Frequency response functions and modal analysis of general nonviscously damped dynamic systems with and without repeated modes

Abstract

This paper seeks to examine some important outstanding theoretical issues of general nonviscously damped vibration systems. Exact frequency response functions (FRFs) have been developed based on Cauchy's residue theorem for the case of repeated eigenvalues with arbitrary multiplicities. The new theory developed has not only extended the classical mode superposition principle, but also laid the necessary theoretical foundation for the modal analysis of nonviscously damped systems whose eigenvalues are nondistinct. Effective numerical methods for the computations of elastic and nonviscous modes are suggested. The unique feature, contribution and significance of nonviscous modes to FRFs have been examined and discussed. Since nonviscous modes are real and are hence similar in characteristics to structural rigid-body modes with zero frequency, a new and accurate method has been developed to lump their contributions to FRFs into a single artificial rigid-body mode, thereby eliminating the necessity of computing them which is numerically challenging. Traditional restrictions of symmetry have not been imposed on system matrices and neither state-space nor additional coordinates have been employed throughout theoretical development. Numerical examples are given to illustrate the new theory and methods developed in the paper.