Explicit analytical expressions for frequency equation and mode shapes of composite beams

A systematic procedure for the derivation of exact expressions for the freq uency equation and mode shapes of composite beams undergoing free vibration is presented by using the symbolic computing package REDUCE. The effect of material coupling between the bending and torsional modes of deformation, which usually exists in composite beams due to ply orientation, is taken in to account while developing the theory. The governing differential equation s of motion of the bending-torsion coupled composite beam are solved analyt ically for bending displacements and torsional rotations in free vibration. For subsequent developments, the important case of the cantilever beam is chosen because of its application to aircraft wings. The boundary condition s for displacements and forces for the cantilever are imposed and the frequ ency equation is obtained, but first in the form of a determinant, and then , in the form of an explicit algebraic expression. The expressions for the mode shapes are also derived in explicit analytical form. The method is dem onstrated by an illustrative example of a composite beam for which some com parative results are available in the literature. The future potential of t his method, particularly in the context of aeroelastic optimisation of comp osite wings, is considerable because it is very accurate, computationally e fficient and importantly, free from ill-conditioning problems usually assoc iated with numerical matrix manipulation. (C) 2001 Elsevier Science Ltd. Al l rights reserved.