Vibration of Continuous Systems, 2nd Edition

11Transverse Vibration of Beams

11.1 INTRODUCTION

The free and forced transverse vibration of beams is considered in this chapter. The equations of motion of a beam are derived according to the Euler–Bernoulli, Rayleigh, and Timoshenko theories. The Euler–Bernoulli theory neglects the effects of rotary inertia and shear deformation and is applicable to an analysis of thin beams. The Rayleigh theory considers the effect of rotary inertia, and the Timoshenko theory considers the effects of both rotary inertia and shear deformation. The Timoshenko theory can be used for thick beams. The equations of motion for the transverse vibration of beams are in the form of fourth‐order partial differential equations with two boundary conditions at each end. The different possible boundary conditions of the beam can involve spatial derivatives up to the third order. The responses of beams under moving loads, beams subjected to axial force, rotating beams, continuous beams, and beams on elastic foundation are considered using the thin beam (Euler–Bernoulli) theory. The free vibration solution, including the determination of natural frequencies and mode shapes, is considered according to these three theories.

11.2 EQUATION OF MOTION: THE EULER–BERNOULLI THEORY

The governing equation of motion and boundary conditions of a thin beam have been derived by considering an element of the beam shown in Fig. 11.1(b), using Newton's second law of motion (equilibrium approach), in Section 3.2. The ...