Continuous Systems

Abstract

This chapter consists of an introduction to the continuous (distributed) systems governed by partial differential equations. The chapter begins with the planar vibration of Euler-Bernoulli beams; the partial differential equation is derived from the equilibrium equations and also from Hamilton’s principle. The effect of axial prestress is discussed. Next, the analytical solution of the free vibration is found for various boundary conditions, using a special set of functions which decouple the boundary conditions; the orthogonality relationships of the mode shapes are established and used to perform a modal decomposition. This chapter also discusses the vibration of a string, the axial vibration of a bar and the bending vibration of thin Kirchhoff plates; the analytical solutions for the free vibration are developed for two frequent configurations: a simply supported rectangular plate and a clamped circular plate. The chapter concludes with a discussion of rotating modes of axisymmetric plates and the response of a disk to a rotating point force. A set of problems is proposed at the end of the chapter.

Le motif seul fait le mérite des

actions des hommes, et le

désintéressement y met la

perfection.

La Bruyère, Les Caractères (1688)