A high-order procedure for dynamics of multi-degree-of-freedom damping systems

The lower order schemes are usually adopted in the traditional time-history integration methods, in which time steps must be selected small enough to meet requirements in the accuracy of computational results. We extend the integral differentiation procedure for dynamics of structures to the multi-degree-of-freedom (MDOF) systems. A high-order dynamic method is developed for the MDOF damping systems without more computational efforts. The duration is divided into a few time intervals consisting of ρ equidistant time steps, and the matrix exponential at each time node over the time interval in consideration is obtained by combining precise time-step integration method (PTIM) with the Qin Jiu-shao algorithm, based upon the dynamic solution of Duhamel's integral for MDOF system. The differential quadrature (DQ) rule is employed in the inverse way to obtain the responses from the matrix exponential solution with convolution form interval by interval. According to this high-order dynamic procedure, only a series of matrix multiplications and their recursions are required in the whole analysis process, without solving the equation and performing extra interpolation. The dynamic responses at ρ discrete time instants can be acquired simultaneously (generally ρ=10~15 may be chosen for dynamic analysis in practice), and it indicates the characteristics of efficient and explicit algorithm. Since it is unnecessary to directly calculate the definite integral term in the integral solution of the dynamic response, the difficulty in special processing of the inhomogeneous term of the dynamic equation would be avoided. Numerical examples further show that this approach owns better numerical stability, the result can quickly converge to an accurate solution, and high calculation accuracy can be still achieved even in the case of large time steps. Copyright ©2021 Engineering Mechanics. All rights reserved.