From shock response spectrum to temporal moments and vice-versa Page: 4 of 7
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Proceedings of the 21" SEM International Modal Analysis Conference (IMAC-XXI).
Kissimmee, Florida, February 3-6, 2003.
a 1ji1] *1aImii
4 [ --:- i
= o - o
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
4iur 7. Procese acelrti a s esr 5.
0 0.5 1 1.5 2 2.5 5 0.5 1 1.5 2 2.5
Time (mns) Tiee (mns)
Figure 7. Processed accelerations at sensor 5.
The response of the system is analyzed from
0.115 to 2.415 milliseconds (ms) to avoid including in
the time signals an early event due to the electrical
impulse of the explosive's detonation. The period
analyzed therefore consists of 2.3 ms of data. Signals
are decimated by a factor 5 to obtain an effective
sampling rate of 200 kHertz. An 8'h-order Butterworth
low-pass filter is applied with cut-off frequency at 50
kHertz. Finally, the signals are normalized by the total
impulse of each test (see table 2) and the mean of each
processed signal is removed.
- - -~ -- ---
0 0.5 1 1.5 2 2.5 50 0.5 1 1.5 2 2.5
00 0.5 1 1.5 2 2.5 00 05 1 1.5 2 25
Time (ms) Time (ms)
Figure 8. Processed accelerations at sensor 6.
The processed signals are pictured in Figure 7
(sensor 5 on the lower mass) and Figure 8 (sensor 6 on
the upper mass) for each one of the four test units.
5. FEATURES FOR TRANSIENT EVENTS
When it comes to the analysis of transient events,
modal analysis is generally not appropriate. This is
because the dynamics of interest is high frequency,
while modal superposition is most appropriate to
identify low-frequency dynamics. Also, non-linearity
Approved for unlimited, public release on October ??, 2002.
might manifest itself in the response (energy coupling,
bifurcation and chaos are well-known examples),
making it questionable to calculate Fourier transforms
or power spectral density estimates. Therefore,
features such as resonant frequencies and modal
damping ratios cannot be used to characterize the
response of the threaded joint.
5.1 Forward Problem: From Signals to Features
The state-of-the-practice for analyzing waveforms
of transient events relies on the following features:
* Peak values. Peaks include absolute peak values
and ranges, that is, the difference between
maximum and minimum values.
* 10% duration of the event. This feature is defined
as the time between the instant of peak response
(for example, shock arrival) and the instant that
the waveform has decayed to 10% of its peak .
* Exponential decrement. This feature is defined as
the scalar exponent d of an exponential decay
z(t)=Aed' best-fitted to the response y(t).
" Statistical moments. A probability density function
and statistics can be estimated from the signal, as
it is assumed to represent realizations of a
* Principal Component Decomposition (PCD). The
PCD is also referred to as the Karhunen-Loeve
decomposition or principal orthogonal modes. It
generalizes the notion of modal superposition to
non-linear systems and has been applied to test-
analysis correlation and model updating [6, 7].
* Fractal dimensions. Fractal analysis characterizes
the "growth" of the signal in time as a function of
various wavelengths or time scales. Numerous
fractal dimensions are defined, that include the
Holder exponent, Lyapunov exponent and the
Higuchi dimension .
* Shock response spectrum (SRS). The SRS
simulates the response "seen" by a single degree-
of-freedom system that would be subjected to the
transient waveform (section 6).
* Temporal moments. Temporal moments are
scalar quantities that condense the information of
the SRS. Computing temporal moments is
analogous to the calculation of the statistical
moments of a random variable (section 7).
For example, features advocated by NASA for the
analysis of pyro-shock events (such as lift-off or stage
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Hemez, F. M. (François M.) & Doebling, S. W. (Scott W.). From shock response spectrum to temporal moments and vice-versa, article, January 1, 2002; United States. (https://digital.library.unt.edu/ark:/67531/metadc930956/m1/4/: accessed April 22, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.