Том 51, № 4 (2016)

We analyze the spatial motion of a rigid body fixed to a cable about its center of mass when the orbital cable system is unrolling. The analysis is based on the integral manifold method, which permits separating the rigid body motion into the slow and fast components. The motion of the rigid body is studied in the case of slow variations in the cable tension force and under the action of various disturbances.We estimate the influence of the static and dynamic asymmetry of the rigid body on its spatial motion about the cable fixation point. An example of the analysis of the rigid body motion when the orbital cable system is unrolling is given for a special program of variations in the cable tension force. The conditions of applicability of the integral manifold method are analyzed.

The motion of a heavy bead on the surface of a parabolic bowl rotating at a constant angular velocity about its axis, which coincides with the vertical, is considered. It is assumed that the dry friction force acts between the bead and the bowl. The sets of nonisolated relative equilibria of the bead on the bowl are determined, and their dependence on the problem parameters is studied. The results are illustrated in the form of bifurcation diagrams.

The paper presents necessary and sufficient conditions whose must be satisfied by the main geometric and dynamic parameters of spherical, ellipsoidal, or parabolic rigid bodies for their physical realization. The main parameters are both the geometric characteristics of the body boundary (radius of the sphere, semiaxes of the ellipsoid, principal curvatures at the vertex, and the paraboloid center location on its symmetry axis) and the body mass and dynamic characteristics (body mass, displacement of the body center of mass from the center on the paraboloid symmetry axis or from the sphere or ellipsoid center of symmetry, the orientation of the principal central axes of inertia with respect to the principal geometric axes of the shell, and the values of the principal central moments of inertia). The physical realization is understood as the existence of an actual distribution of positive masses inside the sphere, ellipsoid, or paraboloid for which the above-listed characteristics of the body are equal to the chosen ones. Several examples from earlier-published papers dealing with the dynamics of spherical, ellipsoidal, or parabolic bodies with physically unrealizable parameters are given.

A method is proposed for obtaining analytic solutions of a set of infinite systems of linear algebraic equations arising in problems of elasticity for stiffened rectangular plates with stiffening ribs. The method is based on a transformation of a set of infinite systems to a single system and on determining a majorant of the function generating the system series with regard to the order of the unknowns. It is proved that the constructed solution satisfies the infinite system for large indices of the unknowns. The amount of computations is decreased, and the reliability of the results increases. Some realization examples are given.

The paper deals with the problem of determining the stress-strain state near the boundary of a one-layer strip made of an orthotropic material subjected to a self-balanced load applied at the end of the strip (Saint-Venant effect, boundary effect, and boundary layer). A comparative analysis of two methods for determining the boundary effect is carried out. The first method, i.e., the solution in stresses (with respect to σ_y), was developed by L. A. Agalovyan and gives good results of one-layer strips. The second approach, i.e., the solution in displacements, was developed by the author, and its results for the one-layer strip practically coincides with the solution in stresses. The obtained results were also verified by FEM. But the solution of the problem of elasticity in displacements is much more promising when analyzing multilayer strips.

The process of propagation of nonstationary waves in a rectangular bar is studied from the viewpoint of three-dimensional elasticity. The motion arises owing to the action of normal impact forces applied at the end face of a half-infinite bar all of whose four lateral surfaces are force-free. Precisely these one-type conditions complicate the solution of this problem. The already known solutions were obtained under the assumption that conditions of mixed type are partially or completely posed on the lateral sides, and precisely this fact permits separating the boundary values of distinct waves on these surfaces. In the absence of this simplifying factor, it is rather problematic to construct a solution satisfying all free lateral conditions.

In the paper, the solutions in integral transforms are guessed for a constant distribution of the impact load and then analytic solutions are constructed at the initial stages of the process.