Digital Processing of Random Oscillations
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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 der...
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 = 2pi/ 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 presentati
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 = 2pi/ 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 presentati
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