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]).     

Experimental and theoretical investigation of the energy dissipation of a rolling disk during its final stage of motion

This paper is concerned with the dominant dissipation mechanism for a rolling disk in the final stage of its motion. The aim of this paper is to present the various dissipation mechanisms for a rolling disk which are used in the literature in a unified framework. Furthermore, new experiments on the ‘Euler disk’ using a high-speed video camera and a novel image analysis technique are presented. The combined experimental/theoretical approach of this paper sheds some more light on the dominant dissipation mechanism on the time-scale of several seconds.     

On the dynamics of a rolling ball actuated by internal point masses

The motion of a rolling ball actuated by internal point masses that move inside the ball’s frame of reference is considered. The equations of motion are derived by applying Euler–Poincaré’s symmetry reduction method in concert with Lagrange–d’Alembert’s principle, which accounts for the presence of the nonholonomic rolling constraint. As a particular example, we consider the case when the masses move along internal rails, or trajectories, of arbitrary shape and fixed within the ball’s frame of reference. Our system of equations can treat most possible methods of actuating the rolling ball with internal moving masses encountered in the literature, such as circular motion of the masses mimicking swinging pendula or straight line motion of the masses mimicking magnets sliding inside linear tubes embedded within a solenoid. Moreover, our method can model arbitrary rail shapes and an arbitrary number of rails such as several ellipses and/or figure eights, which may be important for future designs of rolling ball robots. For further analytical study, we also reduce the system to a single differential equation when the motion is planar, that is, considering the motion of the rolling disk actuated by internal point masses, in which case we show that the results obtained from the variational derivation coincide with those obtained from Newton’s second law. Finally, the equations of motion are solved numerically, illustrating a wealth of complex behaviors exhibited by the system’s dynamics. Our results are relevant to the dynamics of nonholonomic systems containing internal degrees of freedom and to further studies of control of such systems actuated by internal masses.     

Brachistochrone for a rolling cylinder

The motion of a heavy homogeneous cylinder is considered as a no-slip rolling along the desired curve. We obtain a functional in the form of the total time of the cylinder rolling and solve the corresponding variational problem of minimizing this functional. We obtain an algebraic equation for the directional line of steepest descent, brachistochrone, in parametric form. We use the equation of motion of the cylinder with constraint reaction to determine the conditions of implementation of its pure rolling without separation and slip with respect to the brachistochrone.     

Continuous rolling motion of a disk on a vibrating plate

Nonlinear Dynamics - We studied the behavior of the continuous rolling motion (CRM) of a disk placed on a vibrating plate observed in the experiment using numerical simulations. Numerical...     

Motions of a top on a smooth plane without separation

The problem on the motion of a heavy top (a dynamically and geometrically symmetric rigid body) without separation on a smooth horizontal plane is considered. In the case of a spherical top surface (the Thomson top), necessary and sufficient conditions on the parameters and the initial positions and velocities of the body under which the motion is all the time without separation are given. These conditions are represented by explicit analytic formulas that can be used in practice. Keywords: Thomson top, smooth surface, separation conditions.     

Rolling/sliding of a particle on a flat wall in a linear shear flow at finite Re

Recently Lee and Balachandar proposed analytically-based expressions for drag and lift coefficients for a spherical particle moving on a flat wall in a linear shear flow at finite Reynolds number. In order to evaluate the accuracy of these expressions, we have conducted direct numerical simulations of a rolling particle for shear Reynolds number up to 100. We assume that the particle rolls on a horizontal flat wall with a small gap separating the particle from the wall (L = 0.505) and thus avoiding the logarithmic singularity. The influence of the shear Reynolds number and the translational velocity of the particle on the hydrodynamic forces of the particle was investigated under both transient and the final drag-free and torque-free steady state. It is observed that the quasi-steady drag and lift expressions of Lee and Balachandar provide good approximation for the terminal state of the particle motion ranging from perfect sliding to perfect rolling. With regards to transient particle motion in a wall-bounded shear flow it is observed that the above validated quasi-steady drag and lift forces must be supplemented with appropriate wall-corrected added-mass and history forces in order to accurately predict the time-dependent approach to the terminal steady state. Quantitative comparison with the actual particle motion computed in the numerical simulations shows that the theoretical models quite effective in predicting rolling/sliding motion of a particle in a wall-bounded shear flow at moderate Re.     

A novel method for modeling skidding for systems with nonholonomic constraints

In this paper, the effect of wheel skidding on the steering motion of a simple vertical rolling disk is investigated. By modifying the nonholonomic constraints, two novel dynamic models are proposed. The first model rotates the constraints and enforces them along a plane correlated to the skid angle. It then relates the skidding in a wheel to the Lagrange multipliers associated with the kinematic constraints of that wheel. The second model relaxes the no-skidding constraint, allowing its transgression and relates the skidding to the generalized velocities of the actuated degrees of freedom of the system. To validate our model, we compare it to one in the literature and we analyze the motion of the disk on icy and snowy road conditions, where skidding can be significant.     

Dynamics of a powered disk in clay soil

Experiments were conducted with a single powered disk in a laboratory soil bin containing Bangkok clay soil with an average moisture content of 18% (db) and 1100 kPa cone index. The disk was 510 cm in diameter and 560 mm in radius of concavity. During the tests the disk angle was varied from 20° to 35°, ground speed from 1 to 3 km/h and rotational speed from 60 to 140 rpm. The working depth was kept constant at 12 cm. The vertical, horizontal and lateral reactions of the soil were measured by force transducers. The forward and rotational speeds were recorded. It was observed that disk angle, rotational speed and ground speed had significant effects on soil reactive forces and power requirement. With a small disk angle, low ground speed, and high rotational speed, the soil longitudinal reactive force was a pushing force and became a resistive one at larger disk angles and ground speeds. The soil transverse reactive force increased with an increase of rotational and ground speed but decreased with the increase of disk angle, whereas the vertical relative force increased only with the increase of ground speed but decreased with the increase of rotational speed and disk angle. It was found that the powered disk required the least power at a disk angle of 30° and rotational speed between 80 and 100 rpm. Increase in ground speed from 1 to 3 km/h increased the total power requirement by 31.8%. Upon driving the disk forward, the draft reduced considerably compared to that of the free-rolling disk. By driving the disk in the reverse direction, the draft reduced slightly. At a disk angle of 30°, rotational speed of 100 rpm, and ground speed of 3 km/h, the total power requirement of the forward-driven disk was 65% higher than that of the free-rolling disk. The predicted engine power of the forward-driven disk, however, was only 21% higher than that of the free-rolling one owing to the more efficient power transmission through the PTO, as opposed to the drawbar. The effects of reverse driving and free rolling of the disk were also studied.     

Vibration analysis of a circular disk tensioned by rolling using finite element method

Summary The paper proposes a method in finite element analysis for estimating natural frequencies of a disk tensioned by rolling, without the use of eigenvalue analysis. The natural frequencies of a disk vary when the localized plastic deformation caused by roll-tensioning induces residual stresses. Tensioning is used for improving the dynamic stability of circular saws; the optimal condition of rolling can be predicted from natural frequency characteristics. In the proposed method, the natural frequencies after rolling are easily estimated from the mode shapes of the disk before rolling and the stress distribution after rolling. The method is based on ideas similar to thermal stress and sensitivity analysis rather than on eigenvalue analysis. The effectiveness of the method is shown by comparing the natural frequency characteristics obtained by this method with those by eigenvalue analysis. Received 18 June 1998; accepted for publication 8 April 1999     

Validation of actuator disc circulation distribution for unsteady virtual blades model

The actuator disc method is an engineering approach to reduce computer resources in computational fluid dynamics (CFD) simulations of helicopter rotors or aeroplane propellers. Implementation of an actuator disc based on rotor circulation distribution allows for approximations to be made while reproducing the blade tip vortices. Radial circulation distributions can be formulated according to the nonuniform Heyson-Katzoff “typical load” in hover. In forward flight, the nonuniform disk models include “azimuthal” sin and cos terms to reproduce the blade cyclic motion. The azimuthal circulation distribution for a forward flight mode corresponds to trimmed conditions for the disk rolling and pitching moments. The amplitude of the cos harmonic is analysed and compared here with presented in references data and CFD simulations results.     

The impact of the concave distribution of rolling friction coefficient on the seismic isolation performance of a spring-rolling system

A complex yet realistic nonuniform rolling friction force distribution of a spring-rolling isolation system could lead to great complexity in determining its seismic response. This paper investigates the isolation performance of a spring-rolling isolation system assuming that the rolling friction force gradually and linearly increases with the relative displacement between the isolator and the ground. A series of ground motions with different characteristics were applied to this system. The analysis results show that the considered concavely distributed friction force is capable of dissipating the earthquake energy, and it is also able to modify the structural natural period. These merits combined help to improve the isolating efficacy of the spring-rolling isolation system compared with scenarios with uniform distribution pattern, and more importantly lead to a relatively optimum isolation state, avoiding a sudden amplification of the structural seismic response, regardless of the input motion characteristics.     

Poynting oscillations of a rigid disk supported by a neo-Hookean shaft

Simultaneous axial and torsional oscillations of a rigid disk attached to an elastomeric shaft are investigated. Five cases are solved exactly. The uncoupled, small amplitude axial and torsional oscillations of the disk are investigated for neo-Hookean and Mooney-Rivlin shafts with static stretch. The finite torsional vibration of the load superimposed on a static stretch of the shaft is studied for the Mooney-Rivlin model. Solutions for both small and finite amplitude, uniaxial vibrations of the body superimposed on a pretwisted neo-Hookean shaft with static stretch are derived. Simple bounds on the period for the finite motion are provided; and various universal frequency relations for neo-Hookean and Mooney-Rivlin materials are identified.Finally, the main problem of finite, uniaxial vibrations accompanied by a small twisting motion is studied for the neo-Hookean model. The exact periodic solution for the axial response is obtained; and the coupled, small torsional motion is then determined by Hill's equation. A stability criterion for the Mathieu-Hill equation is used to obtain stability maps in a physical parameter space. Geometrical conditions sufficient for universal stability of the motion are read from this graph. Instability of the torsional oscillation, the beating phenomenon and exchange of energies, and the relation of the stability diagram to amplitude bounds on the uncoupled, linearized motion sufficient to assure universal stability predicted for small amplitude vibrations, are discussed and described graphically with the aid of a numerical model. It is shown that an unstable configuration may be stabilized by increasing the diameter of the disk.     

Combined effects of rotation and translation on a MHD flow due to compressible viscous fluid

Summary The modification of an axi-symmetric viscous flow due to a relative rotation of a disk or fluid by a translation of the boundary are studied. The fluid is taken to be compressible and electrically conducting. The equations governing the motion are solved iteratively through a central-difference scheme. The effect of an axial magnetic field and disk temperature on the flow and heat transfer are included in the present analysis. The translation of the disk or fluid generates a velocity field at each plane parallel to the disk (secondary flow). The cartesian components of the velocity due to the secondary flow are oscillatory in nature when a rigid body rotation of the free stream along with a translation of the disk is considered. The magnetic field damps out the velocity field, and reduces the thickness of the boundary layer. The cross component of wall shear due to secondary flow acts in a direction opposite to the rotation of the disk or fluid for all cases of the motion. The rise in disk temperature produces an increment in the magnitude of the wall shear associated with the secondary flow.     

RETRACTED ARTICLE: Exact solutions for the incompressible viscous magnetohydrodynamic fluid of a rotating disk flow

The main interest of the present paper is to generate exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous electrically conducting fluid flow motion due to a disk rotating with a constant angular speed. In place of the traditional von Karman’s axisymmetric evolution of the flow, the rotational non-axisymmetric stationary conducting flow is taken into consideration here. As a consequence, for an external uniform magnetic field applied perpendicular to the plane of the disk, the governing equations allow an exact solution to develop, which is influenced by a fixed point on the disk and also is bounded everywhere in the normal direction to the wall.     

On the instability of a car in the vertical plane under rectilinear motion with friction forces taken into account

We consider the problem on the stability and instability of the equilibrium in the vertical plane for a wheeled vehicle performing a uniform rectilinear motion in the presence of rolling friction forces. We assume that the dependence of the rolling friction coefficients on the motion velocity is known and derive necessary and sufficient conditions on the system parameters under which such equilibria are stable.     

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.     

方形截面弹俯仰振荡对滚转特性的影响

飞行器在大气环境中飞行时,经常受阵风等的干扰,引发非指令的自激振荡,威胁飞行安全.通过建立刚体六自由度运动方程和N-S方程的松耦合求解技术,研究强迫俯仰振荡过程对滚转特性的影响.针对背风区涡流形态及横侧向气动特性复杂的方形截面飞行器,数值模拟研究了其不同攻角下的静态滚转气动特性、自由滚转运动特性,以及俯仰振荡时不同振荡速率对滚转气动和运动特性的影响.结果表明,此飞行器在静态时临界攻角约为13°,当攻角小于临界攻角时,滚转方向是静不稳定的,诱发快速滚转运动;当攻角大于临界攻角时,滚转方向是静稳定,其滚转运动是收敛的.研究发现,俯仰振荡一般会降低飞行器滚转方向静稳定或静不稳定的量值,增强滚转方向的动态稳定性.在俯仰振荡的影响下,即使滚转方向是静不稳定的,如果俯仰振荡的频率足够大,飞行器的滚转运动也可能是收敛的.     

Exact solutions for the incompressible viscous fluid of a porous rotating disk flow

The present paper is concerned with a class of exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous fluid flow motion due to a porous disk rotating with a constant angular speed. The three-dimensional equations of motion are treated analytically yielding derivation of exact solutions with suction and injection through the surface included. The well-known thinning/thickening flow field effect of the suction/injection is better understood from the exact velocity equations obtained. Making use of this solution, analytical formulas corresponding to the permeable wall shear stresses are extracted.Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. As a result, exact formulas are obtained for the temperature field which take different forms depending on whether suction or injection is imposed on the wall. The impacts of several quantities are investigated on the resulting temperature field. In accordance with the Fourier‘s heat law, a constant heat transfer from the porous disk to the fluid takes place. Although the influence of dissipation varies, suction enhances the heat transfer rate as opposed to the injection.