12 - Dynamics and Control of Euler-Bernoulli Beams

Abstract

The chapter presents the governing equations and eigenvalue problem of uniform Euler–Bernoulli beams, and the related MATLAB functions from the toolbox. Eigenvalue problems play an essential role in the dynamic analysis of flexible structures. An eigenfunction associated with a nonzero eigenvalue is called a flexible mode and an eigenfunction associated with a zero eigenvalue is called a rigid-body mode. The chapter provides MATLAB function—setbeam—for specifying the parameters and boundary conditions of an Euler–Bernoulli beam, and for computing the beam eigensolutions. Many approximate methods are available for modeling and dynamic analysis of general structural systems, among which are finite element methods, finite difference methods, and Rayleigh–Ritz methods. The basic idea of the Rayleigh–Ritz method is to approximate the displacement of a nonuniform beam by an N-term series. The admissible functions of a beam are those that are differentiable and satisfy all geometric boundary conditions of the beam. Geometric boundary conditions are those that only involve displacement and rotation.