Interaction of a hollow cylinder of finite length and a plate with a cylindrical cavity with a rigid insert

Two problems of the interaction of a hollow circular cylinder with load-free ends and an unbounded plate with a cylindrical cavity and a symmetrically imbedded rigid insert are considered. Homogeneous solutions are found and the generalized orthogonality of these solutions is used when the modified boundary conditions are satisfied. As a result, we have a system of two integral equations in functions of the displacements of the outer and inner surfaces of the hollow cylinder. These functions are sought in the form of sums of a trigonometric series and a power function with a root singularity. The ill-posed infinite systems of linear algebraic equations obtained are regularized by the introduction of small positive parameters. Since the elements of the matrices of the systems as well as the contact stresses are defined by poorly converging numerical and functional series, an efficient method for calculating of the remainders of the above-mentioned series is developed. Formulae are found for the contact pressure distribution function and the integral characteristic. Examples of the calculation of the interaction of the cylinder and the plate with an insert are given.The method of solving contact problems described here has been used earlier1, 2 and the generalized orthogonality of the solutions found for bodies of finite dimensions, that is, for a rectangle and cylinders of finite length, is its basis. Problems for hollow cylinders with a band ² and an insert reduce to a system of two integral equations, and the problem for a rectangle¹ reduces to one integral equation. Solving these integral equations, ill-posed systems of linear algebraic equations are obtained which are subject to regularization³.     

Permanent rotations of a body with a viscous filler suspended on a rod

Problems of the existence, stability, and branching of the permanent rotations of a heavy, dynamically symmetrical rigid body suspended on a rod and which has an axisymmetric ellipsoidal cavity filled with a fluid are discussed. The phenomenological model of the friction of the fluid against the cavity wall proposed by Samsonov is used. All the trivial permanent rotations of the system and the non-trivial rotations that branch off from the trivial ones are found. Their stability and branching are investigated using a modified Routh's theory. The results obtained are presented in the form of an atlas of bifurcation diagrams.     

Scattering of a plane wave on a wavy surface

Numerical methods are proposed for solving problems of diffraction of a plane electromagnetic wave on a nonhomogeneous interface of two media. Strictly periodic cylindrical surfaces are considered, as well as surfaces that are nonplanar on a finite section. Algorithms are developed for computing the characteristics of the reflected field in the far zone. Numerical results are reported for various electrodynamic models, and their limits of applicability are investigated.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 86–111.     

Extrusion of a plastic material from a circular sector with a small apex angle and a sink at the vertex

The plane inertialess extrusion of a perfect rigid-plastic material from a circular sector, the angle of which serves as a small parameter, is investigated using the asymptotic integration. The flow was initiated by the approach of the two sides of the sector and the existence in it of a sink of specified power. The principal velocity and stress approximations are obtained and the domains of applicability of the asymptotic expansions and the inertialess (quasistatic) solutions are found. Analogies are drawn with the solutions of the classical Prandtl problem and several of its extensions.     

Scheduling a repairman in a finite source system

The scheduling of a single server in a finite source model is considered. TheN customers in the system have different failure and repair rates. Also the costs depend on the customers which are broken down. We give a condition under which the average costs are minimized by a simple list policy, and with a counterexample we show that in the general case no optimal list policy may exist. This motivates us to derive policies which are optimal under low and high traffic conditions. They are again list policies, which behave well numerically.     

On a cycle through a specified linear forest of a graph

Results on the existence of a cycle containing a given linear forest are proved.     

Solution of a regularized problem for a stationary magnetic field in a nonhomogeneous conducting medium by a finite element method

The use of a vector finite element method for solving a regularized stationary magnetic field problem, which is formulated in terms of a vector magnetic potential, is validated. A generalized solutionis approximated using first-order Nedelec vector elements of the second kind on tetrahedrons. The existence and uniqueness of the solution to a discrete regularized problem and its convergence to the generalized solution for the case of an inhomogeneous (in the electromagnetic properties) domain are justified. Some issues of the numerical solution to the discrete regularized problem are discussed. Approaches to optimize the algorithms are shown on a series of numerical experiments.     

The equilibrium positions of an ellipsoid with a cavity containing a liquid on a horizontal plane

The equilibrium positions of an ellipsoid with an ellipsoidal cavity, partially filled with an ideal incompressible liquid, on a horizontal plane in a uniform gravitational field are considered. All trivial and non-trivial equilibrium positions are found and the conditions for their stability are obtained. The results are presented in the form of bifurcation diagrams.     

粘塑性流体在旋转圆盘上的流动

本文对两种情况导出了描述粘塑性流体在旋转圆盘上流动的基本方程.分别用摄动方法和数值方法得到了方程的解.这就有可能去计算薄膜的厚度分布.经计算发现有两种类型的厚度分布.对于粘度和屈服应力都与径向坐标r无关的粘塑性流体,厚度h随r的增加而减小.对于粘度和屈服应力都是时间和r的函数的Bingham流体,厚度h随r的增加而增加.     

The quasistatic motions of a three-body system on a plane

A controlled three-body system on a horizontal plane with dry friction is considered. The interaction forces between each pair of bodies are controls that are not subject to prior constraints but must be chosen in such a way that the motions of the system generated by them are quasistatic, that is, the total force acting on each of the bodies must be close to zero. All motions in which one body moves and the other two are fixed are found in the class of quasistatic motions. The problem of the optimal displacement of a moving body between two specified positions on a plane such that the absolute magnitude of the work of the friction forces along the trajectory is a minimum is solved. The quasistatic controllability of a three-body system is demonstrated and algorithms for bringing it into a specified position are discussed. The system considered simulates a mobile robot consisting of three bodies between which control forces act that can be realized by linear motors. The sizes of the bodies are assumed to be negligibly small compared with the distances between them so that the bodies are treated as particles.     

Unsteady draining of a fluid from a circular tank

Three-dimensional draining flow of a two-fluid system from a circular tank is considered. The two fluids are inviscid and incompressible, and are separated by a sharp interface. There is a circular hole positioned centrally in the bottom of the tank, so that the flow is axially symmetric. The mean position of the interface moves downwards as time progresses, and eventually a portion of the interface is withdrawn into the drain. For narrow drain holes of small radius, the interface above the centre of the drain is pulled down towards the hole. However, for drains of larger radius the portion of the interface above the drain edge is drawn down first, rather than the central section. Non-linear results are obtained with a novel spectral technique, and are also compared against the predictions of linearized theory. Unstable Rayleigh–Taylor type flows, in which the upper fluid is heavier than the lower one, are also discussed.     

Effect of a boundary on the dynamics of a spherical cavity in a liquid

The effect of various kinds of boundaries (solid and elastic walls, free surfaces) on the dynamics of a spherical cavity during an electrical discharge in a liquid is examined. The motion of the liquid is examined in a gas dynamic formulation. Solutions are obtained numerically using grid-characteristic methods. Special moveable grids are developed for calculating long duration processes. It is shown that a free surface has little effect on the maximum radius and period of the pulsations of a cavity. The effect of rigid and elastic walls can be neglected if they are further away than 50 times the maximum radius of the cavity. The computational results are in good agreement with experimental data obtained by others. Donetsk State University. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 29, pp. 175–180, 1999.     

Conditions for a two-dimensional surface in E 5 to be contained in a hypersphere or a hyperplane

Two theorems on conditions under which a two-dimensional surface in Euclidean 5-space is contained in a hypersphere and one theorem on conditions under which such a surface is contained in a hyperplane are proved. The notion of hyperbolic and elliptic domains on a surface are introduced. The conditions in the theorems are expressed in terms of the behavior of the plane of the normal curvature ellipse of the surface and certain boundary conditions. An example which shows that the boundary conditions are essential is constructed.     

Radiation instability of a circular cylinder in a uniform flow of a two-layer fluid

A solution of the plane linear problem of the oscillations of a horizontal circular cylinder in a uniform flow of a two-layer unbounded fluid is obtained using the method of multipole expansions. The direction of the flow is perpendicular to the cylinder axis. The whole cylinder Ges in the upper or lower layer. The fluid is assumed to be ideal and incompressible, the flow in each layer being a potential one. All the components of the radiation load (the apparent masses and damping coefficients) are determined and the regions of existence of radiation instability are found, depending on the flow velocity for a cylinder suspended by horizontal and vertical elastic links. By solving the integro-differential equation numerically the relative oscillations of the body under specified initial conditions are found.     

Evasion of a fixed sphere by a dynamical object driven by a bounded force

A solution is obtained of the problem of synthesizing the control of the motion of a dynamical object (a point mass) evading a fixed spherical obstacle under the action of a bounded force. The set of all points for which evasion is possible is constructed in phase space (of arbitrary dimension), and control modes are constructed for bounded (fixed) and unbounded time intervals. The characteristics of the optimal motion, in particular, the time and minimum distance, are determined for specific initial data. The qualitative properties of the controlled motion are established.     

The steady motions of a gyrostat suspended on a rod in a central gravitational field

The problem of the existence, stability and bifurcation of the steady motions of two bodies in an orbital tethered system, when one of the bodies is a symmetrical satellite with a rotor on the axis of symmetry, is considered. One-parameter families of steady motions are indicated, and their stability and bifurcations are investigated. The conditions which relate the parameters of the system for which stabilization of the families obtained is possible using a rotating rotor are obtained.     

On a Reliable Solution of a Volterra Integral Equation in a Hilbert Space

We consider a class of Volterra-type integral equations in a Hilbert space. The operators of the equation considered appear as time-dependent functions with values in the space of linear continuous operators mapping the Hilbert space into its dual. We are looking for maximal values of cost functionals with respect to the admissible set of operators. The existence of a solution in the continuous and the discretized form is verified. The convergence analysis is performed. The results are applied to a quasistationary problem for an anisotropic viscoelastic body made of a long memory material.     

Boundary value problems of electroelasticity for a plate with an inclusion and a half-space with a slit

Problems of determining the mechanical and electrical fields in a piezoelectric plate reinforced with an inclusion or in a half-space weakened by a cut are considered. Using the methods of the theory of analytic functions these problems are reduced to a system of singular integro-differential equations (for a plate) or to a singular integral equation with a fixed singularity (for a half-space). Approximate and exact solutions of the problems are obtained by the method of orthogonal polynomials and integral transforms.     

Finding a minimum cost path between a pair of nodes in a time-varying road network with a congestion charge

The minimum cost path problem in a time-varying road network is a complicated problem. The paper proposes two heuristic methods to solve the minimum cost path problem between a pair of nodes with a time-varying road network and a congestion charge. The heuristic methods are compared with an alternative exact method using real traffic information. Also, the heuristic methods are tested in a benchmark dataset and a London road network dataset. The heuristic methods can achieve good solutions in a reasonable running time.     

Separating a unital divisor from a rectangular polynomial matrix

It is required to separate a unital divisor with a preassigned characteristic polynomial from a rectangular polynomial matrix over a field. Necessary and sufficient conditions of existence, under certain restrictions, are obtained for such a divisor, as well as a method for constructing it. By the approach used in this paper it is possible to completely solve this problem for rectangular polynomial matrices, all of whose elementary divisiors are pair-wise relatively prime. The results obtained are illustrated by solving matrix equations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1089–1094, August, 1990.