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Digital Processing of Random Oscillations (Hardcover)
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Digital Processing of Random Oscillations (Hardcover)
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This book deals with the autoregressive method for digital
processing of random oscillations. The method is based on a
one-to-one transformation of the numeric factors of the Yule series
model to linear elastic system characteristics. This parametric
approach allowed to develop a formal processing procedure from the
experimental data to obtain estimates of logarithmic decrement and
natural frequency of random oscillations. A straightforward
mathematical description of the procedure makes it possible to
optimize a discretization of oscillation realizations providing
efficient estimates. The derived analytical expressions for
confidence intervals of estimates enable a priori evaluation of
their accuracy. Experimental validation of the method is also
provided. Statistical applications for the analysis of mechanical
systems arise from the fact that the loads experienced by
machineries and various structures often cannot be described by
deterministic vibration theory. Therefore, a sufficient description
of real oscillatory processes (vibrations) calls for the use of
random functions. In engineering practice, the linear vibration
theory (modeling phenomena by common linear differential equations)
is generally used. This theory's fundamental concepts such as
natural frequency, oscillation decrement, resonance, etc. are
credited for its wide use in different technical tasks. In
technical applications two types of research tasks exist: direct
and inverse. The former allows to determine stochastic
characteristics of the system output X(t) resulting from a random
process E(t) when the object model is considered known. The direct
task enables to evaluate the effect of an operational environment
on the designed object and to predict its operation under various
loads. The inverse task is aimed at evaluating the object model on
known processes E(t) and X(t), i.e. finding model (equations)
factors. This task is usually met at the tests of prototypes to
identify (or verify) its model experimentally. To characterize
random processes a notion of "shaping dynamic system" is commonly
used. This concept allows to consider the observing process as the
output of a hypothetical system with the input being stationary
Gauss-distributed ("white") noise. Therefore, the process may be
exhaustively described in terms of parameters of that system. In
the case of random oscillations, the "shaping system" is an elastic
system described by the common differential equation of the second
order: X (t)+2hX (t)+ _0^2 X(t)=E(t), where 0 = 2 / 0 is the
natural frequency, T0 is the oscillation period, and h is a damping
factor. As a result, the process X(t) can be characterized in terms
of the system parameters - natural frequency and logarithmic
oscillations decrement = hT0 as well as the process variance.
Evaluation of these parameters is subjected to experimental data
processing based on frequency or time-domain representations of
oscillations. It must be noted that a concept of these parameters
evaluation did not change much during the last century. For
instance, in case of the spectral density utilization, evaluation
of the decrement values is linked with bandwidth measurements at
the points of half-power of the observed oscillations. For a
time-domain presentation, evaluation of the decrement requires
measuring covariance values delayed by a time interval divisible by
T0. Both estimation procedures are derived from a continuous
description of research phenomena, so the accuracy of estimates is
linked directly to the adequacy of discrete representation of
random oscillations. This approach is similar a concept of
transforming differential equations to difference ones with
derivative approximation by corresponding finite differences. The
resulting discrete model, being an approximation, features a
methodical error which can be decreased but never eliminated. To
render such a presentation more accurate it is imperative to
decrease the discretization interval and to increase realization
size growing requirements for computing power. The spectral density
and covariance function estimates comprise a non-parametric
(non-formal) approach. In principle, any non-formal approach is a
kind of art i.e. the results depend on the performer's skills. Due
to interference of subjective factors in spectral or covariance
estimates of random signals, accuracy of results cannot be properly
determined or justified. To avoid the abovementioned difficulties,
the application of linear time-series models with well-developed
procedures for parameter estimates is more advantageous. A method
for the analysis of random oscillations using a parametric model
corresponding discretely (no approximation error) with a linear
elastic system is developed and presented in this book. As a
result, a one-to-one transformation of the model's numerical
factors to logarithmic decrement and natural frequency of random
oscillations is established. It allowed to develop a formal
processing procedure from experimental data to obtain the estimates
of and 0. The proposed approach allows researchers to replace
traditional subjective techniques by a formal processing procedure
providing efficient estimates with analytically defined statistical
uncertainties.
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