Stability and stabilization of motion of a uniaxial wheel transport platform

We consider a uniaxial wheel transport platform with a single-degree-of-freedom gyroscope moving without slipping either on a plane nonrotating horizontal surface or on the spherical rotating Earth surface. We obtain a general mathematical model which, in a special case, coincides with the model in the form of Chaplygin equations, which permits obtaining a physical interpretation of the Chaplygin equations. In the case of stationary motion where only the balance weight is controlled, we find the minimum value of the gyro angular momentum that ensures the system stability. An example with parameters of the breadboard model is used to consider the problem of the stationary motion stability and stabilization without gyro; the control matrix minimizing the quadratic performance functional is obtained. The characteristic curves of the transient process in the system are given.     

Friction force model and rolling dynamics for a ball on a rough surface with presliding displacement taken into account

A model of sliding and spinning friction forces for a ball in the form of finite relations obtained by integrating the tangential stresses over the contact area whose parameters are determined by Hertz’s theory for the “ball-rough horizontal surface” tribological conjunction pair is supplemented with a model of rolling friction torques. The combined model is peculiar in that the presliding displacement effect in rolling and spinning friction torques is taken into account. It is shown that the ball motions in the presliding displacement zone are of quasilinear character and, under shock perturbations, have the form of damping vibrations in the three orientation angles. The numerical parameters of the rolling and spinning friction model are experimentally determined for the presliding displacement zones, while the sliding friction parameters and partly the spinning friction parameters are calculated. Mathematical modeling permits one to discover new properties of the ball, namely, its deceleration in rolling, the onset of damping vibrations at the beginning and end of motion, and the transient process parameters.     

Using the Dirac approach to simulate the rolling of a wheeled vehicle

The rolling of a wheeled vehicle is considered in the case when the turning angles of the front wheels about the vertical axis are small. The small relative slip is taken into account in the adopted model of contact between the wheels and the road surface. It is shown that, when the stiffness of the wheels tends to infinity, the system of equations of motion may become nonclassical and its form is specified by the no-slip conditions along the longitudinal direction of motion and by the primary Dirac constraints arising because of the degeneracy of the Lagrangian.     

Plastic deformation of a rough surface

Journal of Applied Mechanics and Technical Physics -     

Conditions for pure rolling of a heavy cylinder along a brachistochrone

Algebraic equations for the line of steepest descent of a cylinder are derived in parametric form. Conditions for rolling without slipping and separation of the cylinder along a brachistochrone are established based on the equations of motion with constraint reaction. The important conclusion is drawn that the center of mass of a cylinder moving along a brachistochrone describes a cycloid     

On the motion of a rope on a rough surface

Summary The article deals with dynamical problems of stability of mechanical systems with an infinite number of degrees of freedom, which are acted upon by forces of the dry friction type. Dynamics of a rope moving on a rough flat surface is considered as an example. The article shows that nonhomogeneous properties of friction can result in instability.

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Über die Bewegung eines Seiles auf einer rauhen Fläche

Übersicht Es handelt sich um dynamische Probleme der Stabilität mechanischer Systeme mit unendlich vielen Freiheitsgraden, auf die Kräfte der trockenen Reibung wirken. Die Dynamik eines Seiles, das sich auf einer rauhen Fläche bewegt, wird als Beispiel betrachtet. Es wird gezeigt, daß nichthomogene Eigenschaften der Reibung Instabilität verursachen können.

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This work was supported by the DAAD (der Deutsche Akademische Austauschdienst). I am very grateful to the Director of the Institute B of Mechanics, the University of Stuttgart, Prof. Dr.-Ing. W. Schiehlen for wonderful conditions under which I could work during my stay at his Institute. Without his support, encouragements and useful advice this work could hardly be written.     

Integrability of the motion of a rolling disk of finite thickness on a rough plane

In this article an analytical solution of equations of motion of a rigid disk of finite thickness rolling on its edge on a perfectly rough horizontal plane under the action of gravity is given. The solution is given in terms of Gauss hypergeometric functions. The integrability results are used to construct various bifurcation diagrams of the steady motion of the disk. The bifurcations of the steady motion of a disk on a rough plane complements the author's bifurcation analysis of the steady motion of the disk on a smooth plane ( [M. Batista, Steady motion of a rigid disk of finite thickness on a horizontal plane, Int. J. Non-Linear Mech. 41 (4) (2006) 605–621]).     

Boundary conditions of gas dynamic slip on a rough surface

The phenomenon of gas dynamic slip associated with the flow of a monatomic, slightly rarefied gas over a rough surface is investigated. It is assumed that the characteristic dimensions of the roughness are comparable with the molecular mean free path. It is shown that if there is anisotropy of the surface shape the relation between the slip velocity and the friction stress vector becomes tensorial. In this case for almost any orientation of the gas dynamic flow the so-called cross slip effect is observed. The symmetry of the slip coefficient matrix is proved for fairly general assumptions concerning the type of roughness, the law of reflection of molecules from the surface, and the law of intermolecular interaction. The components of the slip coefficient matrix are calculated by a variational method for a corrugated model of the roughness.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 180–184, January–February, 1987.     

Impact of liquid drops on a rough surface comprising microgrooves

Impact of water drops on a stainless steel surface comprising rectangular shaped parallel grooves is studied experimentally. Geometric parameters of the surface groove structure such as groove depth, groove width and solid pillar width separating any two successive grooves were kept at 7.5, 136 and 66 μm, respectively. The study was confined to the impact of drops in inertia dominated flow regime with Weber number in the range 15–257. Experimental results of drop impact process obtained for the grooved surface were compared with those obtained for a smooth surface to elucidate the influence of surface grooves on the impact process. The grooves definitely influence both spreading and receding processes of impacting liquid drops. A more striking observation from this study is that the receding process of impacting liquid drops is dramatically changed by the groove structure for all droplet Weber number.     

Motion of a sphere colliding with a rough surface

The following problems of inertial motion of a sphere are considered: between two parallel planes, inside another sphere, and inside a circular cylinder. It is assumed that, at the instant of impact, the no-slip condition is fulfilled: the tangential velocity of the contact point of the sphere is equal to zero; in other words, a constraint is imposed and removed instantaneously. It is shown in the above cases that in the limit the motion of the sphere becomes steady in velocity: the angular velocity of the sphere tends to be constant and the velocity of its center becomes periodic in the first case or conditionally periodic in the second and third cases. In certain cases, the coordinates specifying the position and orientation of the sphere also reach the steady-state regime.     

A plane turbulent wall jet on a fully rough surface

The mean velocity field and skin friction characteristics of a plane turbulent wall jet on a smooth and a fully rough surface were studied using Particle Image Velocimetry. The Reynolds number based on the slot height and the exit velocity of the jet was Re = 13,400 and the nominal size of the roughness was k = 0.44 mm. For this Reynolds number and size of roughness element, the flow was in the fully rough regime. The surface roughness results in a distinct change in the shape of the mean velocity profile when scaled in outer coordinates, i.e. using the maximum velocity and outer half-width as the relevant velocity and length scales, respectively. Using inner coordinates, the mean velocity in the lower region of the inner layer was consistent with a logarithmic profile which characterizes the overlap region of a turbulent boundary layer; for the rough wall case, the velocity profile was shifted downward due to the enhanced wall shear stress. For the fully rough flow, the decay rate of the maximum velocity of the wall jet is increased, and the skin friction coefficient is much larger than for the smooth wall case. The inner layer is also thicker for the rough wall case. The effects of surface roughness were observed to penetrate into the outer layer and slightly enhance the spread rate for the outer half-width, which was not observed in most other studies of transitionally rough wall jet flows.     

Elastic indentation of a rough surface by a conical punch

In the contact of a cone with a rough plane the mean pressure in the contact area is constant. In particular, above a critical ratio of the opening angle of the cone with respect to the rms gradient of surface roughness, the mean pressure is the same of that for nominally flat contact, no matter how large is the normal load. We introduce a new variable, namely, the local density of contact area, whose integral over the smooth nominal contact domain gives the real contact area. The results given by the theoretical model agree with the numerical simulations of the same problem presented in the paper.     

Eigenoscillations of an elastic body with a rough surface

Explicit presentations for the initial terms of the asymptotic solution of the spectral problem of the elasticity theory in a plane region with a rapidly oscillating boundary are obtained. Based on asymptotic formulas, two methods for problem modeling are proposed: with the use of Wenzel’s boundary conditions and with the use of the principle of a smooth image of a singularly perturbed boundary. Various approaches to justification of asymptotic presentations are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 103–114, November–December, 2007.     

Statistics of ultrasonic speckles reflected from a rough surface

Summary  First- and second-order statistical properties of ultrasonic speckles reflected from a rough interface are studied theoretically and experimentally in this paper. A theoretical model for predicting statistical properties of ultrasonic speckles is constructed, based on the Fresnel-Huygens principle and three basic assumptions. Distributions of the amplitude and phase of ultrasonic speckles in a scattering space are studied; the study shows that they are in the form of Rayleigh and uniform distribution, respectively. Using the proposed model, the average transverse size of the speckles within a scattering domain, which are received by a focus probe, is investigated. The average transverse size is found to be dependent on the characteristics of the measuring system only, and does not vary with the position in the domain. To verify the applicability of the theoretical model, a special experimental set-up was designed. The corresponding experiments were conducted for measuring the sound pressure of the ultrasonic speckles reflected from a rough interface between water and aluminium alloy. The numerical results are compared with the experimental ones. The comparison demonstrates that the theoretical model and the three related assumptions are suitable for analysing statistical properties of ultrasonic speckles reflected from a rough interface. Received 26 January 2001; accepted for publication 27 November 2001 RID=" ID=" This work was supported by National Natural Science Foundation of China under project 10074017.     

Control of manipulators and wheeled transport robots as systems of rigid bodies

Problems of control of robots (manipulators and wheeled transport robots) that are considered as controllable systems of rigid bodies with holonomic and nonholonomic constraints are reviewed. The basic problems that arise in designing systems of control of such facilities are considered. Namely, the equations of model motion are derived, a specified trajectory is parametrized, and a stabilization algorithm is synthesized (including linear, nonlinear, adaptive, and robust controllers). Some model examples are given to illustrate the efficiency of the algorithms considered. S.P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 35–67, April, 2000.