Continuous Systems

Abstract

The motion of an elastic body can only be described approximately by means of the multibody system method or the finite element method. With an elaborated modelling of infinitesimal volume elements, the elastic continuum has infinitely many degrees of freedom, while its motions are determined locally by partial differential equations. First the local Cauchy equations of motion for a free continuum and for the elastic beam as a continuum with internal constraints are given, both of which must be supplemented with boundary conditions. Global equations of motion are then obtained with the eigenfunctions, which must satisfy the boundary conditions. D’Alembert’s principle is applied again in this context. The global equations of motion now describe the motion of an elastic body exactly. However, this involves solving an infinite-dimensional eigenvalue problem, what is feasible only for geometrically simple bodies. For this reason, continuous systems are not as important in engineering practice as the aforementioned approximation methods. If we restrict ourselves to a finite number of eigenfunctions, such as is the case in engineering modal analysis, then continuous systems represent also an approximation method.