Dynamic stiffness matrix perturbation theory for time-varying structural analysis

Abstract

Matrix perturbation theory (MPT) is widely used in sensitivity analysis and reanalysis of structural eigenvalue problems when small changes are imposed on parameters. However, the conventional MPT based on the finite element method is only applicable to the cases with small parameter variations, and it is difficult to avoid cumulative errors caused by multi-step perturbations. A key issue for the application of MPT is how to evaluate the error of perturbation solution and correct the results during the process of multi-step perturbation solution. By invoking the dynamic stiffness method (DSM), Wittrick-Williams algorithm, and the precise integration method (PIM), in this paper, we propose an improved method with controllable accuracy, self-checking and elimination of accumulated errors, referred to as the dynamic stiffness matrix perturbation theory (DSMPT). In present method, the rationality of the results can be estimated preliminarily according to dynamic stiffness equation, which is not present in the original MPT, can greatly enhance the accuracy of the perturbation results and evaluate the errors. Case studies prove that the DSMPT combined with PIM provides a convenient way to realize accuracy self-inspection and rectification of the perturbation results, and the maximum relative error of eigenvalue and response can be controlled to be smaller than 0.2% and 0.025%, respectively. Besides, DSMPT can also be applied to the analysis of coupled and nonlinear dynamic system with time-varying modal characteristics. Finally, a thermally-induced vibration analysis of tensegrity modules during space deployment is taken as an example to further illustrate the advancements of the DSMPT in solving time-varying dynamic problems. Results show that the DSMPT effectuate an almost sixfold reduction in computational time.