An Improved Energy Envelope Stochastic Averaging Method And Technique To Solve Averaged FPK Equation Based On Integral Transformation

Nonlinear factors of mechanical systems will bring great difficulties to the stochastic vibration research,and the stochastic averaging method is a powerful tool in the nonlinear random vibration study.The stochastic averaging method has been applied more widely inrandom vibration analysis of mechanical and structural systems.The stochastic averaging method is combined with the stochastic averaging principle and the FPK equation method.Stochastic averaging method is divided into the standard stochastic averaging method and energy envelope stochastic averaging method.The standard stochastic averaging method cannot take into account the influence of nonlinear restoring force,while the energy envelope stochastic averaging method can make up this limitation of the standard stochastic averaging method.Nonetheless,the energy envelope stochastic averaging method is directly effective only to the case that the external excitation is ideal white noise.Therefore,the scope of the two stochastic averaging methods will be severely limited.This paperattempts to improve the energy envelope stochastic averaging method of nonlinear single degree of freedom system through combining the characteristics of standard stochastic averaging method.This method can not only taking into account the non-linear restoring force but also can expand the requirement of external incentives,whose application sphere is extensive.The exact solution especially the exact transient solution of the FPK equation governs the system transition probability density which achieved by the stochastic averaging method is very difficult to get.An oscillating system with both nonlinear damping and stiffiiess is taken into account in this paper.We can get the stochastic averaging FPK equation by using the improved energy envelope stochastic averaging method.Under the initial and boundary conditions,the exact transient solution of the FPK equation is worked out by right of a technique based on the integral transformation.Finally,the external excitation of the selected nonlinear system is generated by using the Monte Carlo numerical simulation method in the MATLAB software.We can get the results of the statistical characteristic of the system response by numerical simulation,and then compared the results with them obtained by theoretical calculation.In order to test the accuracy of the FPK equation obtained by using the improved energy envelope stochastic averaging method and 1he exact transient solution of tiie FPK equation worked out by right of a technique based on ttie integral transformatioiL And the influence of parameters of the nonlinear system on the probability characteristics of the response is analyzed.