Abstract: The decay of a Karman vortex street and the formation of a secondary vortex structure in the far wake of a streamlined cylinder are studied. The dynamics of spatially evolving vortex structures is examined in the free flow and in the following ways of external influence on this flow: rotation with a constant velocity and translational and rotational oscillations of the cylinder. The results are obtained by numerically solving the Navier-Stokes equations with two different methods. The corresponding boundary value problems are formulated in the domains extended up to 500 radii of the cylinder.

Abstract: A further development of the homogenization method is proposed to solve the physically nonlinear equilibrium problems for the laminated plates or the plates made of functionally graded materials. In the linear case, according to this method, the corresponding solution is a superposition of the solution to the global problem in the entire domain and the solution to the local problem in a representative domain, e.g., in a periodicity cell. In the nonlinear case, such a superposition is not valid, which complicates the application of the homogenization method. In order to eliminate this difficulty, it is possible to combine the homogenization method and the linearization method when solving a boundary value problem or a variational problem. In the mechanics of deformable solids, the constitutive relations can be considered as equations with respect to velocities or the stress and strain differentials in time or in the loading parameter. When these equations are linear with respect to velocities, it is possible to use the homogenization method. In this paper such an approach is illustrated by the example of a symmetric laminated plate bent under a uniformly distributed time-dependent load.

Abstract: Nonlinear equations of motion for orthotropic plates are proposed on the basis of the three dimensional nonlinear theory of elasticity. The resulting plate model provides a high accuracy, which is confirmed by solving the Vlasov static problem of bending a hinge-supported plate under sinusoidal load.

Abstract: The problem of small-amplitude harmonic wave propagation along the surface of an incompressible nematic liquid crystal is considered when the evolution of the orientation vector is specified by viscous stresses and the orientational elasticity can be ignored. An analytic solution and a dispersion relation are obtained in the case of the planar and homeotropic initial orientation of this vector.

Abstract: A mathematical model of instrument errors of strapdown inertial navigation systems is considered. The parameters of this model are determined by a special algorithm in the case of using a low-accuracy turntable with one degree of freedom. A distinguished feature of this algorithm is that the characteristics of the turntable sensors are not directly used in it. The performance of the algorithm is illustrated by an example of data processing.

Abstract: A variable cross-section bar is considered. The bar is not uniform in length. The bar is compressed by a variable longitudinal force distributed along its axis. The stability loss in the straightline shape of the bar’s equilibrium is discussed when a curved shape is also possible. The critical combination between rigidity and the longitudinal force is a result of using an integral representation for the solution to the original stability equation with variable coefficients with the aid of the solution to a similar equation with constant coefficients. The integral representation contains the Green function of the original equation. This function is determined by the method of perturbations. The numerical results obtained by the derived formulas are compared with the known exact solutions to the stability equations for various particular cases.

Abstract: An analog of Cesaro’s formula and several compatibility conditions are given for the three-dimensional and two-dimensional linear micropolar theory of elasticity in the form different from that used in the literature. A number of formulas are obtained to determine the antisymmetric part of the strain (stress) tensor in terms of the symmetric part of the strain tensor and the symmetric part of the bending-torsion (stress and couple-stress) tensor and to determine the antisymmetric part of the bending-torsion (couple-stress) tensor in terms of the symmetric part of the bending-torsion (couplestress) tensor. Some integro-differential equations of motion expressed in terms of the symmetric parts of the stress and couple-stress tensors are proposed for the micropolar theory of elasticity.

Abstract: Digital filters based on polynomial approximation are commonly used to estimate the low-frequency component and its derivatives for a wideband signal. The simplest digital filter is an arithmetic mean algorithm. The filter properties important for applications are associated with the types of transfer functions. Several formulas are given for the transfer functions of such filters used to estimate the constant component of an input signal and its first and second derivatives on the basis of the second- and third-degree polynomial approximation.

Abstract: A new case of integrability in the spatial problem of motion for a solid body with consideration of the nonconservative moment of forces is discussed. A nonconservative force field of action of the medium on the body is constructed. Contrary to some previous author’s works, a linear dependence of this field on the angular velocity is taken into account, although the introducing of this dependence into the components of such a field is not obvious in advance.

Abstract: The polarization-optical method is used to study stresses and to measure the optical quantities (the isochromatic fringe orders and the isoclinic parameters) and the mechanical quantities (stresses and strains). The dependencies between these quantities are considered. The problem of disk compression along its diameter is solved exactly. These dependencies are analyzed on the basis of this exact solution and the experimental data. The most preferred form to express the basic relations of photoelasticity is discussed.

Abstract: Flows of a viscoplastic incompressible medium in a channel of periodically varying width are numerically simulated. The effect of the yield stress and the boundary perturbation amplitude on the distribution of rigid zones is studied.

Abstract: The possibility of the existence of surface waves in a range of velocities greater than the velocity of transverse waves, but smaller than the velocity of longitudinal waves is shown. It turns out that, in the boundary value problem for an elastic half-space in this velocity range, there are the surface waves whose velocity is constant and equal to \(\sqrt 2 \)b, where b is the velocity of transverse waves. These waves as well as the Rayleigh surface waves have no dispersion. Their velocity is specified only by the elastic constants and density of the medium. It is also shown that the existence of such a velocity is possibly related to the velocity of surface waves that appear as unloading waves under constrained deformation.

Abstract: An algorithm for the bench calibration of strapdown inertial navigation systems is studied. This algorithm was developed at the Moscow Institute of Electromechanics and Automatics. A mathematical model for the calibration process is constructed. A method for representing the original calibration problem in the form of a standard estimation problem is proposed with the aid of a so-called “telescopic” system. On the basis of the original algorithm, a new calibration algorithm is constructed; this new algorithm allows one to improve the estimation accuracy for the parameters of accelerometer and gyroscope units. The maximum attainable estimation accuracy for these parameters is determined.

Abstract: Many problems of Number Theory are connected with investigation of Dirichlet series $$f(s)=\\sum_{n=1}^{\\infty} a_nn^{-s}$$ and the adding functions $$\\Phi(x)=\\sum_{n\\leq x} a_n$$ of their coefficients. The most famous Dirichlet series is the Riemann zeta function $$\\zeta(s),$$ defined for any $$s=\\sigma+it$$ with $$\\Re s=\\sigma> 1$$ as $$\\zeta(s)=\\sum_{n=1}^{\\infty}\\frac{1}{n^s}.$$ The square of zeta function $$\\zeta^{2}(s)=\\sum_{n=1}^{\\infty}\\frac{\\tau(n)}{n^s}, \\,\\, \\Re s > 1,$$ is connected with the divisor function $$\\tau (n)=\\sum_ { d | n } 1,$$ giving the number of a positive integer divisors of positive integer number n. The adding function of the Dirichlet series $$\\zeta^2(s)$$ is the function $$D (x)=\\sum_ { n\\leq x}\\tau(n)$$; the questions of the asymptotic behavior of this function are known as Dirichlet divisor problem. Generally, $$ \\zeta^{k}(s)=\\sum_{n=1}^{\\infty}\\frac{\\tau_k(n)}{n^s}, \\,\\, \\Re s > 1, $$ where function $$\\tau_k (n)=\\sum_{n=n_1\\cdot...\\cdot n_k} 1$$ gives the number of representations of a positive integer number n as a product of k positive integer factors. The adding function of the Dirichlet series $$ \\zeta^k (s)$$ is the function $$D_k (x)=\\sum_ { n\\leq x}\\tau_k(n)$$; its research is known as the multidimensional Dirichlet divisor problem. The logarithmic derivative $$\\frac{\\zeta^{'}(s)}{\\zeta(s)}$$ of zeta function can be represented as $$\\frac{\\zeta^{'}(s)}{\\zeta(s)}=-\\sum_{n=1}^{\\infty} \\frac{\\Lambda(n)}{n^s},$$ $$\\Re s >1.$$ Here $$\\Lambda(n)$$ is the Mangoldt function, defined as $$\\Lambda(n)=\\log p,$$ if $$n=p^{k}$$ for a prime number p and a positive integer number k, and as $$\\Lambda(n)=0,$$ otherwise. So, the Chebyshev function $$\\psi(x)=\\sum_{n\\leq x}\\Lambda(n)$$ is the adding function of the coefficients of the Dirichlet series $$\\sum_{n=1}^{\\infty} \\frac{\\Lambda(n)}{n^s},$$ corresponding to logarithmic derivative $$\\frac{\\zeta^{'}(s)}{\\zeta(s)}$$ of zeta function. It is well-known in analytic Number Theory and is closely connected with many important number-theoretical problems, for example, with asymptotic law of distribution of prime numbers. In particular, the following representation of $$\\psi(x)$$ is very useful in many applications: $$\\psi(x)=x-\\sum_{|\\Im \\rho|\\leq T}\\frac{x^{\\rho}}{\\rho}+O\\left(\\frac{x\\ln^{2}x}{T}\\right), $$ where x=n+0,5, $$n \\in\\mathbb{N},$$ $$2\\leq T \\leq x,$$ and $$\\rho=\\beta+i\\gamma$$ are non-trivial zeros of zeta function, i.e., the zeros of $$\\zeta(s),$$ belonging to the critical strip 0 2, $$T \\geq 2,$$ and $$\\rho=\\beta+i\\gamma$$ are non-trivial zeros of zeta function, i.e., the zeros of $$\\zeta(s),$$ belonging to the critical strip 0 < Res < 1.

Abstract: Some features of the behavior of viscoelastic materials whose existence leads to the choice of nonlinear constitutive relations are discussed. A classification of such constitutive relations is given and a number of requirements imposed by practice on their adequacy are formulated. A nonlinear theory of viscoelasticity is proposed; this theory offers the advantages over the theory in which stresses are expressed in terms of strains by integral operators of increasing multiplicity. By a one-dimensional example, it is shown that the constitutive operator relations are reciprocal.

Abstract: A number of pollution-free sea wave converters installed on ship models are considered The operation of the wave generator used in the hydraulic channel of Moscow University Institute of Mechanics is described The performance evaluation of rectangular and triangular submarine sails is discussed

Abstract: The possibility of control of man-machine (biomechatronical) systems by gaze direction is considered. The basic properties of eye movements being essential for the construction of control are analyzed. In order to study the possibilities of control by gaze, a game program is written. It is shown that, in comparison with control by a computer mouse, the control by gaze requires less time and leads to fewer mistakes on solving some control problems such as specifying a point on a computer screen.

Abstract: One-dimensional nonstationary problems of adiabatic expansion for thick-wall spherical and cylindrical viscoplastic shells are solved exactly under the assumption that, at the initial instant of time, the distributions of radial velocities satisfy the condition of incompressibility of the shell material. The resulting solutions can easily be modified for the case of compression of such shells.

Abstract: It is shown that the minimum value of the ratio of relative yield stresses found according to the energy yield criterion in two variants corresponds to the state with the least resistance to plastic deformation. A distinction in the position of the curves corresponding to the relative yield stresses in these variants allows one to estimate the appearance of tensor nonlinearity in the material. The algorithm in use assumes the nonlinearity of characteristics as functions of stress intensity and of the aspect angle of the stress state.

Abstract: The behavior of a spherical bubble in an ideal fluid, a viscous medium, and an incompressible viscoplastic medium with a yield stress is studied numerically and analytically when a time-varying periodic pressure is exerted at a sufficient distance from the surface of the bubble. Various modes of collapse are examined and classified. The critical values of the key parameters that characterize the behavior of this system are found; one of these parameters is the dimensionless frequency of external pressure fluctuations.

Abstract: The time minimization problem in a linear system with a scalar control under phase constraints is considered. This problem is reduced to the problem of maximum deviations. Necessary conditions of control optimality are obtained. The results are applied to the problem of uniaxial stabilization of a satellite with consideration of a gravitational torque. The stabilization is performed by a flywheel engine with a limited angular momentum.

Abstract: Hamilton’s variational principle for mechanical systems with unilateral constraints is considered. It is shown that the action functional attains its local maximum on the class of variations belonging to the interior of the domain admissible for motion. An example is given.

Abstract: The fracture of two closely spaced elastic beams by an incompressible viscous fluid is considered The beam bending is described in the framework of the Kirchhoff-Love model The self-similar solutions are sought by the group methods The numerical results obtained when solving the corresponding system of differential equations with various boundary conditions are graphically illustrated in the form of pressure and velocity distributions within the fluid and in the form of the distance between the beams in the fracture zone

Abstract: The time of initiation and cessation of a circular flow of a viscoplastic medium in an annular gap is estimated analytically and is verified numerically.

Abstract: The eigenvalue problem for the fourth-order differential equation describing the free transverse oscillations of a strongly inhomogeneous elastic beam is studied numerically and analytically. The lowest modes of oscillations that are of interest in applications are found.

Abstract: The thermocapillary drift of a homogeneous suspension of spherical drops with the constant properties of a carrier fluid and the fluid inside the drops is considered. Several formulas are obtained for the drift velocity and the effective thermal conductivity of the suspension.

Abstract: Hamiltonian systems under small nonautonomous and aperiodic perturbations are considered. Sufficient conditions under which the first integrals of the unperturbed system vary slightly along the solution to the perturbed system are formulated. Some mechanical systems are considered as examples.

Abstract: A model of a planet considered as a homogeneous viscoelastic sphere in the gravitational field of a point mass is discussed. Tidal deformations occur in the process of their mutual motion. The deformation rate tensor and the corresponding dissipative function are found. The time variation of the deformation tensor components accompanied by the heat release at each point of the planet causes the formation of a temperature field described by the inhomogeneous heat conduction equation. The temperature field is determined by averaging with respect to the proper rotation angle and is estimated for the Moon.

Abstract: A crack on the interface between a linear elastic medium and a stress-state dependent physically nonlinear medium is studied. A numerical method to solve such problems is proposed. Some asymptotic distributions of stresses, strains, and displacements near the crack tip are obtained under the assumption that the forces and displacements are continuous on the interface.