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17.4 Testing Procedures

Back

Vibration testing may involve pretesting prior to the main tests. The objectives of pretesting may be

(1) exploratory, in order to obtain dynamic information such as natural frequencies, mode shapes

and damping about the test object; (2) preconditioning, in order to age or pass the “infant-mortality”

stage so that the main test will be realistic and correspond to normal operating conditions. In the present

section, we will describe both pretesting and main testing in an integrated manner.

17.4.1 Resonance Search

Vibration test programs usually require a resonance-search pretest. This is typically carried out at a lower

excitation intensity than that used for the main test in order to minimize the damage potential

(overtesting). The primary objective of a resonance-search test is to determine resonant frequencies of

the test object. More elaborate tests are employed, however, to determine mode shapes and modal

damping ratios in addition to resonant frequencies. Such frequency-response data on the test object are

useful in planning and conducting the main test.

Frequency-response data usually are available as a set of complex frequency-response functions.

There are tests that determine the frequency-response functions of a test object, and simpler tests

are available to determine resonant frequencies alone. Some of the uses of frequency-response data

are given below.

1. A knowledge of the resonant frequencies of the test object is important in conducting the main

test. More attention should be given, for example, when performing a main test in the vicinity

of resonant frequencies. In the resonance neighborhoods, lower sweep rates should be used if

sine sweep is used in the main test, and larger dwell periods should be used if a sine dwell is

part of the main test. Frequency-response data give the most desirable frequency range for

conducting main tests.

2. From frequency-response data, it is possible to determine the most desirable test excitation

directions and the corresponding input intensities.

3. The degree of nonlinearity and the time variance in the system parameters of the test

object can be estimated by conducting more than one frequency-response test at different

excitation levels. If the deviation in the frequency-response functions thus obtained is

Vibration Testing 17-37

sufficiently small, then a linear, time-invariant dynamic model is considered satisfactory in the

analysis of the test object.

4. If no resonances are observed in the test object over the frequency range of interest, as

determined by the operating environment for a given application, then a static analysis will be

adequate to qualify the test object.

5. A set of frequency-response functions can be considered a dynamic model for the test

specimen. This model can be employed in further studies of the test specimen by analytical

means.

17.4.2 Methods of Determining Frequency-Response Functions

Three methods of determining frequency-response functions are outlined here.

17.4.2.1 Fourier Transform Method

If yðtÞ is the response at location B of the test object, when a transient input, uðtÞ; is applied at location A;

then the frequency-response function, Hð f Þ; between locations A and B is given by the ratio of the

Fourier integral transforms of the output, yðtÞ; and the input, uðtÞ:

Hð f Þ ¼

Y ð f Þ

Uð f Þ ð17:79Þ

In particular, if uðtÞ is a unit impulse, then U ðf Þ ¼ 1 and, hence, Hð f Þ ¼ Y ð f Þ:

17.4.2.2 Spectral Density Method

If the input excitation is a random signal, the frequency-response function between the input point and

the output point can be determined as the ratio of the cross-spectral density, Fuy ðf Þ; of the input, uðtÞ;

and the output, yðtÞ; and the PSD, Fuuð f Þ; of the input:

Hðf Þ ¼

Fuy ð f Þ

Fuuð f Þ ð17:80Þ

17.4.2.3 Harmonic Excitation Method

If the input signal is sinusoidal (harmonic) with frequency, f ; the output also will be sinusoidal with

frequency, f ; at steady state but with a change in the phase angle. Then, the frequency-response function

is obtained as a magnitude function and a phase-angle function. The magnitude, lHð f Þl; is equal to the

steady-state amplification of the output signal, and the phase angle, /Hð f Þ; is equal to the steady-state

phase lead of the output signal. This pair of curves, the magnitude plot and the phase angle plot, is called

a Bode plot or Bode diagram.

17.4.3 Resonance-Search Test Methods

There are three basic types of resonance-search test methods. They are categorized according to the

nature of the excitation used in the test; specifically, (1) impulsive excitation, (2) initial displacement, or

(3) forced vibration. The first two categories are free-vibration tests; that is, response measurements are

made on free decay of the test object following a momentary (initial) excitation. Typical tests belonging

to each of these categories are described in the following sections.

17.4.3.1 Hammer (Bump) Test and Drop Test

In a resonance search using the impulsive-excitation method, an impulsive force (a large magnitude

of force acting over a very short duration) is applied at a suitable location of the test object, and the

resulting transient response of the object is observed, preferably at several locations. This is

17-38 Vibration and Shock Handbook

equivalent to applying an initial velocity to the test object and letting it vibrate freely. By Fourier

analysis of the response data, it is possible to obtain the resonant frequencies, corresponding mode

shapes, and modal damping.

Hammer tests and drop tests belong to the impulsive-excitation category. A schematic diagram of the

hammer-test arrangement is shown in Figure 17.18. A schematic diagram of the drop-test arrangement is

shown in Figure 17.19. The angle of swing of the hammer or the drop height of the object determines the

intensity of the applied impulse. Alternatively, the impulse can be generated using explosive cartridges

(for relatively large structures) located suitably in the test object, or by firing small projectiles at the test

object. The response is monitored at several locations of the test object. The response at the point of

application of the impulse is always monitored. Response analysis can be done in real time, or the

response can be recorded for subsequent analysis. A major concern in these tests is making sure that all

significant resonances in the required frequency range are excited under the given excitation. Several tests

for different configurations of the test object might be necessary to achieve this.

Proper selection of the response-monitoring locations is also important in obtaining meaningful test

results. By changing the impulsive-force intensity and repeating the test, any significant nonlinear (or

time-variant-parameter) behavior of the test object can be determined. A common practice is to monitor

the impulsive-force signal during impact. In this way, poor impacts (for example, low-intensity impacts

FIGURE 17.18 Schematic diagram of a hammer test arrangement.

FIGURE 17.19 Schematic diagram of a drop test arrangement.

Vibration Testing 17-39

or multiple impacts caused by the bouncing back of a hammer) can be detected and the corresponding

test results can be rejected.

17.4.3.2 Pluck Test

A resonance search on a test object can be performed by applying a displacement initial condition (rather

than a velocity initial condition, as in impulsive tests) to a suitably mounted test object and measuring its

subsequent response at various locations as it executes free vibrations. By properly selecting the locations

and the magnitudes of the initial displacements, it is sometimes possible to excite various modes of

vibration, provided that these modes are reasonably uncoupled.

The pluck test is the most common test that uses the initial-displacement method. A schematic

diagram of the test set-up is shown in Figure 17.20. The test object is initially deflected by pulling it with a

cable. When the cable is suddenly released, the test object will undergo free vibrations about its staticequilibrium

position. The response is observed for several locations of the test object and analyzed to

obtain the required parameters.

In Figure 17.18 to Figure 17.20, the frequency-response function between two locations (A and B, for

example) is obtained by analyzing the corresponding two signals, using either the Fourier transform

method (Equation 17.79) or the spectral-density method (Equation 17.80). These frequency-domain

techniques will automatically provide the natural-frequency and modal-damping information.

Alternatively, modal damping can be determined using time-domain methods, for example, by

evaluating the logarithmic decrement of the response after passing it through a filter having a center

frequency adjusted to the predetermined natural frequency of the test object for that mode. The accuracy

of the estimated modal-damping value can be improved significantly by such filtering methods.

Often, the most difficult task in a natural-frequency search is the excitation of a single a mode. If two

natural frequencies are close together, modal interactions of the two invariably will be present in the

response measurements. Because of the closeness of the frequencies, the response curve will display a beat

phenomenon, as shown in Figure 17.21, which makes it difficult to determine damping by the

logarithmic-decrement method. It is difficult to distinguish between decay caused by damping and rapid

drop-off caused by beating. In this case, one of the frequency components must be filtered out, using a

very narrowband-pass filter, before computing damping.

The required testing time for the impulsive-excitation and initial-displacement test methods is

relatively small in comparison with forced-vibration test durations. For this reason, these former

(free-vibration) tests are often preferred in preliminary (exploratory) testing before the main tests.

The directions and locations of impact or initial displacements should be properly chosen, however,

so that as many significant modes as possible will be excited in the desired frequency range. If the

FIGURE 17.20 Diagram of a pluck test arrangement.

17-40 Vibration and Shock Handbook

impact is applied at a node point, (of a particular mode, for instance) it will be virtually impossible

to detect that mode from the response data. Sometimes, a large number of monitoring locations are

necessary to accurately determine mode shapes of the test object. This depends primarily on the size

and dynamic complexity of the test object and the particular mode number. This, in turn,

necessitates the use of more sensors (accelerometers and the like) and recorder channels. If a

sufficient number of monitoring channels is not available, the test will have to be repeated, each

time using a different set of monitoring locations. Under such circumstances, it is advisable to keep

one channel (monitoring location) unchanged and to use it as the reference channel. In this manner,

any deviations in the test-excitation input can be detected for different tests and properly adjusted or

taken into account in subsequent analysis (for example, by normalizing the response data).

17.4.3.3 Shaker Tests

A convenient method of resonance search is by

using a continuous excitation. A forced excitation,

which typically is a sinusoidal signal or a random

signal, is applied to the test object by means of a

shaker, and the response is continuously monitored.

The test set-up is shown schematically in

Figure 17.22. For sinusoidal excitations, signal

amplification and phase shift over a range of

excitations will determine the frequency-response

function. For random excitations, Equation 17.80

may be used to determine the frequency-response

function.

One or several portable exciters (shakers) or a

large shaker table similar to that used in the main

vibration test can be employed to excite the test

object. The number and the orientations of the shakers and the mounting configurations and monitoring

locations of the test object should be chosen depending on the size and complexity of the test object, the

required accuracy of the resonance-search results, and the modes of vibration that need to be excited. The

shaker-test method has the advantage of being able to control the nature of the test-excitation input (for

example, frequency content, intensity, and sweep rate), although it might be more complex and costly.

The results from shaker tests are more accurate and more complete.

Test objects usually display a change in resonant frequencies when the shaker amplitude is

increased. This is caused by inherent nonlinearities in complex structural systems. Usually, the

FIGURE 17.21 Beat phenomenon resulting from interaction of closely spaced modes.

FIGURE 17.22 Schematic diagram of a shaker test for

resonance search.

Vibration Testing 17-41

change appears as a spring-softening effect, which results in lower resonant frequencies at higher

shaker amplitudes. If this nonlinear effect is significant, the resonant frequencies for the main test

level cannot be accurately determined using a resonance search at low intensity. Some form of

extrapolation of the test results, or analysis using an appropriate dynamic model, might be necessary

in this case to determine the resonant-frequency information that might be required to perform the

main test.

17.4.4 Mechanical Aging

Before performing a qualification test, it is usually necessary to age the test object to put it into a

condition that represents its state following its operation for a predetermined period under in-service

conditions. In this manner, it is possible to reduce the probability of burn-in failure (infant mortality)

during testing. Some tests, such as design-development tests and quality-assurance tests, might not

require prior aging.

The nature and degree of aging that is required depends on such factors as the intended function of the

test object, the operating environment, and the purpose of the dynamic test. In qualification tests, it may

be necessary to demonstrate that the test object still has adequate capability to withstand an extreme

dynamic environment toward the end of its design life (that is, the period in which it can be safely

operated without requiring corrective action). In such situations, it is necessary to age the test object to

an extreme deterioration state, representing the end of the design life of the test object.

Test objects are aged by subjecting them to various environmental conditions (for example, high

temperatures, radiation, humidity, and vibrations). Usually, it is not practical to age the equipment at the

same rate as it would age under a normal service environment. Consequently, accelerated aging

procedures are used to reduce the test duration and cost. Furthermore, the operating environment may

not be fully known at the testing stage. This makes the simulation of the true operating environment

virtually impossible. Usually, accelerated aging is done sequentially, by subjecting the test equipment to

the various environmental conditions one at a time. Under in-service conditions, however, these effects

occur simultaneously, with the possibility of interactions between different effects. Therefore, when

sequential aging is employed, some conservatism should be added. The type of aging used should be

consistent with the environmental conditions and operating procedures of the specific application of the

test object. Often these conditions are not known in advance, in which case, standardized aging

procedures should be used.

Our main concern in this section is mechanical aging, although other environmental conditions can

significantly affect the dynamic characteristics of a test object. The two primary mechanisms of

mechanical aging are material fatigue and mechanical wearout. The former mechanism plays a primary

role if in-service operation consists of cyclic loading over relatively long periods of time. Wearout,

however, is a long-term effect caused by any type of relative motion between components of the test

object. It is very difficult to analyze component wearout, even if only the mechanical aspects are

considered (that is the effects of corrosion, radiation, and the like are neglected). Some mechanical

wearout processes resemble fatigue aging; however, they depend simultaneously on the number of cycles

of load applications and the intensity of the applied load. Consequently, only the cumulative damage

phenomenon, which is related to material fatigue, is usually treated in the literature.

Although mechanical aging is often considered a pretest procedure (for example, the resonance-search

test), it actually is part of the main test. In a dynamic qualification program, if the test object

malfunctions during mechanical aging, this amounts to failure in the qualification test. Furthermore,

exploratory tests, such as resonance-search tests, are sometimes conducted at higher intensities than what

is required to introduce mechanical aging into the test object.

17.4.4.1 Equivalence for Mechanical Aging

It is usually not practical to age a test object under its normal operating environment, primarily because

of time limitations and the difficulty in simulating the actual operating environment. Therefore, it may

17-42 Vibration and Shock Handbook

be necessary to subject the test object to an accelerated aging process in a dynamic environment of higher

intensity than that present under normal operating conditions.

Two aging processes are said to be equivalent if the final aged condition attained by the two processes is

identical. This is virtually impossible to realize in practice, particularly when the object and the

environment are complex and the interactions of many dynamic causes have to be considered. In this

case, a single most severe aging effect is used as the standard for comparison to establish the equivalence.

The equivalence should be analyzed in terms of both the intensity and the nature of the dynamic

excitations used for aging.

17.4.4.2 Excitation-Intensity Equivalence

A simplified relationship between the dynamic-excitation intensity, U ; and the duration of aging, T; that

is required to attain a certain level of aging, keeping the other environmental factors constant, may be

given as

T ¼

U r ð17:81Þ

in which c is a proportionality constant and r is an exponent. These parameters depend on such

factors as the nature and sequence of loading and characteristics of the test object. It follows from

Equation 17.81 that, by increasing the excitation intensity by a factor n, the aging duration can be

reduced by a factor of nr : In practice, however, the intensity – time relationship is much more

complex, and caution should be exercised when using Equation 17.81. This is particularly true if the

aging is caused by multiple dynamic factors of varying characteristics that are acting simultaneously.

Furthermore, there is usually an acceptable upper limit to n: It is unacceptable, for example, to use a

value that will produce local yielding or any such irreversible damage to the equipment that is not

present under normal operating conditions.

It is not necessary to monitor functional operability during mechanical aging. Furthermore, it can

happen that, during accelerated aging, the equipment malfunctions but, when the excitation is removed,

it operates properly. This type of reversible malfunction is acceptable in accelerated aging.

The time to attain a given level of aging is usually related to the stress level at a critical location of the

test object. Since this critical stress can be related, in turn, to the excitation intensity, the relationship

given by Equation 17.81 is justified.

17.4.4.3 Dynamic-Excitation Equivalence

The equivalence of two dynamic excitations that have different time histories can be represented using

methods employed to represent dynamic excitations (for example, response spectrum, Fourier spectrum,

and PSD). If the maximum (peak) excitation is the factor that primarily determines aging in a given

system under a particular dynamic environment, then response-spectrum representation is well suited

for establishing the equivalence of two excitations. If, however, the frequency characteristics of the

excitation are the major determining factor for mechanical aging, then Fourier spectrum representation

is favored for establishing the equivalence of two deterministic excitations, and PSD representation is

suited for random excitations. When two excitation environments are represented by their respective

PSD functions, F1ðvÞ and F2ðvÞ; if the significant frequency range for the two excitations is ðv1; v2Þ;

then the degree of aging under the two excitations may be compared using the ratio

A2 ¼

ðv2

F1ðvÞdv

ðv2

F2ðvÞdv ð17:82Þ

in which A denotes a measure of aging. If the two excitations have different frequency ranges of interest, a

range consisting of both ranges might be selected for the integrations in Equation 17.82.

Vibration Testing 17-43

17.4.4.4 Cumulative Damage Theory

Miner’s linear cumulative damage theory may be used to estimate the combined level of aging resulting

from a set of excitation conditions. Consider m excitations acting separately on a system. Suppose that

each of these excitations produces a unit level of aging in N1; N2; …; Nm loading cycles, respectively, when

acting separately. If, in a given dynamic environment, n1; n2; …; nm loading cycles, respectively, from

the m excitations actually have been applied to the system (possibly all excitations were acting

simultaneously), the level of aging attained can be given by

A ¼

i¼1

Ni ð17:83Þ

The unit level of aging is achieved, theoretically, when A ¼ 1: Equation 17.83 corresponds to Miner’s

linear cumulative damage theory.

Because of various interactive effects produced by different loading conditions, when some or all of the

m excitations act simultaneously, it is usually not necessary to have A ¼ 1 under the combined excitation

to attain the unit level of aging. Furthermore, it is extremely difficult to estimate Ni; i ¼ 1; …; m: For such

reasons, the practical value of A in Equation 17.83 for using in attaining a unit level of aging could vary

widely (typically, from 0.3 to 3.0).

17.4.5 Test-Response Spectrum Generation

A vibration test may be specified by a RRS. In this case, the response spectrum of the actual excitation

signal, that is, the TRS, should envelop the RRS during testing. It is customary for the purchaser (the

owner of the test object) to provide the test laboratory with a multichannel FM tape or some form of

signal storage device containing the components of the excitation input signal that should be used in the

test. Alternatively, the purchaser may request that the test laboratory generate the required signal

components under the purchaser’s supervision. If sine beats are combined to generate the test excitations,

each FM tape should be supplemented by tabulated data giving the channel number, the beat frequencies

(Hz) in that channel, and the amplitude ðgÞ of each sine-beat component. The RRS curve that is

enveloped by the particular input should also be specified.

The excitation signal that is applied to the shaker-table actuator is generated by combining the

contents of each channel in an appropriate ratio so that the response spectrum of the excitation that is

actually felt at the mounting locations of the test object (the TRS) satisfactorily matches the RRS supplied

to the test laboratory. Matching is performed by passing the contents of each channel through a variablegain

amplifier and mixing the resulting components according to variable proportions. These operations

are performed by a waveform mixer. The adjustment of the amplifier gains is done by trial and error. The

phase of the individual signal components should be maintained during the mixing process.

Each channel may contain a single-frequency component (such as sine beat) or a multifrequency signal

of fixed duration (for example, 20 sec). If the RRS is complex, each channel may have to carry a

multifrequency signal to achieve close matching of the TRS with the RRS. Also, a large number of

channels might be necessary. The test excitation signal is generated continuously by repetitively playing

the FM tape loop of fixed duration.

In product qualification, response spectra are usually specified in units of acceleration due to

gravity ðgÞ: Consequently, the contents in each channel of the test-input FM tape represent

acceleration motions. For this reason, the signal from the waveform mixer must be integrated twice

before it is used to drive the shaker table. The actuator of the exciter is driven by this displacement

signal, and its control may be done by feedback from a displacement sensor. However, if the control

sensor is an accelerometer, as is typical, double integration of that signal will be needed as well. In

typical test facilities, a double integration unit is built into the shaker system. It is then possible to

use any type of signal (displacement, velocity, acceleration) as the excitation input and to decide

17-44 Vibration and Shock Handbook

simultaneously on the number of integrations that are necessary. If the input signal is a velocity time

history, for example, one integration should be chosen and so on.

The tape speed should be specified (for example, 7.5, 15 in./s) when the signals recorded on tapes are

provided to generate input signals for vibration testing. This is important to ensure that the frequency

content of the signal is not distorted. The speeding up of the tape has the effect of scaling up of each

frequency component in the signal. It has also the effect, however, of filtering out very high-frequency

components in the signal. If the excitation signals are available as digital records, then a DAC is needed to

convert them into analog signals.

17.4.6 Instrument Calibration

The test procedure normally stipulates accuracy requirements and tolerances for various critical

instruments that are used in testing. It is desirable that these instruments have current calibration records

that are agreeable to an accepted standard. Instrument manufacturers usually provide these calibration

records. Accelerometers, for example, may have calibration records for several temperatures (for

example, 2 65, 75, 3508F) and for a range of frequencies (such as 1 to 1000 Hz). Calibration records for

accelerometers are given in both voltage sensitivity (mV/g) and charge sensitivity (pC/g), along with

percentage-deviation values. These tolerances and peak deviations for various test instruments should be

provided for the purchaser’s review before they are used in the test apparatus.

From the tolerance data for each sensor or transducer, it is possible to estimate peak error percentages

in various monitoring channels in the test set-up, particularly in the channels used for functionaloperability

monitoring. The accuracy associated with each channel should be adequate to measure

expected deviations in the monitored operability parameter.

It is good practice to calibrate sensor or transducer units, such as accelerometers and associated

auxiliary devices, daily or after each test. These calibration data should be recorded under different scales

when a particular instrument has multiple scales, and for different instrument settings.

17.4.7 Test-Object Mounting

When a test object is being mounted on a shaker table, care should be taken to simulate all critical

interface features under normal installed conditions for the intended operation. This should be done as

accurately as is feasible. Critical interface requirements are those that could significantly affect the

dynamics of the test object. If the mounting conditions in the test set-up significantly deviate from those

under installed conditions for normal operation, adequate justification should be provided to show that

the test is conservative (that is, the motions produced under the test mounting conditions are more

severe than in in-service conditions). In particular, local mounting that would not be present under

normal installation conditions should be avoided in the test set-up.

In simulating in-service interface features, the following details should be considered as a minimum:

1. Test orientation of the test object should be its in-service orientation, particularly with respect to

the direction of gravity (vertical), available DoF, and mounting locations.

2. Mounting details at the interface of the test object and the mounting fixture should represent inservice

conditions with respect to the number, size, and strength of welds, bolts, nuts, and other

hold-down hardware.

3. Additional interface linkages, including wires cables, conduits, pipes, instrumentation (dials,

meters, gauges, sensors, transducers, and so on), and the supporting brackets of these elements,

should be simulated at least in terms of mass and stiffness, and preferably in terms of size as well.

4. Any dynamic effects of adjacent equipment cabinets and supporting structures under in-service

conditions should be simulated or taken into account in analysis.

5. Operating loads, such as those resulting from fluid flow, pressure forces, and thermal effects,

should be simulated if they appear to significantly affect test object dynamics. In particular, the

nozzle loads (fluid) should be simulated in magnitude, direction, and location.

Vibration Testing 17-45

The required mechanical interface details of the test object are obtained by the test laboratory at the

information-acquisition stage. Any critical interface details that are simulated during testing should be

included in the test report.

At least three control accelerometers should be attached to the shaker table near the mounting location

of the test object. One control accelerometer measures the excitation-acceleration component applied to

the test object in the vertical direction. The other two measure the excitation-acceleration components in

two horizontal directions at right angles. The two horizontal (control) directions are chosen to be

along the two major freedom-of-motion directions (or dynamic principal axes) of the test object.

Engineering judgment may be used in deciding these principal directions of high response in the test

object. Often, geometric principal axes are used. The control accelerometer signals are passed through a

response-spectrum analyzer (or a suitably programmed digital computer) to compute the TRS in the

vertical and two horizontal directions that are perpendicular.

Vibration tests generally require monitoring of the dynamic response at several critical locations of

the test object. In addition, the tests may call for the determining of mode shapes and natural

frequencies of the test object. For this purpose, a sufficient number of accelerometers should be

attached to various key locations in the test object. The test procedure (document) should contain a

sketch of the test object, indicating the accelerometer locations. Also, the type of accelerometers

employed, their magnitudes and directions of sensitivity, and the tolerances should be included in

the final test report.

17.4.8 Test-Input Considerations

In vibration testing, a significant effort goes into the development of test excitation inputs. Not

only the nature but also the number and the directions of the excitations can have a significant

effect on the outcomes of a test. This is so because the excitation characteristics determine the

nature of a test.

17.4.8.1 Test Nomenclature

We have noted that a common practice in vibration testing is to apply synthesized vibration excitation to

a test object that is appropriately mounted on a shaker table. Customarily, only translatory excitations as

generated by linear actuators, are employed. Nevertheless the resulting motion of the test object usually

consists of rotational components as well. A typical vibration environment may consist of threedimensional

motions, however. The specification of a three-dimensional test environment is a complex

task, even after omitting the rotational motions at the mounting locations of the test object.

Furthermore, practical vibration environments are random and they can be represented with sufficient

accuracy only in a probabilistic sense.

Very often the type of testing that is used is governed mainly by the capabilities of the test laboratory to

which the contract is granted. Test laboratories conduct tests using their previous experience and

engineering judgment. Making extensive improvements to existing tests can be very costly and timeconsuming,

and this is not warranted from the point of view of the customer or the vendor. Regulatory

agencies usually allow simpler tests if sufficient justification can be provided indicating that a particular

test is conservative with respect to regulatory requirements.

The complexity of a shaker-table apparatus is governed primarily by the number of actuators that are

employed and the number of independent directions of simultaneous excitation that it is capable of

producing. Terminology for various tests is based on the number of independent directions of excitation

used in the test. It would be advantageous to standardize this terminology to be able to compare different

test procedures. Unfortunately, the terminology used to denote different types of tests usually depends on

the particular test laboratory and the specific application. Attempts to standardize various test methods

have become tedious, partly because of the lack of a universal nomenclature for dynamic testing. A

justifiable grouping of test configurations is presented in this section. Figure 17.23 illustrates the various

test types.

17-46 Vibration and Shock Handbook

In test nomenclature, the DoF refers to the number of directions of independent motions that can be

generated simultaneously by means of independent actuators in the shaker table. According to this

concept, three basic types of tests can be identified:

1. Single-DoF (or rectilinear) testing is that in which the shaker table employs only one exciter

(actuator), producing test-table motions along the axis of that actuator. The actuator may not

necessarily be in the vertical direction.

2. Two-DoF testing is that in which two independent actuators, oriented at right angles to each other,

are employed. The most common configuration consists of a vertical actuator and a horizontal

actuator. Theoretically, the motion of each actuator can be specified independently.

3. Three-DoF testing is that in which three actuators, oriented at mutually right angles, are

employed. A desirable configuration consists of a vertical actuator and two horizontal actuators.

At least theoretically, the motion of each actuator can be specified independently.

It is common practice to specify the directions of excitation with respect to the geometric

principal axes of the test object. This practice is somewhat questionable, primarily because it does

not take into account the flexibility and inertia distributions of the object. Flexibility and inertia

elements in the test object have a significant influence on the level of dynamic coupling present in

a given pair of directions. In this respect, it is more appropriate to consider dynamic principal

axes rather than geometric principal axes of the test object. One useful definition is in terms

of eigenvectors of an appropriate three-dimensional, frequency-response function matrix that takes

into account the response at every critical location in the test object. The only difficulty in this

method is that prior frequency-response testing or analysis is needed to determine the test

input direction. For practical purposes, the vertical axis (the direction of gravity) is taken as one

principal axis.

The single-DoF (rectilinear) test configuration has three subdivisions, based on the orientation of

the vibration exciter (actuator) with respect to the principal axes of the test object. It is assumed that

one principal axis of the test object is the vertical axis and that the three principal axes are

Rectilinear

Uniaxial

Two DoF

Biaxial

Principal Axes

Rectilinear

Biaxial

Two DoF

Triaxial

Rectilinear

Triaxial

Three DoF

FIGURE 17.23 Vibration-test configurations.

Vibration Testing 17-47

mutually perpendicular. The three subdivisions are as follows:

1. Rectilinear uniaxial testing, in which the single actuator is oriented along one of the principal axes

of the test object

2. Rectilinear biaxial testing, in which the single actuator is oriented on the principal plane

containing the vertical and one of the two horizontal principal axes (the actuator is inclined to

both principal axes in the principal plane)

3. Rectilinear triaxial testing, in which the single actuator is inclined to all three orthogonal principal

axes of the test object

The two-DoF test configuration has two subdivisions, based on the orientation of the two actuators

with respect to the principal axes of the test object, as follows:

1. Two-DoF biaxial testing, in which one actuator is directed along the vertical principal axis and the

other along one of the two horizontal principal axes of the test object

2. Two-DoF triaxial testing, in which one actuator is positioned along the vertical principal axis and

the other actuator is horizontal but inclined to both horizontal principal axes of the test object

17.4.8.2 Testing with Uncorrelated Excitations

Simultaneous excitations in three orthogonal directions often produce responses (accelerations,

stresses, etc.) that are very different from that which is obtained by vectorially summing the responses

to separate excitations acting one at a time. This is primarily because of the nonlinear, time-variant

nature of test specimens and test apparatus, their dynamic coupling, and the randomness of excitation

signals. If these effects are significant, it is theoretically impossible to replace a three-DoF test, for

example, with a sequence of three single-DoF tests. In practice, however, some conservatism can be

incorporated into two-DoF and single-DoF tests to account for these effects. These tests with added

conservatism may be employed when three-DoF testing is not feasible. It should be clear by now that

rectilinear triaxial testing is generally not equivalent to three-degree-freedom testing, because the

former merely applies an identical excitation in all three orthogonal directions, with scaling factors

(direction cosines). One obvious drawback of rectilinear triaxial testing is that the input excitation in a

direction at right angles to the actuator is theoretically zero, and the excitation is at its maximum along

the actuator. In three-DoF testing using uncorrelated random excitations, however, no single direction

has a zero excitation at all times, and also the probability is zero that the maximum excitation occurs in

a fixed direction at all times.

Three-DoF testing is mentioned infrequently in the literature on vibration testing. A major reason

for the lack of three-DoF testing might be the practical difficulty in building test tables that can

generate truly uncorrelated input motions in three orthogonal directions. The actuator interactions

caused by dynamic coupling through the test table and mechanical constraints at the table supports are

primarily responsible for this. Another difficulty arises because it is virtually impossible to synthesize

perfectly uncorrelated random signals to drive the actuators. Two-DoF testing is more common. In this

case, the test must be repeated for a different orientation of the test object (for example, with a 908

rotation about the vertical axis), unless some form of dynamic-axial symmetry is present in the test

object.

Test programs frequently specify uncorrelated excitations in two-DoF testing for the two actuators.

This requirement lacks solid justification, because two uncorrelated excitations applied at right angles do

not necessarily produce uncorrelated components in a different pair of orthogonal directions, unless the

mean square values of the two excitations are equal. To demonstrate this, consider the two uncorrelated

excitations, u and v; shown Figure 17.24. The components u0 and v0; in a different pair of orthogonal

directions obtained by rotating the original coordinates through an angle u in the counterclockwise

direction, are given by

u0 ¼ u cos u þ v sin u ð17:84Þ

17-48 Vibration and Shock Handbook

v0 ¼ 2u sin u þ v cos u ð17:85Þ

Without loss of generality, we can assume that u

and v have zero means. Then, u 0 and v 0 also will

have zero means. Furthermore, since u and v are

uncorrelated, we have

EðuvÞ ¼ EðuÞEðvÞ ¼ 0 ð17:86Þ

From Equation 17.84 and Equation 17.85, we

obtain

Eðu0v0Þ ¼ E½ðu cos u þ v sin uÞ

􀀐ð2u sin u þ v cos uÞ􀀉

This, when expanded and substituted with

Equation 17.86, becomes

Eðu0v0Þ ¼ sin u cos u½Eðv2Þ 2 Eðu2Þ􀀉 ð17:87Þ

Since u is any general angle, the excitation components u 0 and v 0 become uncorrelated if and only if

Eðv2Þ ¼ Eðu2Þ ð17:88Þ

This is the required result. Nevertheless, a considerable effort, in the form of digital Fourier analysis, is

expended by vibration-testing laboratories to determine the degree of correlation in test signals employed

in two-DoF testing.

17.4.8.3 Symmetrical Rectilinear Testing

Single-DoF (rectilinear) testing that is performed with the test excitation applied along the line of

symmetry with respect to an orthogonal system of three principal axes of the test object mainframe is

termed symmetrical rectilinear testing. In product qualification literature, this test is often referred to as

the 458 test. The direction cosines of the input orientation are ð1=

ffiffi

3 p ; 1=

ffiffi

3 p ; 1

ffiffi

3 p Þ for this test

configuration. The single-actuator input intensity is amplified by a factor of

ffiffi

3 p in order to obtain the

required excitation intensity in the three principal directions. Note that symmetrical rectilinear testing

falls into the category of rectilinear triaxial testing, as defined earlier. This is one of the widely used testing

configurations in seismic qualification, for example.

17.4.8.4 Geometry vs. Dynamics

In vibration testing the emphasis is on the dynamic behavior rather than the geometry of the equipment.

For a simple three-dimensional body that has homogeneous and isotropic characteristics, it is not

difficult to correlate its geometry to its dynamics. A symmetrical rectilinear test makes sense for such

systems. The equipment we come across is often much more complex, however. Furthermore, our

interest is not merely in determining the dynamics of the mainframe of the equipment. We are more

interested in the dynamic reliability of various critical components located within the mainframe. Unless

we have some previous knowledge of the dynamic characteristics in various directions of the system

components, it is not possible to draw a direct correlation between the geometry and the dynamics of the

tested equipment.

17.4.8.5 Some Limitations

In a typical symmetrical rectilinear test, we deal with “black-box” equipment whose dynamics are

completely unknown. The excitation is applied along the line of symmetry of the principal axes of the

mainframe. A single test of this type does not guarantee excitation of all critical components located

inside the equipment. Figure 17.25 illustrates this further. Consider the plane perpendicular to the

direction of excitation. The dynamic effect caused by the excitation is minimal along any line on

v′

u′

FIGURE 17.24 Effect of coordinate transformation on

correlation.

Vibration Testing 17-49

this plane. (Any dynamic effect on this plane is

caused by dynamic coupling among different body

axes.) Accordingly, if there is a component

(or several components) inside the equipment

whose direction of sensitivity lies on this perpendicular

plane, the single excitation might not excite

that component. Since we deal with a black box,

we do not know the equipment dynamics beforehand.

Hence, there is no way of identifying the

existence of such unexcited components. When

the equipment is put into service, a vibration of

sufficient intensity may easily overstress this

component along its direction of sensitivity and

may bring about component failure. It is apparent

that at least three tests, performed in three

orthogonal directions, are necessary to guarantee

excitation of all components, regardless of their

direction of sensitivity.

A second example is given in Figure 17.26.

Consider a dual-arm component with one arm

sensitive in the O – O direction and the second arm

sensitive in the P – P direction. If component

failure occurs when the two arms are in contact, a

single excitation in either the O – O direction or

the P – P direction will not bring about component

failure. If the component is located inside a black

box, such that either the O – O direction or the

P – P direction is very close to the line of symmetry

of the principal axes of the mainframe, a single

symmetrical rectilinear test will not result in

system malfunction. This may be true, because

we do not have a knowledge of component

dynamics in such cases. Again, under service

conditions, a vibration of sufficient intensity can produce an excitation along the A – A direction,

subsequently causing system malfunction.

A further consideration in using rectilinear testing is dynamic coupling between the directions of

excitation. In the presence of dynamic coupling, the sum of individual responses of the test object

resulting from four symmetrical rectilinear tests is not equal to the response obtained when the

excitations are applied simultaneously in the four directions. Some conservatism should be introduced

when employing rectilinear testing for objects having a high level of dynamic coupling between the test

directions. If the test-object dynamics are restrained to only one direction under normal operating

conditions, however, then rectilinear testing can be used without applying any conservatism.

17.4.8.6 Testing Black Boxes

When the equipment dynamics are unknown, a single rectilinear test does not guarantee proper

testing of the equipment. To ensure excitation of every component within the test object that has

directional intensities, three tests should be carried out along three independent directions. The first

test may be carried out with a single horizontal excitation, for example. The second test could then

be performed with the equipment rotated through 908 about its vertical axis, and using the same

horizontal excitation. The last test would be performed with a vertical excitation.

O O

45°

FIGURE 17.26 Illustrative example of the limitation of

several rectilinear tests.

Perpendicular

Plane

Direction of

Excitation

FIGURE 17.25 Illustration of the limitation of a single

rectilinear test.

17-50 Vibration and Shock Handbook

Alternatively, if symmetrical rectilinear tests are

preferred, four such tests should be performed for

four equipment orientations (for example, an

original test, a 908 rotation, a 1808 rotation, and

a 2708 rotation about the vertical axis). These tests

also ensure excitation of all components that have

directional intensities. This procedure might not

be very efficient, however. The shortcoming of this

series of four tests is that some of the components

will be overtested. It is clear from Figure 17.27, for

example, that the vertical direction is excited by all

four tests. The method has the advantage, however,

of simplicity of performance.

17.4.8.7 Phasing of Excitations

The main purpose of rotating the test orientation

in rectilinear testing is to ensure that all components

within the equipment are excited. Phasing

of different excitations also plays an important

role, however, when several excitations are used

simultaneously. To explore this concept further, it

should be noted that a random input applied in

the A – B direction or in the B – A direction has the

same frequency and amplitude (spectral) characteristics.

This is clear because the PSD of u ¼ PSD

of ð2uÞ; and the autocorrelation of u ¼

autocorrelation of ð2uÞ: Hence, it is seen that, if

the test is performed along the A – B direction, it is

of no use to repeat the test in the B – A direction. It should be understood, however, that the situation is

different when several excitations are applied simultaneously.

The simultaneous action of u and v is not the same as the simultaneous action of 2u and v

(see Figure 17.28). The simultaneous action of u and v is the same, however, as the simultaneous action

of 2u and 2v: Obviously, this type of situation does not arise when there are no simultaneous

excitations, as in rectilinear testing.

17.4.8.8 Testing a Gray or White Box

When some information regarding the true dynamics of the test object is available, it is possible to reduce

the number of necessary tests. In particular, if the equipment dynamics are completely known, then a

single test would be adequate. The best direction for excitation of the system in Figure 17.26, for example,

is A – A: (Note that A – A may be lined up in any arbitrary direction inside the equipment housing. In

such a situation, knowledge of the equipment dynamics is crucial.) This also indicates that it is very

important to accumulate and use any past experience and data on the dynamic behavior of similar

equipment. Any test that does not use some previously known information regarding the equipment is a

blind test, and it cannot be optimal in any respect. As more information is available, better tests can be

conducted.

17.4.8.9 Overtesting in Multitest Sequences

It is well known that increasing the test duration increases aging of the test object because of prolonged

stressing and load cycling of various components. This is the case when a test is repeated one or more

times at the same intensity as that prescribed for a single test. The symmetrical rectilinear test requires

four separate tests at the same excitation intensity as that prescribed for a single test. As a result, the

3 4

FIGURE 17.27 Directions of excitation in a sequence

of four rectilinear tests.

A B

–u u

–v

FIGURE 17.28 Significance of excitation phasing in

two-DoF testing.

Vibration Testing 17-51

equipment becomes subjected to overtesting, at least in certain directions. The degree of overtesting is

small if the tests are performed in only three orthogonal directions. In any event, a certain amount of

dynamic coupling is present in the test-object’s structure and, to minimize overtesting in these sequential

tests, a smaller intensity than that prescribed for a single test should be employed. The value of the

intensity-reduction factor clearly depends on the characteristics of the test object, the degree of reliability

expected, and the intensity value itself. More research is necessary to develop expressions for intensityreduction

factors for various test objects.

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