Two Methods Of Calulating The Non-Stationary Random Response For Nonlinear Systems

The research of nonlinear random vibration has engineering background, and great academic value, and it is of theoretical significance as well. But the research of this issue is very difficult, and so far there is not an applicable method for most purposes. The past researches were mostly for single-DOF or less DOF, and the approximation algorithm was limited to stationary random vibration. So it is necessary to develop the research of the calculation method for nonlinear systems non-stationary random response, and improve the precision of the algorithm.Two methods are presented in this paper for the solution of nonlinear non-stationary random response under the Gauss white noise excitation:The first method is that the Runge-Kutta method, which is effective for certainty different equation, is extended to the computation of nonlinear non-stationary random response under random excitation. First, the nonlinear system is statistically linearized. And then, in order to consider the time varying of equivalent linearization system, the equivalent linearization parameters are supposed to keep constant in the minor time distance, but alter at the interval points, by the Runge-Kutta method, the recursion relation of the response of the system is deducted.The second method is based on the high order linearization method. By importing a new variable, the equation is extended higher order, and then the higher order system is statistically linearized to be a series of high order equivalent linearization equations. With the equivalent linearization equations, the moment equations which are closed on the two order moment are deducted, and then response of the system is obtained.As an example, the results of Duffing system, by the two methods above, are compared with the numerical simulation. It can be indicated from the result that: it is feasible that to use the Runge-Kutta method for solving the non-stationary random response of the nonlinear system; by the high order linearization method, the order of the equation is expanded, and the moment equations are deducted, and then the response is gained, which is more exactitude than general equivalent linearization method.