Nonlinear normal modes and proper orthogonal decomposition in inertially-coupled nonlinear conservative systems

In this study, the relationship between the nonlinear normal modes (NNM) and the proper orthogonal decomposition (POD) modes is explored through the nonlinear dynamics of the spring-mass-pendulum system. For this inertially-coupled two degrees-of-freedom nonlinear system, the principle of 'least action' is used to obtain the boundary value problem that governs the NNMs in configuration space. For a specified total energy and other system parameters, shooting method is used to solve the boundary value and numerical approximations to the NNMs are constructed. It is known that various bifurcations arise in the NNMs of the system as the system parameters and the total energy are varied. The proper orthogonal decomposition (POD) modes are then explored. In the case of a linear system with an identity mass matrix, it is seen that as the snapshot number N -> infinity and the total time record length T -> infinity, the POD mode approaches the corresponding linear modal eigenvector. Now, data from simulations of the spring-mass-pendulurn system is used, and it is shown that the POM (or a POD mode) is a linear curve in the configuration space which represents the principle axis of inertia based at the mean of the data in the configuration space. This is the least squares approximation of the data. Finally, the nonlinear generalization of PCA - VQPCA is used to reanalyze the same data for the spring-mass-pendulurn system. In the VQPCA analysis, a modified Linde-Buzo-Gray (LBG) algorithm, suitable for the modal analysis, is developed. The superiority of VQPCA over PCA in capturing the NNM is clearly seen in the simulation results.