Stochastic Jump and Bifurcation of a Slender Cantilever Beam Subject to Narrow-band Random Principal Parametric Resonances

摘要

The nonlinear integro-differential equations of motion for a slender cantilever beam subject to axial narrow-band random excitation are investigated. The method of multiple scales is used to determine a uniform first-order expansion of the solution of equations.According to solvability conditions, the nonlinear modulation equations for the principal parametric resonance are obtained. The FPK (Fokker-Planck-Kolmogrov) equation, which determines the stationary joint probability density of amplitude and phase, associated with its It(o) equation to obtain the statistics of the response is solved numerically by using finite difference method. The stochastic jump and bifurcation is investigated for the first and second mode parametric principal resonance. The stochastic jump phenomena indicate that, under the conditions of system parameters with a smaller bandwidth ?, the most probable motion is around the higher non-trivial stationary response, whereas with a higher bandwidth, the most probable motion is around the trivial stationary response. However, the stochastic jump is sometimes more sensitive to the change of ?, in other words, its small change may induce a series of stochastic jump.