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 = 2p ¿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 d = 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 develo
