Stochastic Response Or Reliability Of Vibro-Impact And Hvsteretic Systems

Mechanical and structural systems under stochastic dynamic loadings are always encountered in engineering applications and attract extensive attention of researchers. Stochastic responses, stability and reliability of traditional stochastic systems, in which conservative components and dissipative components are separated, have been studied sufficiently and many effective traditional analytical methods have been proposed. Unfortunately, some materials such as elastic-plastic, hysteresis and viscoelastic. have the coupling conservation and dissipation components. Stochastic systems with coupling conservation and dissipation components are different from the ordinary systems and traditional analysis methods are not suitable. In the present dissertation, responses and reliability of two typical stochastic systems with coupling conservation and dissipation components, i.e., dissipative vibro-impact system and hysteretic system, are studied.In consideration of the energy dissipative effect, the contact forces are described by modified Hertzian contact model and Wu-Yu impact model, respectively. In case of weak excitations and weak energy dissipation, the vibro-impact systems are researched by energy dissipation balance technique and stochastic averaging method. The contact force can be approximately by conservative component and dissipative component. The equivalent conservative one is represented by the gradient of total potential energy. While the dissipative component is substituted by equivalent quasi-linear damping with energy-dependent damping, which satisfies that energy dissipation due to the inelastic impact in one cycle is equal to the energy dissipation by the quasi-linear damping during the same period. Then, by adopting the stochastic averaging method, stochastic responses can be obtained. Due to the restriction of weak damping and weak excitations, stochastic averaging method is not always effective. Equivalent nonlinear technique is adopted for strong damping and excitations. Based on the vibro-impact system without dissipation, introducing the conservative component correction fact and state-dependent damping coefficient, one class of equivalent nonlinear systems with exact stationary solution is selected. The correction factor and undetermined damping coefficient are determined through minimizing the mean-square difference. By expressing the state-dependent damping coefficient as polynomial functions of system energy, the functional extremum problem is transformed into function extremum problem and finally the responses of the original vibro-impact system are approximately obtained through the equivalent nonlinear system.Many hysteretic models have been developed to describe the hysteretic behavior. Due to the important application of Duhem and Preisach hysteretic models, the first-passage problem of hysteretic systems described by Duhem and Preisach hysteretic models under stochastic dynamics are discussed. Similarly to dealing with inelastic contact force, the conservative component of the hysteretic force is expressed by the gradient of the total potential energy and dissipative component is substituted by quasi-linear damping with energy-dependent damping coefficient. Thus equivalent nonlinear stochastic system with separated conservation and dissipation components is obtained, and then by adopting stochastic averaging method, the averaged Ito stochastic differential equation with respect to system energy is dervied. The establishing and solving of the associated backward Kolmogorov equation yield the reliability function and probability density of first-passage time.