Some Observations on the Random Response of Linear and Nonlinear Dynamical Systems

First examined is the problem of obtaining the nonstationary stochastic response of a nonlinear system subject to deterministically modulated stationary Gaussian random excitation. An extension of the generalized method of equivalent linearization is used to obtain an approximation to this response. The accuracy of this approximate technique is investigated by means of Monte Carlo simulation. Attention is then turned to the first passage problem for the stationary response of a lightly damped linear oscillator excited by white noise. A method is developed to generate approximate values for the limiting decay rate of the corresponding first passage probability density. This method is extended so that an approximate first passage probability distribution may be calculated when the oscillator response is nonstationary. The accuracy of this approximate distribution is examined. As a practical application, it is indicated how this technique may be used to determine an earthquake-like random process which generates a response spectrum consistent with given data. The accuracy and range of validity of the procedure are indicated by a simulation study. The approximate solution of the first passage problem is combined with the equivalent linearization technique to yield a procedure for computing approximate first passage probabilities of a weakly nonlinear oscillator. The errors introduced by this procedure are investigated.