Modelling and control of the motion of a disk rolling on a spherical dome

This work deals with the modelling and control of the motion of a disk rolling without slipping on a rigid spherical dome. It is assumed here that the motion of the disk is controlled by a tilting moment, a directional moment, and a pedalling moment. First, a mathematical model of the motion of the disk rolling on the dome is derived. Then, by using a kind of an inverse control transformation, a control strategy is proposed under which the motion of the disk is stabilized and is able asymptotically to track any smooth trajectory which is located on the spherical dome.     

Modelling and control of the motion of a disk rolling on a sloping plane

This work deals with the stabilization and control of the motion of a disk rolling on a sloping plane. It is assumed here that the motion of the disk is controlled by a tilting moment, a directional moment, and a pedalling moment. By using a kind of an inverse control transformation a control strategy is proposed under which the motion of the disk is stabilized and is able asymptotically to track any smooth trajectory which is located on the sloping plane.     

Evolution of a regular precession of a solid body carrying a visco-elastic disk

The motion of inertia is studied of a system consisting of an axisymmetric solid body with fixed point and a homogeneous visco-elastic disk lying in the equatorial plane of the ellipsoid of inertia of the solid body (the center of disk coincides with the fixed point). In the case of a solid disk immobilized relative to the solid body the system accomplishes a regular precession (the case of Euler motion of a symmetric solid body with a fixed point /1/). The deformation of the disk is taking place in the plane of the disk, and is accompanied by energy dissipation is the cause of the regular precession finishing by steady rotation about the vector of the moment of momentum of the system /2/.     

Vibration suppression of self-excited oscillations by parametric inertia excitation

The main objective of this contribution is to show the phenomenon of full vibration suppression of a simple two degrees of freedom rotary oscillator by interaction between self-excitation and parametric excitation. One disk is under the influence of self-excitation, modelled by a negative damping coefficient, while the moment of inertia of the second disk is periodically varied in time within an open-loop control with a fixed frequency. Both disks are coupled by a linear spring-element. Parametric excitation develops equations of motion with time-periodic coefficients. Using the averaging method for a firstorder approximation general conditions for full vibration suppression are analytically derived for the two degrees of freedom system with harmonic inertia variation. The approximated analytical stability predictions are verified and compared to results obtained from numerical time integration of the original equations of motion. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Feasible command strategies for the control of motion of a simple mobile robot

The concept of feasible command strategies is introduced and its applicability is demonstrated by solving a guidance and control problem. This problem concerns the motion of a system which is composed of a rolling disk and a controlled slender rod that is pivoted, through its endpoint, about the disk center. The motion of the disk-rod system is subjected to state and control constraints, and it serves as a model for the motion of a simple mobile robot. In addition, the concept of path controllability is introduced and a condition is derived for the system motion path controllability. The derivation of this condition enables one to design closed-loop control laws for the system motion.     

Nonaxisymmetric Free Convection Flow over a Rotating Disk in a Viscoelastic fluid with Magnetic Field

The non axisymmetric motion produced by a buoyancy-induced secondary flow of a viscoelastic fluid over an infinite rotating disk in a verticalplane with a magnetic field applied normal to the disk has been studied.The governing Navier Stokes equations and the energy equation admit a self similar solution. The system of ordinary differential equations has been solved numerically using Runge-Kutta Gill subroutine.The turning moment for the viscoelastic fluid is found to be less than that of the Newtonian fluid but the turning moment is increased due to the magnetic parameter. The resultant force due to the buoyancy-induced secondary flow increases with the magnetic parameter but reduces as the viscoelastic parameter increases. The quantity of fluid, which is pumped outwards due to the centrifuging action of the disk, for the viscoelastic fluid is more than that of the Newtonian fluid. The buoyancy-induced secondary flow boundary layer is much thicker than the primary boundary layer thickness. The thermal boundary layer due to the primary flow increases with the magnetic parameter decreases as the viscoelastic parameter increases. The heat transfer increases with the viscoelastic parameter but decreases as the magnetic parameter increases. The effect of the viscoelastic parameter is more pronounced on the secondary flow than on the primary flow.     

The problem of the time-optimal turning of a manipulator

Two types of manipulator that perform three-dimensional motions are considered, and the control problem in which the manipulator rotation is performed in minimum time is studied. The rate of rotation of a rigid body about an axis rises as the moment of inertia about this axis falls. Manipulator control amounts to a problem of the rotation of a system of rigid bodies about an axis. In addition to the angle of rotation, there is a further controlled coordinate, whose variation can vary the moment of inertia about the axis. Assuming that the moment of inertia can be stantaneously “frozen” (that pulse control signals are possible), the in-time-optimal control modes were found in /1, 2/, (see also Akulenko, L.D. et al., “Optimization of the control modes of manipulation robots”, Preprint 218, In-t. Problem Mekhaniki Akad. Nauk SSSR, Moscow 1983). In these modes, the rotation, occurs in the entire time interval with minimum moment of inertia about the axis of rotation. The rotation when there are constraints on the control (pulse control signals are not permitted) was considered in /3/. Numerical studies there led to the false conclusion that, in the optimal motion, with a finite number of control switchings, the moment of inertia is also a minimum throughout the time interval. Below, for a set of extreme configurations, a control is constructed for the two types of manipulator, which satisfies the Pontryagin maximum principle, when there are constraints on the control signals. During its rotation the manipulator section then performs oscillations about a position corresponding to minimum moment of inertia about the axis of rotation. It is shown that the motion considered in /3/, which contains a singular mode with minimum moment of inertia, is not optimal. The motion which satisfies the maximum principle is compared with it. There can be a singular mode in the optimal motion /4/ only when the number of control switchings is infinite.     

Stabilization and optimal decay rate for a non-homogeneous rotating body-beam with dynamic boundary controls

In this paper, we consider the boundary stabilization of a flexible beam attached to the center of a rigid disk. The disk rotates with a non-uniform angular velocity while the beam has non-homogeneous spatial coefficients. To stabilize the system, we propose a feedback law which consists of a control torque applied on the disk and either a dynamic boundary control moment or a dynamic boundary control force or both of them applied at the free end of the beam. By the frequency multiplier method, we show that no matter how non-homogeneous the beam is, and no matter how the angular velocity is varying but not exceeding a certain bound, the nonlinear closed loop system is always exponential stable. Furthermore, by the spectral analysis method, it is shown that the closed loop system with uniform angular velocity has a sequence of generalized eigenfunctions, which form a Riesz basis for the state space, and hence the spectrum-determined growth condition as well as the optimal decay rate are obtained.     

Application of self-adjoint boundary value problems to investigation of stability of periodic delay systems

The problem on determining conditions for the asymptotic stability of linear periodic delay systems is considered. Solving this problem, we use the function space of states. Conditions for the asymptotic stability are determined in terms of the spectrum of the monodromy operator. To find the spectrum, we construct a special boundary value problem for ordinary differential equations. The motion of eigenvalues of this problem is studied as the parameter changes. Conditions of the stability of the linear periodic delay system change when an eigenvalue of the boundary value problem intersects the circumference of the unit disk. We assume that, at this moment, the boundary value problem is self-adjoint. Sufficient coefficient conditions for the asymptotic stability of linear periodic delay systems are given.     

Construction of the Control Realizing the Rotation of a Timoshenko Beam

We consider the linear model of a slowly rotating Timoshenko beam in a horizontal plane whose moment is controlled by the angular acceleration of the disk of the driving motor into which the beam is clamped. This work complements our previous results on the controllability of the beam from a position of rest into another position of rest; we give a method of construction of a piecewise constant control solving the problem.     

The nearly horizontally rolling of a thick disk on a rough plane

In the paper the analytical solution of the partially linearized equations of motion of nearly horizontally rolling of a thick rigid disk on a perfectly rough horizontal plane under the action of gravity is given in terms of Whittaker functions. The solution is used to obtain the asymptotic solutions for a very small inclination angle, the study of unilateral contact between a disk and a plane and the study of disk colliding motion.      

On Synchronization of Control Systems

In the present work the problem of synthesis of synchronizing control is considered. The point is that, instead of stabilization problem we have to construct a special control under which the close‐loop system realized a given programmed motion starting at a fixed moment of time. The theorem on the necessary and sufficient conditions of synchronizability of linear control systems is proved.     

An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane

This paper addresses the problem of an inhomogeneous disk rolling on a horizontal plane. This problem is considered within the framework of a nonholonomic model in which there is no slipping and no spinning at the point of contact (the projection of the angular velocity of the disk onto the normal to the plane is zero). The configuration space of the system of interest contains singular submanifolds which correspond to the fall of the disk and in which the equations of motion have a singularity. Using the theory of normal hyperbolic manifolds, it is proved that the measure of trajectories leading to the fall of the disk is zero.     

Modelling the motion of a disk Rolling on a curve in R: The case where the motion is driven by using two overhead rotors

This work deals with the modelling of the motion of a disk rolling without slipping ona regular curve in R³. The disk's motion is driven by a pedalling torque and by using two overhead rotors.     

Boundary-value problems for an elliptic equation with complex coefficients and a certain moment problem

Elliptic systems of two second-order equations, which can be written as a single equation with complex coefficients and a homogeneous operator, are studied. The necessary and sufficient conditions for the connection of traces of a solution are obtained for an arbitrary bounded domain with a smooth boundary. These conditions are formulated in the form of a certain moment problem on the boundary of a domain; they are applied to the study of boundary-value problems. In particular, it is shown that the Dirichlet problem and the Neumann problem are solvable only together. In the case where the domain is a disk, the indicated moment problem is solved together with the Dirichlet problem and the Neumann problem. The third boundary-value problem in a disk is also investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1476–1483, November, 1993.     

The oscillations of a body with an orbital tethered system

The motion about a centre of mass of a rigid body with a tethered system, designed to launch a re-entry capsule from a circular orbit is considered. In the deployment of the tethered system the direction and value of the tensile strength of the tether vary and, if the point of application of the tensile strength does not coincide with the centre of mass of the body, a moment occurs which leads to oscillations of the body with variable amplitude and frequency. A non-linear equation of the perturbed motion of the body about the centre of mass under the action of the tensile force of the tether and the gravitational moment is derived. Assuming that the change in the value and direction of the tensile force is slow and also that the gravitational moment is small, approximate and exact solutions of the non-linear differential equation of the unperturbed motion are obtained in terms of elementary functions and elliptic Jacobi functions. For perturbed motion, the action integral is expressed in terms of complete elliptic integrals of the first and second kind.     

On boundary-value problems for a second-order differential equation with complex coefficients in a plane domain

We study boundary-value problems for a homogeneous partial differential equation of the second order with arbitrary constant complex coefficients and a homogeneous symbol in a bounded domain with smooth boundary. Necessary and sufficient conditions for the solvability of the Cauchy problem are obtained. These conditions are written in the form of a moment problem on the boundary of the domain and applied to the investigation of boundary-value problems. This moment problem is solved in the case of a disk.     

On the condensed density of the zeros of the Cauchy transform of a complex atomic random measure with Gaussian moments

An atomic random complex measure defined on the unit disk with normally distributed moments is considered. An approximation to the distribution of the zeros of its Cauchy transform is computed. Implications of this result for solving several moment problems are discussed.     

Spatial chaotic vibrations when there is a periodic change in the position of the centre of mass of a body in the atmosphere

The spatial chaotic motion of a blunt body in the atmosphere when there is a periodic change in the position of the centre of mass is considered. A restoring moment, described by a biharmonic dependence on the spatial angle of attack, a small perturbing moment, due to the periodic change in the position of the centre of mass, and also a small damping moment, acts on the body. The motion when the velocity head remains constant is investigated. When there are no small perturbations, the phase portrait of the system can have points of stable and unstable equilibrium. The behaviour of the system in the neighbourhood of the separatrice is investigated using Mel’nikov's method. An analytic solution of the equation of the body motion along the separatrice is obtained. The criteria for the occurrence of chaos are obtained and the results of numerical modelling, which confirm the correctness of the solutions obtained, are presented.