Mysov K. Dynamic torsion problems for an elastic twice-truncated cone.

The dynamic problem of the theory of elasticity for a twice-truncated with a spherical surfaces elastic cone under the external torsional loads applied to the lower spherical surface through an absolutely rigid overlay is considered. Using the G.Ya. The Popov integral transfor-mation problem is reduced to a one-dimensional vector boundaryvalue problem. The solution of the problem in the transform domain is constructed using the apparatus of the fundamental system of solutions, which is then inverted to obtain a continuous solution in the original domain. Dynamic torsion problems for a twicetruncated with a spherical surfaces elastic cone under the external torsional loads weakened by defects of a spherical or conical shape are also conside-red, to solve which a discontinuous solution is constructed. Discon-tinuous solutions for the dynamic torsion equation in unbound space are constructed by applying the integral Legendre transformation directly to the torsion equation. This reduces the equation to onedimensional, which is solved using the Melin integral transform. After successive inversion of the solution in the transformant domain, we obtain a discontinuous solution with unknown jumps of displace-ments and stresses for initial problem. The type of defect is specified to be a crack and the dynamic torsion problems for a twice-truncated with spherical surfaces elastic cone under the external torsional loads weakened by spherical or conical cracks are solved. Their solution is in the form of a superposition of already found continuous solutions and discontinuous solutions. A singular integral or integro-differential equation with a separated singularity is constructed by satisfying the conditions of the absence of stresses at the edges of the crack. They are solved according to the scheme of the method of orthogonal polynomials, which allows to find unknown displacement jumps. An analysis of the first natural frequencies of the cone was carried out depending on various mechanical and geometric charac-teristics of the cone and the crack. The stress intensity factor on the crack banks was also analysed. The dynamic torsion problem for a twice-truncated spherically layered cone is also solved. The solution begins with the application of the G.Ya. Popov integral transformation, which reduces the initial problem to a one-dimensional boundary valueproblem. Fundamental solution is built, using which a general solution is constructed in the transform domain for a one-dimensional boundary value problem with unknown constants for each layer. A system of linear algebraic equations is constructed using general solution and conditions on the edges of the cone and the junctions of the layers. The solution of the system is built in iterative form, allowing to find the unknown layer constants regardless of their number. Next, the inverse integral transformation is used to derive the solution of initial problem. An analysis of the first natural frequencies of the cone was carried out depending on the various mechanical and geometric characteristics of the layers of the cone.