Dynamics of a truncated elastic cone

Abstract

The problem of determining the nonstationary wave field of an elastic truncated cone with nonzero dead weight is formulated in terms of wave functions. The Laplace transform with respect to time and an integral transform with respect to time polar angle are used to reduce the problem to a one-dimensional vector problem in the transform domain. The transforms of the wave functions are expanded into series in inverse powers of the Laplace transform parameter, which makes it possible to study the wave process at the initial instants of interaction. A method is proposed to solve the problem for an elastic cone doubly truncated by spherical surfaces Keywords: truncated elastic cone, nonstationary wave field, Laplace transform, transform with respect to polar angle, cone doubly truncated by spherical surfaces, initial instants of interaction Introduction. The stress state of conical bodies under static loads is studied in many publications. For example, the axisymmetric equilibrium of a circular cone was first addressed in [10]; the equilibrium of a cone subjected to a concentrated force at the vertex was studied in [14]; the equilibrium of a cone subjected to a bending moment at the vertex was studied in [11]; a mixed problem for an infinite cone was solved in [13]; the stress state of a truncated semi-infinite cone was analyzed in [15]; the exact solution of an axisymmetric problem for a cone with a center of rotation at the vertex was first obtained in [6]; the exact solution of an axisymmetric problem for a circular hollow cone was found in [7].Far fewer publications address dynamic elastic problems for conical bodies. Experimental results on the propagation of elastic waves in a truncated elastic cone are presented in [12]; the analytic solution of an axisymmetric dynamic problem for a hemispherical dome, which is a special case of an elastic cone twice truncated by spherical surfaces, was found in [9]; a dynamic problem for an elastic cone under a dynamic load was solved in [2] disregarding the dead weight.1. Problem Formulation. Consider a truncated elastic cone at rest. Its position in a spherical coordinate system is described by 0 0 < < < < -£ £ r a, , q w p j p (Fig. 1, section zOx). A nonstationary axisymmetric load f t ( , ) 0 is applied at time t = 0 to the spherical surface r a = of the cone:

Torsion problem for an elastic twice-truncated cone

“…Next is shown asymptotic procedure which is used to find behavior of sum in (17). Let ( ) be the function for which ( ) it's asymptotic form when k is big enough.…”

“…For example, in [16] the influence of wave propagation in an elastic truncated cone is experimentally investigated. The problem of determining the nonstationary wave field of an elastic truncated cone, taking into account its own weight, is investigated in [17]. In [18] the stressed state of the inhomogeneous thin truncated hollow cone is investigated.…”

Natural frequencies and modes of out-of-plane vibrations a fixed ring

“…In the present paper, we will discuss the natural modes and frequencies of such a ring under zero gravity that were obtained by solving a boundary-value problem for the equations of motion of an elastic ring with direct numerical methods easily implementable on modern computers. A similar problem was addressed in [12].1. Problem Formulation.…”