Analysis Methods Of Nonlinear Stochastic Systems Based On Moment Equations

In the past few decades,the application of computers has expanded into various fields of science and engineering,which makes it possible to analyze more complex and detailed physical models.Although everyone is used to dealing with deterministic static problems,the load on the actual structure not only changes with time,but also has obvious randomness at the same time.The stochastic modeling and solving of engineering problems becomes a trend.Stochastic dynamics have also drawn attention.So far,the development of random vibration theory of linear systems has become more and more mature,and the random response analysis of nonlinear systems is still the challenge of stochastic dynamics.The moment equation method is a commonly used method in the analysis of nonlinear stochastic systems.Since the statistical moment equation of response is an infinite dimensional structure,a truncation scheme must be used to approximate the infinite system with a finite set of equations for easy solution.The authors developed two new methods of moment equations for stochastic response analysis of nonlinear systems.The huge computational cost brought by the high-order moment equation is a major problem faced by the moment equation method.By means of Gaussian mixture model and Gaussian copula function,the author gives the joint density function of the system random response.The assumed joint density is brought into the moment equation of the system,and the parameters in the assumed probability density are determined by minimizing the error of the equation.Thanks to the special properties of the Gaussian mixture model and the Gaussian copula function,the computational cost is greatly reduced.At the same time,the accuracy of the traditional Gaussian truncation method is the lower limit of the method.In addition,due to some special properties of the zero-mean Gaussian distribution,the computational cost is further reduced.Finally,the effectiveness and accuracy of the proposed method are verified by a nonlinear oscillator excited by Gaussian white noise and a Duffing oscillator excited by Gaussian white noise.And compared with the results obtained by the Gaussian closure method and the exact solution.Another major challenge of the moment equation method is that only a very limited response integer moment is involved in the truncation,thereby losing important probability information of the response,resulting in poor accuracy of the truncation.This paper proposes a new fractional moment equation method that overcomes the limitations of existing methods.The method derives the integer order fractional moment equation of the nonlinear system and approximates the PDF of the nonlinear system response by a new fractional moment truncation method.Thanks to the valuable properties of fractional moments,that is,fewer fractional moments can contain a large amount of statistical information to estimate an unknown PDF.Compared with existing methods,this method can make accurate estimation of random response,especially for nonlinear systems with multiple equilibria.Another advantage of this method is that in the context of the fractional moment equation,the joint PDF of the response can be explicitly obtained and can be used for the first time probability analysis of nonlinear systems.Finally,the effectiveness of the proposed method is verified by Duffing oscillator excited by Gaussian white noise and bistable Duffing oscillator.And compared with the exact solution of Gaussian closure method.