Vol 51, No 6 (2016)

We solve a nonlinear orbital stability problem for a periodic motion of a homogeneous paraboloid of revolution over an immovable horizontal plane in a homogeneous gravity field. The plane is assumed to be absolutely smooth, and the body–plane collisions are assumed to be absolutely elastic. In the unperturbed motion, the symmetry axis of the body is vertical, and the body itself is in translational motion with periodic collisions with the plane.

The Poincare´ section surfacemethod is used to reduce the problemto studying the stability of a fixed point of an area-preserving mapping of the plane into itself. The stability and instability conditions are obtained for all admissible values of the problem parameters.

A computational method for obtaining sufficient conditions for the stability of the stationary solution of autonomous conservative systems is proposed in the paper. This method is adapted to linear autonomous gyroscopic systems with three degrees of freedom. It is based on the positive definiteness of a parametric quadratic form composed of the gyroscopic force matrices and the potential function. The control parameters for the stability of the zero solution of the gyroscopic system are the entries of the gyroscopic force matrix. The algorithm of the computational method includes estimating one gyroscopic force parameter in the equation constructed from a necessary stability condition.

A special example is used to demonstrate the application of this algorithm. Comparison is performed with some well-known methods for obtaining sufficient conditions on the basis of an incomplete set of first integrals of motion. It is shown that the positive definiteness of the modified potential energy may result in stable as well as unstable motions.

Some methodological approaches to solving several key problems of dynamic loading and structural strength analysis of Permanent Space Station (PSS)modules developed on the basis of the working experience of Soviet and Russian PSS and the International Space station (ISS) are presented. The solutions of the direct and semi-inverse problems of PSS structure dynamics are mathematically stated. Special attention is paid to the use of the results of ground structural strength tests of space station modules and the data on the actual flight actions on the station and its dynamic responses in the orbital operation regime. The procedure of determining the dynamics and operation life parameters of elements of the PSS modules is described.

Two approaches (classical and nonclassical) of the boundary integral equation method for solving three-dimensional dynamical boundary value problems of elasticity, viscoelasticity, and poroelasticity are considered. The boundary integral equation model is used for porous materials. The Kelvin–Voigt model and the weakly singular hereditary Abel kernel are used to describe the viscoelastic properties. Both approaches permit solving the dynamic problems exactly not only in the isotropic but also in the anisotropic case. The boundary integral equation solution scheme is constructed on the basis of the boundary element technique. The numerical results obtained by the classical and nonclassical approaches are compared.

The process of erection an object under the action of gravity forces in the absence of additional loads is studied together with the technology of application of prestressed structure elements. The mathematically two-dimensional engineering problem of mechanics of gradual building of a heavy semicircular vault from a prestressed viscoelastic homogeneously aging material is solved analytically. The vault fixation on a rigid horizontal base by sliding fixation, which ensures continuous smooth contact between the vault foot and the base, is considered. The performed computations permit demonstrating high efficiency of preliminary stress creation in the material elements added to the vault in the process of its building in order to control its technological stress state. It is shown that this measure permits significantly decreasing the final values of the separating contact stresses on the foot of the built vault and obtaining the final state of the whole structure which is safer with respect to the level of tensile stresses than that obtain by using unstressed elements.