Theory of longitudinal vibrations of an isotropic bar based on the Midlin-Herrmann model

Abstract

Modern theories of one-dimensional bar vibrations account for lateral effects, which are substantial in the case of relatively thick bars. For example, in the Rayleigh-Love and Rayleigh-Bishop models the lateral displacements are supposed to be proportional to the product of longitudinal strain of the bar, its Poisson ratio and the distance from the neutral line of the cross-section. In the Mindlin-Herrmann model the lateral displacements are independent of longitudinal stress and Poisson ratio and proportional to the product of a new dependent function and the distance from the neutral line of the bar. Hamilton’s variational principle is used for correct formulation of the boundary conditions. In this approach a system of equations and possible boundary conditions are obtained. In this case the mathematical model of the bar is described by a system of two partial differential equations of second order, which could be transformed to a single partial differential equation of the fourth order. It is shown how a new Lagrangian may be calculated so as to directly obtain the fourth order equation of the model by application of the Hamilton variational principle. Another major advantage of the variational approach is in the natural formulation of orthogonality conditions for eigenfunctions. Two orthogonal conditions are proven and used to derivation the Green’s function in which the general solution of the problem is formulated. The main theoretical results of the paper are as follows: formulation and proof of two types of orthogonality conditions, presentation of a new Lagrangian in terms of the conventional strain and kinetic energy as well as an energy of accelerations of the bar, and derivation of the general solution in terms of the Green’s function