Dynamic stiffness matrix development and free vibration analysis of a moving beam

The dynamic stiffness matrix of a moving Bernoulli–Euler beam is developed and used to investigate its free flexural vibration characteristics. In order to develop the dynamic stiffness matrix, it is necessary to derive and solve the governing differential equation of motion of the moving beam in closed analytical form. The solution is then used to obtain the general expressions for both responses and loads. Boundary conditions are applied to determine the constants in the general solution, leading to the formation of the frequency dependent dynamic stiffness matrix of the moving beam, relating the amplitudes of the harmonically varying loads to those of the corresponding responses. The application of the resulting dynamic stiffness matrix using the Wittrick–Williams algorithm is demonstrated by some illustrative examples. Numerical results for both simply supported and fixed–fixed end conditions of the beam are discussed, and wherever possible, some are compared with those available in the literature.