A new calculation approach under precise integration for nonlinear dynamic equation

Based on the precise integration method of the exponential matrix developed by Zong Wanxie, we discuss a general dynamic system governed by the state equation v(overdot) = Hv + f(v,t), in which v is an unknown n-dimensional vector, H is a coefficient matrix, Hv and f(v,t) are the linear homogeneous part and nonlinear part in the right of the equation respectively. This paper suggests that the integral of the state equation is evaluated directly through the exponential matrix and its precise algorithm. Thus a closed series solution can be successively obtained and the precision of the solution can be controlled easily, so it is very convenient to study the relations between the nonlinear dynamic response and its physical parameters. This algorithm avoids calculating the inversion of the [H] matrix, the stabilization and efficiency of the computation can be ensured so that this method is especially suitable for the large-scale problems. Two numerical examples are presented to demonstrate the effectiveness of this algorithm.