Choices of the Critical Frequency for φ in TL-φ Algorithms When Applied to Multi-Degree of Freedom Systems

Abstract

TL-φ algorithms are newly developed explicit structure-dependent integration algorithms utilized for solving the temporally discretized equations of motion. In contrast to the existing algorithms, the most significant improvement of TL-φ algorithms is in diminishing the amount of period errors by introducing a precorrection coefficient φ into the integration parameters of TL-φ algorithms, which is related to the critical frequency of a system. In the previous work, the fundamental frequency of the system is deemed to be the critical frequency, so that φ is a constant scaling corresponding to the fundamental frequency for both single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) systems. However, for a MDOF system, the first mode may not contribute to the total response more than other ones under a given external excitation, calculating φ only by the fundamental frequency will underestimate the contribution of the higher-frequency modes to structural dynamics. In this paper, choices of the critical frequency for φ when applying TL-φ algorithms to MDOF systems are investigated thoroughly. By considering the initial structural properties of the system and the frequency characteristics of the external excitation simultaneously, a calculation criterion of φ for MDOF systems under specific external excitations is proposed. Four numerical examples with different initial structure properties and loading conditions are designed, and the results demonstrate that the proposed criterion can be potentially used to solve structural dynamic problems of MDOF systems with a more desirable numerical dispersion performance.