Collocation approaches to the mathematical model of an Euler–Bernoulli beam vibrations

Abstract

Taylor-Matrix and Hermite-Matrix Collocation methods are presented to obtain the modal vibration behavior of an Euler–Bernoulli beam. The methods provide approximate solutions in the truncated Taylor series and the truncated Hermite series by using Chebyshev–Lobatto collocation points and operational matrices. The approaches are applied to the well-known transverse vibration model of a simply-supported Euler–Bernoulli type beam. The beam is assumed under the effect of external harmonic force with spatially varying amplitude. Firstly, the model problem is transformed into a system of linear algebraic equations. Then the approximate solution is computed by solving the obtained algebraic system. The proposed methods are compared with the exact solutions. Mode shapes of the first three fundamental frequencies are obtained for each approach. The convergence behaviors of transverse displacements and absolute errors are calculated for each mode. The numerical results are shown and compared. Based on the given figures and tables, one can state that the methods are remarkable, dependable, and accurate for approaching the mentioned model problems.