RBF-based meshless method for the free vibration of beams on elastic foundations

In this paper, solution is obtained for the free vibration differential equations of motion of an axially loaded beam on elastic foundation using a meshless method. Use is made of the multiquadrics radial basis function (RBF) in obtaining the numerical solution for four different cases: (1) one end clamped, the other end simply supported; (2) both ends clamped; (3) both ends simply supported; and (4) a simple beam on elastic foundation with end rotational springs. The approach is easier to implement and program as compared to grid/mesh-based methods such as the finite difference method (FDM) and the finite element method (FEM). Accuracy of the results obtained using the proposed method was verified using the analytical results available in the literature for the first three cases considered. Numerical results of the fourth case were aimed at justifying the use of the numerical scheme for a problem whose analytical solution is not readily available and to show the high accuracy of the RBF method. The results prove that the method require much less number of nodes to converge to the correct solution as compared to FDM.