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

Abstract

A systematic procedure for the derivation of exact expressions for the frequency 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 into account while developing the theory. The governing differential equations of motion of the bending–torsion coupled composite beam are solved analytically 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 conditions for displacements and forces for the cantilever are imposed and the frequency 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 demonstrated by an illustrative example of a composite beam for which some comparative results are available in the literature. The future potential of this method, particularly in the context of aeroelastic optimisation of composite wings, is considerable because it is very accurate, computationally efficient and importantly, free from ill-conditioning problems usually associated with numerical matrix manipulation.