MOMENT LYAPUNOV EXPONENTS AND STOCHASTIC STABILITY OF A THIN-WALLED BEAM SUBJECTED TO AXIAL LOADS AND END MOMENTS

In this paper, the Lyapunov exponent and moment Lyapunov exponents of two degrees-of-freedom linear systems subjected to white noise parametric excitation are investigated. The method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises. The Lyapunov exponent and moment Lyapunov exponents are important characteristics for determining both the almost-sure and the moment stability of a stochastic dynamic system. As an example, we study the almost-sure and moment stability of a thin-walled beam subjected to stochastic axial load and stochastically fluctuating end moments. The validity of the approximate results for moment Lyapunov exponents is checked by numerical Monte Carlo simulation method for this stochastic system.

Stochastic Stability of Coupled Viscoelastic Systems Excited by Real Noise

The moment stochastic stability and almost-sure stochastic stability of two-degree-of-freedom coupled viscoelastic systems, under the parametric excitation of a real noise, are investigated through the moment Lyapunov exponents and the largest Lyapunov exponent, respectively. The real noise is also called the Ornstein-Uhlenbeck stochastic process. For small damping and weak random fluctuation, the moment Lyapunov exponents are determined approximately by using the method of stochastic averaging and a formulated eigenvalue problem. The largest Lyapunov exponent is calculated through its relation with moment Lyapunov exponents. The stability index, the stability boundaries, and the critical excitation are obtained analytically. The effects of various parameters on the stochastic stability of the system are then discussed in detail. Monte Carlo simulation is carried out to verify the approximate results of moment Lyapunov exponents. As an application example, the stochastic stability of a flexural-torsional viscoelastic beam is studied.

Moment Lyapunov Exponents and Stochastic Stability of a Three-Dimensional System on Elastic Foundation Using a Perturbation Approach

In this paper, the stochastic stability of the three elastically connected Euler beams on elastic foundation is studied. The model is given as three coupled oscillators. Stochastic stability conditions are expressed by the Lyapunov exponent and moment Lyapunov exponents. It is determined that the new set of transformation for getting Ito∧ differential equations can be applied for any system of three coupled oscillators. The method of regular perturbation is used to determine the asymptotic expressions for these exponents in the presence of small intensity noises. Analytical results are presented for the almost sure and moment stability of a stochastic dynamical system. The results are applied to study the moment stability of the complex structure with influence of the white noise excitation due to the axial compressive stochastic load.