A Second-Order Scheme of Precise Time-Step Integration Method for Dynamic Analysis with respect to Long-Term Integration and Transient Responses

In this chapter, a second-order scheme of precise time-step integration (PTI) method is introduced for dynamic analysis with respect to long-term integration and transient responses while spatial discretization is realized with the differential quadrature method. Rather than transforming into first-order equations, a recursive scheme is presented in detail for direct solution of the homogeneous part of second-order differential and algebraic equations. The sine and cosine matrices involved in the scheme are calculated using the so-called N 2 algorithm, and the corresponding particular solution is also presented where the excitation vector is approximated by the truncated Taylor series. The performance and numerical behaviors of the second-order scheme of the PTI method are tested by a series of numerical examples in comparison with the first-order scheme or with the traditional time-marching Newmark-β method as the reference. The issue of spurious high-frequency responses resulting from spatial discretization for shock-excited structural dynamic analysis is also studied in the framework of the second-order PTI method. The effects of spatial discretization, numerical damping and time step on solution accuracy are explored by analyzing longitudinal vibrations of a shock-excited rod with rectangular, half-triangular and Heaviside step impact. Qing-Hua Qin and Hang Ma 146 The chapter is organized as follows: In Section 1, previous research into timemarching algorithms is briefly reviewed. Following a brief introductory overview of the first-order scheme, the second-order PTI method is introduced in detail in Section 2 together with an analysis of the accuracy that can be achieved, the treatment of particular solutions, and solution behavior with the second-order scheme. Numerical examples for smooth responses of various wave equations and wave propagations are presented using the second-order PTI method in Section 3. Numerical examples of impact responses are presented and studied extensively in Section 4 with respect to the effect of spatial discretization, modal types that the discretization can afford, and numerical damping, etc., with some concluding remarks presented in Section 5. Numerical tests show that for the analysis of smooth and shock-excited responses, both efficiency and accuracy can be improved considerably with the second-order PTI method. For the analysis of shock-excited responses, spatial discretization can provide a reasonable number of model types for any given error tolerances. An appropriate spatial discretization scheme for a given structure is usually required in order to obtain an accurate and meaningful numerical solution, especially for describing the fine detail of traction responses with sharp changes. Under the framework of the PTI method, the numerical damping that is often required in traditional integration schemes is found to be unnecessary, and there is no restriction on the size of time steps, because the PTI method can usually produce results with machine-like precision and is an unconditionally stable explicit method.