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SHAKER TABLE SEISMIC TESTING OF EQUIPMENT USING HISTORICAL

STRONG MOTION DATA SCALED TO SATISFY A SHOCK RESPONSE SPECTRUM

Revision C

By Tom Irvine

Email: tom@vibrationdata.com

March 12, 2015

______________________________________________________________________________

The purpose of this paper is to apply the author’s aerospace shock testing experience to

earthquake engineering. The resulting method can be used either for shaker table testing or for

finite element modal transient analysis. The method can also be used for other base input shock

events such as pyrotechnic shock.

Introduction

Consider the following type of equipment:

1. Telecommunication

2. Medical life-support

3. Network servers

4. Nuclear power plant control consoles

Now consider that this equipment is to be installed in buildings in an active seismic zone. The

equipment must be designed and tested accordingly to withstand the dynamic loads.

A typical specification format for the loading is the shock response spectrum (SRS). The testing

is performed on a shaker table. A control computer applies a drive signal to the shaker table to

meet the SRS.

The SRS models the responses of individual single-degree-of-freedom systems to a common

base input. The natural frequency of each system is an independent variable. The damping value

is usually held constant. The SRS calculation retains the peak response of each system as a

function of natural frequency. The resulting SRS is plotted in terms of peak acceleration (G)

versus natural frequency (Hz).

A given time history has a unique SRS. On the other hand, a given SRS may be satisfied by a

variety of base inputs within prescribed tolerance bands.

The SRS format thus offers flexibility. A common testing approach is to use a burst of wavelets

such that the synthesized time history resembles a transient sine sweep with descending

frequency.

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The drawback is that the time history used to satisfy the SRS may be very dissimilar from the

measured source data. This brings us a number of concerns including linearity and multi-modal

response. Some of these concerns can be quantified in terms of the scalar temporal moments

which quantify the energy, RMS, skewness and kurtosis.

These concerns have led to a desire to synthesize a time history which “resembles” the real-

world event while still satisfying the SRS

The purpose of this paper is present a method which uses a measured reference time history as a

basis for synthesizing a time history to meet the SRS specification

Method Description

The following steps use trial-and-error-random number generation with some built-in

convergence. The method is implemented as a function in the Vibrationdata Matlab GUI

package.

The first step is to decompose the reference time history into a series of wavelets using the

method in Reference 1. An acceleration wavelet has zero net velocity and zero net displacement.

A series of wavelets likewise has these properties. The wavelet equation is given in Appendix A.

Wavelets are very amenable to shaker table shock testing. They are also convenient for analysis.

The examples in the paper use a series of 200 wavelets to model a reference time history.

The second step is to randomly vary the wavelet amplitudes so that the modified wavelet series

will have an SRS that matches the specification as closely as possible. The number of iterations

may be 16000 or so. The modified time history will thus have some distortion relative to the

reference, but this is needed to shape the time history so that its SRS meets the specification.

The second step yields an SRS that has some peaks and dips relative to the specification. This is

a consequence of trying to adapt a measured time history to a smoothed SRS.

The third step is to add wavelets so that the resulting SRS meets the specification within, say, + 3

dB tolerance limits.

The third step also adds some distortion. The amount of distortion depends largely on how much

the SRS specification differs from that of the reference data.

These steps are demonstrated in the following example, as well as in another example in

Appendix B.

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Example

Figure 1. El Centro Earthquake Time History

The measured 1940 El Centro North-South time history from Reference 2 is used as a basis. This time

history will be modeled with a series of wavelets.

The magnitude of this earthquake was 7.1. It was the first earthquake for which useful “strong motion

data” was measured in terms of engineering purposes. Note that the North-South component was

horizontal with respect to the ground.

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Figure 2. El Centro North-South Shock Response Spectrum

Note that the amplification factor Q=50 is equivalent to 1% damping. The fundamental mode of

a launch vehicle mounted on a pad at Vandenberg AFB might have 1% damping.

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Figure 3. SRS Specification Curves for Central California

The specification in Figure 3 is from Reference 3. It can be used for ground support equipment

and launch vehicles at Vandenberg AFB.

The 1% curve is used in the following example.

The El Centro SRS plateau in Figure 2 tends to be higher than that of the 1% specification in

Figure 3. Thus the El Centro time history amplitude will need to be scaled downward

accordingly.

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Figure 4. Time History Evolution

The top time history is the measured El Centro NS data. The middle time history is the wavelet

series model. The bottom time history has additional wavelets to improve the SRS match.

The Modified time history appears reasonably similar to the Reference, although no attempt is

made to quantify the “similarity.”

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Figure 5. Modified Acceleration Time History with Corresponding Velocity and Displacement

The velocity and displacement time histories are well-behaved which is important for both

testing and analysis.

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Figure 6. Shock Response Comparison

The SRS of the modified, or synthesized, time history is within + 3 dB of the nominal

specification. The method is thus successful in generating an El Centro-like time history to meet

the specification.

The SRS was calculated using the method in Reference 4.

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Future Work

Additional specification should be considered, including the Bellcore specifications.

Additional historical strong motion accelerograms should also be considered.

A standard library of scaled time histories could then be established with all the permutations.

The synthesis process may be more appropriate for the velocity rather than the time history given

that geophones are often used for source measurements. The resulting acceleration could then be

obtained by differentiation.

It may also be possible to meet all of the goals while still optimizing the pulse for minimum

displacement, which could be a concern for the shaker table displacement limits.

References

1. R.C. Ferebee, T. Irvine, et al., An Alternative Method of Specifying Shock Test Criteria,

NASA/TM-2008-215253, 2008.

2. Pacific Earthquake Engineering Research Center

http://peer.berkeley.edu/research/motions/

3. NASA HDBK-7005, Dynamic Environmental Criteria, 2001.

4. D. Smallwood, An Improved Recursive Formula for Calculating Shock Response

Spectra, Sandia National Laboratories, New Mexico.

5. C. Harris, Shock and Vibration Handbook, Fourth Edition; W.J. Hall, Chapter 24,

Vibrationdata of Structures Induced by Ground Motion, McGraw-Hill, New York, 1996.

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APPENDIX A

Wavelet Equation

The equation for an individual wavelet is

m

mdm

m

mdmdmdmmdm

m

mm

dm

m

f2

Nttfor,0

f2

Ntttfor,ttf2sintt

N

f2sinA

ttfor,0

)t(W

(A-1)

where

)t(Wm = acceleration of wavelet m at time t

mA = wavelet acceleration amplitude

mf = wavelet frequency

mN = number of half-sines

dmt = wavelet time delay

Note that mN must be an odd integer greater than or equal to 3. This is required so that the net

velocity and net displacement will each be zero.

The total acceleration at time t for a set of n wavelets is

)t(W)t(xn

1mm

(A-2)

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-50

-40

-30

-20

-10

10

20

30

40

50

0

0 0.05 0.10 0.15 0.200.012

9

8

7

6

5

4

3

2

1

TIME (SEC)

AC

CE

L (

G)

WAVELET 1 FREQ = 74.6 Hz NUMBER OF HALF-SINES = 9 DELAY = 0.012 SEC

Figure A-2.

A sample, individual wavelet is shown in Figure A-2. This wavelet was a component of a

previous analysis for an aerospace project.

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APPENDIX B

Another Example

0.001

0.01

0.1

1

10

0.1 1 10 100

NATURAL FREQUENCY (Hz)

PE

AK

AC

CE

L (

G)

SRS 2% DAMPING 84.1th PERCENTILE

Figure B-1. Specification

The specification SRS in Figure B-1 is taken from Reference 5. The initial slope is 26.5

dB/octave, which seems unrealistically steep with respect to measured data. Perhaps the reason

was to limit shaker table displacement.

Note that measured shock time histories typically have initial SRS slopes between 6 and 12

dB/octave, corresponding to constant velocity and constant slope respectively.

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Figure B-2. Time History Evolution

The El Centro NS data is again used as a basis. The modified time history departs somewhat

from the Reference in order to meet the SRS specification with its steep initial slope.

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Figure B-3. Modified Acceleration Time History with Corresponding Velocity and

Displacement

The velocity and displacement time histories are again well-behaved.

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Figure B-4. Shock Response Comparison

The SRS of the modified, or synthesized, time history is within + 3 dB of the nominal

specification. The method was thus successful in generating a time history to meet the

specification, loosely based on the El Centro earthquake.