A generalization of the method of lines for the numerical solution of coupled, forced vibration of beams

Abstract

Beam vibrations are encountered in real life and are important to investigate for proper monitoring of the structural health. Closed-form solutions to one-dimensional beam vibration problems are not always available, especially, if the governing equations are nonlinear or have strongly coupled multiple degrees-of-freedom for which, the numerical method is the only solution technique. The method of lines is a numerical technique for solving the initial boundary value problems; however, in the literature, the application is mostly based on simple or lower order linear governing equations with only one degree-of-freedom. Therefore, in this paper, the generalized solution of one-dimensional, axially loaded, coupled, forced beam vibration having multiple degrees-of-freedom is developed by the method of lines which is applicable to any other similar initial boundary value problems. Four different case studies featuring coupled/uncoupled, linear/nonlinear governing equations having single/multiple degree(s)-of-freedom with different types of boundary conditions are presented to demonstrate the applicability of the generalized theory. The models are validated either by theoretical solutions, simulated results, or by published results.