Two-Dimensional Contact Area of a Pneumatic Tire Subjected to a Lateral Force

A mathematical model of a two-dimensional contact patch of pneumatic tires rolling over a rigid flat road surface at arbitrary slip and camber angles has been developed. The model is simple in concept, contains few parameters and is applicable to any tire simulation models. In addition to tire geometric parameters and vertical deflection, the carcass camber angle is introduced in the model. This angle is alone responsible for the asymmetric shape of the tire contact patch when the tire undergoes a lateral force. The computed contact patches agree well with the measured patches of an automotive tire at different slip and camber angles. Lastly, the influence of the contact patch geometry on the tire cornering and aligning properties has been discussed through a computational example. It has been shown that the effect of tire contact patch geometry on the steady state behavior is rather remarkable.     

New Model of Tire Overturning Moment Characteristics and Analysis of Their Influence on Vehicle Rollover Behavior

A newly developed tire model for the Overturning Moment (OTM) characteristics and the analysis of the influence of OTM on vehicle rollover behavior are presented. The new OTM model was developed based on the so-called Magic Formula tire model. The concept of the new model involves identifying the difference between the simple model and the measurements to the newly defined functions. It was seen that the new model agrees very well with the measured data over a wide range of tire vertical loads, slip angles and camber angles. The influence of tire OTM on the vehicle rollover behavior was also investigated by using a full vehicle simulation in which a rather large steering angle was input. The results obtained from the vehicle simulation with three different tire models (model without OTM, simple model and new model) were compared with the experimental results. It was found that the calculated result obtained with the new OTM model agreed best with the experiment.     

New Model of Tire Overturning Moment Characteristics and Analysis of Their Influence on Vehicle Rollover Behavior

A newly developed tire model for the Overturning Moment (OTM) characteristics and the analysis of the influence of OTM on vehicle rollover behavior are presented. The new OTM model was developed based on the so-called Magic Formula tire model. The concept of the new model involves identifying the difference between the simple model and the measurements to the newly defined functions. It was seen that the new model agrees very well with the measured data over a wide range of tire vertical loads, slip angles and camber angles. The influence of tire OTM on the vehicle rollover behavior was also investigated by using a full vehicle simulation in which a rather large steering angle was input. The results obtained from the vehicle simulation with three different tire models (model without OTM, simple model and new model) were compared with the experimental results. It was found that the calculated result obtained with the new OTM model agreed best with the experiment.     

Four-wheel vehicle kinematic and geometric constraints for definition of tire slip angle

When considering vehicle safety, tires and all that they represent are a fundamental topic. Tire studies have received a considerable amount of attention from the research community because their improvement has a direct and strong impact on vehicle handling and braking. Within this eld of analysis lies an important behavioral feature: the tire slip angle, which is a consequence of lateral forces acting on the tire. This characteristic is predicted in some cases and evaluated experimentally in others. This paper addresses another way to assess the slip angle. We propose a mathematical model that describes a constraint linking the slip angle and steering angle that make a vehicle turn. We present a simplied kinematic model (based on the classic bicycle model) and a four-wheel model, which makes all of the angles involved compatible with each other. In our case, the match will be given by the determination of the turning radius. Two different scenarios, understeering and oversteering vehicles, were simulated, and the results and conclusions reached are presented herein.     

Drive control system design for stability and maneuverability of a 6WD/6WS vehicle

This paper describes a drive controller designed to improve the lateral vehicle stability and maneuverability of a 6-wheel drive / 6-wheel steering (6WD/6WS) vehicle. The drive controller consists of upper and lower level controllers. The upper level controller is based on sliding control theory and determines both front and middle steering angle, additional net yaw moment, and longitudinal net force according to the reference velocity and steering angle of a manual drive, remotely controlled, autonomous controller. The lower level controller takes the desired longitudinal net force, yaw moment, and tire force information as inputs and determines the additional front steering angle and distributed longitudinal tire force on each wheel. This controller is based on optimal distribution control and takes into consideration the friction circle related to the vertical tire force and friction coefficient acting on the road and tire. Distributed longitudinal/lateral tire forces are determined as proportion to the size of the friction circle according to changes in driving conditions. The response of the 6WD/6WS vehicle implemented with this drive controller has been evaluated via computer simulations conducted using the Matlab/Simulink dynamic model. Computer simulations of an open loop under turning conditions and a closed-loop driver model subjected to double lane change have been conducted to demonstrate the improved performance of the proposed drive controller over that of a conventional DYC.     

Longitudinal Tire Dynamics

This article begins with a brief review of the traditional concept of lateral relaxation length. The review illustrates that this concept yields a useful approximation which can be used with semi-empirical tire models which assume lateral forces are a function of steady-state slip angles. The article then presents an analogous derivation for longitudinal slip. Like its lateral counterpart, the derivation yields an approximation for transient longitudinal slip which can be used with tire models which assume longitudinal forces are a function of steady-state longitudinal slip. It is shown that, like the relaxation-length-based lateral slip angle, this formulation for longitudinal slip yields the ability to compute shear forces at the tire/road interface for either high or low speed applications, a necessary feature of simulations which support human in the loop driving simulation. Like traditional kinematically-based longitudinal slip, the transient formulation presented here is coupled with the wheel spin equation, and it shares the characteristic that it is very stiff compared to the equations of vehicle motion. This characteristic is a challenge impeding the real-time calculations required for driving simulation. The paper shows that local linearization of the wheel spin equations coupled with analytical solutions of the transient longitudinal slip formulation provide the basis for both insight into the longitudinal dynamics of the tire and for integrating the model in real-time.     

Longitudinal Tire Dynamics

SUMMARY

This article begins with a brief review of the traditional concept of lateral relaxation length. The review illustrates that this concept yields a useful approximation which can be used with semi-empirical tire models which assume lateral forces are a function of steady-state slip angles. The article then presents an analogous derivation for longitudinal slip. Like its lateral counterpart, the derivation yields an approximation for transient longitudinal slip which can be used with tire models which assume longitudinal forces are a function of steady-state longitudinal slip. It is shown that, like the relaxation-length-based lateral slip angle, this formulation for longitudinal slip yields the ability to compute shear forces at the tire/road interface for either high or low speed applications, a necessary feature of simulations which support human in the loop driving simulation. Like traditional kinematically-based longitudinal slip, the transient formulation presented here is coupled with the wheel spin equation, and it shares the characteristic that it is very stiff compared to the equations of vehicle motion. This characteristic is a challenge impeding the real-time calculations required for driving simulation. The paper shows that local linearization of the wheel spin equations coupled with analytical solutions of the transient longitudinal slip formulation provide the basis for both insight into the longitudinal dynamics of the tire and for integrating the model in real-time.     

制动—驱动工况下的轮胎侧偏特性理论研究

本文通过考虑制动和驱动工况下的轮胎印迹内垂直载荷分布的不同，建立了比以往模型更完善的制动－驱动工况下的轮胎侧偏特性理论模型，以６．５０Ｒ１６轮胎为例，分析了滑移率，侧偏角和垂直载荷分布形状参数等对其侧特性的影响。     

大侧偏角下侧偏松弛长度特性的研究

本文在稳态指数统一模型和一阶线性微分方程的基础上，研究了大侧偏角下动态过程中侧偏松弛长度的特性。侧偏松弛长度是由轮胎的侧向弹性决定的。在大侧偏角下，侧偏松弛长度不再是一个常数，而是随着侧向有效滑移率的改变而改变，而且它们之间呈现非线性特性。深入了解侧偏松弛长度的特性对研究轮胎动态特性和建立轮胎动态模型具有重要的作用。     

车辆轮胎动力学仿真模型分析

分析了各种常用轮胎模型的特点和利用范围,介绍了ADAMS中轮胎试验台(tiretesting)这一轮胎参数可视化工具,利用这一工具分析比较一种物理轮胎模型与一种经验-半经验轮胎模型间关于侧向力与纵向力、纵向力与纵向滑移率、回正力矩与纵向滑移率的力学特性,针对一种魔术公式轮胎模型验证了侧向力和纵向滑移率、纵向力和纵向滑移率在不同载荷下的力学关系特性。     

轮胎动力学特性仿真分析研究

分析了各种常用轮胎模型的特点与应用范围,根据汽车操纵动力学研究的需求,在Matlab环境下运用魔术公式建立了轮胎动力学模型,并对汽车轮胎力与纵向滑移率,纵向力、侧向力及回正力矩与纵向滑移率、侧偏角、外倾角、垂直载荷的关系等轮胎特性进行了仿真分析,实验结果表明,魔术公式轮胎动力学模型可以较好地模拟轮胎的动力学特性,适用于车辆动力学研究领域。     

A two-dimensional tire model on uneven roads for vehicle dynamic simulation1

This study proposes a comprehensive analytical tire model for handling and ride comfort in low frequency ranges. A contact algorithm that is developed in this study provides a two-dimensional contact pressure distribution on even and uneven road surfaces with reasonable computational cost. Shear stresses and strains during cornering and braking are estimated by direct measurement of tread deformations. The model is validated against experimental force and moment data for general handling simulations. Cleat tests are also conducted and validated under different forward velocity and vertical load conditions for a tire vibration study.     

Design and evaluation of sideslip angle observer for vehicle stability control

This paper presents a method for estimating the vehicle side slip angle, which is considered as a significant signal in determining the vehicle stability region in vehicle stability control systems. The proposed method combines the model-based method and kinematics-based method. Side forces of the front and rear axles are provided as a weighted sum of directly calculated values from a lateral acceleration sensor and a yaw rate sensor and from a tire model according to the nonlinear factor, which is defined to identify the degree of nonlinearity of the vehicle state. Then, the side forces are fed to the extended Kalman filter, which is designed based on the single-track vehicle model associated with a tire model. The cornering stiffness identifier is introduced to compensate for tire force nonlinearities. A fuzzy-logic procedure is implemented to determine the nonlinear factor from the input variables: yaw rate deviation from the reference value and lateral acceleration. The proposed observer is compared with a model-based method and kinematics-based method. An 8 DOF vehicle model and Dugoff tire model are employed to simulate the vehicle state in MATLAB/SIMULINK. The simulation results shows that the proposed method is more accurate than the model-based method and kinematics-based method when the vehicle is subjected to severe maneuvers under different road conditions.     

用于动力学模拟的STI轮胎模型分析

介绍了用于汽车驾驶模拟器动力学模拟的STI（Systems Technologies Inc）轮胎半经验模型,建立了该轮胎模型的复合滑移率、横向刚度和纵向刚度以及横向力和纵向力的计算方程,并利用Vc＋＋6．0编程计算得到的理论值与实验值对比,对该模型进行了分析和验证。分析结果表明,该轮胎模型用于车辆动力学模拟时有较好的精度。     

Adaptive Yaw Rate Feedback 4WS with Tire/Road Friction Coefficient Estimator

This paper proposes an adaptive yaw rate feedback control system for a four-wheel-steering (4WS) vehicle which involves a tire/road friction coefficient estimator. The adaptive 4WS system has been developed so that the vehicle possesses desirable lateral characteristics even on slippery roads and in critical driving situations. The friction coefficient is estimated on real time from the yaw rate response of the controlled vehicle with the least squares. The control system adopts a two degree of freedom structure which consists of a feedforward compensator and a feedback control subsystem. The feedforward compensator is determined with the estimated friction coefficient to minimize the steady-state and transient vehicle slip angle in spite of changes in tire/road conditions. The feedback subsystem adopts the Internal Model Control (IMC) structure in order to compensate for nonlinearities and to realize robustness against modelling and estimation errors.     

Finite element modeling of static tire enveloping characteristics

To investigate static tire enveloping characteristics, a three dimensional (3-D) finite element model is proposed. The vertical stiffness of the tire is studied on a flat surface with and without cleat. Tire rubber materials and cord layers are represented independently using “rebar” elements available in MSC Marc Mentat. Comparisons of numerical and experimental results are given to show the validity of the proposed model. It is shown that after a certain displacement, the results of the proposed model agree well with experimental results. In addition, the model results show that regardless of the type of the cleat placed under the rim center (hub center), all vertical force curves intersect after a certain displacement, which indicates typical static enveloping characteristics. Moreover, another typical characteristic of the radial tires that is unlike those of bias-ply constructions confirms that the contact patch does not expand laterally after a level vertical load is applied to the tire, which is directly related to fuel consumption and tire tread life.     

轮胎结构有限元分析应用于子午线轮胎帘线受力特性的研究

建立了195／60R14型子午线轮胎三维非线性有限元模型，研究了在静负荷作用下轮胎帘线的受力特征。结果表明，静负荷对轮胎接地区域内帘线受力影响显著；模型带束层结构为2层相交的帘线结构，不能构成力学上的对称；带束层角度的变化对帘线剪应力分布有一定的影响，对胎肩部位帘线剪应力影响较大。     

Dynamic Friction Models for Road/Tire Longitudinal Interaction

Summary In this paper we derive a new dynamic friction force model for the longitudinal road/tire interaction for wheeled ground vehicles. The model is based on a dynamic friction model developed previously for contact-point friction problems, called the LuGre model. By assuming a contact patch between the tire and the ground we develop a partial differential equation for the distribution of the friction force along the patch. An ordinary differential equation (the lumped model) for the friction force is developed, based on the patch boundary conditions and the normal force distribution along the contact patch. This lumped model is derived to approximate closely the distributed friction model. Contrary to common static friction/slip maps, it is shown that this new dynamic friction model is able to capture accurately the transient behaviour of the friction force observed during transitions between braking and acceleration. A velocity-dependent, steady-state expression of the friction force versus the slip coefficient is also developed that allows easy tuning of the model parameters by comparison with steady-state experimental data. Experimental results validate the accuracy of the new tire friction model in predicting the friction force during transient vehicle motion. It is expected that this new model will be very helpful for tire friction modeling as well as for anti-lock braking (ABS) and traction control design.     

Dynamic Friction Models for Road/Tire Longitudinal Interaction

Summary In this paper we derive a new dynamic friction force model for the longitudinal road/tire interaction for wheeled ground vehicles. The model is based on a dynamic friction model developed previously for contact-point friction problems, called the LuGre model. By assuming a contact patch between the tire and the ground we develop a partial differential equation for the distribution of the friction force along the patch. An ordinary differential equation (the lumped model) for the friction force is developed, based on the patch boundary conditions and the normal force distribution along the contact patch. This lumped model is derived to approximate closely the distributed friction model. Contrary to common static friction/slip maps, it is shown that this new dynamic friction model is able to capture accurately the transient behaviour of the friction force observed during transitions between braking and acceleration. A velocity-dependent, steady-state expression of the friction force versus the slip coefficient is also developed that allows easy tuning of the model parameters by comparison with steady-state experimental data. Experimental results validate the accuracy of the new tire friction model in predicting the friction force during transient vehicle motion. It is expected that this new model will be very helpful for tire friction modeling as well as for anti-lock braking (ABS) and traction control design.     

Selfexcited Vibrations of the Tire

In this paper a mechanical work of external forces and torques, acting on the tire has been considered. A theory has been developed for the prediction of necessary conditions for selfexciting vibrations. The theory establishes, that the mechanical work of external forces and torques must be positive, or, what is equivalent, that the tire has to transmit energy from external environment to vehicle. The work of forces and torques has been considered as a convolution of functions, which next has been submitted to Laplace transformation. The condition of selfexcitation of vibrations, obtained from above in the frequency domain, establishes, that the imaginary part of all eigenvalues of frequency response matrix must be negative. As an example, the ranges of selfexcited vibrations of the tire have been calculated. The tire has been treated as a stretched circular string. Four different models have been considered. Three of them have been massles (kinematic), namely Pacejka's model, Von Schlippe's model and single-point model. As fourth model the dynamic Pacejka's model has been considered. The influences of longitudinal deflections on the shimmy tendency have been studied for all models

Frequency response characteristics have been numerically computed and compared for all models. A critical speed and a critical reduced frequency of first and second kind have been defined. Admissible, negative, imaginary parts of the eigenvalues of frequency response matrix have been established. These admissible values of imaginary parts of eigenvalues assure, that the energy absorbing does not excite shimmy vibrations.