobe used in machining the springs’ thin sections is first used to make askin cut across the subplate surface beneath the springs. This ensuresthat the subplate’s mounting surface beneath the springs is exactly par-allel to (1) the axis of the quill head (more correctly, exactly parallel tothe outside cutting surface of the end mill), and (2) is exactly parallel tothe left-right travel of the mill table. Then, when the springs are flippedover to machine the other side, the thin section will turn out to beexactly parallel and uniform in cross-section.

Figure 18.4. Two suspension springsmounted on a subplate in the mill.


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It is assumed that the two 2 in. surfaces on each spring blank areflat and parallel. Any taper between these two surfaces will show up asan identical taper in the thin spring section of the final machined part.The two blanks in a pair also need to be the same thickness. If one blankis 0.001 in. thicker than the other, its thin section will turn out 0.001 in.thinner than the other.

Use a sharp end mill and cutting oil. Lard oil works well. Using cuttingoil on phosphor bronze makes a difference in the depth of cut and thestress of cutting. Use a low cutting speed, about 325 rpm, to eliminatechatter marks (which are actually valleys cut in the material and whichaffect the spring properties in thin sections). Clean all loose particles offthe end mill before it approaches the spring material, or you will get afalse “touch” signal when the loose particles fly off. Start milling at thebottom1 end of the spring, move smoothly to the top1 end, and immedi-ately reverse back to the bottom end without stopping. Keep track ofthe looseness in the milling machine’s table travel when machining right-to-left and left-to-right. If the travel stops while the end mill is cutting, asignificant low spot will occur there due to the end mill whipping aroundin a slightly loose quill bearing. The worst place for this to happen is atthe top end of the thin section. The bottom end is the least critical in thisrespect, so have the end mill go deeper or leave the material at this point,since short pauses while changing the direction of milling are almostinevitable. The pauses should be as short as is practical.

After the first side of the thin section is machined, prick punch oneend of each spring near the top (or bottom) edge to indicate orienta-tion, and turn them over on the subplate. There will now be a rectan-gular hole between the subplate and the thin section area. This holeis filled with wet plaster of paris. The springs are then attached tothe subplate, and the plaster left to dry overnight before machining the second side. The plaster acts as a mechanical support for the thinsection against the machining forces exerted while machining thesecond side. Burrs on the first side, of course, must be removed beforeturning the springs over on the subplate.

As the thin section is machined thinner, the depth of cut per passmust become smaller. A regimen of any individual pass not machiningaway more than one-third of the remaining material thickness workedpretty well. This gave depths of cut on each side of the springs asshown in Table 18.1, assuming spring thicknesses of 0.187 in. (ends) and0.005 in. (thin section).

On cutting the first side of the thin section, my mill consistently cut0.002 in. deeper than the table dials indicated. On cutting the secondside, with the first side backed up with plaster, my mill consistently cut0.005 in. shallower than the table dials indicated. These two numbersvary with the sharpness of the end mill cutter. The effect is almost zerowith a new end mill. The thin section ends up being slightly off-center,

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1 Referring to the way it hangs in the clock.


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which is okay if marked. This is the reason for the prick punch markalong one edge, so that the springs can be used as a pair with theirthin sections located in the same plane. Hard spring material requiresa sharp carbide end mill, or the deeper–shallower effect becomesoverpowering.

The thickness tolerance on the thin section is 0.001 at best. For thisreason the springs are made in pairs, so that when used as a pair, bothwill have the same thickness and stiffness. They should be stored inpairs before use, as well. If you want a specific thickness, plan on mak-ing 2–4 pairs, and selecting the spring pair whose thickness is closest tothe desired value.

After machining, most of the plaster will fall off, and the remainderis gently removed with water and a toothbrush. When the springs areremoved from the subplate after the machining is finished, the springsexhibit a curve in their thin sections due to the machining stresses.Slightly overbend the springs until the thin sections are again flat. Themachining stresses are removed by the stress relieving heat treatmentdescribed in the main part of this chapter. During the heat treatment,the springs are laid on a flat surface.

There are limitations on making suspension springs in a Bridgeportvertical mill. The thinnest spring section I have been able to make with-out tearing is 0.004 in., and that required using a brand new end millcutter. And long springs are harder to make than short ones. Thinsprings 0.5 in. or more in length sometimes varied 0.002 in. in thick-ness along the length of the thin section.

Commercially, the best way to make the thin spring sections is by theelectric discharge machining (EDM) process. I have had mixed successwith the EDM process. One of the claims for it is low distortion becauseit is done under water which conducts the cutting heat away from thepart being cut. Thinner and longer springs are harder to make thanthicker and shorter springs. The two thinnest springs (0.004 in. thick)


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Table 18.1. Cutting depth per machining pass on a

suspension spring

Machining pass Depth of cut (in.)

1 (first) 0.026

2 0.023

3 0.015

4 0.010

5 0.007

6 0.005

7 0.003

8 (last) 0.002

made by the EDM process had a “burnt” look to them, and performederratically on a pendulum. It may be that the EDM cutting current wastoo great for the small cross-sectional area of the springs, even withwater to conduct the heat away. Springs 0.006 in. thick worked okay.Hard 510 phosphor bronze should be stress relieved before using theEDM process, and stress relieved again afterward.

The health risk in machining beryllium copper can be made verysmall by using lots of cutting oil, which will keep the beryllium copperparticles in the cutting oil and out of the air. A facial breathing filtermask will reduce the risk even further. Beryllium copper springs canbe machined commercially without any health risk using the EDMprocess, as EDM is done under water and no metal particles are emit-ted into the air.

The chances of getting a specific spring thickness can be considerablyimproved by sanding the aluminum subplate in the small area beneaththe plaster of paris with 400 grit extra fine sandpaper. Then the plasterof paris will stick to the subplate when the springs are removed. Thisoffers a second chance to machine the thin section closer to the desiredthickness (after measuring its current thickness) by re-mounting thesprings back over the plaster of paris on the subplate, and re-machiningthe second side of the springs’ thin section.

Reference1. E. Atkinson “Dissipation of energy by a pendulum swinging in air,” Proc.

Phys. Soc. 50 (1938), 721–53. Available in NAWCC Library, Columbia,Pennsylvania, 17512, USA.


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chapter 19

Stable connections to apendulum’s suspension spring

What are those things called “chops,” located on the ends of a pendulum’ssuspension spring? Do they do anything useful? Well, the purpose of“chops” is to get a more solid grip on the suspension spring. Witha degree of helpfulness varying from outstanding to useless, chops(1) give a constant fixed length to the suspension spring, and (2) preventthe suspension spring from rocking back and forth on the top edge ofthe crosspin through the top end of the suspension spring. This chapteris about the second item—the rocking of the suspension spring on oneof its crosspins.

Figure 19.1 shows a typical suspension spring with its ends pinned innarrow slots. It has no chops, and the slots are slightly wider thanthe thickness of the spring. As the pendulum swings back and forth, thespring’s ends bend or wiggle back and forth a little in the slots. Thewiggling is more pronounced in the top slot than in the bottom slot.What is happening in Figure 19.1 is that the suspension spring is rock-ing back and forth on the top edge of the top crosspin. This affects thependulum’s timing, and the variability of it makes it undesirable in anaccurate pendulum.

There would be no rocking if the spring’s ends were a tight fit in theslots. One can get this desirable condition by using bolt and nut fastenerson the spring’s ends, as in Figure 19.2, but it is then much harder toremove the pendulum out of the clock.

In Figure 19.3(a), the top and bottom ends of the suspension springhave been widened by fastening or soldering relatively thick pieces ofmetal to the spring. These spring-widening pieces of metal are calledchops. The spring ends, being wider now, provide a longer momentarm to better resist the rocking torques. The thicker the chops, thelonger the moment arm is, and the greater the resistance is to rocking.For the concept to work, the chops must be solidly fastened to thesprings’ ends. Silver soldering is recommended. Any looseness in thejoint between the chops and the spring ends will ruin the concept.

The concept works fine as long as the holes through the chopsand the mating crosspin surfaces are perfectly smooth. But a small

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Figure 19.1. Suspension spring withoutchops.

Figure 19.2. Bolts give a tight grip on thesuspension spring.

Crosspin(1 of 2)

Suspensionspring

Pendulumrod

protruding bump on either of the mating surfaces can cause rocking,depending on the location of the bump. Perfect surfaces are not possi-ble in this world, so the configuration in Figure 19.3(a) can also rock,although it is less than the rocking occurring in Figure 19.1 withoutthe chops.

Figure 19.4 shows a spring with chops that is made out of one pieceof metal. This is a big advantage stability-wise. Here the effective lengthof the spring is solidly fixed and unchanging, as compared to the springwith fastened-on chops in Figure 19.3, where the spring’s length variesa little with the fastening tightness of the chops to the suspensionspring, which can change with temperature. But even a one-piecespring can rock back and forth if there is a protruding bump on one ofthe mating surfaces between a spring end and its crosspin, as shown inFigure 19.4.

The rocking problem can be completely eliminated using the arrange-ment in Figure 19.5. Here each end of the suspension spring is splitdown the middle and spread apart into two separate supports at eachend of the spring. This spring cannot rock as long as the pendulum’sweight, located along the pendulum rod’s central axis X–X, falls withinthe suspension spring’s two support points A and A in Figure 19.5. Evenif the mating surfaces between the spring’s ends and the crosspins arerough and bumpy, the spring still will not rock on the crosspin.

The dimensions of some non-rocking suspension springs made by theauthor for a pendulum with a 2 s period are shown in Figure 19.6. Thependulum’s axis of rotation is measured at 0.12 in. below the top end ofthe spring’s thin section. The pendulum’s maximum half-angle swingthat will keep the pendulum’s weight vector (along the pendulum rod’scentral axis) within the two support points A and A in Figure 19.5 is:

sin1 B2C sin1 0.12

2(0.695) 4.9 half angle.


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T

Chop(1 of 4)

(a) (b)

Figure 19.3. Suspension spring with chops:(a) front and (b) side views.

Figure 19.5. This suspension spring will notrock, even with rough mating surfaces.

B

X

A A

WV

CAxis ofrotation

X

Figure 19.4. Suspension spring and chopsmade from one piece of metal.

This is more than enough swing for an accurate pendulum. If a biggernon-rocking pendulum swing angle is needed, the distance B can beincreased between the two support points at each end of the spring.

A few words on a related issue. For assembly purposes, it is conveni-ent to make the spring’s end width W a loose fit in the support space Vprovided for it, that is, W is slightly less than V in Figure 19.5. Will theswinging of the pendulum cause the top end of the suspension springto slide back and forth on the top crosspin within the oversized space V ?The answer is no if the pendulum’s half angle ( in Figure 19.5) ofswing is less than 19, assuming a brass spring sliding on a steel crosspin.The coefficient of static friction for brass on steel is 0.35, and thetan1(0.35) 19. The coefficient of static friction for steel on steel is0.58, so the maximum half angle of swing ( in Figure 19.5) beforeslipping occurs for a steel spring on a steel crosspin is 30. So some clear-ance space in the slots provided for the spring ends, that is, V greaterthan W in Figure 19.5, will not cause the spring’s top end to slidealong the top crosspin as long as the pendulum’s half angle of swing isless than 19 (brass on steel), or 30 (steel on steel). This result appliesto all configurations of the suspension spring.

In summary, chops provide a more stable connection to the suspen-sion spring. This is important in an accurate pendulum, as its timingwith chops is more constant. For maximum stability, the chops shouldbe thick, so as to provide a longer moment arm (T in Figure 19.3(a) ) toresist any rocking of an end of the suspension spring on its crosspin. Asshown in Figures 19.3(a) and 19.4, the mating surfaces between thecrosspins and the holes in the spring ends need to be perfect, as a slightbump on any of the mating surfaces can cause rocking to occur,depending on the location of the bump. The rocking motion can becompletely eliminated by splitting the suspension spring ends into twospaced sections, as shown in Figure 19.5. Then it makes no difference ifthe holes in the ends of the spring are rough and crude, or if thecrosspins have a rough surface.

chapter 19 | Stable connections to a suspension spring

141

Figure 19.6. Dimensions of suspensionsprings made by the author.

0.006

0.189 ø

0.46

0.70

0.94

0.50

0.70

0.35

0.59

2.14

in.

+

+

– –

– –

38

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18

R18

chapter 20

Stability of suspension springmaterials

Type 642 aluminum silicon bronze, hard temper, is the best material for apendulum’s suspension spring.

A pendulum’s suspension spring is so simple physically, and yet socomplicated in questions of size, what material to use, bending fatigue,length stability, the effects of its Young’s modulus varying with temper-ature, how to solidly grab hold of its top and bottom ends, whether touse one spring or two, etc. This chapter deals with the first four ques-tions. Basically the suspension spring’s length, thickness, and materialare varied here, looking at how these parameters affect the pendulum’stiming and thermal hysteresis. The object of course is to improve thetiming and reduce the thermal hysteresis. Chapter 16 covers the the-oretical aspects of spring length, width, and thickness. Chapter 11 hasinformation on a variety of metals and also on their heat treatment. Thischapter covers what actually happens in practice to a pendulum whenthe suspension spring’s length, thickness, and material are changed.

With a 10–20 lb bob and a 1–1.5 swing (half ) angle, the top end of thesuspension spring operates at a stress level of 10,000–20,000 psi (mostlybending stress), while the rest of the pendulum has a very low stress levelof 200 psi or less. Some spring materials will have to be hardened at leasta little to operate successfully at 10,000–20,000 psi, and the hardeningprocess can add internal stress into the material, which would reducethe spring’s long-term dimensional stability. Low-stress materials can beannealed, do not have to be hardened, and in general are more stableover the long term than high-stress materials. The suspension spring isbending continuously, and needs a long fatigue life so that it can stand upto the continuous bending. The higher stress and continuous bendingput extra requirements on the spring, so the suspension spring is treatedseparately from the other parts of the pendulum.

Varying the spring’s length

In this first series of tests, a pendulum with a 2 s period is cycled overa small 15 C temperature range (22–37 C), using different lengths of

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suspension spring ( to 1 in., free unclamped length). The springs are allthe same width ( in. effective), material (510 phosphor bronze), andapproximately the same thickness (0.005–0.0065 in.). The springs are ofthe solid all-one-piece design, as shown in Figure 19.1(c). The pendulumhas a quartz rod, a 19 lb cylindrical bob (642 aluminum silicon bronze),and is driven electromagnetically with a short electrical pulse at the cen-ter of swing. The pendulum is temperature compensated by a metalsleeve of 642 aluminum silicon bronze 4.71 in. long that surrounds therod and is below and supporting the bob. Figure 10.5 shows theconstruction. The temperature compensation is approximately correctfor a type 510 phosphor bronze suspension spring of size

0.006 in. (L W T). The pendulum has two suspension springs inparallel, each of in. width, giving a total spring width of 0.75 in. Thetemperature compensator is fixed and is not changed throughout all of the tests in this chapter. The thermal hysteresis, that is, the width ofthe thermal hysteresis loop, is measured at (or near) the temperature’shalf-way point (30 C), as shown in Figure 11.1.

Table 20.1 lists both the hysteresis from a 15 C temperature cycleand the change in clock rate when the temperature is raised 15 C(from 22 C to 37 C). Table 20.1 shows that the thermal hysteresisincreases in direct proportion to the length of the suspension spring.With a 0.75 in. increase in spring length and a 15 C temperature rise,the pendulum’s length increases thermally by

0.75 in. (15 C) (17.8 106 in./in. C) 200 in.

And since a 0.001 in. change in pendulum length causes a 1 s/day ratechange,

Table 20.1 shows that with a 1 in. spring length and a 15 C tempera-ture rise the pendulum slowed down by (0.31 0.075) 0.385 s/day.

200 106 in.1 s/day0.001 in. 0.20 s/day.

38

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144

Table 20.1. Varying the suspension spring’s length

Suspension spring Thermal hysteresis,a Change in clock rate after 15 C

Length (free Thickness (in.) Material(s/day) temperature rise (s/day)

unclamped, in.)

0.0065 510 phosphor bronzeb 0.022 0.075


1 0.005 510 phosphor bronzeb 0.084 0.31

Notesa For 15 C temperature cycle.b Hard temper, stress relieved.

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Using the James equation graphed in Figure 16.8, the increase in springlength from 0.25 to 1 in. and the slight decrease in thickness from 0.0065 to0.005 in. together would slow the pendulum down by a calculated 0.19 s/day. Adding the calculated 0.20 s/day slowdown from the thermally longer suspension spring to James’ calculated 0.19 s/dayslowdown from the spring’s thermally softer modulus of elasticity givesa total calculated slowdown of 0.39 s/day. This agrees closely with theexperimentally measured slowdown of 0.385 s/day for a 0.75 in.increase in spring length and a 15 C temperature rise.

Varying the spring’s thickness

In a second series of tests, the same pendulum was cycled over the same15 C temperature range using different thicknesses of suspensionspring (0.004–0.010 in.). The springs were all the same length ( in.),width ( in. total), and material (510 phosphor bronze). Table 20.2 listsboth the hysteresis from a 15 C temperature cycle and the change inclock rate when the temperature is raised 15 C (from 22 C to 37 C).

Table 20.2 shows that the hysteresis is a minimum with a 0.0065 in.spring thickness, and increases with both thinner and thicker springs.This was a surprise. The minimum may possibly be related to the totalstress in the suspension spring. Figures 16.3 and 16.4 both show a min-imum occurring in the total spring stress. But in those figures, the min-imum occurs at a spring thickness of 0.010 in. instead of at the 0.006 in.observed here. The optimum spring thickness (for minimum hysteresis)will most likely vary with bob weight, and with the swing angle as well.

When the spring thickness is increased from 0.004 to 0.010 in.,the measured change in clock rate for a 15 C temperature change

34

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chapter 20 | Stability of suspension spring materials

145

Table 20.2. Varying the suspension spring’s thickness

Suspension spring Thermal Change in clock rate after

Thickness (in.) Length (free Materialhysteresisa 15 C temperature rise (s/day)

unclamped, in.)(s/day)






Notesa For 15 C temperature cycle.b Hard temper, stress relieved.

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is 1.015 (0.144) 1.16 s/day. The James equation graphed inFigure 16.8 gives a calculated slowdown of 1.15 s/day for this changein spring thickness and a 15 C temperature rise, due to thermal soften-ing of the spring’s modulus of elasticity. This agrees closely with thependulum’s experimentally measured slowdown of 1.16 s/day fora spring thickness increase to 0.010 in. and a 15 C temperature rise.

Bending fatigue

If a heavily stressed spring is continuously flexed back and forth, thespring will eventually fatigue and break in two. This occurs at a stresslevel considerably less than the metal’s yield strength. The higher thestress, the fewer are the number of bend cycles before the spring breaks.This fatigue characteristic of springs is usually presented as an S–N curve(stress vs number of bend cycles to failure). Figure 20.1 shows S–Nfatigue curves for several metals. Note that each metal has a lower stresslevel called the fatigue limit, below which the cycle life becomes essen-tially infinite and the spring can be flexed indefinitely and not break.

Each curve in Figure 20.1 is the average of several tests at each stresslevel. A 2 to 1 variation in fatigue life at any stress level is typical. To getan unlimited flex life, one would pick a maximum safe stress level thatis 20–30% below the fatigue limit of the desired metal.

The wrong yield strength

A metal’s yield strength is not a sharply defined stress level where themetal suddenly starts to permanently increase its length. Instead, thereality for most metals is that the permanent length increase starts outmicroscopically small at lower stress levels and increases nonlinearly as


146

120 K

100 K

80 K

60 K

40 K

20 K

0104 2 4 8 105 106 107 108 109

No. of cycles to failure

Stre

ss (

psi)

Fatiguelimits

Spring stress range(10–20 lb bobs) 1 month 1 year 10 years

(Hard)510

172 (Hard)

17-4 (Hard)

729 (Hard)

304 (Annealed)

Figure 20.1. S–N fatigue curves for five springmetals. (From Metals handbook, 10th edn,courtesy of AMS International.) All areflexural bending (R 1) except 304 which isunknown. The time scale shown is for apendulum with a 2 s period.

the stress level increases. The usual way of dealing with this is to definethe yield strength as the stress required to permanently increase thelength of a test piece by 0.2%. This is good enough for most purposes,but not for a pendulum.

A 0.2% change in, say, a in. long suspension spring is 500 in. For apendulum with a 2 s period, a 0.001 in. length change will changeits clock rate by 1 s/day. So 500 in. would change the clock rate by 0.5 s/day. This is too large for an accurate pendulum, which mighthave an accuracy of 1 s/month or better. One s/month is equal to0.033 s/day, which is one-fifteenth of the 0.5 s/day error that would becaused by a stress equal to the material’s yield strength. So for a sus-pension spring, the yield strength needed is the stress level that givesa permanent stretch of one-fifteenth of 0.2% 0.013%. The stresslevel that gives a permanent stretch of 0.01% is not available for mostmetals. Such a yield strength could be anywhere between 20% and 90%of the usually given yield strength for a 0.2% permanent stretch. Thepractical solution to this problem is to pick (if possible) a suspensionspring material whose listed yield strength at a 0.2% permanent stretchis 2–3 times higher than the actual operating stress in the spring.

A metal’s fatigue limit is roughly proportional to its tensile strength.Hardening a metal will raise both its fatigue limit and its tensilestrength.

Varying the spring’s material

In a third series of tests, the suspension spring’s material was variedwhile keeping the spring’s length, width, and thickness constant. Thegeneral properties of eight spring metals are listed in Table 20.3. Themetals are listed by their full UNS numbers, which for convenience ineveryday usage are commonly shortened to three-digit numbers, suchas 304, 172, or 510. All of the metals in Table 20.3 are nonmagnetic andwill not rust, except for 902 nickel iron and 17-4 stainless steel, both ofwhich are magnetic, and 902 nickel iron which does rust (slightly). Fiveare copper alloys, two are stainless steels, and one is a special nickel ironalloy. Several of the springs were made from rod stock, as sheet stock inthe desired thickness ( in.) was not available.

The two stainless materials are types 304 and 17-4. Both are annealed,which is the most commonly available state for stainless. The 300 seriesstainless materials cannot be hardened by heat treatment, which wouldraise their fatigue limit. The 400 series stainless materials can be hardenedby heat treatment, but unfortunately they rust (slightly). Type 17-4 willnot rust and can be hardened by heat treatment. Its annealed yield strength of 75,000 psi is high enough to give a fatigue strength of 28,000 psi(estimated), so that a hardening heat treatment is optional. Type 17-4

316

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147

stainless does have the drawback of containing many ingredients, whichmay affect its long-term dimensional stability. This drawback is offset bybeing an annealed material, which gives it lower internal stresses and bet-ter dimensional stability over time.

Type 902 nickel iron (Ni Span C) is unique in that its modulus of elas-ticity has a very low temperature sensitivity of zero 36 104 %/Cmaximum. Its maximum sensitivity of 36 104 %/C is about anorder of magnitude less than that of a copper alloy or a stainless steel.Ni Span C is expensive to use, as it requires heat treatment (aftermachining) at 1000–1300 F for 3–5 h in a vacuum furnace to attain itslow temperature sensitivity. The heat treatment costs about $200. BothNi Span C and the stainless steels require carbide tooling to machinethem to size. Type 43-PH, made by Carpenter Technology Corp., is the


148

Table 20.3. General properties of nine spring metals

Material Compositiona % Young’s Thermal Coefficient Yield Fatigue

modulus of coefficient of thermal strength limitb for

elasticity of modulus expansion (psi) 108 cycles

( 106 psi) of elasticity ( 106 (psi)

(%/C) in./in. C)

S17400 stainless steelc 72Fe17Cr4Ni4Cu 28.5 0.049 10.8 75K–170Kd 64Ke

S30400 stainless steelc 68Fe19Cr10Ni2Mn 28.0 0.13f 17.2 30K–110Kd 27Kg

N09902 nickel iron (Ni Span C)h 48Fe42Ni5.3Cr2.4Ti 27.8i–28.5j 0 0.0036 7.6–8.1 126K–170Kd 50Ke

C17200 beryllium copperk 98Cu1.9Be 18.5 0.029f 17.5 45K–195Kd 45Ke

C51000 phosphor bronzek 95Cu5Sn0.2P 16 0.042f 17.8 47K–107Kd 33Ke

C64200 alum. silicon bronzek 91Cu7AL2Si 16 — 18 35K–102Kl 50Ke

C65500 silicon bronzek 97Cu3Si 15 — 18 21K–60Kl 29Ke

C72900 nickel tin bronzek 77Cu15Ni8Sn 18.5 — 16.4 75K–170Kd 40Ke

Notesa Components 1% or less are not listed.b It takes 6.3 years for 108 cycles at 2 s/cycle.c Data from ASM metals reference book and Metals handbook, desk edn. (courtesy ASM International.)d Annealed and hardened values for 0.2% permanent strain.e Hard temper.f H. Imai and K. Iizuka “Temperature dependence of Young’s modulus of materials used for elastic transducers,” Proc. 10th

Conf. IMEKO TC-3 Meas. Force Mass, Kobe, Japan, September 1984, pp. 29–31.g Annealed.h Data from Special Metals Corp.i In strip form and heat treated.j In rod and plate forms and heat treated.k Data from website www.copper.org (courtesy Copper Development Association.)l Annealed and hardened values for 0.5% permanent strain.

www.copper.org

same material as Ni Span C, made by Special Metals Corp. (formerlyINCO Alloys International).

The copper alloys include the two normally best spring metals forgeneral applications: type 172 beryllium copper (best) and type 510phosphor bronze (second best), plus three others that looked like theymight have good spring properties and low thermal hysteresis. Thehealth risk in machining beryllium copper is discussed in Chapter 18.

Table 20.4 shows the pendulum’s thermal hysteresis after a 15 Ctemperature cycle for eight different spring metals. The metals are listedin the order of increasing hysteresis. The change in clock rate after a15 C temperature rise is also shown in Table 20.4. In each case, the pen-dulum is the same as described in the first section except for the sus-pension spring material. The springs all have the same length ( in. ) andwidth (0.75 in. total), and their thicknesses are as close to 0.006 in. ascould be made in small quantities. Table 20.4 shows that springs madeof types 902 nickel iron and 642 aluminum silicon bronze have the leasthysteresis, that is, they were the most repeatable after a temperaturecycle.

With a suspension spring made of 304 stainless steel (annealed), thependulum kept slowing down more and more over time, indicating thatthe spring was either continuously weakening or lengthening. The test-ing of the 304 stainless spring was discontinued after 13 days, to avoid

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149

Table 20.4. Varying the suspension spring’s material

Suspension spring Change in clock Thermal

Material Temper Thickness (in.)rate after 15 C hysteresisa

temperature rise (s/day) (s/day)

902 nickel iron (Ni Span C) Hardb,c 0.0065 0.61 0.018

642 alum. silicon bronze Hardd,c 0.0055 0.08 0.023

172 beryllium copper Hardd,c 0.006 0.15 0.042

510 phosphor bronze Hardd,c 0.006 0.01 0.044

17-4 stainless steel Annealedc 0.006 0.09 0.055

655 silicon bronze hardd,c 0.006 0.01 0.063

729 nickel tin bronze hardd,c 0.0062 0.06 0.105

304 stainless steel Annealedd,c 0.006 e e

Notesa For 15 C temperature cycle.b Heat treatment: 565 C (1050 F) for 3 h, furnace cooled.c And temperature cycled 5–7 times: 12 h at 40 F, then 12 h at 200 F.d Stress relieved.e Spring continuously weakened or lengthened during test. Not recommended for pendulum

suspension springs.

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the spring breaking and dropping the pendulum. The S–N fatigue curvefor 304 stainless (annealed) in Figure 20.1 indicates that the materialshould be able to take the pendulum’s load stress of 10,000–20,000 psi.But S–N curves are based on the actual breaking of the springs, and noton their constancy. Type 304 stainless (annealed) stretches 40% before itbreaks, and this stretching may be the cause of the discrepancy. A pendu-lum of course cannot tolerate any stretching of this kind.

A suspension spring’s torque on the pendulum is directly propor-tional to the spring’s modulus of elasticity (Young’s modulus). A copperalloy spring has about a 2 to 1 advantage over a steel spring in minimiz-ing the torque and the torque’s effects on the pendulum, because theelastic modulus of a copper alloy is only about half that of a steel alloy.If the elastic modulus changes slightly with temperature, as it does inmost metals, the effect on the pendulum is not trivial. The effect ina steel suspension spring can easily exceed the thermal expansion ofa whole invar pendulum rod (see Chapter 16). Type 902 nickel iron isspecifically designed for its elastic modulus to be constant and inde-pendent of temperature. However, any uncertainty or variation in itsspring torque due to stress relaxation (or whatever) will still couple overinto the pendulum’s clock rate at almost twice the amplitude of thesame uncertainty or variation in a copper alloy spring, because of thealmost 2 to 1 difference in their moduli of elasticity. Because of this dif-ference, type 642 aluminum silicon bronze is a better spring materialthan 902 nickel iron. Type 642 aluminum silicon bronze is also easier toobtain, and is much cheaper than 902 nickel iron because of 902’s heattreatment cost.

In Chapter 11, type 642 aluminum silicon bronze turned out to bethe best material (lowest hysteresis) for the bob and the temperaturecompensator. So maybe it should be no surprise that it also turns out tobe the best material for the suspension spring.

With the 902 nickel iron suspension spring, Table 20.4 shows that thependulum is over-compensated for temperature change. The temper-ature compensator, which is 4.71 in. long, should be shortened by about0.5 in. for that spring material.

Conclusions

On spring length, the thermal hysteresis increased as the suspensionspring’s length increased. On spring thickness, hysteresis was a min-imum at 0.006 in., and increased with both thicker and thinner springs.On material, type 642 aluminum silicon bronze was the best of allmaterials tested.

What makes 642 aluminum silicon bronze more stable than othermetals? After months of digging in libraries and coming up almost


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empty-handed, the most I found is that it is probably due to how theindividual grains of aluminum, silicon, and copper interlock with eachother in the metal. A grain is a bundle of atoms, all of one kind (alu-minum, silicon, or copper) that coalesced together as the metal cooled,with the bundle forming a single crystal of that kind of metal. Grainsvary in size but are in the 0.01 in. ballpark. Epprecht [1] describes thetheory. Indeed, there is very little literature available on the wholesubject of the dimensional stability of metals. Most of it is theory andspeculative, with almost no data. See the references in Chapter 11.

In sum, for best performance from the suspension spring:

1. Keep the spring’s length as short as practical.2. There is an optimum spring thickness, the optimum being

0.006 in. for a 19 lb bob and a 1 swing (half ) angle.3. Make the suspension spring out of 642 aluminum silicon bronze,

hard temper, stress relieved, and temperature cycled.

Reference1. W. Epprecht. “Behavior of complex alloys under thermal cycling,” (in

German), Zeit. Metallkd. 59(1) (1968), 1–12. English translation availablefrom Copper Dev. Assoc., Accession no. 4787, www.copper.org.


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part i i i

Pendulum rod

chapter 21

Pendulum rod materials

Over time, several different materials have been used for the pendulum rod—steel, wood, and invar. The best one is quartz, because of its proven stability and low thermal expansion.

Steel

Steel is used for the pendulum rod in simple ordinary clocks becausesteel is cheap and because its relatively low thermal expansion of 10.5

106 in./in.C is only a small part of the total error in an ordinary clock.

Wood

Wood is sometimes recommended because of its low linear thermalexpansion coefficient along the grain (6.6 106 in./in.C for walnut,2.6 106 in./in.C for beech. Thermal expansion across the grain is5–25 times larger). But no one seems to mention that wood is an inher-ently unstable material. It warps, splits, and exhibits a high mechanicalcreep under load. Worst of all, wood expands and contracts consider-ably with relative humidity (RH).

Wood expands approximately linearly with its internal moisture con-tent (IMC) over an IMC range of 0–30%. Wood shows negligible lengthchange when the IMC goes above 30%. Wood stabilizes to a 6.2% IMCin a 30% RH environment and to a 13.1% IMC in a 70% RH. The linearshrinkage of wood parallel to the grain from green to oven dry is0.1–0.2%. Taking an average shrinkage of 0.15%, the change in lengthdue to a change in RH from 30% RH to 70% RH is:

This corresponds to a pendulum rate change of 15 s/day. This is bad.Awful, in fact.

The modulus of elasticity parallel to the grain is 1.68 106 psifor black walnut (dry) and 1.72 106 psi for American beech (dry).

13.1 6.230 0.15% 0.035%.

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The modulus changes 0.24%/C with temperature, and 1.57% for every1% change in IMC. Let us assume a 0.5 in. diameter rod and a 15 lb bob.The stress in the rod is a low 76 psi, and a 40 in. length stretches0.0018 in. At 76 psi, the mechanical creep is only 10% of the rod’s initialstretching, occurring over a 1-month to 1-year period. The creepincreases to 200% of the initial stretching at 2000 psi. Back at our lowactual stress of 76 psi, the creep amounts to 180 in. in a 40 in. length,or a pendulum rate change of 0.19 s/day due to creep.

With a 10 C temperature change, a wood (beech) pendulum rod willchange 1.1 s/day due to linear thermal expansion and 0.047 s/day dueto the thermal change in modulus. It also changes 0.21 s/day due to RHchange (30–70% RH) in the modulus. All of these effects are secondary,being an order of magnitude less than the simple expansion or shrink-age of wood with moisture (RH).

Invar

Invar is a mixture of 36% nickel and 63% iron. It is magnetic and rusts ina humid environment. There are three types available. Regular invarcomes in both normal and free machining grades, and then there is superinvar. Regular invar (both grades) has a nominal thermal expansion coef-ficient of 1.5 106 in./in. C, which is about one-sixth that of ordinarysteel. By adding 5% cobalt, you get super invar, which has about a threetimes lower thermal expansion coefficient of 0.63 106 in./in.C.

The two grades of regular invar cost about the same. Super invar costsabout 30% more than regular invar. Both super invar and the normalgrade of regular invar are very difficult to machine if you want to cutthreads on one end for a rating nut. Selenium and more manganese areadded to regular invar to make the free machining grade. From bitter per-sonal experience, I can heartily recommend buying the free machininggrade of normal invar. However, the free machining grade does have adrawback—it has about twice the carbon content of the other two invars.

Invar is not a perfectly stable material. It has some instability [1],which is related to the amount of carbon in it. The invar made today ismore stable than it was years ago, because it now has less carbon in it.

In the United States, invar is available from Scientific Alloys,Westerly, Rhode Island, or from Fry Steel Co., Santa Fe Springs,California. A 4-ft long piece of free machining invar in. in diameterwill cost about US$45.

Carbon fiber

A new material of interest for the pendulum rod is carbon fiberrod, because its tempco is nominally zero or slightly negative (0 to

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0.5 ppm/C). The carbon fibers are bonded together with a cyanateester (an epoxy with low moisture absorption). The SmithsonianAstrophysical Observatory, Cambridge, Massachusetts, is developing it.

The carbon fiber rod may not work too well as a pendulum rod, theObservatory says, as the epoxy absorbs moisture, changing the rod’slength and its weight as well. A change in RH from 20 to 30% givesabout % moisture absorption, changing the rod’s length by about 5 ppm and also changing the rod’s weight by %. An uncoated rod’slength is several times more sensitive to moisture than it is to temper-ature. Plating of a moisture barrier (a metal eutectic) is being tried, butthe coating is never perfect and there are always pinholes. Their work sofar has been with unloaded carbon fiber rods, and they also mentionedthat epoxy creeps under load.

Jacobs at the University of Arizona has made stability measurementson carbon fiber rod. He said that although it is still early in its develop-ment, the initial measurements on carbon fiber rods show a good sta-bility of a few ppm per year. Carbon fiber rod is available from WEEBEE Enterprises, Yemassee, So. Carolina. A 4-ft long rod of , , , or

in. diameter can be obtained for a minimum charge of US$50.

Quartz

Although not mentioned very often for pendulum rods, fused quartz isan almost ideal material in several respects. It is extremely stable (farbetter than invar), and its linear coefficient of thermal expansion is only0.55 106 in./in.C, which is slightly less than that of super invar. Andit is nonmagnetic, of course. Holes are easily drilled in it using diamonddrills.

Grinding threads on one end of a fused quartz rod for a rating nut isexpensive. An alternate length adjustment scheme of inserting a pincrosswise through the quartz rod and then stacking washers on top ofthe pin is both better and cheaper. Quartz does have the disadvantage ofbreakage, and requires care in handling. Going to a larger diameter rod,like or in., makes the rod much stronger and less prone to breakage.An extra fused quartz rod could be kept on hand, if desired.

Quartz rod is available from GM Associates, Oakland, California andQuartz Scientific, Fairport Harbor, Ohio. Quartz Scientific has bigovens for annealing quartz rod. A 4-ft long quartz rod in. in diameterwill cost about US$35 (in 2003).

Reference1. J. Steele and S. Jacobs. “Temperature and age effects on the temporal

stability of invar,” SPIE 1752 ( January 1992), 40–51.

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chapter 22

The heat treatment of invar

For the best stability, invar must be heat treated before use.

Even though invar is the most common material used for the pen-dulum rod in a good clock, it is still a poor material for the purpose,because of its relatively poor dimensional stability over time (relativelypoor here means when compared with other materials that are knownto be stable, such as quartz or platinum). Invar is usually picked for itslow thermal expansion coefficient, and not for its dimensional stability.Quartz, however, is an ideal material for a pendulum rod, if you can getaround the glass breakage problem.

There are three types of invar available: regular invar, regular invarfree machining, and super invar. Each has a different thermal expansioncoefficient (hereinafter called tempco). For more information on thethree types, see Chapters 21 and 23.

The tempco of each type of invar is dependent on its heat treatmentand any coldworking or machining that the part has received. Whatnever gets mentioned and is not widely known is how big the changesfrom heat treating and machining really are. The change in tempco isabout 4–1, or about 2–1 in both the plus and minus directions from thenominal rated tempco value.

In plain English, invar’s rated tempco is only a ballpark number, andvaries over about a 4–1 range depending on the heat treatment (nothing,one-step, three-step) of the material. This is true of all three types ofinvar. Invar is normally sold in an annealed state, that is, air cooled with-out heat treatment of any kind. What is not understood is that the enduser is expected to provide his own heat treating, whatever is appropriate forhis application. As a practical matter, the rated tempco of invar is close towhat you will get with the one-step heat treatment. Heat treating shouldnormally occur after machining, so as to eliminate the machining stresses.

The tempco of invar as it comes from the supplier is normally high,about 1 times the rated value. Machining it (threads for a rating nut, mak-ing a hook for hanging, etc.) will make it even higher, up to two times therated value. A one-step heat treat (1500 F in a salt bath for 15–30 min, aircool) will reduce the tempco to near its rated value. The three-step heattreat was developed by Lement at the Massachusetts Institute ofTechnology (MIT), and gives the maximum dimensional stability over

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Some of this information came from Mr. Les Harner at Carpenter TechnologyCorp., who is known as “Mr. Invar” for hisknowledge of and experience with invar.Other sources were: a publication ofScientific Alloys on invar; Mr. Ralph Berg atHoneywell, Inc., who as a user has testedmuch super invar over the years; and thewriter’s personal experience with invar.

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time. It typically gives a low tempco of about half the rated value. Becauseof the pendulum’s need for dimensional stability, the three-step heat treatis the one of interest to clockmakers. As a general rule, machining raisesthe tempco, and heat treating (with water quenching) lowers it.

The MIT three-step heat treat for maximum dimensional stability (inall three types of invar) is as follows:

1. 1500 F for 30 min per inch of thickness, water quenched. Heatingin a vacuum or inert atmosphere will prevent oxidation of the invar.

2. Reheat to 600 F for 1 h, air cool.3. Reheat to 200 F for 24 h, air cool.

The 1500 F temperature puts the carbon back into solution in theinvar, with the fast quench attempting to “freeze” the carbon in a uni-form distribution before it can “lump up.” The 600 F step removes thestress of the preceding water quench. The 200 F is an aging step, lead-ing to dimensional stability over time. It is thought that the “aging”allows the carbon, which is in invar in small amounts, to diffuse andform micro-sized precipitates, which change the volume very slightly.To save on heat treating costs, the third step can be skipped, letting theinitial relaxations and changes occur in normal time during the initialuse of the part. Initial use of the part will not be as stable, however.

The water quench at the end of the 1500 F step in the three-stepheat treat is critical in how much the tempco is reduced by heat treat-ing. For the lowest possible tempco, the invar should be cooled as fastas possible via a water quench in the coldest possible water. Ice water isbetter than warm water. If the invar is allowed to slow cool (i.e. aircool) from 1500 F, the tempco will increase rather than decrease. Invarcannot be hardened by heat treating or quenching.

There is disagreement as to the best heat treatment for invar. Everyoneagrees that the heat treatment is important, and has a big effect on invar’sperformance. Harner (Mr. Invar) at Carpenter Technology Corp. recom-mends the MIT three-step. The water quench in the first step gives thelowest tempco. Others argue that the thermal shock of the water quenchhas to put stress into the material, and why put stress (that can cause insta-bility) into a material that you want to be stable? So skip the waterquench, accept the higher tempco that results from not quenching, andas a result, get a material that is more stable over time.

A first alternate heat treat is to use the same MIT three-step, but withthe water quench replaced with an air cool. A second alternate takes noteof the fact that the invar, as delivered from the forge or extruder, alreadyhas been allowed to air cool. So the second alternate heat treat consists ofdoing just the second and third steps of the MIT three-step. Jacobs’ testdata at the University of Arizona indicate that the second alternate heattreat gives the best stability over time (there are quibbles, ifs, ands, andbuts on this) for both the regular and free machining invars [1].


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Figure 22.1. The tempco test certificate.

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Invar’s dimensional instability appears to be largely proportional tothe amount of carbon present in the invar. The amount of carbon intoday’s invar is much less than that of 20 or 30 years ago, so today’sinvar should be more dimensionally stable than that of yesteryear.

The writer recommends using the free machining type of regularinvar, as the other two types of invar are very difficult to machine.

The “rated tempco” test certificate (Figure 22.1) furnished with theinvar by the manufacturer is not very useful to the user. The manufac-turer makes a tempco measurement on a test sample (after a specificone-step heat treatment for regular invar and regular invar free machin-ing, and a specific three-step heat treatment for super invar) from eachinvar melt. The measured tempco is given on the test certificate as the“rated tempco,” and is used by the manufacturer as a cross check onwhether this particular melt of invar matches his invar recipe, that is,the right ingredients in the right proportions, etc. So it is mainly usefulto the manufacturer as a process control.

The writer ran into invar’s heat treat problem while doing temperaturecompensation experiments on a pendulum. The length of the pendulum’saluminum temperature compensator was calculated as 2.98 in., using thegiven tempcos of aluminum and invar (regular invar, free machining). Thecompensator’s correct length by actual test on an invar pendulum rod witha two-step heat treat (the first two steps of the three-step heat treat) meas-ured only 1.37 in., which is much lower than the calculated value. And thecorrect length by actual test on another pendulum rod using the same typeof invar but that had received no heat treating whatsoever measured 3.62in., which is much higher than the calculated value. The tempco of thependulum’s suspension springs was included in the calculated value, andamounted to 3.3% of the total expansion of the invar rod by itself. Thewide discrepancy between the two temperature compensator lengths of1.37 and 3.62 in. in actual temperature tests of heat treated and non-heat-treated pendulum rods forced me to dig into the heat treatment of invar.

Some conclusions can be drawn from the above. First, the large 4–1variation in tempco of each of the three types of invar makes it uselessto attempt calculating the size of the pendulum’s temperaturecompensator. It must be determined experimentally.

Second, if you are using a non-heat-treated invar pendulum rod,which I suspect many people are (did anyone know about the largeeffects of heat treating?), your invar tempco is 1 –2 times higher thanthe rated tempco, and 2–3 times higher than it would be if given thethree-step heat treatment. And third, using a non-heat-treated pendu-lum with its greater dimensional instability will give larger clock timingerrors than a pendulum rod with the three-step heat treat would give.

Reference1. D. Schwab, S. Jacobs, and S. Johnston. “Isothermal dimensional instability

of invar,” 29th Natl. SAMPE Symp. (April 1984), 169–84.

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chapter 23

The instability of invar

Information about invar has increased continuously over the years. Thischapter provides an overview of invar as of 1995. It is based on publishedarticles and on phone conversations with two invar experts, Stephen Jacobsand Les Harner.

Almost since its invention by Guillaume in 1896, invar has beenknown to be dimensionally unstable. A few articles have been publishedover the years, trying to get a handle on the instability and also trying toeliminate it. Guillaume himself published data in 1927 showing a dimen-sional growth of 50 ppm over a 27 year interval [1]. The growth wasexponential, gradually slowing down with time. Invar’s growth todaystill follows the same exponential pattern, although shrinkage is occa-sionally observed. Most of the growth Guillaume measured occurred inthe first 6 years (35 ppm), but the invar never stopped growing. In 1950,invar’s instability was tied to the presence of impurities, carbon espe-cially [2]. The lower the level of impurities, the more stable the invar is.

Invar’s impurity level has been reduced over the years, so that today’sinvar, using the traditional furnace melt process, is more stable than thatof 20 years ago. Jacobs measured the stability of today’s regular invar asbeing 2–27 ppm per year, at room temperature [3]. And instabilities ashigh as 11 ppm per day have been reported by others at various temper-atures (20–70 C), chemical compositions, and thermomechanical condi-tions [3, 4]. A 10 ppm per year (5 ppm average) change in the length of a1 s beat pendulum corresponds to a time rate error of 0.20 s/day/year,or a total accumulated time error of 73 s at the end of 1 year.Unfortunately, invar from one of today’s biggest suppliers (CarpenterTechnology Corp.) has never been tested for stability at the University ofArizona, the key place for hard test data on dimensional stability. I sayunfortunate, because my invar came from Carpenter Technology.

There are three types of invar available today. There is regular invar,free machining invar (it is regular invar with 0.2% selenium added toimprove machinability), and super invar (regular invar with 5% cobaltadded). Super invar’s advantage is that its thermal expansion coefficient(hereinafter called tempco) is three times smaller than that of regularinvar.

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As to stability, Jacobs’ test data shows that regular invar is the moststable of the three types of invar [1]. Free machining invar increasesdimensionally at roughly twice the rate of regular invar [1], so its stabil-ity is roughly only half that of regular invar. The 0.2% selenium in thefree machining invar apparently acts as a destabilizing impurity,although it is desirable from a machining viewpoint. And super invar isunstable temperature-wise (due to its 5% cobalt “impurity”?). Superinvar shows a very low-dimensional growth of about one-fifth that ofregular invar at room temperature, but this low-growth is easilydegraded by rather small changes in temperature [5]. Super invar is alsovery sensitive to its environment—Jacobs says “don’t heat it, magnetizeit, or drop it” [5].

Invar’s heat treatment is discussed in Chapter 22.The most recent (1992) improvement in invar is high purity regular

invar [4]. The idea is to get rid of all those pesky impurities that causedimensional growth over time. The key is to use a powder metallurgyapproach in making the invar, instead of the traditional furnace meltapproach. The advantage is that the iron and nickel ingredients can bemuch purer in the powder metallurgy approach. Powder metallurgyproduces the most stable regular invar made to date, and with a lowertempco than that produced by the melting process. Jacobs’ test datashows a time stability of better than 1 ppm/year and a tempco of0.2–0.8 ppm/C for high purity regular invar.

The drawback to high purity regular invar is that the powder metal-lurgy process is considerably more expensive than the traditionalfurnace melt process. And as far as I can determine, the powder metal-lurgy process for invar is not offered commercially (as of 1995) excepton a custom basis. Spang Specialty Metals, Butler, Pennsylvania, willmake high purity regular invar billets using the powder metallurgyprocess, in a minimum lot size of 1000 lb. Scientific Alloys, Westerly,Rhode Island, will extrude the billets into (pendulum) rods or whatever,for a $500 minimum charge.

According to Harner at Carpenter Technology, only a very smallnumber of Carpenter’s invar customers care about invar’s stability.Many of their customers buy the free machining version and do noteven bother to heat treat it. The biggest application of invar is in tem-perature sensors that use a bi-metal sensing element. Invar is used as thelow thermal expansion material in the bi-metal element. Based on thesmall market interest in a stable version of invar, there is little reason forthe market status to change.

I still recommend using free machining invar rather than regularinvar for the pendulum rod, if you want to use a threaded rating nut,in spite of its poorer stability. If you are not going to cut threads inthe invar, but instead drill a crosswise hole, put a dowel pin in the holeand adjust pendulum timing by stacking washers above the crosswise


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dowel pin, or by putting coarse weights in a weight pan, then goahead and use regular invar. Bateman has recommended the crosswisedowel pin approach in the past, and it has its advantages as the follow-ing points out.

First, no type of invar really machines well. Even threads cut in freemachining invar are below average, but threads cut in regular invar aremuch worse: they are just awful—chunks of thread missing, and athread surface so rough it looks like a bulldozer did it. Second, roughthread surfaces on the mating male and female threads will prevent anyfine adjustment of clock rate (1 s/day 0.001 in. of axial nut travel ona pendulum with a 2 s period). Third, the high contact pressures on thehigh points of the thread surfaces can raise the question of axial stability.Even with free machining invar, the author finds that the threads mustbe lapped to get them smooth enough for 1 s/day adjustability.

Measuring the stability of a material is extremely difficult, as itinvolves making measurements to much less than a micro-inch, and themeasurements must remain accurate over many months of testingtime. Many people involved in the stability (or instability) of “stable”materials went to see Stephen Jacobs (now retired) at the University ofArizona, Optical Sciences Center, in Tucson, Arizona. Jacobs designedand built some highly specialized equipment for measuring stability. Heused an optical multiple-beam interference technique, which gavesharper fringes by a factor of 100–1000 over those obtained by the stand-ard two-beam interference technique. Using two helium neon lasers(one as a frequency stabilized standard), this gave him a 0.001 ppm pre-cision of length measurement [5].

A hole is drilled through the material to be tested for the laser beamto go through. A mirror is attached to each end of the test material, cov-ering both ends of the laser hole. Optically, this is called an etalon, withthe test material being used as the spacer between the two end mirrors.Jacobs used two reference length standards: a frequency stabilized laserand a piece of fused quartz (Homosil).

Any pendulum builder wanting a more stable material than invar forhis pendulum rod should note what Jacobs considered to be a really sta-ble material, so stable that it could be used as a reference length in hisstability work—fused quartz [5, 6].

References1. J. Steele et al. “Temperature and age effects on the temporal stability of

invar,” SPIE 1752 ( January 1992), 40–51.2. B. Lement, B. Averbach, and M. Cohen. “Dimensional behavior of invar,”

Trans. ASM 43 (1950), 1072–97.3. E. Schwab, S. Jacobs, and S. Johnston. “Isothermal dimensional instability

of invar,” 29th Natl. SAMPE Symp. (April 1984), 169–84.

4. W. Sokolowski et al. “Dimensional stability of high purity invar 36,” SPIE1993 (February 1993), 115–26.

5. J. Berthold, S. Jacobs, and M. Norton. “Dimensional stability of fused sil-ica, invar, and several ultra-low thermal expansion materials,” Metrologia13 (1977), 9–16.

6. S. Jacobs. “Variable invariables—dimensional instability with time andtemperature,” SPIE Crit. Rev., Optomech. Des. C.R. 43 ( July 1992), 181–203.

7. C. Marschall and R. Maringer. Dimensional instability, Pergamon Press,London and New York, 1977.

8. S. Jacobs. “Dimensional stability of materials useful in optical engineer-ing,” Optica Acta 33(11) (1986), 1377–88.

9. R. Paquin. “Dimensional stability,” SPIE 1335 (1990), 2–19.10. J. Wittenauer (ed.) “The invar effect,” Minerals, Metals, and Materials Soc.,

Warrendale, Pennsylvania, USA, (1996).


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chapter 24

Position sensitivity along thependulum rod

Every clock’s pendulum needs a fine trim to adjust its rate to the desiredvalue. This is frequently done by adding small weights to a weight pan,which is usually located about one-third of the way down the pendu-lum rod.1 The clock literature says that the effect of adding a smallweight to a pendulum will vary, depending on the weight’s locationalong the pendulum rod. The literature also says that the weight (1) willhave maximum effect on the pendulum’s rate if placed halfwaybetween the bob and the suspension spring, and (2) will have zero effectif placed at the center of the bob or at the suspension spring. For thesuspension spring, it would be more exact to say that the zero effectoccurs where the pendulum’s axis of rotation is located, along the finitelength of the suspension spring.

I have never seen a position sensitivity curve for a pendulum rod, anddecided to measure one on a real pendulum. The primary reason fordoing so was that I wanted to put a fine rate adjustment near the top ofa pendulum rod—threading the rod end and adjusting the position of anut on the threaded segment. The threaded segment is 1.4–2.4 in.below the pendulum’s axis of rotation. There is an advantage2 to put-ting a fine rate trim at the top of the rod, but the position sensitivitywas not known for this location.

The experimental technique used to measure the position sensitivitywas to clamp a 24 g weight on the pendulum rod at a given location,and measure the change in clock rate. The weight was then moved toanother location about 4 in. down the rod, and the change in clock ratewith and without the weight attached was again measured. The fulllength of the rod was covered in this fashion, resulting in the positionsensitivity curve shown in Figure 24.1. The pendulum is shown directlybeside the sensitivity curve, and is drawn to the same scale, so as tomake it easier to correlate the measured sensitivity with position alongthe pendulum rod. Since the 24 g weight has no effect on clock ratewhen placed at the axis of rotation (suspension spring) and bob centerpositions, the sensitivity curve shows a zero change in clock rate atthose two locations.

1 This location is apparently based ongetting the weight pan as high as possible onthe pendulum rod while still keeping theweight pan below the clock dial, so that theweights on the weight pan can be easilychanged.

2 The advantage is that the nut can beadjusted here while the clock is running, withlittle or no disturbance of the pendulum. Andif the thread is lapped smooth, it will have anadjustment sensitivity of about 0.001 s/day(1.8 rotation of a 50 g nut). The weight panapproach has approximately equal sensitivity,with 0.001 s/day corresponding roughly to a0.001 g weight, which is the usual minimumsize available.

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Results

To repeat, position sensitivity means how much does the clock ratechange when a small weight is added at various positions along the pen-dulum rod. And as the literature indicated, a maximum change in clockrate does occur at the halfway point (point H in Figure 24.1) betweenthe suspension spring and the center of the bob. The clock speeds up ifthe weight is added above the bob, and slows down if the weight isadded below the bob. But what the literature does not say is that if thependulum rod extends for more than a short distance (6–7 in. for a 1 sbeat pendulum) below the bob, you will get an even higher sensitivityat the bottom end of the rod.

But if the weight is moved along the rod, instead of being added toor removed from the rod, the situation is quite different, as is shown inFigure 24.2. When the weight is moved along the rod, the position sens-itivity varies with the slope of the line in Figure 24.1. More particularly,it varies with the tangent of the angle between the vertical axis inFigure 24.1 and a line drawn tangent to the data line at each point alongthe data line. The actual sensitivity of moving a 27 g nut one revolution(32 threads/in.) at the top of the pendulum rod (2 in. below the pendu-lum’s axis of rotation) was measured as 0.11 s/day/revolution. So for a24 g nut, the same weight as was used for Figure 24.1, the sensitivitywould be 0.097 s/day/revolution at this position. Combining this scalefactor with the slope of the line in Figure 24.1 then gives Figure 24.2,which turns out to be a straight line.

Figure 24.2 shows the position sensitivity when a 24 g weight ismoved along the pendulum rod. The maximum sensitivity, that is, themaximum change in clock rate per unit of distance traveled by theweight along the rod, occurs at the top and bottom ends of the rod.And there is good sensitivity right at the pendulum’s axis of rotationand at the bob’s position. Contrast this with the first case wherein theweight was being added to the rod, and where the sensitivity was zeroat these two positions.

Moving the weight downward along the rod makes the clock speed upif the weight is above the halfway point (point H in Figures 24.1 and24.2) between the suspension spring and the bob. If the weight is belowthe halfway point (point H), moving the weight downward makes theclock slow down. The sensitivity is zero at the halfway point (point Hin Figure 24.2), and this is obviously not a good location to put a ratetrimming device that moves along the rod.

One good result of this test is that the top end of the rod turns outto be a good place to put a threaded nut for clock rate adjustment.Using a thread of 32 threads/in., one revolution of a 50 g nut willchange the clock rate by the desired sensitivity of 0.2 s/day.


168

–30 0 +30

Change in clockrate (s/day)

Suspensionspring

Halfwaydown

to bobcenter

H

+Bob

10

20

30

40

50

Dis

tan

ce fr

om a

xis

of r

otat

ion

(in

.)

Figure 24.2. Effect of moving a 24 g weightalong the pendulum rod.

–0.15 0 +0.15

Change in clock rate(s/day/revolution downward)

Suspensionspring

Halfwaydown

to bobcenter

+Bob

10

20H

30

40

50

Dis

tan

ce fr

om a

xis

o

f rot

atio

n (

in.)

Figure 24.1. Effect of adding a 24 g weight atdifferent positions on the pendulum rod.

The nut’s clock rate sensitivity depends on the ratio of the nut’sweight to that of the pendulum’s bob. The data here was taken with abob weighing 18.4 lb. If a different bob weight is used, the position sens-itivity numbers in Figures 24.1 and 24.2 should be changed in inverseproportion to the new bob weight.

chapter 24 | Position sensitivity along the rod

169

chapter 2 5

Fasteners for quartzpendulum rods

This chapter describes five different ways of fastening things to a quartzpendulum rod.

Connecting to a metal rod is easy. Just drill a hole in it or cut a threadon it. But how do you fasten something to quartz? Quartz is like glass—it is brittle and breaks easily. Five types of fasteners are described here,with some pros and cons on each. The five fasteners are:

Cemented sleeve Clamp ring Solder joint Dowel pin Split sleeve.

Cemented sleeve

It is assumed here that the piece to be fastened to the quartz rod is madeof metal. An obvious cement to pick is epoxy, for its strength. Figure 25.1shows a sleeve cemented around a quartz rod. Epoxy is a plastic, and assuch is not a very stable material. Its thermal expansion coefficient is quitehigh (45 106 to 65 106 in./in.C), it moves and generates stress during the curing process (which is exothermic), it relaxes under contin-uous stress, and it absorbs moisture so its weight changes with humidity.Exothermic means it gives off heat during the curing process. A 1 in. cubeof epoxy gets so hot during curing that you cannot touch it.

All of this is bad news, and says that epoxy is not a suitable material foran accurate pendulum. But, but, but. What if only a very small amount ofit is used, and that only in a thin layer? Then all of these bad effects shouldbe small as well, should they not be? At least that is the rationale that peo-ple offer for using epoxy anyway, although I have never heard of any exper-imental data to prove or disprove it. In any case, epoxy is sometimes usedin such situations by reputable people. And if you would have troublesleeping nights, worrying about the breakage of the epoxy’s bond to thequartz or sleeve surfaces, with the pendulum consequently crashing downon the floor, there is a fix. Just rough up the quartz surface with sandpaperFigure 25.1. Cemented sleeve.

Quartz rod

Rating nut

Epoxy joint

Roughened surfaces

Threaded sleeve,32–40 TPI

171

or grind grooves in the quartz surface to a level deeper than the maximumshrinkage of the epoxy, and cut a shallow groove in the sleeve’s inner sur-face. Then if the epoxy does break loose, the thickness of the epoxy layerin these grooves will prevent the sleeve and bob from sliding off the pen-dulum rod and falling on the floor.

Clamp ring

In this fastener, shown in Figure 25.2, a close fitting ring is slid onto thequartz rod and an axial squeeze force is exerted on the ring by a lockingnut, to compress the ring tightly against the rod. To reduce the axialforce needed to do this, the ring is cut through radially in one spot. Thefastener holds its axial position on the rod by means of the resultingfriction force between the ring and the rod.

This fastener has the small advantage of being easily moved to a newposition. It has a larger disadvantage in that the axial position of the topring (see Figure 25.2) changes with the tightening (and loosening) of thelock nut. This is undesirable for long-term stability, as the clampingstresses in the fastener will relax over time. The split ring’s inner surfacethat is up against the quartz rod must be smooth and have no sharppoints that would “dig” into the quartz and cause the quartz to fracture.To avoid sizable changes in the frictional clamping force on the rod whenthe temperature changes, and the possible resultant slippage of the fast-ener down the rod, the fastener should be made out of invar. Invar’sthermal coefficient of expansion (0.8 106 to 3 106 in./in.C,depending on heat treatment) more closely matches that of quartz(0.5 106/C) than that of any other metal.

The radial and axial clamping forces generate a lot of stress in thefastener and in the quartz rod under the ring. The axial clamping forceneeded in the fastener is inversely proportional to the fit of the uncom-pressed split ring on the quartz rod. The better the fit, the less axialforce needed from the locking nut. Smooth surfaces on the four taperedclamping surfaces would also reduce the amount of axial clampingforce needed. This fastener concept was originally proposed by DallasCain. It worked okay when Paul Hopkins tried it, with the proviso thathis quartz rod broke off below the clamp.

Solder joint

In this fastener, shown in Figure 25.3, a groove is cut in and around thequartz rod, near its end. This end is inserted in a blind metal hole that istinned on the inside and partly filled with hot liquid solder. When thesolder cools, the solder in the groove forms a mechanical wedge, lockingthe end of the quartz rod in the hole. This is certainly a simple and fastproduction technique. The concept works because quartz can withstand


172

Figure 25.2. Clamp ring.

Quartz rod

Top ring

Split ring

Wrenchingshoulder

Lock nut

Figure 25.3. Solder joint.

Suspensionspring (2)

Blind hole

Solder joint

Indent groove

Quartz rod

the 193 C temperature of liquid solder (60/40 tin/lead) withoutcracking or breaking. Donald Hendrickx replaced a broken quartz pen-dulum rod (2 s period) in a German clock that used this type of fastenerat both the top and bottom ends of the rod. Hendrickx thinks the holemetal was brass. The thermal expansion coefficient for 60/40 tin/leadsolder is 25 106 in./in.C; for brass it is 20 106 in./in.C.

I like this fastener—it meets the KISS rule (keep it simple, stupid) forgood design. The fastener’s axial stability depends on the stability of thesolder under shear stress. It could be improved by using the strongerand much more rigid silver solder (98% tin, 2% silver), with only a slightincrease in the melting point (232 C) over that of the 60/40 tin/leadsolder. This silver solder is Alpha Metals #53982, and is commonlyavailable at hardware stores. Its thermal expansion coefficient was notavailable from Alpha Metals.

Dowel pin

The dowel pin fastener is shown in Figure 25.4. It consists of a sleevearound the quartz rod and a horizontal dowel pin passing through boththe sleeve and the rod. The hole through the quartz is made with a dia-mond drill, which costs only $6 for the small 3.5 mm diameter size, andis available at lapidary supply stores. The small cross-section of the

in. diameter dowel pin increases the compressive stress in the quartz,but this is okay as quartz’s compressive design stress limit (10,000 psi) issix times higher than its tensile design stress limit (1500 psi). This fast-ener concept was tried by the writer, and it works well. More detail onthe quartz drilling technique is given in the Appendix.

By making the sleeve and dowel pin out of brass and stainless steel,respectively, the fastener can be made part (or all, by increasing the axialdimension) of the pendulum’s temperature compensator. If they aremade out of invar, they become a very small part of the pendulum’stemperature compensation. I like this fastener’s design also. I think it isthe best of the five concepts presented here.

Split sleeve

The split sleeve clamp was proposed by Roger Irving [1] who used itsuccessfully on a half second pendulum. The basic idea is shown inFigure 25.5(a–c). With a glassblower’s torch, the end of the quartzpendulum rod is softened and pushed into a ball whose diameter issignificantly larger than the rod’s diameter. The height H of this ball inFigure 25.5(a) should be at least equal to the rod’s diameter, to preventthe ball’s surface from spalling off the end of the rod under stress.A sleeve whose inner diameter is slightly larger than the quartz rod’s

18

chapter 2 5 | Fasteners for quartz rods

173

Figure 25.4. Dowel pin.

Quartz rod

Sleeve

Dowel pin

diameter is cut in half lengthwise and the two halves of the sleeve areplaced around the rod just above the balled end. A loose-fitting metalsecuring ring is placed around the split sleeve to hold the sleeve in placeon the quartz rod without binding. The securing ring’s inner diameteris a little larger than the quartz ball’s diameter so that the securing ringcan pass over the quartz ball and fit loosely around the split sleeve. It isalso possible to eliminate the securing ring and use the central hole inthe bob or compensating sleeve to secure the split sleeve.

Figure 25.5(a) shows the quartz ball, split sleeve, securing ring, andbob spaced apart, to give a better view of the individual pieces. The boband/or temperature compensator rests on top of either the split sleeveor the securing ring, as shown in Figure 25.5(b and c). The central holethrough both the bob and the temperature compensator is a little largerthan the quartz ball’s diameter so that they can pass over the quartz balland onto the quartz pendulum rod.

There is one critical joint. And that is the one between the quartz balland the split sleeve, where a sharp point or edge on the split sleeve’sbottom surface can raise the stress level in the quartz ball and causefracture. Irving reduced this problem by filling the split sleeve’s jointswith liquid shellac (an adhesive) and letting the shellac harden in place.Shellac is a liquid at 120 C and is hard at room temperature. Using anadhesive has the drawback that the bob cannot be removed from thependulum rod without dissolving or melting the adhesive. This pre-sumes that there is a split sleeve fastener at both ends of the pendulumrod, preventing bob removal off the rod’s top end.

Two alternatives to using an adhesive come to mind: (1) do not useany adhesive and just closely watch the critical joint’s mating surfaces toeliminate any sharp points or edges, or (2) put a metal washer betweenthe split sleeve and the quartz ball. The washer would be cemented tothe quartz ball, and none of the other parts would be cemented. The


174

Figure 25.5. Split sleeve fastener: (a) spacedapart, (b) bob resting on split sleeve, and(c) bob resting on securing ring.

Quartzrod

Quartzrod

Bob

Bob Bob

Quartz ball

H

Quartz ball

Securingring

Securingring

Splitsleeve

Splitsleeve

(a) (b) (c)

washer’s outer diameter would be slightly less than that of the quartzball, so that the securing ring, bob, and temperature compensator couldpass over the washer.

Comments

Any of the fasteners described here can be used at either the top or bot-tom of a quartz rod, for connecting to a suspension spring, bob, ratingnut, or whatever. As examples, Figure 25.1 shows a cemented sleevefastener connecting to a rating nut, and Figure 25.3 shows a solder jointfastener connecting to a double suspension spring. In most cases the fas-tener material should be invar, because of its low thermal expansioncoefficient. The reader is cautioned that the cemented sleeve concept inFigure 25.1 has not yet been tried in practice.

Appendix: Drilling holes in quartz

Holes can be drilled in quartz in a home workshop, and at low cost. Itdoes require one shop tool—a drill press with a vise. The vise needs tobe firmly clamped or bolted to the drill press table.

When putting a quartz rod in a vise, put two layers of paper betweenthe quartz rod and the sides and bottom of the vise. This is to preventany small sharp metal points on the vise from fracturing the quartz rod.This assumes that the sides and bottom of the vise already have smoothsides. If there are any high spots (watch for a raised edge around anydents) on the vise, they need to be filed and sanded down beforehand,or you will crack the quartz rod when you tighten the vise (only to amedium clamping pressure).

Since the hole(s) to be drilled will be near the rod ends, the far end ofthe rod away from the drill press will need to be supported vertically.This will most likely require making a temporary wooden supportstand for that purpose. It should allow for some horizontal movementof the quartz rod, with fixed stops, so that the rod cannot roll off thesupport and go bang on the floor. Figure 25.6 shows the idea.

If a hole is drilled all the way through a rod from one side, there mostlikely will be some large breakout chips when the drill comes out the farside. Drilling halfway through from opposite sides and meeting in themiddle gets around the breakout problem. To do this and get the two“half through” holes to line up on each other requires accurately rotatingthe rod 180 in the vise, so as to drill the hole in from the opposite side.This can be done by making use of three simple mechanical align-ments: (1) a repeatable axial position stop for the rod, (2) an anglepointer for the 180 rotation, and (3) putting the hole through the exactaxial centerline of the rod, without an offset.


175

Figure 25.6. Top end of rod support stand.

Horizontalmovement (2–3 in.)

Limit stop (1 of 2)

Wooden supportstand

Quartz rod

The repeatable axial position stop is provided by simply holding asmall flat plate against the side of the vise, and moving the quartz rodaxially in the vise until the rod’s end butts up solidly against the plate,as shown in Figure 25.7. The rod’s end face will most likely be roughand uneven, so the flat plate must cover the rod’s whole end face for theaxial stop to be accurate.

An accurate 180 rotation is obtained by temporarily mounting a 2 or3 in. long radial pointer on the quartz rod, as shown in Figures 25.7 and25.8. The pointer is made out of scrap sheet metal, and is shown in boththe 0 and 180 positions by the solid and dotted lines in Figure 25.8.The rod is rotated slightly at both the 0 and 180 positions until theheight of the pointer’s end, W and X in Figure 25.8 is set equal to theheight D of the rod’s axial centerline. The longer the pointer, the moreaccurate the 180 rotation will be.

The drill is positioned directly over the rod’s axial centerline by measur-ing the distances Y and Z on each side of the drill bit, that is, between the


176

Figure 25.7. Axial position stop, and anglepointer for 180 rotation.

Rotation angle pointer

Quartz rod

Vise

Axial stop plate

Drill press table

Figure 25.8. Angle pointer for 180 rotation,and drill bit centered over rod’s axialcenterline.

Paper (2 layers)

Diamond drill

Rotation angle pointer

Vise

Quartz rod Y Z

W DX

+

Mot

ion

Drill press table

paper on each side of the vise and the closest side of the drill bit. Whenthe two distances Y and Z are the same, the drill is centered over the rod’saxial centerline. To make them the same, the vise clamps (or bolts) to thedrill press table are loosened, the vise is moved back and forth until the Yand Z distances are the same, and the vise clamps (or bolts) are then re-tightened. The measurements of Y and Z are made with an accurateruler and a magnifying glass (to get sufficient resolution), or better yet,with a dial caliper. The drill bit is lowered to just below the top edge of thevise to make the measurements easy to do, as shown in Figure 25.8.

Diamond drills use a special drilling technique. You do not just pressdown continuously on the drill bit. Instead, the quartz hole area is firstflooded with water and is kept flooded with water. A plastic water bottlewith a squirt tube (not a spray) works well. Then you lightly press thedrill down against the quartz for a few seconds, and then lift the drill upfor 1 s, in a repetitive “down and up” drilling technique. The periodiclifting of the drill bit allows the water to get in under the drill bit, tolubricate and also cool things down. The drill and the quartz hole areaabsolutely must be kept flooded with water at all times, to keep the heatfrom destroying the drill or cracking the quartz rod.

For diamond drills of the small size involved here (approximately in. diameter), the drill press is set at its highest speed setting

(3000–5000 rpm). One drill manufacturer recommends an even highersmall drill speed of 5000–30,000 rpm. For a in. diameter dowel pin, Iused a slightly larger 3.5 mm (0.138 in.) diameter diamond drill, givinga hole slightly larger than needed. This allowed a small 0.013 in. toler-ance on drilling the hole correctly.

The diamond drills in the lapidary store are designed to be low cost,and consequently do not last long. The diamond coating on my drillwore through after drilling only two holes through a 0.640 in. diameterquartz rod. Depending on the job, you may want to buy a second drill(they are cheap, only $6 apiece). And remember that diamond drillinggoes slow. It will take 15–30 min to drill halfway through a 0.640 in.diameter quartz rod.

I would not recommend drilling a hole any closer than a in. fromthe end of a quartz rod. Like glass, quartz is rather unpredictable inwhat the drilling stresses will do to the material. If you need to drillcloser than a in. from a rod end, I would recommend practicing onceor twice beforehand on a spare piece of quartz rod.

Reference1. R. Irving. “Silica rod for a seconds pendulum,” Hor. Sci. Newslett. NAWCC

chapter 161 (September 2000). Available in NAWCC Library, Columbia, PA17512, USA.

12

12

18

18


177

chapter 26

Effect of the pendulum rod on Q

A pendulum rod’s air drag has a significant effect on the pendulum’s Q.

Many years ago, Bateman published a good article [1] on the effect ofbob shape on a pendulum’s Q. He showed that a football-shaped bob,pointed horizontally in the direction of swing, had the highest Q (leastair drag) of any bob shape tested. That highest Q shape had a 2 : 1length-to-diameter ratio. Spherical bobs had a little lower Q. And rightcircular cylinders, with their cylindrical axis parallel to the pendulumrod’s axis, had even lower Q.

Bateman used the sphere as a reference point for all of the bobshapes he tested. For his Q tests, the bobs were suspended by two finewires (0.006 in. diameter), so as to make the Q measurements predom-inantly a function of bob shape, and minimize any air-drag effects fromthe bob’s suspension.

In a real clock, of course, the pendulum rod’s air drag would affectthe pendulum’s overall Q. The relevant question is—by how much?

I have a pair of cylindrical and spherical bobs of almost equal weight,and an assortment of rods of different diameters. So I decided to findout what the rod’s effect was on Q. The pertinent rod variable is therod’s diameter.

Three rods were tested, all of circular cross-section, and with dia-meters of (invar), (invar), and 0.64 in. (quartz). The use of two rodmaterials has no effect on the Q measurements, of course, except for thesmall difference in total pendulum weight. Only their diameters are ofconcern here. Both bobs are made of leaded brass. The spherical bob is 4.90 in. in diameter and weighs 18.4 lb. The cylindrical bob is 3.65 in. indiameter and 6.00 in. long, and weighs 18.2 lb. The test pendulums wereassembled using the traditional method of temperature compensationfor an invar rod: supporting the bob on an external sleeve around therod. The sleeve is the temperature compensator (brass), and is locatedbelow the bob. The pendulums all have a 2 s period. There is no escape-ment, as my pendulums are normally driven electromagnetically. A typ-ical pendulum assembly is shown in Figure 26.1. With the quartz rod,the sleeve spacers in Figure 26.1 are quartz tubing, not invar.

38

14

Figure 26.1. Typical pendulum layout, shown with cylindrical bob. * Depends on bob length. d2 d1 (0.12–0.20) in.

Top of clock case

5.5

3.00

6.00

in.

1.0

Rod, invar

Bushing,brass

Bob,brass

3.65

d1

d2

Spacer, invar


Spacer, invar

Rating nut

0or4.0*

1.5to2.0

~ ~39

179

The Q measurements were made inside a plywood clock case, withinternal dimensions of 66.5 17.8 13.5 in. (H W D). The frontdoor of the clock case was removed and left off, and the pendulumswere hung in the center of the case (front to back, as well as side toside). The pendulum’s drive coils were shut off and removed from thecase. With the two smaller rod diameters, the rod’s hole through thebob was bushed inward at the top and bottom of the bob, so as to pro-vide a reasonably smooth bob surface in those two areas. Q is calculatedfrom the expression Q 4.53 times the number of full swings neededfor the amplitude to decay by 50%.

The measured Q values are listed in Table 26.1. The Q decreases asthe rod diameter increases, which is what one would expect. The pen-dulums lose 2–9% of their in. diameter Q value with the middle in.rod diameter, and 15–25% of their Q value with the large 0.64 in. roddiameter. What is happening is that increasing the rod’s diameterincreases the surface area which increases the pendulum’s total air drag,thereby reducing the Q. The Q values for the 0.64 in. rod diameter maybe a few percent lower than they should be, because the rod was 4 in.longer at the bottom than the other two rods, creating extra air drag.But I was reluctant to cut it to size because of other upcoming tests.

The cylindrical bob has a length to diameter ratio of 1.64. Bateman’sdata for this ratio (interpolated) shows that this cylinder’s Q should beabout 70% of that of a sphere of equal weight and volume (using a0.006 in. diameter wire suspension). The new data shows that thecylinder-to-sphere Q ratio is 75–77% for the in. rod diameter, 80–83%for the in. diameter, and 90–94% for the 0.641 in. diameter. As therod’s cylindrical surface increases, the more efficient spherical bob’ssurface becomes a smaller percentage of the total air drag.

My pendulums always exhibit two Q values: a higher one for swingangles less than 1 (half angle), and a lower one for swing angles greaterthan 1 (half angle). Figure 26.2 is a typical case, and shows the decay ofthe pendulum’s amplitude over time, for a in. diameter rod with aspherical bob. Note that the vertical scale is logarithmic. Figure 26.2

38

38

14

38

14


180

Table 26.1. Q vs diameter of pendulum rod

Pendulum rod Q

diameter (in.)For half angle 1 For half angle 1

Cylindrical bob Spherical bob Cylindrical Q/ Cylindrical bob Spherical bob Cylindrical Q/

Spherical Q Spherical Q

24,000 31,300 0.77 19,200 25,500 0.75

23,600 28,500 0.83 19,000 23,800 0.80

0.64 20,400 22,700 0.90 14,900 15,900 0.94

38

14

clearly shows the two slopes in the plotted line, each representing adifferent Q. Why are there two slopes and two Q values for eachpendulum? I do not know. Probably nonlinear air effects. At the 1 halfangle crossover point between the two values (the crossover is always at1 half angle), the bob is moving 1.4 in. peak to peak. None of the pen-dulum dimensions are anywhere near 1.4 in.

To conclude, the data shows that the Q decreases as the rod’s diameterincreases. It also shows that the pendulum’s Q with a spherical bob is6–20% better than with a cylindrical bob, for practical rod diameters of

to in. The better Q means proportionately better timekeeping. Youwill have to decide for yourself whether the additional cost and effort ofmaking a spherical bob, as compared with the cheaper and easier tomaking cylindrical one, is worth the increase in Q.

Reference1. D. Bateman. “Is your bob in better shape?” Clocks 2 ( June 1988), 34–7. Also

in A. Rawlings. Science of clocks and watches, 3rd edn, Brit. Hor. Inst.,Upton, England, pp. 89–94, 1993.

58

38

chapter 26 | Effect of the rod on Q

181

Figure 26.2. Pendulum amplitude decay overtime: in. diameter rod, spherical bob.

38

2.0

1.5

1.0

0.8

0.6

0.40 1 2

Time (h)3 4 5

Am

plit

ude

(de

g) (

hal

f an

gle)

part iv

Air and clock case effects

185

chapter 27

Correcting the pendulum’s airpressure error

This chapter describes how to calculate and remove the air pressure error from a clock’s time error vs time chart.

Depending on your geographic location and the density of yourpendulum materials, air pressure variations [1, 2] can cause 2 to 18 s oferror in a clock in a year’s time interval. This is because the pendulum“floats” in a sea of air, and variations in the air pressure make the pen-dulum slow down or speed up. In this chapter, the pressure error itselfwill be described first. Then an actual clock data run will be used toshow how the pressure error is calculated and then removed from aclock’s time error vs time curve.

Correcting for the pressure error corrects the clock’s time to a con-stant average pressure at the clock site. The error is the product of thepressure difference (actual average) times time. More specifically, it isthe integral of the pressure difference (actual average) with respect totime. Mathematically, the pressure error is

where

E(m) time error in seconds after m hours.E(0) time error in seconds at start of run.K pressure sensitivity factor in s/day/in. Hg.P local air pressure in inches of mercury.Pav value of P averaged over the run.t variable of integration in hours.t time interval in hours between readings of clock data.

Note that the time error is measured in seconds, but the elapsed timeis measured in hours. Clock data is normally recorded at periodic timeintervals (t) rather than continuously. The integral of pressure overtime can be approximated digitally by algebraically summing up the

E(m) E(0) K24

m

0(PPav) dt,

products of the pressure differences (actual average) times the timeinterval between readings:

The air pressure varies as a crude sine wave of varying amplitude,with a period of 2–5 days. To get any sort of accuracy, 9 or 10 datapoints are needed across the period of the sine wave. So the air pressureneeds to be recorded at intervals of 5 h or less. The pressure correctionE(0) at the beginning of the run is not known. One can either set theinitial pressure correction E(0) equal to zero or select it so that the aver-age pressure error over the run is zero.

Figure 27.1 shows the data recorded for a pendulum clock that washeated to a constant temperature above ambient over a 6-day interval.On day zero, the electric heating blankets wrapped around the clock

E(m) E(0) K24

tm

t0[(PPav)t].


186

Figure 27.1. Recorded clock data: (a) uncorrected clock time error, (b) averageair pressure, (c) actual air pressure, (d) temperature, and (e) heat input.

1.0

0.8

0.6

0.4

29.5

28.5

27.5110

90

70

1

0

0 2 4Time (days)

6

(a)

Clo

ck t

ime

erro

r (s

)A

ir p

ress

ure

(in

.Hg)

Hea

t Te

mpe

ratu

re (

˚F)

(e)

(d)

(b)

(c)

were turned on and raised the clock’s temperature about 30 F (17 C).At the end of 6 days, the heat was shut off and the clock allowed to cooldown to room temperature. The clock’s time error and its temperaturewere recorded and plotted about every 8 h. The air pressure was recordedevery 15 min, but only its value at about 8-h intervals is plotted inFigure 27.1.

The clock’s time error curve in Figure 27.1 shows a sinusoidal vari-ation that appears unrelated, shape-wise, to either the pressure or thetemperature curves. The clock’s temperature is relatively constantthroughout the 6-day interval, and could not have caused the sinusoidalwaveshape in the clock’s time error. It may have come from the vari-ation in pressure. To calculate the pressure error, assume (1) that thepressure error E(0) at the beginning of the run is zero, and (2) that thisis a new pendulum whose sensitivity to pressure variations is unknown.The pendulum’s pressure sensitivity, which we do not have, is neededfor the error calculation. So we will make a scientific wild assedguess (SWAG) at the pressure sensitivity factor and use that to calculatean estimated pressure error curve, which is shown in Figure 27.2(a).This is subtracted from the uncorrected clock error curve, repeated inFigure 27.2(b) from Figure 27.1, with the corrected error curve shownin Figure 27.2(c). Note that the Figure 27.2(a) curve is plotted upsidedown, to make any similarities between the Figure 27.2(a and b) curvesmore apparent.

chapter 27 | Correcting the air pressure error

187

Figure 27.2. (a) Estimated pressure error,(b) uncorrected clock time error, (c) clocktime error minus the estimated pressure error,and (d) clock time error minus the finalcorrected pressure error.

1.0

(b)

(c)

(d)

(a)

Clo

ck t

ime

erro

r (s

)P

ress

ure

err

or (

s)

0.8

0.6

0.4

–0.2

+0.2

0

0 2Time (days)

64

Some of the estimated pressure error’s sine wave still remains in theestimated clock error curve in Figure 27.2(c). Increasing the estimatedpressure sensitivity factor about 50% will remove the rest of the sinewave. Figure 27.2(d) shows the clock’s time error with the sine waveeliminated. Figure 27.2(d) was obtained by trial and error, varying thepressure sensitivity factor until the sine wave was eliminated fromFigure 27.2(d). Empirically, the pressure sensitivity value that eliminatesthe most “wiggles and bumps” (a sine wave in this case) from the clock’serror curve is the correct pressure sensitivity factor.

Summary

In summary, trying to determine if there is any pressure error in theclock’s time error curve by comparing it with the air pressure curve isalmost a wasted effort, as Figure 27.1(a and b) shows. There are no sim-ilarities between the two curves, the presence of which would indicatethat a pressure change was affecting the clock’s time error data inFigure 27.1(b). What is needed alongside the clock’s time error curve isthe pressure error curve, that is, the integral of the pressure difference(actual average) over time, as Figure 27.2(a and b) shows. Here it isreadily apparent from the similarities between the two curves(Figure 27.2(a and b) ) that the air pressure is showing up in the clock’stime error data, and needs to be subtracted out of it.

The digital approach is the easy way to generate the pressureerror curve—adding up the pressure differences (actual average) inm sequential increments of time and multiplying by both (1) the timeinterval t between data readings, and (2) the pressure sensitivity fac-tor. Subtracting the pressure error curve from the clock’s uncorrectedtime error curve then gives a clearer and truer picture of the clock’sactual performance.

The need for frequent pressure readings almost requires an auto-matic pressure recording system of some sort, as it is rather tiring tomanually take pressure readings every 5 h or less.

It may be worth noting that the technique described above can alsobe applied to temperature errors. Recording the temperature at inter-vals over time and then digitally summing the product of the temper-ature difference (actual average) times the time interval betweenreadings will give a temperature error curve similar to the pressureerror curve. This would correct the clock to the average temperature atthe clock site.

If the pendulum has an electromagnetic drive, the two correction con-cepts can be carried even further. Inserting electric drive pulses at the endsof pendulum swing will advance or delay the pendulum by t incrementsof time. This would amount to a closed servo loop correction of all


188

pressure and temperature errors. Of course, the calculation of thepressure and time errors would have to be done by electronic circuitryrather than by hand. But that would not be too hard with the digitalsumming technique described herein.

References1. R. Matthys. “Time error due to air pressure variations,” Hor. J. ( January

1996), 16–18.2. J. Bigelow. “Barometric pressure changes and pendulum clock error,” Hor.

J. (August 1992), 62–4.

chapter 27 | Correcting the air pressure error

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chapter 28

Pendulum air movement: A failed experiment

This is a short description of a failed experiment—an attempt to shape theinside surface of a clock case to provide less air drag on the pendulum, andthereby improve the pendulum’s performance. This report might be of interestto someone else wanting to measure or modify the pendulum’s air movement.

In a previous experiment (see Chapter 21), I had learned that thewalls of my clock case slow down the pendulum (via air drag) byapproximately 1 s/day. The pendulum has a 2 s period. Now I wantedto find out if the walls’ drag on the pendulum could be reduced ormade more constant by shaping the walls’ inside surface for easier airflow. The inside wall shape I had in mind is that shown in horizontalcross-section in Figure 28.1. The concept basically involved roundingthe square corners inside the clock case. For a cylindrical bob with itsaxis along the pendulum rod, the horizontal cross-section shown inFigure 28.1 would be the same at all elevations in the region of the bob.

Six foot high case walls of transparent plastic (plexiglass) were madeup, and the internal curved wall surfaces of Figure 28.1 were bent upout of sheet metal. A transparent antistatic coating was applied to theplastic walls, to avoid electrostatic charge effects on the pendulum,which also has a 2 s period. To see the air currents, white smoke wasgenerated by burning the 0.5 1.5 in. “incense cones” commonly available in knickknack stores. The smoke from one cone will last about

Figure 28.1. Cross-section of clock case andinsert intended to smooth the internal airflow.

Clock casewalls

Proposedairflow

18 in.

Bob motion

Curvedsheet metalinsert (1 of 2)

14 in

.

20 min. The incense cones were usually burned two at a time, and werelocated about 12 in. below the bob and about 1 in. in front of and 1 in.in back of the pendulum rod. The heated smoke from each cone wouldrise smooth and undisturbed at about 10 in./s in a straight and unmov-ing column ( in. in diameter) until it struck the bottom surface of theswinging bob. With the spherical bob, the smoke then spread out acrossthe bob’s moving bottom surface in a thin layer approximately 0.03 in.thick, went up all around the bob’s sides in a thicker layer, and on topflowed back toward the pendulum rod in the center of the bob in a halfinch thick disturbed layer.

A circular and very disturbed smoke column 10 in. in diameter (whichvaried with bob size and shape) rose above the bob, up and out of thetall plastic clock case. In no instance did the disturbed smoke column gonear or touch the walls (14 18 in. horizontal spacing, wall to wall).

In fact, removing the case walls had no visible effect on the size orshape of any part of the smoke column, either near the bob or above it.And moving the smoke cones over next to the case walls revealed novisible air movement within 1–2 in. of the case walls. In all of thesetests, the pendulum was driven electromagnetically, and the bob wasswinging 1.4 in. peak to peak (0.9 half angle).

From the above, it was apparent that all or almost all of the air move-ment was taking place adjacent to or above the bob’s surface, and thatvery little or no air movement was occurring near the case walls. Thewalls may slow the pendulum down by 1 s/day, but that is still only onepart in 86,400—an amount that is not visible to my naked eye. The curvedinternal surfaces made of sheet metal in Figure 28.1 were removed asbeing essentially worthless for changing the airflow. And the conclusionwas drawn that anything done with the airflow inside the clock caseshould be done at the bob’s surface and not at the clock case walls.

Three bobs were used in this test: a large brass sphere, 4.9 in. dia-meter, 18 lb; a large bronze cylinder, 3.7 in. diameter 6.0 in. long, 19 lb;and a small brass cylinder 2.0 in. diameter 6.2 in. long, 5.3 lb. Bothcylindrical axes were located along the pendulum rod axis. With thelarge spherical bob, the airflow has already been described. With thesmall cylindrical bob, there was about 2 in. of disturbed airflow radiallyfrom the bob in the plane of swing, and about in. of disturbed airflowin front of and behind the bob and the plane of swing. The diameter ofdisturbed air above the small cylindrical bob was about 8 in.

The large cylindrical bob disturbed the most air, extending out 3 in.radially from the bob. The diameter of the disturbed air column abovethis bob was about 12 in. Only a little of the air surrounding this bobmoved with the bob’s swing frequency. A small bubble of air (smoke)collected behind the temperature compensator (a in. O.D. 2 in. longsleeve surrounding the pendulum rod below the bob) in each directionof swing.

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192

Airflow around the large cylindrical bob was not continuouslysmooth. A big bubble of air would “flip” around each side of the cylin-drical bob each time the bob was about halfway back to center fromeach end of its swing. This air bubble was approximately in. thick(radially from bob) in. long (around bob) 3 in. high. This air bub-ble was not noticed flipping around the other two bobs, but it may havebeen present and been overlooked.

To sum up, the spherical bob had the smoothest airflow. The largecylindrical bob disturbed the most air. There was very little if any airmovement near the clock case walls. Any further attempt to affect theairflow should be aimed at the bob’s surface-to-air interface, and not atthe case walls.

And finally, a word of caution. The smoke from the incense conescoated everything it touched with a light brown tarry substance. Itrequired effort and an alcohol solvent to remove the gooey coating.

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chapter 28 | Air movement: A failed experiment

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195

chapter 29

Pendulum air movement: A second try

The movement of air around four different pendulum bobs is shown by whitesmoke in a series of pendulum photographs.

Air drag is the biggest energy loss in a pendulum. It would be help-ful to know what the airflow around the pendulum looks like, as a pre-liminary step toward minimizing this energy loss. The bob generatesmost of the air drag, because of its large size and its location at the endof the pendulum rod. White smoke, rising from small incense cones,makes the airflow visible. The airflow can then be photographed,although with some difficulty (see Photography in the Appendix).

Test setup

The photographs in Figures 29.1–29.6 show a pendulum with a 2 speriod hanging inside at the center of four 6-ft high transparent plastic(Plexiglas) walls. The walls provide an internal rectangular 14 in.

18 in. (front-to-back left-to-right) space for the pendulum. The plasticwalls represent the walls of a clock case, and provide whatever effect theclock case’s walls have on air movement, which except for the 1 s/dayslowdown (see Chapter 31) of the pendulum is very little, as was shownduring the first try at observing the airflow (see Chapter 28).

Pendulum

The pendulum design is the standard temperature compensatedarrangement used for invar pendulum rods. There is a rating nut at thebottom of the pendulum rod. A loose-fitting temperature compensat-ing sleeve around the pendulum rod’s bottom end rests on the ratingnut, with the bob resting on top of the temperature compensatingsleeve. The only nonstandard item is that the temperature compensat-ing sleeve is not located up inside the bob, but is instead placed belowthe bob out in the open so as to get a faster temperature compensation.

Four different bobs are shown in the photographs. Two are cylinders:one with a large diameter, and one with a smaller diameter about half


196

Figure 29.1. Large cylindrical bob at left end of swing, different amplitudes. Front views: (a) pendulum stopped, (b) at 2 peak amplitude, (c) at 4 peak amplitude, (d) at 8 peak amplitude.

(a) (b)

(c) (d)

Table 29.1. Bob dimensions and weights

Type Diameter Length Oak bob weight Equivalent

(in.) (in.) (lb) volume weight in

brass (lb)

Cylinder, large 3.84 6.05 2.20 21

Cylinder, small 1.87 6.00 0.48 5.0

Sphere, large 5.08 — 2.41 21

Sphere, small 3.19 — 0.53 5.2

that of the larger one. The other two bobs are spheres, one large and onesmall, with volumes and weights approximately equal to that of the twocylindrical bobs. Bob dimensions and weights are given in Table 29.1.

Photographs

All of the photographs were taken from the front or the right side ofthe clock case. The front photographs were taken with the pendulumat two positions: at the left end of swing, and at the center of swing (thebob is always going to the right). The side photographs were all takenwith the bob at the center of swing, and with the bob moving awayfrom the reader (camera) and into the paper. The photographic timingdepended on my eye-to-hand coordination, which was not perfect, as

chapter 2 9 | Air movement: A second try

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Figure 29.2. Large cylindrical bob at center ofswing, different amplitudes. Front views: (a) pendulum stopped, (b) at 2 peakamplitude, (c) at 4 peak amplitude, (d) at8 peak amplitude.

(a) (b)

(c) (d)

can be seen in some of the photos. Photographs were taken at swingamplitudes of 0, 1, 2, 4, and 8 (half angle).

The front photos show a swing amplitude scale, located just belowthe pendulum and marked in 1 increments (half angle). The 5 and 10

amplitude marks are shown with white triangles.

Comments on the photographs

The thing to look for is what happens or does not happen to the smokecolumns in the photographs. The smoke movement is a little differentthan it was on the earlier first try.


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Figure 29.3. Large cylindrical bob at center ofswing, different amplitudes. Side views:(a) pendulum stopped, (b) at 2 peakamplitude, (c) at 4 peak amplitude, (d) at8 peak amplitude.

(a) (b)

(c) (d)

First, some limits. The front and side view photographs were takenat different times, and are of different pendulum swings. As a result thefront and side views show slightly different images. Also, the side viewphotos are magnified about 30% more than the front view photos. Forcost reasons, the magnification difference was not corrected, althoughit would have been nice to do so. The smear marks below the bob insome of the photos are not smoke effects, and should be ignored. Theyare due to spurious reflections from the smeary antistatic coating on theplastic “case” walls.

To limit the number of photographs, only the large cylindrical bob isshown over a range of swing amplitudes, in Figures 29.1–29.3. All fourbobs are shown together at one commonly used amplitude, 1 halfangle, for easy comparison in Figures 29.4–29.6.


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Figure 29.4. Four bobs at left end of 1 swing(half amplitude). Front views: (a) largecylinder, (b) small cylinder, (c) large sphere,(d) small sphere.

(a) (b)

(c) (d)

Except for above the pendulum, the air disturbed by the pendulumbob’s passage lies within about in. of the pendulum’s surface. Air far-ther than about in. away from the pendulum’s surface is not disturbedor disturbed very little by the bob’s passage. Notice how the temper-ature compensator below the bob passes within in. of the smokecolumns with essentially no disturbance of the smoke columns. Over alonger time period (15 min) at the larger 4 and 8 swing ampli-tudes, the whole internal wall-to-wall space becomes smoky. This indic-ates that at these larger swing amplitudes, the large cylindrical bob isslowly stirring all of the air in the clock case.

With a cylindrical bob most of the air disturbance takes place at thebob’s cylindrical section, rather than at the bob’s top and bottom ends.With a spherical bob most of the air disturbance takes place at the

38

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200

Figure 29.5. Four bobs at center of 1 swing(half amplitude). Front views: (a) largecylinder, (b) small cylinder, (c) large sphere,(d) small sphere.

(a) (b)

(c) (d)

equator or just above it. The upward velocity of the warm smokecolumns probably causes the air disturbance to take place above thesphere’s equator.

You may think that when the pendulum reverses direction, it swingsback into its own wake. But in reality, that does not happen. At least itmostly does not happen. The smoke columns show that the air behindthe bob re-forms very quickly back into smooth and relatively calm air.The time taken by the pendulum to reverse direction at the end ofswing allows the previously disturbed air enough time to smooth outinto a relatively calm condition, so that the bob in its reverse swing


201

Figure 29.6. Four bobs at center of 1 swing(half amplitude). Side views: (a) large cylinder,(b) small cylinder, (c) large sphere, (d) smallsphere.

(a) (b)

(c) (d)

again moves through relatively calm air. Note that “relatively calm”does not mean “dead calm.”

Even with the pendulum stopped, the smoke columns do not alwaysrise straight up. The warm smoke columns rise vertically at 5–10 in./s.At random intervals they will change direction and rise at an angle fora minute or so, as shown in Figure 29.2(a), before resuming theirstraight upward path again. This indicates that there is a sizable amountof very slow (minutes) air movement in the clock case, probably due tosmall temperature gradients.

The photos show that large bobs disturb more air than small bobs,and that cylindrical bobs disturb more air than spherical ones of equalvolume. Nothing new here. The first is common sense, and the secondhas been shown before [1].

In sum, the photos provide general information on airflow aroundthe pendulum bob, but not much specific information that can be usedto reduce the air drag and the pendulum’s energy losses. Changingfrom a cylindrical to a spherical bob shape will reduce the energy lossesand improve the pendulum’s Q by 6–20% (see Chapter 26). But it takesabout twice as much material (doubling the cost) and about four timesas much effort to make a spherical bob as it does to make a cylindricalone. One has to decide whether the 6–20% improvement is worth theextra time and money.

The best solution is to operate the clock in a vacuum chamber, andthis has been done in the past. But vacuum chambers have problems oftheir own. First is long-term air leakage into the vacuum chamber,which raises the vacuum pressure and changes the clock rate. The clockrate is usually adjusted by changing the vacuum pressure up and down.A secondary problem at higher vacuum levels is that friction levelsincrease and weird frictional effects show up. I have been told that twopieces of metal touching each other in a high vacuum will tend to sticktogether. Oil and grease can be used sparingly at low vacuum levels, butnot at all at higher vacuum levels.

Appendix

Photography

The white smoke from the incense cones is very thin and “ghost like.”And the bob’s surface must be dark if the thin white smoke is to be seenagainst the bob’s surface. In addition, the scattering of light by small par-ticles (smoke) is strongest in the ongoing direction of the illuminatinglight, and is down 1000 to 1 at 90 to the illuminating light. Unfortunately,the most convenient camera position is at 90 to the illuminating light. In sum, it is a low light level situation.


202

The pendulum is illuminated from both directly overhead and frombelow. The illumination from below is at a 45 angle, from a light sourcelocated below the camera. The plastic “clock case” walls generateunwanted light reflections, most of which are eliminated by the appro-priate use of matte black cardboard and the hanging of black cloths.A Polaroid filter might be useful here in eliminating these unwantedreflections, but it was not tried.

Both the white smoke and the dark bob can be picked up by a 35 mmcamera with an f/1.4 lens operated wide open, with and smokeexposure times, and using Kodak T-Max ASA 400 film operated at ASA800 speed and developed at one-third more development time than nor-mal. The overall picture quality is poor, but the smoke and the bob arevisible in the resulting photographs.

The camera is deliberately located 5 in. below the bob’s bottom edge,so that the airflow on the bob’s bottom surface will show up on thecamera’s film. The camera is not tilted upward. Instead, the camera’sfilm plane is kept vertical so that everything in the pendulum’s verticalplane is rendered in a true recti-linear format on the film, to make it eas-ier to figure out what the airflow is doing.

Pendulum

In the first try at observing the pendulum’s air movement, a metal boband a metal pendulum rod were used, and the smoke coated them witha gooey coating that took effort to remove. This time I made the pen-dulum out of wood, so it could be thrown away at the end of testing.The pendulum rod is a in. diameter wooden dowel with a 1-ft piece of-20 threaded metal rod at the bottom of it.

The four wooden bobs were made from oak tree limbs, provided freefrom a friend’s fireplace woodpile. The bobs were painted a matte darkred (antirust primer paint). The suspension spring is a soldered metal unitwith two beryllium copper springs that had not been used in some time.

With plastic clock case walls, I have previously observed electrostaticforces on the pendulum changing the clock rate by up to 100 s/day.Antistatic sprays will reduce (eliminate?) this, but their effect is gener-ally not permanent. The antistatic spray I used lasts several months.One company has now come out with a supposedly permanent anti-static spray (ACL, Inc., Elk Grove Village, Illinois, USA).

Incense cones

Three incense cones were used in the front view photos, with the coneslocated at the 4, 0 (center of swing) and 4 swing angle positions. Allthree cones are located about in. in front of the pendulum rod’s center-line, so that their rising smoke plumes will pass by the temperature

58

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14

1125 s1

60


203

compensating sleeve (0.5 in. diameter 3.5 in. long) located below thebob. Only two incense cones were used in the side view photos, with bothcones located at the center of swing (0). The two cones are positionedabout in. directly in front of and in back of the pendulum rod’s center-line, again so their rising smoke plumes will pass by the temperature compensator below the bob. The cones can be seen at the bottom of thephotographs.

For some reason different incense perfumes generate different amountsof white smoke. The Patchouli perfume worked well. The 0.5 1.5 in.incense cones are commonly available in knickknack stores.

As a more expensive alternative to using white smoke, Sage Action Co.in Ithaca, New York, USA makes bubble generating equipment specifi-cally for observing airflow. The bubbles have neutral buoyancy in air,and will stay in the bob area instead of continuously rising vertically likethe warm white smoke does. The bubbles are 0.04–0.08 in. in diameterand last about 2 min, when another batch of bubbles can be releasedinto the area. The bubbles can also be released continuously. The bubblegenerating equipment can be leased for 2 months for $577. This equip-ment was not used because of its cost.

Reference1. D. Bateman. “Is your bob in better shape?,” Clocks II ( June 1988), 34–7.

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chapter 30

Time error due to air pressurevariations

Depending on the geographic location, air pressure variations can cause atleast 2.5 s of time error in a pendulum clock.

It is well known that the rate of a pendulum clock is affected by air pres-sure. As the air pressure increases, the clock slows down, and vice versa.The basic cause is that the pendulum floats in a sea of air, and when thedensity of the air changes, the effective weight of the pendulum changesby a small but significant amount. A pendulum’s sensitivity to air pressuredepends on the bob’s shape and density, and is in the range of 0.2–0.4 s/day/in. of mercury. My bob’s shape has a relatively low drag, beingan ellipse of revolution 7 in. long by 3.5 in. in diameter. Its long axis is hor-izontal and in the plane of swing. The bob is made of brass, has a densityof 7.36 g/cm3, and weighs 13.3 lb. The pendulum has a 2 s period, and itspressure sensitivity has been measured as 0.26 s/day/in. Hg (10%).

A clock is normally set to run true over some length of time, mean-ing a nominally zero time error is obtained at the average air pressureduring that time period. So the relevant question is—what are theeffects of plus and minus variations from that average air pressure?

A local meteorologist says that there are big airflows between theearth’s northern and southern hemispheres, and also from over landmasses to over water masses (and vice versa). The average air pressureover land is highest in winter and lowest in summer, but the pressurevariations over a short 2–5-day interval at any location are much biggerthan the average winter-to-summer difference, about 5–10 times bigger.

Hourly pressure readings over a 3–4-year interval at a given locationare available on a 3.5 in. computer disk from the U.S. Weather Bureau’sClimate Centers for about $30. Four years of hourly readings amountto 35,064 data points.

The clock time error is the product of air pressure times time. Morespecifically, it is the integral of air pressure with respect to time. So thetime error T at time t m is

Ttm K24

tm

t0(P P0) dt C,

I want to acknowledge the help of my sonNeal Matthys, who processed the pressuredata on a Sun workstation computer. Thehelpful comments of meteorologist BruceWatson are also acknowledged.

where:

T time error in secondsK pressure sensitivity coefficient 0.26 s/day/in. HgP local pressure in inches of HgP0 average local pressure in inches of Hgm a specific time in hoursC constant of integrationt time in hours.

This is easy to do with a computer, as

The constant of integration is the time error at the start of integra-tion (t 0), and is not known. An artificial value is assumed for C,selected so that the average time error is zero over the 4-year time span.The value for the average pressure P over the 4-year data span is takenas the average of the 35,064 pressure readings on the data disk obtained.The average pressure over another or a longer time span will mostlikely have a slightly different value, but using the average of the dataset removes any excess slope out of the time error curve.

Figure 30.1 shows 4 years of pressure data for Minneapolis, Minnesotaat the clock’s altitude (1991–4 inclusive). Figure 30.1 also shows the result-ing clock time error, obtained by integrating the pressure variations fromthe average pressure and multiplying by the sensitivity coefficient K.

Notice in Figure 30.1’s time error curve that the maximum error is4 s peak-to-peak and that it is cyclic, repeating its pattern at 1-yearintervals. The amplitude and shape of the time error curve vary some

TtmK24

tm

t0(PP0)C.


206

Figure 30.1. Air pressure variations and the resulting time error at Minneapolis,Minnesota for 4 years, 1991–4.

30.5

Pressure

Time error

30

29.5

29

28.5

27.5

26.5

0 0.5 1.5 2.5Time (year)

3.51 2 3 4

27

28

+1.5C

lock

tim

e er

ror

(s)

Pre

ssu

re (

in.H

g)

+0.5

–0.5

–1.5

–2.5

+1

0

–1

–2

from year to year, but the basic pattern repeats every year. Figure 30.2shows 1 year’s worth (1991) of the same data on an expanded timescale, to better show the seasonal changes during a 1-year time period.And Figure 30.3 shows the first 3 months of the same 1991 data on aneven more expanded time scale, to better show the short-term vari-ations in barometric pressure.

Figure 30.3 shows that the period of a pressure cycle varies in lengthfrom 2 to 5 days, with the average length being about 3 days. The pres-sure cycles are rough sine waves, which convert into cosine waves whenintegrated into time error. The integration gives a 90 phase shift to thewaveform. Because of this phase shift, it is much easier to see any pres-sure effects in a clock’s performance record if the integrated time errorcurve is plotted alongside the clock’s error vs time curve, instead of

chapter 30 | Time error and air pressure variations

207

Figure 30.2. Air pressure variations and theresulting time error at Minneapolis,Minnesota for 1 year, 1991.

30.5

29.5

28.5

27.5

26.5

28

27

30

29

10 2 3 4 5 6 7 8 9 10 11 12

+1.5

+0.5

–0.5

–1.5

–1

–2

–2.5

+1

0

Pressure

Time (months)

Time errorP

ress

ure

(in

.Hg)

Clo

ck t

ime

erro

r (s

)

Figure 30.3. Air pressure variations and theresulting time error at Minneapolis,Minnesota for the first 3 months of 1991.

30.5

30

29.5

29

28.5

28

27.5

27

26.5

+1.5

+1

+0.5

0

–0.5

–1

–1.5

–2

–2.5

0 10 20 30 40 50 60 70 80 90

Pressure

Time error

Time (days)

Pre

ssu

re (

in.H

g)

Clo

ck t

ime

erro

r (s

)

plotting the pressure curve alongside the clock’s error vs time curve.Woodward has mentioned this earlier, as well [1, 2].

To get any sort of accuracy, one should have at least 9 or 10 data pointsacross a sine wave of pressure. For a wavelength of 2 days, this meansa data point at least every 5 h or so. The curves in Figures 30.1–30.3 arebased on barometric pressure readings taken every hour.

Bigelow has also published some pendulum time error curves due tothe variations in air pressure [3]. Figure 30.4 shows his time error curve


208

Figure 30.4. Clock time error due to airpressure variations at Albany, New York for 2 years, 1989 and 1990 (after Bigelow).

– 0

– 4

– 8

– 12

– 16

– 200 100 200 300 400 500 600 700

Days

Err

or (

s)

Figure 30.5. Clock time error due to airpressure variations at Malvern, England for 2 years, mid-1983 to mid-1985 (after Bigelow).

0

–2

–4

–6

–8

–120 100 200 300 400 500 600 700

Days

Err

or (

s)

for Albany, New York over a 2-year interval (1989 and 1990), andFigure 30.5 shows his time error curve for Malvern, England over a 2-year interval (mid-1983 to mid-1985). His data are quite different fromthe Minneapolis data. Using a somewhat larger sensitivity factor of0.4 s/day/in. Hg, he found much larger time errors of 19 s peak-to-peakat Albany, and 10 s peak-to-peak at Malvern. In addition, his time errorplot at Albany shows essentially no 1-year cycle, but instead exhibits apredominant cycle length equal to or greater than 2 years, the max-imum length of the time error data presented. His time error plot atMalvern does show some cycling at a 1-year interval, but the predom-inant repeat cycle interval is equal to or greater than 2 years, the max-imum length of the time error data presented.

Conclusions

Two conclusions can be drawn from the above data. First, a clock’s timeerror varies considerably with location. It seems to matter a great dealwhether the clock is located near the center of a continent, near theedge of a continent, or on an island. In fact, location appears to be thesingle most important factor in determining the effect of air pressureon a clock’s time error. Minneapolis, Minnesota is only 400 miles fromthe center of the North American continent, which is located in thenorthern part of the neighboring state of North Dakota. InMinneapolis, the time error has a relatively low amplitude of 4.5 s peak-to-peak, and its cyclic period is 1 year. At Albany, New York, which isonly 200 miles inland from the edge of the North American continent,the time error is much larger at 12 s peak-to-peak, and its cyclic periodis 2 years or longer. And at Malvern, England, which is located onan island, the time error has an intermediate amplitude of 6.5 s peak-to-peak, and its cyclic period is 2 years or longer. (For an apples-to-apples comparison here, Bigelow’s pressure sensitivity coefficienthas been changed to the same value used in the Minneapolis data,i.e. 0.26 s/day/in. Hg.)

The second conclusion is that the predominant effect of air pressureis long-term time error, not short term, as any effects of 1 year or morein duration are considered long term. This is shown in Figures 30.1and 30.2. And as Figure 30.1 also shows, the time error repeats onlyapproximately from one cycle to the next. So although most of the 4 s(1.5 s, 2.5 s?) peak-to-peak time error accumulated during a year atMinneapolis, for instance, would cancel out at the end of a year, about0.5 s or so of the time error may remain, to be canceled out (or added to)in the following year. The variability of this is due to the variability ofthe weather, of course.

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Accuracy

A few words about the accuracy of the U.S. Weather Bureau’s data maybe relevant here. Although the barometric pressure is recorded to a res-olution of 0.01 in. Hg, which is very good (that is 1 part in 2900 on bothscale factor and resolution), it is at best mediocre for calculating clocktime errors. If the Weather Bureau’s pressure sensor were to shift its nullor its scale factor by the very small amount of its resolution (0.01 in. Hg),then the indicated clock time error at the end of a year’s cycle would be

Such a large time error arising from such a small pressure error! Thissays we should not try to read too much truth into any answers calcu-lated from the pressure data.

In addition, the pressure data is initially recorded at the local weatherstation to a resolution of 0.01 in. Hg. But farther down the line, the datagets “massaged” into a better resolution of 0.005 in. Hg, and some of thepressure readings get changed up or down by up to 0.02 in. Hg beforebeing recorded in the official data book of the Weather Bureau. Therationale behind this is not known to the author, but an initial pressureresolution of 0.01 in. Hg cannot truthfully be “improved” to 0.005 in. Hg.

The Weather Bureau tries to maintain the accuracy of its instru-ments. Currently, they are replacing the temperature sensors in theirautomated weather stations in the state of Minnesota, because their cal-ibrations have drifted off by up to 3 F. I have no idea of what is donewith the corrupted data, but you certainly cannot go back in time andre-take the data. The reason for mentioning this is that the same type ofmaintenance problem undoubtedly occurs with their pressure sensors.

References1. P. Woodward. “Stability analysis,” Part 3, Hor. J. (March 1987), 15–16.2. P. Woodward. “Analysis of performance records,” Part 2, Hor. J. (May

1990), 370–1, 388.3. J. Bigelow. “Barometric pressure changes and pendulum clock error,” Hor.

J. (August 1992), 62–4.

0.26 s/day1 in. Hg (0.10 in. Hg) (365 days) 2 1.9 s.


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chapter 31

Effect of the clock case walls on a pendulum

What effect do the walls of a clock case have on a pendulum? I couldnot find anything published on the issue, so I decided to find out byactual experiment. A test setup was built, using a pendulum with a 2 speriod. The pendulum was at the center and surrounded by four adjust-able walls, walls that could be moved in increments closer to (or fartheraway from) the pendulum. The four walls form a rectangle centeredabout the pendulum, with the walls parallel and perpendicular to theplane of swing. The basic experimental approach was to measure thependulum’s clock rate and drive force, with the walls set at different dis-tances from the pendulum. All measurements were made at atmosphericpressure.

The walls affect the pendulum via the air between the walls and thependulum. The bob is the biggest part of a pendulum, so its size andpossibly its shape are important in determining the walls’ effect on thependulum. To determine the effect of bob size and shape, two differentbob sizes (5.3 lb and 19 lb) and two different bob shapes (a sphere, anda cylinder with its main axis along the pendulum rod) were tested. Thependulum is compensated for temperature, but not for atmosphericpressure variations. The pendulum rod is a in. diameter invar rod.Information on the three bobs used is given in Table 31.1.

The test setup

The arrangement of the test setup was dictated by the need foradjustable walls and for moving them up close to the pendulum. Thetest setup was located in my garage, with the pendulum hung from oneof the roof rafters and swinging in the direction of the rafter for max-imum rigidity in suspension. The space around the pendulum is clearfor 4.5–6 ft radially in all directions. The roof rafter suspension, beingsomewhat flexible, introduced an additional suspension loss that thependulum drive had to provide. This extra suspension loss obscuredtwo pendulum drive force relationships, as will be discussed later.

38

211

Although the extra suspension loss adds to the drive force, the losssubtracts out of and does not affect the walls’ clock slowdown rate whenthe loss is constant. But unfortunately, the stiffness of the garage walland roof structure varies with the outside temperature, particularly withthe sun angle and the cloud cover. The variation in outside temperaturein the time interval between a measurement with walls and a meas-urement without walls made it difficult to get consistent sets of data.Any change in the rafter’s suspension stiffness with outside temperatureduring this interval does not subtract out of the data, and this becamethe largest source of error. The temperature inside the garage is heldreasonably constant with a thermostatically controlled gas furnace.

The suspension rafter is slightly over 8 ft above the concrete floor,and the pendulum is a little over 3 ft long. A 4 4 ft plywood platform2 ft high was built underneath the pendulum, raising the effective floorlevel to 2.7 ft below the pendulum bob.

The walls are not really adjustable, but instead consist of a collectionof walls of different widths that can be bolted together to form any wallspacing desired. The walls are of wood, sheets of in. thick orientedstrand board 6 ft high and of assorted widths from 4 to 36 in. No flex-ing of the walls was observed with pendulum motion. Even so, thewidest walls (36 in.) were reinforced with two external in. angle alu-minum bars (see Figure 31.1), more so to obtain wall straightness thanwall stiffness. The narrower walls (7 in. wide or less) have viewing portsin them, to make sure the pendulum inside is centered between the

34

14


212

Table 31.1. Bob dimensions

Type Material L (in.) D (in.) Wt (lb) Cross-sectional Density

areaa (in.) (lb/in.)

Small Brass 6.16 1.98 5.3 12.2 0.308cylinder

Large Bronze 6.00 3.70 19.0 22.2 0.306cylinder

Large Brass — 4.89 18.4 18.8 0.313

sphere

Notea In the plane perpendicular to the direction of swing.

L

D

L

D

D

walls. During tests the viewing ports are plugged, with the plugs beingflush with the inside wall surface.

The 6 ft high walls stand on the raised plywood platform 2.7 ft belowthe pendulum bob, and extend upward to 2 in. short of the pendulum’ssuspension spring. The walls’ outer surface is similar to rough sawnwood. The walls’ inner surface finish next to the pendulum is abouthalfway between rough sawn and sanded. There is no “ceiling” abovethe walls. The “ceiling” is open to the rest of the garage interior.

The test setup is shown in Figure 31.1, with the pendulum swinging left-right in the photograph. The four walls shown in the photo are spacedapart 7 35.75 in. To make the inside wall surfaces free of disruptions andas smooth as possible, the walls are held together with external cornerbrackets, using flat head screws flush mounted on the walls’ inside sur-faces. Any air gaps at the wall corners are covered over with tape.

The pendulum is driven electromagnetically with a continuous sinewave drive (see Chapter 33), using two movable rod-type magnetsattached to the bottom of the pendulum rod, and two fixed electricalcoils. The magnets are mounted horizontally, and extend halfwaythrough the coils, with each magnet intruding into one coil. The twoelectrical coils are attached to the top of a rigid 1.9 in. diameter pipewith a heavy three-point mounting base resting on the garage floorimmediately below the pendulum. The pipe passes through a hole in

Figure 31.1. Adjustable wall test setup,showing perpendicular wall-to-wall spacingsof 7 35.75 in.2 The pendulum hanginginside the walls does not show in thisphotograph.

chapter 31 | Effect of clock case walls

213

the plywood platform without touching the platform itself. The coiland magnet arrangement is shown in Figure 31.2. The coils shown areone of three interchangeable pairs of coils used in the setup. The use ofa sine wave drive for the testing is not significant, other than measuringthe electrical drive coil current gives a measure of the drive forcerequired to maintain the pendulum at a constant amplitude. The sinewave drive holds the pendulum at a constant sinusoidal velocity. To theextent that the pendulum’s length holds constant, the pendulum’sswing amplitude will also be held constant.

At the start of testing, the walls were made of acrylic plastic. Butacrylic has an extremely high electrical surface resistance, and will holdan electrostatic charge for many days. The electrostatic charge on theacrylic walls made the pendulum’s clock rate erratic, and also changedit by up to 100 s/day. The walls were changed to wood (oriented strandboard, a mixture of glue and 2–4 in. long wood shavings). Wood willnot hold an electrostatic charge, and the pendulum then settled downto a constant rate. Testing could then actually begin. Oriented strandboard is cheaper, flatter, and less warped than regular plywood. Thelesson here is that anyone using a plastic (like acrylic) with a highelectrical surface resistance for a clock case needs to put an electricallyconductive coating on the case’s inside surfaces, and ground it to avoidelectrostatic charge effects on the pendulum.

Test data

The test data on wall spacing was taken independently on each oppos-ing wall pair. This was done by leaving one pair of opposing walls farapart at 35.75 in. spacing, while moving the other pair of opposing wallsinward in increments toward the pendulum. Then that pair of opposingwalls was moved far apart to 35.75 in. spacing, while the first pair ofopposing walls was moved inward in increments toward the pendulum.The pendulum is always at the middle between the opposing walls. Ateach wall spacing, the pendulum’s drive current is measured, and itsclock rate measured over a 1 h interval, comparing it to the WWVradio time signal, which is the official time standard in the UnitedStates. To sort out the effect of just the walls on the pendulum,the clock rate is measured without walls over a 1 h interval just beforeor after the measurement with walls. The effect of the walls, of course,is the difference in clock rate with and without the walls. All of this datawas taken at the same swing half angle of 0.89, and is shown inFigures 31.3 and 31.4 for all three bobs.

Figure 31.4 shows a wall slowdown effect of 8.6 s/day with a 4 in.wall-to-wall spacing. With the cylindrical bob’s 3.70 in. diameter in themiddle of the 4 in. wall-to-wall spacing, the 8.6 s/day slowdown comes

Figure 31.2. Showing the velocity sensing andmagnetic drive coils, located below the bob.


214

Figure 31.3. Pendulum slowdown vs wallspacing in the plane of swing.

Figure 31.4. Pendulum slowdown vs wallspacing perpendicular to the plane of swing.


215

Large sphere

Large cyl.

Sm. cyl.

pk–pk swing

pk–pk swing

+ pk–pkswing

bob dia. +

bob dia. +

bob dia.

6.12 in.

4.93 in.

3.23

Eff

ect

of w

alls

(s/

day) –4

–2

00 5 10

Wall-to-wall spacing (in.) (in plane of swing)15 20 25

0 5 10 15 20 25Wall-to-wall spacing (in.) (⊥ to plane of swing)

Eff

ect

of w

alls

(s/

day)

–8

–6

–4

–2

0

Lg. cyl.

Lg. spherebob dia.

bob dia.3.70 in.

4.89 in.

1.98

Sm.cyl.bobdia.

from the 0.15 in. clearance distance on each side of the bob. Figures 31.5and 31.6 show how the pendulum’s drive current changed with thedifferent wall spacings, again for the same swing half angle of 0.89. Inall of the figures, data taken with the small cylindrical bob is marked .,data taken with the large cylindrical bob is marked , and that with thelarge spherical bob is marked . Properly scaled bob diameters andswing amplitudes are shown on the graphs for reference, so that theradial bob-to-wall clearance distances can be determined easily.

Figure 31.5. Pendulum drive current vs wallspacing in the plane of swing.

0 5 10 15 20 25Wall-to-wall spacing (in.) (in plane of swing)

Dri

ve c

urr

ent

(mA

) pk

–pk

2

1

0

Large cyl.bob dia. +

pk–pk swing

Large spherebob dia. +

pk–pk swing

Sm. cyl.bob dia.+ pk–pk

swing

6.12 in.

3.23

4.93 in.

Assuming the plane of swing is left-right, the front-back walls can bemoved inward until they actually touch the pendulum bob. The left-right walls can be moved in until they touch the two electrical coils(see Figure 31.2), which means a minimum wall to wall spacing of 7 in.in the plane of swing.

Figure 31.6. Pendulum drive current vs wallspacing perpendicular to the plane of swing.


216

0 5 10 15 20 25Wall-to-wall spacing (in.) (⊥ to plane of swing)

Dri

ve c

urr

ent

(mA

) pk

–pk

4

3

2

1

0

Lg. cyl.

Lg. sphere

bob dia.

bob dia.

3.70 in.

4.89 in.

Sm.cyl.bobdia.1.98

Table 31.2. Individual and combined effect of perpendicular wall spacings on clock rate,

using a large 19 lb cylindrical bronze bob. Because of wall overlap at the corners, a in. is

lost on one of the wall spacings in each setup, as indicated

Wall spacings (in.) Individual measured wall Combined measured

effects, wall spacing (s/day) wall effect (s/day)

6 in. front-back 1.37 2.867 in. left-right 1.39

7 in. front-back 0.93 1.928 in. left-right 1.03

9 in. front-back 0.52a 1.03

10 in. left-right 0.50

Bob motion (plane of swing)

Notea Interpolated from curve.

14

5.75

7

6.75

8

8.75

10

The combined effect of simultaneously moving the walls inward inboth the front-back and left-right directions was also measured, forthree cases where the effects of the individual wall pairs were largeenough for the combined effects not to get lost in measurement error.Again the pendulum’s clock rate without walls was measured eitherimmediately before or after the measurement with walls. The differ-ence in clock rate with and without walls is the effect of the walls onthe pendulum. In each case the clock rate was measured over a 1 h inter-val, and at the same swing half angle of 0.89. This combined-wall-pairsdata is shown in Table 31.2.

The effect of the walls was also measured at various pendulum swingangles. This was done with just the large cylindrical bob and one wallspacing: 6 in. front-back (perpendicular to the plane of swing) by 35.75 in.left-right (in the plane of swing). This data is shown in Figure 31.7. Thependulum’s drive force changes with the angle of swing, and this isshown in Figure 31.8 under the same conditions as the data in Figure 31.7.

Results and conclusions

1. The walls of a clock case affect both the pendulum’s clock rate andthe pendulum’s drive force. In all cases, the walls made thependulum run slower and increased the drive force.

2. The combined effect of the front-back and left-right wall spacingsis the sum of their individual effects. This was found to be true forthree different sets of wall spacings, as shown in Table 31.2.


217

Figure 31.7. Pendulum slowdown vs swingangle at a fixed wall spacing.

0 0.5 1.0 1.5 2.0Swing half angle (deg)

Eff

ect

of w

alls

(s/

day)

0

–2

–4

Figure 31.8. Pendulum drive current vs swingangle at a fixed wall spacing.

0 0.5 1.0 1.5 2.0Swing half angle (deg)

Dri

ve c

urr

ent

(mA

) pk

–pk

0

1

2

With walls

No walls

If a 7 in. front-back wall spacing independently slowed thependulum by 0.93 s/day, and an 8 in. left-right wall spacingindependently slowed the pendulum by 1.03 s/day, then combiningthese clock case walls with a 7 in. front-back spacing and an 8 in.left-right spacing will cause a total clock slowdown of the sum ofthese two: 1.96 s/day (1.92 s/day measured).

In contrast, there was no apparent pattern to the change in thependulum’s drive force when the walls in both perpendiculardirections were moved inward together. Instead, the average drive force increased 8–24% when the walls in both perpendiculardirections were moved inward, as compared to when the walls in only one direction were moved inward. If there was a patternhere, I believe the extra support losses in the rafter suspensionobscured it.

3. The walls’ slowdown effect on the pendulum is independent of theangle of swing. This is shown in Figure 31.7. The pendulum’s driveforce increased as the swing angle increased, but the walls’ effecton the pendulum’s clock rate did not change.

4. When the swing angle was increased, the pendulum’s drive forceincreased approximately as the 1.4 power of the swing’s half angle.This is an odd relationship, believed due to the extra losses in thegarage rafter suspension adding to the normal pendulum losses.Because of this, I believe that this particular drive force relationship does not apply to pendulums in general.

5. Surprisingly, the small bob has a bigger wall slowdown effect thanthe two large bobs, at least at wall-to-wall spacings of 12 in. ormore. This was verified by repeated testing of all three bobs at 14 35.75 in.2 wall-to-wall spacings, where the small bob’s wallslowdown effect averaged 0.5 s/day higher than that of either ofthe two large bobs. This was true whether the 14 in. wall widthwas parallel or perpendicular to the plane of swing.

This is not new information. In 1832, Baily’s experiments [1]showed among other things that the air drag has a bigger effect onsmall bobs than on large ones. And in 1850, in a theoreticalanalysis, Stokes [2] also found that the effect of air drag is greateron small bobs than on large ones.

6. If the walls are more than 2–5 in. away but less than 9–10 in. awayfrom the bob’s outer surface, the total slowdown effect (the sum ofwhat is shown in Figures 31.3 and 31.4) in this region is relativelyconstant at 0.5–1.1 s/day, and does not change much when the wallspacing changes.

Two things should be kept in mind in this region. First, the totalclock rate measurement accuracy is estimated at 0.4 s/day. Thisincludes a time measurement error of 0.1 s/day, based onmeasuring the pendulum’s clock time against WWV to 0.001 s


218

accuracy at the beginning and end of a 1 h interval, and measuringit again a second time without the walls around the pendulum.And stiffness variations in the rafter suspension are included as0.3 s/day. Second, the wall effect at these wall spacings wasobtained by subtracting the clock rate with the so-called “no-walls,” that is, with the walls at 4.5–6 ft (9–12 ft wall spacing)away from the pendulum. So, in reality, the wall effect is actuallynot constant in this region, but keeps on slowly decreasing out to a much wider wall spacing.

7. The dividing line between a large and a small slowdown effect (on a given bob) occurs where the clock case walls are 2–5 in.away from the outer surface of the bob. From Figures 31.3 and31.4, that is 2–2.5 in. away for both the small 1.98 in. diametercylindrical bob and the large 4.89 in. diameter spherical bob, and2.5–5 in. away for the large 3.70 in. diameter cylindrical bob. Ifthe walls are closer than that, the walls’ slowdown effect on thependulum’s clock rate increases sharply. At greater distances, theslowdown effect is low and “relatively” constant. For a cylindricalbob at the dividing line between, the clock case walls are about 1.3 bob diameters away from the outer surface of the bob at allpositions in its swing. For a spherical bob at its dividing linebetween large and small wall effects, the case walls are about half the bob diameter away from the outer surface of the bob at all positions in its swing. These minimum wall spacings would be a good empirical recommendation for a clock case for an accurate pendulum.

Wall stability

Wooden clock case walls expand and contract much more with humid-ity than with temperature. Since the walls’ position affects the clockrate, how much does the walls’ expansion and contraction affect theclock rate? Getting the answer to this question was the real reasonbehind finding the walls’ effect on the pendulum.

Let us pick three cases, using the large cylindrical bob as an example,where the wall-to-bob clearance distances are 0.5, 1.0, and 1.5 times thebob diameter. Let us further assume that the walls contract 2% acrossthe wood grain, corresponding to a 24% change in relative humidity(i.e. 64–40% RH). At a wall-to-bob clearance of half the bob diameter,the slope of the slowdown curves in Figures 31.3 and 31.4 for the largecylindrical bob is (0.25 0.23) 0.48 s/day/in., at wall-to-wall spacingsof 8.6 and 7.4 in., respectively. A 2% contraction in the average wallspacing of 8.0 in. then changes the clock rate by 0.02 8.0 0.48

0.076 s/day, or 28 s/year. Similarly at wall-to-bob clearances of


219

1.0 and 1.5 bob diameters, a 2% contraction in the wall-to-wall spacingschanges the clock rate by 0.030 s/day (11 s/year) and 0.020 s/day(7.3 s/year), respectively.

If the clock case walls warp out-of-flat or bow inward by 2% of theirwidth, the same change in clock rate as above can occur. Two percentof a wall 8–14 in. wide is 0.16–0.28 in.

These changes in clock rate with humidity are rather large for anaccurate clock. For such a clock one might want to consider making theclock case out of a more stable material such as plastic, glass and metal,or even plywood. The length change with humidity in wood alongthe grain is only th to th of that across the grain. The effect ofhumidity in plywood is almost halved compared to non-plywood,because with the 90 orientation of the alternate ply layers in plywood,the low expansion along the grain in one ply reduces the high expansionacross the grain in the next ply.

References1. F. Baily. “On the correction of a pendulum for the reduction to a vac-

uum,” Philos. Trans. Royal Soc. 122(part 2) (1832), 399–492.2. G. Stokes. “On the effect of the internal friction of fluids on the motion of

pendulums,” Trans. Cambridge Philosophical Soc. 9(part 2) (1856), 8–106.

140

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220

part v

Electronics

223

chapter 32

An electronically driven pendulum

Some features of an electromagnetically driven pendulum clock are described.

I believe that an electromagnetically driven pendulum is moreaccurate than a mechanically (escapement) driven pendulum. This isbecause a pendulum is disturbed less by an electromagnetic drive pulsethan by hitting and dragging a pallet across an escape wheel’s tooth.This is empirically based on Q—the less you disturb a pendulum, themore accurate it will be. I also believe that a short drive pulse at thecenter of swing is superior to the continuous sine wave drive approach.This is due to the difficulty in avoiding spurious electrical drive currentsat the ends of swing, where unwanted low level electrical currents ina continuous sine wave drive can cause significant time errors over longtime intervals. In this clock, the pendulum is electronically driven bya short current pulse in each drive coil at the center of swing.

The pendulum’s mechanical layout is shown in Figure 32.1. The twolarge drive coils in Figure 32.1 are deliberately spread far apart to makeit easy to remove or replace the pendulum. I wanted no part of anenclosed hard-to-take-apart magnet structure, with close fussy coilclearances and possible dragging of the coil on the magnet structure(inside on the back, of course, where you cannot see it).

To get the deliciously large coil clearances that are shown in Figure 32.1,one has to pay. The price is that the drive coils’ magnetic efficiency is downby about 2–3 orders of magnitude. This is compensated for by usinga larger and more powerful current pulse into the pendulum’s drive coils.This was an easy tradeoff to make.

The pendulum has a 2 s period and a in. diameter polycrystallinequartz rod. The in. rod diameter is a good size for a 2 s periodpendulum. It makes for a strong stiff rod that has stood up to much hand-ling without breaking. Currently the bob is a 19 lb vertically orientedcylinder. Both the bob and the temperature compensator are made oftype 642 aluminum silicon bronze. Two alnico 5 magnets, each in. diameter by 1.4 in. long, are buried in the bob, flush beneath the surfacehalfway up on opposite sides. This puts the drive force reasonably close

14

58

58

to the pendulum’s center of mass, so that the pendulum rod’s internalmodes of vibration receive only a minimum amount of excitation fromthe drive force.

The pendulum’s swing amplitude is controlled by two verticallyoriented light beams, located below the bob as shown in Figure 32.1.The two light beams are reference points for the ends of swing. A horizontal metal strip at the bottom of the pendulum swings throughand slightly beyond each light beam. The swing distances beyond thetwo light beams are digitally time averaged, so that the net amplitudesignal is independent of any tipped clock case effects, or of any off-centering of the light beams. When the swing amplitude falls below thedesired level, a current drive pulse is sent into the drive coils, whichslightly increases the swing amplitude. This type of amplitude controlservo is called a bang-bang servo.

The two light beams are amplitude modulated (on and off ) at 100 kHz,making circuit operation independent of the ambient room light. Thelight beams are normally round, in. in diameter. For better accuracy, thelight beams are narrowed to 0.010 in. in the plane of swing by means oftwo 0.010 in. wide slits placed in each light beam (see Chapter 34). Theslits are made from small pieces of a razor blade’s sharp edge, cementedin place just in front of each light source and each light detector.

The drive coils are dual purpose. They drive the pendulum and alsoprovide a velocity readout signal. The velocity signal goes through zerovolts at the ends of swing—plus to minus volts at one end of swing, andminus to plus volts at the other end of swing. The velocity signal goingthrough zero means that the pendulum has stopped moving for aninstant, so going through zero electrically defines the pendulum’s end-of-swing. This signal remains accurate even if the clock case tips or sitsoff at an angle.

18


224

Figure 32.1. Mechanical layout.

2 flat springs (Be Cu)

Quartzrod

Magnet,1 of 2

1 of 2 pendulumdrive coils

1 of 2 verticallite beams

10 in.

Invar tie-rodbetweenlite beams

Bob

4.0

in.

N S N S

The electrical drive pulse into a drive coil overloads the velocity read-out amplifier on each coil by about 300,000 to 1. To avoid burning outthe readout amplifiers (one on each drive coil), 10,000 20 W resistorsare placed in series between the drive coils and the readout amplifiers,and shunting short-to-ground switches are installed between the resis-tors and the readout amplifiers’ input terminals. The inputs to thevelocity amplifiers are shorted to ground during each drive pulse.

The pendulum’s electrical drive pulse is generated as a near-critically-damped discharge pulse from a charged capacitor into a combined resis-tive and inductive load. Two 50 F oil-filled paper dielectric capacitorsare charged up to 300 VDC over a roughly 2 s interval. Each capacitoris discharged through one drive coil in 0.02 s, providing a 1.9 A peakdrive pulse in each drive coil. The drive coils’ inductance and resistanceare selected so that the capacitors’ discharge current pulses into thedrive coils are near-critically damped, with a relatively slow rise and fallwithin the 0.02 s interval. Using the capacitor discharge approach madeit possible to use a much smaller electrical power supply, and reducedthe radiated electrical noise considerably.

The best location for the drive pulse is at the center of the pendu-lum’s swing, where it will have no effect on the pendulum’s timing.The center of swing, of course, is halfway between the two ends ofswing, that is, halfway between the two points where the velocitysignal goes through zero. The center position is found electrically bycounting 100 kHz pulses (from a quartz crystal oscillator) as the pen-dulum swings left to right, from one end of a swing to the other. Thena new count is made as the pendulum swings back from the right endtoward the left. When the new count reaches half of the old count, thecenter of swing has been reached. This is implemented in hardware bycounting up 50 kHz pulses in a digital counter as the pendulum swingsfrom left to right, and then subtracting 100 kHz pulses from the samecounter as the pendulum swings back from right to left. When thecount in the counter drops down and reaches zero, the center of swinghas been reached, and a pendulum drive pulse can be released ifneeded. This approach accurately locates the center of swing within alittle more than 30 s, and the drive pulse is centered on this mark.To cancel any remaining off-center drive errors that might possiblycouple in as timing errors, the circuitry is further arranged so thatconsecutive drive pulses alternately push the pendulum left, then right,left, right, etc.

Variations in atmospheric pressure cause a pendulum’s clock rate tospeed up or slow down a little. In this clock, a silicon pressure sensor isused to measure the atmospheric pressure variations, and the variationsare integrated over time into a T total time error. When the Ttotal time error exceeds the value of one time correction pulse, adrive pulse is applied to the pendulum at one of the ends-of-swing.

chapter 32 | An electronically driven pendulum

225

This time-correcting drive pulse will advance or delay the pendulum inangle and in time by about 0.0014 s. The time value of the pulse (about0.0014 s) is then subtracted from the T total time error. This circuitis described in more detail in Chapter 36.

The electronically driven clock briefly described here is currentlyrunning. Its performance over a long time interval has not yet beenrecorded.


226

227

chapter 33

Sinusoidal drive of a pendulum

In this chapter, four electronic circuits for sinusoidally or semi-sinusoidally driving a pendulum are analyzed, and their pros and cons discussed.

Several writers have commented that a sinusoidal drive would be idealfor a pendulum, but gave no data. A semi-sinusoidal pendulum driveconsisting of a sinusoid with its peaks clipped off was first reported byBush and Jackson in 1960 [1]. They used a pendulum-mounted coilmoving in a fixed magnetic field to generate a sinusoidal velocity signal.The peaks of this sinusoidal signal were then clipped off at a fixed ampli-tude, using a parallel reversed set of two diodes and two batteries. Thepeak-clipped sine wave was then fed to a second pendulum-mounted coilmoving in another fixed magnetic field. The magnetic force generatedby the peak-clipped sine wave current in the second coil drove thependulum and kept it running.

In 1993, Bigelow [2] reported a very simple semi-sinusoidal drive circuitthat had only one op-amp and one coil-and-magnet structure instead oftwo. The one coil both senses and drives the pendulum. The pendulum’sdrive force is a peak-clipped sine wave, like the Bush and Jackson’s. Itis an extremely simple drive system, requiring only a few resistors, one op-amp, two diodes, and two 9 V batteries for a power supply.

Electronic circuits

It is difficult to write about electronic circuit design and servomech-anism characteristics for what I suspect is a largely nonelectronic audi-ence. I will try, but certain technical issues must be covered. If you feelsnowed under, skip over to the conclusions where the issues are sum-marized and discussed from a clock performance standpoint.

I was very interested in Bigelow’s circuit, as it is much simpler thanwhat I am currently using to drive my own pendulum. So I built hiscircuit to try it out. My version of his circuit is shown as Figure 33.1,which is the same as his except that I used two 5.1 V zener diodes foramplitude clipping, and Bigelow used two ordinary diodes. My feed-outof the drive signal to an external clock face and a WWV time com-parator is also different from Bigelow’s, but that does not matter where

the pendulum’s sense and drive circuitry is concerned. In addition,I built two separate coil-and-magnet structures into the test setup, forreasons of mechanical symmetry and because I wanted to do sometesting with separate sensing and drive coils, as Bush and Jackson did.The two coils are connected in series in Figure 33.1, and operate asa single coil. The issues of how many coil windings and how manymagnet structures are used are significant, and will be covered later.

The arrangement of the two coils and magnets is shown in Figure 33.2.The two moving magnets are attached to the pendulum below the bob,and are oriented horizontally in the plane of swing on each side of thependulum rod. The voltage induced in a fixed coil is reasonably propor-tional to the moving magnet’s velocity as long as the end of the magnetstays within the length of the coil. When the end of the magnet goesoutside the coil, the induced voltage drops toward zero. The alnico5 magnets are in. in diameter and 1.3 in. long. The outer end of eachmagnet is at the center of its coil when the pendulum is at rest. Each coilhas a length of 1.2 in. (1.5 in. including the coil form ends), an outer dia-meter of 1.2 in. and an inner clearance diameter of in. Each coil has1400 turns of #28 wire, a resistance of 25 , an inductance of 0.026 H,and a self-resonant frequency of 93 kHz. The maximum pendulum swingangle is determined by the length of the coils. In this case, a coil length of1.2 in. at a radius of 45 in. gives a maximum swing angle of 0.76 halfangle. There is a large in. radial clearance between the magnets and theirencircling coils, as I wanted no chance of one hanging up or dragging onthe other. This is an absolute no-no. Any touching or dragging of one onthe other is easily seen and corrected for in this open-style construction.

Operation of the circuit in Figure 33.1 is as follows. The swinging ofthe pendulum with a 2 s period generates a 0.5 Hz sine wave Ei in themagnet coils, which then shows up as part of the total voltage E3 atsumming point E3. The op-amp increases E3 to a same-phase voltage E0

at point E0, which then puts a current back through the magnet coils

14

1116

316

Figure 33.1. Constant force drive withcombined sense and drive coils.

Figure 33.2. Physical layout of the magnetson the pendulum, below the bob, with theiradjacent fixed-mount coils.


228

Voltagecomparator

Timepulse

+5 V

+5.6 V

–5.6 V

20K

100K

R 2 Am

p. gain

C 1 0.00

68–0.0

33

1N46

25(2

) 5.1V 0

+

–LM311

100E0

E3

Pendulum sense/drivecoils

2 s

100

51K

5.1

K

U

V

1KR6

200K10K

R1

LT 1001

– +

Drive gain

through the series resistor R1. This coil current generates a magneticforce that pushes the pendulum and increases the swing amplitude. Theseries resistor R1 controls the magnitude of the drive current, and hencecontrols the amplitude of swing. Resistors R2 and R6 control the op-amp gain, giving an op-amp output voltage E0 of about 7 V peak, whichis then clipped off by the zener diodes at 5.6 V peak (5.1 V zener volt-age plus a 0.5 V diode drop gives a clipping level of 5.6 V). The zenersprovide an order of magnitude improvement in the constancy of theclamping voltage over that provided by an ordinary diode alone. The5.1 V zener voltage level is picked because it has the lowest temperaturecoefficient of any zener voltage.

The inductance of the drive coil and inductive coupling betweenthe drive and sense coils cause phase shifts which make the op-amposcillate at a high frequency (about 30 kHz). This spurious oscillation iseliminated by capacitor C1, and is discussed further in ParasiticOscillation in the Appendix. Unfortunately capacitor C1 also introducesa 0.0015–0.015 s time delay in the op-amp, which shows up in theop-amp’s output voltage E0 and also in the drive current into the magnetcoils. The end result is that the clipped-peak sine wave of drive currentgoing into the magnet coils is always 0.0015–0.015 s (0.27–2.7) behindthe pendulum’s physical position, as it swings to and fro.

The circuit works. But I was uneasy with it—for several reasons. First,at some settings of the resistor R1, the op-amp goes into self-oscillationdue to the positive-phased feedback around the op-amp through R1. Thiswas expected, but it was still an irritation. Second, I did not understand allthe implications of feeding back the pendulum’s drive signal into thesame winding that the velocity signal came out of. Things were happen-ing as a function of the coils’ resistance, and other things were happeningas a function of the coils’ inductance. I would have felt better if the drivesignal had been fed back into a separate coil winding on the same or a dif-ferent magnet structure than that used for the velocity sensing. If into thesame magnet structure, then you have to worry about transformercoupling between the windings, with the induced voltage being 90 out-of-phase with the drive current, and about what that does to the sensingsignal. Third, too much was happening in too little space. There were notenough adjustments for all of the variables, and the existing adjustmentsinteracted with each other and affected multiple variables. And fourth,there was the time delay between the pendulum’s angular swing positionand its drive current, the significance of which I did not know as yet.

For these reasons I decided to spread the circuit out some, resultingin the circuit in Figure 33.3. It contains three op-amps instead of just theone op-amp in Figure 33.1. Figure 33.3 uses the same two coils andmagnet structure as Figure 33.1, except that the sensing and drivingfunctions are separated, each with its own coil and magnet structure(the Bush and Jackson approach). The drive coil is current driven

chapter 33 | Sinusoidal drive of a pendulum

229

instead of voltage driven, so that the coil’s inductance does not delaythe magnetic drive force. To minimize load loss, the only load on thevelocity sense coil is the input impedance of an op-amp (greater than15 M for the LT1001 op-amp). And there is no capacitor C1, so thependulum’s drive current is exactly in-phase with the velocity signal andthe pendulum’s angular position. The gain control pot R2 has beenpushed into the second op-amp stage, so that the gain of the first op-amp stage is fixed at a constant value. Thus, the first op-amp’soutput signal always gives a true measure of the pendulum’s velocityand amplitude, without worrying about whether the op-amp’s gain hasbeen changed since the last measurement. And last, there are two gaincontrols on the pendulum’s drive current. At a convenient input signallevel, control pot R1 is adjusted (to 30 k) so that the drive current isjust enough to hold the pendulum at an approximately constant ampli-tude. Then control pot R3 is used to adjust the pendulum’s drive force.This is quite helpful, but there is still a strong interaction between gainpots R2 and R3.

When gain pot R3 is reduced to set the pendulum amplitude at a lowerlevel, the velocity signal into the amplifier goes down as well. Then gainpot R2 in the middle op-amp stage has to be increased, to raise the veloc-ity signal going into the peak clipping stage back up, so that the zenerscan still clip off the sine wave peaks at the same voltage level as before.

The strong interaction between gain pots R2 and R3 can be eliminatedby resorting to a variable peak clipping level, as shown in Figure 33.4.I doubt if it is worth the extra parts, since the main application is long-term operation at one fixed amplitude. For testing purposes, however, itwould be useful. The circuits in Figures 33.1, 33.3, and 33.4 were all builtand tested. Figure 33.4 was not built, its advantage being obvious fromlooking at its schematic.

After testing Figure 33.3, two things became clear. First, Figures 33.1,33.3, and 33.4 are not controlling the pendulum to a constant velocity,but instead are providing a constant force drive to the pendulum.This “constant” force drive is in the form of a clipped-peak sine waveof magnetic force being applied to the pendulum. And second, the

Figure 33.3. Constant force drive withseparate sense and drive coils.


230

1N4625(2)

5.1 V

5.1 K5.1 K

Sensegain

Sensecoil

50 K

R2

To voltage comparator

+LT

1001

Drivecoil

Drivegain

100

K10

K

10 KR3

––

LT 1001

3.3

K10

0K

+

+LT

1001–

S

T

R1

U

V

magnitude of the drive force varies significantly with the velocitysignal’s amplitude going into the peak clipper. Figure 33.5 shows twodifferent amplitudes of the velocity sine wave being clipped at the samepeak level. The parallel-lined areas show the extra drive force providedby a larger sine wave input to the peak clipper. One is forced to thereluctant conclusion that a clipped-peak sine wave drive is not a verygood drive.

Let us back off and regroup here. What we are really looking for isa pendulum amplitude control that uses a sine wave drive. A sine wavedrive would give a reasonably close match to the pendulum’s energylosses just as they occur. A clipped-peak sine wave really does not do that.A genuine sine wave drive is needed. In electrical terms, we want to con-trol the amplitude of a carrier frequency, with the carrier’s frequencybeing independently controlled by something else (a pendulum). Thisrequires an analog multiplier circuit. There are a variety of analog multi-plier circuits available, but they are all complex, slightly nonlinear (1–2%at best), somewhat temperature sensitive, and lack any guarantee of long-term stability. But what if we do the multiplying thermally, and incorpo-rate the multiplier and its nonlinearities inside an error correcting servo?

Figure 33.6 is a block diagram of a servo containing a thermal multi-plier for controlling the amplitude of a sine wave carrier. The carrier’sfrequency is independently controlled by a pendulum. The multiplica-tion is accomplished by a thermistor whose resistance changes withtemperature. The thermistor acts as a gain control resistor on an

Figure 33.4. Constant force drive withindependent clip level adjustment.


231

To voltage comparator

U

V

+LT

1001–

+LT

1001–

+LT

1001

Cliplevel

Sensecoil

10 VRef.

10 K

Sensegain

Drivegain

Drivecoil

–

–LT

1001+

10 K

1N645(2)

2K

510

100

K

100

K10

K

10 K

R3

R2

R4

R1

15

+10 VDC

1 M S

T

Figure 33.5. Parallel-lined areas show theadditional pendulum drive force provided by a large input sine wave, compared to a small one.

Fixedclippinglevels

Figure 33.7. Constant velocity servo amplifier with separate coil windings for sense and drive.

op-amp. The thermistor’s resistance is changed by heat radiated froma 1 W heater resistor mounted adjacent to the thermistor.

The servo will linearize and stabilize the multiplication process as longas the servo’s feedback elements are linear, that is, as long as the velocitysensing coil, op-amp A2, and the AC-to-DC converter in Figure 33.6 areall linear and stable. This is a reasonable expectation, except for a smalltemperature coefficient in the velocity sensing coil, which will be ignored.The temperature coefficient comes from the alnico 5 magnets, whosemagnetic strength decreases slowly with increasing temperature at a rateof 0.015%/C. The amplitude of the magnetically sensed velocity signalwill have the same temperature coefficient, which as mentioned abovewill be conveniently ignored.

Figure 33.7 shows the electronics for the velocity control servo. Threethermistor-controlled gain stages are cascaded in series to get enoughamplitude control range. A maximum input of W into the 1 W heaterresistor raises the resistor’s temperature about 35 F above ambient, and

14

Figure 33.6. Block diagram of a velocity (or amplitude) control servo containing a thermally operated multiplier.


232

Pendulum

Drivecoil Sense

coilBob

N NS S

MagnetsThermistorFixedresistor

Heaterresistor

AmpA2

AmpA1

Pendulum velocity or amplitudewanted(VDC)

AmpHeat

+

–

VACVDC AC-to-DC

converter

+–

To voltage comparator (time pulse)

U

VVelocity

sensecoil

SensegainR2

R4

R5ServoloopgainPendulum

velocity

10VPP

5VPP

10 K5 K

Full wave rectifier

Thermistor (3)

Drivecoil

Drivegain R1

LT1001

A2

+

–

LT1001 10 K 10 K

10 K

20K

100

K47

0

10 K0.5 M(3)

1 M1 M

10 K

10K

10 K

0.5 M10K

IN914 2N

2222

1 K

+5 V501 W

+

–

LT1001

+

–

LT1001

+

–

LT1012

A3

2 µF (3)

1N91

4(2)

2 µF

A1

–15 V

KeystoneKC009G

–4.4%/°C10 KΩ at 25 °C

LM385–2.5Ohmite 41 F

Ripplefilter

+

–

LT1001

–

–LT1001

–

+

LT1001

+

–

3.3

K10

0K

1

2

T

S

2

1

~~~~

–2.5 VDC≈

gives a total amplitude change of 11.7 to 1. Three thermistors, one fromeach gain stage, are held against the side of a common 1 W heaterresistor by a short piece of heat shrink tubing.

The pendulum’s magnetic drive force is exactly in phase with thependulum’s position and its velocity signal, as (1) there are no phaseshifts (no high frequency attenuating capacitors) in the amplifiers, and(2) the magnetic drive coil is current driven, not voltage driven, to elim-inate any inductive time delay effects. The LM385-2.5, which is used forcontrolling the pendulum’s velocity, is a voltage reference, which is anextremely stable version of a zener diode in amplitude, temperature,and time. The drive sensitivity control R1 is set at 5 k, which givesenough drive current to maintain a constant pendulum amplitude withabout a 5 V peak-to-peak sinusoidal input signal. The sense coil’s gainpot R2 is adjusted to give about a 10 V peak-to-peak sine wave out of thefirst amplifier stage. The 2–1 drop in signal amplitude going throughthe three thermistor-controlled amplifier stages provides a good controlrange, both up and down, for both the thermistors and the servo towork with without exceeding the 20 V peak-to-peak linearity limits ofthe op-amps. The AC-to-DC converter in Figure 33.6 is the precisionfull wave rectifier and ripple filter in Figure 33.7.

New coils and magnets are used in Figure 33.7. The new coilsand magnets are longer, and allow a bigger pendulum swing angle of2.0 half angle. Each new coil is 3.25 in. long (3.62 in. including the coilform ends) with an outer diameter of 1.3 in. and an inner clearancediameter of 0.75 in. The alnico 5 magnets are still in. in diameter, but3.75 in. long. These are the coils and magnets actually shown in thephotograph in Figure 33.2. The 1 downward tip of the inner end ofeach coil to better clear the magnets’ circular swing path is justdetectable in the photo. The coils are wound with two separate wind-ings in each, one for driving and one for sensing. The drive winding con-sists of only two layers of #32 wire for a total of 710 turns per coil. Thisleaves more room for a larger sense winding of 9000 turns of #32 wireper coil. The drive windings measure 28 of resistance and 2.8 mH ofinductance per coil. The sense windings measure 446 of resistanceand 600 mH of inductance per coil. The sense windings have a self-resonance frequency of 9.4 kHz. Even with the reduced number ofturns in the drive winding, it takes a current of only 0.5 mA peak todrive the pendulum.

Servo loop

In one way, Figure 33.7 is an interesting servo to watch operating.In another way it is very boring, because nothing seems to be happen-ing. By turning up the loop gain, the servo becomes underdamped, and

316


233

it oscillates about the correct thermistor amplifier gain setting neededto maintain a constant pendulum amplitude. Because of the servo’slong time constants, just looking at the sine wave drive shows that noth-ing is happening. But the servo’s resonant frequency is 0.003 Hz, and ittakes 6 min(!) for one complete cycle of oscillation. So if you watch thedrive waveform for 10 min, you can watch its amplitude oscillate ever soslowly (slow to you, but not to the servo) above and below the correctamplitude. Then by reducing the servo’s loop gain back down to givecritical damping, which in my case meant reducing the gain of op-ampA3 to 100X by means of the servo gain pot R4, you can watch the drivesine wave slowly and regally rise (or fall, as appropriate) to its correctamplitude, and then calmly stay at that amplitude as if the servo werenot doing anything to make it happen. It takes 5–20 min for the servo(critically damped) to settle out at a “constant” sine wave drive ampli-tude, which is very slow for the human observer but just right for theservo with its inherently long time constants. The “constant” driveamplitude is not really constant, of course, as the drive is actually a sinewave, and the servo is slowly and continuously adjusting the sine wave’samplitude up and down as needed so as to obtain a constant averagependulum velocity.

Since Figure 33.7 is a servo, some attention must be paid to the timeconstants (phase shifts) around the control loop, to keep the loop stableand prevent it from oscillating. The biggest time constant is the pen-dulum itself, which with its physical mass (5 lb bob) and its slow rate ofamplitude decay, introduces a 90 phase shift in the servo loop. Thependulum has a Q of 10,000 and a time constant of 1.9 h, since it takes1.9 h for its swing amplitude (without electronic drive) to decay to 37%or 1/e of its initial value. The second largest time constant is the heattransfer time (37–85 s) between the heater resistor and the thermistors,which introduces another 90 phase shift into the loop. The third largesttime constant (2 s) is in the full wave rectifier’s ripple filter.

When I first built the servo in Figure 33.6, a 5 W resistor was used forthe heater resistor, which had a thermal time constant of 1.4 min heat-ing and 2.4 min cooling. For maximum stable gain in a servo of thistype, you want the largest possible ratio between the servo’s biggest andsecond biggest time constants. The 5 W heater resistor was replacedwith a 1 W resistor having a smaller thermal mass and a consequentlysmaller thermal time constant of 0.62 min heating and 1.4 min cooling.The 1 W heater’s smaller time constant gave a larger servo loop gainand more accurate amplitude control.

The servo provided by the electronics in Figure 33.7 gives a constantpendulum velocity, and does it with a non-clipped sine wave drive. Ingeneral terms, the pendulum’s peak-to-peak amplitude is its velocitytimes half its period, or its velocity divided by twice its frequency. Theservo is also a constant amplitude servo, if the pendulum’s period


234

remains constant. If the clock rate varies by 1 s/day, that is an error of1 part in 86,400. The servo then provides a constant pendulum amplitudewithin the same error of one part in 86,400, a very respectable tolerance(0.08 arc seconds in a swing of 2 half angle). This does not include thepermanent magnets’ negative temperature coefficient of 0.015%/C,which was discussed earlier.

Magnet coils and magnetic structure

The type of magnetic force generator used here is commonly called a“voice coil” type, because they are used in audio loudspeakers. One oftheir characteristics is that the magnetic flux from the coil is in parallelwith and adds on top of the magnetic flux from the permanent magnet.Since the force generated by the coil current is the product of the coilcurrent times the total magnetic flux, the magnetic force scale factor inpounds per amp of coil current increases or decreases as the currentincreases, depending on whether the coil’s magnetic flux adds toor opposes the permanent magnet’s magnetic flux. As a result, themagnetic force generated is nonlinear with coil current, and dependson the current’s polarity and magnitude. The effect can be large. Onanother system, I observed a 2–1 difference in the magnetic force scalefactor for the two directions of current through a “voice coil” forcegenerator. On a third system, the difference was only about 5% (asI remember it). The effect can be minimized by using large magnets, sothat the ratio of coil flux to magnet flux is small. The effect can beeliminated by orienting the coil flux to be perpendicular to the magnetflux, as is done in the D’Arsonval panel meter. The effect can also beeliminated by using two voice coil structures in force parallel, with thecoil flux oriented to add to the permanent magnet flux in one magneticstructure and subtract from it in the other. The total force generatedby the two voice coil structures is then linear with coil current, withinthe limits of symmetry of the two structures. The section in theAppendix, Measuring the Linearity of a “Voice Coil”, describes amethod of measuring a voice coil’s current-to-force linearity.

If the same magnet structure is used for both generating force andsensing velocity, the sensing signal is slightly affected by the drive cur-rent, through transformer coupling between windings on the same mag-netic core, which in this case is mostly air. Let us assume that the senseand drive currents are in separate windings, so that the voltage dropfrom the drive current across a common winding resistance does notdirectly couple over into the sensing circuit. The transformer coupledvoltage in the 9000 turn sense winding from a 2 mA peak-to-peak cur-rent in the 710 turn drive winding is 0.0002 Vpp at 0.5 Hz, measured byextrapolation from higher frequencies. It actually measured 0.4 Vpp at


235

1000 Hz, 0.04 Vpp at 100 Hz, and 0.004 Vpp at 10 Hz, which extrapolatesto 0.0002 Vpp at 0.5 Hz. The presence or absence of the permanentmagnet inside the coil made no difference in the induced voltage, whichis not surprising with most of the magnetic circuit being air.Transformer-induced voltage is always 90 out-of-phase with its excita-tion current, making the induced voltage also 90 out-of-phase with thevelocity signal. With a 1 s beat pendulum swinging through 1.6 (halfangle), the total signal in the 9000 turn sense winding is 0.44 Vpp ofvelocity signal at 0 phase angle plus 0.0002 Vpp of transformer-inducedvoltage at 90 phase angle. The transformer-induced voltage effect isthere, but it is pretty small.

Conclusions

The electronic circuits in Figures 33.1, 33.3, and 33.4 do not provide aconstant velocity drive to the pendulum. Instead, they do provide a con-stant force drive, letting the pendulum seek its own swing amplitude inresponse to the constant drive force. The constant force drive consists ofa clipped-peak sine wave, which continuously drives the pendulumthroughout its entire swing. The word “constant” as used here refers tothe average force or velocity over a complete swing of the pendulum.The constant force drive is really not very constant, as the parallel-linedareas in Figure 33.5 show.

The advantage of Figure 33.1 is its extreme simplicity—it uses onlyone op-amp. It ignores a lot of circuit complexities, giving it rather poorperformance and making it difficult to work with. Figure 33.3 is spreadout more, has independent sense and drive coil windings, constant cur-rent drive, and is easy to work with except for a strong interactionbetween gain controls R3 and R2. The performance of Figure 33.3 isbetter than that of Figure 33.1, but it is still not good. Figure 33.4 isthe same as Figure 33.3 but without the strong interaction betweengain controls R3 and R2. It does this by using an adjustable clippinglevel instead of the fixed clipping level used in Figures 33.1 and 33.3.Figure 33.4’s performance is the same as that of Figure 33.3: not good.Figure 33.4 is easier to work with and adjust, however.

Figure 33.7 does provide a constant velocity drive to the pendulum. Itdoes this by means of a servo and a thermally operated multiplier. Itssine wave driving the pendulum is of reasonably good waveform, andis not peak-clipped. It also provides a constant amplitude drive within avery close tolerance, because of the very small variation in the periodof an accurate pendulum. Figure 33.7 performs well and is easy towork with, except for a strong interaction between gain controls R2

and R4. It uses eight op-amps, compared to the one op-amp used inFigure 33.1.


236

The use of separate magnetic circuits for driving and sensing has abig advantage in avoiding any time delay between the pendulum’sswing angle and the drive current. Two magnetic circuits does meanmore pendulum parts, however, which unfortunately means more pen-dulum parts and joints to move around and cause instability.

In my opinion, the above circuits are good enough for low andmedium accuracy pendulums, but are not good enough for a high accu-racy pendulum. For a high accuracy pendulum, I do not think a sinewave drive will work good enough—precisely because the sine wavedoes provide drive over the whole pendulum swing angle.

I have another pendulum clock, driven by a short electromagneticpulse, wherein the drive pulse can be moved around to any point in thependulum’s swing angle. Pulsing this pendulum at the center of swingincreases the swing amplitude with no effect on the clock’s time (exceptfor circular error). Pulsing this pendulum at either end of its swingadvances or delays the pendulum by a small increment of time, andthere is no effect on the amplitude of swing.

At the ends of swing, a sine wave driving force will affect the pendu-lum’s timing, unless some very accurate cancelations of the drive forceoccur here. And some good force cancelations do occur here. A timedelay from the sine wave drive pushing the pendulum outward justbefore the end of swing cancels the time advance from the sine wavepushing the pendulum inward just after the end of swing. A higher scalefactor of magnetic force per amp of coil current for pushing the pen-dulum rather than pulling it cancels out when the pendulum movesfrom one side of center to the other. An amplifier phase delay betweenthe sensed velocity signal (i.e. the pendulum’s angular position) and thependulum’s drive force will advance the pendulum (angle-wise andtime-wise) on one side of center and delay the pendulum an equalamount on the other side of center. These individual force cancelationshave to be highly accurate, but only for 2 s of time. But the totalnumber of highly accurate force cancellations needed is mind boggling:there are 31 million seconds in a year.

What about distortion in the sine wave’s magnetic drive force? Whatabout noise in the drive signal? A 1% noise level in an analog signal isconsidered very good. A 0.1% noise level is extremely hard to get, andrarely occurs. In sum, with a sine wave drive, good timing accuracylooks very hard to get, with lots of nebulous high accuracy cancellationeffects to consider. And do not forget—the linearity of the sensed veloc-ity signal is limited by the uniformity of the number of winding turnsalong the length of the magnet coil. The same is true for the magneticdrive force.

I think the idea of driving a pendulum only at (or near) the center ofswing has a big advantage over continuously driving the pendulum overits whole circle of swing, where one has to deal with the pendulum’s


237

Figure 33.8. This magnet arrangement givessome positive feedback between coils, andcaused unwanted oscillation.

sensitivity to time errors at the ends of swing. The only advantagesI can see in favor of continuously driving the pendulum over the wholeangle of swing are (1) that it is less complex, and (2) that it is a lowerpowered approach, compared to the necessarily higher amplitude of adrive pulse of narrow width delivered at (or near) the center of swing.

Three end notes. First, if someone builds one of the circuitsdescribed herein for long-term use, particularly Figure 33.7 because ofits better performance, they should use 1% resistors having long-termstability (Vishay thick metal film type S102C or equivalent). Suchresistors are expensive, about $10 to $15 apiece in small quantities.Not all of the resistors need to be of this quality. Second, poten-tiometers are okay for short-term experiments, but for long-term usethey should be replaced by two fixed resistors or by a string of fixedresistors mounted on a multi-point rotary switch. Potentiometers arenot reliable for accurate long-term use. And finally, I would bedelighted if someone would dig in and make a high performance sinewave drive, blowing my prognostications of mediocre performanceinto the waste basket.

Appendix

Parasitic oscillation

I did not find out until after all the testing was done, but with themagnet orientation shown in Figure 33.8, the pendulum drive currentin one coil magnetically induces a positive feedback voltage in the other(sensing) coil, and this can cause a parasitic oscillation in the amplifier.With constant current excitation of the drive coil, the transformercoupling increases with frequency, making the parasitic oscillationoccur at a high frequency (30 kHz region). With both the drive andsense windings in the same magnet structure, or with both drive andsense in the same winding, the magnetic coupling again causes positivefeedback and possible parasitic oscillation.

This undesired oscillation can be eliminated by adding a high fre-quency attenuating capacitor to the servo amplifier. In Figure 33.1, theattenuation is provided by capacitor C1. This introduces an unwantedtime delay in the amplifier, which shows up as an angular displacementbetween the pendulum’s angular position and its sinusoidal drivingforce. A more desirable alternative is to put the drive and sense coils inseparate magnetic structures where they cannot couple together mag-netically and cause parasitic oscillation.

The possibility of parasitic oscillation can be eliminated by reversingthe polarities of both one magnet and its coil, either drive or sense, asshown in Figure 33.9. The pendulum’s drive current then induces


238

Pendulum rod

N NS S

+ – +Magnets –

Pendulum rod

N NS S

+ – +–

Figure 33.9. This magnet arrangement givesnegative feedback between coils, andeliminates the oscillation

a negative feedback voltage in the sense coil, and parasitic oscillationwill not occur. Additionally, shielding the wiring to the sense coil orkeeping its wiring 2–3 in. away from the drive coil wiring will minimizethe capacitive coupling between the amplifier’s output and input stages,and further reduce any tendency to parasitic oscillation at high servoloop gains.

Measuring the linearity of a “voice coil”

The current-to-force linearity of a voice coil force generator can bemeasured using the pendulum it is mounted on as the measuring stick.The basic technique is to (1) increase the pendulum’s displacementsensitivity to applied force by removing the bob (weight removal),(2) increase the coil current to a much higher test value, in both () and() directions, and (3) extrapolate the difference in the resulting oversize() and () pendulum deflections back down to the actual operatingcurrent level.

Removing the bob (19 lb) reduces the pendulum weight to just thatof the pendulum rod and its attached voice coil magnet (1.6 lb total).This weight reduction increases the pendulum’s displacement sensitiv-ity by (19 1.6)/1.6 or 13 times. The voice coil’s resistance measured30.5 . Putting the voice coil across a 5 VDC power supply in both theforward and reverse directions gave horizontal deflections of 0.37 in.(push the magnet out) and 0.36 in. (pull the magnet in) at the bottomof the pendulum rod, with a coil current of 5.0 V/30.5 164 mA.

The pendulum’s horizontal deflection is proportional to the voicecoil’s force. Half the difference in deflection occurs in each direction,and represents the amount of nonlinear force in each direction.Extrapolating this back down to the 0.5 mA peak actual current levelgives an operating nonlinearity of

In other words, the current-to-force nonlinearity is one part in 23,900at the 0.5 mA operating current level. Obviously, in this case, the lin-earity of just one voice coil is more than sufficient to drive the pendu-lum. The deflection was measured using a 3.00 in. long alnico 5 magnetwith one magnet end coaxially located at the center of a 3.25 in. long

0.37 0.36

2 0.37 0.36

2 0.5

1641

23,900.

12 Difference in deflection

Deflection Operating currentTest current


239

coil. With such a long coil, axial deflections of 0.37 and 0.36 in. atthe center of the coil will not change the current-to-force scale factorvery much. The long coil was picked for this test for this reason.

References1. V. Bush and J. Jackson. “The amateur scientist,” Sci. Amer. ( July and

August 1960).2. J. Bigelow. “Ideal pendulum drive,” Hor. Sci. Newslett. NAWCC chapter 161,

(April 24, 1993 and September 1993). Available from NAWCC Library,Columbia, PA 17512, USA.


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chapter 34

Photoelectronics for pendulums

Some ideas are offered on applying photoelectronics to pendulums.

Photoelectronics make good sensors for pendulum clocks, becausethey add no power losses to the pendulum. A swinging pendulum inter-rupts a light beam, and a light detector provides an electrical signal—tocompare to a time standard or to incrementally drive the second handon a clock face. This is done without putting a load on the pendulum.(The photons of light do have a microscopically small impact on thependulum, but it is nonexistent on a clock’s energy scale.)

The simple circuit

Most photoelectronic circuitry is aimed at very simple applications,such as counting slow-moving cans or boxes on a production line. Or asmentioned above, detecting the passage of a slow-moving pendulum.For the pendulum application, where the light source and light detectorare about a half-inch or so apart, the most suitable light source is aninfrared light emitting diode (LED). And the most suitable light detec-tor is a silicon transistor or diode, depending on the speed of responseneeded. (All silicon semiconductor junctions are sensitive to light, andcan be used as light detectors. Unless sensitivity to light is desired, allsilicon semiconductors are either painted black or immersed in opaquematerial to eliminate the effect.)

The light source and detector come in an assortment of packages, butthe most convenient and recommended one for mechanical mounting iswhat is called the “TO-18 can” with an epoxy lens in the top of the can.

A typical simple photoelectronic circuit is shown in Figure 34.1.The light beam that the pendulum interrupts is round, about in. indiameter. The detector’s output signal gradually (and roughly propor-tionally) starts going from maximum output down toward zero outputas the pendulum starts into and moves across the in. diameter lightbeam, interrupting it. The detector’s output signal does not reach zerooutput until the pendulum has interrupted all of the light beam. Thekey point here is that the pendulum’s position is not resolved to betterthan in. with the simple circuit shown in Figure 34.1. And if the3

16

316

316

swing amplitude of a pendulum with a 2 s period is 1 in. at the lightbeam’s location, then the pendulum’s timing is not electrically resolvedto better than ths of a second. (A constant pendulum velocity isassumed for discussion purposes.)

The above describes the limits of what can be done with simple photo-electronic circuitry. It is good enough to electrically drive the second handon a clock face, or compare against WWV’s radio time standard to thssecond accuracy. But suppose we want to measure a pendulum’sisochronism, that is, how much the pendulum’s period varies as itsswing amplitude decays. Then we need better time resolution. Or sup-pose we want to use light beams to define the ends of swing and/orcontrol the pendulum’s swing amplitude. Then we need better mechan-ical position resolution and better time resolution.

Basic improvements

There are four basic things that can be done to improve the dimensionaland time resolutions of a pendulum:

Narrow the light beam down to just a slit width Use a voltage comparator on the light detector’s output signal Better stray light reduction Use a faster light detector.

With a narrower light beam, the detector’s output signal goes frommax to min in a shorter pendulum distance, giving better dimensionalresolution of the pendulum’s position. The pendulum also crossesthe narrower light beam in a shorter time interval, giving better timeresolution. One good way to narrow the light beam is to cement tworazor blades with a small (0.01 in.?) gap between them just in front of

316

316

Figure 34.1. Simple photoelectronic circuit.


242

Pendulumpath thrulite beam

+5 VDC

Photo-transistor

Vout

1 k

Litebeam

100mA

LED

30

+5 VDC

316

in.

316

s

the light source, and cement another two razor blades with a 0.01 in.gap just in front of the detector. Figure 34.2 shows the arrangement.Razor blades are about 0.01 in. thick, so a piece of razor blade can alsobe used to size the gaps and keep them uniform in width along theirlength while the cement is hardening.

The slits need to be closely parallel to the pendulum edge that inter-rupts the light beam. This is hard to do. Any angular misalignment ofthe slits to the pendulum edge cutting the light beam effectively widensthe beam to larger than the 0.01 in. slit width of the razor blades.

The narrower beam width drastically reduces the detector’s signalamplitude by 100–300 times. A wideband amplifier is added to bring thedetector’s signal back up to its original amplitude. Figure 34.3 shows agood amplifier arrangement. Its gain is adjusted by adjusting the feedbackresistor R2 until about a 5 V signal at the amplifier’s output is obtained.The wideband amplifier’s input impedance, which is the load resistancefor the light detector, is approximately zero ohms. Since the discharge ofthe detector’s capacitance through its load resistance limits the detector’sspeed of response, a zero load resistance gives the fastest detectorresponse, limited only by the detector’s internal (emitter) resistance andthe detector’s internal capacitance. The amplifier in Figure 34.3 is capac-itively coupled to the detector, making the amplifier’s output signal Eout

independent of any constant stray light impinging on the detector.The second basic area of improvement is adding a voltage compara-

tor to the detector’s output signal. A voltage comparator compares two

Figure 34.2. Narrowing the light beam forbetter pendulum location accuracy.

Figure 34.3. This amplifier arrangementprovides an approximately zero ohm load tothe detector for fastest detector response. Theamplifier is capacitively coupled to thedetector.

chapter 34 | Photoelectronics for pendulums

243

Epoxy

LED

Narrowlite beam

Litedetector

Razor blade slits (2)

+5 V +5 V

Pendulum path

Litedetector

30

LED

Litebeam

100

C2 15 K

1.0 µF LM 318 Eout

+5 V0 V

Pendulumwidth (in.)

?

0.01 in. (2)

R2

Gain ≅ –200

input voltages, and when one is just a few millivolts higher or lowerthan the other, the comparator’s output will flip hard over very quickly(0.2 s) from off to on, or vice versa. This gives a 1000 to 1 theoreticalimprovement in both the pendulum’s position and the time it interruptsthe light beam. In practice you only get somewhere between 10 to 1 and100 to 1 improvement, because noise riding on top of the incoming signal makes the comparator trip earlier or later than it should. Thevoltage comparator helps most with slow-changing light signals, likewhat the simple circuit in Figure 34.1 provides. The comparator tripshard over every time the slow-changing input signal reaches or passesthe voltage reference level on its other input line, within plus or minusa few millivolts, out of a total range of 10 V.

Figure 34.4 shows a voltage comparator circuit with automatic ampli-tude compensation. The comparator is designed to trip hard over at theexact same mechanical location, such as when the pendulum reaches thecenter of the light beam, regardless of the overall brightness of the lightbeam. This makes the circuit give an accurate mechanical location forthe pendulum, independent of the aging of the LED light source, whoselight output decreases with time. More on LED aging later.

The automatic amplitude compensation comes from the comparatorcomparing the detector’s output signal against a heavily filtered half-amplitude version of the same detector’s output signal. The circuit takesadvantage of the light beam and the detector signal being “on” most ofthe time, and that the light beam is interrupted by the pendulum only asmall part of the time. Heavy filtering of the light signal then eliminatesthe beam interruption of the signal, giving a continuous smoothrunning average of the detector’s light signal. Feeding this smooth aver-age signal to the comparator at half amplitude, as in Figure 34.4, to becompared against the unfiltered detector output signal at full amplitudemeans that the comparator always trips hard over every time thedetector’s light signal reaches half amplitude. This will happen even ifthe light intensity, that is, the full signal amplitude, decays way downover time to 10% or less of its original amplitude.

Figure 34.4. Voltage comparator withautomatic amplitude compensation.


244

Vin

0 V

0 V

+5 V

+5 V

+5 V0 V

R1 200 K

9 Amp

LF355

200 3 K 3 K

200

20K

2K

Voltage comparator

LM311 +5 V

Vout

Vin

0.5 Vin

Pendulum width(not to scale)

Gain = +10.01in.

0.01in.

F

+

–

–

+

C1

AVG

Vin

2

The third basic improvement area is stray light reduction, that is, keep-ing unwanted ambient light out of the detector. A certain amount ofshielding is needed even with the simple circuit in Figure 34.1, as brightdaylight or an overhead lamp can completely saturate the detector’s out-put signal. Even more stray light reduction is needed to take advantage ofthe voltage comparator’s high resolution, as the stray light acts as electri-cal noise on top of the detector’s output signal—noise that makes the com-parator trip earlier or later than it should. Stray light reduction starts withsimple things like shielding the detector from outside light, and paintingthe surrounding surfaces with a nonreflecting flat black paint. There aresome electronic things that can be done too, the most important of whichis to capacitively couple the light detector to its amplifier. This will stop anysignal from a constant ambient light. Electric lighting has a strong 120 Hzcomponent, so the detector’s amplifier should pass the (mostly) higher fre-quencies contained in the light beam’s cutoff signal by the pendulum, andreject the lower frequencies like the 120 Hz component of electric lighting.

One of the best ways to do that is with the synchronousmodulator–demodulator concept. This concept involves modulating(on and off ) the LED light source at a high frequency rate, say 50–100 kHz,and demodulating (rectifying) the detector’s output signal in synchro-nism with the modulation of the LED light source. With a low pass filterattached to the demodulator’s output, only signals close to or at themodulating frequency will pass through the low pass filter. Since ambientlight has little or no frequency content at or near the 50–100 kHz modu-lation frequency, the synchronous modulation–demodulation concept isvery effective in eliminating any ambient light effects.

Figure 34.5 shows the synchronous modulation–demodulation con-cept applied to a pendulum. Two 74HCO4 digital inverters in parallelturn the LED light source on and off at a 100 kHz rate. The detector’s


245

Figure 34.5. Modulation of the LED light source and synchronous demodulation of the detector’s output signal.

100 kHz

74HCO4 (2)

74CO4

Modulatedon/offLED

R3

C3

+5 V

Litedetector Full wave

synchronousdemodulator

S–

S–

S

+5 V

Pendulum path

(VDC) outto Voltagecomparator

–

+Amp

–

+Amp

LM318

Gain = –1

DG411

S

R4

C4

Low pass filterR4C4 = 19 s

output signal is run through a full wave rectifier (demodulator) usingtwo analog switches, and then goes through a low pass filter. The twoanalog switches provide full wave rectification by simultaneously closingone switch and opening the other, and then, again simultaneously, open-ing the first switch and closing the second, both being done in synchro-nism at 100 kHz with the on/off switching of the LED light source.

The fourth basic improvement area is a faster light detector. LED lightsources have turn-on and turn-off times of 0.1–0.5 s. Darlington two-transistor light detectors have high signal gain, but are slow with turn-onand turn-off times of 250–300 s. Single-transistor light detectors have alittle signal gain with medium speed turn-on and turn-off times of 7–8 s.The fastest silicon light detectors are PIN diodes, with no signal gain andwith turn-on and turn-off times of 1 ns (109 s) to 1 s (106 s). All diodedetectors require an external amplifier to amplify their small signal level.

With all of the silicon light detectors, the biggest (or one of the biggest)speed factor is the discharge time of the detector’s internal capacitancethrough its load resistance, and the response times given above assume avery low load resistance, down near zero ohms. The response times are10–100 times longer with larger load resistances of 1000–10,000 .

As to cost, infrared LEDs cost $0.40 to $1.70, one- and two-transistordetectors cost $0.40 to $1.60, and diode detectors cost $2 to $50.

What about timing accuracy? A single-transistor detector feeding intoa (near) zero ohm load will give 8 s accuracy of reading on a singlependulum pass, and 2–3 s accuracy of reading when averaging over10 passes. If you use a high load resistance such as 1000 or 10,000 onthe detector, all you will get is 80–800 s accuracy at best, even if youaverage over 1000 pendulum passes. The reason is that the large tem-perature coefficient of the detector’s internal capacitance changes thedetector’s internal capacitance and response time so much that you willend up measuring more of the detector’s response time variations withtemperature than you will of any variations in a pendulum’s swing time.There is no substitute for a fast detector response time, that is, for shortturn-on and turn-off times.

There is one other very tough subject that should be mentioned, andthat is the location stability of the light source and detector. To meas-ure the pendulum’s swing position to 0.001 in. and 0.001 s assume thatthe light source and detector positions are stable to at least 0.001 in. dur-ing the measurement interval. To measure the pendulum’s positioneven more accurately to, say, 10 in. and 10 s assume that the lightsource and detector positions are stable to at least 10 in. during themeasurement interval. Stability of mounting to these low levels is veryhard. Spring clips and scotch tape are out. The light source and detec-tor mountings must be solid hefty metal brackets, and not just a skimpypiece of bent sheet metal. The moving pendulum generates air currentsthat will push springy-mounted things around some. Temperature


246

effects, vibration and shock effects when your house’s front door isslammed, etc. The stability problem is much easier, or course, if themeasurement interval is just 1 s, instead of 1 month or 1 year.

Detectors and light sources

Some words on individual light detectors and light sources might beworthwhile. As mentioned earlier, the recommended package for bothdetectors and sources is the “TO-18 can” with an epoxy lens on top ofit. The recommended detector is the one-transistor #L14G3 or itsequivalent. Always pick the highest sensitivity, and the narrowest field-of-view. A narrower field-of-view means a smaller silicon chip inside,a smaller detector capacitance, and a faster response time.

Silicon detectors, both the transistor and diode types, are very stableover time. Silicon diode detectors in particular are so stable that theycan be used as secondary laboratory standards.

Transistor light detectors are very much an economic compromise.As such, their optical field-of-view is only roughly centered to themechanical axis of the TO-18 can, and their sensitivity varies consider-ably (2 or 3 to 1) from unit to unit. It pays to test individual detectors,and rotate them in their mounting hole for maximum signal output.Some manufacturers do not bother to list a detector’s field-of-view.

As for LED light sources, the infrared ones are much brighter thanthe visible ones. Select for maximum power output into a minimumfield-of-view. In pendulum applications, the light beam has a narrowfield-of-view. LED light that is emitted into a wide field-of-view iswasted—only a narrow part in the center of the field-of-view will arriveat the detector. An LED’s radiated light beam is not too well centeredon the mechanical axis of the TO-18 can. And the radiated lightintensity varies widely (at least 2 to 1) from unit to unit, so it pays to testindividual LEDs and rotate them in their mounting holes.

LEDs have a limited life. Their light output falls off with time. Therate of falloff is quite variable from unit to unit, falling to 50% of the initiallight output in anywhere from 6 months to several years. It helps only alittle to operate them below their rated current of 100 mA. In pastyears, those made by General Electric had the best life. The #LED55C wasrecommended. But GE sold their LED line to Harris Semiconductor, whoresold it to Quality Technologies Corp. Unfortunately, the knowledge ofmaking the LEDs long-lived did not survive the two transfers. Motorolaput out a technical report on an LED lifetime improvement program theyinstalled, but I have no data on its results. Currently, I am using TRW LEDs(Optron #OP-133). The OP-133 LEDs have held up well so far (4 years).

Many companies make silicon diode light detectors. Motorola makessome low cost ($2) ones that look appropriate for pendulum use, like


247

the #MRD500 with 1 ns (109 s) response time. And although they costmore, the diode detectors from UDT Sensors (formerly UnitedDetector Technology) are extremely good, and have given me out-standing results in several demanding applications.

The detector amplifier should have a 0.5–1 MHz power bandwidth,such as the LM318 (National Semiconductor) or the LT1122 (LinearTechnology). Good wideband amplifiers cost only $2 to $4. Junk 741smay cost only 50 cents, but their low bandwidth will not cut it here.

Final comments

Figure 34.6 shows how the various concepts described here can be inte-grated together into one circuit. The light detector in Figure 34.6 has a144 phase lag in it at 100 kHz. To compensate for this, the LED’s phase islagged (180 144) 36. The why of this and why the 36 LED phaselag circuit is designed the way it is are above the level of this chapter.

The circuit in Figure 34.6 has been used in my clock for 4 years nowto detect the pendulum’s minimum swing amplitude, that is, when itneeds another push—a short electromagnetic “kick in the pants” at thecenter of swing.

In addition to what has been described here, an experienced circuitdesigner would do some more things, like reverse biasing a diode detec-tor to reduce its internal capacitance, and adjusting the lead network R3C3

in Figures 34.5 and 34.6 for optimum square wave response out of thelight detector. But as mentioned earlier, this chapter is mainly intended togive some basic ideas on applying photoelectronics to pendulums.

Figure 34.6. Photoelectronic circuit to detect light beam interruption.


248

OptronOP-133

1N4148

100 kHz

LED

+5 V

Pendulum path +5 V

15 K

LM318

1 K 1 K

0.001LM318

LM318

DG411

51 K Low passfilter

2 K 8.1 K 30 K

330pF680

pF2200pF

200 K

9.0µF

3 K 3 K

Avg

200 20

020

K2

K

S

RC = 1.8 s

LF 355

Vc

Amp–

+ Amp–

+

Amp–

+ Amp+

–

LM311

Voltagecomparator

+5 V

Lite Beaminterrupt

–

+

1.0

Lite detectorhoneywellS - 5443 - 3

74HCO4(2)

200 K 12 K

.1

74CO4(4)33pF

S–

S–

S

36 °74HCO4

2K 20K .1

100kHz

Phaselag

100pF

8 mA

510

0.00

22

100

5.1K

5.1K

Vc

2

249

chapter 35

Check your clock against WWV

An electronic method of accurately comparing a clock’s time against WWV’s time is described.

In the United States, WWV is the obvious time standard to checkyour clock against. WWV’s claimed time accuracy as transmitted inBoulder, Colorado is 10 s short term, and 1 s in 3000 years long term.The received short-term accuracy is reduced to 0.001 s, because of varia-tions in the signal’s transit time. An exception is the better received short-term accuracy of 100 s in WWV’s 60 kHz ground wave signal, whichcan be improved even further to 10 s by proper averaging techniques.

A shortwave radio receiver is needed to pick up WWV’s signal. I usea Radio Shack receiver that receives only WWV and the weather sta-tions. It receives WWV at 5, 10, or 15 MHz at the flip of a switch. Thisparticular receiver is no longer available, but Radio Shack does offeranother WWV receiver.

WWV broadcasts a lot of information in their signal. For clock-checking purposes, the important parts are the 1 s “ticks” and the 1 min“beeps.” These have to be separated out from the rest of WWV’s signal.The 1 s “tick” consists of 5 cycles of a 1000 Hz sine wave, and is 0.005 slong. The instant of time identified by each tick is the instant at the startof the tick.

For coarse time comparisons, you can just listen to the ticks andbeeps while visually watching the second hand on your clock. This willgive 1 s accuracy. For finer accuracy to fractions of a second, electroniccomparison circuitry is resorted to.

To facilitate finding the tick in WWV’s signal, there is a 0.010 s deadzone of no signal immediately before the tick, and a 0.025 s dead zoneof no signal immediately after the tick. Figure 35.1(a) shows whatWWV’s 1 s “tick” looks like, as received. Notice the extra negative-going blip that appears at the beginning of the 5-cycle tick, whoseapparent purpose is to sharpen up the first instant of the first cycle ofthe tick. Figure 35.1(b) shows the 1 s pulse in my WWV comparatorthat is triggered by the start of the 1 s tick. The 1 min “beep” consistsof 800 cycles of a 1000 Hz sine wave, and is 0.8 s long. It replaces the 1 stick at the start of each minute.

The basic concept in all of the electronic time comparison schemesis the same, and is shown in Figure 35.2. Two digital signals are neededto implement any of them. One digital signal needed is from WWV torepresent each second’s tick. The other digital signal needed is fromyour clock to represent each second from your clock. Fundamentally,the fractional-second time difference between your clock and WWV ismeasured by a digital counter that counts the constant flow of pulsescoming out of a quartz crystal oscillator, with one digital signal (yourclock or WWV) starting the counting interval, and the other digitalsignal ending it. If the pulses coming out of the oscillator come every0.001 s, or at a rate of 1000 pulses/s, then the actual count in the digi-tal counter represents how many 0.001 s increments there are in thetime difference between your clock and WWV. For instance, a count of604 in the digital counter would tell you that your clock is 0.604 s aheadof (or behind, more on this later) WWV.

To make the counter easy to use, some “bells and whistles” are addedto the basic concept. Three of these are shown in Figure 35.3. The first“bell and whistle” is to add a visual digital display to show what thecount is in the digital counter. Second, every counter has an inherentuncertainty of plus or minus one count in its counting. So the second“bell and whistle” is to increase the crystal oscillator’s pulse rate by10 times, up to 10,000 Hz, so that the oscillator pulses come out every0.0001 s. The inherent 1 count uncertainty then becomes a

Figure 35.1. (a) WWV’s 1 s “tick,” and (b) comparator’s 1 s pulse triggered by the“tick.” Horizontal scale: 0.001 s/div.

Figure 35.2. Basic measurement concept.


250

(a)

(b)

1000 Hz oscillator pulses

Your clock’s 1 s pulses

WWV’s 1 s pulses

Count input

Start

Stop

Digital counter

negligible 1 count in the fourth decimal place, or 0.0001 s in yourtime comparison. This fourth decimal place may or may not beincluded in the visual display. I chose to include it in mine.

The third “bell and whistle” is to put a reversing switch in the startand stop lines to the counter. An extra contact is also added to thereversing switch, as shown in Figure 35.2, so that the count display willvisually show the two switch positions as () and (). Now you caneither start the digital counter with your clock’s 1 s pulse and stop thecounter with WWV’s 1 s pulse, or you can do the reverse by means ofthe newly installed switch, that is, start the counter with WWV’s 1 spulse and stop the counter with your clock’s 1 s pulse. The difference isthat now the digital counter will count how far your clock is behindWWV as well as how far your clock is ahead of WWV. For instance, thedigital count display would show that your clock is behind WWV by0.396 s (say), as well as (by throwing the reversing switch) showingthat your clock is ahead of WWV by 0.604 s.

The advantage of this becomes apparent if your clock changes frombeing ahead of WWV to being behind WWV (or vice versa). Thenwhen you plot your clock’s time error from WWV on a graph, you canplot the displayed time error directly, instead of having to subtract0.604 s from 1.000 s every time, to get the correct number to be plotted(0.396 s). Figure 35.4 is a graph of a short 8-day run on my pendulumclock, just after it was built. The clock is mostly following local baro-metric pressure changes in Figure 35.4, as the barometric correctioncircuit was not tied in yet.

Now the 1 s pulses from your clock and WWV, that are so simplemindedly shown in the basic measurement concept in Figure 35.2, are

Figure 35.3. Adding bells and whistles to thebasic concept.

Figure 35.4. Clock time error vs time.

chapter 35 | Check your clock against WWV

251

10,000 Hz oscillator pulses

+ 5 V

Your clock’s 1 s pulses

WWV’s 1 s pulses

Count input

Start

Stop

Digital counter

0–1.0

0

+1.0

2 4 6 8Time (days)

Clo

ck a

hea

d of

WW

V (

s)

not really so simple minded. The 1 s pulses from your clock should comeeither at the ends of pendulum swing or at the center of swing, so thatthe pulses’ timing will not vary with the swing amplitude. And separat-ing out WWV’s 1 s pulses from the rest of the stuff on WWV’s signaltakes some effort, and can take a sizeable amount of circuitry. TheWWV signal varies widely in amplitude over time, periodically fades outfor intervals varying from 1 min to many hours, and is corrupted by allkinds of noise. Noise is an old and nasty problem in radio, and a zilliontechnical papers have been published on the subject over the years.

The best, simplest, and cheapest solution to fading and noise prob-lems is to put up an outside longwire antenna (for your WWV receiver)that is as long as possible, as high as possible, and in the clear as muchas possible. Even a 10 foot antenna will help, but a 100 footer is much,much better. The watchword is to do what you can; anything is betterthan nothing.

A second but more expensive solution is to receive WWV on severalfrequencies in parallel (WWV broadcasts in parallel on 60 kHz, and 2.5,5.0, 10, 15, and 20 MHz), and pick the one with the strongest signal andleast noise. When one frequency is skipping over your location, anothermay be landing right on you. The skip varies with frequency, time ofday, and distance from the transmitter. In Minneapolis, 5 MHz isgenerally the best frequency during the day, and 10 MHz is best at night.But I also find myself switching back and forth between them over a 5-min interval, as they fade in and out. Reception can be a sometimesthing, so you do your WWV comparisons when you can get the signal.The WWV signal is usually available locally over most if not all of theday and night, every day.

Once you get WWV’s signal, there is a wide assortment of tech-niques available to pull the 1 s ticks and 1 min beeps out of the signal.Two of the simpler ones will be described here. The simplest is to usea 1000 Hz bandpass filter to separate out the 1000 Hz components ofthe ticks and beeps. I use this technique to sort out the 1 min beeps.

When the 1000 Hz bandpass filter receives a 1 s tick, the filter’s out-put amplitude gradually builds up over time, due to resonance consid-erations, with the rate of buildup depending on the filter’s bandwidth,which is related to the filter’s Q. Unfortunately, the ticks do not last longenough to build up much amplitude at the filter’s output. And the typ-ical diamond-shaped waveform that comes out of the filter, as shown inFigure 35.5, smears out the timing accuracy to 0.005 s. In addition, themany radio noise spikes coming in make the bandpass filter “ring” at itscenter frequency of 1000 Hz, creating many false 1 s pulses at the filteroutput. The beeps last much longer than the ticks, so that with a filterQ of 20, the beeps come through just fine, with only a low levelresponse to the noise spikes. However, the timing accuracy of the 1 minbeeps is smeared out to 0.1–0.8 s, depending on the filter’s Q.

Figure 35.5. 1000 Hz. bandpass filter: 1 s“tick” input and (b) filter output response.


252

0.005 s

(a)

(b)

A second technique for pulling the 1 s ticks out of WWV’s signal isto use a narrow timing gate, which opens just before and closes justafter each tick. In this way all of the noise and extraneous signals out-side of the timing gate are eliminated. The gate is moveable in time,and is referenced to and triggered by the clock’s 1 s pulses. This tech-nique requires the use of an oscilloscope, which the first technique doesnot need. The scope is used to locate the 1 s ticks in the WWV signal,and to see that the moveable gate does indeed straddle the ticks.

If your clock’s time rate differs from WWV’s, the ticks will slowly“walk” away from the timing gate on the scope, as the gate is referencedto your clock, and not to WWV. The gate’s position must then be peri-odically trimmed time-wise, to keep WWV’s 1 s tick within the gate.The scope provides a continuous monitoring that the tick really iswithin the gate and hence passing through it. I use this timing gate tech-nique to sort out the 1 s ticks. If the WWV receiver and the clock are inthe same room, you can keep track of any whole second differencesbetween your clock and WWV by looking at your clock’s second handwhile simultaneously listening to WWV. In my case, the clock is inanother room 40 ft away from my workshop where the scope, WWVreceiver, and antenna are located. So it is necessary to electronicallymeasure whole seconds as well as fractions of a second. This is donewith a fourth “bell and whistle,” which uses the 1 min pulses from theclock and WWV to start and stop the same digital counter, which nowcounts the number of 1 s clock pulses occurring between the two 1 minpulses that are starting and stopping the counter.

I built my own WWV–clock comparator. Its size is 3 9 10 in.3 (H

W D). It contains two 4 6 in. circuit boards, one for analog circuitryand one for digital circuitry. Figure 35.6 shows the front panel. Three ofthe test points on the front panel are signals needed for the scope.

The 0.001 s time resolution provided by the WWV comparator isa very big timesaver to anyone involved in testing or developing apendulum clock. One second/day corresponds to 0.001 s in 1.44 min.A pendulum that is way out of adjustment can be trimmed to correctlength (within 1 s/day) in 1 h. The effect of leaving the clock case door

12

chapter 35 | Check your clock against WWV

253

Lite

100 kHz

WWV

RCVR

WWV WWV

Led

SEC.

CLK

Led

SEC.

CLK

Led

MIN.

WWV

Led

MIN.SEC.

CLK PART

SEC. WHOLE SEC. GATE

DELAY+

–

Test points

10 turnpot. and dial

ON

OFFPOWER GND 1 kHz MIN. MIN. GATE AHEAD OF WWV

Seconds

WWV–Clock comparator

Figure 35.6. Front panel of WWV–Clockcomparator.

open (the clock slows down 0.08 s/day when the door is closed) can bedetermined in 2 h. A temperature compensation test on a pendulumcan be done in a few weeks. Tests on a series of new suspension designscan be done in a few days, instead of over months or a year. The 0.001 stime resolution makes the practical difference between doing and notdoing many things with your pendulum clock.

As a final note, other countries have standard stations similar toWWV, and at least some (if not all) of them are coordinated togetherto provide a common time standard. This chapter describes somecharacteristics of WWV’s transmitted signal which may be different inother signals. There is a good brochure available1 (free!) from NIST thatdescribes WWV’s services in detail.

Reference1. “NIST time and frequency services,” NIST Special Publication No. 432, is

available free from U.S. Department of Commerce, National Institute ofStandards and Technology, Gaithersburg, Maryland, 20899.


254

255

chapter 36

Electronic correction for airpressure variations

An electronic circuit that corrects for a pendulum’s air pressure variations isdescribed.

At atmospheric pressure, a pendulum is slightly buoyant in a “sea” ofair. This buoyancy causes the pendulum’s timing to be a little sensitiveto the air’s density, and consequently to its pressure. My pendulum’stiming changes 0.26 or 0.71 s/day/in. Hg change in the air pressure,depending on whether the pendulum rod is invar or quartz. This can becorrected mechanically by putting a small bellows-supported weight onthe pendulum [1, 2]. As the air pressure increases, the bellows shrinksand lowers the weight resting on top of the bellows, which speeds upthe pendulum to compensate for its natural slowdown with increasingpressure. This assumes that the bellows’ weight is located in the pen-dulum’s upper half. If located in the lower half, the bellows-supportedweight must hang below the bellows instead of sitting on top of it (seeChapter 24).

Several people have used bellows on pendulums [2–4]. The bellowsapproach has the advantage that it does not require electronicknowhow; only mechanical skill is needed. But this mechanicalapproach adds 4–6 piece parts to the pendulum, decreasing its dimen-sional stability. Two more limits to this approach are the mechanicaland thermal hysteresis (repeatability) of the bellows. The overall accu-racy of this approach should be reasonably good, but I have not seenany data on its accuracy or stability.

Concept

The effect of the air pressure variations can also be corrected forelectronically, using a silicon-based pressure sensor, some electroniccircuitry, and an electromagnetic (coil and magnet) pendulum drive.A block diagram of the concept is shown in Figure 36.1.

The analog pressure sensor measures the total air pressure, which isconverted into a relative pressure change ( P ) by subtracting out the

average pressure, which is 29.00 in. Hg at my location. The relativepressure (P) is integrated over time and digitized using an integ-rating type of analog-to-digital converter. Every 15 min the value of therelative pressure is added to the count in counter A.

When the count in counter A exceeds the equivalent count value ofone electromagnetic pulse to the pendulum, which is stored in counter B,then a short pulse 0.025 s long is sent to the pendulum when thependulum is at the far end of a swing. The pulse advances (or delays)the pendulum by a small increment of time T, about 0.0014 s. At thefar end of a swing, a drive pulse will change the pendulum’s time with-out affecting the pendulum’s amplitude or rate. The T time advance(or delay) corrects for the slowdown (or speedup) in the pendulum’srate caused by an increase (or decrease) in air pressure. Each time thependulum is pulsed, the pulse’s equivalent count value, which is storedin counter B, is subtracted from the count stored in counter A.

The electromagnetic pendulum drive has the usual “voice coil”arrangement with the magnet’s north–south axis along the coil’s centralaxis (see Figure 32.1). There are two moving permanent magnetsattached to the pendulum and two fixed coils attached to the clock case,


256

Interval Xtal.Osc.

100 kHztimer

1/4 hour

Clk

Add (±)Counter A

A/D converter(Integrator)

SubtractAmpPressure

sensor

Pavg.

Comparator Is count A > count B?

Counter B

Switch array(Advance)

Switch array(Delay)

Count per(+) pulse

Count per(–) pulse

Yes 1 T

Pulse

29.00in. Hg.

+

– Subtract

±∆P

Tope

ndu

lum

Polarity:

Advance (Hi pressure)

or delay (Lo pressure)

Figure 36.1. Block diagram of the electronicpressure correction circuit. The equivalent count value of one pendulum drive pulse ismanually entered into the switch arrays.

with one coil and one magnet each on the left and right sides of thependulum. The pendulum’s advance and delay time increments are notthe same. This is normal in the voice coil type of construction, becauseof the way the voice coil’s magnetic field adds to and subtracts fromthe permanent magnet’s magnetic field. In this case, the time advanceincrement is about 30% bigger than the delay increment. To accommo-date the difference, the count equivalents of the pendulum’s advanceand delay pulses are stored on separate manual switch arrays, and the polarity of the count in counter A is used to preset the appropriateequivalent count into counter B, as shown in Figure 36.1.

The signal needed to generate the T changes in pendulum time isnot just the variation in air pressure but the integration of the variationin pressure over time. This is obtained almost for free by using the inte-grating type of analog-to-digital converter. The integration is obtainedby digitally converting the analog pressure variations (P) at regularperiodic intervals, and continuously adding up the digital valuesobtained in counter A. In practice, all we have to do is remove the“reset to zero” signal that normally goes into counter A at the start ofevery analog-to-digital conversion. One advantage of the integratingtype of analog-to-digital converter is its greater accuracy. Its disadvan-tage of a longer conversion time (0.031 s here) is not a problem in thisapplication.

The pendulum already has a 0.025 s timer (not shown in Figure 36.1),which is used to generate the pendulum’s amplitude maintaining pulsesat the center of swing. This timer is also used to convert the “Yes” levelchange signal coming out of the comparator in Figure 36.1 into a short0.025 s electromagnetic pulse that actually advances (or delays) thependulum at the end of a swing.

Pressure sensor

The pressure sensor is a small in. sized container with one wall madeof silicon. The silicon wall is thin and flexes with the pressure differencebetween the inside and outside of the container. The sensor has tworesistors imbedded in the silicon wall. These two resistors increase anddecrease their resistance as the wall flexes, thereby providing two resis-tors whose resistance varies with pressure.

The sensor is available two ways: (1) vacuum sealed, which makes itsense the outside ambient pressure, or (2) with a tube connected to thecontainer’s inside space, so that you can feed in any pressure you wantto measure. Both versions are available with and without temperaturecompensation. The sensor is made of silicon and (I believe) pyrex, bothof which are very stable materials, making a very stable sensor. Thesensor’s biggest drawback is its high sensitivity to temperature, with its

14

chapter 36 | Electronic correction for air pressure variations

257

sensing resistance values changing about 2% for only a 10 C change intemperature. The temperature compensated units are much better,changing about 0.005 in. Hg/C in the 15 psi full scale units.

I ended up using two pressure sensors, one Honeywell 136PC15A1Land one Motorola MPX2100A, both vacuum sealed and temperaturecompensated. One sensor had a positive temperature coefficient andthe other had a negative coefficient. The two sensor outputs were elec-trically summed together in a ratio that provided a net temperaturecoefficient of zero. Limited testing showed that the combined pressurereading was repeatable within 0.015 in. Hg. As to long-term stability,8 years later, the sensors’ combined pressure reading still agreed withthe local Weather Bureau’s reading within 0.01 in. Hg.

Test run

The electronic pressure correction circuit was built and connected to atemperature compensated pendulum with a 2 s period, a 19 lb bob, anda quartz pendulum rod. With the servo loop open, that is, with no drivepulses going to the pendulum, the air pressure’s integrated time error(see Chapter 27) appears in counter A in Figure 36.1 as counts of a con-stant 100 kHz frequency from a quartz crystal oscillator. Dividing thecount in counter A by 190,000 counts/s gives the pressure’s integratedtime error in seconds of pendulum time.

With the servo loop closed and drive pulses going to the pendulum,the pendulum’s time correction for variations in the air pressure isobtained by algebraically summing the number of () and () pulses,each multiplied by its time value of 0.00161 s/() pulse and0.00122 s/() pulse. The pressure correction circuit contains twoextra counters (not shown in Figure 36.1) that count the number ofadvance and delay pulses sent to the pendulum.

Figure 36.2 shows the air pressure over an 11-day interval. Figure 36.3shows the pressure’s resulting integrated time error, which is fed backto the pendulum as a time correction by the drive pulses applied atthe far end of swing.

The scale factor for counter A of 190,000 counts/s of pendulumdelay (or advance) time is found empirically. When the air pressure dif-fers from the average pressure, applying a 190,000 counts/s scale factorto the count in counter A provides the best time correction to the pen-dulum; that is, it gives the straightest straight line, i.e., the least varia-tion (due to pressure change) in the pendulum’s “time error vs time”curve when the curve is corrected by the pressure’s integrated timeerror stored in counter A.

How much a single pulse affects the pendulum’s timing is foundexperimentally. A hundred time advance pulses were quickly sent to


258

the pendulum over a short 3-min time interval, and the resultingchange in pendulum timing was measured as 0.00161 s/pulse.And after 100 time delay pulses, the change in pendulum timing wasmeasured as 0.00122 s/pulse.

Results

The pendulum’s time error with the pressure correction circuit operat-ing is shown in Figure 36.4(a). The maximum time error during the11 days is 0.065 s. If the pendulum is running a little fast (or slow), thetwo curves in Figure 36.4 will have a rising (or falling) slope to them.Because of this, the maximum time error is defined as the maximumdeviation of the curve from a straight line drawn between the curves’two end points.

Figure 36.3. Time correction given to thependulum by electronic drive pulses at the farend of swing.

chapter 36 | Electronic correction for air pressure variations

259

Figure 36.2. Barometric pressure.

29.5

29.0

28.5

1 2 3 4 5Time (days)

Bar

omet

er (

in.)

Hg

6 7 8 9 10 11

+0.4

+0.2

–0.2

01 2 3 4 5

Time (days)Cor

rect

ion

(s)

6 7 8 9 10 11

Figure 36.4. Pendulum’s net time error (a) with pressure correction, and (b) withpressure correction and using the revised pulse scale factors.

+0.6

+0.4

+0.2

1 2 3 4 5Time (days)

Pen

dula

m t

ime

erro

r (s

)

6

(a)

(b)

7 8 9 10 11

The calibration numbers inserted into the manual switches inFigure 36.1 are (190,000 counts/s) (0.00161 s/pulse) 306 counts/()pulse, and (190,000 counts/s) (0.00122 s/pulse) 232 counts/()pulse. Over the 11-day run there were 417 pendulum advance pulses and315 pendulum delay pulses. Calculation after the 11-day run showed thatincreasing the calibration numbers by 15% would reduce the maximumtime error to 0.020 s. This is shown in Figure 36.4 ( b). The revised calib-ration numbers are 360 counts/() pulse and 273 counts/() pulse.

The pendulum’s time error is measured with respect to a radio timesignal from WWV, which provides the frequency and time standards inthe United States.

Conclusions

The electronic pressure correction circuit works well. Without it, theair pressure variations would have caused a 0.43 s maximum error inthe pendulum’s “time error vs time” curve in Figure 36.3. With it, themaximum error was reduced to 0.065 s, as shown in Figure 36.4(a).With the revised calibration numbers, the maximum error would bereduced even further to 0.020 s. This is a 21–1 reduction in the effect ofair pressure variations on the pendulum.

Electronic pressure correction provides an alternative to the mechan-ical bellows approach for pressure correction. Which is better? Nothaving tried the bellows approach (yet), I do not know enough aboutthe accuracy of a bellows pressure correction to say which is better. Butthis chapter at least gives some information on an electronic approach.

References1. C. Bartrum. “A barometic compensator for clock pendulums,” J. Brit.

Astronomical Assoc. 44(6) (1934), 233–8.2. C. Bartrum. “Barometic compensation for clocks,” Practical Watch and

Clock Maker, (March 15, 1929), 55–57.3. W. Notman. “Barometric compensation of precision clocks,” Hor. J.

(March 1959), 154–6.4. D. Bateman. “Electronically maintained precision pendulum clock—

longer term performance,” Hor. J. (October 1975).


260

1 inch = 2.54 cm1 foot = 0.3048 m

1 pound (lb) = 0.4536 kg1 cm = 0.3937 inch

1 m = 3.281 feet1 kg = 2.204 lb

CONVERSION TABLE

INDEX

Air dragbob 82, 195clock case 191pendulum rod 179

Air pressureeffect on pendulum 185, 205electronic correction 255

Allan, D. 37Allan variance 37

Bateman, D. 27, 31, 32, 35, 180Bigelow, J. E. 208, 227Bob shape 87Bohannon, W. 82Bottom, V. 29

Cain, D. 27, 28Case, clock

effect on pendulum 191, 195, 211

Hysteresis, thermal 70, 74

Invartypes 159

heat treatment 159stability 163

James, K. 121

Leeds, L. 28, 121

Materialsbob 66pendulum rod 65suspension spring 99, 147

Photoelectronics 241

Q 27, 179Quartz

drilling holes in 175fastening to 171

Riefler, S.air drag 82bob shape 79clocks 112, 114temperature compensation 8

Shape, bob 87Shortt clock ix, 112, 114Sine wave drive 227Stability, dimensional

bob 57, 74materials 74suspension spring 143, 149

Suspension springdesign 97, 121, 134, 139materials 99, 147, 148stability 139, 143, 149

Temperature compensationbimetal 11gridiron 8history 7invar 10mercury 7

Thermal hysteresis 70, 74

Wallman, H. 29Woodward, P. 27, 35, 40, 41WWV 249

0198529716 Accurate Clock Pendulums

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