Thorough understanding of dynamic system behavior is of major importance in many fields in science and engineering. When the spectral content of a process is changing with time, neither the time nor the frequency domain alone is sufficient to describe the process accurately. It is clearly recognized that a method to analyze excitation and response processes in both time and frequency domains is needed. This dissertation introduces a method to estimate the response of nonlinear systems to non-stationary excitations described by their wavelet coefficients. Time-frequency localization properties of wavelets are utilized to capture the evolutionary behavior of the spectral characteristics of the non-stationary processes and to describe the time dependent behavior of a nonlinear system. The wavelets-based approach developed in this thesis employs harmonic wavelets due to their concise form in the frequency domain, where each point in the frequency domain belongs to one particular scale only, thereby allowing one to use the terms "scale" and "frequency band" interchangeably. Utilizing this attractive property of the harmonic wavelets, an explicit relationship between the harmonic wavelet coefficients of a process and its time dependent spectral content is derived. To estimate the response of a nonlinear system, a wavelets-based equivalent linearization method is introduced, where the nonlinear system is replaced by its linear equivalent with time varying stiffness and damping coefficients. The applicability and the physical soundness of the procedure developed in this dissertation are demonstrated with examples from systems with elastic and hysteretic nonlinearity using data pertaining to simulated and recorded earthquake accelerograms. The applications presented in this dissertation show that the wavelet based linearization method is a quite promising tool for "real world" vibration problems
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