Stochastic bifurcation in non-linear structural systems near 1 : 1 internal resonance

Abstract

This paper examines the problem of stochastic bifurcation and response statistics of a clamped-clamped beam in the neighborhood of 1 : 1 internal resonance. The beam is initially exposed to an axial static load and subjected to a lateral wide band random excitation. The static load exceeds the Euler buckling load and thus results in the internal resonance condition 1 : 1 between the first two modes. The Fokker-Planck equation is used to generate a general first-order differential equation in the joint moments of response coordinates. Gaussian and non-Gaussian closure schemes are used to close the infinite coupled moment equations. The closed equations are then solved for response statistics and bifurcation boundaries in terms of system and excitation parameters. The theoretical results are compared with those determined by Monte Carlo simulation. The numerical simulation is also used to estimate the marginal probability density of response coordinates. It is found that the numerical simulation results are in good agreement with those obtained by the Gaussian closure, particularly in predicting stochastic bifurcation of the second mode. Under relatively higher excitation levels the Monte Carlo simulation exhibits the occurrence of snap-through phenomenon in the first mode. Both Gaussian and non-Gaussian closures fail to predict the snap-through.