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.     

Identification of the mechanical properties of bicycle tyres for modelling of bicycle dynamics

Advanced simulation of the stability and handling properties of bicycles requires detailed road–tyre contact models. In order to develop these models, in this study, four bicycle tyres are tested by means of a rotating disc machine with the aim of measuring the components of tyre forces and torques that influence the safety and handling of bicycles. The effect of inflation pressure and tyre load is analysed. The measured properties of bicycle tyres are compared with those of motorcycle tyres.     

Geometric-based tyre vertical force estimation and stiffness parameterisation for automotive and unmanned vehicle applications

Advanced empirical, and physical-based tyre models have proven to be accurate for simulating tyre dynamics; however, these tyre models typically require expensive and intensive tyre parameterisation. Recent research into wheeled unmanned ground vehicles requiring vertical force analysis has shown good results using a simple linear spring model for the tyre which demonstrate the continued use for simple tyre models; however, parameterisation of the tyre still remains a challenge when load test equipment is not available. This paper presents a cost-effective tyre vertical stiffness parameterisation procedure using only measured tyre geometry and air pressure for applications where high-fidelity tyre models are unnecessary. Vertical forces calculated through an air volume optimisation approach are used to estimate tyre vertical stiffness. Nine tyres from the literature are compared to evaluate the performance of the vertical force estimation and stiffness parameterisation algorithms. Experimental results on a pair of ATV tyres are also presented.     

Analysis of vehicle rollover dynamics using a high-fidelity model

Recent data show that 35% of fatal crashes in sport utility vehicles included vehicle rollover. At the same time, experimental testing to improve safety is expensive and dangerous. Therefore, multi-body simulation is used in this research to improve the understanding of rollover dynamics. The majority of previous work uses low-fidelity models. Here, a complex and highly nonlinear multi-body model with 165 degrees of freedom is correlated to vehicle kinematic and compliance (K&C) measurements. The Magic Formula tyre model is employed. Design of experiment methodology is used to identify tyre properties affecting vehicle rollover. A novel, statistical approach is used to link suspension K&C characteristics with rollover propensity. Research so far reveals that the tyre properties that have the greatest influence on vehicle rollover are friction coefficient, friction variation with load, camber stiffness and tyre vertical stiffness. Key K&C characteristics affecting rollover propensity are front and rear suspension rate, front roll stiffness, front camber gain, front and rear camber compliance and rear jacking force.     

The effect of the inflation pressure of tyres on motorcycle weave stability: experiments and simulation

Increasing the stability of a motorcycle requires an understanding of the optimal conditions of the tyre. The inflation pressure is one of the main parameters that directly affects the tyre properties, which in turn influences motorcycle stability and safety. This paper focuses on the effect of the inflation pressure of the tested tyres on motorcycle weave stability. Experimental data are collected from tests carried out in straight running at constant speed. The data analysis is based on stochastic subspace identification methods. Simulations are performed using an advanced motorcycle multi-body code with parameters measured from the tested vehicle. Finally, the comparison between simulations and experimental tests is discussed. The research results show an agreement between experimental tests and simulations where weave stability increases with inflation pressure for the specified range of tyre pressure.     

Longitudinal wheel slip during ABS braking

Anti-lock braking system (ABS) braking tests with two subcompact passenger cars were performed on dry and wet asphalt, as well as on snow and ice surfaces. The operating conditions of the tyres in terms of wheel slip were evaluated using histograms of the wheel slip data. The results showed different average slip levels for different road surfaces. It was also found that changes in the tyre tread stiffness affected the slip operating range through a modification of the slip value at which the maximum longitudinal force is achieved. Variation of the tyre footprint length through modifications in the inflation pressure affected the slip operating range as well. Differences in the slip distribution between vehicles with different brake controllers were also observed. The changes in slip operating range in turn modified the relative local sliding speeds between the tyre and the road. The results highlight the importance of the ABS controller's ability to adapt to changing slip–force characteristics of tyres and provide estimates of the magnitude of the effects of different tyre and road operating conditions.     

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.     

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.     

Unscented Kalman filter for vehicle state estimation

Vehicle dynamics control (VDC) systems require information about system variables, which cannot be directly measured, e.g. the wheel slip or the vehicle side-slip angle. This paper presents a new concept for the vehicle state estimation under the assumption that the vehicle is equipped with the standard VDC sensors. It is proposed to utilise an unscented Kalman filter for estimation purposes, since it is based on a numerically efficient nonlinear stochastic estimation technique. A planar two-track model is combined with the empiric Magic Formula in order to describe the vehicle and tyre behaviour. Moreover, an advanced vertical tyre load calculation method is developed that additionally considers the vertical tyre stiffness and increases the estimation accuracy. Experimental tests show good accuracy and robustness of the designed vehicle state estimation concept.     

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.     

Tyre rolling kinematics and prediction of tyre forces and moments: part I – theory and method

A new method to describe tyre rolling kinematics and how to calculate tyre forces and moments is presented. The Lagrange–Euler method is used to calculate the velocity and contact deformation of a tyre structure under large deformation. The calculation of structure deformation is based on the Lagrange method, while the Euler method is used to analyse the deformation and forces in the contact area. The method to predict tyre forces and moments is built using kinematic theory and nonlinear finite element analysis. A detailed analysis of the tyre tangential contact velocity and the relationships between contact forces, contact areas, lateral forces, and yaw and camber angles has been performed for specific tyres. Research on the parametric sensitivity of tyre lateral forces and self-aligning torque on tread stiffness and friction coefficients is carried out in the second part of this paper.     

New tyre-road contact model for applications at low speed

Most of the tyre models have been developed for high speed, combined forces, etc., however, in certain tests it is necessary to know tyre behaviour at very low speed in order to evaluate different systems. So, during vehicle inspection and maintenance of the steering and brake system, by means of sideslip tester and roller brake tester respectively, the forces transmitted by the tyres are measured; all of these inspections are carried out at low speeds. Furthermore, usually, automobile vehicles run at low speeds during an important part of their operating life (less than 60 km/h), mainly during urban traffic, and in steady state conditions. Therefore, it is particularly interesting to develop an accurate model of the contact patch tyrepavement for low speeds without the complexity of models that cover a wide speed range but provide less precision at very low speeds. The dynamometer plate has proved to be an appropriate test equipment to characterise the tyre-pavement contact at low speed and the steering geometry and wheel alignment. It has the feature of being able to carry out tests with the tyre installed in the vehicle as in completely real conditions. The main aim of this research is to set up a contact model between tyre and pavement at very low speed based on the measurement of longitudinal and lateral forces. A test methodology that allows carrying out the experimental tests in a systematic and controlled way with the dynamometer plate has also been developed. From this model it will be possible to estimate the forces that tyres are capable of transmitting in different situations to act in the parameters which affect these forces and maximize them.     

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.     

Estimation of longitudinal speed robust to road conditions for ground vehicles

This article seeks to develop a longitudinal vehicle velocity estimator robust to road conditions by employing a tyre model at each corner. Combining the lumped LuGre tyre model and the vehicle kinematics, the tyres internal deflection state is used to gain an accurate estimation. Conventional kinematic-based velocity estimators use acceleration measurements, without correction with the tyre forces. However, this results in inaccurate velocity estimation because of sensor uncertainties which should be handled with another measurement such as tyre forces that depend on unknown road friction. The new Kalman-based observer in this paper addresses this issue by considering tyre nonlinearities with a minimum number of required tyre parameters and the road condition as uncertainty. Longitudinal forces obtained by the unscented Kalman filter on the wheel dynamics is employed as an observation for the Kalman-based velocity estimator at each corner. The stability of the proposed time-varying estimator is investigated and its performance is examined experimentally in several tests and on different road surface frictions. Road experiments and simulation results show the accuracy and robustness of the proposed approach in estimating longitudinal speed for ground vehicles.     

Environmental effects on Pacejka’s scaling factors

A set of scaling factors has been introduced by Pacejka [Pacejka, H.B., 2002, Tyre and Vehicle Dynamics (Oxford: Butterworth Heinemann Editions)] into his Magic Formula tyre model to take into account the influence of a number of external overall parameters such as road roughness, weather conditions, suspension characteristics and so on. These scaling factors are important for a correct prediction of tyre–road contact forces, but are not a function of the tyre itself. Changing the point of view, one could say that scaling factors should remain constant for different tyres on the same circuit, with the same weather conditions and with the same car. After characterizing different tyres through indoor tests (that do not consider external overall parameters) and after having identified Pacejka’s coefficients with scaling factors equal to one, several outdoor experimental tests have been carried out to assess the influence of vehicle and road surface conditions on scaling factors. These experimental data allowed us to identify, through a minimization approach, the ‘best’ set of Pacejka’s scaling factors for that vehicle and for that tyre on that track. Scaling factors for equal track and vehicle but different tyres were compared to check whether their values remained constant. To access the validity of scaling factors, a comparison between experimental data, collected on an instrumented passenger car, and MB simulations considering unity and identified scaling factors’ values, were carried out. All experimental data shown in this article come from tests carried out within the VERTEC project, a European founded research project (Task 2.a and 2.b) that puts together knowledge coming from vehicle manufacturers (Volvo, Porsche and Centro Ricerche Fiat CRF), tyre manufacturers (Pirelli and Nokian Tyres), control logic manufacturers (Lucas Varity GmbH), road maintenance experts (Centres d’Études Techniques de l’Équipement CETE), transport research organizations (Transport Research Laboratory TRL, Swedish National Road and Transport Research Institute VTI) and universities (Helsinki University of Technology HUT, Politecnico di Milano and University of Florence UNIFI).     

Design of safety-oriented control allocation strategies for overactuated electric vehicles

The new vehicle platforms for electric vehicles (EVs) that are becoming available are characterised by actuator redundancy, which makes it possible to jointly optimise different aspects of the vehicle motion. To do this, high-level control objectives are first specified and solved with appropriate control strategies. Then, the resulting virtual control action must be translated into actual actuator commands by a control allocation layer that takes care of computing the forces to be applied at the wheels. This step, in general, is quite demanding as far as computational complexity is considered. In this work, a safety-oriented approach to this problem is proposed. Specifically, a four-wheel steer EV with four in-wheel motors is considered, and the high-level motion controller is designed within a sliding mode framework with conditional integrators. For distributing the forces among the tyres, two control allocation approaches are investigated. The first, based on the extension of the cascading generalised inverse method, is computationally efficient but shows some limitations in dealing with unfeasible force values. To solve the problem, a second allocation algorithm is proposed, which relies on the linearisation of the tyre–road friction constraints. Extensive tests, carried out in the CarSim simulation environment, demonstrate the effectiveness of the proposed approach.     

Investigation of Features of Tyre Rolling at Non-Small Velocities on the Basis of a Simple Tyre Model with Distributed Mass Periphery

SUMMARY

On the basis of the brush-type tyre model the paper considers the interaction between steady-state rolling deformable wheel and flat road surface as well as corresponding force and moment characteristics of the wheel.

At least two zones of sliding, anisotropic dry friction, sliding friction coefficient speed-dependent and instantaneous leap of the friction coefficient when transition from sliding to adhesion zone occurs, have been taken into account, as well as distributed peripheral mass of tyre, elasticity, pseudo-dry friction and damping properties in radial, tangential and lateral directions of the elements at the wheel periphery, including a visco-elastic belt. Vertical force distribution in the contact area is not supposed to be known in advance and follows from the calculation. As a result, sliding zone lengths, distributed forces in contact area, six components of generalized road reaction reduced to the wheel center, and rolling resistance moment are found as functions of vertical load, movement velocity, longitudinal and side slip, friction in contact area with road, stiffnesses, dry friction and damping in the tyre model elements and of distributed peripheral mass.

A computer program developed in Fortran and results of calculations are of particular interest for qualitative analysis including steady rolling of studded tyre and also racing car and aircraft tyres which peripheral mass shows itself in a special way because of great movement velocities.     

Sliding mode-based lateral vehicle dynamics control using tyre force measurements

In this work, a lateral vehicle dynamics control based on tyre force measurements is proposed. Most of the lateral vehicle dynamics control schemes are based on yaw rate whereas tyre forces are the most important variables in vehicle dynamics as tyres are the only contact points between the vehicle and road. In the proposed method, active front steering is employed to uniformly distribute the required lateral force among the front left and right tyres. The force distribution is quantified through the tyre utilisation coefficients. In order to address the nonlinearities and uncertainties of the vehicle model, a gain scheduling sliding-mode control technique is used. In addition to stabilising the lateral dynamics, the proposed controller is able to maintain maximum lateral acceleration. The proposed method is tested and validated on a multi-body vehicle simulator.     

Tyre rolling kinematics and prediction of tyre forces and moments: part II – simulation and experiment

There are two aims for the second part of this paper: verifying the theory presented in the first part through parameter variation and comparison between simulation and experiment, and to study the effect of the belt structure on the cornering properties of radial tyres. Research has been carried out with a passenger car radial tyre and two different kinds of truck or bus radial tyres using both simulation and experiment. This second part of the paper shows that belt structure plays an important role in the generation of tyre forces and moments in addition to the effects of the tread stiffness and friction coefficients. The theory and method presented in this paper opens a new robust way to predict the tyre forces and moments from the tyre design and provides a reliable model for a generation mechanism.