Higher mode natural frequencies of stepped beam using spectral finite elements

Abstract

The dynamic behaviour of a structure is of great importance in engineering which is necessary to to accurately predict the dynamic characteristics of the structure. The finite element method (FEM) has been used extensively in structural dynamics. The finite element model may provide accurate dynamic characteristics of a structure if the wavelength is large compared to the mesh size. However, the finite element solutions become increasingly inaccurate as the frequency increases. Although the accuracy can be improved by refining the mesh, this is sometimes prohibitively expensive. The conventional finite element (mass and stiffness) matrices are usually formulated from assumed frequency-independent polynomial shape functions. Because the vibrating shape of a structure varies with the frequency of vibration in reality, the FEM requires subdivision of the structure into finite elements (or a mesh) for accurate solutions. Alternatively, if the shape functions are frequency dependent, then the subdivision may not be necessary. The spectral element method gives frequency dependent dynamic element stiffness matrix regard less of the length or size of the element. Once this Stiffness Matrix for an element is formulated the global Dynamic Stiffness Matrix is obtained by following the procedure similar to that of the Finite Element Method (FEM). The great advantage of such a matrix is that even higher frequencies of a structure can be obtained by considering only few elements thus minimizing the computational cost. In this thesis the higher mode natural frequencies of a stepped beam are obtained. The natural frequency of a stepped beam was found up to the tenth mode by just by considering two spectral elements.