A novel approach in analysing the vibration of elastic body systems is presented and discussed in this paper. This method uses a meshless spatial approximation based only on nodes, which constitutes an advantage over the "nite element method. As it is applicable to an elastic body with arbitrary shape, it also has advantages over the classical Rayleigh}Ritz method and its extensions. The paper is organized as follows: in Section 2, Hamilton's principle is used for obtaining the equations of motion for a three-dimensonal simply connected elastic body. In Section 3, the development of weighted base functions for the moving leastsquares interpolant is described. In Section 4, the mass and sti!ness matrices are derived using the previous results. In Section 5, a system synthesis description is given, in which we also develop the approach used for integrating the boundary conditions. Finally in Section 6, the theoretical results are validated by case studies chosen to highlight various features of the approach, including optimization of the shape function parameters. The particle method has already been used for structural dynamics (Liu et al., International Journal for Numerical Methods in Engineering 1995;38:1655}1679), but for free vibration of beams and plates, to the authors knowledge, this paper is the "rst application of the element-free method to modal analysis. Figure 2. Frequency parameters for SS beam, with k"2, using an exponential function 6.1.1. One-dimensional tests. (a1) Optimization of parameters k, and : In order to perform this optimization, the problem of a beam simply supported at both ends, and for which an analytical solution is forthcoming, was examined. In a "rst analysis, parameter k was set to 2, which corresponds to a Gauss distribution, and we varied parameters and , solving the Galerkin equation (45) for each ( , ) pair. Figure 2 shows the solution surfaces for the "rst 16 NUMERICAL MODAL ANALYSIS 9 Figure 2. (Continued)frequencies: we observe that for all modes, there is a region of ( , ) for which the solutions tend towards a constant approximately equal to the analytical solution, and hence an in"nity of ( , ) values are acceptable. We chose for the value 1, and taking a vertical section of the surface at "1, Figure 3 results providing the evolution versus of the "rst 16 frequencies. Analysis of these curves shows that, from "3 onwards, a good solution is obtained. The lowest possible 10