correlation of sinusoidal and random vibrations

Correlation of Sinusoidal and Random Vibrations B. M. HALLt AND L. T. WATERMAN:~ MISSILES AND SPACE SYSTEMS E N G I N E E R I N G D E P A R T M E N T DOUGLAS AIRCRAFT COMPANY, INC., SANTA MONICA, CALIFORNIA

T H E R E is probably no subject in the field of shock and vibration

which is as controversial or as promi- nent as the subject of random vs sine- wave vibration testing. The purpose of this paper is to present detailed equations showing a method of finding the equivalence between random and sine- wave testing. On this basis one may then proceed rationally to the use of either the sine wave or the random vibration technique.

Before diving into the mathematics of the problem, it would be advisable to review some of the circumstances which make this investigation such an urgent one. In the field of aircraft and missiles, the rapid transition from propeller-driven vehicles to jet- and rocket-powered vehicles has resulted in a vibration environment which is essentially random. For the thousands of items of structure and equipment which were designed to military speci- fications such as MIL E 5272, there immediately arises the question of their suitability for operation in random vibration fields. This question could of course be answered by a gigantic requalification program along with a complete switch from sine-wave facilities to random facilities. The cost of such a program is staggering and upon closer examination the complete switch is not particularly desirable, since sine-wave qualification tests are still a valuable tool for locating and examining the nature of resonances in equipment. The common sense approach to the problem is therefore a program which maintains the sine- wave test technique but has in addi- tion random tests on large assemblies

and structural specimens. These test units would usually be assembled at one of the large companies or at a large government test center where random facilities are rapidly becoming available. The common sense approach also needs a suitable theory to correlate the levels to be used in each type of testing.

PRINCIPLE OF CORRELATION

The principle of correlation de- veloped in this paper is that each type of test produces the same damage on a second-order system model. The equations are derived to show the relationship between a sinusoid at resonance, a sinusoidal sweep, and random vibrations. The sinusoids are defined in term of zero-to-peak accelerations, whereas the magnitude of the random vibration is given in power spectral density. The work done by internal damping forces is used as a basis for establishing equal fatigue damage.

A dynamical system subjected to a random fatigue environment will de- teriorate with time. This is usually ex- plained in terms of the S-N curve for the material being stressed. However, if the environment is stopped short of failure, it is necessary to use Miner's hypothesis, or the equivalent, to determine the amount of the accumulated damage. Miner's hypothesis es- tablishes a linear relationship between the number of cycles and the percent of fatigue damage. This suggests that the work done on the system can be related to the fatigue damage. Further, the only source of dissipative energy

in a linear system is the damping forces. Hence, using the work done by the damping forces as a basis for the damage criterion is equivalent to using Miner's hypothesis for accumulated damage.

SINUSOIDAL WORK PERFORMED BY A SINGLE MODE

Although the formulas for the work performed by a simple spring-mass- damper combination are well known, a short derivation will be given in order to introduce the symbols and nomenclature necessary for the en- suing discussion. Consider for instance a single normal mode of a lightly damped dynamical system.

The equation of motion is

Mij+ (Moa.@Q) fl+ Moa~'2Y = Force.

The increment of work done by this system is

dW-- Fedy = (F~dy/gt)dt (1)

and Fd= (Mo~N/Q)~). (2)

Substituting Eq. (2) into Eq. (t) and integrating over N cycles gives

NMwx (2,~/~ w=---Q-do (3)

where

Fd= force due to damping y = amplitude coefficients of the

mode f~= amplitude at any station along

the mode shape

July/August 1961 25

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y i=y f i M = ~ Mifi2= generalized mass cox= natural frequency of the mode

in radians per second Q is the reciprocal of twice i', where ~" is the percent of critical damping

of the system w= frequency in radians/sec.

If the system is excited by a force at a sinusoidal constant frequency and constant amplitude, the response is

[ (Force/M~x~)e ~'~t "~ y = R - ~ (4)

= R(iwei<~,+| F~ M~o~ ] ' (5)

where R indicates the real part,

and

~ = tan-l[ (oa/ Qwx) / (1--~/OjN2) ].

Equation (5) may be integrated in Eq. (3) to find the work done for N cycles. I t will be noticed that the phase angle drops out by virtue of the integration.

W=Tr]V-MwN2 ( w

jVorcyV

Assume that the structure is being vibrated by a uniform rigid-body acceleration. This situation occurs when equipment or structure is fixed to the head of a shake table. Consider, for example, Fig. 1. The force on the specimen is given by:

Force = ~ Mif,Gg. (7)

Substituting (7) into (6) gives

7r,TVm2G2 [ H I 2 w W - g2, (8)

QMwN 2 ~o~

where r n = ~ M ~ f ~ and G=zero-to- peak acceleration of the table in g units.

A useful form of Eq. (8) occurs when the forcing frequency is at resonance. This is

W(re~o,a,,c,)='n'Nm2O~Qg2/MwN 2. (9)

RANDOM WORK PERFORMED BY A SINGLE MODE

For the case of random excitation where S denotes the power spectral density in g2/cycle/second the mean- square acceleration response of a general damped second-order system is

'f0 a2=--- S[H[2do~, (10) 2r

where d 2 is the mean-square acceleration response in g's.

If the power spectral density is constant, the result is response to "white noise." Hence, it may be shown that

a~=Q,,.,~s/4. (11)

This is the total mean-square acceleration response of a general second- order system. Although there are re- fined mathematical techniques available by which one may compute the work performed by white noise, a more intuitive approach will be used here. To determine the work done by random vibration, it is convenient to represent the excitations by a suffi-

/NUMBER OF~/COUPLING] [ INPUT WORK PERFORMED ~ '/T~ CYCLES ] ~ FACTOR ] ~ACCELERATION ] [AMPLIFICATION AT ANY FORCING | ~

FREQUENCY / I SPRING ~ ~ FACTOR ) ~CONSTANT] ' IANT]' ~ / ~ - SPECIMEN BEING

~ o o ~ H A K E N AT 2NO MODE FREQ.

SPECIMEN HOLDING JOG:

J , ~ INPUT ACCELERATION

Fig. 1. Dynamic model.

ciently large number of discrete forces to approximate the continuous state. This is accomplished in the following manner.

I t has been established that 90 to 95~o of the response of a normal mode is obtained from excitation within a bandwidth, say 15% for tile Q range of interest, bracketing the resonant frequency. This bandwidth is divided into a number of equal-width frequency slots where discrete excitations wilt be applied at the center frequency of each slot. If these slots are suffi- ciently small, the random components will have essentially the same frequency but with randomly varying phase and amplitude.

The problem is to devise an equivalent sinusoid for each slot which will produce the same rms response as the random environment. Starting with the basic definition of power spectral density, the mean-square excitation is given by

Ok2=Sao~W2,n (12)

where 2x~o1~= width of slot. Now since Acok is constant, 0k 2 is

also constant. The work performed by a sinusoid in the kth slot is from Eq. (8).

2xWk= rcm2g2Gk2 ~o~ Nk - - }H , ] 2, (13)

QMcoN 2 a~x

where Gk is the zero-to-peak value of the sinusoidal excitation. The mean square value of G~ is �89 2 which is equal to G~2 in Eq. (12). Since the equivalent sinusoidal excitation will be applied at each slot for the same length of time and ~ok varies between slots, clearly the number of stress re- versals Nk must also vary between slots. In order to take care of this problem the following technique will be employed. Let N be the number of cycles in the slot containing the resonant frequency, then

N,~= (,,,d~N)X. (14)

Equation (13) may now be rewritten with the aid of Eqs. (12) and (14), and summed over all the frequency slots.

Nm~Sg ~ 12[~k \~ W=~_MooN2 s t 2 ) AWk" (15)

If the limiting value of the above sum-

26 NOISE Control


60 x 10- ~

50-

~

20

103 104 I 0 $ 106 107

CYCLES

Fig. 2. S-N curve for 1020 steel.

marion is taken, it may be shown that

look \ 2 lira&o--+ 0 ~ [ Hklzt~-~N )

rQ~x Xawk= (16)

2

Notice that this is the same value as given by fo ~ S/Hl~&0in Eq. (11). The total work now becomes

W=~'NgZm2S/2o~NM, (17)

Consider a single normal mode of the dynamical system of frequency w~. If random excitation is continued for a time T~ the number of accumulated fatigue cycles is approximated by

N=~o~-TI/2r:; (18)

hence the total work done can be written, from Eq. (17),

W= T~g2m2S/4M. (19)

I t is to be emphasized that work is independent of the damping in the system. On the basis that the fatigue damage is proportional to the work done by the internal damping forces, the work done by a sinusoid at resonance [-Eq. (9)-] is set equal to the work done by "white noise" [Eq. (17)-]; this gives

G= (5~/2Q)}, (20)

where G= zero-to-peak acceleration of sinusoidal excitation at resonance g's, S= power spectral density g2/cps, wx =resonant frequency (radians/sec), and Q = damping magnification factor. Hence the equivalent power spectral density is

S= 2QG2/~oN. (21)

I t is to be noted that this relationship is identical with that derived by equating the rms response of a sinusoidal resonance to that of the rms response due to a white-noise random excitation. This equivalence is not surpris- ing, since the work equations are essentially mean-square response equations.

W O R K P E R F O R M E D B Y A

S I N U S O I D A L S W E E P

For the sinusoidal sweep, it is necessary to devise a scheme for adding work done at different levels of magnification as the frequency is swept through the frequency spectrum. As- sume that the frequency spectrum may be divided into a number of bands each zXf cps wide. Assume also that the number of cycles of work performed by the sweeping sinusoid may be approximated by Af(f / f ' ) , where f is the center frequency of the band and / , is the sweep rate in cps/sec. From Eq. (8), the work performed in the kth band may be written as

m2G2g 2 (wk~ ~ W= QMwx~-~[-\WN---2] I H~ I2A~ok. (22)

Summing and passing to the limit as A~0~ ~ 0 and the number of segments increases without limit, the integral

again results and its value is �89 Substituting this result into Eq.

(22) gives

Work due to sweep =m2C2g2/affM. (23)

Now equating this to Eq. (19) to get

the random and sinusoidal equivalence we get

1/TIf'=2S/G ~, (24)

Notice that the equivalence equations are independent of the system damping and resonant frequency.

APPLICATION OF T H E EQUIVALENCE EQUATIONS

A sweeping sine-wave test is very cumbersome to apply if one is required to sweep at a constant rate through- out the entire spectrum as is implied by Eq. (24). It is noted, however, that all the work performed by a sweeping sinusoid is done at or near resonance.

I ~ RESPONSE DUE TO SWEEP

- | i �9

I RESPONSE DUE TO I I

~///?/ff////////;///////X//// T RANOOM t~Va

F--+ + 4 FIRST SECONO THIRD

OCTAVE OCTAVE OCTAVE

Fig. 3. Response curve with resonance in only one octave.

] I ,l ,lr, l ~ / / / Y ~ f / / / ~ i U I SWEEP LEVEL

I IA AI ,I YCt////Y~///Ng///Y.r T ,ANooM ~VEL *L 3. v v

FIRST SECOND THIRD OCTAVE OCTAVE OCTAVE

Fig. 4. Response curve with random noise the same as in Fig. 3, but with resonances

in all octaves.

In view of the foregoing and in the discussion that follows, Eq. (24) may be rewritten by replacing f ' with the time necessary to sweep an octave and the octave bandwidth.

To~T1= 2SAfo/G ~, (25)

where Af0=octave bandwidth and To= time to sweep an octave.

The above approximation to the equivMence formula is easier to apply and is backed by physical reasoning as seen in the following discussion. This form of the equivalence equation is a good approximation to the log- arithmic sweep which is available on most shake tables.

J u l y / A u g u s t 1961 27


in applying Eq. (25) to an actual system, some careful considerations must be applied. First of all, the equations must be general for a system containing many modes, and second, due consideration must be given to the nonlinearities of an S-N curve (Fig. 2). The latter consideration may be satis- fied by setting To/Tt equal to unity. Tha t is to say that the time taken to sweep through one octave must be equal to the total time the random noise is applied. As experimental data become available it may well indicate that some number other than unity be chosen for T~/T1. For the purpose of this discussion it will be assumed that this number is absorbed in the constant 2.

This follows from the fact that the sweep can do work only at the frequencies it is sweeping while the random is doing work on all frequencies of the system. The effect is to boost the random levels until they are caus- ing rms stress responses in the same order of magnitude as the sweeping sinusoid. If one were to reduce the interval of consideration to, say, �89 octaves, he would have an even closer equivalence of stress levels.

To demonstrate the applicability of Eq. (25) to a system with many resonances, consider Fig. 3. Assume there is one mode in a given octave. The random noise is applied to all octaves for Tr but it is doing work only in the octave containing the resonance. The sweep is of course only doing work in the octave containing the resonance. The levels are adjusted so that each does the same amount of work in time T1.

Now consider modes in the other octaves, Fig. 4. The random noise is applied at the same level for the same time as previously but now it is doing work in all octaves since there are resonances there. Now it is necessary that equivalence be established for each mode independently (or octave as in this example). This equivalence is an approximation for high (2 systems, since in the derivation the inte- grals were evaluated for a constant level from zero to infinity. Therefore the sweep time T~ for the first octave must remain the same as before, since the random is doing the same work as in the previous case. By the same reasoning, then, the sweep time must be adjusted in octaves 2 and 3 so that

101,

,g

Q = ~ 0 �9

S/G2 = Q / g T f N / ~

10

EQUIVALENT R E S O N A ~

100 1,000 flCPSI Fig. 5. Equivalence curves for the ratio of S/G ~.

they are each swept in time T1. This equivalence is plotted in Fig. 5 as S/G 2 vs frequency. This chart also contains resonance equivalents for reference.

In using the graph it is assumed that the slope of 6 db/octave is pret ty well established from theoretical considerations. The vertical position of the curve is determined by the constant 2 in Eq. (25) and the equivalence time factor. I t is assumed that experimental data will become available to better determine the actual value of these constants.

Figure 5 is replotted as S vs G curves (Fig. 6) for easier reference. To use the curves one would first determine the time duration of test and then enter the curves to find the octave levels for the sweeping sinusoid and the white-noise inputs.

CONCLUSION

The theoretical analysis given in this paper is intended as a basis for a rational approach to the equivalence of sine-wave and random testing. A test program is currently underway at Douglas to determine experimentally some of the theoretical constants derived in the paper. Due to the diffi- culties of fatigue testing, these con-

stants will be determined as statistical quantities only. �9 �9 �9

R e f e r e n c e s

* Presented to the 29th Symposium on Shock, Vibration and Associated Environ- ments.

t Supervisor, Flutter and Vibration. Group Engineer, Shock and Vibration.

1. Stephan FI. Crandall, Notes for the M.I.T. Special Summer Program on Random Vibration (Technology Press, Cambridge, Massachusetts, 1958).

2. J. G. Truxal, Automatic Feedback Con- trol System Synthesis (McGraw-Hill Book Company, Inc., New York, 1955).

3. W. B. Davenport and W. L. Root, An Introduction to the Theory oJ Random Signals and Noise (McGraw-Hill Book Company, Inc., New York, 1958).

10

%

10-1

1~ 2

g

~io-3

10 4 1 10 G, O-PEAK ACCELERATION IN g's

100

Fig. 6. Equivalence curves for random noise vs sweep.

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correlation of sinusoidal and random vibrations

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