Chapter 14 - Multi-degree-of-freedom systems: Forced vibrations

Abstract

This chapter integrates the dynamic analysis of the MDOF systems. It presents the mode superposition method for solving the equations for the forced vibrations. It elucidates the participation of the eigenmodes in the mode superposition method and investigates the error due to the truncation of higher modes. It describes the static correction method, which reduces the task of the dynamic analysis of structures. It presents the reduction of the dynamic degrees of freedom by eliminating the rotational degrees of freedom (static condensation) or taking into account the kinematic constraints, such as the diaphragmatic behavior of slabs in buildings and the omission of the axial deformation. The Rayleigh-Ritz method and its application to the reduction of the degrees of freedom are discussed and methods for the selection of Ritz vectors are presented. The advantages of the Rayleigh-Ritz method over the mode superposition method are discussed by comparing results obtained by the two methods. The multiple and uniform support excitation are presented, providing the necessary knowledge to study the response of the structure due to ground motion (seismic excitation). The response spectrum method is analyzed and it is applied to determine the dynamic response of structures. The chapter concludes by presenting the most efficient methods for the numerical solution of the equations of motion, linear and nonlinear. All issues are illustrated by appropriately chosen and well-corroborated examples. A related bibliography recommended for further study is also included. The chapter is enriched with problems to be solved.