# Multiple Shaker Random Vibration Control--An Update Page: 4 of 12

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SAND 98-2044C

also fall into this category, as does most seismic

testing. This testing will be discussed first.

In a sense sine testing is a special case of waveform

control discussed below. Sometimes the same

control strategies are used, but in some cases a

significantly different strategy is used. Sine testing

will be discussed very briefly in this paper.

Random testing uses nondeterministic waveforms

that are generated as the test progresses and are

described in probabilistic terms. This form of

testing will be discussed last.

NOTATIONAL NOTES

Before I start the discussion on control some

notational items will be discussed.

Functions of time are defined in blocks at a discrete

set of points using the inverse fast Fourier

transform, FFTf', of a corresponding frequency

domain description. Auto and cross-spectral

densities are typically estimated using the Welch

algorithm. This algorithm divides the time history

into blocks, sometimes overlapping, multiplies the

blocks by a window, and computes the FFT.

Multiple blocks are averaged to arrive at estimates

of the spectra.

Lowercase letters will be used to denote samples in

time. Each element in a vector is a time history.

The vector is in reality a matrix, time in one

dimension and spatial location in the other

dimension. But for convenience the time dimension

is not included. The corresponding frequency

domain description is denoted with the

corresponding upper case letter. Similarly for

convenience the frequency dimension is excluded.

This avoids the requirement for 3-dimensional

tensors later in the development. Matrices are

denoted in bold.

An important change in the notation for the cross-

spectrum is used. Bendat and Piersol (1986) define

the cross-spectrum between x and y as<D (f)= lim EX*(f)Y(f),-T

(1)

where E[ ] is the expected value, T is the record

length, the superscript * is the complex conjugate.

This was the notation used in my previous papers.With matrices it is much more convenient to define

the cross-spectrum asDxy(f)= li -E[X(f)Y*(f).

** oT(2)

Using this notation the cross-spectral density matrix

of a column vector becomes(3)

we Xx(f)= lim E X(f )X'(f

where X' is the conjugate transpose of X.The only difference is that

f)= A y f.(4)

I will use the notation of Equations (2) and (3) in

this paper.

WAVEFORM CONTROL

Multiple shaker waveform control was the first test

procedure to be implemented successfully (Fisher,

1973, Fisher and Posehn, 1977, for example). The

basic concept of waveform control is shown in

Figure 1. The vector desired waveforms, {x(t) 1, is

defined. The waveforms usually transformed into

the frequency domain, {X(w)}. If the waveform is

too long to be conveniently described in the

frequency domain, the waveforms are broken into

blocks and overlap and add methods (Gold and

Rader, 1969) are used to reproduce the waveforms.

Next the system must be identified. A matrix of

frequency response functions, H, is identified. The

elements of H can be identified by exciting the

system one input at a time (other inputs zero) and

measuring the response. The system can also be

excited with a vector of independent or partially

correlated inputs to identify the system frequency

response matrix.

H = (,( (5)

where

rr. = the cross-spectral density matrix of the

return signals with the test item attached to

the vibration system.2

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Smallwood, D.O. Multiple Shaker Random Vibration Control--An Update, article, February 18, 1999; Albuquerque, New Mexico. (digital.library.unt.edu/ark:/67531/metadc676130/m1/4/: accessed July 18, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.