DIMENSION REDUCTION OF FPK EQUATION FOR VELOCITY RESPONSE ANALYSIS OF STRUCTURES SUBJECTED TO ADDITIVE NONSTATIONARY EXCITATIONS

The nonlinear response analysis of structures subjected to random excitation is a highly challenging problem. The solution of FPK equation provides exact solutions to these problems. However, for nonlinear multi-degree-of-freedom systems, the direct solutions of the FPK equation is prohibitively difficult. Actually, the numerical solutions are strictly limited by the dimension of the equation, while the analytical solutions are available only for very few specific systems, and most of them are steady-state solution. Therefore, reducing the dimension of the FPK equation is an important way to solve the high-dimensional nonlinear dynamic response analysis problem. In the present paper, for the nonstationary response analysis of multi-degree-of-freedom nonlinear structures subjected to amplitude-modulated additive white noise, the high-dimensional FPK equation in terms of the joint probability density function is reduced in dimension. For the probability density function of velocity response, it is converted to a one-dimensional FPK-like equation through introducing the equivalent drift coefficient. The method of conditional mean function estimate is suggested to construct the equivalent drift coefficient. Afterwards, the numerical results of probability density function of velocity can be obtained by applying the path integration method to solve the dimension-reduced FPK-like equation. The accuracy and efficiency of the proposed method are discussed and verified through the numerical examples, including the non-stationary response analysis of velocity of a single-degree-of-freedom Rayleigh oscillator, a ten-story linear shear frame structure and a nonlinear shear frame structure subjected to amplitude-modulated additive white noise.