Algorithm Design Of Time Integration For Structural Dynamics Based On Discrete Control Theory

Development of numerical methods for structural dynamic analysis provides a powerful tool for analyzing dynamic performance and conducting dynamical design of structure. Developing new and efficient time integration algorithms for structural dynamic is an important research topic in this field. This paper proposes two families of explicit time integration algorithms for motion equation of structural dynamics by utilizing the discrete control theory.For the two families of new algorithms, the recursive formula of velocity and displacement of CR algorithm and Chang algorithm are adopted, respectively, and the transfer function of algorithms with coefficients using Z transformation of discrete control theory are obtained. Further, the specific expressions of coefficients of recursive formula are derived according to the pole condition. Then, a variable s in the coefficients to control the period elongation is introduced, which is applied to adjust the accuracy of new algorithms.Theoretical analysis indicates that the two families of new proposed explicit time integration algorithms for structural dynamics have same poles, so the accuracy and stability of them are same. The new algorithms possess the properties of second accuracy, zero amplitude decay and self-starting, and their period elongation can be controlled by the variable s, two familes of algorithms without overshoot are proved when s respectively takes any values and 4. Moreover, CR algorithm and Chang (2002) algorithm, Chang (2009) algorithm are respectively special case of the first family of algorithms and the second family of algorithms. Then, the stability limits of the new algorithms applying to linear and nonlinear systems are determined. When the variable s satisfies certain relationships, two families of algorithms for linear systems, nonlinear stiffness softening system are unconditionally stable, greatly improving the computational efficiency for solving the structural response. For nonlinear stiffness hardening system, the time integration algorithms are conditionally stable. Variable interval corresponding to the higher accuracy of the new algorithms is presented. Numerical examples of SDOF system and MDOF system demonstrate that in this interval of variable s, the accuracy of new algorithms are superior to that of Newmark constant average acceleration, CR algorithm, Chang (2002) algorithm and Chang (2009) algorithm.Finally, two families of new algorithms applied to dynamic responses analysis of frame structure under near-fault ground motions illiustrate the effectiveness and stability of the new algorithms. Moreover, the seismic response chracteristics of frame structure subjected to near-fault records with forward directivity and fling-step effects and without pulse are investigated.