Chapter 12 - Multi-degree-of-freedom systems: Free vibrations

Abstract

In this chapter, the free vibrations of MDOF systems are discussed. First, the undamped free vibrations are studied. The relevant eigenvalue problem is formulated and its solution is presented to obtain the eigenfrequencies (natural frequencies) and eigenmodes (mode shapes). The standard as well the generalized linear eigenvalue problems are in-depth studied and the conditions imposed on the mass and stiffness matrices, which ensure positive eigenfrequencies, are elucidated. The orthogonality properties of the eigenmodes are derived for distinct as well as multiple eigenfrequencies and their application to the decoupling of the equations of motion is presented. The Rayleigh method for the approximate computation of the eigenfrequencies is discussed and the properties of the Rayleigh quotient are studied. The damped free vibrations are also discussed and the solution of the ensuing quadratic eigenvalue problem is presented. The free vibration problem for undamped and damped vibrations is solved using the mode superposition method. The concept of proportional damping is introduced, its usefulness in the analysis of the damped free vibrations of structures is demonstrated, and methods for its construction are presented. All presented material is illustrated by appropriately chosen examples. The pertinent bibliography with recommended references for further study is also included. The chapter is enriched with problems to be solved.