Applications of the singular value decomposition in dynamics

• Published in: Computer methods in applied mechanics and engineering Vol. 103; no. 1; pp. 161 - 173
• Main Author: Meijaard, J.P.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.03.1993; Elsevier
• Subjects: Exact sciences and technology; Fundamental areas of phenomenology (including applications); Physics; Solid dynamics (ballistics, collision, multibody system, stabilization...); Solid mechanics; N body system; Dynamical system; Singular value decomposition
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/0045-7825(93)90044-X

The singular value decomposition of a general rectangular matrix can serve as a means for determining its numerical rank and for solving underdetermined and overdetermined linear systems in a least-squares sense. This decomposition is used in all kinds of applications e.g. in control theory and statistics. This article presents some applications of this readily available tool for determining solutions of underdetermined systems which arise in the study of dynamical mechanical systems. First, the theory of the singular value decomposition is exposed briefly. Then, it is shown how periodic solutions of Hamiltonian, conservative systems can be determined in a direct way. Finally, a path following method for the continuation of stationary and periodic solutions and bifurcation points is described. The usefulness of these procedures is shown in examples of a rotor with non-linear support stiffness and a system which consists of an array of ten non-linearly coupled oscillators.