The Equivalent Static Accelerations

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NAVAL RESEARCH LABORATORY WASHINGTON, .C .



NRL REPORT N O . 0-3302

T H E E Q U I V A L E N T S T A T I C A C C E L E R A T I O N S O F S H O C K M O T I O N S J . P . W alsh and R. . Blake

June 21, 1948

n V O R P BEI Approved by :

O r. H. M . Trent, Head, Applied M athematics Section O r. E . B . Stephenson, Superintendent, M echanics Division

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C ON T E N T S

Abstract i

Problem Status i

Authorization i

I N TRODUCT ION

R E S P ON S E O F SINGLE-DEGREE-OF-FREEDOM SY S T EM S T O SHOCK MOT IONS 1

THE DERIVAT ION O F SHOCK S P E C T R A ROM R EED - G AG E DATA EX AM P L E S O F SHOCK S P E C T R AS IGNIF ICANCE O F HOCK S P E C T R A

NAVY SHOCK -MACH INE P E C T R A

S U MM A R Y

A P P END IX I NO T E S O N T HE R E ED GAGE 0



A BS T RA CT T he use of the reed gage as a method of determining important char-

acteristics o f a shock motion is discussed. t is shown that he'shock spectrum," erived rom th e reed-gage record, provides design condi- t ions for "shockproof" equipment in th e form o f equivalent static accelerations f he hock motion, an d a rational method for comparing th e severity

o f different

shock

motions .

Shock spectra are presented for certain theoretical motions as well

as for th e motions of Navy shock-testing machines. ifferences between jthe machines are pointed o ut an d it is noted that th e predominant characteristic o f th e shock-machine motion is a sudden velocity change.

P R O B L EM S T A T U S

This is an interim report o n this problem^ wor k is continuing.

AUTHOR I ZATI ON N R L Problem No . F 03-06 R .

vi



THE QUIVALENT STAT IC ACCELERAT IONS OF SHOCK MOTIONS

INTRODUCTION

When contemplating th e design of a structure which must be shock-proof," he designer seeks design conditions which can be used to calculate th e sizes of th e members of th e structure. Because ot training and past experience, t s ikely that he will ttempt to transform he ynamic nertia oads cting n he tructure, nto equivalent tatic loads oads which, f polied tatically, will roduce imilar eflections, tresses, ana lailures. n th e reports of investigations on laboratory shock machines and aboard ship, ecords of th e acceleration imparted to various parts of these tructures under shock conditions are found. n general, these records show that th e acceleration reached peak values many times the acceleration due to gravity - in some cases thousands of times as much. nowing that systems can no t be designed to carry several thousand times their own weight, the designer chooses some arbitrary number (say 50, 00 , or 50) and proceeds on the assumption that the equivalent static load acting on th e structure is its tatic weight multiplied by this arbitrary number.

It is th e purpose of this paper to point out: a) that th e effect of a given shock motion is not th e ame fo r all structures and b), that th e load fo r which a structure should be designed depends not only upon th e hock motion to which it is to be subjected, but also upon its natural frequencies; and to present data useful in th e design of structures.

RESPONSE OF S INGLE -DEGREE -OF -FREEDOM

SYSTEMS

TO SHOCK

MOT IONS

T he concept of the response of a single-degree-of-freedom system to th e hock motion under consideration is of fundamental importance. That this is true will be apparent later. A single-degree-of-freedom-system (Figure ) is defined as a rigid mass (M ) supported by a massless linear spring. he motion or response of th e ystem can be completely described by th e displacement of the mass in a direction parallel with th e axis of th e spring.

X = x 1-x 6 Fig. A single-degree-of-freedom system



NAVAL RESEARCH LABORATORY single-degree-of-freedom system has o ne natural frequency. t is often described as a "simple system "and will be so designated in th e remainder o f this report. necessary assumption throughout this report is that the motion (Xj) o f th e foundation is unaffected by th e motion o f the structure which it supports.

There are several methods o f finding th e response o f a simple system (Figure ) to shock excitations. f an analytic expression for the shock excitation is known, the response o f th e simple system ca n be obtained by Duhamel's integral.

1 or each natural frequency assigned to th e simple system, a response to the excitation ca n be computed.

Wh en th e excitation is given only in graphical form (an acceleration time curve taken aboard ship for example) there are tw o convenient methods of determining th e response o f simple systems to such excitations. he first is a graphical method of evaluating Duhamel's integral, described in detail in reference

. B y this method, th e maximum value of the

acceleration o f th e mass (M ) can be found for an y natural frequency o f th e system when X j, th e acceleration o f th e base o f the simple system, s given as a graphical record or an analytical expression which ca n be plotted.

T he econd method is by use o f a torsion pendulum. Blot* has sho wn that if th e support o f a torsion pendulum (Figure 2) s given an angular displacement d proportional to X j (the acceleration o f the base o f th e simple system (Figure then t th e absolute angular displacement of th e bo b o f th e torsion pendulum) is proportional to the relative displacement (X ) in the simple system. In th e discussion o f th e design o f earthquake-proof buildings. Blot found the maximum value o f th e acceleration o f th e mass o f a simple system (x max)

subjected to a given earthquake acceleration-time record. his maximum acceleration w as plotted against th e natural frequency of th e simple system, an d this p l o t w as called an "earthquake spectrum."

Fig. 2 - Diagram o f torsion pendulum

If a series o f simple systems could be constructed an d subjected to an actual shock motion, an d if th e relative motions (X ) were recorded, «shock spectrum could be plotted similar to th e earthquake spectrum. T he reed gage Figure 3) , designed at the David Taylor M o d e l Basin approximates such an instrument. he reed gage, as its name implies, is composed o f a series of cantilevers carrying masses at their

free ends.

Each

cantilever

has a different

natural frequency in th e first m od e . he maximum value o f th e displacement o f th e free end, elative to th e fixed end, s recorded by scribing on waxed paper, TH E DE R I VATI ON O F SHOCK S P E C T R A FR OM R E E D - G A G E DATA

In order to obtain th e shock spectrum of a given shock by us e of th e reed-gage, th e excitation is applied to th e

1 T. V on Karman nd M . A. Blot , Mathematical methods in engineering, McGraw Hill, 1940), . 403.

2 G . E. Hudson, A method of estimating equivalent static loads in simple elastic structures, David W . Taylor M o d e l Basin Report N o. 07 , June 1943. 3 M . A. Blot, Bulletin o f th e seismological society of America, 31 : 151-171, (1941)



NAVAL RESEARCH LABORATORY

base f he age, nd he maximum eflections of he eeds elative o he ase re recorded.

If th e e « > d s are considered to be simp'e systems, fo r each reed there is a static deflection ös(due to th e action of gravity alone) whicnis related to th e natural frequency fn of th e reed by th e amiliar equation

f = .1 3 n

l/fcps (1)

Let X be th e deflection of a reed tip relative to th e base of th e instrument under shock.

Then max 5 .

= N (2 ) W here N is th e "equivalent static acceleration." T he equivalent static acceleration is th e gradually applied acceleration, expressed as a multiple of th e acceleration due to gravity, to which th e reed must be subjected in order to produce th e same deflection of th e tip as was produced under shock.

Th i s value of N is plotted against th e primary natural frequency of th e reed. his plot of N against fn is called th e shock spectrum" of th e shock to which th e reed gage was subjected.

EXAMPLES O F SHOCK S P ECTRA

Figures 4 and 5 show th e shock spectra for some simple motions which can be expressed and analyzed mathematically. hese and many other imilar curves have been published by several writers.

«a- Curve , igure 4 , is th e shock spectrum fo r a half-sine pulse of acceleration defined by equations (3).

x

a sin

2 T r t

fo r

fo r

oS.S-I and

t< 0 andt>Y (3 ) N/a is th e atio of th e peak acceleration of th e mass of a simple system to peak acceleration, a, of th e applied shock motion nd fn is th e natural frequency of th e simple system. A ss igning values to a and T reduces th e coor- dinates to th e conventional ones (i.e., N and fn).

Curve n shows th e effect on th e shock spectrum of having th e in e pulse of curve I

Fig. Reed gage, (cover removed), show- in g reeds and records scribed on wax paper




s i li.

i - * A -in

e «

ft r ^ n -li W u

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\ * \ r C

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V 1

» .1 «i MA L FMO XI 1 MRUV • I OMO I MILK

continue on as a damped transient vibration. Thi s motion is defined by

xj = 0

and x e -bt

fo r t<0

a sin y) fort O (4 )

In th e curve shown, b is appropriate fo r 0.5

percent of critical damping and th e phase angle y is practically zero. h is curve is very similar to a steady-state vibration trans- missibility curve.

Curve in is th e shock spectrum fo r a "square"pulse of acceleration defined by

Xj or K0, t>|-

" ^ or O^t^-y

and

(5 ) Figure 5 is th e shock spectrum fo r a step

velocity change defined by

0 or <0 and

Fig. 4 - Shock spectra fo r three simple hocks

n wtmti * • » * *

Fig. 5 - Shock spectra fo r velocity change

^ V or > 0 (6 ) Various values of th e velocity change (V)

have been used to obtain th e family of curves shown. T he equation of this family of curves is given by

v{ V N= — ' 7)

SIGNIFICANCE O F SHOCK S P ECTRA

Shock spectra are important because they can be used to calculate whether or no t a given shock will

damage a specific strurture. n order to us e th e shock spectra in designing a structure, it is necessary to assume that th e structure is elastic and linear up to th e point of failure, and is subject to th e same shock motion as th e reed gage.

T he kinds of damage to a structure which are predictable from th e us e of shock spectra are: he yielding of ductile materials, the fracture of brittle materials, and th e collision of adjacent parts through elastic distortion of their supports. Types of failure not predictable from shock spectra include: reaking of a friction bond, as in th e loosening of screws or opening of knife witches; pilling of liquids; fracture of

ductile materials; and collision of unsecured parts.

When a structure can be assumed to be a simple system (i. e., a single degree of freedom ys tem) and




th e shock spectrum is available, it is an easy matter to calculate its natural frequency and to determine th e stress and deflection which will be caused by th e shock. ld s calculation is simple because th e shock spectrum gives th e equivalent static acceleratipn from which th e equivalent static load may be computed. ince static stress is proportional to static load, th e shock spectrum enables on e to predict stress, and hence damage .

Similar calculations of stress and deflection are possible fo r a multi-degree-of-freedom structure if th e flexible members of th e structure are linear and elastic. he method of calculation s given in reference (4 ) and will be discussed in detail in a future report. he

important parts of th e method are th e following: irst, t is necessary to find th e shapes of th e modes of vibration of th e structuce, and ihejiatural frequencies associated with them. T he amplitude of each mode excited by a shock motion is proportional to th e number of g in the shock spectrum corresponding to th e natural frequency of th e mode . ince each moae is nalogous to a simple system, th e total stress and deflection of th e structure can be found by superposing th e response of th e various modes .

T he hock spectrum is an excellent index of th e damaging capacity of a shock, fo r if th e vulnerability of a structure to two different shock motions is under consideration, th e spectrum with th e greater value at th e natural frequency of th e structure is th e on e which will produce th e greater stress or likelihood of damage . he shock spectrum is also an excellent index of th e relative protective abilities of two or more shock mounts. ne hock mount is considered th e best of a lot in th e performance of its protective function, if th e motion of a mass mounted on it has , during shock, a lower shock spectrum than that of any

of th e others. I T he hock spectrum presents a shock motion in a graphical form which can be easily understood. nthe other hand, by looking at th e velocity-meter records. igures 1 and 12 , on e can see that the motion at th e center of th e 4A plate is different from that at th e edge; but it s not apparent what the effect of this difference will be upon structures to be tested. T he shock spectra Figures 6 through 10 show the significance of this difference, which will be discussed in th e following section.

NAVY SHOCK-MACHINE S P E C TRA

i

Reed-gage records were obtained on th e Navy Light-W eight Shock M achine, prior to its modification in 94 7 ? by mounting th e gage on th e 4A plate and subjecting th e anvil plate to 1ft, 2 ft , ft , 4 ft , and ft back blows. he shock spectra obtained of th e motion of th e

center of th e 4A plate are shown as dotted lines in Figures 6 through 0. T he reed gage

was also placed at th e edge of the 4A plate directly over th e channel support of th e A plate. T he shock spectra of the motion at this location are shown in Figures 6 through 10 as solid lines. hese spectra were obtained by connecting th e observed points with straight lines. T he motion at the two locations on th e 4A plate as recorded by British type velocity meter are shown in Figures 11 and 12 . eaks occurring in th e shock spectra are due to near- resonance of particular reeds with transient vibrations of th e anvil and 4 A plate. he width and height of these peaks are no t well defined ecause of th e small number of reeds used. Since th e frequency of th e transient vibration probably d id not coincide with ny reed frequency, th e true peaks are probably higher than shown and resemble, in shape, th e peak of Figure 4, urve ü.

Reed-gage records obtained on th e Navy M edium-W eight Shock M achine are shown in Figures 13 through 15 he important features of the motion of the anvil plate of this machine M . A . Blot , Natl. Acad . Sei. (U.S.) 9: 62-66 1933)



6 NAVAL RESEARCH LABORATORY Figures 6 through 10 indicate shock spectra fo i he Navy Light-W eight Shock M achine.

Dotted line-center of 4A plate. Solid line edge of 4A plate over channel.

WOO 9

3 i ft s

/

A r

A / f s

y

h J s

\ i i L y y

200 00 •00 00 00 0 fn Niturol » q. (eft)

ttoo

t*oo J f

r

1 J / / tN8 ..00

i

//

/

K j | y i

t

\ j *

I r j i \

/ /

/ s

i

s y. y

to o 00 fn ««ral n «. ( »M l

Fig. 6 1-foot back blow Fig. 7 - 2-foot back blow

\ uoo /

/

/ /1 , ' 1« 1 } f s

s

—-

/

hi j i

i \ \ 1 1 1

1 i *00

I f \I 1

I j i

f A \S ;

IO 0 00 00 00 000

Fig. 8 - 3-foot back blow Fig. -foot back blow




;/| ^ 1/ - J

i \ :/ i \ /

/ i < BO00 * - I i .

j t-

/r

r ~ 7 / j r

J i

i a oo; i /i i KJ i

r i —

400-- v-/ h A v

in

Fig. 10 5-foot back blow

are tw o successive velocity changes - o ne due to th e impact of the hammer, th e second to the impact o f th e anvil plate against rigid stops. he effect o f tu e second impact (velocity change) upon th e reeds (already vibrat- ing from the first impact) depends upon th e phase o f th e reeds at th e m om e n t o f th e econd impact. he ampli- tudes o f some reeds are increased by the econd velocity change while those o f other reeds m ay be unaffected or decreased; hence th e ecord shows some deviations from th e straight-line characteristic of an instantaneous velocity change.

A comparison o f Figure 5 with th e hock spectra o f th e hock machines shows that th e predominate characteristic o f the shock-machine motion is an instantaneous velocity change. These records demonstrate that designing for some fixed value o f g without regard for the natural frequency o f th e structure being designed is no t correct an d cannot be justified. Wh en it is desired to reduce the description of a shock machine motion to a simple equivalent, this ca n be done with ess error by finding the velocity change th e spectrum o f which lies just over the shock spectrum.

The spectra o f th e motion of the center of th e 4A plate n the Navy Light-Weight Shock Machine shows clearly an important characteristic of th e motion at this point—the relatively high

aequivalent-static-accelerations "to which

th e 100 cp s systems are subjected. This high acceleration is a result of th e transient [^•dTCB OF 4A P. ATE

M iiin.auilHU titilU

&" lACK BLOW t-'fipn iinwtMTii* , .

imiuiituKitUuitiuMHbUiU iiiu RACiC

J 9 * :« I 0 •

Fig. 1 Velocity time record o f motion o f center o f 4A plate Navy Light-Weight Shock Machine



VlLOC.TV NCTC« »r tD3E Of -iJiT-l

C* ACH »tO«

l«CK BLOW

3 HACK -.,. :

5" PACK BL 0 .r .a ^r .'.««« -a

TrMr Ih'fflVJiL BETWEEN TPACF IN'EHh

Fig. 2 - Velocity t ime record of motion of 4A plate over supporting channel Navy Light-W eight Shock M achine

1 8 A

HI KOO

f \/

v

I IIOO /

/

i ,r— J /

40C

y i : M * ) t *

ihm f 1 0 66

1 I oo /- .

A N s

/

v X

/ /

A V

/ /

100 00 00. •00 00 0 Html r., I OM I

Fig. 3 - ühock spectra of th e motion of th e ig . 4 - Shock spectra of th e motion of th e M edium-W eight Shock M achine, -inch table edium-W eight Shock M achine, -inch table travel -foot hammer crop, 00-pound load ravel, - foot hammer drop, 00-pound oad on table n table



NAVAL RESEARCH LABORATORY9

/

ssoo /

d * 8 400 £ S at

/:

:/i 1

N 1 i i,

A i

/\J V

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•00 000 NOlUrdl rr»q tcp»)

Fig. 15 Shock spectra o f th e motion o f the Medium-Weight Shock Machine, L54nch table travel, 1-foot hammer drop, 500-pound load o n table

vibration o f the 4A plate, which is sho wn on the velocity-time curve o f this motion,(Fig- ure 2). It has been pointed o ut earlier that the maximum height o f the peak in the vicinity o f 100 cp s is no t known exactly. owever , this peak, if found exactly, would have a shape similar to curve II o f Figure 4. It has been shown, then, that the Light-Weight Shock Machine, because

o f the vibration

o f th e 4A

plate, an cause systems having a natural frequency in the vicinity o f 100 cp s to be subjected to at least f ive times the acceleration o f systems having a natural frequency o f 50 cps. T he shock spectra for locations at th e edges o f the 4 A plate over the channels do no t exhibit this peak at 100 cps; neither do the shock spectra for th e Medium-Weight Shock Machine. When the response o f systems having approximately 350-cps natural frequencies are examined, other marked differ- ences are seen. t the center o f the 4A plate o n th e Light-Weight Shock Machine, th e equivalent static acceleration o f th e 350-cps system has about he same value as th e 5 0- cps system, several times less than that o f th e 100-cps system. However , th e equivalent- static-acceleration for the 350-cps system o n the Medium-Weight Shock Machine or over the channels o n the Light-Weight Shock Machine, has continued to rise from the value for th e 100-cps system with the general characteristic o f an instantaneous velocity change.

Now since the spectra are measures o f th e damaging effect o f a shock motion, they demonstrate that different results from the shock test machines ca n be obtained, depending no t only o n which type machine is used but, also, in th e case o f th e Light-Weight Shock Machine, where th e component is mounted. SU MMA RY

T he importance o f th e shock spectrum," which ca n be derived from a reed gage record o f a shock motion, has been discussed. It has been pointed o ut that th e shock spectrum o f a shock motion gives th e gradually applied accelerations which will produce deflections and stresses in a simple system equal to those produced by th e shock. These equivalent static accelerations are a function o f th e natural frequency o f th e simple system an d are th e proper basis for design o f structures to withstand shock. T he practice o f designing structures fo r a single, arbitrarily chosen, equivalent static acceleration, without regard to natural frequency, s sho wn to be incorrect by presenting shock spectra obtained o n Navy High-Impact Shock Machines. These spectra show that th e best approximation o f th e shock machine motion is a sudden e ocity change.

Shock spectra give unique basis ior comparing th e relative damge capacity o f differ- en t shock motions. Shock spectra are being used to study th e performance o f shock mounts.



10 AVAL RESEARCH LABORATORYComparison o f the shock-machine spectra shows that the damage capacity o f the tw o types o f Navy Shock Machines differ an d that in th e case o f the Light-Weight Shock Machine signifi- cantly ifferent spectra are obtained for different locations o n the test-specimen mounting fixture.

A PPENDIX I NOTE S ON TH E R E E D G A G E

T he present reed gage (Figure 3) diverges from the theoretical ideal in several respects. Corrections ca n be made for some deviations o n a theoretical basis, bu t other errors are inherent in th e instrument. T he following observations are the result o f experience in using and applying th e reed gage.

1. he accuracy with which a deflection ca n be measured is no t satisfactory for the high frequency reeds. It is believed from experience, that reed deflections cannot be measured closer than 0.01 inches. This ca n amount to an error o f the following number o f ~gdepending upon the frequency.

Reed Natural Frequency rror ( in g 's ) 20 40 40 .39

100 .8 2 210 7.70 34 5 20 430 85 570 16

These errors are du e largely to th e use o f rounded point styli o n waxed paper to record deflections. T he line scribed is broad an d poorly defined. Suggested improvements have no t yet obtained the desired simplicity, freedom from friction drag, an d ability to operate during shock. There is also an error because the scribed deilection mark is an arc rather than a straight line. This is no t important, since it predominates in the lo w frequency reeds an d is a small error in terms o f equivalent static acceleration. Deflection is customarily measured as th e perpendicular distance from the en d o f the arc to the zero-deflection line.

Occasionally, shifting o f the recording paper, when it is removed from under th e reeds after a shock, makes the zero position o f the reeds difficult to locate.

2 . he mass o f an actual reed is distributed rather than concentrated at th e tip. T he proper point at which to measure the deflection is no t thecenter o f the weight at thetip o f th e reed, bu t some effective center o f mass. It is possible to find the ratio of th e deflection o f a reed at th e point measured, to th e deflection which a simple system o f the same frequency would undergo.

5 he following table gives numbers, corresponding to standard reeds o f th e gage, which should be multiplied by th e measured reed deflections to find th e deflections which a simple system would have recorded.

9 E. Z. Stowell, t al. Bending and shear stresses developed by th e instantaneous arrest o f the root of a cantilever"beam with a mass at its tip. N ACA M R N o. 14K30




Reed Frequency orrection Factor 20 97 40 845 10 0 860 210 93 345 98 430 97 5 70 945

T he above effect of distributed mass is quite different from th e effect of higher modes of vibration of th e reeds. T he deflection of th e reed tip due to th e second and higher modes is, except in rare cases of resonance, a negligibly small fraction of th e tip deflection due to th e first mode.

3 . Friction of th e scribers on th e waxed paper causes damping of th e reeds . Thi s is certainly no t th e same amount and type of damping to be found in th e practical structures which th e reed gage is supposed to represent, though it would be difficult to specify an ideal damping characteristic. Examination of th e transient, traced upon a special moving paper, as a result of a velocity-change shock shows th e damping to be coulomb (rubbing) friction. T he decrement of amplitude per cycle was uniform regardless of amplitude and is tabulated in th e following table.

Reed Decrement P er Decrement P er Frequency (cps) Cycle (inches) Cyc le (g)

20 .132 5.28 40 .115 16.0 100 .014 12.4 210 .004 15.1

F or shocks which will build up th e simple system amplitude by resonance, amping will cause a large error. Except in such cases, th e error is negligible. In th e case of resonant build-up of vibration, t is no t y et clear whether or no t a simple undamped system represents an actual structure any better than th e friction-damped reed. f a practical structure is to fail by successive increases in its amplitude of motion because of resonance with a transient vibration in th e shock motion, it must, on some cycle, pass from wholly elastic deformation

into a slight plastic deformation. This orovides a considerable increase in th e damping an d may thus prevent th e deformation from exceeding an acceptable value. urther investigation of this important question is required.

4. T he number of reeds which can be accommodated in th e reed gage frame is to o small to define satisfactorily a shock spectrum. he present gage is already too large fo r some applications. T he difficulty is largely inherent in th e nature of th e gage since a large space is required fo r otions of th e low- frequency reeds. Shorter low-frequency reeds would be more non-linear than th e present reeds since their deflection would be a larger fraction of their length, and since lighter reeds are affected more by scriber friction. Perhaps th e solution lies I n having several reed gage designs - ach fo r a special type of application.

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The Equivalent Static Accelerations

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