Pragmatic Tyre Model for Short Wavelength Side Slip Variations

The first-order relaxation length concept is often used to model transient tyre behaviour. It gives a rather good representation of the lateral force response, also at shorter wavelengths. The self aligning moment appears to behave rather differently and its responses cannot be represented by a relaxation length system only. Based on the analytical frequency response functions of the pneumatic trail of the brush type tyre model, a new pragmatic approach is developed. The model consists of a phase leading system in series with the first-order model for the lateral force. The results are compared to the responses of a discrete brush type simulation model. It appears that the aligning moment can be represented by this new pragmatic model very well.     

String tyre model for evaluating steering agility performance using tyre cornering force and lateral static characteristics

The cornering force and lateral static characteristics of a tyre are fundamental factors that describe the steering feel for handling performance. However, it is difficult to justify the contribution of each factor when the tyre’s cornering motion is evaluated through subjective assessment. Currently, the relaxation length of Pacejka’s tyre model is close to describing these tyre motions. Therefore, this paper proposes a string tyre model based on the relaxation length in order to represent the steering performance. The proposed method provides a more accurate modelling of the steering agility performance. Therefore, it is possible to use this model to predict the steering response performance, and this is validated through comparison with a real relaxation length.     

Effects of model response on model following type of combined lateral force and yaw moment control performance for active vehicle handling safety

This paper presents a comparison study of the effect of model response on the performance of the model following type combined lateral force and yaw moment control. The combined controls aim to maximize stability limit as well as vehicle responsiveness. In order to realize this aim, two types of model responses are proposed to introduce the required lateral force and yaw moment control. The model responses (a) is the side-slip angle and yaw rate vehicle response of the two degree of freedom vehicle motion (bicycle model). The model responses (b) is an intentional modification from the model responses (a) to the side slip angle converging to zero and first order yaw rate. Three different cases of combining lateral force and yaw moment control have been investigated using the two types of model responses. The effect of model responses is proved by computer simulations of the vehicle response to a single sine wave steering input with braking for the combined control methods proposed. It is found that the influence of the model response has a significant effect on the combined control performance.     

Study of the Pneumatic Tyre Behaviour on Dry and Rigid Road by Finite Element Method

The paper presents a physical tyre model capable of describing the complete pneumatic tyre behaviour during steady and transient states. Given the radial deflection, the longitudinal and lateral slip, the camber angle, the inner pressure and the mechanical parameters describing the tyre structure, the model returns the vertical load, the longitudinal and lateral forces, the self aligning torque. Particular attention has been devoted to the computation (by f.e.m.) of tyre carcass and tread deformations; it is explained how side force increases by moderate braking at constant slip angle. An experimental verification validates the model, although more studies could be needed to improve model effectiveness.     

Study of the Pneumatic Tyre Behaviour on Dry and Rigid Road by Finite Element Method

SUMMARY

The paper presents a physical tyre model capable of describing the complete pneumatic tyre behaviour during steady and transient states. Given the radial deflection, the longitudinal and lateral slip, the camber angle, the inner pressure and the mechanical parameters describing the tyre structure, the model returns the vertical load, the longitudinal and lateral forces, the self aligning torque. Particular attention has been devoted to the computation (by f.e.m.) of tyre carcass and tread deformations; it is explained how side force increases by moderate braking at constant slip angle. An experimental verification validates the model, although more studies could be needed to improve model effectiveness.     

Dynamic tyre friction models for combined longitudinal and lateral vehicle motion

An extension to the LuGre dynamic friction model from longitudinal to longitudinal/lateral motion is developed in this paper. Application of this model to a tyre yields a pair of partial differential equations that model the tyre-road contact forces and aligning moment. A comparison of the steady-state behaviour of the dynamic model with existing static tyre friction models is presented. This comparison allows one to determine realistic values of the parameters for the new dynamic model. Via the introduction of a set of mean states we reduce the partial differential equations to a lumped model governed by a set of three ordinary differential equations. Such a lumped form describes the aggregate effect of the friction forces and moments and it can be useful for control design and online estimation. A method to incorporate wheel rim rotation is also proposed. The proposed model is evaluated by comparing both its steady-state as well as its dynamic characteristics via numerical simulations. The results of the simulations corroborate steady-state and dynamic/transient tyre characteristics found in the literature.     

Experimental analysis of tyre-enveloping characteristics at low speed

In this study, experiments are conducted to investigate tyre-enveloping characteristics. Four different types of tyres are tested. Parameters such as different tyre inflation pressures, vertical loads and types of obstacles (cleats) are considered. In addition to vertical stiffnesses of all tyres, vertical and horizontal force variations while traversing different obstacles at low speed are studied. The effects of inflation pressure and vertical load on variations of force and moment are investigated. Static test results showed that after a certain vertical displacement, all curves in force–deflection diagrams plotted with and without cleat intersect regardless of cleat and tyre types, depending on the inflation pressure of the tyre, which can be called typical static tyre-enveloping characteristics. Test results at low speed show that there is a considerable influence of the vertical load on vertical and lateral force responses of a tyre.     

Shear forces in the contact patch of a braked-racing tyre

This article identifies tyre modelling features that are fundamental to the accurate simulation of the shear forces in the contact patch of a steady-rolling, slipping and cambered racing tyre. The features investigated include contact patch shape, contact pressure distribution, carcass flexibility, rolling radius (RR) variations and friction coefficient. Using a previously described physical tyre model of modular nature, validated for static conditions, the influence of each feature on the shear forces generated is examined under different running conditions, including normal loads of 1500, 3000 and 4500 N, camber angles of 0° and?3°, and longitudinal slip ratios from 0 to?20%. Special attention is paid to heavy braking, in which context the aligning moment is of great interest in terms of its connection with the limit-handling feel. The results of the simulations reveal that true representations of the contact patch shape, carcass flexibility and lateral RR variation are essential for an accurate prediction of the distribution and the magnitude of the shear forces generated at the tread–road interface of the cambered tyre. Independent of the camber angle, the contact pressure distribution primarily influences the shear force distribution and the slip characteristics around the peak longitudinal force. At low brake-slip ratios, the friction coefficient affects the shear forces in terms of their distribution, while, at medium to high-slip ratios, the force magnitude is significantly affected. On the one hand, these findings help in the creation of efficient yet accurate tyre models. On the other hand, the research results allow improved understanding of how individual tyre components affect the generation of shear forces in the contact patch of a rolling and slipping tyre.     

A novel semi-empirical tyre model for combined slips

A new tyre-force model for simultaneous braking and cornering is presented, which is based on combining existing empirical models for pure braking and cornering with brush-model tyre mechanics. The aim is to offer an easy-to-use, accurate model for vehicle-handling simulations. On a working tyre the contact patch between the tyre and the road is, in general, divided into an adhesion region where the rubber is gripping the road and a sliding region where the rubber slides on the road surface. The total force generated by the tyre is then composed of components from these two regions. The brush model describes this in a mechanical framework. The proposed model is based on a new method to extract adhesion and sliding forces from empirical pure-slip tyre models. These forces are then scaled to account for the combined-slip condition. The combined-slip self-aligning torque is also described. A particular feature of the model is the inclusion of velocity dependence, even if this is not explicitly present in the empirical pure-slip model. The approach is quite different from most previous combined-slip models, in that it is based on a rather detailed mechanical model in combination with empirical pure-slip models. The model is computationally sound and efficient and does not rely on any additional parameters that depend on combined-slip data for calibration. It can be used in combination with virtually any empirical pure-slip model and in this work the Magic Formula is used in examples. Results show good correspondence with experimental data.     

A 3D Brush-type Dynamic Tire Friction Model

The use of advanced dynamic friction models can improve the brush-type tire friction models. This paper presents a 3D dynamic brush model based on the LuGre friction model. The model describes the dynamics of longitudinal and lateral tire friction forces, as well as the self aligning torque dynamics. It has been originally derived in a distributed-parameter form, and then transformed to a simpler lumped-parameter form with only three internal states. Both uniform and non-uniform normal pressure distributions are considered. The model has analytical solution for steady-state conditions. The steady-state behavior is validated with respect to “magic” formula static model, which served as an “ideal” benchmark. The lumped model dynamic behavior is validated by comparing its time-responses with original distributed model responses. The model parameterization with respect to normal force and other tire/road parameters is considered as well.     

A 3D Brush-type Dynamic Tire Friction Model

The use of advanced dynamic friction models can improve the brush-type tire friction models. This paper presents a 3D dynamic brush model based on the LuGre friction model. The model describes the dynamics of longitudinal and lateral tire friction forces, as well as the self aligning torque dynamics. It has been originally derived in a distributed-parameter form, and then transformed to a simpler lumped-parameter form with only three internal states. Both uniform and non-uniform normal pressure distributions are considered. The model has analytical solution for steady-state conditions. The steady-state behavior is validated with respect to “magic” formula static model, which served as an “ideal” benchmark. The lumped model dynamic behavior is validated by comparing its time-responses with original distributed model responses. The model parameterization with respect to normal force and other tire/road parameters is considered as well.     

3D brush model to predict longitudinal tyre characteristics

A 3D tyre brush model, which aims to predict the longitudinal tyre characteristic under steady-state conditions by modelling the occurring physical effects in the tyre–road contact patch, is presented. The model includes an analytical method to describe the tyre footprint geometry, the pressure distribution, the slip due to the lateral tyre contour, the slip due to braking or traction and the longitudinal as well as the lateral shear stresses on a flattened tyre. The presented development tool offers a method to investigate different rubber friction data (caused by different tread compounds and/or surface textures) and to analyse its influence on longitudinal tyre characteristics. The tyre design is fixed (same casing, dimension and pattern). The results include the shear stresses as well as the different sliding velocities in the contact patch for different slip conditions. The model was developed for a standard summer pattern design and a standard tyre dimension (205/55R16). It can also be adapted to other tread designs and tyre dimensions. To offer a good comparability between model results and test bench measurements, the surface curvature of an internal test rig is considered.     

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.     

Vehicle steering returnability with maximum steering wheel angle at low speeds

In this paper, an analytical model with suitable vehicle parameters, together with a multi-body model is proposed to predict steering returnability in low-speed cornering with what is expected to be adequate precision as the steering wheel moves from lock to lock. This model shows how the steering response can be interpreted in terms of vertical force, lateral force with aligning moment, and longitudinal force. The simulation results show that vertical steering rack forces increase in the restoring direction according to steering rack displacement for both the inner and outer wheels. As lateral forces due to side-slip angle are directed toward the medial plane of the vehicle in both wheels, the outer wheel pushes the steering wheel in the returning direction while the inner wheel does not. In order to improve steering returnability, it is possible to increase the total steering rack force in both road wheels through adjustments to the kingpin axis and steering angle. This approach is useful for setting up a proper suspension geometry during conceptual chassis design.     

轮胎制动力—滑移率特性的动态模型研究

轮胎制动力及滑移率对制动力矩的响应存在着显著的低频衰减振荡现象，这从制动力－滑移率特性曲线上看，表现为一系列的以稳态点为中心的旋涡状过渡曲线。这是稳态的轮胎模型所不能描述的。本文在稳态ｂｒｕｓｈ轮胎模型基础上，通过印迹前端蹼处胎冠一阶变形模型，建立了轮胎制动力－滑移率的动态模型，并根据转鼓试验台上的轮胎制动试验，验证了该模型的仿真效果。     

Lateral Tyre Force by a Milliken Test on a Flat Track Roadway Simulator

The lateral tyre force versus slip angle and vertical load is obtained through specific tests on the single tyre or with a theoretical-empirical analysis with physical models. This study is about the possibility to use a third way: Executing a particular handling test, known as Force-Moment test, using a Flat Track Roadway Simulator (the Fiat-Elasis MTS FTRS). The understanding and the project of vehicle handling is strongly based on the knowledge of lateral tyre force response [1, 2, 3, 7].     

Lateral Tyre Force by a Milliken Test on a Flat Track Roadway Simulator

The lateral tyre force versus slip angle and vertical load is obtained through specific tests on the single tyre or with a theoretical-empirical analysis with physical models. This study is about the possibility to use a third way: Executing a particular handling test, known as Force-Moment test, using a Flat Track Roadway Simulator (the Fiat-Elasis MTS FTRS). The understanding and the project of vehicle handling is strongly based on the knowledge of lateral tyre force response [1, 2, 3, 7].     

A combined-slip predictive control of vehicle stability with experimental verification

In this paper, a model predictive vehicle stability controller is designed based on a combined-slip LuGre tyre model. Variations in the lateral tyre forces due to changes in tyre slip ratios are considered in the prediction model of the controller. It is observed that the proposed combined-slip controller takes advantage of the more accurate tyre model and can adjust tyre slip ratios based on lateral forces of the front axle. This results in an interesting closed-loop response that challenges the notion of braking only the wheels on one side of the vehicle in differential braking. The performance of the proposed controller is evaluated in software simulations and is compared to a similar pure-slip controller. Furthermore, experimental tests are conducted on a rear-wheel drive electric Chevrolet Equinox equipped with differential brakes to evaluate the closed-loop response of the model predictive control controller.