On the methods of Galerkin, Ritz and Krylov-Bogoliubov in the theory of non-linear vibrations

Abstract

The conditions for equivalence of the Galerkin method and the Ritz minimizing method are reviewed. It is then shown that Galerkin's method may also lead to a result which, for steady state vibrations, is the same as the first approximation of Krylov-Bogoliubov. Both the Ritz method and the first approximation of Krylov-Bogoliubov may therefore be thought of as special cases of the general Galerkin method. The difference between them lies in the different ways in which the describing differential equations are expressed, in the different forms of the approximate solution used, and in the different choice of Galerkin weighting functions. As an example of these differences, the free vibration of a centrifugal pendulum is considered. This is a two degrees-of-freedom problem with a known exact solution. The exact solution is compared with approximate solutions by the Ritz minimizing method and the method of Krylov-Bogoliubov. It turns out that the two methods give results which are practically indistinguishable from each other, and very close to the exact answer.