Keywords: free vibration, dynamic stiffness method, Wittrick-Williams algorithm, higher-order shear deformation theory.

Using a higher order shear deformation theory, the dynamic stiffness element for a beam is developed to investigate its free vibration characteristics. Hamilton's principle is applied to derive the governing partial differential equations in free vibration utilizing the kinetic and potential energies of the beam. The potential energy is based on a higher order shear stress distribution of generic type which dispenses with the approximation of shear correction factor used in the Timoshenko theory. The natural boundary condition obtained from Hamiltonian formulation gives the expressions for axial force, shear force, bending moment and higher order moment that are required in the dynamic stiffness formulation. For harmonic oscillation, the partial differential equations are transformed into ordinary ones and then solved in closed analytical form. Next the dynamic stiffness matrix is constructed by relating the amplitudes of forces and displacements at the ends of the beam. The Wittrick-Williams algorithm is used to solve the eigenvalue problem resulting from the dynamic stiffness matrix of the beam. The theory developed is sufficiently general and can be extended to any order of polynomial or other types of shear stress distributions. However, for illustrative purposes a parabolic shear distribution is used in this paper.

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