Chapter 4 - Energy methods in vibrations

Abstract

In this chapter, the energy methods explained in the previous chapters are extended to deal with vibration problems and formulating the Eigen value problem. The methods will be presented and the formulation of total energy is achieved from which the frequency equations are deduced. Different examples are chosen to demonstrate the effectiveness of using energy methods in vibration analysis. The presented examples cover the application of Rsyleigh’s principles to two degrees of freedom system, axial free vibration of beams, and cantilever beam bending vibrations. The frequency equations are formulated using Rayleigh Ritz from which the natural frequencies are obtained directly. The kinetic and strain energies were presented for plate bending applications presented in Chapter 3, Application of Energy Methods to Plate Problems. The applications cover rectangular plate shapes with different boundary conditions using Galerkin’s, Galerkin’s Vlasov, and Ritz methods. The frequency equations will be derived for a vibrating taper beam and a non-uniform circular shaft carries a disk as practical applications. This chapter also includes problems covering different applications.