Free vibration of rods, beams and frames using spectral element method

When the structure vibrates with high frequency, the finite element method needs to be modeled with very large number of elements to capture all necessary frequencies of higher modes. The exact solution of the governing differential equations for the vibration problems with higher modes of vibration can be formed by using the frequency dependent shape functions and thereby formulating the frequency dependent Dynamic Stiffness Matrix for an element from which the global Dynamic Stiffness Matrix is obtained by following the procedure similar to that of the Finite Element Method (FEM). From this global DSM the higher modes of free vibration frequencies have been obtained for rods and beams using only 2 spectral elements and for frames considering each member as an element