Probabilistic Analysis Method Of Typical Nonlinear Stochastic Systems

The nonlinear stochastic dynamics theory takes the nonlinear stochastic system as the research object,develops a set of probability analysis methods to study the systems'dy-namics behaviors,and has been widely used in the communication,aerospace,mechanical,civil and Marine engineering practices.However,their structures are very different due to the different nonlinear modeling mechanisms.Therefore,there is no universal probability analysis methods to study their dynamic responses.The main purpose of this paper is to propose novel probabilistic analysis methods for two kinds of typical nonlinear systems under different random excitations.On the one hand,the moment Lyapunov exponent method is applied to the nonlinear stochastic system.Through analyzing the moment Lyapunov exponent of the nonlinear stochastic system,and then its stochastic stability is studied.On the other hand,a quasi-conservative stochastic averaging method based on the equilibrium point of bistable system is developed,then the stochastic probability response of the nonlinear bistable vibration energy harvesting(BVEH)system is analyzed.In chapter 2,firstly,the motion equation of a class of new nonlinear stochastic system is established,that is,the coupled nonlinear stochastic system with non-viscous damping structure under Gaussian white noise excitation,whose non-viscous damping structure fol-lows the exponential integral form.Using the coordinate transformation,the coupled Ito stochastic differential equations of the norm of the response and angles process are obtained.Then the problem of the moment Lyapunov exponent is transformed to the eigenvalue prob-lem,and then the second-perturbation method is used to derive the moment Lyapunov exponent of coupled stochastic system.Finally,the effects of various physical quantities of stochastic coupled system on the stochastic stability are discussed by moment Lyapunov ex-ponent method in detail.The effectiveness of moment Lyapunov exponent method to obtain analytical results is verified by Monte Carlo numerical method.In chapter 3,firstly,the dynamic model of the nonlinear bistable vibration energy har-vesting system is constructed,its external excitation is filtered white Gaussian noise.The noise has the characteristics of frequency dependence and can represent more true random vibration in the environment than the Gaussian white noise.Next,through a new transfor-mation based on the equilibrium points of bistable system,the nonlinear electromechanical coupling system can be approximated by an equivalent single degree of freedom bistable sys-tem,which contains the energy-dependent frequency functions and the equilibrium points of the equivalent system.Then the analytic expressions of the stationary probability den-sity function of the mechanical system states are obtained through the developed quasi-conservative stochastic averaging method based on the equilibrium points of system.And applying the relationship between voltage and state variables,the mean-square voltage and the mean output power are also given.Finally,the effects of the excitation intensity and the peak frequency of seismic motion of the stochastic excitation,the parameters of the vibration system and the electromechanical coupling coefficients on the mean-square volt-age and the mean output power of the stochastic BVEH system are also analyzed in detail.The effectiveness of devoloped quasi-conservative stochastic averaging method based on the equilibrium point of bistable system to obtain analytical results is verified by Monte Carlo numerical method.