Establishment of Model for Tire Steady State Cornering Properties using Experimental Modal Parameters

Modeling of tire cornering properties using experimental modal parameters is studied. With tire lateral experimental modal parameters, the distribution of side force and lateral deformation under total adhesive and sliding conditions are obtained. Side force, self-aligning, cornering stiffness and relaxation length under different loads and friction coefficients are also calculated. The calculated results are in correspondence to experimental results in the references qualitatively. The non-dimensional side force obtained from the calculated results is compared with the Fiala model, its modified expression by experiments and also the widely used empirical Magic Formula model. The calculated results tally well with the fitted results obtained using Magic Formula model. It can be seen that the tire steady state cornering model using experimental modal parameters proposed in this paper is better for an in-depth study of tire natural characteristics, and the labored experimental work can be avoided.     

Study on Tire Dynamic Cornering Properties Using Experimental Modal Parameters

Based on the modeling of tire vertical characteristics and steady cornering properties, the model of tire nonsteady cornering is established in this paper using tire modal parameters extracted experimentally. In order to consider the influence of tire rolling effect on its cornering properties, the induced transfer function is introduced. The analytical formulae for calculation of transfer-function of lateral force and self-aligning torque with respect to lateral displacement and yaw angle are derived. The nonsteady cornering characteristics of tire under different loads can be calculated. The results of calculation are consistent with the experimental ones in the literature. The influence of rolling speed on tire cornering properties including the static case is analyzed. The nonsteady cornering properties in high frequency range are given by the model. The study demonstrates the consistency between steady and nonsteady tire modeling by using modal parameters.     

Study on Tire Dynamic Cornering Properties Using Experimental Modal Parameters

Based on the modeling of tire vertical characteristics and steady cornering properties, the model of tire nonsteady cornering is established in this paper using tire modal parameters extracted experimentally. In order to consider the influence of tire rolling effect on its cornering properties, the induced transfer function is introduced. The analytical formulae for calculation of transfer-function of lateral force and self-aligning torque with respect to lateral displacement and yaw angle are derived. The nonsteady cornering characteristics of tire under different loads can be calculated. The results of calculation are consistent with the experimental ones in the literature. The influence of rolling speed on tire cornering properties including the static case is analyzed. The nonsteady cornering properties in high frequency range are given by the model. The study demonstrates the consistency between steady and nonsteady tire modeling by using modal parameters.     

轮胎滚动速度效应对轮胎侧偏动特性的影响

在利用试验模态参数建立轮胎侧偏非稳态模型基础上，本文重点分析了轮胎滚动速度效应对侧偏动特性的影响。解析模型计算结构的变化趋势和文献的试验研究结果一致。表明计入轮胎滚动所带来的速度对侧偏动特性的影响是不可忽略的，它会对不同速度下行驶汽车的操稳性及横向振动产生影响。滚动速度效应对轮胎侧偏特性影响的成功建械表明了利用模态参数对轮胎建模的优越性。     

利用模态参数对轮胎纯侧偏特性建模

在轮胎静生趣特怀建模的基础上，探讨了利用轮胎模态参数对轮胎纯侧偏特性的建模。通过侧向激振试验提取了轮胎的侧向模态参数并计算了轮胎在不同载荷下的偏离特性，计算结果与以往轮胎的试验结果具有很好的一致性。计算结果转化为无量纲形式并与经验模型进行了比较，建模和计算结果充分说明了模型的合理并显示了建模方法的优越性。     

Magic Formula轮胎模型参数辨识的一种混合优化方法

Magic Formula(MF)轮胎模型能够准确描述轮胎的侧偏特性,广泛应用于车辆动力学的研究。由于MF轮胎模型参数多,且高度非线性,从大量的试验数据中准确辨识这些参数相当困难。提出一种基于遗传算法和数值优化算法的混合优化方法,采用由粗到精的辨识过程,先利用遗传算法得出近似最优解,再利用数值优化算法辨识出精确的参数。利用辨识出的参数计算轮胎的侧偏特性,计算结果与试验数据吻合良好,表明该方法是辨识MF轮胎模型参数的有效手段。     

利用试验模态参数建立轮胎非稳态侧偏模型

在对轮胎垂直特性和稳态侧偏特性建模的基础上，利用由轮胎模态试验提取的试验模态参数建立了轮胎非稳态侧偏模型，该模型考虑了印迹的动态变形和胎宽的影响，对印迹进行离散化并初步计入了速度对非稳态性的影响，推导出侧向力和回正力矩关于侧向位移和摆动角的传递函数的解析公式，可以计算不同载荷下的非稳态特性，计算结果与文献中的试验结果相符，建模和计算结果说明利用试验模态参数可以方便地建立轮胎的非稳态特性，计算结果与     

A study of the relationship between Magic Formula coefficients and tyre design attributes through finite element analysis

Pacejka's Magic Formula Tyre Model is widely used to represent force and moment characteristics in vehicle simulation studies meant to improve handling behaviour during steady-state cornering. The experimental technique required to determine this tyre model parameters is fairly involved and highly sophisticated. Also, total test facilities are not available in most countries. As force and moment characteristics are affected by tyre design attributes and tread patterns, manufacturing of separate tyres for each design alternative affects tyre development cycle time and economics significantly. The objective of this work is to identify the interactions among various tyre design attributes-cum-operating conditions and the Magic Formula coefficients. This objective is achieved by eliminating actual prototyping of tyres for various design alternatives as well as total experimentation on each tyre through simulation using finite element analysis. Mixed Lagrangian–Eulerian finite element technique, a specialized technique in ABAQUS, is used to simulate the steady-state cornering behaviour; it is also efficient and cost-effective. Predicted force and moment characteristics are represented as Magic Formula Tyre Model parameters through non-linear least-squares fit using MATLAB. Issues involved in the Magic Formula Tyre Model representation are also discussed. A detailed analysis is made to understand the influence of various design attributes and operating conditions on the Magic Formula parameters. Tread pattern, tread material properties, belt angle, inflation pressure, frictional behaviour at the tyre–road contact interface and their interactions are found to significantly influence vehicle-handling characteristics.     

基于自适应模型预测的智能汽车横向轨迹跟踪控制

当路面附着情况和车辆行驶状态不断变化时,基于恒定侧偏刚度的模型预测控制(MPC)不能考虑轮胎非线性特性的影响,难以保证车辆轨迹跟踪的适应性。为此,提出一种考虑轮胎侧向力计算误差的自适应模型预测控制(AMPC),以提高智能汽车在不确定工况下的轨迹跟踪性能。分析了路面附着系数和垂向载荷对轮胎侧向力的影响,基于平方根容积卡尔曼滤波(SCKF)算法,设计了利用侧向加速度和横摆角速度作为测量变量的前后轮胎侧向力估计器。利用轮胎侧向力线性计算值与估计值的差值计算得到侧偏刚度修正因子,设计了前后轮胎侧偏刚度的自适应修正准则,进而提出了一种基于时变修正刚度的AMPC控制方法。基于CarSim与MATLAB/Simulink联合仿真和硬件在环测试平台,对AMPC控制的有效性和实时性进行了验证。研究结果表明：在不同的路面附着情况和车辆行驶状态下,AMPC控制都能够降低横向位置偏差和航向角偏差,有效提高车辆的轨迹跟踪精度,其控制效果明显优于基于恒定侧偏刚度的标准MPC控制。尤其在低附着工况下,标准MPC控制会因为线性轮胎力的计算误差过大而导致车辆在轨迹跟踪时严重失稳,而AMPC控制通过估计轮胎力修正侧偏刚度依然能够保证车辆稳定有效的跟踪参考轨迹。所提出的AMPC控制在保证控制精度的同时具有良好的实时性,对智能汽车控制系统的设计与优化具有重要参考价值。     

Effect of the front and rear weight distribution ratio of a Formula car during maximum-speed cornering on a circuit

Because Formula cars are lighter than ordinary cars, the optimal settings for this type of car are thought to be different from those of a ordinary car. The front and rear weight distribution ratio of a vehicle is an important parameter that exerts a significant influence on critical cornering. The tendency of a ordinary car to under-steer during critical cornering is determined by the front and rear weight distribution ratio of the vehicle. Specifically, when the front of an ordinary FR (front-engine, rear wheel drive) vehicle is slightly heavier than the rear, the car tends to understeer during critical cornering. However, the optimal weight distribution ratio for critical cornering is not obvious for a formula car because of its lightness. This observation was investigated using a driving course similar to a real driving course to perform a maximum speed cornering simulations. It was found that a front to rear weight distribution ratio of 40:60 resulted in the fastest lap time. This ratio also gave the best results in the maximum-speed driving experiment performed using a driving simulator. Moreover, the maximum lateral acceleration during turning, the driving force, and the load movement of the inside and outside wheels was calculated using experimental driving force data and the concept of a tire friction circle. As a result, driving mechanics have been determined for a vehicle having a front/rear weight distribution ratio of 40:60 while traveling at maximum speed.     

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.     

Vertical Load-Deflection Behaviour of a Pneumatic Tire Subjected to Slip and Camber Angles

In this paper the vertical load-deflection behaviour of a pneumatic tire has been studied theoretically. A simple mathematical model which is especially suitable for a tire applied by a side force has been developed. Researches carried out in the past show that the tire vertical stiffness varies neither proportionally nor symmetrically as the slip angle of a cambered tire is changed. This effect can be explained by the theory developed here.

The model predictions have been verified using experimental results obtained from literature. Moreover, tire cornering characteristic curves obtained under different test conditions, i. e. during increasing of the slip angle the vertical load is kept constant or not, have been discussed through a simulation example. This study shows that the characteristic curves vary rather considerably under the different conditions.     

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.     

轮胎非稳态侧偏特性在空间域内的仿真研究

本文根据胎体复杂变形的轮胎非稳态侧偏特性理论模型在空间域内的表达式,推导出侧向力和回正力矩关于转动角与侧向位移的积分表达式,将其离散化并实现了非稳态侧偏特性在空间域内的仿真,最后给出了轮胎在几种典型输入时的仿真结果。     

An easy procedure to determine Magic Formula parameters: a comparative study between the starting value optimization technique and the IMMa optimization algorithm

In 2004, a new searching algorithm for Magic Formula tyre model parameters was presented. Now, a summary of the results, for pure and combined slip, that this algorithm is able to achieve is presented. The Magic Formula tyre model needs a set of parameters to describe the tyre properties. The determination of these parameters is dealt with in this article. A new method, called IMMa Optimization Algorithm (IOA), based on genetic techniques, is used to determine these parameters. Here, we show the computational cost that has been used to obtain the optimum parameters of every characteristic of the Magic Formula tyre model, called Delft Tyre 96. The main advantages of the method are its simplicity of implementation and its fast convergence to optimal solution, with no need of deep knowledge of the searching space. Hence, to start the search, it is not necessary to know a set of starting values of the Magic Formula parameters (null sensitivity to starting values). The search can be started with a randomly generated set of parameters between [0, 1]. Nowadays, MF-Tool, an application developed by TNO, uses an optimization technique to fit Magic Formula parameters from Matlab toolbox [van Oosten, J.J.M. and Bakker, E., 1993, {Determination of magic tyre model parameters}. Vehicle System Dynamics, 21, 19–29; van Oosten, J.J.M., Savi, C., Augustin, M., Bouhet, O., Sommer, J. and Colinot, J.P., 1999, {Time, tire, measurements, forces and moments, a new standard for steady state cornering tyre testing}. EAEC Conference, Barcelona, 30 June–2 July.]. We refer to that algorithm as the starting value optimization technique. The comparison between the optimization technique employed by TNO and the proposed IOA method is discussed in this article. In order to give a relative idea of adjustment accuracy, the sum-squared error and the mean-squared error, from the curves of the tyre model with the parameters optimized by both applications compared with test data are evaluated.     

动态载荷下轮胎侧偏特性的理论及试验研究

提出了在动载和时变印迹长度下轮胎的接触历程概念,导出了计算动载时轮胎接地印迹内各点接触历程的入迹方程,阐明轮胎动载侧偏力学特性建模机理,克服了动载侧偏特性理论建模的一个基本障碍,建立了考虑胎体平移弹性小幅动载时的理论模型,并提出了轮胎动载侧偏特性的半经验模型,进行了相应的试验研究,给出了试验结果与模型计算结果的比较。     

A comparison of various algorithms to extract Magic Formula tyre model coefficients for vehicle dynamics simulations

Tyre models are a prerequisite for any vehicle dynamics simulation. Tyre models range from the simplest mathematical models that consider only the cornering stiffness to a complex set of formulae. Among all the steady-state tyre models that are in use today, the Magic Formula tyre model is unique and most popular. Though the Magic Formula tyre model is widely used, obtaining the model coefficients from either the experimental or the simulation data is not straightforward due to its nonlinear nature and the presence of a large number of coefficients. A common procedure used for this extraction is the least-squares minimisation that requires considerable experience for initial guesses. Various researchers have tried different algorithms, namely, gradient and Newton-based methods, differential evolution, artificial neural networks, etc. The issues involved in all these algorithms are setting bounds or constraints, sensitivity of the parameters, the features of the input data such as the number of points, noisy data, experimental procedure used such as slip angle sweep or tyre measurement (TIME) procedure, etc. The extracted Magic Formula coefficients are affected by these variants. This paper highlights the issues that are commonly encountered in obtaining these coefficients with different algorithms, namely, least-squares minimisation using trust region algorithms, Nelder–Mead simplex, pattern search, differential evolution, particle swarm optimisation, cuckoo search, etc. A key observation is that not all the algorithms give the same Magic Formula coefficients for a given data. The nature of the input data and the type of the algorithm decide the set of the Magic Formula tyre model coefficients.     

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.     

轮胎胎面特性对侧偏模型预测精度影响分析

胎面花纹是影响轮胎性能的重要因素。文中以模态参数模型的稳态侧偏模型为基础，从理论推导和实际计算两方面说明胎而花纹刚度的重要性，其对侧偏模型计算结果的影响要大大高于对垂直模型的影响。因此，在确定轮胎解析模型中胎面花纹参数时应首先考虑其对侧偏模型的影响。     

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。