Multimodal nonlinear spectral response of a beam with impact under random input

A linearization procedure for estimating the spectral response of a randomly excited beam—stop system is proposed. The elastic stop is replaced by a spring with a stiffness depending on the amplitude of the deflection at the impact location. The probability density function of the amplitude is obtained using the stochastic averaging principle. Next, an estimate of the nonlinear response spectrum is derived providing the expectation of the spectral density function of the random spring linear system with respect to the probability density function of the amplitude response (assumed to be a random variable). The efficiency of the method is checked by comparing results with those of numerical simulations.