Random Response and Crossing Rate of Fractional Order Nonlinear System with Impact

Yang Zheng,Jiayin Wu,Ming Xu,Davood Younesian

doi:10.1155/2021/6666843

Yang Zheng, Jiayin Wu + Show 2 more

Open Access

https://doi.org/10.1155/2021/6666843

Copy DOI

Abstract

The random response and mean crossing rate of the fractional order nonlinear system with impact are investigated through the equivalent nonlinearization technique. The random additive excitation is Gaussian white noise, while the impact is described by a phenomenological model, which is developed from the actual impact process experiments. Based on the equivalent nonlinearization technique, one class of random nonlinear system with exact probability density function (PDF) solution of response is selected. The criterion of the appropriate equivalent nonlinear system is the similarity with the original system on the damping, stiffness, and inertia. The more similar, the higher the precision. The optimal unknown parameters of the equivalent random nonlinear system in the damping and stiffness terms are determined by the rule of smallest mean-square difference. In the view of equivalent nonlinearization technique, the response of the original system is the same as that of the equivalent system with the optimal unknown parameters in analytical solution manner. Then, the mean crossing rate is derived from stationary PDF. The consistence between the results from proposed technique and Monte Carlo simulation reveals the accuracy of the proposed analytical procedure.

Highlights

Based on the impact experiments of elastic-plastic structures, a phenomenological impact model is developed
Due to time duration of the elastic-plastic deformation process, the impact model of the velocity jump at collision moment is not appropriate in this case [16, 17]. By using this phenomenological impact model, the stochastic averaging method has been adopted to obtain the stationary probability density function (PDF) of the vibro-impact system subjected to Gaussian white noises [18]
The mean crossing rate, which always relates to predicting the extreme response statistics and the system reliability, is another important quantity which should be considered. e evaluation of the mean crossing rate has been of great interest among researchers

Summary

Introduction

Based on the impact experiments of elastic-plastic structures, a phenomenological impact model is developed. It can be expected that the equivalent nonlinearization technique is applicable to deal with the random systemincorporated inelastic impact described by the phenomenological impact model.

Results

Conclusion