The Transition Conditions In The Dynamics Of Elastically Restrained Beams And Plates.

Abstract

This paper deals with two problems:
1) 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.
2) The free transverse vibration of anisotropic plates of different geometrical, generally restrained
boundaries which is restrained against translation along an intermediate line and has an internal hinge
elastically restrained against rotation.
A rigorous and complete development is presented. First, a brief description of several papers
previously published is included. Second, the Hamilton´s 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 mechanical systems is treated. Finally, the
method of separation of variables is used for the determination of the exact frequencies and mode
shapes and/or 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.