General Formulation of Eliminating Unusual Amplitude Growth for Structure-Dependent Integration Algorithms

Abstract

An unusual amplitude growth of the steady-state response for structures with high natural frequencies was found in explicit or semi-explicit structure-dependent integration algorithms with unconditional stability, second-order accuracy and no overshoot. This paper proposes a general formulation for eliminating such an unusual amplitude growth, by incorporating the load-dependent term into the displacement recursive formula without changing the numerical properties of the integration algorithm. Compared with the existing formulation using the local truncation error for eliminating the unusual amplitude growth, the proposed formulation has the advantages of less symbolic operations, while naturally including the existing formulation. In addition, it is observed that the coefficients of the load-dependent term are proportional to the limit values of the displacement coefficients in the displacement recursive formula as the natural frequency of the system tends to infinity. Then, the general formulation of the load-dependent term is tested for 15 structure-dependent integration algorithms to verify its correctness. Finally, numerical examples of linear and nonlinear multiple-degrees-of-freedom systems illustrate that the general formulation can remove effectively and conveniently the unusual amplitude growth for dynamic response analyses.