Vol 51, No 1 (2016)

We propose new differential equations describing the perfect operation of strapdown intertial navigation systems (SINS) intended to determine the apparent, gravitational, and relative velocities of amoving object and the geographic coordinates of the object position. The superposition principle is used to construct the equations.

We construct analytical solutions of differential equations for the apparent and gravitational velocities in the case of an object immovable with respect to the Earth, which can be used to analyze the accuracy of operation of SINS mounted on an immovable base.

L. A. Galin’s contact model for a narrow beam bending on an elastic half-space and Melan’s contact model for a stringer are used to consider two problems of contact interaction between one or two identical symmetrically loaded stringers with small rectangular cross-sections and an elastic half-space. The basic characteristics of these problems are expressed by explicit formulas, and the results of their numerical analysis are given as well.

The exact solution of the problem of coupled seismic vibrations of an underground pipeline and an infinite elasticmediumis given. A method dramatically simplifying the solution of the exterior problem for themedium is proposed on the basis of the established theorem on the separation of the boundary conditions for the wave potentials on the surface of the cylinder. The obtained results permit improving the incorrect consideration of the problem accepted in the literature.

The exact statement of the problem allows one to use the solution of the problem as a test for estimating the accuracy of appropriate approaches and solutions in seismodynamics of extended underground structures.

The results of comparison show that the solutions practically coincide both in the subsonic operation mode (when the seismic wave velocity is smaller than the rod velocity of wave propagation in the pipeline) and in the supersonic operation mode, where a resonance is possible. Thus, the high accuracy of the significantly simpler theory of one-dimensional deformation is confirmed.

We consider problems on transient wave processes in linearly viscoelastic piecewise homogeneous bodies in the case of small strains, a bounded perturbation propagation domain, and bounded creep of the materials forming the homogeneous components of the bodies. We study problems related to the construction of solutions of such problems by the method of Laplace integral transform with respect to time and the subsequent inversion. We state assertions about the properties of Laplace transforms of the solutions, which simplify the process of determining the original functions. We also consider relations of correspondence between relaxation kernels that belong to different function classes but still affect transient wave processes in a similar way.

The structural-probabilistic approach to the modeling of combined crack formation and material deformation processes is used to develop a technique for solving bifurcation stability problems for thin-walled structural members made of damageable materials under single and repeated loadings. The example of a uniformly compressed spherical shell is used to show that, under repeated loading, thin-walled structural members made of shear damageable materials can lose stability under loads smaller than the upper critical loads. The ambiguity of the critical loads for various damage accumulation processes in the material of thin-walled structures depends on the level and character of loading. This phenomenon can be one possible cause of the experimental data spread and the discrepancy between theoretical and experimental results used to determine the critical loads for spherical and cylindrical shells.

Static elasticity problems for a half-plane and a strip weakened by a rectilinear transverse crack are studied. In each case, the upper boundary of the body is reinforced by a flexible patch. Various versions of conditions on the lower boundary are considered in the case of the strip. The crack is maintained in the open state by distributed normal forces. The method of generalized integral transforms reduces solving the problem for the equations of equilibriumto solving a singular integral equation of the first kind with the Cauchy kernel with respect to the derivative of the crack opening function. The solutions of the integral equation are constructed by the small parameter and collocation methods for various combinations of the geometric and physical parameters of the problem, and the structure of the solutions is analyzed. The values of the stress intensity factor (SIF) near the crack vertex are obtained.