The transition conditions in the dynamics of elastically restrained beams

This paper deals with the free transverse vibration of a non-homogeneous tapered beam subjected to general axial forces, with arbitrarily located internal hinge and elastics supports, and ends elastically restrained against rotation and translation. A rigorous and complete development is presented. First, a brief description of several papers previously published is included. Second, the Hamilton principle is rigorously stated by defining the domain D of the action integral and the space Da of admissible directions. The differential equations, boundary conditions, and particularly the transitions conditions, are obtained. Third, the transition conditions are analysed for several sets of restraints conditions. Fourth, the existence and uniqueness of the weak solutions of the boundary value problem and the eigenvalue problem which, respectively, govern the statical and dynamical behaviour of the mentioned beam is treated. Finally, the method of separation of variables is used for the determination of the exact frequencies and mode shapes and a modern application of the Ritz method to obtain approximate eigenvalues. In order to obtain an indication of the accuracy of the developed mathematical model, some cases available in the literature have been considered. New results are presented for different boundary conditions and restraint conditions in the internal hinge.