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. Dynamic airframe loads due to air turbulence, cinematic effects inside a car driving on rough roads, ship vibrations caused by sea waves etc. have quite an irregular and random nature. Therefore, a sufficient description of real oscillatory processes (vibrations) calls for the use of random functions. As a rule, vibration theory problems are formulated in terms of differential equations describing the system state: L X(t) = E(t) The operator L relates an external effect E(t) (system input) to the system state X(t). It is set by appropriate equations for which suitable initial and boundary conditions have been defined. E(t) is a stochastic process when oscillations are random. In engineering practice, the linear vibration theory (described 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 various technical applications. A great development of linear vibration theory, the generality of its laws and their physical interpretation motivate the researchers’ intention to transfer the studied phenomena to linear models. To do this, different linearization techniques were developed to allow notions of the linear theory such as the logarithmic decrement and natural oscillation frequency to be applied to non-linear oscillations with close-to-sine wave form. In technical applications, two types of research tasks exist: direct and inverse. The former consists of determining stochastic characteristics of the system output X(t) resulting from a random process E(t). The operator L (object model) is assumed to be 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 even at the design stage. Thus, using an appropriate operator L to describe such a design, solutions may be found that most closely meet specified requirements. The inverse task is aimed at evaluating the operator L based on known processes E(t) and X(t), i.e. finding factors of differential equations. This task is usually met at the R&D tests of prototypes to check whether some chosen designs comply with the requirements and when the parameters of operator L must be identified (or verified) experimentally. This information is often obtained from the observations of an operated object under specified conditions, sometimes, from natural "noise" generated by the tested object. 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: , where ω0 = 2π/Т0 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 decrementas well as the process varianceor its root-mean-square (r.m.s.) deviation. Evaluation of these parameters is subjected to experimental data processing based on frequency or time-domain representations of oscillations. It must be noted that data processing concepts 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. 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 and their spectral/covariance functions. To render the digital processing more accurate it is imperative to increase the sampling rate and realization size and as a result, to meet the growing requirements for computing power. This approach is based on a concept of transforming differential equations to difference equations 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. 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 that is based on 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. This transformation allows for the development of a formal processing procedure from experimental data to obtain the estimates of d and ω0. A straightforward mathematical description of the procedure is made possible by optimizing the discrete representation of oscillations and obtaining efficient estimates. The proposed approach allows researchers to replace traditional subjective techniques by a formal processing procedure providing efficient estimates with analytically defined statistical uncertainties. weiterlesen