Path Integral Solutions For Three Kinds Of Dynamical Systems

In nature,there are a class of vibration sources that induce random vibration of mechanical structure systems,such as turbulence,wave motion,road roughness and seismic motion,etc.They cannot be described by deterministic time or spatial coordinates,but by the characteristics of the probability and statistics.For random vibration,due to the uncertainty of its frequency,it is easy to cause the unpredictability of structural vibration,so it is difficult to prevent and cure the effects.In order to improve the reliability of mechanical systems working in random environments,studying the law of random motion has important theoretical significance and application value in natural science and engineering technology.The response of a stochastic dynamic system is usually discussed by the Fokker-Planck-Kolmogorov(FPK)equations corresponding to the probability density function(PDF),and then the response statistical characteristics of the system are studied by using the probability density functions.In this paper,the evolutions of probability density functions(PDFs)over time for turbulent tower system,random variable-mass system and vibratory energy harvesting system excited by Gaussian white noise are studied by using path integration(PI)method.The results are compared with those of finite difference method and Monte-Carlo numerical simulation and verified.The details are as follows:(1)The statistical characteristics of turbulent towers excited by Gauss white noise and parametric excitation are studied.The motion equations of above systems are established.The It? stochastic differential equations and the corresponding FPK equations are given,then the corresponding second-order moment equations are obtained by Gaussian truncation scheme.Based on path integral method of Gauss-Legendre formula,the steady state solutions and the transient solutions at different time of path integration of the turbulent tower model are obtained,and are compared with the results of Monte-Carlo numerical simulation,of which the validity is verified.(2)The process of probability density evolution for variable-mass Duffing oscillator excited by Gaussian white noise is considered.Both small mass disturbance and large mass disturbance are discussed,and the corresponding FPK equations are derived,which are calculated and analyzed by path integral method.The validity of the method is verified.(3)The random responses of vibratory energy harvesting systems under the excitation of Gaussian white noise are studied.The transient probability density functions andstationary probability densities of FPK equations for three different types of nonlinear vibratory energy harvesting systems are calculated by path integral method.The results are in good agreement with those of the Monte-Carlo numerical simulation method.(4)Finite difference method is proposed to solve the general two-dimensional problem,which is verified by the path integral method in this paper.Based on the implicit finite difference method,the random responses of four nonlinear stochastic dynamical systems under Gaussian white noise and harmonic excitation are discussed.The nonstationary probability density functions of cubic nonlinear oscillator,quadratic-cubic type oscillator,cubic-quintic type oscillator and bistable oscillator are investigated,respectively.The evolutions of marginal probability density functions and joint probability density functions with time are analyzed.