Dynamic Stiffness Method for Free Vibration of Beams and Frameworks Using Higher-Order Shear Deformation Theory

Abstract

Abstract. The dynamic stiffness method for free vibration of beams and frameworks is developed using a higher-order shear deformation theory. Starting with the displacement field, the potential and kinetic energies of the beam in flexural vibration are first formulated. Then, Hamilton's principle is applied to derive the governing differential equations and the associated natural boundary conditions. Next, the differential equations are solved to obtain the expressions for flexural displacement, bending rotation, and the first derivative of the flexural displacement. The expressions for the shear force, bending moment, and the higher-order moment are obtained from the natural boundary conditions resulting from the Hamiltonian formulation. Finally, the force vector comprising the amplitudes of the shear force, bending moment, and the higher-order moment is related to the amplitudes of the displacement vector comprising the flexural displacement, bending rotation, and the first derivative of the flexural displacement through the frequency-dependent dynamic stiffness matrix. The dynamic stiffness matrix for axial motion that is uncoupled from the flexural motion is now implemented in the dynamic stiffness matrix in flexural motion to analyze individual beams and frameworks for their free vibration characteristics by applying the Wittrick–Williams algorithm. Illustrative examples are given, and significant conclusions are drawn.