Stochastic stability of viscoelastic systems

Abstract

Many new materials used in mechanical and structural engineering exhibit viscoelastic properties, that is, stress depends on the past time history of strain, and vice versa. Investigating the behaviour of viscoelastic materials under dynamical loads is of great theoretical and practical importance for structural design, vibration reduction, and other engineering applications. The objective of this thesis is to find how viscoelasticity affects the stability of structures under random loads.

The time history dependence of viscoelasticity renders the equations of motion of viscoelastic bodies in the form of integro-partial differential equations, which are more difficult to study compared to those of elastic bodies.

The method of stochastic averaging, which has been proved to be an effective tool in the study of dynamical systems, is applied to simplify some single degree-of-freedom linear viscoelastic systems parametrically excited by wide-band noise and narrow-band noise. The solutions of the averaged systems are diffusion processes characterized by Itô differential equations. Therefore, the stability of the solutions is determined in the sense of the moment Lyapunov exponents and Lyapunov exponents, which characterize the moment stability and the almost-sure stability, respectively. The moment Lyapunov exponents may be obtained by solving the averaged Itô equations directly, or by solving the eigenvalue problems governing the moment Lyapunov exponents.

Monte Carlo simulation is applied to study the behaviour of stochastic dynamical systems numerically. Estimating the moments of solutions through sample average may lead to erroneous results under the circumstances that systems exhibit large deviations. An improved algorithm for simulating the moment Lyapunov exponents of linear homogeneous stochastic systems is presented. Under certain conditions, the logarithm of norm of a solution converges weakly to normal distribution after suitably normalized. This property, along with the results of Komlós-Major-Tusnády for sums of independent random variables, are applied to construct the algorithm. The numerical results obtained from the improved algorithm are used to determine the accuracy of the approximate analytical moment Lyapunov exponents obtained from the averaged systems. In this way the effectiveness of the stochastic averaging method is confirmed.

The world is essentially nonlinear. A single degree-of-freedom viscoelastic system with cubic nonlinearity under wide-band noise excitation is studied in this thesis. The approximated nonlinear stochastic system is obtained through the stochastic averaging method. Stability and bifurcation properties of the averaged system are verified by numerical simulation. The existence of nonlinearity makes the system stable in one of the two stationary states.