Stochastic stability of SDOF linear viscoelastic system under wideband noise excitation

The objective of this paper is to present a higher-order stochastic averaging method to study stability of a single-degree-of-freedom (SDOF) ordinary viscoelastic system subject to stochastic fluctuation.
• Higher-order stochastic averaging is performed to obtain analytical expressions for the moment Lyapunov exponent and the Lyapunov exponent.
• Monte Carlo simulation of moment Lyapunov exponents and Lyapunov exponents of the SDOF viscoelastic system.
• The stochastic stability of a viscoelastic column under two main wideband noise excitation is discussed and explained.
• Both analytical and simulation results show that higher-order stochastic averaging improves the accuracy compared with the first-order stochastic averaging. However, It is advisable to consider a balance between accuracy achievement and calculation endeavor when using higher-order stochastic averaging.It is found the intensity of white noise σ would de-stabilize the system. For real noise, large α and small σ would stabilize the system. For both white and real noise, the stronger the viscoelasticity (i.e. larger γ), the larger the relaxation (i.e. smaller κ), the more stable the system.

The stochastic moment stability and almost-sure stability of a single-degree-of-freedom (SDOF) viscoelastic system subject to parametric fluctuation is investigated by using the method of higher-order stochastic averaging. The stochastic parametric excitation is modeled as a wideband noise, which is taken as Gaussian white noise and real noise. The viscoelastic material is assumed to follow ordinary Maxwell linear constitutive relation. For small damping and weak stochastic fluctuation, analytical expressions are derived for the moment Lyapunov exponent and the Lyapunov exponent, which indicate moment stability and almost-sure stability respectively. The effects of various system and loading parameters on the stochastic stability are discussed. Both analytical and simulation results show that higher-order stochastic averaging improves the accuracy compared with the first-order stochastic averaging. However, results of the third-order averaging are almost overridden by those of second-order averaging and the third-order averaging involves far more calculation. It is advisable to consider a balance between accuracy achievement and calculation endeavor when using higher-order stochastic averaging.