Frequency equation and mode shape formulae for composite Timoshenko beams

Exact expressions for the frequency equation and mode shapes of composite T imoshenko beams with cantilever end conditions are derived in explicit anal ytical form by using symbolic computation. The effect of material coupling between the bending and torsional modes of deformation together with the ef fects of shear deformation and rotatory inertia is taken into account when formulating the theory (and thus it applies to a composite Timoshenko beam) . The governing differential equations for the composite Timoshenko beam in free vibration are solved analytically for bending displacements, bending rotation and torsional rotations. The application of boundary conditions fo r displacement and forces for cantilever end condition of the beam yields t he frequency equation in determinantal form. The determinant is expanded al gebraically, and simplified in an explicit form by extensive use of symboli c computation. The expressions for the mode shapes are also derived in expl icit form using symbolic computation. The method is demonstrated by an illu strative example of a composite Timoshenko beam for which some published re sults are available. (C) 2001 Elsevier Science Ltd. All rights reserved.