A Numerically Efficient Frequency Domain Method for Analysis of Non-Linear Multi Degree of Freedom Systems

Abstract

Linear structural models contain mass, stiffness and damping matrices as well as a forcing vector. Once these matrices and the forcing vector are known, the response can be calculated through the methods of linear algebra. The system matrices of the linear model do not contain any terms that depend on the system response vector, therefore the calculation of the system response do not require an iterative procedure. On the other hand, frequency domain analysis of non-linear structural models generally contain terms that are non-linear functions of the system response vector. Therefore these terms make the system matrices become dependent on the non-linear system response vector itself. This forces one to employ an iterative solution. The disadvantage of the iterative solution is more pronounced when large degree of freedom systems are analyzed. For each frequency value, the iterative procedure requires solution of large system of equations to be evaluated several times until the iterative procedure converges to a value. Therefore the numerical cost significantly increases as the model size gets larger. This study introduces a method which requires one-time calculation of response of the linear part of the system. After finding the response of the linear part, no further matrix inversions are needed to iteratively find the non-linear system response. In the case studies it will be shown that the proposed method has significant computation time advantages to conventional time domain and frequency domain methods for solution of large non-linear structural models. In order to employ this numerically efficient method, describing function theory is used to obtain a non-linearity matrix which contains the non-linear terms. Also, a computationally efficient recursive method is used to evaluate the inverse of the sum of the linear system impedance and the non-linearity matrix. In order to employ this recursive method the non-linearity matrix is decomposed into summation of rank-one matrices.