Two-step Component Mode Synthesis with convergence for the eigensolution of large-degree-of-freedom systems

Abstract

Component Mode Synthesis (CMS) is a dynamic substructuring technique for the approximate eigensolution of large-degree-of-freedom (dof) systems divisible into two or more components. System synthesis using component modes results in approximate eigenparameters; in general, using more component modes in synthesis improves the approximation. A two-step CMS approach is developed in this research. The first step involves system synthesis using the minimum number of component modes required to obtain approximate eigenvalues up to a preselected cut-off frequency, and the second step introduces additional component modes in a convergence scheme operating on the system eigenparameters calculated in the first step.

The method is developed using constraint modes for system synthesis. The eigenvectors resulting from the initial eigenproblem solution are used to transform the system matrices; this results in a non-linear eigenvalue problem which is solved by a modified shooting method. A perturbation approach is adopted to derive a convergence scheme in which successive iterations are performed for the eigenvalue and eigenvector in each step. A procedure for selecting initial values for the convergence scheme is presented. A condensation procedure is also developed for economical synthesis of systems with many connection coordinates compared to normal mode coordinates. Advantages of the present method include minimal order of system matrices, savings in computation time and a knowledge of the accuracy of the eigenparameters. Numerical examples of spring-mass systems and systems constructed with beam and shell elements are included to demonstrate the applicability of the method and results are compared with full-system eigensolution.

The method is also applicable to the synthesis of generally damped systems. Conventional state-space formulation is used to cast the equations of motion in first-order form for each component. Complex modes are combined with an appropriately defined set of constraint modes to give the mode superset for each component. The method evolves similar to the method for undamped systems. A numerical example is included to demonstrate the method and results from CMS are compared with those obtained by full-system eigensolution. Also, the applicability of the method in solving non-linear eigenvalue problems in structural dynamics is discussed, and examples are included for demonstration.