Dynamics of transversely vibrating beams using four engineering theories

In this paper, the full development and analysis of four models for the transversely vibrating uniform beam are presented. The four theories are the Euler-Bernoulli, Rayleigh, shear and Timoshenko. First, a brief history of the development of each beam model is presented. Second, the equation of motion for each model, and the expressions for boundary conditions are obtained using Hamilton's variational principle. Third, the frequency equations are obtained for four sets of end conditions: free-free, clamped-clamped, hinged-hinged and clamped-free. The roots of the frequency equations are presented in terms of normalized wave numbers. The normalized wave numbers for the other six sets of end conditions are obtained using the analysis of symmetric and antisymmetric modes. Fourth, the orthogonality conditions of the eigenfunctions or mode shape and the procedure to obtain the forced response using the method of eigenfunction expansion is presented. Finally, a numerical ex:ample is shown for a. non-slender beam to signify the differences among the four beam models. (C) 1999 Academic Press.