TABLE OF CONTENTS There are several popular methods that make use of a set of modes or other functions to approximate the dynamic behavior of systems. In this section, without going into details on the theories associated with this subject, we will illustrate within the framework already established how one can use a truncated set of modes or other set of functions to obtain an approximate solution. Details of the theories behind modal approximation methods may be found in texts that treat structural dynamics at the graduate level. The two main approaches are Galerkin's method, applied to ordinary or partial differential equations, and the Ritz method, applied to the principle of virtual work. These two methods yield identical results in certain situations. Thus, if time is limited it would only be necessary to discuss one of the two methods to give the student an introduction to the method and an appreciation of results that can be obtained this way. The Ritz method is to be preferred in the present context because of the ease with which it can be presented within the framework of Lagrange's equations. Nevertheless, both of these methods will be presented at a level suitable for undergraduate students.
Building on the earlier treatment, we start with Lagrange's equations, given by
| (2.280) |  |
where the Lagrangean, L = K P, the total kinetic energy is K, the total potential energy is P, n is the number of generalized coordinates...
