Deterministic and random vibrations of systems with frequency-dependent parameters or fractional derivatives

Frequency and time domain analyses of system models containing frequency dependent parameters or fractional derivatives are considered. Both model types result in integro-differential equations in the time domain, and in algebraic equations in the frequency domain. The deterministic and random vibrations of these systems are considered. A deterministic solution technique is developed and its ability to accurately approximate a system response is shown. Techniques for the random vibration analysis are presented using the Monte Carlo simulation method. It is assumed that the power spectral density of the random excitations is known. An AR method is used to generate an ensemble of random excitations records compatible with the given spectral density. The deterministic solution technique is employed to obtain the corresponding ensemble of system responses. Numerical results are presented which show that the Monte Carlo Simulation method yields good estimates of statistical information for the system response.