Wavelet-based excitation representation and response determination of linear and nonlinear systems
Spanos, Pol D.
Doctor of Philosophy thesis
The dissertation determines the evolutionary spectrum of non-stationary stochastic processes and describes the time-varying characteristics of the response of linear and nonlinear dynamic systems subject to non-stationary inputs by using the wavelet transform. Models of non-stationary stochastic processes most realistically describe inputs to dynamic systems such as earthquake, wind, and sea-wave excitations. A time-frequency analysis is necessary to describe the evolutionary characteristics of non-stationary processes, which cannot be captured by an analysis in the frequency domain. The wavelet transform offers a unique time-frequency representation that captures the important characteristics of stochastic processes. This work presents a theoretical basis that relates the wavelet representation with the spectral representation of stochastic processes. The Harmonic, the Morlet and the Daubechies wavelet families are employed to assess the applicability of the proposed technique for spectrum estimation of stochastic processes. The Harmonic wavelet scheme yields most successful approximations to the evolutionary spectrum due to the appealing spectral properties of its basis functions. A time-frequency system representation is derived for linear systems by converting the frequency response function to a wavelet transfer function based on the Harmonic scheme. Input/output relationships are derived in the wavelet domain based on linear system theory, the spectral representation of stochastic processes, and the Harmonic wavelet scheme. These relationships successfully approximate the response of a linear system. Next, the problem of determining the nonlinear system response to stationary and non-stationary excitation is considered by performing a wavelet-based stochastic linearization. The method yields an equivalent linear system, which approximates the nonlinear system at every wavelet scale. Exploratory studies related to the analysis of classical nonlinear oscillators are presented. The method captures the nonlinear features of the output response as well as its time and frequency dependent characteristics. The results presented in this study have shown that the wavelet transform is an appealing tool for dynamic analysis of systems due to its accuracy, mathematical simplicity, ease of implementation, and computational efficiency.
Mathematics; Electronics; Electrical engineering; Mechanical engineering