Rough Horizontal Plane Research Articles

The theory of the generalized magnus effect for non-holonomic mechanical systems

The motion of a carriage with two wheeled pairs over a rough horizontal plane is investigated in the following cases: (1) inertial motion, (2) when there is an elastic constraint which produces a restoring moment of the forces when the axis of rotation of the leading wheeled pair deviates from the unperturbed position, and (3) when there is a small harmonic moment between the leading wheeled pair and the platform. The properties of the exact solution of the system is analysed in the first case. In the second, using the method of averaging, it is shown that for small oscillations of the leading wheeled pair with respect to the platform, after a transition process, motion of the centre of mass of the system with constant velocity, proportional to the initial amplitude of the oscillation, occurs. In the third case, the average motion of the centre of mass occurs with a constant acceleration, the value of which is estimated using the asymptotic multiscale method.

Decomposition of the problem of controlling a mobile one-wheel robot with an unperturbed gyrostabilized platform

The design of the mobile robot under consideration is based on the analysis of the classical problem of rolling a heavy disc that carries a flywheel and whose rotation axis Cz is perpendicular to the disc plane and passes through its center of mass C on an absolutely rough horizontal plane [1] (see Fig. 1). The nonholonomic mechanical system “disc + flywheel” has four degrees of freedom and is a gyrostat. Its position is uniquely specified by six generalized coordinates: Euler angles ψ, θ, and φ between the trihedron rigidly bound to the principal central axes of inertia of the disc and stationary trihedron OX*Y*Z*, angle γ of the flywheel rotation with respect to the disc about the Cz axis, and coordinates x and y of the projection of the disc’s center of mass in the plane OX*Y*, where the disc rolls. The problem involves three cyclic coordinates ψ, θ, and φ and the energy integral. Thus, the analytical analysis of the equations of motion of a disc with a flywheel reduces to integrating one second-order linear differential equation of the form

The equilibrium conditions of a rod on a rough plane

The problem of the equilibrium of a rod on a rough horizontal plane when there are dry friction forces is considered. The equilibrium conditions which ensure that the rod remains at rest are determined by solving the problem of an extremum. The results obtained are compared with the well-known result for a Zhukovskii bench.

Nonlinear oscillations in the rolling motion of Euler’s disk

The fixed points of the dynamical system describing the rolling motion of a uniform disk or a uniform circular hoop on a rough horizontal plane without dissipation are analyzed. All fixed points of the system are stated to be steady motion states which are either saddle points or centers. The spectral analysis of the sound amplitude time series emitted by a toy of the rolling disk shows that the similar nonlinear oscillations are expected in the motion with dissipation. If under adiabatic approximation a finite-time singularity occurs, the bifurcation in the disk motion can appear.

The stability of the steady motions of a symmetrical gyrostat on an absolutely rough horizontal plane

The problem of the motion, without slipping, of a dynamically symmetrical gyrostat on a fixed horizontal plane is investigated. It is shown that Chaplygin's equations express, in projections on the axes of a semimobile system of coordinates, the theorem on the angular momentum of the gyrostat about the point of contact with the plane. All possible steady motions of a heavy symmetrical gyrostat on an absolutely rough plane including the case when the nutation angle is equal to zero are investigated using the generalized Routh theorem. The effect of the rotor on the stability of steady motions is also examined. The case when the body of the gyroscope is a solid with a circular base, in particular, a disc with a rotor, is considered. It is shown that the rotating rotor has a stabilizing influence on equilibrium of the gyrostat and a destabilizing influence on the rolling of the gyrostat about a straight line, provided the axial angular momentum of the gyrostat is zero.

Periodic Folding of Thin Sheets

When a thin sheet of a flexible material such as paper is fed from a horizontal spool towards a rough horizontal plane below it, the sheet folds on itself in a regular manner. We model this phenomenon as a free boundary problem for a nonlinearly elastic sheet, taking into account the stiffness and weight of the sheet and the height of the spool above the plane. By using a continuation scheme we solve the problem numerically and follow the evolution of one period of the fold for various values of the parameters. The results are found to agree well with observations of the folding of paper sheets.

The steady motions of a disc on an absolutely rough plane

The branching of the steady motions of a heavy circular disc on an absolutely rough horizontal plane is investigated. The motions corresponding to critical points of the energy integral at fixed levels of two other integrals having the form of hypergeometric series are considered.

Skateboard dynamics

Dynamics of the controlled motion of a skateboard over rough horizontal plane is considered. The control is realized by means of purpose of change of displacements and turnings of a sportsman's body. Propulsion force that provides the acceleration of skateboard and overcomes friction drag is produced by reaction forces of non-holonomic constraints. Differential equations of the motion are derived in Appell's form. Dependence of the skateboard velocity and its trajectory on the motions of a sportsman's body is obtained.

Periodic Folding of Thin Sheets

When a thin sheet of a flexible material such as paper is fed from a horizontal spool towards a rough horizontal plane below it, the sheet folds on itself in a regular manner. We model this phenomenon as a free boundary problem for a nonlinearly elastic sheet, taking into account the stiffness and weight of the sheet, and the height of the spool above the plane. By using a continuation scheme we solve the problem numerically and follow the evolution of one period of the fold for various values of the parameters. The results are found to agree well with observations of the folding of paper sheets.

The motion of a sphere with weak nutation on a rough horizontal plane

For the motion of a sphere on a rough horizontal plane, in the previous paper [1], the author aimed at providing approximate analytical solutions while the nutation is neglected. In this paper, the control equations for the sphere with nutation have been deduced on the basis of paper [1]. Through the medium of solving these equations, the conclusion for the velocity of contact point in paper [1] is still proved true for the case with nutation. What is more, some interesting results are gained, for example, the velocity of centre and contact point is relative to the angular velocity of spin and nutation; the direction of velocity of centre and contact point is constant. Under the condition which is supposed to be weak nutation, the approximate analytical solutions are obtained, so that the results of paper [1] is proved to be true.

Specific features of the application of Routh's theory to systems with differential constraints

Specific points connected with the application of Routh's theory to the study of steady motions in systems with differential constraints are discussed, on the assumption that first integrals exist. The need to eliminate all dependent variables initially from the expressions for the first integrals is pointed out. The general conclusions are illustrated by investigating, as an example, the steady motions of a dynamically symmetric sphere on an a absolutely rough horizontal plane.

The stability of a fluid-filled top rotating on a horizontal plane

The stability of a fluid-filled top rotating on a rough horizontal plane about the erect axis is discussed and a stability criterion in analytical form is obtained. A sufficient condition of stability for a fluid-filled Chaplygin sphere is proved by using Lyapunov's direct method, and a more satisfactory explanation to Kelvin's question considering the influence of gravity and sliding friction is given. The fluid-filled Lagrange gyro is discussed as a special case when the plane is absolutely smooth.

The Motion of a Sphere on a Rough Horizontal Plane

The Motion of a Sphere on a Rough Horizontal Plane

On the motion of a particle over a rough rotating plane

The motion of a particle over a rough horizontal plane that rotates with constant angular velocity about a vertical axis is studied. The particle is acted on by dry friction with the plane that satisfies Coulomb's law. The equations of motion are solved asymptotically close to the point where the particle stops. Various types of motion with stopping are found. The motions are simulated numerically over the full range of variation of the parameters and initial conditions. As a result, typical trajectories of particle motion are constructed. The integral manifold which isolates in phase space the motions with stopping and the motions in which the particle moves away to infinity is found. The phase trajectories are constructed in this manifold.

The stability of a top rotating on a rough horizontal plane

In the present paper the stability of permanent rotation of a symmetric top rotating on a slightly rough horizontal plane about the erect polar or transverse axis is discussed via the perturbation method. As a perturbation factor, the frictional force is an arbitrary nonlinear function of the sliding velocity. The same stability criterion as that in the Contensou-Magnus linear theory is obtained, but the hypothesis of the linearity for friction can be omitted. It shows that the direction of the sliding velocity at the contact point is an important factor, which influences the stability of the top. And then an explicit physical explanation is given.

Stability of the uniform rotation of a gyrostat round the vertical main axis on an absolutely smooth horizontal plane

The motion of a gyrostat on an absolutely smooth plane is discussed. A Hamilton function which gives the canonical equations of motion is obtained. This admits of particular solutions, namely uniform rotations round a vertical axis which are identical with that of the uniform rotations of the rotor. A transition to a system with two degrees of freedom is realized, and the expansion of the Hamiltonian in the vicinity of the corresponding position of equilibrium, with an accuracy to within fourthorder terms, is obtained. In the region of admissible values of the parameters the domain of the necessary stability conditions, and the domains where the Hamiltonian functions are of fixed sign and alternating, are examined. In those cases where the Hamiltonian is not fixed sign, its normalization is performed, both a non-resonance situation and resonances of the first, second and fourth order being considered. The sufficient conditions for stability of uniform gyrostat rotation in terms of constraints on the coefficients of normal forms are obtained. For a clear interpretation of the results, special cases where the values of all the parameters except two are fixed, are given. The plane domain of the necessary stability conditions and resonance curves are constructed, and using computer results stability on the curves is discussed. The stability of uniform rotations of a heavy solid around the vertical principal and minor axes on an absolutely smooth, and on an absolutely rough horizontal plane, and also on a plane with viscous friction is discussed in /1–4/. The stability of uniform rotations of a gyrostat round the vertical principal axis on absolutely smooth and absolutely rough horizontal planes was considered in /5, 6/. Investigations of the motion of a solid on an absolutely rough plane, the body being perturbed with respect to rotation round the principal axis (in particular with respect to the steady position of equilibrium), are described in /7, 8/. The stability of two types of rotation of a homogeneous ellipsoid on an absolutely smooth horizontal plane, in particular the stability of the uniform rotations of an ellipsoid round the vertical principal axis is discussed in /9/.

On realizing nonholonomic constraints by viscous friction forces and celtic stones stability

It is shown that equations of motion of nonholonomic systems can be obtained from the equations of motion of systems freed of nonholonomic constraints subjected to suitably chosen dissipative forces, if in the latter the dissipation coefficient is assumed infinite. It is shown that on fairly general assumptions motions of non-holonomic systems represent limits of motions of corresponding holonomic systems, as the dissipation coefficient approaches infinity. The stability of rotation on a horizontal plane of a heavy asymmetric rigid body (Celtic stone) about the vertical, with allowance for friction, is investigated. The obtained stability conditions are compared with previously published papers about rotation of a body on an absolutely rough horizontal plane around the vertical.

On permanent rotations of a heavy solid body on an absolutely rough horizontal plane

Conditions of existence and stability of permanent rotations of a heavy solid body on an absolutely rough horizontal plane are obtained. A well-defined analogy of the investigated problem to problems of permanent rotations of a body on an absolutely smooth plane and of a body with a fixed point is pointed out, as well their essential differences. In the case of the absolutely rough plane an asymptotic stability of permanent rotations is possible with respect to a part of variables, although the system is conservative. Stability then depends on the direction of rotation.

On the stability of steady motions of a heavy body of revolution on an absolutely rough horizontal plane

On the stability of steady motions of a heavy body of revolution on an absolutely rough horizontal plane

Three Interesting Instances of Rocking

(1) Two equal thin circular rings are firmly jointed together at one point so that their planes are inclined to each other. The combination is placed on a rough horizontal plane and made to rock to-and-fro. The period is calculated and confirmed by experiment. Repetition is made with jointed disks. (2a) A right solid cone is suspended by the vertex from a fixed point on a rough vertical plane. It is made to rock through small oscillations. The period is calculated and confirmed by experiment. (2b) A right solid cone results in equilibrium on a rough plane inclined to the horizontal. It is set rocking and the period calculated and confirmed by experiment. Repetition is performed with a thin conical shell. (3) A thin circular ring is placed standing on a rough horizontal plane and leaning against a rough vertical wall. It is set rocking to-and-fro and the period calculated and confirmed by experiment. Repetition is made with leaning disks and wide bands. Variants of (3) are also performed.