Stochastic Stability Of Multi-Degree-of-Freedom Svstem With Viscoelasticitv Or Time Delay

Stochastic stability is always an important issue in stochastic dynamics. Due to the continuous advent of new techonology and new material, lots of system dynamics equations depend on its history status, such as structures with viscoelastic material and controlled systems with time delay. In this paper, stochastic stability of multi-degree-of-freedom (MDOF) system with viscoelasticity or time delay is studied.The response and stability of a single degree-of-freedom (SDOF) viscoelastic system with strongly nonlinear stiffness under the excitations of wide-band noise are studied. Viscoelastic term in the equation of motion is described through the generalized Maxwell model. First, terms associated with the viscoelasticity are approximately equivalent to damping and stiffness terms with the application of the properties of the generalized harmonic functions; the viscoelastic system is approximately transformed to a SDOF system without viscoelasticity. Then, by using the method of stochastic averaging based on generalized harmonic function, the stationary response and the largest Lyapunov exponent can be analytically expressed. The asymptotic stability with probability one of the system is determined.The stochastic stability of a MDOF system with time-delayed feedback control is studied. First, a quasi-integrable Hamiltonian stochastic system with time-delayed feedback control is studied by using Lyapunov functions and stochastic averaging method. Through the appropriate relations between the current states and the delayed states, the system is approximated by a quasi-integrable Hamiltonian stochastic system without time delay. Stochastic averaging method for quasi-integrable Hamiltonian system is adopted to reduce the dimension of the equivalent system and the averaged lto stochastic differential equations of the independent first integrals in involution are obtained. The Lyapunov function is taken as the optimal linear combination of the first integrals and its averaged lto stochastic differential equation is derived. The sufficient condition for the asymptotic stability with probability one of the system is determined by evaluating the eigenvalues and eigenvectors of the linearized drift coefficient matrix. Furthermore, a more general method for determining the stochastic stability of a MDOF system is proposed. A specific linear combination of subsystems’energies is constructed as Lyapunov function. The stochastic stability is directly determined. It is unlike the previous method. Herein, the stochastic averaging is not used. Besides, through this method, the effect of coupled damping can be considered and it can be applied to the system with strong damping and excitation.The method isextended to a MDOF system with time-delayed feedback control.A procedure for designing a feedback control to asymptotically stabilize, with probability one, quasi-generalized Hamiltonian systems subject to stochastically parametric excitations is proposed. First, the motion equattings of controlled systems are reduced to lower-dimensional averaged Ito stochasuc differential equations by using the stochastic averaging method. Then, a dynamic programming equation for the averaged system with an appropriate performance index (with undetermined parameters in cost function) is established based on the dynamic programming principle, and the optimal control law is derived from minimization condition with respect to control. In order to verify the effectiveness of the optimal feedback control, Lyapunov function method is adopted to evaluate the stability boundary of asymptotic stability with probability one for the uncontrolled/controlled systems. Numerical results show that the domain of stochastic stability of the controlled system is obviously larger that that of the uncontrolled one. This strategy is successfully applied to a practical viscoelastic beam.