Dynamic Stiffness Formulation and Free Vibration Analysis of a Moving Timoshenko Beam

Abstract

A moving Timoshenko beam is analyzed for its free vibration characteristics using the dynamic stiffness method. First the governing differential equations of motion in free vibration of a moving Timoshenko beam are derived using Hamilton’s principle. The derivation gives the expressions for shear force and bending moment from the natural boundary conditions which are consequential of the Hamiltonian formulation. Next the dynamic stiffness matrix is developed by solving the governing differential equations of motion and then eliminating the arbitrary constants from the general solution so as to form the force-displacement relationship of the harmonically vibrating moving Timoshenko beam. The resulting dynamic stiffness matrix, which turns out to be a Hermitian matrix, is used in conjunction with the Wittrick-Williams algorithm to determine the natural frequencies and mode shapes of some examples. In the analysis, simply supported, fixed-simply supported and fixed-fixed end conditions are considered. Results using the Timoshemko theory are compared and contrasted with the corresponding results obtained from the Bernoulli-Euler theory. The critical speeds for all three end conditions are illustrated. Representative mode shapes are presented for different moving speeds. Finally, some conclusions are drawn.