Dynamic instability of structures under the excitation of stochastic forces can be
found in many engineering practices, such as earthquake engineering, wind
engineering, and off-shore engineering. The moment Lyapunov exponent is a
fundamental characteristic quantity for studying stochastic stability. We investigate
the stochastic instability of a nonlinear system subjected to parametric excitation by
real noise. The real noise is modeled by the Ornstein–Uhlenbeck (real) process, a
stationary process that is often used to approximate realizable wide-band
excitation. For small damping and weak random fluctuation, the moment
Lyapunov exponents are determined approximately by using the method of
stochastic averaging and an appropriate mathematical transformation based
on the system’s energy. We also incorporate linear viscoelastic effects into
the stochastic stability framework. By introducing a hereditary kernel of
the Maxwell type into the equation of motion, viscoelastic contributions
are explicitly embedded in the averaged drift. This extension enables the
assessment of how memory-dependent energy dissipation modifies the stability
boundaries of nonlinear systems under wide-band noise. Parametric studies reveal
that the viscoelastic amplitude enhances stochastic stability, while a larger
decay rate reduces the effectiveness of memory and weakens the stabilizing
effect. Monte Carlo simulation is carried out to provide numerical results
and to verify the approximate analytical results from stochastic averaging,
demonstrating good agreement and thereby supporting the validity of the
theoretical derivations and the applicability of the proposed viscoelastic
extension.
Keywords
nonlinear oscillator, stochastic instability, moment
Lyapunov exponent, stochastic averaging, real noise
excitation
