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17.2 Representation of a Vibration Environment

Back

A complete knowledge of the vibration environment in which a device will be operating is not available to

the test engineer or the test program planner. The primary reason for this is that the operating

environment is a random process. When performing a vibration test, however, either a deterministic or a

random excitation can be employed to meet the test requirements. This is known as the test environment.

Based on the vibration-testing specifications or product qualification requirements, the test

environment should be developed to have the required characteristics of (1) intensity (amplitude), (2)

frequency content (effect on the test-object resonances and the like), (3) decay rate (damping), and (4)

phasing (dynamic interactions). Usually, these parameters are chosen to represent conservatively the

worst possible vibration environment that is reasonably expected during the design life of the test object.

So long as this requirement is satisfied, it is not necessary for the test environment to be identical to the

operating vibration environment.

Vibration Testing 17-3

In vibration testing, the excitation input (test environment) can be represented in several ways. The

common representations are (1) by time signal, (2) by response spectrum, (3) by Fourier spectrum, and

(4) by PSD function. Once the required environment is specified by one of these forms, the test should be

conducted either by directly employing them to drive the exciter or by using a more conservative

excitation when the required environment cannot be exactly reproduced.

17.2.1 Test Signals

Vibration testing may employ both random and deterministic signals as test excitations. Regardless of its

nature, the test input should conservatively meet the specified requirements for that test.

17.2.1.1 Stochastic vs. Deterministic Signals

Consider a seismic time-history record. Such a ground-motion record is not stochastic. It is true that

earthquakes are random phenomena and the mechanism by which the time history was produced is a

random process. Once a time history is recorded, however, it is known completely as a curve of response

value versus time (a deterministic function of time). Therefore, it is a deterministic set of information.

However, it is also a “sample function” of the original stochastic process, the earthquake, by which it was

generated. Hence, valuable information about the original stochastic process itself can be determined by

analyzing this sample function on the basis of the ergodic hypothesis (see Section 17.2.3). Some may

think that an irregular time-history record corresponds to a random signal. It should be remembered that

some random processes produce very smooth signals. As an example, consider the sine wave given by

a sinðvt þ fÞ: Let us assume that the amplitude a and the frequency v are deterministic quantities

and the phase angle f is a random variable. This is a random process. Every time this particular

random process is activated, a sine wave is generated that has the same amplitude and frequency but,

generally, a different phase angle. Nevertheless, the sine wave will always appear as smooth as a

deterministic sine wave.

In a vibration-testing program, if we use a recorded time history to derive the exciter, it is a

deterministic signal, even if it was originally produced by a random phenomenon such as an earthquake.

Also, if we use a mathematical expression for the signal in terms of completely known (deterministic)

parameters, it is again a deterministic signal. If the signal is generated by some random mechanism

(whether computer simulation or physical) in real time, however, and if that signal is used as the

excitation in the vibration test simultaneously as it is being generated, then we have a truly random

excitation. Also, if we use a mathematical expression (with respect to time) for the excitation signal for

which some of the parameters are not known numerically and the values are assigned to them during the

test in a random manner, we again have a truly random test signal.

17.2.2 Deterministic Signal Representation

In vibration testing, time signals that are completely predefined can be used as test excitations. They

should be capable, however, of subjecting the test object to the specified levels of intensity, frequency,

decay rate, and phasing (in the case of simultaneous multiple test excitations).

Deterministic excitation signals (time histories) used in vibration testing are divided into two broad

categories: single-frequency signals and multifrequency signals.

17.2.2.1 Single-Frequency Signals

Single-frequency signals have only one predominant frequency component at a given time. For the entire

duration, however, the frequency range covered is representative of the frequency content of the vibration

environment. For seismic-qualification purposes, for example, this range should be at least 1 to 33 Hz.

Some typical single-frequency signals that are used as excitation inputs in vibration testing of equipment

are shown in Figure 17.2. The signals shown in the figure can be expressed by simple mathematical

expressions. This is not a requirement, however. It is acceptable to store a very complex signal in a storage

device and subsequently use it in the procedure. In picking a particular time history, we should give

17-4 Vibration and Shock Handbook

proper consideration to its ease of reproduction and the accuracy with which it satisfies the test

specifications. Now, let us describe mathematically the acceleration signals shown in Figure 17.2.

17.2.2.2 Sine Sweep

We obtain a sine sweep by continuously varying the frequency of a sine wave. Mathematically,

uðtÞ ¼ a sin½vðtÞt þ f􀀉 ð17:1Þ

T12

T11

Pause Pause

(a)

Time

Acceleration Acceleration Acceleration Acceleration Acceleration

(b) Frequency = w1 Frequency = w 2 Frequency = w3

Frequency = w1 Frequency = w2 Frequency = w3

T1 T2 T3

(c)

(d)

(e)

Frequency = w1 Frequency = w2

Frequency = w3

FIGURE 17.2 Typical single-frequency test signals: (a) sine sweep; (b) sine dwell; (c) sine decay; (d) sine beat;

(e) sine beat with pause.

Vibration Testing 17-5

The amplitude, a, and the phase angle, f, are usually constants and the frequency, vðtÞ; is a function of

time. Both linear and exponential variations of frequency over the duration of the test are in common

usage, but exponential variations are more common. For the linear variation (see Figure 17.3), we have

vðtÞ ¼ vmin þ ðvmax 2 vminÞ

Td ð17:2Þ

in which

vmin ¼ lowest frequency in the sweep

vmax ¼ highest frequency in the sweep

Td ¼ duration of the sweep

For the exponential variation (see Figure 17.3), we have

log

vðtÞ

vmin

􀀒 􀀓

log

vmax

vmin

􀀒 􀀓

ð17:3Þ

vðtÞ ¼ vmin

vmax

vmin

􀀒 􀀓t=Td

ð17:4Þ

This variation is sometimes incorrectly called logarithmic variation. This confusion arises because of its

definition using Equation 17.3 instead of Equation 17.4. It is actually an inverse logarithmic (i.e.,

exponential) variation. Note that the logarithm in Equation 17.3 can be taken to any arbitrary base. If

base ten is used, the frequency increments are measured in decades (multiples of ten); if base two is used,

the frequency increments are measured in octaves (multiples of two). Thus, the number of decades in the

frequency range from v1 to v2 is given by log10ðv2=v1Þ; for example, with v1 ¼ 1 rad/sec and

v2 ¼ 100 rad/sec, we have log10ðv2=v1Þ ¼ 2; which corresponds to two decades. Similarly, the number of

octaves in the range v1 to v2 is given by log2ðv2=v1Þ: Then, with v1 ¼ 2 rad/sec and v2 ¼ 32 rad/sec we

have log2(v2/v1) ¼ 4, a range of four octaves. Note that these quantities are ratios and have no physical

units. The foregoing definitions can be extended for smaller units; for instance, one-third octave

represents increments of 21/3. Thus, if we start with 1 rad/sec and increment the frequency successively by

one-third octave, we obtain 1, 21/3, 22/3, 2, 24/3, 25/3, 22, and so on. It is clear, for example, that there are

four one-third octaves in the frequency range from 22/3 to 22. Note that v is known as the angular

frequency (or radian frequency) and is usually measured in units of radians per second (rad/sec).

Linear Sine

Sweep

Frequency

Sine Dwell

Time

O Td t

Exponential

Sine Sweep

wmin

wmax

w(t)

FIGURE 17.3 Frequency variation in some single-frequency vibration-test signals.

17-6 Vibration and Shock Handbook

The more commonly used frequency is the cyclic frequency which is denoted by f. This is measured in

hertz (Hz), which is identical to cycles per second (cps). It is clear that

f ¼

2p ð17:5Þ

This is true because there are 2p radians in one cycle.

So that all important vibration frequencies of the test object (or its model) are properly excited, the

sine sweep rate should be as slow as is feasible. Typically, rates of one octave per minute or slower are

employed.

17.2.2.3 Sine Dwell

Sine-dwell signal is the discrete version of a sine sweep. The frequency is not varied continuously but is

incremented by discrete amounts at discrete time points. This is shown graphically in Figure 17.3.

Mathematically, for the rth time interval, the dwell signal is

uðtÞ ¼ a sinðvr t þ fr Þ; Tr21 # t # Tr ð17:6Þ

in which vr ; a, and f are kept constant during the time interval ðTr21; Tr Þ: The frequency can be

increased by a constant increment or the frequency increments can be made bigger with time

(exponential-type increment). The latter procedure is more common. Also, the dwelling-time interval is

usually made smaller as the frequency is increased. This is logical because, as the frequency increases, the

number of cycles that occur during a given time also increases. Consequently, steady-state conditions

may be achieved in a shorter time.

Sine-dwell signals can be specified using either a graphical form (see Figure 17.3) or a tabular form,

giving the dwell frequencies and corresponding dwelling-time intervals. The amplitude is usually kept

constant for the entire duration ð0; TdÞ; but the phase angle, f, may have to be changed with each

frequency increment in order to maintain the continuity of the signal.

17.2.2.4 Decaying Sine

Actual transient vibration environments (e.g., seismic ground motions) decay with time as the vibration

energy is dissipated by some means. This decay characteristic is not present, however, in sine-sweep and

sine-dwell signals. Sine-decay representation is a sine dwell with decay (see Figure 17.2). For an

exponential decay, the counterpart of Equation 17.6 can be written as

uðtÞ ¼ a expð2lr tÞ sinðvr t þ fr Þ; Tr21 # t # Tr ð17:7Þ

The damping parameter (the inverse of the time constant), l, is typically increased with each frequency

increment in order to represent the increased decay rates of a dynamic environment (or increased modal

damping) at higher frequencies.

17.2.2.5 Sine Beat

When two sine waves having the same amplitude but different frequencies are mixed together (added or

subtracted), a sine beat is obtained. This signal is considered to be a sine wave having the average

frequency of the two original waves, which is amplitude-modulated by a sine wave of frequency equal to

half the difference of the frequencies of the two original waves. The amplitude modulation produces a

transient effect which is similar to that caused by the damping term in the sine-decay equation (Equation

17.7). The sharpness of the peaks becomes more prominent when the frequency difference of the two

frequencies is made smaller.

Consider two cosine wave having frequencies ðvr þ Dvr Þ and ðvr 2 Dvr Þ and the same amplitude a/2.

If the first signal is subtracted from the second (that is, it is added with a 1808 phase shift from the first

wave), we obtain

uðtÞ ¼

2 ½cosðvr 2 Dvr Þt 2 cosðvr þ Dvr Þt􀀉 ð17:8Þ

Vibration Testing 17-7

By straightforward use of trigonometric identities, we obtain

uðtÞ ¼ aðsin vr tÞðsin Dvr tÞ; Tr21 # t # Tr ð17:9Þ

This is a sine wave of amplitude, a; and frequency, v, modulated by a sine wave of frequency Dvr : Sinebeat

signals are commonly used as test excitation inputs in vibration testing. Usually, the ratio vr =Dvr is

kept constant. A typical value used is 20, in which case we obtain 10 cycles per beat. Here, cycles refer to

the cycles at the higher frequency, vr ; and a beat occurs at each half cycle of the smaller frequency, Dvr :

Thus, a beat is identified by a peak of amplitude a in the modulated wave and the beat frequency is 2Dvr :

As in the case of a sine dwell, the frequency, vr ; of a sine-beat excitation signal is incremented at

discrete time points, Tr ; so as to cover the entire frequency interval of interest ðvmin; vmaxÞ: It is a

common practice to increase the size of the frequency increment and decrease the time duration at a

particular frequency, for each frequency increment, just as is done for the sine dwell. The reasoning for

this is identical to that given for sine dwell. The number of beats for each duration is usually kept

constant (typically at a value over seven). A sine-beat signal is shown in Figure 17.2(d).

17.2.2.6 Sine Beat with Pauses

If we include pauses between sine-beat durations, we obtain a sine-beat signal with pauses.

Mathematically, we have

uðtÞ ¼

aðsin vr tÞðsin Dvr tÞ; for Tr21 # t # T0r

;

0; for T0r

# t # Tr

(

ð17:10Þ

This situation is shown in Figure 17.2(e). When a sine-beat signal with pauses is specified as a test

excitation, we must give the frequencies, the corresponding time intervals, and the corresponding pause

times. Typically, the pause time is also reduced with each frequency increment.

The single-frequency signal relations described in this section are summarized in Table 17.1.

17.2.2.7 Multifrequency Signals

In contrast to single-frequency signals, multifrequency signals usually appear irregular and can have

more than one predominant frequency component at a given time. Three common examples of

multifrequency signals are aerodynamic disturbances, actual earthquake records, and simulated road

disturbance signals used in automotive dynamic tests.

TABLE 17.1 Typical Single-Frequency Signals Used in Vibration Testing

Single Frequency Acceleration Signal Mathematical Expression

Sine sweep uðtÞ ¼ a sin½vðtÞt þ f􀀉

vðtÞ ¼ vmin þ ðvmax 2 vmin Þt=Td (linear)

vðtÞ ¼ vmin

vmax

vmin

􀀏 􀀐t=Td

ðexponentialÞ

Sine dwell uðtÞ ¼ a sinðvr t þ fr Þ Tr21 # t # Tr ; r ¼ 1; 2; …; n

Decaying sine uðtÞ ¼ a expð2lr tÞ sinðvr t þ fr Þ Tr21 # t # Tr , r ¼ 1; 2; …; n

Sine beat uðtÞ ¼ aðsin vr tÞ ðsin Dvr tÞ Tr21 # t # Tr ; r ¼ 1; 2; …; n;

vr =Dvr ¼ constant

Sine beat with pauses uðtÞ ¼

aðsin vr tÞðsin Dvr tÞ; for Tr21 # t # T 0 r

¼ 0; for T0r

# t # Tr

(

17-8 Vibration and Shock Handbook

17.2.2.8 Actual Excitation Records

Typically, actual excitation records such as overhead guideway vibrations are sample functions of random

processes. By analyzing these deterministic records, however, characteristics of the original stochastic

processes can be established, provided that the records are sufficiently long. This is possible because of the

ergodic hypothesis. Results thus obtained are not quite accurate, because the actual excitation signals are

usually nonstationary random processes and hence are not quite ergodic. Nevertheless, the information

obtained by a Fourier analysis is useful in estimating the amplitude, phase, and frequency-content

characteristics of the original excitation. In this manner, we can choose a past excitation record that can

conservatively represent the design-basis excitation for the object that needs to be tested.

Excitation time histories can be modified to make them acceptably close to a design-basis excitation by

using spectral-raising and spectral-suppressing methods. In spectral-raising procedures, a sine wave of

required frequency is added to the original time history to improve its capability of excitation at that

frequency. The sine wave should be properly phased such that the time of maximum vibratory motion in

the original time history is unchanged by the modification. Spectral suppressing is achieved, essentially,

by using a narrowband reject filter for the frequency band that needs to be removed. Physically, this is

realized by passing the time history signal through a linearly damped oscillator that is tuned to the

frequency to be rejected and connected in series with a second damper. The damping of this damper is

chosen to obtain the required attenuation at the rejected frequency.

17.2.2.9 Simulated Excitation Signals

Random-signal-generating algorithms can be easily incorporated into digital computers. Also, physical

experiments can be developed that have a random mechanism as an integral part. A time history from

any such random simulation, once generated, is a sample function. If the random phenomenon is

accurately programmed or physically developed so as to conservatively represent a design-basis

excitation, a signal from such a simulation may be employed in vibration testing. Such test signals are

usually available either as analog records on magnetic tapes or as digital records on a computer disk.

Spectral-raising and spectral-suppressing techniques, mentioned earlier, also may be considered as

methods of simulating vibration test excitations.

Before we conclude this section, it is worthwhile to point out that all test excitation signals considered

in this section are oscillatory. Though the single-frequency signals considered may possess little

resemblance to actual excitations on a device during operation, they can be chosen to possess the

required decay, magnitude, phase, and frequency-content characteristics. During vibration testing, these

signals, if used as excitations, will impose reversible stresses and strains on the test object, whose

magnitudes, decay rates, and frequencies are representative of those that would be experienced during

actual operation during the design life of the test object.

17.2.3 Stochastic Signal Representation

To generate a truly stochastic signal, a random phenomenon must be incorporated into the signalgenerating

process. The signal has to be generated in real time, and its numerical value at a given time is

unknown until that time instant is reached. A stochastic signal cannot be completely specified in advance,

but its statistical properties may be prespecified. There are many ways of obtaining random processes,

including physical experimentation (for example, by tossing a coin at equal time steps and assigning a

value to the magnitude over a given time step depending on the outcome of the toss), observation of

processes in nature (such as outdoor temperature), and digital-computer simulation. The last procedure

is the one commonly used in signal generation associated with vibration testing.

17.2.3.1 Ergodic Random Signals

A random process is a signal that is generated by some random (stochastic) mechanism. Generally, each

time the mechanism is operated, a different signal (sample function) is generated. The likelihood of any

two sample functions becoming identical is governed by some probabilistic law. The random process is

Vibration Testing 17-9

denoted by XðtÞ; and any sample function by xðtÞ: It should be remembered that no numerical

computations can be made on XðtÞ because it is not known for certain. Its Fourier transform, for

instance, can be written as an analytical expression but cannot be computed. Once a sample function,

xðtÞ; is generated, however, any numerical computation can be performed on it because it is a completely

known function of time. This important difference may be somewhat confusing.

At any given time, t1; Xðt1Þ is a random variable that has a certain probability distribution. Consider a

well-behaved function, f {Xðt1Þ}; of this random variable (which is also a random variable). Its expected

value (statistical mean) is denoted E½f {Xðt1Þ}􀀉: This is also known as the ensemble average because it

is equivalent to the average value at t1 of a collection (ensemble) of a large number of sample functions

of XðtÞ:

Now, consider the function f {xðtÞ} of one sample function xðtÞ of the random process. Its temporal

(time) mean is expressed by

lim

T!1

ðT

f {xðtÞ}dt

Now, if

E½f {Xðt1Þ}􀀉 ¼ lim

T!1

ðT

f {xðtÞ}dt ð17:11Þ

then the random signal is said to be ergodic. Note that the right-hand side of Equation 17.11 does not

depend on time. Hence, the left-hand side also should be independent of the time point t1:

As a result of this relation (known as the ergodic hypothesis), we can obtain the properties of a random

process merely by performing computations using one of its sample functions. The ergodic hypothesis is

links the stochastic domain of expectations and uncertainties and the deterministic domain of real

records and actual numerical computations. Digital Fourier computations, such as correlation functions

and spectral densities, would not be possible for random signals if not for this hypothesis.

17.2.3.2 Stationary Random Signals

If the statistical properties of a random signal, XðtÞ; are independent of the time point considered, it

is stationary. In particular, Xðt1Þ will have a probability density that is independent of t1; and the

joint probability of Xðt1Þ and Xðt2Þ will depend only on the time difference, t2 2 t1: Consequently, the

mean value E½XðtÞ􀀉 of a stationary random signal is independent of t; and the autocorrelation function

defined by

E½XðtÞXðt þ tÞ􀀉 ¼ fxx ðtÞ ð17:12Þ

which depends on t and not on t: Note that ergodic signals are always stationary, but the converse is not

always true.

Consider Parseval’s theorem:

ð1

x2ðtÞdt ¼

ð1

lXð f Þl2df ð17:13Þ

This can be interpreted as an energy integral, and its value is usually infinite for random signals. An

appropriate measure for random signals is its power. This is given by its root-mean-square (RMS) value

E½XðtÞ2􀀉: PSD Fðf Þ is the Fourier transform of the autocorrelation function fðtÞ and, similarly, the latter

is the inverse Fourier transform of the former. Hence,

fxx ðtÞ ¼

ð1

Fxx ðf Þexpðj2pf tÞdf ð17:14Þ

17-10 Vibration and Shock Handbook

Now, from Equation 17.12 and Equation 17.14, we obtain

RMS value ¼ E½XðtÞ2􀀉 ¼ fxx ð0Þ ¼

ð1

Fxx ðf Þdf ð17:15Þ

It follows that the RMS value of a stationary random signal is equal to the area under its PSD curve.

17.2.3.3 Independent and Uncorrelated Signals

Two random signals XðtÞ and Y ðtÞ are independent if their joint distribution is given by the product of

the individual distributions. A special case is that of uncorrelated signals, which satisfy

E½Xðt1ÞY ðt2Þ􀀉 ¼ E½Xðt1Þ􀀉E½Y ðt2Þ􀀉 ð17:16Þ

Consider the stationary case, with mean values

mx ¼ E½XðtÞ􀀉 ð17:17Þ

my ¼ E½Y ðtÞ􀀉 ð17:18Þ

The autocovariance functions are given by

wxx ðtÞ ¼ E½{XðtÞ 2 mx }{Xðt þ tÞ 2 mx }􀀉 ¼ fxx ðtÞ 2 m2

x ð17:19Þ

wyy ðtÞ ¼ E½{Y ðtÞ 2 my }{Y ðt 2 tÞ 2 my }􀀉 ¼ fyy ðtÞ 2 m2y

ð17:20Þ

and the cross-covariance function is given by

wxy ðtÞ ¼ E½{XðtÞ 2 mx }{Y ðt 2 tÞ 2 my }􀀉 ¼ fxy ðtÞ 2 mxmy ð17:21Þ

For uncorrelated signals (Equation 17.16)

fxy ðtÞ ¼ mxmy ð17:22Þ

and from Equation 17.21 it follows that

wxy ðtÞ ¼ 0 ð17:23Þ

The correlation-function coefficient is defined by

rxy ðtÞ ¼

wxy ðt ffiffiffiffiffiffiffiffiffiÞffiffiffiffiffiffi

wxx ð0Þwyy ð0Þ

p ð17:24Þ

which satisfies

21 # rxy ðtÞ # 1 ð17:25Þ

For uncorrelated signals, rxy ðtÞ ¼ 0: This function measures the degree of correlation of the two signals.

The correlation of two random signals, XðtÞ and Y ðtÞ; is measured in the frequency domain by its

ordinary coherence function

g 2

xy ðf Þ ¼

lFxy ð f Þl2

Fxx ð f ÞFyy ðf Þ ð17:26Þ

which satisfies the condition

0 # g 2

xy ð f Þ # 1 ð17:27Þ

17.2.3.4 Transmission of Random Excitations

When the excitation input to a system is a random signal, the corresponding system response will also be

random. Consider the system shown by the block diagram in Figure 17.4(a). The response of the system

Vibration Testing 17-11

is given by the convolution integral

Y ðtÞ ¼

ð1

hðt1ÞU ðt 2 t1Þdt1 ð17:28Þ

in which the response PSD is given by the Fourier transform

Fyy ð f Þ ¼ I{E½Y ðtÞY ðt þ tÞ􀀉} ð17:29Þ

Now, by using Equation 17.28 in Equation 17.29, in conjunction with the definition of Fourier transform,

we can write

Fyy ð f Þ ¼

ð1

dt expð2j2pf tÞE

ð1

dt1hðt1ÞUðt 2 t1Þ

ð1

dt2hðt2ÞU ðt þ t 2 t2Þ

􀀒 􀀓

which can be expressed as

Fyy ð f Þ ¼

ð1

dt1 hðt1Þ

ð1

dt2 hðt2Þ

ð1

dt expð2j2pf tÞfuuðt þ t1 2 t2Þ

Now, by letting t 0 ¼ t þ t1 2 t2, we can write

Fyy ð f Þ ¼

ð1

hðt1Þexpðj2pft1Þdt1

􀀒 􀀓ð1

hðt2Þexpð2j2pft2Þdt2

􀀒 􀀓ð1

fuuðt 0Þexpð2j2pf t 0Þdt 0

􀀒 􀀓

Note that UðtÞ is assumed to be stationary.

Next, since the frequency-response function is given by the Fourier transform of the impulse response

function, we obtain

Fyy ð f Þ ¼ Hpð f ÞHðf ÞFuuð fÞ ð17:30Þ

(a)

(b)

(c)

U(t)

Excitation Y

Response

Excitations

U1(t)

Ur(t)

Uf (t)

Combined

Response

Overall

Response

Excitation

Delay

h(t)

^(s)

1(s)

^ (s)

r(s)

e−ts

FIGURE 17.4 Combined response of a system to various random excitations: (a) system excited by a single input;

(b) response to several random excitations; (c) response to a delayed excitation.

17-12 Vibration and Shock Handbook

in which Hp ðf Þ is the complex conjugate of Hð f Þ: Alternatively, if lHð f Þl denotes the magnitude of the

complex quantity, we can write

Fyy ð f Þ ¼ lHð f Þl2Fuuð fÞ ð17:31Þ

By using Equation 17.30 or Equation 17.31, we can determine the PSD of the system response from the

PSD of the excitation if the system-frequency-response function is known.

In a similar manner, it can be shown that the cross-spectral density function may be expressed as

Fuy ð f Þ ¼ Hð f ÞFuuðfÞ ð17:32Þ

Now, consider r stationary, independent, random excitations, U1; U2; …; Ur ; (which are assumed to have

zero-mean values, without loss of generality) applied to r subsystems, having transfer functions

H^ 1ðsÞ;H^ 2ðsÞ;…; H^ r ðsÞ; as shown in Figure 17.4(b). The total response, Y ; consists of the sum of individual

responses, Y1; Y2; …; Yr : It can be shown that Y1; Y2; …; Yr are also stationary, independent, zero-mean,

random processes. By definition, we have

fyy ðtÞ ¼ E½{Y1ðtÞ þ · · · þ Yr ðtÞ}{Y1ðt þ tÞ þ · · · þ Yr ðt þ tÞ}􀀉 ð17:33Þ

Now, for independent, zero-mean Yi; Equation 17.33 becomes

fyy ðtÞ ¼ E½Y1ðtÞY1ðt þ tÞ􀀉 þ · · · þ E½Yr ðtÞYr ðt þ rÞ􀀉 ð17:34Þ

Since Yi are stationary, we have

fyy ðtÞ ¼ fy1 y1 ðtÞ þ · · · þ fyr yr ðtÞ ð17:35Þ

On Fourier transformation, we obtain

Fyy ð f Þ ¼ Fy1 y1 ðf Þ þ · · · þ Fyr yr ð fÞ ð17:36Þ

In view of Equation 17.31, it can be written

Fyy ð f Þ ¼

i¼1

lHið f Þl2Fui ui ðfÞ ð17:37Þ

from which the response PSD can be determined if the input PSDs are known.

If all inputs, UiðtÞ; have identical probability distributions (for example, when they are generated by

the same mechanism), the corresponding PSDs will be identical. Note that this does not imply that the

inputs are equal. They could be dependent, independent, correlated, or uncorrelated. In this case,

Equation 17.37 becomes

Fyy ð f Þ ¼

i¼1

lHiðf Þl2

" #

Fuuð fÞ ð17:38Þ

in which Fuuð f Þ is the common input PSD.

Finally, consider the linear combination of two excitations, Uf ðtÞ and Ur ðtÞ; with the latter excitation

delayed in time by t but otherwise identical to the former. This situation is shown in Figure 17.4(c). From

Laplace transform tables, it is seen that the Laplace transforms of the two signals are related by

Ur ðsÞ ¼ expð2tsÞUf ðsÞ ð17:39Þ

From Equation 17.39, it follows that (see Figure 17.4(c)):

Y ðsÞ ¼ ðA1 expð2tsÞ þ A2ÞHðsÞUf ðsÞ ð17:40Þ

Consequently, we have

Fyy ð f Þ ¼ lðA1 expð2j2pf tÞ þ A2ÞHð f Þl2FuuðfÞ ð17:41Þ

From this result, the net response can be determined when the phasing between the two excitations is

known. This has applications, for example, in determining the response of a vehicle to road disturbances

at the front and rear wheels.

Vibration Testing 17-13

17.2.4 Frequency-Domain Representations

In this section, we shall discuss the Fourier spectrum method and the PSD method of representing a test

excitation. These are frequency-domain representations.

17.2.4.1 Fourier Spectrum Method

Since the time domain and the frequency domain are related through Fourier transformation, a time

signal can be represented by its Fourier spectrum. In vibration testing, a required Fourier spectrum may

be given as the test specification. Then, the actual input signal that is used to excite the test object

should have a Fourier spectrum that envelops the required Fourier spectrum. The generation of a signal

to satisfy this requirement might be difficult. Usually, digital Fourier analysis of the control sensor signal

is necessary to compare the actual (test) Fourier spectrum with the required Fourier spectrum. If the

two spectra do not match in a certain frequency band, the error (i.e., the difference in the two spectra) is

fed back to correct the situation. This process is known as frequency-band equalization. Also, the sample

step of the time signal in the digital Fourier analysis should be adequately small to cover the frequency

range of interest in that particular vibration testing application. Advantages of using digital Fourier

analysis in vibration testing include flexibility and convenience with respect to the type of the signal that

can be analyzed, availability of complex processing capabilities, increased speed of processing, accuracy

and reliability, reduction in the test cost, practically unlimited repeatability of processing, and reduction

in the overall size and weight of the analyzer.

17.2.4.2 Power Spectral Density Method

The operational vibration environment of equipment is usually random. Consequently, a stochastic

representation of the test excitation appears to be suitable for a majority of vibration-testing situations.

One way of representing a stationary random signal is by its PSD. As noted before, the numerical

computation of the PSD is not possible, however, unless the ergodicity is assumed for the signal. Using

the ergodic hypothesis, we can compute the PSD of a random signal simply by using one sample function

(one record) of the signal.

Three methods of determining the PSD of a random signal are shown in Figure 17.5. From

Parseval’s theorem (Equation 17.13), we notice that the mean square value of a random signal may be

obtained from the area under the PSD curve. This suggests the method shown in Figure 17.5(a) for

estimating the PSD of a signal. The mean square value of a sample of the signal in the frequency

band, Df ; having a certain center frequency is obtained by first extracting the signal components in

the band and then squaring them. This is done for several samples and averaged to obtain a high

accuracy. It is then divided by Df : By repeating this for a range of center frequencies, an estimate for

the PSD is obtained.

Approximate

psd

Approximate

psd

Approximate

psd

(a)

(b)

(c)

Tracking Filter

Bandwidth Δf

Square Law

Circuit

Signal

Averaging

Network Δf

ADC

Digital

Correlation

Function

Signal Digital

Fourier

Transform

Averaging

Software

ADC

Signal

Display/Recording

Unit

Display/Recording

Unit

Display/Recording

Unit

FFT

Processor

Averaging

Software DAC

FIGURE 17.5 Some methods of PSD determination: (a) the filtering, squaring, and averaging method; (b) using an

autocorrelation function; (c) using direct FFT.

17-14 Vibration and Shock Handbook

In the second scheme, shown in Figure17.5(b), correlation function is first computed digitally. Its

Fourier transform (by fast Fourier transform, or FFT) gives an estimate of the PSD.

In the third scheme, shown in Figure 17.5(c), the PSD is computed directly using FFT. Here, the

Fourier spectrum of the sample record is computed and the PSD is estimated directly, without first

computing the autocorrelation function.

In these numerical techniques of computing PSD, a single sample function will not give the required

accuracy, and averaging results for a number of sample records is usually needed. In real-time digital

analysis, the running average and the current estimate are normally computed. In the running average, it

is desirable to give a higher weighting to the more recent estimates. The fluctuations about the local

average in the PSD estimate could be reduced by selecting a larger filter bandwidth, Df (see Figure 17.6),

and a large record length T. A measure of this fluctuation is given by

1 ¼

1 ffiffiffiffiffiffi

Df T

p ð17:42Þ

It should be noted that increasing Df results in reduction of the precision of the estimates while

improving the appearance. To offset this, T must be increased further, or averaging must be done for

several sample records.

Generating a test-input signal with a PSD that satisfactorily compares with the required PSD can be a

tedious task if one attempts to do it manually by mixing various signal components. A convenient

method is to use an automatic multiband equalizer. By this means, the mean amplitude of the signal in

each small frequency band of interest can be made to approach the spectrum of the specified vibration

environment (see Figure 17.7). Unfortunately, this type of random-signal vibration testing can be more

costly than testing with deterministic signals.

17.2.5 Response Spectrum

Response spectra are commonly used to represent signals associated with vibration testing. A given signal

has a certain fixed response spectrum, but many different signals can have the same response spectrum.

For this reason, as will be clear shortly, the original signal cannot be reconstructed from its response

spectrum (unlike in the case of a Fourier spectrum). This is a disadvantage. However, the physical

significance of a response spectrum makes it a good representation for a test signal.

FIGURE 17.6 Effect of filter bandwidth on PSD results.

Vibration Testing 17-15

If a given signal is applied to a single-degree-of-freedom (single-DoF) oscillator (of a specific natural

frequency), and the response of the oscillator (mass) is recorded, we can determine the maximum (peak)

value of that response. Suppose that we repeat the process for a number of different oscillators (having

different natural frequencies) and then plot the peak response values thus obtained against the

corresponding oscillator natural frequencies. This procedure is shown schematically in Figure 17.8. For

an infinite number of oscillators (or for the same oscillator with continuously variable natural

frequency), we get a continuous curve, which is called the response spectrum of the given signal. It is

obvious, however, that the original signal cannot be completely determined from the knowledge of its

response spectrum alone. As shown in Figure 17.8, for instance, another signal, when passed through a

given oscillator, might produce the same peak response.

Note that we have assumed the oscillators to be undamped; the response spectrum obtained using

undamped oscillators corresponds to z ¼ 0: If all the oscillators are damped, however, and have the same

damping ratio, z; the resulting response spectrum will correspond to that particular z. It is, therefore,

clear that z is also a parameter in the response-spectrum representation. We should specify the damping

value as well when we represent a signal by its response spectrum.

Random Number

Generator

Random Signal

Constructor

Shaping (Intensity)

Function

Automatic Multiband Filter Circuit

Equalizer

Probability Distribution

Simulated Vibration

Environment

Reference Power Spectrum

FIGURE 17.7 Generation of a specified random vibration environment.

FIGURE 17.8 Definition of the response spectrum of a signal.

17-16 Vibration and Shock Handbook

17.2.5.1 Displacement, Velocity, and Acceleration Spectra

It is clear that a motion signal can be represented

by the corresponding displacement, velocity, or

acceleration. First, consider a displacement signal,

uðtÞ: The corresponding velocity signal is u_ ðtÞ and

the acceleration is u€ ðtÞ:

Now consider an undamped simple oscillator,

which is subjected to a support displacement uðtÞ;

as shown in Figure 17.9. As usual, assuming that

the displacements are measured with respect to a

static equilibrium configuration, the gravity effect

can be ignored. Then, the equation of motion is

given by

my€d ¼ kðu 2 ydÞ ð17:43Þ

y€d þv2

nyd ¼ v2

nuðtÞ ð17:44Þ

where the (undamped) natural frequency is given by

vn ¼

ffiffiffiffi

ð17:45Þ

Suppose that the support (displacement) excitation, uðtÞ; is a unit impulse dðtÞ: Then, the corresponding

(displacement) response y is called the impulse-response function, and is denoted by hðtÞ: It is known

that hðtÞ is the inverse Laplace transform (with zero initial condition) of the transfer function of the

system (Equation 17.44), as given by

HðsÞ ¼

ðs2 þ v2

nÞ ð17:46Þ

The impulse-response function (to an impulsive support excitation) for an undamped, single-DoF

oscillator having natural frequency vn is given by

hðtÞ ¼ vn sin vnt ð17:47Þ

The displacement response yd ðtÞ of this oscillator, when excited by the displacement signal uðtÞ; is given

by the convolution integral

ydðtÞ ¼ vn

ð1

uðtÞsin vnðt 2 tÞdt ð17:48Þ

The “velocity” response of the same oscillator, when excited by the velocity signal, u_ ðtÞ; is given by

yv ðtÞ ¼ vn

ð1

u_ ðtÞ sin vnðt 2tÞdt ð17:49Þ

and the “acceleration” response when excited by the acceleration signal, u€ ðtÞ; is

yaðtÞ ¼ vn

ð1

u€ ðtÞ sin vnðt 2tÞdt ð17:50Þ

These results immediately follow from Equation 17.44. Specifically, differentiate Equation 17.44 once to

obtain

y€v þv2

nyv ¼ v2

nu_ ðtÞ ð17:51Þ

u(t)

FIGURE 17.9 Undamped simple oscillator subjected

to a support excitation.

Vibration Testing 17-17

and differentiate again, to obtain

y€a þv2

nya ¼ v2

nu€ ðtÞ ð17:52Þ

in which

yv ¼

dyd

dt ð17:53Þ

ya ¼

d2yd

dt2 ¼

dyv

dt ð17:54Þ

If the peak value of ydðtÞ is plotted against vn; we get the displacement – spectrum curve of the displacement

signal, uðtÞ: If the peak value of yv ðtÞ is plotted against vn; we get the velocity – spectrum curve of the

displacement signal, uðtÞ: If the peak value of yaðtÞ is plotted against vn; we get the acceleration – spectrum

curve of the displacement signal, uðtÞ: Now consider Equation 17.49. Integration by parts gives

yv ðtÞ ¼ ½vnuðtÞsin vnðt 2 tÞ􀀉10 þ v2

ð1

uðtÞcos vnðt 2 tÞdt ð17:55Þ

The initial and final conditions for uðtÞ are assumed to be zero. It follows that the first term in

Equation 17.55 vanishes. The second term is vn½ydðt þ p=2vn 2 tÞ􀀉; which is clear by noting that

sin vnðt þ p=2vn 2 tÞ is equal to cos vnðt 2 tÞ;; thus

yv ðtÞ ¼ 2vnyd t þ

2vn

􀀏 􀀐

ð17:56Þ

If we integrate Equation 17.50 by parts twice, and apply the end conditions as before, we obtain

yaðtÞ ¼ 2v2

nydðtÞ ð17:57Þ

By taking the peak values of response time histories, we see from Equation 17.56 and Equation 17.57 that

vðvnÞ ¼ vndðvnÞ ð17:58Þ

aðvnÞ ¼ v2

ndðvnÞ ð17:59Þ

in which dðvnÞ; vðvnÞ; and aðvnÞ represent the displacement spectrum, the velocity spectrum, and the

acceleration spectrum, respectively, of the displacement time history, uðtÞ: It follows from Equation 17.58

and Equation 17.59 that

aðvnÞ ¼ vnvðvnÞ ð17:60Þ

17.2.5.2 Response-Spectra Plotting Paper

Response spectra are usually plotted on a frequency – velocity coordinate plane or on a frequency –

acceleration coordinate plane. Values are normally plotted in logarithmic scale, as shown in Figure 17.10.

First, consider the axes shown in Figure 17.10(a). Obviously, constant velocity lines are horizontal for this

coordinate system. From Equation 17.58, the constant-displacement line corresponds to

vðvnÞ ¼ cvn

By taking logarithms of both sides, we obtain

log vðvnÞ ¼ log vn þ log c

It follows that the constant-displacement lines have a slope of þ1 on the logarithmic frequency – velocity

plane. Similarly, from Equation 17.60, the constant-acceleration lines correspond to

vnvðvnÞ ¼ c

17-18 Vibration and Shock Handbook

Hence,

log vðvnÞ ¼ 2log vn þ log c

It follows that the constant-acceleration lines have

a slope of negative one on the logarithmic

frequency – velocity plane. Similarly, it can be

shown from Equation 17.59 and Equation 17.60

that, on the logarithmic frequency – acceleration

plane (Figure 17.10(b)), the constant-displacement

lines have a slope of þ2, and the constant-velocity

lines have a slope of þ1.

On the frequency – velocity plane, a point

corresponds to a specific frequency and a specific

velocity. The corresponding displacement at the

point is obtained (Equation 17.58) by dividing the

velocity value by the frequency value at that point.

The corresponding acceleration at that point is

obtained (Equation 17.60) by multiplying the

particular velocity value by the frequency value.

Any units may be used for displacement, velocity,

and acceleration quantities. A typical logarithmic

frequency – velocity plotting sheet is shown in

Figure 17.11. Note that the sheet is already

graduated on constant displacement, velocity,

and acceleration lines. Also, a period axis

(period ¼ 1/cyclic frequency) is given for convenience

in plotting. A plot of this type is called a

nomograph.

17.2.5.3 Zero-Period Acceleration

Frequently, response spectra are specified in terms of accelerations rather than velocities. This is

particularly true in vibration testing associated with product qualification, because typical operational

disturbance records are usually available as acceleration time histories. No information is lost because the

logarithmic frequency – acceleration plotting paper can be graduated for velocities and displacements as

well. It is, therefore, clear that an acceleration quantity (peak) on a response spectrum has a

corresponding velocity quantity (peak), and a displacement quantity (peak). In vibration testing,

however, the motion variable that is in common usage is the acceleration. Zero-period acceleration (ZPA)

is an important parameter that characterizes a response spectrum. It should be remembered, however,

that zero-period velocity or zero-period displacement can be similarly defined.

ZPA is defined as the acceleration value (peak) at zero period (or infinite frequency) on a response

spectrum. Specifically,

ZPA ¼ lim

vn!1

aðvnÞ ð17:61Þ

Consider the damped simple oscillator equation (for support motion excitation):

y€ þ 2zvny_ þv2

ny ¼ v2

nuðtÞ ð17:62Þ

By differentiating Equation 17.62 throughout, either once or twice, it is seen, as in Equation 17.51

and Equation 17.52, that if u and y initially refer to displacements, then the same equation is valid

when both of them refer to either velocities or accelerations. Let us consider the case in which

u and y refer to input and response acceleration variables, respectively. Consider a sinusoidal

Acceleration (Log) Velocity (Log)

Displacement =Constant

Acceleration =Constant

Format 1

Format 2

Frequency (Log)

Velocity =Constant

Displacement =Constant

Acceleration = Constant

Frequency (Log)

(a)

(b)

Velocity = Constant

FIGURE 17.10 Response-spectra plotting formats:

(a) frequency – velocity plane; (b) frequency – acceleration

plane.

Vibration Testing 17-19

signal, uðtÞ; given by

uðtÞ ¼ A sin vt ð17:63Þ

The resulting response, yðtÞ; neglecting the transient components (that is, the steady-state value), is

given by

yðtÞ ¼ A

ffiffiffiffiffiffiffiffiffiffiffiffiffiffinffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðv2

n 2 v2Þ2 þ 4z2v2

nv2

p sinðvt þ fÞ ð17:64Þ

FIGURE 17.11 Response-spectra plotting sheet or nomograph (frequency – velocity plane).

17-20 Vibration and Shock Handbook

Hence, the acceleration-response spectrum, given

by aðvnÞ ¼ ½yðtÞ􀀉max; for a sinusoidal signal of

frequency, v; and amplitude, A; is

aðvnÞ ¼ A

ffiffiffiffiffiffiffiffiffiffiffiffiffiffinffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðv2

n 2 v2Þ2 þ 4z2v2

nv2

p ð17:65Þ

A plot of this response is shown in Figure 17.12.

Note that að0Þ ¼ 0: Also,

ZPA ¼ lim

vn!1

aðvnÞ ¼ A ð17:66Þ

It is worth observing that at the point vn ¼ v

(i.e., when the excitation frequency, v; is equal to

the natural frequencies, vn; of the simple

oscillator), we have aðvnÞ ¼ A=ð2zÞ; which corresponds to an amplification by a factor of 1=ð2zÞ

over the ZPA value.

17.2.5.4 Uses of Response Spectra

In vibration testing, response-spectra curves are employed to specify the dynamic environment to which

the test object is required to be subjected. This specified response spectrum is known as the required

response spectrum (RRS). In order to satisfy conservatively the test specification, the response spectrum of

the actual test input excitation, known as the test response spectrum (TRS), should envelop the RRS.

Note that, when response spectra are used to represent excitation input signals in vibration testing, the

damping value of the hypothetical oscillators used in computing the response spectrum has no bearing

on the actual damping that is present in the test object. In this application, the response spectrum is

merely a representation of the shaker-input signal and, therefore, does not depend on system damping.

Another use of response spectra is in estimating the peak value of the response of a multi-DoF or

distributed-parameter system when it is excited by a signal whose response spectrum is known. To

understand this concept, we recall the fact that, for a multi-DoF or truncated (approximated)

distributed-parameter system having distinct natural frequencies, the total response can be expressed as a

linear combination of the individual modal responses. Specifically, the response yðtÞ can be written

yðtÞ ¼

i¼1

aiaðviÞ exp

2zivit ffiffiffiffiffiffiffiffi

1 2 z2i

sinðvit þ fiÞ ð17:67Þ

in which the spectrum, aðviÞ, is comprised of the amplitude contributions from each mode (simple

oscillator equation), with “damped” natural frequency, vi: Hence, aðviÞ corresponds to the value of the

response spectrum at frequency vi: The linear combination parameters, ai; depend on the modalparticipation

factors and can be determined from system parameters. Since the peak values of all terms in

the summation on the right-hand side of Equation 17.67 do not occur at the same time, we observe that

½yðtÞ􀀉peak ,

i¼1

aiaðviÞ ð17:68Þ

It follows that the right-hand side of the inequality (Equation 17.68) is a conservative upper-bound

estimate (i.e., the absolute sum) for the peak response of the multi-DoF system. Some prefer to make the

estimate less conservative by taking the square root of sum of the squares (SRSS):

½yðtÞ􀀉SRSS ¼

i¼1

a2i

a2ðviÞ

" #1=2

ð17:69Þ

Acceleration

Spectrum

ZPA

a(wn)

Oscillator Natural Frequency

2ζ

wn = w wn

FIGURE 17.12 Response spectrum and ZPA of a sine

signal.

Vibration Testing 17-21

The latter method, however, has the risk of giving an estimate that is less than the true value. Note

that, in this application, the damping value associated with the response spectrum is directly related

to modal damping of the system. Hence, the response spectrum, aðviÞ, should correspond to the

same damping ratio as that of the mode considered within the summation of the inequality

(Equation 17.68). If all modal damping ratios, zi; are identical or nearly so, the same response

spectrum could be used to compute all terms in the inequality 17.68. Otherwise, different responsespectra

curves should be used to determine each quantity, aðviÞ; depending on the applicable modal

damping ratio, zi:

17.2.6 Comparison of Various Representations

In this section, we shall state some major advantages and disadvantages of the four representations of the

vibration environment that we have discussed.

Time-signal representation has several advantages. It can be employed to represent either

deterministic or random vibration environment. It is an exact representation of a single excitation

event. Also, when performing multiexcitation (multiple shaker) vibration testing, phasing between the

various inputs can be conveniently incorporated simply by delaying each excitation with respect to the

others. There are also disadvantages to time-signal representation. Since each time history represents

just one sample function (a single event) of a random environment, it may not be truly representative of

the actual vibration excitation. This can be overcome by using longer signals, which, however, will

increase the duration of the test, which is limited by test specifications. If the random vibration is truly

ergodic (or at least stationary), this problem will not be as serious. Furthermore, the problem does not

arise when testing with deterministic signals. An extensive knowledge of the true vibration environment

to which the test object is subjected is necessary, however, in order to conclude that it is stationary or that

it could be represented by a deterministic signal. In this sense, time-signal representation is difficult to

implement.

The response-spectrum method of representing a vibration environment has several advantages. It is

relatively easy to implement. Since the peak response of a simple oscillator is used in its definition, it is

representative of the peak response or structural stress of simple dynamic systems; hence, there is a direct

relation to the behavior of the physical object. An upper bound for the peak response of a multi-DoF

system can be conveniently obtained by the method outlined in Section 17.2.5.4. Also, by considering the

envelope of a set of response spectra at the same damping value, it is possible to use a single response

spectrum to conservatively represent more than one excitation event. The method also has disadvantages.

It employs deterministic signals in its definition. Sample functions (single events) of random vibrations

can be used, however. It is not possible to determine the original vibration signal from the knowledge of

its response spectrum, because it uses the peak value of response of a simple oscillator (more than one

signal can have the same response spectrum). Thus, a response spectrum cannot be considered a

complete representation of a vibration environment. Also, characteristics such as the transient nature and

the duration of the excitation event cannot be deduced from the response spectrum. For the same reason,

it is not possible to incorporate information on excitation-signal phasing into the response-spectrum

representation. This is a disadvantage in multiple excitation testing.

Fourier spectrum representation also has advantages. Since the actual dynamic environment signal can

be obtained by inverse transformation, it has the same advantages as for the time-signal representation.

In particular, since a Fourier spectrum is generally complex, phasing information of the test excitation

can be incorporated into Fourier spectra, in multiple excitation testing. Furthermore, by considering an

envelope Fourier spectrum (like an envelope response spectrum), it can be employed to represent

conservatively more than one vibration environment. Also, it gives frequency-domain information (such

as information about resonances), which is very useful in vibration testing situations. The disadvantages

of Fourier spectrum representation include the following. It is a deterministic representation but, as in the

response-spectrum method, a sample function (a single event) of a random vibration can be represented

by its Fourier spectrum. Transient effects and event duration are hidden in this representation. Also, it is

17-22 Vibration and Shock Handbook

somewhat difficult to implement, because complex procedures of multiband equalization might be

necessary in the signal synthesis associated with this representation.

PSD representation has the following advantages. It takes the random nature of a vibration

environment into account. As in response-spectrum and Fourier-spectrum representations, by taking an

envelope PSD, it can be used to represent conservatively more than one environment. It can display

important frequency-domain characteristics, such as resonances. Its disadvantages include the following.

It is an exact representation only for truly stationary or ergodic random environments. In nonstationary

situations, as in seismic ground motions, significant error could result. Also, it is not possible to obtain

the original sample function (dynamic event) from its PSD. Hence, the transient characteristics and

duration of the event are not known from its PSD. Since mean square values, not peak values, are

considered, PSD representation is not structural-stress-related. Furthermore, since PSD functions are real

(not complex), we cannot incorporate phasing information into them. This is a disadvantage in multiple

excitation testing situations, but this problem can be overcome by considering either the cross spectrum

(which is complex) or the cross correlation in each pair of test excitations.

Random vibration testing is compared with sine testing (single-frequency, deterministic excitations) in

Box 17.1. A comparison of various representations of test excitations is given in Box 17.2.

Box 17.1

RANDOM TESTING VS. SINE TESTING

Advantages of Random Testing:

1. More realistic representation of the true environment

2. Many frequencies are applied simultaneously

3. All resonances, natural frequencies, and mode shapes are excited simultaneously

Disadvantages of Random Testing:

1. Needs more power for testing

2. Control is more difficult

3. More costly

Advantages of Sine Testing:

Appropriate for:

1. Fatigue testing of products that operate primarily at a known speed (frequency) under

in-service conditions

2. Detecting sensitivity of a device to a particular excitation frequency

3. Detecting resonances, natural frequencies, modal damping, and mode shapes

4. Calibration of vibration sensors and control systems

Disadvantages of Sine Testing:

1. Usually not a good representation of the true dynamic environment

2. Because vibration energy is concentrated at one frequency, it can cause failures that would

not occur in service (particularly single-resonance failures)

3. Since only one mode is excited at a time, it can hide multiple-resonance failures that might

occur in service

Vibration Testing 17-23

In practice, the generation of an excitation signal for vibration testing may not follow any one of the

analytical procedures and may incorporate a combination of them. For example, a combination of sinebeat

signals of different frequencies with random phasing is one practical approach to the generation of a

multifrequency, pseudo-random excitation signal. This approach is summarized in Box 17.3.

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