Mathematical Problems in Engineering

Volume 2018, Article ID 6532305, 16 pages

https://doi.org/10.1155/2018/6532305

## Stochastic Stability Analysis of Coupled Viscoelastic Systems with Nonviscously Damping Driven by White Noise

^{1}School of Mathematics, Shanxi University, Taiyuan 030006, China^{2}Applied Science College, Taiyuan University of Science and Technology, Taiyuan 030024, China

Correspondence should be addressed to Di Liu; moc.liamtoh@ual-id

Received 27 May 2018; Revised 1 August 2018; Accepted 15 August 2018; Published 16 September 2018

Academic Editor: Xue-Jun Xie

Copyright © 2018 Di Liu 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

Nonviscously damped structural system has been raised in many engineering fields, in which the damping forces depend on the past time history of velocities via convolution integrals over some kernel functions. This paper investigates stochastic stability of coupled viscoelastic system with nonviscously damping driven by white noise through moment Lyapunov exponents and Lyapunov exponents. Using the coordinate transformation, the coupled Itô stochastic differential equations of the norm of the response and angles process are obtained. Then the problem of the moment Lyapunov exponent is transformed to the eigenvalue problem, and then the second-perturbation method is used to derive the moment Lyapunov exponent of coupled stochastic system. Lyapunov exponent also can be obtained according to the relationship between moment Lyapunov exponent and Lyapunov exponent. Finally, the effects of various physical quantities of stochastic coupled system on the stochastic stability are discussed in detail. These results are validated by the direct Monte Carlo simulation technique.

#### 1. Introduction

Viscoelastic materials are widely used in aerospace, construction, textile, and other industries, because they have a series of excellent properties, including light weight, high strength, wide source, and good shock absorption. These materials’ stress depends on the past time history of stain; that is, the stress will increase correspondingly with the increase of time and the bucking will be more likely to occur. Therefore, the research of dynamical behavior of viscoelastic system has received a lot of interests in recent years [1–4].

To better understand the dynamical behavior of the viscoelastic system, many methods had been put forward. McTavish et al. [5] presented GHM method to analyze the linear multiple degree of freedom viscoelastic damping systems. Li et al. [6] studied the decay rate of energy functional of nonlinear viscoelastic full Marguerre-von Kármán shallow shell system. Later, Li [7] also showed the existence and uniqueness of weak solution of this system by applying the Galerkin finite element method. Nieto and Ahmad [8] used the generalized quasi-linearization method to obtain the explicit approximate solutions of an initial and terminal value problem for the forced Duffing equation with nonviscous damping. Recently, Li and Du [9] studied general energy decay of a degenerate viscoelastic Petrovsky-type plate equation with boundary feedback through a priori estimate and analysis of Lyapunov-like functional.

The dynamic stability as an important field of nonlinear dynamics has attached increasing attention in recent years. Guo et al. [10] derived a new simple sufficient condition of the global asymptotic stability of the integrodifferential systems with delay through constructing suitable Lyapunov functionals combined with the analytical technique. And then Meng et al. [11, 12] obtained a sufficient condition of uniform stability for nonlinear integrodifferential system. However, when we consider the dynamical behavior of the system in a real environment, the stochastic excitation cannot be ruled out, such as wind gusts, earthquakes, and ocean waves [13–15]. Stochastic stability of the system under the stochastic excitation has attracted more and more attention [16–20]. The Lyapunov exponent specially has been researched by many scholars as an important indication for judging the stability of stochastic system. For example, Potapov [21] derived the sufficient condition of the almost-sure asymptotic stability of elastic and viscoelastic systems by the Lyapunov’s direct method. He [22] also studied the stability of elastic and viscoelastic systems under the non-Gaussian excitation. For stochastic stability of high dimensional coupled system, Pavlović et al. [23] proposed the direct Lyapunov method to investigate the almost-sure stochastic stability of a viscoelastic double-beam system under parametric excitation. Then they used the same method to analyze the instability of coupled nanobeam systems subject to compressive axial loading [24]. These works only indicate that the system is almost-surely stable.

According to the theory of large deviation, which was first proposed by Baxendale and Stroock [25], the almost-sure stability of stochastic system does not mean the moment stability. Therefore, it is necessary to study the moment Lyapunov exponent of a stochastic system. The moment Lyapunov exponent can judge not only almost-sure stability but also the moment stability [26]. Nevertheless, analytic solutions of the moment Lyapunov exponent are very difficult to derive. To overcome this, many scholars developed different approximation methods to study the moment Lyapunov exponents of the stochastic system. For example, Sri Namachchivaya and Van Roessel [27] studied the moment Lyapunov exponents of two-degrees-of-freedom coupled elastic oscillators under real noise excitation by combining an asymptotic approximation for the moment Lyapunov exponents. Kozić and his associates [28–30] developed the first-order perturbation approach to obtain weak noise expansion of moment Lyapunov exponents and Lyapunov exponents for a stochastically coupled double-beam system and Timoshenko beam system, respectively. Subsequently, Stojanović and Petković [31] used a perturbation method to study the moment Lyapunov exponents and the Lyapunov exponents of the three elastically connected Euler beams. More recently, Deng [32] applied the averaging method to establish the moment Lyapunov exponent of coupled gyroscopic stochastic system under bounded noise excitation. But it is rare to see studies of the moment Lyapunov exponent of a viscoelastic coupled system with nonviscous damping. Deng [33] investigated stochastic stability of two-degrees-of-freedom coupled viscoelastic system under white noise through moment Lyapunov exponent, but its damping term is viscous.

The purpose of this paper is to study stochastic stability of a viscoelastic coupled system with nonviscous damping subject to Gaussian white noise excitation through moment Lyapunov exponents and Lyapunov exponents, in which the nonviscous damping term is expressed by Boltzmanns superposition integral with a hereditary relaxation kernel. The paper is organized as follows. Section 2 derives the governing equations of motion of a stochastic viscoelastic system with a nonviscous damping. And then solving the problem of the moment Lyapunov exponent is changed into the eigenvalue and eigenfunction problem through a suitable transformation. In Section 3, the zeroth-order and the first-order and second-order perturbation solutions of the moment Lyapunov exponents are derived, respectively, based on the second-order perturbation method. The effects of different physical quantities on the stochastic stability of the coupled system are discussed under the given parameters. Correspondingly, the results are verified by means of the direct Monte Carlo simulation in Section 4. Finally, conclusions will be drawn in Section 5.

#### 2. Problem Formulation

Considering a coupled viscoelastic system with nonviscous damped structure driven by white noise excitation, the governing equations can be expressed aswhere and are generalized displacements. and are natural frequencies. is a small parameter. and are nonviscous damping coefficients. and are constants. is a Gaussian white noise with zero mean and noise intensity . is a relaxation function of nonviscous damping, which can be expressed by in which is the relaxation parameter. Note that the limit case , the nonviscous damping force reduces to classical viscous damping.

Let , , , ; system (1) can be represented aswhere

Using a coordinate transformation,where represents the norm of the response. and are the angles process of the coupled stochastic system. denotes the coupling or exchange of energy between the coupled system. Equation (3) can be rewritten asTherefore, the solution forms of can be expressed as from (6), where is quickly varying process relative to the process due to . Therefore, the following approximations will be obtained:Substituting (7) into the nonviscous damping, one obtainsSimilarly, the following results also can be derived:whereSubstituting (8) and (9) into (6) and letting , (6) can be written as the following Itô stochastic differential equations:where is the standard Wiener process; the drift and diffusion coefficients can be rewritten as

To obtain the eigenvalue problem for the th moment Lyapunov exponents of a four-dimensional linear Itô stochastic system, a linear stochastic transformation [34] is introducedwhere is a new norm process depending on the transformation function . The transformation function is defined on the independent stationary phase processes , , and in the ranges , , . Applying Itô formula, one can obtain

If the transformation function is bounded and nonsingular, both processes and have the same stability behavior. Therefore, transformation function is chosen so that the drift term of the Itô differential equation (14) does not depend on the phase processes , , and , so one can obtain

Comparing (14) and (15), such transformation function can be written by the following equation:where and are the following first-order and second-order differential operators:in which

In the following investigation, the perturbation theory will be used to obtain the solution of (16).

#### 3. Moment Lyapunov Exponents

The method of deriving the eigenvalue problem for the moment Lyapunov exponents of a two-dimensional linear Itô stochastic system was first applied by Wedig [34]. The eigenvalue problem for a differential operator of three independent variables , , and will be identified from (16), in which is the eigenvalue and is the associated eigenfunction. Meanwhile, the eigenvalue is seen to be the Lyapunov exponent of the th moment of system (1) from (15). Applying the perturbation theory, both the moment Lyapunov exponent and the eigenfunction are expanded in power series of ; that is,

Substituting (19) into (16) and equating terms of the equal powers of lead to the following equations:where must be positive and periodic in the range , .

##### 3.1. The Zeroth-Order Perturbation

Substituting (17) into (20), the zeroth-order perturbation equation isThrough , thus, , will be obtained. Then (21) does not contain , and the eigenvalue is independent of . Therefore, will lead to for arbitrary . At this point, (21) will be reduced toUsing the method of separation of variables, so we can express aswhere , , and are functions about , , and , respectively.

Substituting (23) into (22), (22) will become Solving the above differential equation (24), one can easily obtain In addition, since function is periodic about and , the following boundary conditions can be obtained:Substituting (23) and (25) into (26), we can determine the constants . Therefore, is a function about only to be determined; that is, .

The adjoint equation of (22) is Applying the method of separation of variables to solve (27), one obtainswhere is an arbitrary function.

##### 3.2. The First-Order Perturbation

From (20), the first-order perturbation equation isBased on the analysis results in Section 3.1, one obtainsThen the solvability condition of (30) isApplying the sense of the arbitrary of function , (31) will becomewhere then (32) is simplified as the following equation:Then the boundary conditions of (34) are determined by considering the adjoint equation for the case of ,where is the Fokker-Planck operator and and are the entrance boundaries [27]. The eigenfunction satisfies zero Neumann boundary condition, and is the largest eigenvalue of (34) with zeros Neumann boundary. Therefore, the solution of (34) can be calculated from an orthogonal expansion [34]. Under zeros Neumann boundary conditions, can be expressed by a Fourier cosine series [27]; that is,Substituting (36) into (34), multiplying by in both sides and integrating for , we can obtainwhere In order to guarantee the existence of the nontrivial solution for each , the coefficient matrix must equal zero. Therefore, the problem of calculating is translated into calculating the leading eigenvalue of . The sequence of approximations will be constructed by the eigenvalues of a sequence of the following submatrices:

Obviously, the set of approximate eigenvalues obtained by this method converges to the associated true eigenvalues as . In general, we can obtain the approximate eigenvalues by truncating , which can be written as and

The comparison results of approximate analytical moment Lyapunov exponents obtained by truncating and the direct Monte Carlo simulation results are shown in Figure 1. It is easily found that the approximate results agree well with the simulation results. Therefore, the first-order perturbation will be reduced toWithout loss of generality, there is a relationship between two frequencies of the form , in which and are integers. The second frequency can be rewritten as . Then the function in (41) can be rewritten aswhere function is the periodic function on and given asNote that the general solution of (41) can not be obtained except in some special cases. Considering the nature of the coefficients of (41), the series expansion of function can be presented in the following form: