Stochastic P-bifurcation in a nonlinear viscoelastic beam model with fractional constitutive relation under colored noise excitation

Abstract

In this paper, we study the stochastic P-bifurcation problem for an axially moving bistable viscoelastic beam with fractional derivatives of high-order nonlinear terms under colored noise excitation. Firstly, using the principle for minimal mean square error, we show that the fractional derivative term is equivalent to a linear combination of the damping force and restoring force, so that the original system can be simplified to an equivalent system. Secondly, we obtain the stationary probability density function of the system amplitude by the stochastic averaging and the singularity theory, we find the critical parametric conditions for stochastic P-bifurcation of the system amplitude. Finally, we analyze different types of the stationary probability density function curves of the system qualitatively by choosing parameters corresponding to each region divided by the transition set curves. We verify the theoretical analysis and calculation of the transition set by showing the consistency of the numerical results obtained by Monte Carlo simulation with the analytical results. The method used in this paper directly guides the design of the fractional order viscoelastic material model to adjust the response of the system.

Stochastic bifurcation analysis of a bistable Duffing oscillator with fractional damping under multiplicative noise excitation

“…Liu et al investigated a Duffing oscillator system with fractional damping under combined harmonic and Poisson white noise parametric excitation, and then the asymptotic Lyapunov stability with probability one of the original system was analyzed based on the largest Lyapunov exponent [15]. Li et al studied the stochastic P-bifurcation problem for an axially moving bistable viscoelastic beam with fractional derivatives of high-order nonlinear terms under colored noise excitation and obtained the stationary probability density function of the system amplitude by the stochastic averaging method and the singularity theory [16]. Chen et al proposed a stochastic averaging technique which can be used to study the randomly excited strongly nonlinear system with delayed feedback fractional-order proportional-derivative controller, and obtained the stationary PDF of the system [17].…”