Higher-order stochastic averaging for a SDOF fractional viscoelastic system under bounded noise excitation

The moment Lyapunov exponents and stochastic stability of a single-degree-of-freedom (SDOF) fractional viscoelastic system under bounded noise excitation are studied by using the method of higher-order stochastic averaging. A realistic example of such a system is the transverse vibration of a viscoelastic column under the excitation of stochastic axial compressive load. The excitation is modeled as a bounded noise, which is a realistic model of stochastic fluctuation in engineering applications. The viscoelastic material is assumed to follow a fractional Kelvin-Voigt constitutive relation. The method of higher-order stochastic averaging is used to approximate the fractional stochastic differential equation of motion, and then moment Lyapunov exponents are determined for the system with small damping and weak random fluctuation. The approximate results are confirmed by Monte-Carlo simulations. It is found that convergence of moment Lyapunov exponents depends on the width of power spectral density of the bounded noise process. For this viscoelastic structure, second-order averaging analysis is adequate for stability analysis. The effects of various parameters on the stochastic stability of the system are discussed and possible explanations are explored.