Vibration of Continuous Systems, 2nd Edition

6Solution Procedure: Eigenvalue and Modal Analysis Approach

6.1 INTRODUCTION

The equations of motion of many continuous systems are in the form of nonhomogeneous linear partial differential equations of order 2 or higher subject to boundary and initial conditions. The boundary conditions may be homogeneous or nonhomogeneous. The initial conditions are usually stated in terms of the values of the field variable and its time derivative at time zero. The solution procedure basically involves two steps. In the first step, the nonhomogeneous part of the equation of motion is neglected and the homogeneous equation is solved using the separation‐of‐variables technique. This leads to an eigenvalue problem whose solution yields an infinite set of eigenvalues and the corresponding eigenfunctions. The eigenfunctions are orthogonal and form a complete set in the sense that any function that satisfies the boundary conditions of the problem can be represented by a linear combination of the eigenfunctions. This property constitutes what is known as the expansion theorem. In the second step, the solution of the nonhomogeneous equation is assumed to be a sum of the products of the eigenfunctions and time‐dependent generalized coordinates using the expansion theorem. This process leads to a set of second‐order ordinary differential equations in terms of the generalized coordinates. These equations ...