Theoretical and experimental investigation of tyre dynamics

The lateral vibration of towed wheels - also called shimmy - is related to the elasticity of the suspension system or to the elasticity of the wheel. The second case is studied here since the majority of the vehicles has pneumatic tyres. A one degree-of-freedom wheel model is considered with perfect rolling and elastic tyre. To verify the theoretical stability charts, an experimental rig is built and one of the stability boundaries is identified. Close to a double Hopf bifurcation point quasiperiodic vibrations are observed. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The vibrations of a free and loaded tyre

Using the model of a wheel with a reinforced tyre [Vil’ke VG, Kozhevnikov IF. A model of a wheel with a reinforced tyre. Vestnik MGU. Ser.1. Matematika Mekhanika 2004;4:37–45], the natural frequencies and natural forms of vibrations of a free or loaded tyre in the neighbourhood of the equilibrium position are determined. The spectrum of natural frequencies and natural forms of vibration are found analytically for a free (unloaded) tyre with a fixed disc. A similar problem is solved numerically in the case of a loaded tyre. The results of this analysis can be used to estimate the level of noise which occurs when a vehicle moves on an uneven surface.     

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).     

A model of a reinforced tyre with a rod protector

A tyre design consisting of a steel-cord-reinforced rigid bond with sides connected to the wheel disc and a protector(tread) in contact with the road is examined. The tread is in the form of a set of rods connected by one end to the band, with the other end either free or in contact with the road. The rod end in contact with the road is acted upon by a force applied from the road, represented by a force normal to the road plane and a shear force due to dry friction. If the modulus of the shear force does not exceed the magnitude of the normal force multiplied by the dry friction coefficient, there is no slip at the contact point. In the opposite case, the rod end will be displaced along the road by an amount sufficient to distribute the normal and shear forces. The dynamics of longitudinal and transverse strains of the rods in contact with the road is analysed using the motion separation method in the quasi-static approximation. The behaviour of the tread rods as a function of the vertical displacement of the wheel centre is investigated, the contact area is found and the conditions are determined under which the contact area is divided into parts in which either slip of the rod ends occurs or does not occur, depending on the magnitude of the longitudinal displacement of the wheel centre or its turning relative to the horizontal axis. An analogue of a continuous model of a rod-like tread is considered, and the magnitudes of the forces and moments are found as a function of the wheel disc displacements. The equations of wheel rolling are obtained, and the conditions under which steady motions exist are found.     

The rolling of a wheel with a pneumatic tyre on a planet

A model of a pneumatic tyre as a system with an infinite number of degrees of freedom is proposed, when its surface is represented by the deformed surface of a torus. Using a number of hypotheses a functional of the potential energy of the deformations of the tyre is obtained as a function of the deformations of its tread. A complete system of equations of motion is obtained, assuming that the wheel rolls without slipping in the area of contact of the tread with the plane, with respect to the previously unknown part of the tread. In two special cases of the rolling of a wheel with breakaway and on a banking, all the characteristics of the motion (the contact area, the tyre deformation, and the forces and moments applied to the disc of the wheel) are obtained.     

Optimal Control of an Elastic Tyre-Damper System with Road Contact

We study an elastic tyre with a wheel rim that is suspended at the chassis of a car by means of a spring-damper element. This quarter car model may be controlled by varying the damping constant of the electrorheological damper. Our mathematical model yields a coupled ODE-PDE problem with a free boundary at the tyre-road contact. In this study we approximate the tyre by the Hertz contact stress formula. The resulting optimal control problem with control constraints is solved numerically. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fuzzy-hybrid modelling of an Ackerman steered electric vehicle

Physical system modelling with known parameters together with 2-D or high order look-up tables (obtained from experimental data), have been the preferred method for simulating electric vehicles. The non-linear phenomena which are present at the vehicle tyre patch and ground interface have resulted in a quantitative understanding of this phenomena. However, nowadays, there is a requirement for a deeper understanding of the vehicle sub-models which previously used look-up tables. In this paper the hybrid modelling methodology used for electric vehicle systems offers a two-stage advantage: firstly, the vehicle model retains a comprehensive analytical formulation and secondly, the ‘fuzzy’ element offers, in addition to the quantitative results, a qualitative understanding of specific vehicle sub-models. In the literature several hybrid topologies are reported, sequential, auxiliary, and embedded.In this paper, the hybrid model topology selected is auxiliary and within the same hybrid model, the first paradigm used is the vehicle dynamics together with the actuator/gearbox system. The second paradigm is the non-linear fuzzy tyre model for each wheel. In particular, conventional physical system dynamic modelling has been combined with the fuzzy logic type-II or type-III methodology. The resulting hybrid-fuzzy tyre models were estimated for a-priori number of rules from experimental data. The physical system modelling required the available vehicle parameters such as the overall mass, wheel radius and chassis dimensions. The suggested synergetic fusion of the two methods, (hybrid-fuzzy), allowed the vehicle planar trajectories to be obtained prior to the hardware development of the entire vehicle. The strength of this methodology is that it requires localised system experimental data rather than global system data. The disadvantage in obtaining global experimental data is the requirement for comprehensive testing of a vehicle prototype which is both time consuming process and requires extensive resources. In this paper the authors have proposed the use of existing experimental rigs which are available from the leading automotive manufacturers. Hence, for the ‘hybrid’ modelling, localised data sets were used. In particular, wheel-tyre experimental data were obtained from the University tyre rig experimental facilities. Tyre forces acting on the tyre patch are mainly responsible for the overall electric vehicle motion. In addition, tyre measurement rigs are a well known method for obtaining localised data thus allowing the effective simulation of more detailed mathematical models. These include, firstly, physical system modelling (conventional vehicle dynamics), secondly, fuzzy type II or III modelling (for the tyre characteristics), and thirdly, electric drive modelling within the context of electric vehicles. The proposed hybrid model synthesis has resulted in simulation results which are similar to piece-wise ‘look-up’ table solutions. In addition, the strength of the ‘hybrid’ synthesis is that the analyst has a set of rules which clearly show the reasoning behind the complex development of the vehicle tyre forces. This is due to the inherent transparency of the type II and type III methodologies. Finally, the authors discussed the reasons for selecting a type-III framework. The paper concludes with a plethora of simulation results.     

The steady vibrations and resistance of a railway track to the uniform motion of an unbalanced wheel

A model of a railway track, in the form of an infinite Timoshenko beam resting on equally spaced massive visco-elastic supports, is considered. Steady vertical vibrations of the track due to a harmonic force moving along it at a constant velocity are investigated. The vertical displacement of the track is represented in a moving system of coordinates by a generalized Fourier series. The steady vertical vibrations of a massive rigid wheel rolling along the track at a constant velocity and loaded by a vertical harmonic force are investigated. The track-wheel interaction force is expressed as a generalized Fourier series whose coefficients are determined using an equality relating the vertical displacements of the wheel and the track. Vibrations of the wheel due to centrifugal force and periodic changes in the track parameters are considered. Parametric vibrations of a wheel moving at a constant velocity under a static load due to periodic variation in the stiffness of the track are investigated. The force with which the track resists the uniform motion of an unbalanced wheel is computed.     

Research on lateral stability and rollover mechanism of articulated wheel loader

A seven-degrees-of-freedom (DOF) dynamic model was established based on the Lagrange equation to analyse the lateral stability and instability mechanism of an articulated wheel loader. A scale wheel loader was designed and manufactured to validate the dynamic model in two conditions, namely turning on slopes and passing over obstacles. Experimental data and simulated data fitted well on the whole, so the developed dynamic model was proved to be useful and could serve as an important tool to analyse the stability of wheel loaders. At last, the lateral stability of one ZL50 wheel loader was analysed by using this dynamic model. The results showed that there is a phenomenon of sudden stability losing during the wheel loader cornering due to the relative rotation between the subframe and the rear axle. The subframe can enhance the stability when the wheel loader is passing over obstacles but reduces the stability when it is turning.     

Simulation of stiffness analysis of rubber material applied in resilient wheel

Resilient wheels are well-known for their contribution to prevent squealing, impact noise and dynamic stresses on unsprung masses. In this study, a FE model is established in ABAQUS to analyze the stiffness of resilient wheels based on the Mooney-Rivlin hyperelastic material theory. The modes of each part in the resilient wheel and the whole wheel are calculated. The static and dynamic stiffness of the resilient wheel are analyzed. The simulation shows that the resilient wheel can reduce the dynamic loads both in the lateral and vertical directions between the vehicle and the track. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic response of a rotating ball screw subject to a moving regenerative force in grinding

This paper investigates the dynamic response of a rotating ball screw subjected to a moving regenerative force. The rotating ball screw is modeled as a rotating Timoshenko shaft with simply supports. The moving regenerative force describes the nonlinear interactions including the effects of wheel wear, time-delay, and the possibility of contact loss between the grinding wheel head and screw. The assumed mode method together with Runge–Kutta method is employed to analyze the system dynamic response. The total grinding depth is suggested to divide into several passes from rough grinding to fine grinding for obtaining fine surface. The effect of parameters such as the depth of cut and the rotational speeds of grinding wheel and screw are discussed for each pass. The numerical results show that the critical depth of cut depends on the rotational speeds of screw and grinding wheel. If the more the depth of cut is smaller than the critical depth of cut, the earlier the chatter occurs and the faster the vibration grows.     

The rolling of a wheel along a corrugated rail

The rolling without detachment of a rigid massive wheel, carrying a static load, along a rail with undulations on the running surface, which arises as a result of non-uniform wear, is investigated. The rail is supported by an elastoviscous base. Because of the inertia of the wheel and the carriage the horizontal component of the velocity of the wheel centre differs only slightly from a constant quantity, and hence the motion of the wheel along the rail is assumed to be uniform. Steady vertical vibration of the wheel is considered. The vertical coordinate of the wheel centre, and also the difference between the longitudinal coordinates of the wheel centre and the point of contact of the wheel and the rail, are periodic and, correspondingly, even and odd functions of the longitudinal coordinate of the wheel centre, and their period is equal to the wave length on the rail surface. The periodic force of interaction of the wheel and the rail is given in the form of a Fourier series. Short waves, the amplitude of which is much less than their length, are often observed on the rail surface, and this length is much less than the wheel radius. In this case the coefficients of the Fourier series are expressed in terms of Bessel functions of the first kind of integer order. Observations show that the depth of the short wave on the rail surface increases until the radius of curvature in the rail trough approximates to the wheel radius, and hence it is assumed that these radii are close to or equal to one another. In this case the trajectory of the wheel centre differs considerably from the wave on the rail surface.     

Handling characteristics and stability of the steady-state powerslide motion of an automobile

Powerslide of an automobile may be defined as a steady-state cornering motion at a large side slip angle of the vehicle, considerably large traction forces and a large negative steering angle of the handwheel. In this case the front wheels direct towards the outside of the turn. As this extrem driving condition, which can be seen e.g. in Rallye sports, is hardly addressed in literature so far, this paper investigates the respective handling characteristics. Therefore, a nonlinear four-wheel vehicle model is applied including nonlinear tyre characteristics, the load transfer between inner and outer wheels and the influence of the traction forces on the lateral tyre forces. A basic stability analysis reveals the unstable nature of the steady-state powerslide motion of a certain test vehicle. To approve the numerical findings, measurements have been performed with a sports utility vehicle with rear-wheel drive at various speeds on a wet circular test track.     

Efficient Calculation of Tread Block Vibrations

A model is presented that allows to calculate the dynamics of tyre tread blocks. The numerical efficiency is achieved by the combination of static and important modal ansatz functions, also known as Hurty/Craig/Bampton transformation. Nonlinear single point contacts provide the contact forces and consider the surface roughness. The non-smooth friction characteristic is approximated an arctan function and a parameter study of the slope parameter is conducted. Typical stickslip limit cycles of the tread block are shown. These phenomena can have a part in the tyre/road noise. Experimental results qualitatively agree with the modelled tread block vibrations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于预测控制的直接侧向力气动力复合控制方法研究

针对大气层内拦截导弹直接侧向力与气动力复合控制系统设计问题, 首先, 根据发动机的配置建立了复合控制系统模型;其次, 提出了复合控制策略, 包括动态分配算法、直接力控制子系统、气动力控制子系统3部分; 然后, 在考虑两套执行机构动态特性差别的情况下,基于预测控制思想给出了过载误差动态分配算法; 在此基础上, 根据直接侧向力的离散特性, 基于预测控制方法设计了直接力控制规律, 考虑到直接力控制作用对弹体产生的扰动, 基于自抗扰方法设计了气动力子系统; 最后, 通过仿真验证了直接力气动力复合控制策略与方法的有效性.     

Influence of the rail pad on a vehicle travelling on a straight track

In the following text, the influence of the stiffness of rail fastening systems of high‐speed railway lines on the dynamic stability of a vehicle travelling along a straight track is investigated. The dynamic behaviour of the rail head in lateral direction is incorporated in the model by means of mechanical models derivedfrom a preceding frequency analysis of the track. For the wheel‐rail interface, linear contact mechanics and kinematic relations are applied. The investigation leads to the conclusion that on high‐speed lines the critical velocity of vehicles is reduced as a result of the increased elasticity of the rail pads. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Painlevé’s paradox and dynamic jamming in simple models of passive dynamic walking

Painlevé’s paradox occurs in the rigid-body dynamics of mechanical systems with frictional contacts at configurations where the instantaneous solution is either indeterminate or inconsistent. Dynamic jamming is a scenario where the solution starts with consistent slippage and then converges in finite time to a configuration of inconsistency, while the contact force grows unbounded. The goal of this paper is to demonstrate that these two phenomena are also relevant to the field of robotic walking, and can occur in two classical theoretical models of passive dynamic walking — the rimless wheel and the compass biped. These models typically assume sticking contact and ignore the possibility of foot slippage, an assumption which requires sufficiently large ground friction. Nevertheless, even for large friction, a perturbation that involves foot slippage can be kinematically enforced due to external forces, vibrations, or loose gravel on the surface. In this work, the rimless wheel and compass biped models are revisited, and it is shown that the periodic solutions under sticking contact can suffer from both Painlevé’s paradox and dynamic jamming when given a perturbation of foot slippage. Thus, avoidance of these phenomena and analysis of orbital stability with respect to perturbations that include slippage are of crucial importance for robotic legged locomotion.     

矩形板的侧屈

本文研究矩形板的侧向屈曲问题.文中分别讨论了有集中力,均布荷载及集中力偶作用之下矩形板发生侧向屈曲时的最小临界荷载.文中使用了能量法.