12.10 Control by a Shock Response Spectrum

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12.10.1 Principle

The exciters are actually always controlled by a signal that is a function of time. An acceleration – time

signal gives only one SRS. However, there is an infinity of acceleration – time signals with a given

spectrum. The general principle thus consists in searching out one of the signals, x€ðtÞ; having the specified

spectrum.

Historically, the simulation of shocks with spectrum control was first carried out using analog and

then digital methods (Smallwood and Witte, 1973; Smallwood, 1974).

Mechanical limitations of electrodynamic shakers for shocks:

* Maximum stroke of the coil-table unit: 25.4 to 75 mm peak-to-peak.

* Maximum acceleration, related to the maximum force: according to the author, # 4 times

the maximum force in sine mode, with the proviso of not exceeding 300g on the armature

assembly, more than eight times the maximum force in sine mode in certain cases (very

short shocks; 0.4 msec, for example).

* Maximum velocity: 1.5 to 3 m/sec in sine mode.

12-52 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

From the data of selected points on the shock spectrum to be simulated, the calculator of the control

system uses an acceleration signal with a very tight spectrum. For that, the calculation software proceeds

as follows (Lalanne, 2002b):

* At each frequency, f0, of the reference shock spectrum, the software generates an elementary

acceleration signal, for example, a decaying sinusoid. Such a signal has the property of having a

SRS presenting a peak of the frequency of the sinusoid whose amplitude is a function of the

damping of the sinusoid (Figure 12.58). With an identical shock spectrum, this property makes it

possible to realize shocks on the shaker that would be unrealizable with a control carried out by a

temporal signal of simple shape. For high frequencies, the spectrum of the sinusoid tends roughly

towards the amplitude of the signal.

* All the elementary signals are added by possibly introducing a given delay (and variable) between

each one of them in order to control to a certain extent the total duration of the shock, which is

primarily due to the lower frequency components (Figure 12.59).

* The total signal being thus made up, the software proceeds to processes correcting the amplitudes

of each elementary signal so that the spectrum of the total signal converges towards the reference

spectrum after some iterations.

Time

(a)

Acceleration

Shock response spectrum

(b) Frequency f0

FIGURE 12.58 Elementary shock (a) and its SRS (b). (Source: Lalanne, Chocs Mecaniques, Hermes Science

Publications. With permission.)

60

50

40

30

20

10 Shock response spectrum (

m/s2)

0

0 100 200 300

Frequency (Hz)

400 500

FIGURE 12.59 SRS of the components of the required shock. (Source: Lalanne, Chocs Mecaniques, Hermes Science

Publications. With permission.)

Mechanical Shock 12-53

© 2005 by Taylor & Francis Group, LLC

The operator must provide to the software with, at each frequency of the reference spectrum:

* The frequency of the spectrum

* Its amplitude

* A delay

* The damping of sinusoids or other parameters characterizing the number of oscillations of the

signal

When a satisfactory spectrum time signal has been obtained, it remains to be checked that the

maximum velocity and displacement during the shock are within the authorized limits of the test facility

(by integration of the acceleration signal). Lastly, after measurement of the transfer function of

the facility, one calculates the electric excitation which will make it possible to reproduce on the table

the acceleration pulse with the desired spectrum, as in the case of control from a signal according to the

time (Powers, 1974).

12.10.2 Principal Shapes of Elementary Signals

12.10.2.1 Decaying Sinusoid

The shocks measured in the field environment are very often responses of structures to an

excitation applied upstream, and are thus composed of the superposition of several modal

responses of a damped sine type (Smallwood and Witte, 1973; Crimi, 1978; Boissin et al., 1981;

Smallwood, 1985). Electrodynamic shakers are completely adapted to the reproduction of this

type of signal. According to this, one should be able to reconstitute a given SRS from such signals

of the form:

aðtÞ ¼ A e2hVt sin Vt t $ 0

aðtÞ ¼ 0 t , 0

9=

; ð12:43Þ

where V ¼ 2pf , f ¼ frequency of the sinusoid, and h ¼ damping factor.

Velocity and displacement are not zero at the

end of the shock with this type of signal. These

nonzero values are very awkward for a test on a

shaker. Compensation can be carried out in

several ways:

1. By truncating the total signal until it

is realizable on the shaker. This correction

can, however, lead to an important degradation

of the corresponding spectrum

(Smallwood and Witte, 1972).

2. By adding to the total signal (sum of all

the elementary signals) a highly damped

decaying sinusoid at low frequency,

shifted in time, defined to compensate

for the velocity and the displacement

(Smallwood and Nord, 1974; Smallwood

1975, 1985).

3. By adding to each component two

exponential compensation functions,

with a phase in the sinusoid (Nelson

and Prasthofer, 1974; Smallwood, 1975).

0

150

100

50

0

−50

−100

−150

−200

−250

−300

−350

20 40 60 80 100

Time (ms)

m/s2

FIGURE 12.60 Shock pulse generated from decaying

sinusoids, compensated by a decaying sinusoid.

(Source: Lalanne, Chocs Mecaniques, Hermes Science

Publications. With permission.)

12-54 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Example

The reference SRS is that of Figure 12.24 (Section

12.5.2.5). Examples of acceleration signals generated

from decaying sinusoids and having approximately

the same SRS are shown in Figure 12.60

and Figure 12.61 in the cases of a compensation by

a decaying sinusoid and by two exponential

functions.

12.10.2.2 ZERD Function

The use of a decaying sinusoid with its compensation

waveform modifies the response spectrum at

the low frequencies and, in certain cases, can harm

the quality of simulation. Fisher and Posehn (1977)

proposed using a waveform, which they named

“ZERD” (Zero Residual Displacement), defined by

the expression:

aðtÞ ¼ A e2hVt 1

V

sin Vt 2 t cosðVt þ fÞ

􀀒 􀀓

ð12:44Þ

where f ¼ arc tanð2h=1 2 h2Þ: This function

resembles a damped sinusoid and has the advantage

of leading to zero velocity and displacement at the

end of the shock (Figure 12.62).

Example

The reference SRS is that of Figure 12.24

(Section 12.5.2.5). Figure 12.63 shows an

example of acceleration signal generated from

ZERD functions having approximately the

same SRS.

12.10.2.3 WAVSIN Waveform

Yang (1970, 1972) and Smallwood (1974, 1975,

1985) proposed (initially for the simulation of the

earthquakes) a signal of the form:

aðtÞ ¼ am sin 2pbt sin 2pft 0 # 0 # t

aðtÞ ¼0 elsewhere

)

ð12:45Þ

where

f ¼ Nb ð12:46Þ

t ¼

1

2b ð12:47Þ

0

100

50

0

−50

−100

−150

−200

−250

−300

10 20 30 40 50 60 70 80

Time (ms)

m/s2

FIGURE 12.61 Acceleration signal generated from

decaying sinusoids, compensated by two exponentia

functions. (Source: Lalanne, Chocs Mecaniques, Hermes

Science Publications. With permission.)

1.5

1.0

0.5

0.0

−0.5

−1.0

−1.5

0 4 8 12 16 20

Time (s)

Acceleration (m/s2)

f = 1 Hz

h = 0.05

A = 1

FIGURE 12.62 ZERD waveform of D.K. Fisher and

M.R. Posehn (example).

Mechanical Shock 12-55

© 2005 by Taylor & Francis Group, LLC

where N is an integer (which must be odd

and higher than on 1). The first term of aðtÞ is

a window of half-sine form of half-period t: The

second describes N half-cycles of a sinusoid of

greater frequency ðf Þ; modulated by the preceding

window (Figure 12.64).

This function leads also to zero velocity and

displacement at the end of the shock.

Example

Figure 12.65 shows an example of acceleration

signal generated from WAVSIN functions having

approximately the same SRS as the reference SRS

of Figure 12.24 (Section 12.5.2.5).

12.10.3 Comparison of WAVSIN,

SHOC Waveforms, and Decaying

Sinusoid

The cases treated by Smallwood (1974) seem to

show that these three methods give similar results.

It is noted, however, in practice, that, according to

the shape of the reference spectrum, one or other

of these waveforms allows a better convergence.

The ZERD waveform very often gives good

results.

12.10.4 Criticism of Control by a

Shock Response Spectrum

Whatever the method adopted, simulation on a

test facility of shocks measured in the real world

requires the calculation of their SRSs and the

search for an equivalent shock.

If the specification must be presented in the

form of a time-dependent shock pulse, the test

requester must define the characteristics of shape,

duration, and amplitude of the signal, with the

already quoted difficulties.

If the specification is given in the form of an SRS,

the operator inputs in the control system the given

spectrum, but the shaker is always controlled by a

signal according to the time calculated and

according to procedures described in the preceding

sections. It is known that the transformation shock

spectrum signal has an infinite number of solutions,

and that very different signals can have

identical SRSs. This phenomenon is related to the

loss of most of the information initially contained

in the signal, x€ðtÞ; during the calculation of the

spectrum (Metzgar, 1967).

200

150

100

50

0

−50

−100

−150

0 10 20 30 40

Time (ms)

m/s2

50 60 70 80 90

FIGURE 12.63 Acceleration signal generated from

ZERD functions.

0.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

0.5

Acceleration (m/s2)

1.0

Time t (s)

WAVSIN (N = 5 f = 1 t = 2.5)

1.5 2.0 2.5

FIGURE 12.64 Example of WAVSIN waveform.

(Source: Lalanne, Chocs Mecaniques, Hermes Science

Publications. With permission.)

0

−150

−100

−50

0

m/s2

50

100

150

20 40 60 80 100

Time (ms)

120 140 160 180

FIGURE 12.65 Acceleration signal generated from

WAVSIN functions.

12-56 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

The oscillatory shock pulses have a spectrum that presents an important peak to the frequency of the

signal. This peak can, according to choice of parameters, exceed by a factor of five the amplitude of the

same spectrum at the high frequencies; that is, five times the amplitude of the signal itself. Being given a

point of the specified spectrum of amplitude, S; it is thus enough to have a signal vs. time of amplitude

S=5 to reproduce the point. For a simple shaped shock, this factor does not exceed two in the most

extreme case. All these remarks show that the determination of a signal of the same spectrum can lead to

very diverse solutions, the validity of which one can question.

If any particular precaution is not taken, the signals created by these methods have, in a general

way, one duration much larger and an amplitude much smaller than the shocks that were used to

calculate the reference SRS (a factor of about ten in both cases). Figure 12.66 and Figure 12.67 give

an example.

In the face of such differences between the excitations, one can legitimately wonder whether the SRS is

a sufficient condition to guarantee a representative test. It is necessary to remember that this equivalence

is based on the behavior of a linear system that one chooses the Q factor a priori. One must be aware of

the following.

0

−60

−40

−20

0

m/s2

20

40 Shock A

60

0.5 1 1.5 2 2.5

Seconds

Shock B

3 3.5 4 4.5 5

× 10E-2

FIGURE 12.66 Example of shocks having spectra near the SRS.

0

0

10

20

30

40

m/s2

50

60

70

80

90

0.2 0.4 0.6 0.8 1

Hz

1.2 1.4 1.6 1.8 2

Shock B

Q = 10

× 10E3

Shock A

FIGURE 12.67 SRS of the shocks shown in Figure 12.66.

Mechanical Shock 12-57

© 2005 by Taylor & Francis Group, LLC

* The behavior of the structure is in practice far from linear and that the equivalence of the

spectrum does not lead to stresses of the same amplitude. Another effect of these nonlinearities

appears sometimes by the inaptitude of the system to correct the drive waveform to take account

of the transfer function of the installation.

* Even if the amplitudes of the peaks of acceleration and the maximum stresses of the

resonant parts of the tested structure are identical, the damage by the fatigue generated by

accumulation of the stress cycles is rather different when the number of shocks to be applied is

significant.

* The tests carried out by various laboratories do not have the same severity.

These questions did not receive a really satisfactory response. By prudence rather than by rigorous

reasoning, many agree, however, on the need for placing parapets, while trying to supplement the

specification defined by a spectrum with complementary data (DV ; duration of the shock, require SRS at

two different values of damping, etc.; Favour, 1974; Smallwood, 1974, 1975, 1985).

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