Stochastic Stability of a Fractional Viscoelastic Plate Driven by Non-Gaussian Colored Noise

In this paper, the moment Lyapunov exponent and stochastic stability of a fractional viscoelastic plate driven by non-Gaussian colored noise is investigated. Firstly, the stochastic dynamic equations with two degrees of freedom are established by piston theory and Galerkin approximate method. The fractional Kelvin–Voigt constitutive relation is used to describe the material properties of the viscoelastic plate, which leads to that the fractional derivation term is introduced into the stochastic dynamic equations. And the noise is simplified into an Ornstein-Uhlenbeck process by utilizing the path-integral method. Then, via the singular perturbation method, the approximate expansions of the moment Lyapunov exponent are obtained, which agree well with the results obtained by the Monte Carlo simulations. Finally, the effects of the noise, viscoelastic parameters and system parameters on the stochastic dynamics of the viscoelastic plate are discussed.

DYNAMIC BEHAVIOR OF TWO ELASTICALLY CONNECTED NANOBEAMS UNDER A WHITE NOISE PROCESS

This paper investigates the almost-sure and moment stability of a double nanobeam system under stochastic compressive axial loading. By means of the Lyapunov exponent and the moment Lyapunov exponent method the stochastic stability of the nano system is analyzed for different system parameters under an axial load modeled as a wideband white noise process. The method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises.

Stochastic Stability of a Planner Gyropendulum System with Synchronous Motor Driven by Gaussian White Noises

In this paper, the stochastic moment stability and almost-sure stability of a planner gyropendulum system with synchronous motor under the white noises are investigated. By applying the theory of diffusion process, an eigenvalue problem for the moment Lyapunov exponent is formulated. Then, through a perturbation method and a Fourier cosine series expansion, the second-order expansion of the moment Lyapunov exponent is solved, which is just the leading eigenvalue of an infinite matrix. Finally, the convergence and validity of the procedure are numerically verified, and the effects of system and noise parameters on the moment Lyapunov exponent are discussed. It was found that the increase in both the noise intensity and coefficient of the synchronous motor torque will weaken the stability of the gyropendulum system, and when they reach certain values, the system becomes unstable. In addition, according to the relationship between the moment Lyapunov exponent and maximal Lyapunov exponent, the stable thresholds are also given.

Numerical Simulation of Dynamic Stability of Fractional Stochastic Systems

The modern theory of stochastic dynamic stability is founded on two main exponents: the largest Lyapunov exponent and moment Lyapunov exponent. Since any fractional viscoelastic system is indeed a system with memory, data normalization during iterations will disregard past values of the response and therefore the use of data normalization seems not appropriate in numerical simulation of such systems. A new numerical simulation method is proposed for determining the [Formula: see text]th moment Lyapunov exponent, which governs the [Formula: see text]th moment stability of the fractional stochastic systems. The largest Lyapunov exponent can also be obtained from moment Lyapunov exponents. Examples of the two-dimensional fractional systems under wideband noise and bounded noise excitations are presented to illustrate the simulation method.

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

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.

Moment Lyapunov Exponent and Stochastic Stability of Binary Airfoil under Combined Harmonic and Non-Gaussian Colored Noise Excitations

Both periodic loading and random forces commonly co-exist in real engineering applications. However, the dynamic behavior, especially dynamic stability of systems under parametric periodic and random excitations has been reported little in the literature. In this study, the moment Lyapunov exponent and stochastic stability of binary airfoil under combined harmonic and non–Gaussian colored noise excitations are investigated. The noise is simplified to an Ornstein-Uhlenbeck process by applying the path-integral method. Via the singular perturbation method, the second-order expansions of the moment Lyapunov exponent are obtained, which agree well with the results obtained by the Monte Carlo simulation. Finally, the effects of the noise and parametric resonance (such as subharmonic resonance and combination additive resonance) on the stochastic stability of the binary airfoil system are discussed.