Journal of Sound and < ibration (1999) 224(3), 519 } 539 Article No. jsvi.1999.2198, available online at http://www.idealibrary.com on A TECHNIQUE FOR THE SYSTEMATIC CHOICE OF ADMISSIBLE FUNCTIONS IN THE R AYLEIGH }RITZ METHOD M. AMABILI AND R. GARZIERA Department of Industrial Engineering, ;niversity of Parma, < iale delle Scienze, Parma, 43100 Italy (Received 4 December 1998, and in ,nal form 1 February 1999) A simple and systematic choice of admissible functions, which are the eigenfunctions of the closest, simple problem extracted from the one considered, is proposed. The extracted problem must be &&less-constrained'' than the original one; in other words it must be a problem where some constraints or other complications (e.g. added masses) are eliminated. Elastic constraints replace the eliminated rigid ones. The convergence is also analyzed. This approach has practical applications when it is possible to extract a problem with eigenfunctions expressed in closed form. It also allows a very simple calculation of the potential energy of the system. Solutions for several cases involving beams are given in order to show the power of the method. Application of the method to circular plates and shells is also addressed.  1999 Academic Press 1. INTRODUCTION The Rayleigh}Ritz method [1] assumes de#ection shapes in the form of a linear combination of functions which satisfy at least the geometrical boundary conditions of the vibrating structure. Courant [2] addressed the method for solving problems having rigid boundaries by treating such problems as limiting cases of free boundary problems, for which the admissible functions can be simpler. This technique introduces arti"cial translational and rotational springs at the free boundaries; the sti!ness of these springs can be assumed su$ciently high to simulate rigid constraints with the required accuracy. Applications of this technique were made, e.g., by Kao [3], Mizusawa et al. [4], Yuan and Dickinson [5] and Cheng and Nicolas [6]. Additional references are given by Laura [7]. Warburton and Edney [8], Gorman [9], Gelos and Laura [10], and Laura and Gutierrez [11] applied the Rayleigh}Ritz method to structures with elastic constraints. In these studies, a large number of di!erent admissible functions were used. In fact, the most critical aspect of the Rayleigh}Ritz method is regarding the choice of appropriate admissible functions. If these functions form a complete set, computed natural frequencies converge to actual ones from above. The nature of natural frequencies obtained by using the Rayleigh}Ritz method and their dependence on the nature of the assumed shape functions was investigated by Bhat [12]. Dickinson and Li [13] introduced a set of admissible functions derived from 0022-460X/99/280519#21 $30.00/0  1999 Academic Press