Oscillator Response to Modulated Random Excitation

A closed-form expression for the response power spectral density function is derived for the case of exponentially modulated stationary excitation as an extension of the Heaviside case. A simply analytical expression is obtained for the mean-square response due to broad-band input when the excitation modulation can be expressed as the sum of exponential terms. Examples are presented to graphically compare the response for oscillator parameters typical of structures and two modulations approximating earthquake excitation. Numerical calculation of the first passage probabilities based on the two-state Markov approach permits a comparison of the stationary, Heaviside, and exponential excitation envelopes. For the examples presented, stationary or quasi-stationary approximations do not represent the reliability accurately.