Discrete Dynamics in Nature and Society

Volume 2018, Article ID 6935095, 10 pages

https://doi.org/10.1155/2018/6935095

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

^{1}Department of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China^{2}Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, Tianjin University, Tianjin 300072, China

Correspondence should be addressed to Zhiqiang Wu; nc.ude.ujt@uwqihz

Received 28 July 2018; Revised 26 October 2018; Accepted 5 November 2018; Published 2 December 2018

Academic Editor: Dorota Mozyrska

Copyright © 2018 Yajie Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. 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. Due to the limitation of the definition of integer-order derivative, it cannot express the memory property of viscoelastic substances. The definition of fractional derivative contains convolution, which can well express the memory effect and show the cumulative effect over time. Compared with the traditional integer-order calculus, fractional calculus has more advantages and is a suitable mathematical tool for describing the memory characteristics [1–11] and in recent years, it has become the powerful mathematical tool in many disciplines, especially in the study of viscoelastic materials.

The fractional derivative can accurately describe the constitutive relation of viscoelastic materials with fewer parameters, so the studies of fractional differential equations on the typical mechanical properties and the influences of fractional order parameters on the system are very necessary and have important significance. In recent years, 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]. Liu et al. introduced a transfer entropy and surrogate data algorithm to identify the nonlinearity level of the system by using a numerical solution of nonlinear response of beams, the Galerkin method was applied to discretize the dimensionless differential governing equation of the forced vibration, and then the fourth-order Runge-Kutta method was used to obtain the time history response of the lateral displacement [13]. Liu et al. investigated the stochastic stability of coupled viscoelastic system with nonviscously damping driven by white noise through moment Lyapunov exponents and Lyapunov exponents, obtained the coupled Itô stochastic differential equations of the norm of the response and angles process by using the coordinate transformation, and discussed the effects of various physical quantities of stochastic coupled system on the stochastic stability [14]. Nutting, Gemant and Scott-Blair et al. [15–17] first proposed the fractional derivative models to study the constitutive relation of viscoelastic materials and the research on the viscoelastic materials with fractional derivative is also increasing, and so far, it is still a research hotspot [18–25]. Rodr Guez et al. calculated the correlation function of transverse wave in linear and homogeneous viscoelastic liquid by the Generalized Langevin Equation (GLE) method and the influence of fractional correlation function on the dynamic behavior of the system is analyzed [26]. Bagley and Torvik used fractional calculus to study the dynamic behavior of viscoelastic damping structure and the responses of the system under general load as well as step load are analyzed respectively [27, 28]. Pakdemirli and Ulsoy studied the primary parametric resonance and combined resonance of the axial acceleration rope based on the discrete perturbation method and the multiscale method [29]. Zhang and Zhu analyzed the stability and dynamic response of viscoelastic belt under parametric excitation by the multiscale method [30, 31]. Chen et al. studied the dynamic behavior and steady-state response of axially accelerating viscoelastic beam by the Galerkin method [32–35], derived the differential equation of nonlinear vibration for axially moving viscoelastic rope, and then pointed out that the damping of viscoelastic rope only exists in the nonlinear term [36, 37]. Leung et al. studied the steady-state response of a simply 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, then the bifurcation behavior of the system caused by the order of the fractional derivative is analyzed [38]. Ghayesh and Moradian developed the Kelvin-Voigt viscoelastic model of the axially moving and the tensile belt, and then found the existence of nontrivial limit cycle in this system [39]. Liu et al. studied the dynamic response of an axially moving viscoelastic beam under random disorder periodic excitation, the first order expression of the solution is obtained by the multiscale method, and the stochastic jump phenomenon between the steady-state solutions is carried out [40]. Yang and Fang derived the system equation based on Newton’s second law and the fractional Kelvin constitutive relation and then studied the stability of the axially moving beam under the parametric resonance condition [41]. 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]. Galucio et al. obtained the fractional derivative model to describe the viscoelasticity of the system based on the Timoshenko theory and Euler-Bernoulli hypothesis and proposed a finite element formula for analyzing the sandwich beam of viscoelastic material with fractional derivative and the results were verified numerically [43].

Due to the complexity of fractional derivative, the analysis method of it becomes more difficult, the study on the vibration characteristics of the parameters can only be qualitatively analyzed, and the critical conditions of the parametric influences cannot be found, which affect the analysis and design of such systems, as well as the stochastic P-bifurcation of bistability for the viscoelastic beam with fractional derivatives of high order nonlinear terms under random noise excitation has not been reported. In view of the above situation, the nonlinear vibration of viscoelastic beam with fractional constitutive relation under Gaussian white noise excitation is taken as an example, the transition set curve of the fractional order system as well as the critical parametric condition for stochastic P-bifurcation of the system is obtained by the singularity theory, and then the types of stationary PDF curves of the system in each region in the parametric plane divided by the transition set are analyzed. 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 thus the correctness of the theoretical analysis in this paper is verified.

#### 2. Equation of Axially Moving Viscoelastic Beam

There are many definitions of fractional derivatives, and the Riemann-Liouville derivative and Caputo derivative are commonly used. The initial conditions corresponding to the Riemann-Liouville derivative have no physical meanings, however, the initial conditions of the systems described by the Caputo derivative have clear physical meanings and their forms are the same as the initial conditions for the differential equations of integer order. So in this paper, the Caputo-type fractional derivative is adopted as follows:where , , is the Euler Gamma function, and is the m order derivative of .

For a given physical system, due to the fact that the initial moment of the oscillator is , the following form of the Caputo derivative is often used:

In this paper, the transverse vibration of a viscoelastic simply supported beam under lateral excitation as shown in Figure 1 is considered; applying the d’Alambert principle, the governing equation can be written as [42]where is the mass density of the beam, is the area of cross-section, is the bending moment, is the lateral force, and is the horizontal force. From (3), we haveAssuming the material of the beam obeys a fractional derivative viscoelastic constitutive relation:where is the order of fractional derivative as is defined in (2), is the material modulus ratio, and is axial strain component.