Obtaining mode shapes through the Karhunen-Loève expansion for distributed-parameter linear systems

Abstract

The Karhunen-Loève expansion is a powerful spectra technique for the analysis and synthesis of dynamical systems. It consists in decomposing a spatial correlation matrix, which can be obtained through numerical or physical experiments. The decomposition produces orthogonal eigenfunctions or proper orthogonal modes, and eigenvalues that provide a measurement of how much energy is contained in each mode. The relation between KL modes and mode shapes of linear vibrating systems has already been derived and demonstrated for two and three dofs mass-spring-damper systems. The purpose of this paper is to extend this investigation to more complex distributed-parameter linear systems. A plane truss and a simply supported plate subjected to impulsive forces, commonly used in modal analysis are studied. The resulting KL modes are compared to the analytical mode shapes. Damping and random noise effects in the procedure performance are evaluated. Two methods for indirectly obtaining natural frequencies are also presented.