Modes with switchings of increasing frequency in the problem of controlling a robot

Trajectories that are optimal with respect to high-speed response are constructed for a system for controlling a two-component manipulator (a robot). It is shown that when the initial conditions lie within a certain open region of the phase space, all optimal trajectories will have a segment of switchings of increasing frequency (SIF), i.e. a segment in which the control will undergo an infinite number of switchings in a finite time interval.

The synthesis of the optimal control in the R² plane containing the mode of SIF was first constructed by Fuller /1/. It was shown in /2/ that the synthesis is structurally stable in the sense that adding terms of higher order of smallness to the integrand and to the right-hand sides of the system of differential constraints does not affect the qualitative pattern of the optimal synthesis in the neighbourhood of the origin of coordinates.

The present paper explains that the synthesis in the problem of optimal control (relative to the high speed response) of the motion of the robot appears, in a certain sense, a direct product of the synthesis appearing in the Fuller problem and of the synthesis in the simplest problem of high-speed response (/3/, pp.38–47). The special aspect of the present paper consists of the proof of the proposition that switching surface is a piecewise-smooth manifold. The presence of the SIF mode is connected only with the fact that every trajectory intersects this surface an infinite number of times. In existing papers, the piecewise smoothness of the switching curve was proved for the two-dimensional problems using the SIF mode only for problems admitting of a one-parameter group of symmetries /1, 4–6/. A proof of the presence of SIF was given in /7, 8/.     

The contact problem when there are friction forces in the contact area for a three-component cylindrical base

The plane contact problem of elasticity theory on the interaction when there are friction forces in the contact area of an absolutely rigid cylinder (punch) with an internal surface of a cylindrical base, consisting of two circular cylindrical layers rigidly connected to one another and with an elastic space, is considered. The layers and space have different elastic constants. A vertical force and a counterclockwise torque, act on the punch, and the punch – base system is in a state of limiting equilibrium,. An exact integral equation of the first kind with a kernel represented in an explicit analytical form, is obtained for the first time for this problem using analytical calculation programs. The main properties of the kernel of the integral equation are investigated, and it is shown that the numerator and denominator of the kernel symbols can be represented in the form of polynomials in products of the powers of the moduli of the displacement of the layers and the half-space. A solution of the integral equation is constructed by the direct collocation method, which enables the solution of the problem to be obtained for practically any values of the initial parameters. The contact stress distributions, the dimensions of the contact area, the interconnection between the punch displacement and the forces and torques acting on it are calculated as a function of the geometrical and mechanical parameters of the layers and the space. The results of the calculations in special cases are compared with previously known results.     

Analytical solution of the quasi-static thermoelasticity problem in a pressurized thick-walled cylinder subjected to transient thermal loading

Thermoelasticity problem in a thick-walled cylinder is solved analytically using the finite Hankel transform. Time-dependent thermal boundary conditions are assumed to act on the inner surface of the cylinder. For the mechanical boundary conditions two different cases are assumed: Traction–displacement problem (traction is prescribed on the inner surface and the fixed displacement boundary condition on the outer one) and Traction–Traction problem (tractions are prescribed on both the inner and outer surfaces of the hollow cylinder). The quasi-static solution of the thermoelasticity problem is derived analytically, i.e., the transient thermal response of the cylinder is derived and then, quasi-static structural problem is solved and closed form relations are extracted for the thermal stresses in the two problems. The results show to be in accordance with that cited in the literature in the special cases.     

权函数法研究高速旋转厚壁筒的应力强度因子

采用厚壁筒在内压作用下的应力强度因子作为参考载荷的应力强度因子,通过权函数方法推导出了内壁带二维径向边裂纹的旋转厚壁筒的应力强度因子公式．这些公式可用于计算旋转厚壁筒在不同裂纹深度、转速、材料和尺寸情况下的应力强度因子．算例表明该文的公式具有良好的精度．同时还研究了旋转圆筒应力强度因子随裂纹深度和内外径比之间的变化规律,方便了工程应用．     

Free vibrations and stability of hydraulic cylinder fixed elastically on both ends

The boundary value problem of the stability and free vibration of a hydraulic cylinder has been formulated and solved in this paper. The considered hydraulic cylinder has been elastically fixed at both ends and loaded by Euler force. An elastical fixation has been modelled by rotational springs. The mentioned above hydraulic cylinder consists of a piston rod and cylinder replaced by rods. There are conditions of continuity between the rods. The boundary value problem has been formulated on the basis of minimum potential energy (static problem) and on the basis of Hamilton's principle (free vibration problem). In this paper example results of numerical calculations of the stability and free vibration have been presented. Experimental research has been performed in order to verify the correctness of the assumed mathematical model. Professional measuring apparatus and special stand for research into the slender systems have been used in experiment. Natural frequencies have been measured in dependence on the values of an external load. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Methods of analysing ropes. The extension–torsion method

Two new approaches are used for calculating the stress–strain state of a rope and its stiffnesses. The first approach relies on the theory of fibrous composites and Saint-Venant's solution for a cylinder with helical anisotropy. The second approach is based on the solution by the finite element method of the three-dimensional problem of elasticity theory for a solid inhomogeneous cylinder formed by a finite number of elastic fibres arranged in helical lines and connected by a weak filler (in the sense that its Young's modulus is several orders of magnitude less than the Young's modulus of the fibre). The behaviour of the stiffness when the modulus of elasticity of the filler tends to zero is analysed, and the results of the limiting transition are discussed. The numerical results obtained are compared with calculations by other well-known applied theories.     

Mathematical modelling and the study of the influence of initial stresses on the SIF and ERR at the crack tips in a plate-strip of orthotropic material

A plate-strip fabricated from the orthotropic material and containing a crack whose edges are parallel to the face planes of the plate is considered. It is assumed that the strip is stretched (or compressed) initially along the crack edges by uniformly distributed external normal forces acting on the simply supported ends of the plate-strip. After this initial stretching (or compression) the crack edges are loaded by additional uniformly distributed normal (opening) forces. As a result of the action of these additional forces the stress concentration characterized by the stress intensity factor (SIF) of mode I or by the energy release rate (ERR) of mixed mode arises at the crack tips. In this paper, the influence of the initial stresses on the SIF or ERR is modelled mathematically by the use of the three-dimensional linearized theory of elasticity. The aim of the present investigations is to study the effect of the mechanical–orthotropic properties of the plate-strip material on this influence by the use of the finite element method (FEM) modelling of the corresponding boundary-value problem.     

The scattering of electromagnetic wave at oblique incidence in a homogeneous chiral medium

We consider the scattering of a time-harmonic electromagnetic wave by a perfectly and imperfectly conducting infinite cylinder at oblique incidence respectively. We assume that the cylinder is embedded in a homogeneous chiral medium and the cylinder is parallel to the z axis. Since the x components and y components of electric field and magnetic field can be expressed in terms of their z components, we can derive from Maxwell's equations and corresponding boundary conditions that the scattering problem is modeled as a boundary value problem for the z components of electric field and magnetic field. By using Rellich's lemma and variational approach, the uniqueness and the existence of solutions are justified.     

The method of homogeneous solutions in an axisymmetric problem of statics for a finite cylinder with homogeneous conditions on the ends

Using the property of Papkovich generalized orthogonality of eigenfunctions, we develop a method of satisfying the boundary conditions on the lateral surface of a cylinder. The stresses and displacements in a finite cylinder with homogeneous conditions on the ends are represented in terms of the axial displacement. The solution is constructed as an expansion in a series of eigenfunctions of the corresponding homogeneous boundary-value problem. We find a class of boundary conditions that admits a solution of the problem without reduction to an infinite system of algebraic equations. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 135–139.     

Simulation and study of stratified flows around finite bodies

The flows past a sphere and a square cylinder of diameter d moving horizontally at the velocity U in a linearly density-stratified viscous incompressible fluid are studied. The flows are described by the Navier–Stokes equations in the Boussinesq approximation. Variations in the spatial vortex structure of the flows are analyzed in detail in a wide range of dimensionless parameters (such as the Reynolds number Re = Ud/ν and the internal Froude number Fr = U/(Nd), where ν is the kinematic viscosity and N is the buoyancy frequency) by applying mathematical simulation (on supercomputers of Joint Supercomputer Center of the Russian Academy of Sciences) and three-dimensional flow visualization. At 0.005 < Fr < 100, the classification of flow regimes for the sphere (for 1 < Re < 500) and for the cylinder (for 1 < Re < 200) is improved. At Fr = 0 (i.e., at U = 0), the problem of diffusion-induced flow past a sphere leading to the formation of horizontal density layers near the sphere’s upper and lower poles is considered. At Fr = 0.1 and Re = 50, the formation of a steady flow past a square cylinder with wavy hanging density layers in the wake is studied in detail.     

Static plane-strain deformation of transversely isotropic magneto-electro-elastic and layered cylinders to general surface loads

Anisotropic and layered cylinders are important composite structures; however, their system of governing equations is usually solved numerically due to the complicated geometry and material anisotropy involved. In this paper, we analytically solve the plane-strain equations for general static deformation of a cylindrically anisotropic, layered magneto-electro-elastic (MEE) cylinder. We assume that the layers are perfectly bonded at the interfaces. We solve the equations through separation of variables and eigenfunction expansion. Results for each mode shape (2π periodic) are solved independently. Because the eigenspace of the mode shapes is the set of all continuous functions on the interval, any continuous loading can be applied and the corresponding solution can be found analytically through superposition of the mode-shape results. To check our formulation, we consider a cylinder with two isotropic-elastic layers under simple radial loading and reproduce the known, exact results. Then, we compare our formulation to an FEA solution for a layered piezo-electric (PE) cylinder. Finally, we apply a radial stress to three comparable MEE cylinders (one uniform MEE cylinder and two layered cylinders made of alternating piezo-electric (PE) and piezo-magnetic (PM) materials). Deformation and stress amplitudes are plotted for the first six mode shapes of each cylinder as benchmarks for further reference.     

Nonlinear flexural vibrations of composite shallow open shells

This paper addresses geometrically nonlinear flexural vibrations of open doubly curved shallow shells composed of three thick isotropic layers. The layers are perfectly bonded, and thickness and linear elastic properties of the outer layers are symmetrically arranged with respect to the middle surface. The outer layers and the central layer may exhibit extremely different elastic moduli with a common Poisson's ratio ν. The considered shell structures of polygonal planform are hard hinged supported with the edges fully restraint against displacements in any direction. The kinematic field equations are formulated by layerwise application of a first order shear deformation theory. A modification of Berger's theory is employed to model the nonlinear characteristics of the structural response. The continuity of the transverse shear stress across the interfaces is specified according to Hooke's law, and subsequently the equations of motion of this higher order problem can be derived in analogy to a homogeneous single-layer shear deformable shallow shell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Trapped modes in a channel containing three layers of fluids and a submerged cylinder

The problem of existence of trapped waves in fluids due to a cylinder is investigated for the hydrodynamic set-up which involves a horizontal channel of infinite length and depth and of finite width containing three layers of incompressible fluids of different constant densities. The set-up also contains a cylinder which is impermeable, fully immersed in the bottom (lower-most) fluid layer of infinite depth, and extends across the channel with its generators perpendicular to the side walls of the channel. When the ratios of the densities of the adjacent fluids differ from unity by sufficiently small quantities, the underlying mathematical problem reduces to a generalized nonlinear eigenvalue problem involving a cubic polynomial-cum-operator equation. The perturbation analysis of this eigenvalue problem suggests existence of three distinct modes with different frequencies: one of the order of one persisting at the free surface, and the other two of the order of the density ratio (except for modulo one) persisting at the two internal interfaces. The correlation between these results for the three-layer case and very recent numerical results of other authors in the two-layer case has also been addressed. Received: March 3, 2005     

Method of homogeneous solutions in the mixed problem for a finite circular cylinder : PMM vol. 43, no. 6, 1979, pp. 1073–1081

The mixed axisymmetric problem of elasticity theory on the torsion of a finite circular cylinder by a stamp is considered. The stamp is fixed rigidly to one plane face of the cylinder, the other plane face is fixed, and conditions for no displacements or stresses are given on the cylinder surface. The problem is investigated by the method of homogeneous solutions [1], which permits obtaining its approximate solution for practically any values of the parameters. Such efficiency of the method is determined by the fact that the solution of the problem reduces to investigating an infinite algebraic system of the Poincaré — Koch normal systems type. When the ratio of the cylinder height to the radius of the stamp is sufficiently large, the system coefficients, the contact stresses, and the other characteristics of the problem are evaluated to any degree of accuracy, and effective asymptotic expressions are obtained for small values of this ratio. Results of numerical computations are presented.

A solution of the problem for the case of a large value of the ratio (R − a) /h and small values of the ratio λ = h / a is obtained in [2].     

Spinup from Rest of Immiscible Fluids

We investigate the spinup from rest of two immiscible fluids of different densities and kinematic viscosities in a vertically mounted cylinder. Our attention is restricted to small internal rotational Froude numbers, in which case the interface remains essentially horizontal. By requiring small enough Ekman numbers, Wedemeyer's (1964) approximation may be used to obtain partial differential equations describing the inward convection and diffusion of the azimuthal velocity in both layers. Solutions are found illustrating the effects of varying the density, viscosity, and height of each layer. Qualitative agreement with experiment is reported.     

Exponential decay for the solutions of nonlinear elliptic systems posed in unbounded cylinders

We study the asymptotic behavior at infinity of the solutions of a nonlinear elliptic system posed in a cylinder of infinite length. The problem is written in a variational formulation, where we ask the derivative of the solutions to be in Lp. We show that an exponential decay at infinity for the second member implies exponential decay for the derivative of the solutions. We also give an application of this result to the study of boundary layers problems.     

Missing boundary data reconstruction by the factorization method

We consider the data completion problem for the Laplace equation in a cylindrical domain. The Neumann and Dirichlet boundary conditions are given on one face of the cylinder while there is no condition on the other face. This Cauchy problem has been known since Hadamard (1953) to be ill-posed. Here it is set as an optimal control problem with a regularized cost function. We use the factorization method for elliptic boundary value problems. For each set of Cauchy data, to obtain the estimate of the missing data one has to solve a parabolic Cauchy problem in the cylinder and a linear equation. The operator appearing in these problems satisfy a Riccati equation that does not depend on the data. To cite this article: A. Ben Abda et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Designing a nonscattering cylindrical structure

We consider how an ideally conducting circular cylinder can be transformed into a body that does not scatter the field of a plane TE-polarized wave in a certain frequency range. The transformation is accomplished by using an appropriate system of resonators and narrow longitudinal slits in the cylindrical body (a “transparent” body synthesis problem). Sufficient conditions for the solvability of the synthesis problem are derived. The synthesis problem is solved numerically for a number of primary field frequencies, and the frequency dependence of the decay coefficient is obtained. __________ Translated from Prikladnaya Matematika i Informatika, No. 20, pp. 16–39, 2005.     

Packing cylinders on a cylinder with contact constraints

The problem of cylinder packing is investigated. The specific problem is to determine the maximum number of congruent cylinders that can be packed around a core cylinder of arbitrary dimensions. The constraint is that their circular face must keep in contact with the core cylinder and there may be no overlapping. Only right circular cylinders are considered. Mathematically, a lower and upper bound is determined. A quantitative result is also found using a modified genetic algorithm. The algorithm was found to reproduce the published results for the top and bottom circular faces of the core which reduces to the problem of packing congruent circles within a circle. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)