I. INTRODUCTIONThe purpose of this article is to investigate

the following questions: What is the ideallocation to impulse a pendulum? What arethe consequences of not impulsing at theideal location from the viewpoint of accuratetimekeeping?

Over the years a lot of attention has beengiven to circular error and not enough to theso-called “escapement” error, which is actual-ly impulse error. Impulse error is the changein the period of a pendulum due to the act ofimpulsing. Impulsing is required to maintainthe amplitude of the pendulum and to take asynchronizing signal from it.

In our space-time continuum three dimen-sions are required to define a point. The ideallocation for impulsing a pendulum is definedby the swing plane, the center of percussion,and the position of equilibrium of the pendu-lum. The position of equilibrium is the posi-tion the pendulum takes when it is at rest.

II. CIRCULAR ERRORAn item to consider for precision timekeep-

ing with a pendulum is maintaining a con-stant amplitude of swing. Most present-daypendulums follow a circular path, so theyhave what is called circular error.

Circular error is the change in the period ofa pendulum due to the fact that a freelyswinging pendulum, following a circular pathunder the influence of gravity, takes longer totraverse a large arc than a small one.

As can be seen in Table 1, the rate ofchange of circular error increases with anincreased amplitude. Thus, to minimize theproblems with circular error, it is best to keepthe amplitude of the pendulum as small aspossible. Later we will use the results of Table1 to try to cancel out the effect of the changein circular error with the change in theimpulse error. It should also be noted that inboth cases (i.e., circular and impulse errors),the “error” itself causes no problem in time-keeping because the pendulum length can beadjusted to compensate for them; the realproblem is the change in the error due toslight variations in the amplitude of the pen-dulum.

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Impulsing the Pendulum: Escapement Errorby Dr. George Feinstein (NY)

Table 1. Circular Error for 1-SecondNondamped Pendulum, Series Solution

DESCRIPTION OF TERMS

Nominal period - The desired period of a pendulum in seconds, the period istwice the time of the pendulum designation (i.e., for a 1-second pendulum theperiod is 2 seconds).

g - Acceleration due to gravity, in centimeters per second per second.Length - Theoretical pendulum length for the nominal period, in centime-

ters. See Eq. (B-1).1 & 2. Semiamplitude - The chosen angle of swing from the position of equi-

librium to the position of maximum displacement, given in degrees and min-utes.

3. Period - The actual period of the pendulum, including the circular error.See Eq. (E-3).

4. Circular error - The increase in the period of a circular pendulum due toan increase in the amplitude of swing, given in seconds per day. See Eq. (E-2).

5. Linearity of circular error - Compared to the 1-minute value.6. Rate of Change - The rate of change of the circular error, which is the slope

of the circular error vs. semiamplitude curve, in seconds per day per second ofarc. See Eq. (E-4).

7. Linearity of rates of change - Rate of change of circular error compared tothe semi amplitude of the pendulum.

8. Maximum velocity - The maximum velocity of the pendulum, which occursat the position of equilibrium, in centimeters per second. See Eq. (B-3).

A version of this article was originally published in the NAWCCHorological Science Chapter 161 Newsletter—Issue 2005-3, June 2005.

III. SWING PLANEThe impulsing force should be perpendicular to the

axis of the pendulum and in the swing plane of the pen-dulum. The axis of the pendulum is the line perpendic-ular to its axis of rotation and passing through its cen-ter of mass. The swing plane is the plane generated byrotating the axis of the pendulum about the axis ofrotation.

Any force that is not in the swing plane and perpen-dicular to the axis of the pendulum produces one ormore of the following effects: twisting, bending, andstretching or shortening of the pendulum. These effectscause extraneous movements of the pendulum thataffect the regularity of its motion.

IV. CENTER OF PERCUSSION(CENTER OF OSCILLATION)

A ball hitting the “sweet spot” of a baseball bat ortennis racket causes no sting or jarring to the handsgripping the bat or racket. The sweet spot is at the cen-ter of percussion of the bat or racket.

If a pendulum is impulsed at its center of percus-sion, there is no reaction at the pivot point or center ofrotation. For a compound pendulum (a real-life pendu-lum) the distance from the center of rotation to the cen-ter of percussion is equal to the length of a theoreticalsimple pendulum of the same period. Another relation-ship between the center of rotation and the center ofpercussion is that they are conjugate to each other.That is, if the pendulum is suspended from its center ofpercussion, its period remains the same, and the origi-nal center of rotation becomes the new center of per-cussion. For this reason it is also called the “center ofoscillation.”

Any impulsing force applied at other than the centerof percussion will disturb the pendulum by causing thependulum rod to flex and vibrate. The closer the loca-tion of the impulsing is to the center of percussion thelighter the required pendulum rod.

V.“Q” IS FOR QUALITY?1

Q is called the Quality factor and is a measure of theefficiency of a pendulum in maintaining its amplitude.From the literature it would seem that a “free” pendu-lum requiring no impulsing and therefore having aninfinite Q would be the perfect pendulum. But from thetimekeeping point of view this is not true.

It is true that “other things being equal” the higherthe Q, the more constant the period of the pendulum,but the corollary that any pendulum with a high Q hasa more constant period than any other pendulum witha lower Q is untrue.

Q is a measure of the relative magnitude of theimpulse required to maintain pendulum amplitude;thus, it is also a measure of the magnitude of the vari-ous energy losses the pendulum experiences. There are

many factors, not measured by Q, that affect the time-keeping ability of a pendulum. For example, Q wouldnot be directly affected by temperature error, errorsdue to buoyancy effects, circular error, or errors due tophase angle changes in the impulsing.

From the above we see that although it is desirableand maybe even necessary to have a large Q for goodtimekeeping, it is not sufficient. A pendulum with ahigh Q can be a good or bad timekeeper depending onits attributes not measured by Q, but a pendulum witha low Q will never be a great timekeeper.

VI.“ESCAPEMENT” - IMPULSE ERROR2,3,4,5,6,7

Escapement error was first described by GeorgeBiddell Airy as Example 7 in his 1826 article “On theDisturbances of Pendulums and Balances, and on theTheory of Escapements.”

There have been qualitative discussions of the prob-lem since then, but no exact and comprehensive quan-titative presentations, due to the extensive computa-tions required.

VII. IMPULSE ERROR, QUALITATIVE REVIEWTo understand the cause of impulse error we will

examine five cases of impulsing the pendulum:A. Impulse at the position of equilibriumB. Impulse at the end of swing, away from the posi-

tion of equilibriumC. Impulse at the end of swing, toward the position

of equilibriumD. Impulse at intermediate phase angle, away from

the position of equilibriumE. Impulse at intermediate phase angle, toward the

position of equilibrium

For this exercise use a 1-second isochronous pendu-lum (i.e., no circular error).

A. The velocity of the pendulum is increased by theimpulse at the position of equilibrium. This increasesthe semiamplitude of the pendulum, but there is nochange in the period because the pendulum is isochro-nous.

B. The pendulum takes 0.5 second to travel from theposition of equilibrium to the end of the swing, atwhich point it is impulsed in the direction away fromthe position of equilibrium. This causes the pendulumto travel some additional distance from the position ofequilibrium, before stopping again, which takes someadditional time; let’s call it ΔtB. After the pendulumreaches this point, it reverses direction and returns tothe position of equilibrium in 0.5 second because it isisochronous. Thus, the total time of travel is 1.0 second+ ΔtB. The term + ΔtB is the impulse error for this case.

C. The pendulum takes 0.5 second to travel from theposition of equilibrium to the end of the swing, atwhich point it is impulsed in the direction toward theposition of equilibrium. This causes the pendulum to

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start traveling toward the position of equilibrium atsome finite velocity rather than zero speed, as if it hadstarted from some point further from the position ofequilibrium. Thus, the time to return to the position ofequilibrium is 0.5 second - ΔtC, and the total travel timeis 1.0 second - ΔtC. The term - ΔtC is the impulse errorfor this case.

D. The pendulum swings from the position of equi-librium to the impulse phase angle, at which point it isimpulsed in the direction away from the position ofequilibrium and continues to the end of the swing.

After the pendulum reaches this end point, it revers-es direction. Because the pendulum is isochronous, itreturns to the position of equilibrium in 0.5 secondHowever, the initial travel from the position of equilib-rium to the end of the swing took 0.5 second + ΔtD

because the semiamplitude traveled was for the pen-dulum with the impulsed energy state; however, ini-tially the pendulum was traveling at the lower pre-impulse velocity until it reached the point of impulsing.Thus, the total time of travel is 1.0 second. + ΔtD. Theterm + ΔtD is the impulse error for this case.

E. The pendulum swings from the position of equi-librium to the end of the swing in 0.5 second Afterreaching this point the pendulum reverses directionand continues to the impulse phase angle, at whichpoint it is impulsed in the direction toward the positionof equilibrium and returns to the position of equilibri-um. The trip from the end of the swing to the positionof equilibrium took less than 0.5 second by - ΔtE

because the semiamplitude traveled was for the pen-dulum with the preimpulsed energy state; however,after the impulse the pendulum was traveling at thehigher postimpulse velocity from the point of impuls-ing until it reached the position of equilibrium. Thus,the total time of travel is 1.0 second - ΔtE; the term - ΔtE

is the impulse error for this case.Even though the above discussion was done for a 1.0

second isochronous pendulum, the conclusions arevalid for all pendulums, balance wheels, and torsionpendulums.

Impulse (escapement) error is due to the time differ-ence in the travel of the pendulum from the position ofequilibrium to the point of impulse, with the initial andimpulsed energy states.

All interference to the motion of a pendulum is aform of impulsing. There is both positive and negativeimpulsing. Positive impulsing (in the direction ofmotion of the pendulum) is energy transferred to thependulum to maintain its energy level, whereas nega-tive impulsing (opposed to the direction of motion ofthe pendulum) is energy taken from the pendulum.

It should be noted that any forces applied to the pen-dulum and acting toward the position of equilibriumreduce the period of the pendulum, whereas forces act-

ing away from the position of equilibrium increase theperiod.

Because damping forces act throughout the motionof a pendulum and are generally symmetric (i.e., actinghalf the time toward the position of equilibrium andthe other half away from the position of equilibrium),they have practically no net effect on the period of thependulum.

However, the positive impulsing force generally actsrepetitively over a small portion of the pendulum’smotion and usually in the same direction, eithertoward or away from the position of equilibrium,depending on the escapement, therefore producing animpulse error.

VIII. IMPULSE ERROR, QUANTITATIVE EVALUATIONThe values of the impulse errors and rates of change

were calculated on a Quattro Pro spreadsheet by usingthe equations derived in the Appendices. The QuattroPro spreadsheet carries 15 significant figures. It wouldhave been better to use a program that carries moresignificant figures, such as FORTRAN, but with carethe Quattro Pro program was adequate for the job.

First, the impulse errors and rates of change werecalculated for the following four cases (see Table 2):

1. Nondamped circular path pendulum using aseries solution of elliptic integrals. This was the firstcase used to calculate impulse error; the equation isthe only pendulum equation of motion that has anexact solution and is the easiest to use. It is the classicequation of motion of a pendulum following a circularpath. The one deficiency of this equation is that it is amathematical equivalent to a perpetual motionmachine. Because there is no damping to reduce theamplitude of swing, any impulse will increase theamplitude of the pendulum. There was a questionwhether this would affect the results.

2. Linearly damped circular path pendulum usingfourth-order Runge-Kutta method. This case includesdamping and a reduction in amplitude, which is coun-tered by impulsing. But this case requires numericalintegration for solution and thus a much greateramount of calculation.

3. Nondamped cycloidal path pendulum usingnumerical linear integration

4. Linearly damped cycloidal path pendulum usingan algebraic solution. Cases 3 and 4 are for a cycloidalpath pendulum, which is isochronous and thus has nocircular error and were used to ensure that circularerror was correctly eliminated in cases 1 and 2.

These calculations were done for Q = 10,000 andsemiamplitude = 240 minutes. All cases were calculat-ed at the same energy levels, and the circular pathsolutions were corrected for circular error. The calcula-tions for the four cases used totally different mathe-matical techniques; however, the results came out vir-

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tually identical (see Table 2). Because of this a decisionwas made to perform all the rest of the calculations byusing the equations for the nondamped circular pathpendulum because it was the simplest to work with.

The impulse errors and rates of change of theimpulse error (RCIE) were calculated for five values ofthe quality factor Q, from Q = 100 to Q = 1,000,000. Foreach value of Q, calculations were done for five valuesof the semiamplitude, from 30 minutes to 480 minutesof arc, which covers the range for most presently usedpendulums.

The impulse is provided once per swing by increas-ing the velocity at the required point in the swing.

From these results it was found that impulse error(see Figure 1) is a function of two variables: (1) thequality factor Q, which defines the magnitude of theimpulse, and (2) the relative impulse location. The rel-ative impulse location is obtained by dividing thephase angle of the impulse by the semiamplitude of thependulum (phase angle/semiamplitude).

Because of this, even though the impulse error isindependent of the semiamplitude, the RCIE isinversely proportional to the semiamplitude (seeFigure 2 and Table 3). We see this by noting that for

any of the lines of data inTable 3, for a particular rela-tive impulse location (column1) the impulse error (column2) is constant for all semi-amplitudes, whereas theRCIE (columns 4, 6, 8, 10, and12) is reduced with anincrease in semiamplitude.That is, the greater the semi-amplitude the smaller theRCIE. We also see that theRCIE grows very rapidly nearthe very end of the swing.This is the reason that pendu-lums impulsed there are verypoor timekeepers.

During 90 percent of theswing of the pendulum, theimpulse error is nearly linear-ly inversely proportional to Qwith a deviation of less than 5percent; however, near theend of the swing the relationbecomes nonlinear andchanges to a relation of thesquare root of the inverse of Qat the very end of the swing.

The reason for the above isas follows: The impulse, nomatter where in the swing itis regularly applied, has toprovide the same amount of

energy to maintain the amplitude of the pendulum. Themoving pendulum has two forms of energy: potentialand kinetic. The potential energy is due to the locationof the center of mass above some fixed datum, whereasthe kinetic energy is due to the motion of the pendulumand is a function of the square of the velocity of the cen-ter of mass of the pendulum. The impulse increases thekinetic energy of the pendulum. When the pendulum istraveling at a relatively high velocity (e.g., near theposition of equilibrium), the change in velocity to pro-vide the impulse energy is very small, and it has a verysmall effect on the period of the pendulum. But whenthe pendulum is at or near the ends of the swing (i.e.,moving very slowly), the change in velocity is muchgreater and so is its effect on the period of the pendu-lum. The kinetic energy of the pendulum, as stated ear-lier, is proportional to the square of the velocity of thependulum, whereas the period of the pendulum is pro-portional to the velocity. This is the cause for theimpulse error at the ends of the swing being propor-tional to the inverse of the square root of Q.

For over 90 percent of the swing, mentioned earlier,the average impulse error is essentially the samewhether the escapement impulses the pendulum once

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Table 2. Comparison of Impulse Error for Circular and CyclodialPendulums, Both Linearly Damped and Nondamped

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Figure 1. Escapement error for one impulse/swing. Figure 2. Relative rate of change of escapement error.

Table 3A. Impulse Error for 1-Second Nondamped Circular Pendulum,Eliptic Integral - Series Solution

in every swing, as in thesecalculations, or once everyother swing, or even once in30 swings as in some escape-ments. It is important to notethat even though the averageimpulse error is practicallythe same, the actual impulseerror only occurs in the swingthat the impulse is applied.

Impulse error can be posi-tive or negative, dependingon the direction of theimpulsing force. An impuls-ing force acting away fromthe position of equilibriumproduces a positive impulseerror, whereas one toward theposition of equilibrium pro-duces a negative impulseerror. A positive impulseerror increases the period ofthe pendulum, but a negativeone reduces the period.

IX. IMPULSE ERROR, CONCLUSIONSBy comparing the rate of change of the circular error

(RCCE) (see Table 1) with the RCIE (see Table 3) wesee that the optimum semiamplitude is dependent onthe value of Q for the clock and the relative impulselocation because increasing the amplitude increasesthe RCCE but decreases the RCIE.

The ideal would be to eliminate or at least to mini-mize the total change in the circular error plus impulseerror by matching the RCCE and RCIE by adjustingthe semiamplitude for a pendulum with a specificvalue of Q.

We can theoretically eliminate the combined changein circular error plus the impulse error. To do this we

would need an escapement with two attributes: (1) Theimpulse must be toward the position of equilibriumand (2) The relative impulse location must beadjustable.

Then we could choose a combination of amplitudeand phase angle (relative impulse location) such thatwhen the amplitude varied, the increase or decrease inthe circular error would be corrected by the decrease orincrease in the impulse error.

For example from Table 1 the RCCE for a semi-amplitude of 4o is 0.00366 sec/day/asec, whereas fromTable 3B for a Q = 10,000, a semiamplitude of 240' (4o)for an impulse toward the position of equilibrium anda phase angle of 216' the RCIE is -0.00362 sec/day/asec;

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Table 3B. Impulse Error for 1-Second Nondamped Circular Pendulum,Eliptic Integral - Series Solution

a slight increase in the phase angle or decrease in thesemiamplitude will make the two changes exactly can-cel out each other.

X. APPENDIXFor the equations used to obtain the numerical

results in this article and their derivation with a com-plete set of results, see the online appendix atwww.nawcc.org/pub/bulsupp.htm or contact theNAWCC Library and Research Center.

About the AuthorA retired civil engineer by profession, Dr. George

Feinstein has spent his career in various phases ofmass transportation. He is very active in ElectricalHorology Chapter 78 where he has been the co-editor ofthe chapter newsletter since 1987. He has contributedseveral articles to that publication, including descrip-tions of the Fedchenko clock as well as a list of U.S.patents in electrical horology.

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Table 3C. Impulse Error for 1-Second Nondamped Circular Pendulum,Eliptical Integral - Series Solution

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Notes1. D. A. Bateman, “Accuracy of Pendulums and Many

Factors that Influence It,” NAWCC BULLETIN, No. 290(June 1994): pp. 300-312.

2. G. B. Airy, On the Disturbances of Pendulums andBalances, and on the Theory of Escapements,Transactions of the Cambridge Philosophical Society,Vol. 3, 1830: pp. 105-128.

3. Prof. R. A. Sampson, Studies in Clocks and Time-keeping, Proceedings of The Royal Society ofEdinburgh.

a. No. 1. Theory of the Maintenance of Motion, Vol. 38,1918: pp. 75-114.

b. No. 2. Tables of the Circular Equation, Vol. 38,1918: pp. 169-218.

c. No. 3. Comparative Rates of Certain Clocks, Vol.44, 1924: pp. 56-76.

d. No. 4. The Present-day Performance of Clocks, Vol.48, 1928: pp. 161-166.

e. No. 5. The Suspended Chronometer, Vol. 56, 1935:pp. 13-25.

f. No. 6. The Arc Equation, Vol. 57, 1936: pp. 55-63.4. Prof. David Robertson, “The Theory of Pendulums

and Escapements,” Horological Journal.1928: Sept., pp. 11-16; Oct., pp. 45-48; Nov., pp. 70-73;

Dec., pp. 101-105.1929: Jan., pp. 128-131; Feb., pp. 160-163; March, pp.

194-196; Apr., pp. 215-218; May, pp. 236-240; June,pp. 254-256; July, pp. 271-274; Aug., pp. 293-295;Nov., pp. 51-54.

1930: Feb., pp. 116-119; March, pp. 139-140; Apr., pp.153-156; June, pp. 191-193; Aug., pp. 219-222;Sept., pp. 16-18; Nov., pp. 52-54.

1931: March, pp. 129-130; Apr., pp. 147-149; Sept., pp.16-18; Nov., pp. 40-42; Dec., pp. 68-72

1932: Feb., pp. 96-97.5. E. C. Atkinson, The Proceedings of the Physical

Society, London.a. Escapement Errors of Pendulum Clocks, Vol. 42,

1930: pp. 58-70.b. The Relation Between Rate and Arc for a Free

Pendulum, Vol. 48, 1936: pp. 606-617.c. On Airy’s Disturbance Integrals and Knife-edge

Supports for Pendulums, Vol. 48, 1936: pp. 618-625.d. The Dissipation of Energy by a Pendulum

Swinging in Air, Vol. 50, 1938: p. 721-741.e. The Amplitude Deviation of Rate of a Pendulum: A

Second Experiment, Vol. 50, 1938: p. 742-756.6. Paul F. Gaehr, Airy’s Theorum and the

Improvement of Clocks, American Journal of Physics,Vol. 16, 1948: pp. 336-342, 519-520.

7. A. L. Rawlings, The Science of Clocks & Watches(British Horological Institute, 3rd Ed., 1993): pp. 163-185.

8. V. M. Faires & S. D. Chambers, Analytic Mechanics(The Macmillan Co., 3rd Ed., 1952): pp. 359-363.

9. H. W. Reddick & F. H. Miller, AdvancedMathematics for Engineers (John Wiley & Sons, Inc.,3rd Ed., 1955): pp. 128-133.

10. I. S. Gradshteyn & I. M. Ryzhik, Table ofIntegrals, Series, and Products (Academic Press, Inc.,5th Ed., 1994): pp. 907-913.

11. B. O. Peirce, A Short Table of Integrals (Ginn &Co., 3rd Ed., 1929): pp. 66-67.

12. C. H. Edwards & D. E. Penney, DifferentialEquations & Linear Algebra (Prentice Hall, 2001): pp.564-566.

13. S. C. Bloch, Classical and Quantum HarmonicOscillators (John Wiley & Sons, Inc., 1997): pp. 93-106.

14. W. E. Boyce & R. C. DiPrima, ElementaryDifferential Equations and Boundary Value Problems(John Wiley & Sons, Inc. 5th Ed., 1992): pp. 444-445.

15. D. Greenspan, Discrete Numerical Methods inPhysics and Engineering (Academic Press, Inc., 1974):p. 40.

16. F. B. Hildebrand, Introduction to NumericalAnalysis (McGraw-Hill Book Co., 2nd Ed., 1974): pp. 291-292.

17. P. A. Tipler, Physics for Scientists and Engineers(W. H. Freeman & Co. 4th Ed., 1999): p. 423-424, 43 1-432.

18. S. L. Loney, An Elementary Treatise on theDynamics of a Particle and of Rigid Bodies (Cambridgeat the University Press 1950): pp. 108-110.

19. W. D. MacMillan, Theoretical Mechanics; Staticsand the Dynamics of a Particle (Dover Publications,Inc., 1958): pp. 319-321.

20. L. A. Pars, Introduction to Dynamics (Cambridgeat the University Press, 1953): pp. 214-217.

21. B. Williamson, An Elementary Treatise on theDifferential Calculus (Longmans, Green, & Co., 9thEd., 1899): pp. 335-339.

22. L. N. Hand & J. D. Finch, Analytical Mechanics(Cambridge University Press, 1998): pp. 90-94.

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