A series solution for the free vibrations of thin-walled composite beams

Based on series solutions, the free vibration analysis of thin-walled composite beam with channel sections is performed. For this, the dynamic stiffness matrix is presented using the homogeneous form of the simultaneous ordinary differential equations. The general theory for the vibration analysis of composite beam with arbitrary lamination including the restrained warping torsion is developed by introducing Vlasov's assumption. Next, the equations of motion and force-displacement relationships are derived from the energy principle and explicit expressions for displacement parameters are presented based on power series expansions of displacement components. Then the exact dynamic stiffness matrix is determined using force-displacement relationships. In addition, the finite element model based on Hermitian interpolation polynomial is developed. In order to verify the accuracy and validity of this study, numerical solutions are presented and compared with the analytical solutions from other researchers and the finite element solutions using the Hermitian beam elements. Particularly, the effects of modulus orthotropy and boundary conditions on the vibrational behavior of thin-walled composite beam with various fiber orientations are investigated.