Methods of Analysis for Vibration Problems
This chapter considers a linear combination of the several assumed functions satisfying some boundary condition with the hope to obtain a closer approximation to the exact values of the natural modes of vibration. It shows the methodology for finding frequency by this method for a string. Methods like finite element methods, boundary integral equation methods, finite difference methods, and the method of weighted residuals have made it possible to handle a variety of vibration problems. The displacement of the structure under forced excitation is approximated by the sum of a limited number of normal modes of the system multiplied by normal coordinates. The maximum kinetic and potential energies of the system must be equal since no energy is lost and no energy is fed into the system over one cycle of vibration. This gives us a quotient known as Rayleigh quotient.