Straight-ahead running of road vehicles – analytical formulae including full tyre characteristics

ABSTRACT

During straight-ahead running, the longitudinal axis of road vehicles, notably cars, is not parallel to road axis. This occurrence is general and is due both to road cross slope (road banking) and to tyre characteristics, particularly ply-steer and conicity. In order to describe such a phenomenon, the paper develops a new and relatively simple analytical model. Despite the model is linear, the solution which is provided is exact, since straight-ahead motion occurs with small angles and both the elastokinematics of suspension system and tyre characteristics can be modelled by linearised equations. The Handling Diagram theory is updated and completed by introducing the actual shifts of tyre characteristics. The validation of the analytical expressions is performed by using a MSC Adams^TM full model of a car. A subjective-objective experimental test campaign provides preliminary substantiation of the ability of the derived formulae to describe tyre performance. By means of the unreferenced analytical formulae developed in the paper, we allow, given the vehicle, the proper tyre design specification and vice-versa. In particular, a formula is given to make null the steering torque during straight-ahead driving. The derived analytical formulae may provide a sound understanding of the straight-ahead running of road vehicles.     

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.     

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.     

Experimental and numerical analysis of the influence of tyres’ properties on the straight running stability of a sport-touring motorcycle

The behaviour of a motorcycle on the road is largely governed by tyre properties. This paper presents experimental and numerical analyses dealing with the influence of tyre properties on the stability of weave and wobble in straight running. The final goal is to find optimal sets of tyre properties that improve the stability of a motorcycle. The investigation is based on road tests carried out on a sport-touring motorcycle equipped with sensors. Three sets of tyres are tested at different speeds in the presence of weave and wobble. The analysis of telemetry data highlights significant differences in the trends of frequency and damping of weave and wobble against speed. The experimental analysis is integrated by a parametric numerical analysis. Tyre properties are varied according to the design of experiments method, in order to highlight the single effects on stability of lateral and cornering coefficient of front and rear tyres.     

The simulation of in-plane tyre modal behaviour: a broad modal range comparison between analytical and discretised modelling approaches

In-plane tyre modal behaviour determines the response of tyres to ride excitations and braking/traction manoeuvres. In many studies, the interest is limited to relatively low frequencies and a detailed investigation into the ability of models to accurately simulate higher-order responses is unnecessary. In cases where an in-plane model is to be used for the generation of the contact deformation and stresses, or where modal reduction methods are implemented, a detailed knowledge of the modal response is desirable. The present work forms a study on the ability of a number of frequently used modelling approaches to generate realistic modal data throughout a wide frequency range. The analytical ring on elastic foundation model is used as a benchmark throughout the paper. Its predictions are compared with those of two discretised models, namely a truss- and a beam-based model. The sensitivity of the ring’s response to a number of physical parameters is discussed. The results are used to inform the comparison between the analytical ring and the discretised models, providing explanations for any discrepancies observed. The limited applicability of the truss model is pointed out, while the accuracy of the beam-based model is enhanced by a circumferential inextensible string element. Both the ring and the enhanced beam models are further improved with the addition of a nonlinear string-based sidewall that accounts for the change in sidewall stiffness with inflation pressure. The findings may offer a reference when setting up in-plane models, including the stage of planning modal tests for parameter identification.     

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.     

Tyre friction behaviour under abrupt wheel torque transients on slippery road surfaces: experimental analysis and modelling

The paper shows that, during abrupt wheel torque transients for ice surface and low vehicle speeds, the tyre can develop significantly larger longitudinal force than the peak value of the tyre static curve. This so-called dynamic tyre friction potential (DTFP) effect has many influencing factors such as the rate of change of the wheel torque, the vehicle speed, and the tyre dwell time. The paper presents a detailed analysis of the DTFP behaviour based on the experimental data collected by using an in-wheel motor-based tyre test vehicle. The analysis results and an insight into the brush structure of a tyre model lead to the hypothesis that the different influencing factors may be predominantly explained by the bristle dwell time (BDT) effect. Following this hypothesis, the LuGre model of the tyre friction dynamics is extended with a physical BDT sub-model. The experimental validation results show that the proposed model can accurately capture the low-speed tyre–ice friction behaviour during abrupt wheel torque transients.     

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.     

Energy analysis of the Von Schlippe tyre model with application to shimmy

Shimmy is an engineering example of self-excited vibrations. Much research on shimmy has considered the tyre as a positive feedback or negative damping to introduce instability of the entire system. In this context, we focus on the behaviour of the tyre under periodic excitations. The Von Schlippe tyre model is selected and the energy flow method is applied to illustrate the energy transfer by the tyre during shimmy. The energy flow method evaluates the tyre performance with a prescribed sinusoidal motion and provides a novel evaluation method for tyre models. With the help of straight contact line assumption in the Von Schlippe tyre model, the relative motion between the contact line and the wheel centre is studied to understand the path dependency of the energy transfer. It turns out that the tyre is extracting energy from the forward motion to induce unstable lateral and yaw vibrations when the motion or orientation of the contact line has a phase lead with respect to the wheel centre.     

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.     

Estimation of the tyre–road maximum friction coefficient and slip slope based on a novel tyre model

In this article, a new approach to estimate the vehicle tyre forces, tyre–road maximum friction coefficient, and slip slope is presented. Contrary to the majority of the previous work on this subject, a new tyre model for the estimation of the tyre–road interface characterisation is proposed. First, the tyre model is built and compared with those of Pacejka, Dugoff, and one other tyre model. Then, based on a vehicle model that uses four degrees of freedom, an extended Kalman filter (EKF) method is designed to estimate the vehicle motion and tyre forces. The shortcomings of force estimation are discussed in this article. Based on the proposed tyre model and the improved force measurements, another EKF is implemented to estimate the tyre model parameters, including the maximum friction coefficient, slip slope, etc. The tyre forces are accurately obtained simultaneously. Finally, very promising results have been achieved for pure acceleration/braking for varying road conditions, both in pure steering and combined manoeuvre simulations.     

Time-optimal control of the race car: influence of a thermodynamic tyre model

The robustness of an existing numerical method for the time-optimal control of the race car is demonstrated through its application to a model of a Formula 1 car equipped with a simplified thermodynamic tyre model. The tyre model includes a temperature- and frequency-dependent model of road/tyre friction. A lumped parameter approach is used to model the thermodynamics of the various parts of the tyre such as the tread, carcass and inflation gas. The influence of tyre, track surface and ambient temperatures on time-optimal manoeuvring is presented.     

Application of the Pacejka Magic Formula Tyre Model on a Study of a Hydraulic Anti-Lock Braking System for a Light Motorcycle

The object of the study is to apply the Pacejka magic formula tyre model on a study of a hydraulic anti-lock braking system, especially applied to a light motorcycle. A sliding mode PWM controller is designed and tested. Both simulation and experimental studies of an anti-lock braking system are undertaken. The paper presents an analytical approach for estimating the longitudinal adhesive coefficient between a tyre and the road through the magic formula tyre model, the parameters of which are identified by a genetic algorithm. A dynamic analysis of a light motorcycle is carried out in detail. The experimental results show that the antilock braking system designed in the study is effective to prevent wheels locking during emergency braking. The proposed simulation results match experimental data well.     

Application of the Pacejka Magic Formula Tyre Model on a Study of a Hydraulic Anti-Lock Braking System for a Light Motorcycle

The object of the study is to apply the Pacejka magic formula tyre model on a study of a hydraulic anti-lock braking system, especially applied to a light motorcycle. A sliding mode PWM controller is designed and tested. Both simulation and experimental studies of an anti-lock braking system are undertaken. The paper presents an analytical approach for estimating the longitudinal adhesive coefficient between a tyre and the road through the magic formula tyre model, the parameters of which are identified by a genetic algorithm. A dynamic analysis of a light motorcycle is carried out in detail. The experimental results show that the antilock braking system designed in the study is effective to prevent wheels locking during emergency braking. The proposed simulation results match experimental data well.     

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).     

Noise,vibration, harshness model of a rotating tyre

ABSTRACT

The tyre plays a fundamental role in the generation of acoustically perceptible driving noise and vibrations inside the vehicle. An essential part of these vibrations is induced by the road excitation and transferred via the tyre into the vehicle. There are two basic ways to study noise, vibration, harshness (NVH) behaviour: Simulations in time and frequency domains. Modelling the tyre transfer behaviour in frequency domain requires special attention to the rotation of the tyre. This paper shows the approach taken by the authors to include the transfer behaviour in the frequency range up to 250?Hz from geometric road excitations to resulting spindle forces in frequency domain. This paper validates the derived NVH tyre model by comparison with appropriate transient simulations of the base transient model.     

Identifying tyre models directly from vehicle test data using an extended Kalman filter

Individual tyre models are traditionally derived from component tests, with their parameters matched to force and slip measurements. They are imported into vehicle models which should, but do not always properly provide suspension geometry interaction. Recent advances in Global Positioning System (GPS)/inertia vehicle instrumentation now make full state measurement viable in test vehicles, so tyre slip behaviour is directly measurable. This paper uses an extended Kalman filter for system identification, to derive individual load-dependent tyre models directly from these test vehicle state measurements. The resulting model therefore implicitly compensates for suspension geometry and compliance. The paper looks at two variants of the tyre model, and also considers real-time adaptation of the model to road surface friction variations. Test vehicle results are used exclusively, and the results show successful tyre model identification, improved vehicle model state prediction – particularly in lateral velocity reproduction – and an effective real-time solution for road friction estimation.