Random Vibrations of a Nonlinear Elastic Beam

Abstract

The theory of the Markoff process and the associated Fokker Planck equation is used to investigate the large vibrations of a hinged, axially restrained, elastic beam driven by white noise. An expression for the joint probability-density function of the first N modal amplitudes is obtained and used to derive an approximate expression for the mean-squared displacement. Numerical integration of the exact expression for the mean-squared displacement indicates a reduction of this quantity as compared in the linear theory. Calculations also show a range of applicability of the derived approximate expression. Furthermore. computations show that the first node still gives a good estimate of the mean-squared displacement but the coupling effects of the nodes are so strong, for sufficiently large deflections. that the effect of the higher modes must be taken into account when computing the mean-squared value of the first mode.