Akademik Veri Yönetim Sistemi
International Journal of Mechanical Engineering Education, 2024 (ESCI)
The method known as Galerkin's method, developed in the early twentieth Century for problems in elasticity is still used today, especially for the approximate solution of boundary value problems. In order to apply this method, which is more general and powerful than the Rayleigh-Ritz method, the trial functions used must satisfy all geometric and dynamic boundary conditions accompanying the differential equation. This could be considered a disadvantage of the method. This led with the contributions of H. Leipholz, to the development of the “extended Galerkin's method”. This modification generalized the method to problems where the trial functions do not satisfy some of the conditions, especially dynamic boundary conditions. While the classical Galerkin's method is widely described in the textbooks, the extended Galerkin's method is unfortunately found less frequently. Moreover, with the exception of two reference works, the book of Leipholz and partly an another book of Wunderlich and Pilkey, only the expression is given in all of them. The aim of this paper is two-fold. Firstly, to draw the reader's attention to the extended Galerkin's method, an important and very useful method, in the opinion of the authors of this paper. Secondly, they wish to discuss some important issues, including a critique of the literature. The derivation of the method, its application and related general information will be described with reference to the example of a longitudinally vibrating elastic rod carrying a mass at its free end.