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.     

Stabilization and control of the motion of a rolling disk

This work deals with the stabilization and control of the motion of a disk rolling on the horizontal 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 given smooth ground trajectory.     

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/.     

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.     

Diffraction of a plane wave by a circular disk

In this paper we solve the problem of diffraction of a normally incident plane wave by a circular disk. We treat both the hard and soft disk. In each case we obtain the solution as a series which converges when the product of the wave number and the radius of the disk is large. Our construction leads directly to asymptotic approximations to the solution for large wave number.     

The rolling of a wheel with a reinforced tyre along a plane without slip

A model of a wheel with a reinforced tyre, whose surface is simulated by a flexible strip (tread) attached to parts of two tori (the sidewalls of the tyre) is considered. The disk of the wheel (a rigid body) has six degrees of freedom and is in contact with the plane along part of the tread. Based on several assumptions, the potential energy functional of the deformed wheel is found as a function of the deformations of the centre line of the tread. On the assumption that the wheel is rolling without slip in the region of contact of the tread with the plane along a previously unknown section of the tread, the complete system of equations of motion is obtained. The equilibrium of the wheel and the steady state of rolling in a straight line with given swivel and tilt are investigated, and all characteristics of the motion are found (the contact region, the tyre deformation, and the forces and torques applied to the wheel disk).     

On the dynamics of a rigid body carrying a material point

In this paper we consider a system consisting of an outer rigid body (a shell) and an inner body (a material point) which moves according to a given law along a curve rigidly attached to the body. The motion occurs in a uniform field of gravity over a fixed absolutely smooth horizontal plane. During motion the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. We present a derivation of equations describing both the free motion of the system over the plane and the instances where collisions with the plane occur. Several special solutions to the equations of motion are found, and their stability is investigated in some cases. In the case of a dynamically symmetric body and a point moving along the symmetry axis according to an arbitrary law, a general solution to the equations of free motion of the body is found by quadratures. It generalizes the solution corresponding to the classical regular precession in Euler??s case. It is shown that the translational motion of the shell in the free flight regime exists in a general case if the material point moves relative to the body according to the law of areas.     

Velocity and pressure distribution in the gap of a disk extruder

The axisymmetric two-dimensional flow of a polymer melt in the plane gap of a disk extruder produced by the normal stress effect is considered. The polymer is assumed to be a nonlinear viscoelastic medium, whose strain history is expressed by means of kinematic matrices. A rheological equation of state of the medium, in which all the invariants of the kinematic matrices are function of strain rate intensity, is established. The laws of distribution of the radial and tangential velocity components over the gap are found from the solution of the equations of motion, and expressions are obtained for the radial pressure distribution and the integral thrust.Volgograd Polytechnic Institute. Translated from Mekhanika Polimerov, No. 3, pp. 515–521, May–June, 1971.     

Hydromagnetic forced flow against a porous rotating disk

This paper deals with the steady forced flow of a viscous, incompressible and electrically conducting fluid against a porous rotating disk when a uniform magnetic field acts perpendicular to the disk surface. For small suction the equations of motion are integrated numerically by Kármán-Pohlhausen method, but for large suction a series solution in the inverse powers of the suction parameter is obtained. The effects of disk porosity and magnetic field on the various flow parameters are discussed in detail.     

Non-linear dynamic analysis of a flexible rotor supported on porous oil journal bearings

In the present paper, the non-linear dynamic analysis of a flexible rotor with a rigid disk under unbalance excitation mounted on porous oil journal bearings at the two ends is carried out. The system equation of motion is obtained by finite element formulation of Timoshenko beam and the disk. The non-linear oil-film forces are calculated from the solution of the modified Reynolds equation simultaneously with Darcy’s equation. The system equation of motion is then solved by the Wilson-θ method. Bifurcation diagrams, Poincaré maps, time response, journal trajectories, FFT-spectrum, etc. are obtained to study the non-linear dynamics of the rotor-bearing system. The effect of various non-dimensional rotor-bearing parameters on the bifurcation characteristics of the system is studied. It is shown that the system undergoes Hopf bifurcation as the speed increases. Further, slenderness ratio, material properties of the rotor, ratio of disk mass to shaft mass and permeability of the porous bush are shown to have profound effect on the bifurcation characteristics of the rotor-bearing system.     

Computation of a rotating flow past a permeable bed

Summary The rotating flow of a viscous incompressible fluid between two disks is studied when there is a porous layer on the lower disk. The motion relative to a rotating frame is caused by a differential rotation of the disks. Generalised Darcy's law represents the flow. The numerical solution is obtained using a shooting method.     

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.     

On Shunkov groups with a strongly embedded subgroup

The problem of reconstructing a locally Euclidean metric on a disk from the geodesic curvature of the boundary given in the sought metric is considered. This problem is an analog and a generalization of the classical problem of finding a closed plane curve from its curvature given as a function of the arc length. The solution of this problem in our approach can be interpreted as finding a plane domain with the standard Euclidean metric whose boundary has a given geodesic curvature.     

Integral equation solution for the flow due to the motion of a body of arbitrary shape near a plane interface at small Reynolds number

The problem of determining the slow viscous flow due to an arbitrary motion of a particle of arbitrary shape near a plane interface is formulated exactly as a system of three linear Fredholm integral equations of the first kind, which is shown to have a unique solution. A numerical method based on these integral equations is proposed. In order to test this method valid for arbitrary particle shape, the problem of arbitrary motion of a sphere is worked out and compared with the available analytical solution. This technique can be also extended to low Reynolds number flow due to the motion of a finite number of bodies of arbitrary shape near a plane interface. As an example the case of two equal sized spheres moving parallel and perpendicular to the interface is solved in the limiting case of infinite viscosity ratio.     

Plane brownian motion in a region with holes

We consider a fixed family of balls with decreasing radii in the plane. We establish a relationship between a Dirichlet problem in a region without the balls and the solution of a Schroedinger equation in the complete region. Then we find upper bounds for the probability that a brownian motion exits the region without touching these balls. This is used to study harmonic measure and entire functions.     

On Two-Phase Flow in a Rotating Boundary Layer

The two-phase flow induced by a rotating disk in a stationary unbounded mixture is considered. The generalized similarity assumption of von Karman reduces the averaged equations of motion with a linear drag between the phases to a system of ordinary differential equations. These are investigated by asymptotic and numerical techniques. The equations display a nontrivial behavior in a sublayer near the boundary, whose thickness is of the order of the particle size. The volume fraction of the dispersed phase is singular unless a small suction is applied on the disk or a small diffusion term is added to the continuity equations. Outside this sublayer, the velocity field is quite similar to a rescaled classical von Karman flow. Good agreement between asymptotic and numerical solution is obtained, although there is considerable stiffness in the equations. The motion of a solid particle in a von Karman flow is also discussed, but the present investigation is restricted to small radii because the shear-lift force is neglected.     

A qualitative analysis of the dynamics of a disc on an inclined plane with friction

The problem of the motion of a disc on an inclined plane with dry friction is investigated. It is shown that, if the friction coefficient is greater than the slope of the plane, the disk will come to rest after a certain finite time, and its sliding and rotation will cease simultaneously. The limit position of the instantaneous centre of velocities is indicated. The limit motions of the disc in the case when the ratio of the friction coefficient to the slope of the plane is equal to or less than unity: uniform sliding (in the case of a general position) and equiaccelerated sliding (always) of the disc along the line of greatest slope of the plane, respectively, are obtained. The case when the friction coefficient is equal to the slope, while the initial sliding velocity is directed upwards along the line of greatest slope, is an exception. In this case, the disc comes to rest after a finite time, and the sliding velocity and the angular velocity of the disc vanish simultaneously.     

Motion of a rolling disk by a new generalized Hamilton-Jacobi method

Summary The rolling motion of a disk on a horizontal plane in a gravitational field is studied by applying a new generalized Hamilton-Jacobi method. This method covers systems subject to nonholonomic Chetaev's constraints and is based on a variational principle for the integral of action. The particular case of the uniform circular rolling of the disk is discussed in detail.

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Résumé Le roulement d'un disque sur un plan horizontal dans un champ gravitationnel est étudié en utilisant une nouvelle méthode généralisée de Hamilton-Jacobi. Cette méthode s'applique à des systèmes soumis à des contraintes non-holonomes du type de Chetaev et est basée sur un principe variationnel pour l'intégrale de l'action. Le cas particulier du roulement uniforme circulaire du disque est discuté en détail.

     

On detachment conditions in the problem on the motion of a rigid body on a rough plane

The classical mechanical problem about the motion of a heavy rigid body on a horizontal plane is considered within the framework of theory of systems with unilateral constraints. Under general assumptions about the character of friction, we examine the question on the possibility of detachment of the body from the plane under the action of reaction of the plane and forces of inertia. For systems with rolling, we find new scenarios of the appearing of motions with jumps and impacts. The results obtained are applied to the study of stationary motions of a disk. We have showed the following.

The effects of lateral–torsional coupling on the nonlinear dynamic behavior of a rotating continuous flexible shaft–disk system with rub–impact

This study investigates the lateral–torsional coupling effects on the nonlinear dynamic behavior of a rotating flexible shaft–disk system. The system is modeled as a continuous shaft with a rigid disk in its mid span. Coriolis and centrifugal effects due to shaft flexibility are also included. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed mode method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work include time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The main objective of the present study is to investigate the torsional coupling effects on the chaotic vibration behavior of a system. Periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for cases with and without torsional effects. As demonstrated, inclusion of the torsional–lateral coupling effects can primarily change the speed ratios at which rub–impact occurs. Also, substantial differences are shown to exist in the nonlinear dynamic behavior of the system in the two cases.