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 February 23, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.