A precise time-step integration method by step-response and impulsive-response matrices for dynamic problems

In this paper, a precise time-step integration method for dynamic problems is presented. The second-order di¤erential equations for dynamic problems are manipulated directly. A general damping matrix is considered. The transient responses are expressed in terms of the steady-state responses, the given initial conditions and the step-response and impulsive-response matrices. The steady-state responses for various types of excitations are readily obtainable. The computation of the step-response and impulsive-response matrices and their time derivatives are studied in this paper. A direct computation of these matrices using the Taylor series solutions is not e¦cient when the time-step size *t is not small. In this paper, the recurrence formulae relating the response matrices at t"*t to those at t"*t/2 are constructed. A recursive procedure is proposed to evaluate these matrices at t"*t from the matrices at t"*t/2m. The matrices at t"*t/2m are obtained from the Taylor series solutions. To improve the computational e¦ciency, the relations between the response matrices and their time derivatives are investigated. In addition, these matrices are expressed in terms of two symmetric matrices that can also be evaluated recursively. Besides, from the physical point of view, these matrices should be banded for small *t. Both the stability and accuracy characteristics of the present algorithm are studied. Three numerical examples are used to illustrate the highly precise and stable algorithm.