Constraint implementation in wheeled apparatus rolling problems

The models obtained from equations of rolling of an apparatus with small slips of wheels with respect to the supporting plane are considered by passing to the limit of infinite values of rigidity of contact forces (zero values of slip speeds). The conditions under which these equations become the classical nonholonomic wheel no-slip model are discussed. It is shown that, for small angles of turn of the apparatus front wheels about the vertical axis, neglect of slips in the transverse direction is not correct; namely, the limit model is determined by the wheel no-slip conditions in the longitudinal direction and by the primary Dirac constraints arising owing to the degeneration of the Lagrangian of the system. The methods used to decrease the order of equations, where small slips of wheels are taken into account, can be useful for qualitative analysis of motion of wheeled systems and for solving the problems of estimation and control in real time.     

活力板运动的动力学分析

作为一种特殊的滑板运动,活力板依靠乘员自身的摆动和扭动前进. 本文分析活力板运动的动力学 原理. 当乘员的动作引起前后板的周期性摇摆时,地面对轮缘的摩擦力可产生向前 的推动效应. 导出推力平均值的计算公式, 并解释蛇形轨迹的产生原因.     

A method for predicting the internal motion resistance of rubber-tracked undercarriages,Pt. 3. A research on bending resistance of rubber tracks

Optimizing the efficiency of rubber-tracked undercarriages requires models for calculating external and internal motion resistance, including the resistance resulting from bending of rubber tracks. The experiments on the bending resistance of rubber tracks and a new model of this phenomenon are discussed in this article. An empirical model of friction in bearings typically implemented in driving and idler wheels of rubber-tracked undercarriages is also presented. According to the sample computations carried out on the basis of these models, the efficiency of rubber-tracked undercarriages might be improved by minimizing the number and maximizing the diameter of idler wheels. Furthermore, it has been shown that increase in the initial tension and driving force transmitted by rubber tracks does not significantly affect bending resistance of these tracks; however, it results in increased friction in the driving and idler wheels’ bearings. Nevertheless, the higher the driving force transmitted by the rubber tracks, the higher the efficiency of rubber-tracked undercarriages. Consequently, since track systems of vehicles operating at relatively small drawbar pull will manifest exceptionally low efficiency, there is a serious need for optimizing them in terms of energy consumption.     

A method for predicting the internal motion resistance of rubber-tracked undercarriages,Pt. 2. A research on the motion resistance of road wheels

In spite of an increasing number of rubber-tracked crawlers, the literature provides few guidelines and calculation models suitable for minimizing their internal motion resistance. This article presents a model where the internal resistance of double-flanged road wheels for rubber-tracked vehicles is calculated as a sum of the losses resulting from the indentation of the wheels into the track surface and friction of the wheels against the track guide lugs. The model allows for vertical and lateral load of the wheels, the non-uniform distribution of the wheel pressure on the track, and the relationship between the friction coefficient and normal reaction force in the interface between the wheel and track guide lugs. The model has been verified by experiments. According to the results of model computations and experiments discussed in the article, the internal losses of a given rubber-tracked undercarriage might be reduced if: the road wheels are covered with a material that exhibits low friction coefficient and mechanical hysteresis, the vehicle suspension system features oscillating bogie wheels, the undercarriage is fitted with the largest possible number of road wheels, and the vehicle weight is evenly distributed to all of the road wheels.     

Theory of the shimmy phenomenon

The shimmy phenomenon is the appearance of angular self-excited vibrations of the carriage wheels. Such self-excited vibrations provide a serious safety hazard for motion, which explains the great interest of scientists in this phenomenon [1–6]. This problem is most serious for the aircraft fore wheels.     

Perturbation-based prediction of vibration phase shift along fluid-conveying pipes due to Coriolis forces,nonuniformity, and nonlinearity

This paper investigates the influence of tread conicity variation on hunting dynamical changing features for railway vehicle. Nonlinear contact force between wheels and rail is estimated by Vermeulen–Johnson creep force law. And a piecewise linear function is employed to appropriate the collision between wheel flange and rail. Hunting asymmetrical motions are analyzed by lateral bifurcation behaviors between maximum and minimum of car body lateral displacement. The result shows that the critical speed decreases with the increase in tread conicity, while the speed gap between linear and nonlinear speeds is narrowing. Compared with wheelsets, lateral amplitudes of the bogies are vulnerable to the tread conicity and decrease gradually. Besides, more asymmetrical motions are put into consideration with regard to tread conicity variation. Similarly, one asymmetrical behavior with small amplitude difference originates from the same chaotic attractor at both sides. And small interactive amplitude jumps in two sides at chaotic or periodic occasions are revealed. Distinguishingly, the other new asymmetrical type is found at a certain tread conicity that amplitudes of the hunting motion in positive and reverse directions no longer coincide and go away in opposite directions when the tread conicity increases to a certain value. And this particular asymmetrical motion disappears with further growth of tread conicity.

     

OPTIMAL MOTION PLANNING FOR A RIGID SPACECRAFT WITH TWO MOMENTUM WHEELS USING QUASI-NEWTON METHOD

An optimal motion planning scheme based on the quasi-Newton method is proposedfor a rigid spacecraft with two momentum wheels. A cost functional is introduced to incorporatethe control energy, the final state errors and the constraints on states. The motion planning fordetermining control inputs to minimize the cost functional is formulated as a nonlinear optimalcontrol problem. Using the control parametrization, one can transform the infinite dimensionaloptimal control problem to a finite dimensional one that is solved via the quasi-Newton methodsfor a feasible trajectory which satisfies the nonholonomic constraint. The optimal motion planningscheme was applied to a rigid spacecraft with two momentum wheels. The simulation results showthe effectiveness of the proposed optimal motion planning scheme.     

PIV measurement of the flow past a generic car body with wheels at LES applicable Reynolds number

Experiments by using 2D–2C Particle Image Velocimetry (PIV) were carried out and reported concerning the flow field past a generic car body (modified Ahmed body) which is equipped with wheels and wheel-arches. The Reynolds number was chosen to not exceed 2E+5 based on the height of the Ahmed body which makes it possible to investigate the same configuration by means of Large Eddy Simulation (LES). The wheels were rotating but the ground was stationary. The wheel-ground contact was realized by means of small rectangular openings below the wheels in the ground plane in which the wheels were immersed. The transition contour of the immersed wheels and the ground, as well as the rectangular openings below the wheels were properly sealed to prevent parasite flow and to provide well defined boundary conditions for an upcoming LES investigation.The flow field was measured in several planes with normal vectors pointing towards the directions normal to the free stream. Statistical characteristics of the flow are provided and discussed.     

Mobile robots with two independent coaxial drive wheels: Dynamics and the schemes of control

A two-wheeled robot and a robot composed of a snakelike chain of two-wheeled objects are considered. The problem of trajectory synthesis and the problem of motion planning and control are studied for these mobile robots. The mobile robots with two independently controlled coaxial wheels and with one or several passive wheels are called robots with differential drive. These problems are formulated in such a way that the construction of dynamically correct and precise control becomes possible for such robots.     

Dynamic Stability and Random Vibrations of Rigid and Elastic Wheelsets

A stability analysis and vibration studies are presented for a passenger coach model which is equipped with rigid or elastic wheels. The elastic components between wheel rim and disc act as third suspension reducing the unsprung mass and isolating the passenger coach from the high frequency motion of the wheel rim. The vertical and lateral motion by such a design requires a thorough analysis of the system dynamics. The excitation of the vertical vibrations by stochastic track irregularities results in acceleration amplitudes of the carbody that may generate droning noise. A parameter study of spring and damper coefficients of the system with radialelastic wheels leads to considerably reduced droning noise and lower force level between wheel and rail. Furthermore, the eigenmotion of a rigid and an elastic wheelset rolling on a track, the so-called hunting, is investigated. The variation of the spring and damper coefficients shows limits to guarantee the stability of the system.     

Dynamics equations of a mobile robot provided with caster wheel

Kinematics and dynamics of a mobile robot, consisting of a platform, two conventional wheels and a crank that controls the motion of a free rolling caster wheel, are analyzed in the paper. Based on several matrix relations of connectivity, the characteristic velocities and accelerations of this non-holonomic mechanical system are derived. Using the principle of virtual work, expressions and graphs for the torques and the powers of the two driving wheels are established. It has been verified the results in the framework of the second-order Lagrange equations with their multipliers. The study of the dynamics problems of the wheeled mobile robots is done mainly to solve successfully the control of the motion of such systems.     

On path planning for a compound wheeled vehicle

The path-planning problem for a compound robotic vehicle with three steerable wheels is solved. In the kinematic approximation, this problem is formulated as a variational problem with conditions prescribed at three points. The case of inertial motion is analyzed. The motion of a robot in an L-shaped holding alley is considered as an illustration of the algorithm proposed     

The motion equations of rotors,gyrostats and gyrodesics: application to powertrain-vehicle mechanical systems

The kinematics of wheels and rotors is described using a new auxiliary frame called gyrodesic frame or simply gyrodesic. By this, the absolute motion of the wheel becomes a serial composite of two motions: (1) the gyrodesic motion and (2) the wheels eigenmotion (or spin), i.e., the motion relative to the gyrodesic. The eigenmotion is described by an equation called rotor-equation. Gyrodesic coordinates turn out to be a particular useful tool in powertrain- and vehicle-dynamics as well as for general multibody systems. They allow a proper separation of the rotor- from the vehicle-equations and provide a rigorous method of coupling the powertrain-model into full spatial multibody-systems vehicle model. Some common misconceptions regarding this subject are identified and dispelled. The method is generalized to be applicable to the study of motion of general systems of rigid bodies with gyrostats or rotors as subsystems. The usefulness of the formalism is demonstrated by means of an illustrative example of non-trivial nature: the gyrostatic chain. Gyrodesic coordinates lead to a better grasp and deeper understanding of the structure of the dynamic equations of spatial vehicles in particular and of the motion of multibody-systems with rotors in general. The investigation reveals an interesting analogy to concept of parallel transport of vector fields in the sense of Levi-Civita. Dedicated to Prof. J. Wittenburg at the occasion of his 70th birthday.     

Reaction Forces under Static Loading of a Wheel Pair with Camber

The static loading of a wheel pair with a deformable periphery is considered when the wheels are mounted on a common axle with a non-zero camber angle. A tire tread (protector) is modeled by a set of elastic rods interacting with the motion plane according to the dry friction law. The wheel camber effect on the slip in the contact zone and on the magnitude of reaction forces from the road is studied. An analog of the continuous model for a rod tread is discussed. The normal and tangential reaction forces are found depending on the vertical displacement of the center of the mechanical system under discussion.     

Precise measurement of soil deformation and fluctuation in drawbar pull for steel and rubber-coated rigid wheels

The travelling performance of rigid wheels on sand stratum is measured using two kinds of surface material, i.e. steel and steel coated with rubber. A new method for measuring the displacement of soil beneath the wheel has been developed using small polyester film markers. The trajectories of soil particles beneath the wheels are approximated by an exponential function and the fluctuations in the drawbar pull are represented by a sinusoidal function. The amplitude and basic wavelength of the fluctuation in the drawbar pull are discussed for both types of wheels.     

Swing-up and positioning control of an inverted wheeled cart pendulum system with chaotic balancing motions

This paper explores the problem of swinging-up an inverted pendulum formed by a rod attached to a wheeled cart with a hanging bob at its opposite end. The system is driven by the wheeled cart platform system, which is formed by a cart, wheels with counterbalance and connecting-rods. The model of the system is initially obtained under the assumption of rolling without slipping of the wheels, which is then verified by computing the reaction forces. The motion of the wheeled cart is initially oscillating, whereas the rod can move freely giving rise to an under-actuated mechanical system. From the harmonic prescribed motion for the wheeled cart, necessary conditions for chaotic rod motion are deduced by means of the Melnikov function. Once the chaotic oscillation has been reached and the rod is close to the upright position, the force over the wheeled cart is commutated to a control law based on the pole-placement plus integrator technique. This procedure allows driving the rod and the wheeled cart system to the upright position and to a prescribed set point respectively. The onset of strange attractors is crucial in the design of the control law, whose performance to obtain rolling without slipping is researched by means of sensitive dependence, power spectral density, Lyapunov exponents and reaction forces. The results of the analytical calculations are verified by full numerical simulations.     

Studies of dynamics of a lightweight wheeled mobile robot during longitudinal motion on soft ground

The paper is concerned with the study of longitudinal motion of a lightweight wheeled mobile robot on soft ground. The study is focused on the influence of the desired longitudinal velocity of a robot on both the longitudinal slip of the wheels and the ratio of wheel-terrain contact angles. Design of the four-wheeled skid-steered robot and research environment are described. Experimental investigations were conducted on a dedicated test stand with dry sand. A dynamics model of the robot-ground system taking into account properties of soft ground is presented. The classical terramechanics models of Bekker and Janosi-Hanamoto are used. Results of simulation research of robot motion and of the analogous experimental investigations are presented. Actual motion parameters of the robot and the values of longitudinal slip ratio of the wheels are determined. The results of simulation and experimental investigations are compared and discussed. A formula to describe front-to-back wheel-terrain contact angle ratio dependency on the desired velocity is proposed.     

Theoretical analysis of the forces and torques acting on a tractor during ploughing

This paper presents a physical model of a four-wheeled tractor during ploughing. In this model, the wheels are approximated by ideal non-oscillating harmonic oscillators. The velocity of the tractor is considered to be constant, i.e., the tractor is in an equilibrium state of forces and torques. The equilibrium state is described by Euler’s laws of motion. Finally, an infinite approximation of the wheel stiffness is performed and an exact solution of the forces is presented.     

Modeling motion resistance of rigid wheels

A mathematical model was developed to describe the motion resistance of rigid wheels under self-propelled conditions where no drawbar pull is developed. The modeling was based on the concept that the work required to overcome the motion resistance is equal to the energy dissipated in compacting the soil beneath the wheel. Slippage of the wheel was also considered in the modeling. Finally the validity of the proposed model was discussed.