Abstract :
The relationship between the Helmholtz equations, describing steady-state vibrations of elastic bodies, and the Sturm-Liouville problem [1, 2] predetemines the transformation of [he boundary conditions to an infinite system of linear algebraic equations [3,4]. The computational efforts needed for the numerical implementation of the analytical solution for bodies and structures approximated by canonical bodies are the less the more complete the preliminary analytical transformations. In many cases, the construction of infinite systems by analytical methods is possible only when based on additional orthogonality conditions for a restricted class of Sturm-Liouville problems and the boundary conditions in the form corresponding to this orthogonality. These orthogonality conditions and boundary conditions are given in the present paper. In particular, such an approach is very efficient when using the first of the two possible schemes of approximation of bodies of revolution by cylindrical bodies [5]. In this case, it is possible to extend the class of structures the natural frequencies of which can be defined by methods of the theory of elasticity. Natural frequencies of a body of revolution computed in accordance with both schemes of approximation of this body by cylindrical bodies are compared. This comparison defines the application area of each approximation scheme depending on the body shape. Solutions of unsteady problems can be obtained by expansion into series in terms of natural modes. However, there are also other familiar effective methods for solving steady and unsteady problems [6-8].
