Stochastic P-Bifurcation of a Bistable Viscoelastic Beam with Fractional Constitutive Relation under Gaussian White Noise

Yajie Li,Guoqi Zhang,Zhiqiang Wu,Feng Wang

doi:10.1155/2018/6935095

Abstract

In this paper, we study the stochastic P-bifurcation problem for axially moving of a bistable viscoelastic beam with fractional derivatives of high order nonlinear terms under Gaussian white noise excitation. First, using the principle for minimum 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. Second, we obtain the stationary Probability Density Function (PDF) of the system’s amplitude by stochastic averaging, and using singularity theory, we find the critical parametric condition for stochastic P-bifurcation of amplitude of the system. Finally, we analyze the types of the stationary PDF curves of the system qualitatively by choosing parameters corresponding to each region within the transition set curve. 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.

Highlights

Fractional calculus is a generalization of integer-order calculus, it extends the order of calculus operation from the traditional integer order to the case of noninteger order, and it has a history of more than 300 years as so far
Many scholars have done a lot of work and achieved fruitful results in this field: Li and Tang studied the nonlinear parametric vibration of an axially moving string made by rubber-like materials, a new nonlinear fractional mathematical model governing transverse motion of the string is derived based on Newton’s second law, the Euler beam theory, and the Lagrangian strain, and the principal parametric resonance is analytically investigated via applying the direct multiscale method [12]
Leung et al studied the steady-state response of a supported viscoelastic column under the axial harmonic excitation based on the fractional derivative constitutive model of cubic nonlinear and derived the generalized Mathieu-Duffing equation with time delay by the Galerkin discrete method, the bifurcation behavior of the system caused by the order of the fractional derivative is analyzed [38]

Summary

Introduction

Fractional calculus is a generalization of integer-order calculus, it extends the order of calculus operation from the traditional integer order to the case of noninteger order, and it has a history of more than 300 years as so far. Leung et al studied the steady-state response of a supported viscoelastic column under the axial harmonic excitation based on the fractional derivative constitutive model of cubic nonlinear and derived the generalized Mathieu-Duffing equation with time delay by the Galerkin discrete method, the bifurcation behavior of the system caused by the order of the fractional derivative is analyzed [38]. Leung et al studied the single mode dynamic characteristics of the nonlinear arch with the fractional derivative, the steady-state solution of the system is obtained based on the residual harmonic homotopy method, and the influence of the parametric variation on the dynamic behaviors of the viscoelastic damping material is analyzed [42]. By the method of Monte Carlo simulation, the numerical results are compared with the analytical results obtained in this paper, it can be seen that the numerical solutions are in good agreements with the analytical solutions, and the correctness of the theoretical analysis in this paper is verified

Equation of Axially Moving Viscoelastic Beam

The Stationary PDF of Amplitude

Stochastic P-Bifurcation

Bistable region

Conclusion