Free Vibration Analysis of Continuous Beams

Abstract

An analytical method for determining eigenvalues of continuous beams is developed by using a general solution for the Bernoulli‐Euler differential equation. This method results in an eigenvalue problem in which the solution of a transcendental equation containing trigonometric and hyperbolic functions is obtained, and it leads to an exact solution. Also, the approximate method based on the finite element approach is presented. The mathematical relationship between the exact and the approximate methods is discussed, and the accuracy of the eigenvalues obtained by these methods is investigated. Some typical continuous beams are analyzed to illustrate the applicability of the lumped, consistent, and continuous mass methods, and the computed results are given in tabular form.