Chapter 3 - Methods of Analysis

Abstract

This chapter discusses several analytical methods to determine the frequencies and mode shapes of laminated composite structures. The Rayleigh and Ritz methods are among the most common approximate methods used in the vibration analysis of continuous systems. A displacement field is assumed in both methods. The coefficients of the displacement field are completely determined beforehand in the method of Rayleigh. In the Ritz method, undetermined coefficients are used in the displacement field. The displacement field is then substituted in the energy functional that is Lagrangian. The Lagrangian is then minimized by taking its derivatives with respect to these coefficients and making them equal to zero. The Galerkin method is a special case of the general weighted residual methods. In these methods, the governing differential equations and corresponding boundary conditions are needed. The finite element method (FEM) overcomes the difficulties that the Ritz and Galerkin methods have in dealing with various boundary conditions and relatively complex shapes.