Stochastic bifurcation of one flexible beam subject to axial Gauss white noise excitation

One stochastic nonlinear dynamical model has been proposed to describe the vibration of flexible beam under axial excitation considering the influence of the environment random factors. Firstly, the model has been simplified applying the stochastic average theory of quasi-integral Hamilton system .Secondly, we utilize the methods of Lyapunov exponent and boundary classification associated with diffusion process respectively to analyze the stochastic stability of the trivial solution of system. Thirdly, we explore the stochastic Hopf bifurcation of the vibration model according to the qualitative changes in stationary probability density of system response. It is concluded that the stochastic Hopf bifurcation occurs at two critical parametric values. Finally, some explanations are given in a simply way on the potential applications of stochastic stability and bifurcation analysis.