Spectral Element Approach for the Homogeneous Continuum Representation of a Periodic Lattice Structure

Abstract

Motivated by the apparent accuracy of the spectral element in comparison with the conventional finite element, a new dynamic continuum modeling method is developed by using the spectral element to represent a large periodic lattice beam as an equivalent continuum beam model. As the first step, the transfer matrix for a representative lattice cell of the periodic lattice beam is numerically derived in terms of the continuum degrees of freedom introduced in this paper. The global dynamic stiffness matrix is then obtained by assembling the spectrally formulated dynamic stiffness matrices for the structural elements within the lattice cell. As the second step, the transfer matrix for an equivalent continuum beam element is analytically derived in terms of the unknown equivalent continuum structural properties. Lastly, the two transfer matrices are forced to be equal to each other to determine the equivalent continuum structural properties of the continuum beam model. In this paper, the equivalent continuum structural properties and vibration characteristics of an equivalent continuum beam model by the present method are compared with those by the other existing continuum methods to show that the present continuum model gives vary reliable bending vibration characteristics of the original lattice structures compared to others.