Stochastic stability of gyroscopic viscoelastic systems and applications in axially moving bands

This paper investigates the stochastic stability of gyroscopic viscoelastic systems subjected to parametric wide-band noise excitation. The analysis focuses on both moment stability, using moment Lyapunov exponents, and almost-sure stability, via the largest Lyapunov exponent. The wide-band noises considered include Gaussian white noise and Ornstein–Uhlenbeck noise. The Stratonovich stochastic differential equations governing the system with small damping and weak excitation are first converted to Itô stochastic differential equations through stochastic averaging techniques. An elegant mathematical framework is then introduced to approximate the moment Lyapunov exponents through stochastic transformations and an eigenvalue problem. The largest Lyapunov exponent is subsequently derived based on its relationship with the moment Lyapunov exponents. An application example involves deriving the stochastic equations of motion for an axially moving band system with fluctuating tension, analyzing its stochastic stability. The analytical approximations are validated via Monte Carlo simulations and compared with results from the literature. The study also discusses the influence of various parameters on the system’s stability, with potential applications in engineering fields.