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Abstract

The tire and vehicle setup definition, able to optimise grip performance and thermal working conditions, can make the real difference as for motorsport racing teams, used to deal with relevant wear and degradation phenomena, as for tire makers, requesting for design solutions aimed to obtain enduring and stable tread characteristics, as finally for the development of safety systems, conceived in order to maximise road friction, both for worn and unworn tires. The activity discussed in the paper deals with the analysis of the effects that tire wear induces in vehicle performance, in particular as concerns the consequences that tread removal has on thermal and frictional tire behaviour. The physical modelling of complex tire–road interaction phenomena and the employment of specific simulation tools developed by the Vehicle Dynamics UniNa research group allow to predict the tire temperature local distribution by means of TRT model and the adhesive and hysteretic components of friction, thanks to GrETA model. The cooperation between the cited instruments enables the user to study the modifications that a reduced tread thickness, and consequently a decreased SEL (Strain Energy Loss) and dissipative tread volume, cause on the overall vehicle dynamic performance.

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... Such complex structure is necessary for supporting the interaction forces with the road. In literature there are different models able to describe tyre behavior both based on an physical approach [1][2][3][4][5][6][7][8][9][10][11][12] and based on an empirical one [13][14][15][16] . In order to parametrize physical based models [1][2][3][4][5][6][7][8][9][10][11][12] , it is very significant the knowledge about the thickness of the different tyre layers. ...
... In literature there are different models able to describe tyre behavior both based on an physical approach [1][2][3][4][5][6][7][8][9][10][11][12] and based on an empirical one [13][14][15][16] . In order to parametrize physical based models [1][2][3][4][5][6][7][8][9][10][11][12] , it is very significant the knowledge about the thickness of the different tyre layers. This aspect is particularly important as it concerns the physical models describing the tyre thermal behavior [8][9][10] , the tyre grip behavior [1,2,6,7] and the tyre wear [11] . ...
... In order to parametrize physical based models [1][2][3][4][5][6][7][8][9][10][11][12] , it is very significant the knowledge about the thickness of the different tyre layers. This aspect is particularly important as it concerns the physical models describing the tyre thermal behavior [8][9][10] , the tyre grip behavior [1,2,6,7] and the tyre wear [11] . ...
Article
In this work is exploited the possibility to use two optical techniques and combining their measurements for the 3D characterization of different tyres with particular attention to the tyre's section. Electronic Speckle Pattern Interferometry (ESPI) and Laser Scanner (LS) based on principle of triangulation have been employed for investigating and studying the tyre's section and 3D shape respectively. As case studies two different racing tyres, Michelin S9H and Pirelli Diablo respectively, have been considered. The investigation has been focused at the aim to evaluate and measure the section's components in order to add to the 3D model obtained by Laser Scanning accurate information about the different layers along through the tyres sections. It is important to note that the assessment about the different layers along the section is a very difficult task to obtain by visual inspection or classical microscopy and even with the LS. Here we demonstrate that the different layers can be easily highlighted and identified by mean of the ESPI.
... Vehicle's behavior is strongly influenced by tire-road interaction and is linked to a series of complex phenomena which occur within tire contact patches [1]. In the recent years, the growing need to reproduce with a high level of detail the interaction between the vehicle and the external environment in simulation platform has led to the development of physical models through which to deeply investigate thermodynamics and wear influences on vehicle performance [2,3]. ...
... Thanks to the availability of the internal tire layers temperatures, it is possible to obtain much more useful correlations, as the ones reported in Fig. 2(a), from which the optimal thermal range and the expected grip vs temperature bell-shaped curve can be identified. In particular, the temperature adopted in the -axis is a proper ''mix'' of the temperatures of different layers, with a significant weight of the tread core one [2,20], able to reproduce a clear trend, useful to identify the optimal working range. ...
... In Fig. 18(a) it is possible to note that the tire in test 11, characterized by a lower initial value of compound thickness, shows less thermal inertia, warming up faster than the other ones, whereas in the stabilized phase, where the heat generated due to cyclic deformation prevails (SEL) the temperature and pressure reach lower values if compared to the other two tests. With reference to the cited case, it has been demonstrated in [2] that the highlighted behavior (a faster transient, and a lower temperature at stabilized conditions, for a thinner tread) is replicable at any thermal range, because it is due to the sole phenomena related to the thermal generations due to SEL effects, and therefore independent from the ambient temperature. The pie charts in Figs. ...
Article
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The handling behavior of a vehicle is one of its most important properties because of its relation to performance and safety and to its deep link with concepts such as “over-steer” or “under-steer”. Tire-road interaction models play a fundamental role in the vehicle system modelling, since tires are responsible for the generation of forces arising within contact patches, fundamental for both handling and ride/comfort. Among the models used to reproduce such forces, Pacejka’s Magic Formula (MF) is undoubtedly one of the most used ones in real-time automotive simulation environments because of its ability to fit quite easily a large amount of experimental data, but its original formulation did not take into account of the tire thermodynamics and wear conditions, which clearly affect tire and vehicle dynamics and are not negligible, especially for high level applications, such as motorsport competitions. Exploiting a multiphysical tire model, which consists in an evolved version of the standard MF model (MF-evo), and a vehicle model properly validated throughout experimental data acquired in outdoor testing sessions carried out with an industrial partner, the current work presents a study on vehicle behavior variation induced by thermodynamic and wear parameters, defining a series of metrics to analyze and show results. One of the elements of interest on which the focus is placed is the possibility to highlight how under-over-steering behavior of a car changes according to different thermodynamic states of tires; to do this, a commercial software VI CarRealTime has been used to perform a series of objective steady-state maneuvers and long runs, exploiting the logic of a lap time optimizer.
... In [12], the author has successfully extended Kalker's theory to rubber tyres starting from a semi-analytical model normally employed in studies related to the wheel-rail contact mechanics. In [13], the effect of the thermal and frictional effects on ground vehicle performance has been studied by employing a simplified model for tire wear estimation. Finally, other works [14][15][16] have been substantially aimed at estimating the road friction coefficient from forces and slip measurements by assuming the tyre behaviour following some enhanced brush models. ...
... Generally speaking, the rolling speed V r (t) depends on time and an analytical solution to (13) is very hard to provide. Thus, we only limit our analysis to the case in whichV r = 0. We can solve (13) by means of the method of characteristics to get ...
... On the other hand, at high rolling speed, we can neglect the partial derivative taken with respect to the time in Equation (13) and express the generalised tangential forces as a function of the slip quantities and the tyre carcass displacement and speed, according to ...
Article
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This paper deals with unsteady-state brush tyre models. Starting from tyre-road contact theory, we provide a full analytical solution to the partial differential equations (PDEs) describing the bristle deformation in the adhesion region of the contact patch. We show that the latter can be divided in two different regions, corresponding to two different domains for the solution of the governing PDEs of the system. In the case of constant sliding speed inputs, the steady-state solution coincides with the one provided by the classic steady-state brush theory. For a rectangular contact patch and parabolic pressure distribution, the time trend of the shear stresses is investigated. For the pure interactions (longitudinal, lateral and camber), some important conclusions are drawn about the relaxation length. Finally, an approach to derive simplified formulae for the tangential forces arising in the contact patch is introduced; the tyre formulae obtained by using the proposed approach are not based on the common slip definition, and can be employed when the rolling speed approaches zero. The outlined procedure is applied to the cases of linear tyre forces and parabolic pressure distribution. ARTICLE HISTORY
... Later, the original model, designed for vehicle handling applications, has been reformulated to include the internal pressure effect [21,22] and to extend the applicability in dynamic scenarios with higher frequency [23]. The MF model has been further enhanced in [24], where the authors have proposed an advanced multiphysical MF-based (MF-evo) realtime tyre model with the aim to extend the Pacejka's Magic Formula tyre model in the whole range of the tyre operating conditions, taking into account its internal temperature distribution [25,26], inflation pressure [22], tread wear [27,28], compound viscoelastic characteristics and road roughness [29,30]. The potential risks, related with the employment of empirical models, are linked with their parametrisation and the quality of data, since the adoption of numeric data-based techniques makes it possible to completely misinterpret the tyre behaviour even in case of a good fitting towards experimental results. ...
... The vehicle has been equipped with a significant amount of instrumentation to evaluate the kinematics and the dynamics at each corner, as well as the thermodynamics of each tyre: The kinematic and dynamic quantities concerning a single corner can be reliably acquired and estimated [26,28,40]. Compound and carcass temperatures, inner air pressure and wear level are among the additional inputs of the MF-evo model, and they can be provided by means of both additional sensors and physical predictive models. ...
... Indeed, once properly calibrated and validated towards experimental data, the MF-evo-based real-time co-simulation tyre system can be employed within offline vehicle setup optimisation routines, advanced data analysis algorithms, hard real-time simulation environments for driver-in-the-loop, software-inthe-loop and hardware-in-the-loop, and, finally, embedded onboard model-based control logics. In [25,26], the authors describe the real-time tyre thermal model, able to calculate the temperatures governing the grip and stiffness properties, whereas the procedure to take into account of tread wear and compound degradation thanks to physical grip model is presented in [28]. ...
Article
To cite this article: Aleksandr Sakhnevych (2021): Multiphysical MF-based tyre modelling and parametrisation for vehicle setup and control strategies optimisation, Vehicle System Dynamics, ABSTRACT Starting from the earliest phases of design of the vehicle and its control systems, the understanding of tyres is of fundamental importance to govern the overall vehicle dynamics. A properly charac-terised tyre-road interaction model is essential to achieve a reliable vehicle dynamics model on which more design variations can be studied directly in simulation environment optimising both cost and time. The possibility to count on computationally efficient and reliable formulations represents nowadays a great advantage, and the multiphysical Pacejka's Magic Formula (MF-evo) tyre model presented is one of the best trade-off solutions to meet the strict real-time requirements and to reproduce multiphysical variations of the tyre dynamic behaviour towards temperature, pressure and wear effects. A specific methodology has been developed to characterise and to identify the MF-evo parameters with a high grade of accuracy and reliability directly from experimental data. The proposed technique is based on a pre-processing procedure to remove non-physical outliers and to cluster the data, which allows to optimise the multidimensional parameterisation process. To the purpose of validation of the parametrisation routine, data from a motorsport case, exceptionally difficult to reproduce in simulation due particularly significant variations of the tyre dynamics during a single test, have been employed demonstrating the MF-evo model potential and robustness. ARTICLE HISTORY
... Many phenomena occurring in the tyre/road interaction are largely dependent on the viscoelastic properties of the tyre tread [4][5][6][7][8][9][10], thus influencing the vehicle dynamics. Grip performance, rolling resistance and wear are some examples of events in which the viscoelasticity of the tread plays a fundamental role. ...
... The possibility to obtain the tyre tread viscoelastic response by means of a ND procedure, preserving the tyre integrity, represents an innovative topic that would define new scenarios for stakeholders, such as the monitoring of the tread characteristics during the entire tyres' lifecycle. A portable instrument, which can give a real-time characterization, allows to provide fundamental information to study the direct effects of tyre temperature variation [9] and tread wear [6,21], and it would also allow to select the compounds based on track and road dynamics, with the aim of improving performance and safety. The final target is the identification of the stiffness and damping parameters of the tested polymer, emphasizing that the instrument has been developed for tyre applications but the proposed methodology is general and can be potentially applied to any kind of material. ...
Article
The non-destructive quantification of the mechanical properties of solid materials is a growing research topic for many applications. It could be used for monitoring material performance during the whole lifecycle of a component, for when sampling is limited or impossible or for applications that require sample identification. In this paper a portable instrument for non-destructive viscoelastic characterization of polymers, based on instrumented indentation, is presented, its aim is to allow a real-time assessment of viscoelastic storage and loss moduli (E’ and E”) directly in-situ. The designed architecture of the device is described in details alongside the procedure adopted for testing polymers, the corresponding signal processing procedure for the identification of materials stiffness and damping parameters is also described. For the scope of this paper the aforementioned procedure was tested on two different rubber compounds. Finally, the storage and loss moduli are calculated based on the linear viscoelasticity theory and the results are compared to the ones obtained with the standard Dynamic Mechanical Analysis technique (DMA): both these approaches show the same relative ranking between the compounds and a different trend in temperature due to the reached indentation depth. To overcome this limitation of linear viscoelasticity theory, normally valid for low indentation depths, a generalized formulation is proposed that takes into account the indentation depth on the moduli estimation. The results obtained with the generalized formulation show that this approach allows to evaluate accurately the trends and rankings of viscoelastic moduli, giving reliable results.
... To perform the study, the standardized DLC maneuver, currently employed for the validation of virtual driver and advanced driving assistance systems (ADAS) [16,19,20], is implemented in Matlab/Simulink virtual environment. The vehicle and tyre models have been characterized and validated for a reference GT vehicle, identifying the requisite complex tyre-road coupled phenomena concerning the temperature, wear, and road pavement dependencies [21,22]. Four different roads, i.e., dry, wet, snowy, and icy, two diverse tyre mileages, i.e., new and worn, and three thermal tyre conditions are combined and analyzed in the study to understand which could be the advantages of the employment of the model-based controllers, aware of the tyre instantaneous characteristics, boundary operating and weather conditions, and overall vehicle dynamic potential [23]. ...
... All of them are deeply interconnected and dependent on the specific tyre working conditions, in terms of sliding velocity, temperature, and pressure distributions, arising at the tyre contact patch as a result of different excitation spatial frequency spectra, representative of diverse types of road pavement [35]. Furthermore, tyres may deeply modify their dynamic behavior over time due to ageing effects, influencing the dynamic potential of the overall vehicle [22]. ...
Article
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In recent years the increasing needs of reducing the costs of car development expressed by the automotive market have determined a rapid development of virtual driver prototyping tools that aims at reproducing vehicle behaviors. Nevertheless, these advanced tools are still not designed to exploit the entire vehicle dynamics potential, preferring to assure the minimum requirements in the worst possible operating conditions instead. Furthermore, their calibration is typically performed in a pre-defined strict range of operating conditions, established by specific regulations or OEM routines. For this reason, their performance can considerably decrease in particularly crucial safetycritical situations, where the environmental conditions (rain, snow, ice), the road singularities (oil stains, puddles, holes), and the tyre thermal and ageing phenomena can deeply affect the adherence potential. The objective of the work is to investigate the possibility of the physical model-based control to take into account the variations in terms of the dynamic behavior of the systems and of the boundary conditions. Different scenarios with specific tyre thermal and wear conditions have been tested on diverse road surfaces validating the designed model predictive control algorithm in a hardware-in-the-loop real-time environment and demonstrating the augmented reliability of an advanced virtual driver aware of available information concerning the tyre dynamic limits. The multidisciplinary proposal will provide a paradigm shift in the development of strategies and a solid breakthrough towards enhanced development of the driving automatization systems, unleashing the potential of physical modeling to the next level of vehicle control, able to exploit and to take into account the multi-physical tyre variations.
... In this scenario, the development of predictive realtime physical models, able to provide the real-time onboard estimation of the tyre compound bulk temperature, linked with the tyre dynamic behaviour in terms of grip and stiffness variations, is an important task [17,18]. Furthermore, these models have to be full three-dimensional, taking into account of the composite tyre structure, rim geometry and peculiar sidewalls' characteristics. ...
... In [20] the authors have included the sidewalls and the wheel rim within the tyre model node layout, which represents the final model configuration adopted for this work. Since the thermal model can be employed for both the offline Software-in-the-Loop routines for vehicle performance analyses and the online hard real-time Driver-in-the-Loop applications, the number of nodes n can vary, depending on the particular tyre geometry under the investigation (motorsport, truck/bus or passenger tyre) and on the presence of eventual additional thermal sources (brakes, specifically designed cooling and warming systems or diffusers) to be considered, as addressed by the authors in [17,20]. At the current stage, the average thermal model configuration of requires 0.0004-0.0005 ...
Article
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The characterization and reproduction of tyre behaviour for vehicle modelling is a topic of particular interest both for real-time driver in the loop simulations and for offline performance optimization algorithms. Since the accuracy of the tyre forces and moments can be achieved by the accurate physical modelling of all the phenomena concerning the tyre-road interaction, the link between the tyre thermal state and the tyre frictional performance turns into a crucial factor. An integrated numerical methodology, allowing to couple the full 3D CFD (Computational Fluid Dynamics) flux within the internal chamber of the tyre with an equivalent discrete 3D structure model, is proposed with the aim to completely represent the tyre thermodynamic convective behaviour in the steady-state operating conditions. 3D CFD model enables the evaluation of the internal distribution of the gas temperature and of the thermal powers exchanged at each sub-wall in detail. This allows to increase the reliability of the tyre thermodynamic modelling with a particular reference to the proper managing of the aero-thermal flow of the brake disc impact on the rim temperature and therefore on the internal gas dynamics in terms of temperature and pressure, being able to optimize the tyre overall dynamic performance in both warm-up and stabilized thermal conditions. The steady RANS (Reynolds Averaged Navier-Stokes) simulations have been performed employing the 3D CFD model in a wide range of angular velocities with the aim to calculate the convective thermal flux distributions upon rim and inner liner surfaces. The simulation results have been then exploited to derive the convective heat transfer coefficients per each sub domain to be employed within the real-time tyre physical thermal model, with the peculiar advantage of an enhanced model reliability for thermal characteristics. To validate the proposed methodology, the tyre thermal model outputs, in terms of temperatures of internal and external layers, have been validated towards the acquired ones within the specific routine performed on tyre force and moment test bench, confirming an excellent agreement with the experimental data in the entire range of operating conditions explored.
... Moreover, polymers are widespread in the transport sector since tires are made of polymers composite, assuming a crucial role in terms of safety, reliability and performance. Indeed, tire compounds are responsible for the tire/road interaction phenomena that significantly affect the vehicle dynamics; friction performance [Lorenz et al., 2015;Arricale et al., 2020;Genovese et al., 2019], rolling resistance and wear [Moore, 1980;Braghin et al., 2006;Farroni et al., 2017;Genovese et al., 2020b] are some examples of events in which the viscoelasticity of the tread is a key factor. It is evident how their great diffusion poses the study of the mechanics of polymers as an active and important area of research. ...
Article
A procedure of non-destructive experimental tests aimed at determining the viscoelastic characteristics of the rubbers is described. After recalling the basic principles of viscoelastic theory, the experimental setup and the measurements necessary for the test are described. The procedure consists in hitting the surface of the object under test with a specially instrumented indenter. The techniques for processing the acquired signals to identify the characteristic starting and ending points of indentation in the time-histories are then illustrated. By analyzing separately the phases of the free drop of the indenter and the phase of contact between indenter and rubber, the main mechanical characteristics, such as stiffness and damping of both the instrumented indenter and the bulk of rubber under test are estimated. The methods of calculating the viscoelastic parameters of the rubber starting from the knowledge of the above mechanical parameters are then illustrated. The determined results, in terms of storage and loss moduli, are compared with those provided by classic DMA type tests. The qualitative agreement is excellent. The quantitative agreement falls in the scattering range typical for this kind of measurement. The paper is completed with a discussion of the main causes of measurement uncertainty.
... Hence, the Vehicle Dynamics research group of the Department of Industrial Engineering of the University of Naples Federico II has designed and developed an innovative and portable device, defined as Viscoelasticity Evaluation System Evolved (VESevo), which allows users to characterise the tire tread viscoelasticity and its variations due to cooling or heating, due to wear phenomena [10] [11], aging or different compounding [12] depending on vehicle applications. Thus, engineers, especially Motorsport ones, will be capable of analysing more useful information about confidential tires and improving the vehicle performances and safety. ...
Chapter
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The evaluation of the viscoelastic properties is a key topic for the analysis of the dynamic mechanical behaviour of polymers. In vehicle dynamics field, the knowledge of the viscoelasticity of tread compound is fundamental for tire-road contact mechanics modelling and friction coefficient prediction for the improvement of vehicle performance and safety, i.e. motorsport field. These properties are usually characterised by means of Dynamic Mechanical Analysis, which implies testing a compound sample obtained by destroying the tire of interest or a slab manufactured in different conditions respect to the final product provided by tiremakers. In this scenario, the non-destructive procedures are an advantageous solution for the analysis of the tread viscoelasticity, without affecting the tire integrity, allowing a great number of tests in the shortest possible time. For this reason, the authors propose an innovative instrument, called VESevo, for viscoelasticity evaluation by means of non-destructive and user-friendly technique. The purpose of the following work is the preliminary analysis of the dynamic response of the tires tested employing the VESevo in order to determine viscoelastic behaviour indexes for mechanical properties evaluation.
... The characterization of this sub-system is quite difficult and delicate because tyre forces show a complex and strongly nonlinear dependence on slip ratio, slip angle, camber angle, vertical load, tyre structure, inflation pressure, inner and surface temperature, wear [9][10][11][12]. Road conditions and properties are very relevant too [13]. ...
Article
Tyres play a key role in ground vehicles' dynamics because they are responsible for traction, braking and cornering. A proper tyre-road interaction model is essential for a useful and reliable vehicle dynamics model. In the last two decades Pacejka's Magic Formula (MF) has become a standard in simulation field. This paper presents a Tool, called TRIP-ID (Tyre Road Interaction Parameters IDentification), developed to characterize and to identify with a high grade of accuracy and reliability MF micro-parameters from experimental data deriving from telemetry or from test rig. The tool guides interactively the user through the identification process on the basis of strong diagnostic considerations about the experimental data made evident by the tool itself. A motorsport application of the tool is shown as a case study.
... The use of a dynamic model can lead to a good VSA estimation, yet results are accurate only if the tyre model truly reflects the actual conditions. Unmodeled effects such as road conditions and tyre wear can dramatically worsen the reliability of the estimation [125]. Attempts to address this issue include algorithms providing an online update of tyre parameters (e.g., Pacejka coefficients, cornering stiffness, and rational tyre model coefficients, respectively [16,120,122]). ...
Article
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Typical active safety systems that control the dynamics of passenger cars rely on the real-time monitoring of the vehicle sideslip angle (VSA), together with other signals such as the wheel angular velocities, steering angle, lateral acceleration, and the rate of rotation about the vertical axis, which is known as the yaw rate. The VSA (also known as the attitude or “drifting” angle) is defined as the angle between the vehicle’s longitudinal axis and the direction of travel, taking the centre of gravity as a reference. It is basically a measure of the misalignment between vehicle orientation and trajectory; therefore, it is a vital piece of information enabling directional stability assessment, such as in transience following emergency manoeuvres, for instance. As explained in the introduction, the VSA is not measured directly for impracticality, and it is estimated on the basis of available measurements such as wheel velocities, linear and angular accelerations, etc. This work is intended to provide a comprehensive literature review on the VSA estimation problem. Two main estimation methods have been categorised, i.e., observer-based and neural network-based, focussing on the most effective and innovative approaches. As the first method normally relies on a vehicle model, a review of the vehicle models has been included. The advantages and limitations of each technique have been highlighted and discussed.
... Koketsu et al. [10] proposed a distributional multi-mass tire model for investigation of transient wear behavior. Farroni et al. [11] developed a physical modeling method that reflects complex tire-road interaction phenomena and considers the thermal effect on tread wear. A nonlinear lateral dynamic model of a multi-axle steering vehicle was presented by Chen et al. [12] to calculate tire wear considering the effects of the hydropneumatics suspension, steering system and toe angle. ...
Article
Polygonal wear seriously decreases the lifespan of a tire of a passenger car and adversely affects vehicle dynamic safety. The present paper builds a model that reflects the dynamic contact characteristics of the tire and reveals the mechanism and conditions of polygonal wear of a tire. The model describes the dynamic contact behavior of the tread block and considers the characteristics of dynamic friction between the road and tread of a rolling tire. Conducting numerical bifurcation analysis, the paper reveals the conditions for self-excited vibration of the tread, i.e., the improper combination of the vertical load, wheel slip angle, tire pressure and vehicle speed considerably strengthen the lateral self-excited vibration of the tread, which is the direct vibrational source of abnormal circumferential polygonal wear. The polygonal wear of a tire occurs when a vehicle travels for a certain long distance at a so-called polygonal wear speed. The polygonal wear speed should induce lateral self-excited vibration on the contact tread of the tire and the frequency of the lateral self-excited vibration should be divisible by the rolling frequency of tire that is determined by the polygonal wear speed. Visible polygonal wear requires that the vehicle travels at a certain polygonal wear speed for a minimal distance to produce a stably developing polygonal wear pattern even for subsequent driving at variable speed.
... It is also worth to mention that the RCH-tire could definitely provide a relevant physical infrastructure to completely understand the tire behavior in dynamic transient conditions, in particular, taking into account the influence of other connected physical phenomena as thermal effect on tire stiffness and compound characteristics or wear impact. 36 ...
Article
A tire is an extremely integrated and multi-physical system. From only a mechanical point of view, tires are represented by highly composite multi-layered structures, consisting of a multitude of different materials, synthesized in peculiar rubber matrices, to optimize both the performance and the life cycle. During the tire motion, due to the multi-material thermodynamic interaction within the viscoelastic tire rubber matrix, the dynamic characteristics of a tire may alter considerably. In the following paper, the multibody research comfort and handling tire model is presented. The main purpose of the research comfort and handling tire is to constitute a completely physical carcass infrastructure to correctly transmit the generalized forces and torques from the wheel spindle to the contact patch. The physical model structure is represented by a three-dimensional array of interconnected nodes by means of tension and rotational stiffness and damper elements, attached to the rim modeled as a rigid body. Research comfort and handling tire model purpose is to constitute a structural physical infrastructure for the co-implementation of additional physical modules taking into account the modification of the tire structural properties with temperature, tread viscoelastic compound characteristics, and wear degradation. At the stage, the research comfort and handling tire discrete model has been validated through both static and dynamic shaker test procedures. Static test procedure adopts contact sensitive films for the contact patch estimation at different load and internal pressure conditions, meanwhile the specifically developed sel test regards the tire dynamic characterization purpose at the current stage. The validation of the tire normal interaction in both static and dynamic conditions provided constitutes a necessary development step to the integration of the tangential brush interaction model for studying the handling dynamics and to the analysis of the model response on the uneven surfaces.
... In [9], the author, starting from a semianalytical model normally employed in wheel-rail contact representations, successfully extended Kalker's theory to rubber tyres. In [10] the effect of the thermal and frictional effects on ground vehicle performance has been studied by employing a simplified model for tire wear estimation. Finally, other studies [11][12][13] were substantially aimed at estimating the road friction coefficient from forces and slip measurements by assuming the tyre behaviour following some improved brush models. ...
Article
Full-text available
In this investigation, a double brush model, which aims at predicting both the longitudinal and the lateral tyre characteristics during transient phases, is developed. The solution of the tyre-road contact equation is provided by using the method of characteristics and a time delay of the bristles deformation with respect to the time is also introduced by modelling both the tyre tread and the carcass by means of viscoelastic and elastic elements, respectively. The temporal trend of some quantities of interest such as the adherence length and the critical slip value is then obtained as explicit function of the time or, equivalently, of the travelled distance. A preliminary analysis is carried out with reference to the transition from a pure rolling condition to accelerating or braking ones. The tyre response to a constant lateral slip input is also comprehensively discussed. Finally, in the case of consecutive manoeuvre , the model shows that all the generalised forces exerted by the road on the tyre vary continuously by introducing a finite increase in the slip parameter. Several examples are presented in order to demonstrate the applicability of the proposed model to severe braking or handling dynamic scenarios.
... In [9], the author, starting from a semianalytical model normally employed in wheel-rail contact representations, successfully extended Kalker's theory to rubber tyres. In [10] the effect of the thermal and frictional effects on ground vehicle performance has been studied by employing a simplified model for tire wear estimation. Finally, other studies [11][12][13] were substantially aimed at estimating the road friction coefficient from forces and slip measurements by assuming the tyre behaviour following some improved brush models. ...
Article
Full-text available
In this investigation, a double brush model, which aims at predicting both the longitudinal and the lateral tyre characteristics during transient phases, is developed. The solution of the tyre-road contact equation is provided by using the method of characteristics and a time delay of the bristles deformation with respect to the time is also introduced by modelling both the tyre tread and the carcass by means of viscoelastic and elastic elements, respectively. The temporal trend of some quantities of interest such as the adherence length and the critical slip value is then obtained as explicit function of the time or, equivalently, of the travelled distance. A preliminary analysis is carried out with reference to the transition from a pure rolling condition to accelerating or braking ones. The tyre response to a constant lateral slip input is also comprehensively discussed. Finally, in the case of consecutive manoeuvre, the model shows that all the generalised forces exerted by the road on the tyre vary continuously by introducing a finite increase in the slip parameter. Several examples are presented in order to demonstrate the applicability of the proposed model to severe braking or handling dynamic scenarios.
... Moreover, several studies report that tire tread exhibits optimal grip depending on the temperature reached in its core layer [12,13], not reachable by measurement instruments for thermal monitoring, and that global tire stiffness is highly influenced by the thermodynamic conditions of the carcass [14], also not directly measurable. For this reason, and for the increasing request of tools able to reproduce with high reliability the contact with road in a vehicle dynamics environment, the development of a tire thermal model has become a necessity for the players requiring the predictivity in challenging simulation scenarios. ...
Article
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While in the automotive field the relationship between road adherence and tire temperature is mainly investigated with the aim to enhance the vehicle performance in motorsport, the motorcycle sector is highly sensitive to such theme also from less extreme applications. The small extension of the footprint, along with the need to guarantee driver stability and safety in the widest possible range of riding conditions, requires that tires work as most as possible at a temperature able to let the viscoelastic compounds-constituting the tread and the composite materials of the whole carcass structure-provide the highest interaction force with road. Moreover, both for tire manufacturing companies and for single track vehicles designers and racing teams, a deep knowledge of the thermodynamic phenomena involved at the ground level is a key factor for the development of optimal solutions and setup. This paper proposes a physical model based on the application of the Fourier thermodynamic equations to a three-dimensional domain, accounting for all the sources of heating like friction power at the road interface and the cyclic generation of heat because of rolling and to asphalt indentation, and for the cooling effects because of the air forced convection, to road conduction and to turbulences in the inflation chamber. The complex heat exchanges in the system are fully described and modeled, with particular reference to the management of contact patch position, correlated to camber angle and requiring the adoption of an innovative multi-ribbed and multi-layered tire structure. The completely physical approach induces the need of a proper parameterization of the model, whose main stages are described, both from the experimental and identification points of view, with particular reference to non-destructive procedures for thermal parameters definition. One of the most peculiar and challenging features of the model is linked with its topological and analytical structure, allowing to run in real-time, usefully for the application in co-simulation vehicle dynamics platforms, for performance prediction and setup optimization applications.
... Hence, a real-time compounds characterization, which could be performed by means of a portable device [2], allows engineers to analyse directly the effects due to tire heating and cooling [3,14,15], tread wear [16,17], aging or winter and summer season compound choice on road so that the safety and handling performances can be improved by taking into account tire viscoelastic behaviour in friction coefficient evaluation for vehicle dynamics onboard algorithms. ...
Chapter
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The knowledge of tires tread viscoelastic behaviour plays a fundamental role in automotive to optimize vehicle performance and safety. These properties are usually characterized by means of Dynamic Mechanical Analysis [1] which implies the testing of compound sample that can be obtained by destroying the tire of interest or manufactured in different condition respect to the final product provided by tiremakers. Nowadays, the non-destructive analysis procedures are an attractive solution. These techniques are essentially advantageous for being employed in testing the whole tire, allowing the analysis of a great number of them without affects costs. The purpose of this work is the experimental analysis of the tire tread response, in different working condition, evaluated by a commercial dial indicator [2] considering the measured displacement values of the device. Experimental tests have been carried out on different tread compounds and, being the tire performance strictly affected by the working temperature, additional tests have been performed by heating and cooling each sample in a range of interest [3, 4]. Moreover, the effects of aging on a tire has been studied. The comparison of the testing activity results shows the reliability of the dynamic dial indicator to capture the tires tread different behaviour within the operating condition of interest. These encouraging results lead to next step of the research activity which will focus on the evaluation of properties characterizing the hysteretic behaviour of tires.
... The output of the tool provides a sort of "virtual telemetry" containing forces and slips estimation, so that tires interaction characteristics can be obtained. The big potential of the tool described lies in the possibility to integrate it with other software capable to predict tire thermal and wear behavior [13]. ...
Chapter
The tire behavior optimization is crucial for the definition of the best setup of the whole vehicle; in fact, its interface with the ground is constituted by the sum of small surfaces in which tire-road interaction forces are exchanged. The fundamental role that in the last years tires have played in automotive industry and the growing need to reproduce with a high level of detail the phenomena concerning with vehicle dynamics have given a strong impulse to the research in the field of vehicle systems and modelling.
... Moreover, other researchers have been developed a tire models that taking into account also the thermodynamic effect [17] and tire wear [18] which are fundamental phenomena in tire-road interaction. ...
Chapter
The implementation of a tire model in a simulation environment is fundamental to characterize the vehicles and to predict the dynamic behaviour during the design phase, e.g. to test automotive control systems like ADAS [1] or different parameters or working conditions like tire compound, pressure, and speed. Moreover, the output of a tire model can be employed also to predict its temperature distribution [2]. This paper deals with the comparison between different Pacejka formulations, differing for the sensitivity to physical factors, like inflation pressure and slight analytical variations. Since the discussed tire models are different version of the same formulation, the microparameters concerning specific physical effects have been zeroed, in order to make the comparison more reliable. In particular, a Pacejka’s MF 5.2 has been compared towards the MF 6.1 tire model employing a tire vehicle in specific dynamic manoeuvres. Some longitudinal (braking and acceleration) and lateral manoeuvres (spiral, steering pad, fishhook, line change, constant speed curve) have been adopted to compare the results of the implemented tire model influence on the overall vehicle dynamics. Finally, to evaluate the effect of different tire configurations, a sensitivity test was carried out.
... Recently the original Thermo Racing Tyre model has been revised to include also the thermal phenomena linked not only with the tire structure but also with the entire wheel (including rim and brake disks [11]), with local contact pressure distributions (fundamental, as an example, in motorcycle dynamics [12]) and with tread wear [13]. The model advanced structure enables to include particular aero-design solutions and specific cooling/heating configurations within its interaction with both the external environment and the internal air. ...
Chapter
The characterization and reproduction of tire behaviour for vehicle modelling is a topic of particular interest both for real-time driving simulations and for offline performance optimization algorithms. In such contexts, the Pacejka’s Magic Formula (MF) tire model [1] represents a standard that gained in the last 25 years a role of high relevance due to its low computational request and attitude to allow an efficient parameterization for a wide range of tires working conditions. Nevertheless, the original MF formulation was conceived with the aim to provide tire/road interaction forces and moments as a function of vertical load, longitudinal and lateral slip, and inclination (or camber) angle; such variables are fundamental but not totally satisfying in the description of the complex multi-physical phenomena occurring at tire/road interface [2]. In particular, the relationship between interaction forces and the cited input variables is highly influenced by further effects, linked to tire temperature, tread wear, compound viscoelastic characteristics and road roughness. Among these, the influence that the thermal conditions of the different layers constituting the global thickness of tires have on the friction and on their stiffness characteristics, is highly significant and definitely not negligible in case a full reliability of the vehicle dynamics simulations is required, especially in motorsport applications [4]. The paper illustrates the basic concepts linked to the development of a novel version of a MF-based formulation, able to take into account uncommon factors affecting tire/road interaction. Once described the structure and the parameters identification process [4], some results obtained with the MF-evo employed in a simulative loop with a thermal model are reported.
... Even though researchers have investigated tyre wear modeling in details with analytic and experimental studies [6][7][8][9], the optimisation towards tyre wear minimisation has not been widely discussed. In the literature so far, the majority of these works consider finite element models for evaluating the tyre wear levels. ...
Chapter
Due to increased environmental issues and for economic reasons, the automotive industry intensifies the research towards energy efficient driving, by also taking into consideration the comfort and road holding aspects. In this direction, efforts of modeling and minimising tyre wear, one of the main non-exhaust traffic related sources, are investigated in the literature. This work investigates the suspension and tyre optimisation for tyre wear minimisation while the vehicle is driving on different road surfaces. Nevertheless, given that the road holding and ride comfort are important design criteria in the design process, they are incorporated in the optimisation as objectives by configuring a multi-objective problem with all of the above objectives. Conclusions are extracted about the behavior of tyre wear regarding ride comfort and road holding, and also about where the optimum design variables have converged trying to compromise the above performance aspects under different road roughness profiles.
... These restrictions mainly concern limiting combustion emissions coming from internal combustion engines, both gasoline and diesel [2,3]. Another important source of pollution from motor vehicles is the chassis system, specifically the wear products from tires [4,5]. For several years, environmentalists have been pointing out that in addition to the emission of harmful substances from internal combustion engines, there is also the emission in the braking systems. ...
Article
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This paper presents the results of tests on the railway disc brake with regard to the weight wear of friction pads. The tests were carried out at a certified brake test bench where the friction-mechanical characteristics of the railway brake were determined. The test stand was additionally equipped with a thermal imaging camera to observe the contact between the brake pads and the brake disc. The scientific goal of the test is to evaluate the relationship between the weight wear of friction pads and the quantities characterizing the braking process. The quantities characterizing the braking process included pad-to-disc contact area, friction pad thickness, pad-to-disc pressure, and braking speed. A regression model to estimate the friction pad wear on the basis of a single braking with the given input quantities was determined. The greatest influence on the increase in weight wear of friction pads has the braking velocity, which was confirmed by the value of the correlation coefficient of the regression model at value 0.81. The pressure of the friction pad to the disc and the friction pad thickness do not have a significant effect on the weight wear described by the regression model, and the obtained correlation coefficient for these parameters was lower than the value of 0.2.
... In this way, the dual nature of the tyre is mimicked by a series system that behaves as a spring at low rolling speed and as a damper at high speeds. This pragmatic approach leads to a very straightforward model, which generally shows a good agreement with experimental evidence and -combined with MF, which is currently able to take into account physical phenomena connected to tyre inflation pressure, temperature and wear [45][46][47] -can also handle the presence of large camber angles and steering speeds. However, two major disadvantages of the single contact point models consist in that they neglect nonstationary phenomena connected to the tread rubber flow inside the contact patch, i.e. the transient of the bristles, and do not come with important dynamical properties, like, e.g., asymptotic and input-to-state stability, that are appetible for control applications. ...
Article
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This paper presents a novel tyre model which combines the LuGre formulation with the exact brush theory recently developed by the authors, and which accounts for large camber angles and turning speeds. Closed-form solutions for the frictional state at the tyre-road interface are provided for the case of constant slip inputs, considering rectangular and elliptical contact patches. The steady-state tyre characteristics resulting from the proposed approach are compared to those obtained by employing the standard formulation of the LuGre-brush tyre models and the exact brush theory for large camber angles. Then, to cope with the general situation of time-varying slips and spins, two approximated lumped models are developed that describe the aggregate dynamics of the tyre forces and moment. In particular, it is found that the transient evolution of the tangential forces may be approximated by a system of two coupled ordinary differential equations (ODEs), whilst the dynamics of the self-aligning moment may described by combining two systems of two coupled ODEs. Given its stability properties and ease of implementation, the lumped one may be effectively employed for vehicle state estimation and control purposes.
... The mechanics of contact taking place at the interface of rough soft solids is attracting a wide scientific interest due to its implications in a number of applications, ranging from biology and biomedical devices to machine elements (Persson, 2006). As an example, passenger cars maneuverability, comfort and energy efficiency in the era of autonomous/assisted driving and novel tires design are strictly related to the ability to adopt advanced physically-based tire friction and wear models during car engineering (Farroni et al., 2017;Fortunato et al., 2017). Moreover, an accurate prediction of contact mechanics, combined with an increasingly realistic physical description of the rubber behavior, plays a key role in term of tire durability, driving safety and sustainability (Lu, 2010), as well as in the prediction of the driving range in electric vehicles (Farfan-Cabrera, 2019). ...
Article
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Contact mechanics models based on linearity assumptions, often using the viscoelastic half space theory and numerically implemented with the boundary element method, are known to provide accurate results for small mean square slope of the surface roughness. For large mean square slope, models accounting for finite deformations, often implemented with the non-linear finite element method, are more accurate but lead to a prohibitive computational cost. We propose a new hybrid multiscale approach able to account for the finite deformations arising due to large mean square slope, while keeping a computational cost similar to that associated to linear approaches. The basic strategy is a decomposition of the surface roughness power spectrum into a discrete number of waves, whose spectral range is partitioned into a high mean square slope range and a low mean square slope range. The contact mechanics in the former is accurately solved with the kinematically non-linear model and the results averaged out at the larger wavelength scale in terms of an effective interface interaction law. This law is then applied in the linear simulation involving the scales within the low mean square slope range. The proposed approach is a more accurate alternative to fully linear and a computationally faster alternative to fully non-linear contact mechanics approaches.
... Inflation pressure usually has a moderate effect on the cornering properties of a tyre, but in general, cornering stiffness increases with an increase of the inflation pressure. However, the relationship between the cornering force and the normal load is non-linear; it means that the transfer of load from the inside to the outside tyre during a turning manoeuvre will (Jazar, 2019) reduce the total cornering force that a pair of tyres can perform, making so possible to act on the under/over steering behaviour of the whole vehicle modifying the value of the roll stiffness, able to manage the load transfers (Connelly & Huston, 1994;Farroni et al., 2017). ...
Chapter
This chapter deals with tyre mechanics and it has a particular focus on thermal effects on its dynamical behaviour. In the first part the typical tyre structure is introduced together with the tyre mechanical/dynamical behaviour according to a classical approach, so recalling the main kinematic and dynamic quantities involved in tyre pure and combined interactions. The core of this chapter is the description of a physical-analytical tyre thermal model able to determine the thermal status in each part of the tyre useful for vehicle dynamics modelling and driving simulations in order to take into account thermal effects on tyre interactions and consequently on vehicle dynamical behaviour. Successively also the tyre wear modelling is faced, after a brief introduction to the different models available in literature some considerations are reported concerning the thermal effects on wear.
... Several authors have already tackled the problem describing different techniques and modelling approaches [63][64][65]. The authors plan also to include the impact of the road bank angle and slope, the tire combined interaction characteristics, as well as, the variations of the vehicle dynamic behaviour due to the tire intrinsic multi-physics (i.e., wear and temperature effects) [66,67]. Informed Consent Statement: Informed consent was obtained from all subjects involved in the study. ...
Article
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In recent years, autonomous vehicles and advanced driver assistance systems have drawn a great deal of attention from both research and industry, because of their demonstrated benefit in reducing the rate of accidents or, at least, their severity. The main flaw of this system is related to the poor performances in adverse environmental conditions, due to the reduction of friction, which is mainly related to the state of the road. In this paper, a new model-based technique is proposed for real-time road friction estimation in different environmental conditions. The proposed technique is based on both bicycle model to evaluate the state of the vehicle and a tire Magic Formula model based on a slip-slope approach to evaluate the potential friction. The results, in terms of the maximum achievable grip value, have been involved in autonomous driving vehicle-following maneuvers, as well as the operating condition of the vehicle at which such grip value can be reached. The effectiveness of the proposed approach is disclosed via an extensive numerical analysis covering a wide range of environmental, traffic, and vehicle kinematic conditions. Results confirm the ability of the approach to properly automatically adapting the inter-vehicle space gap and to avoiding collisions also in adverse road conditions (e.g., ice, heavy rain).
... On the other hand the carcass temperature reacts more slowly making it more difficult to manage-this is particularly true in cases where carcass-temperature reductions are required. In order to recognise the long-run impact of tyre degradation, the grip is modelled as a function of both tyre wear [67] and the tread temperature [68]. Simulations on the Circuit de Catalunya show that with new tyres the first lap tends to be slow reflecting the combined influences of a low operating temperature and tyres that have not yet been 'rubbed in'. ...
Article
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The paper begins with a survey of advances in state-of-the-art minimum-time simulation for road vehicles. The techniques covered include both quasi-steady-state and transient vehicle models, which are combined with trajectories that are either pre-assigned or free to be optimised. The fundamentals of nonlinear optimal control are summarised. These fundamentals are the basis of most of the vehicular optimal control methodologies and solution procedures reported in the literature. The key features of three-dimensional road modelling, vehicle positioning and vehicle modelling are also summarised with a focus on recent developments. Both cars and motorcycles are considered.
... The trend results from the literature are presented in Figure 115. These trends from 153 experiment results are coherent with the trends from literature [36]. It validates our assumption and method to calculate the rolling resistance. ...
Thesis
CO2 emissions from road transport account for a significant share of global greenhouse gas emissions and therefore contribute to on-going climate change. In fact, fuel consumption is influenced by one of major source of energy loss among others, represented by the rolling resistance linked to tyre/road contact. Studies show that energy losses due to rolling resistance represent approximately 20% for a light vehicle. It is mainly due to the repetitive deformation of the tyre. It is highly dependent on tyre parameters such as inflation pressure, load, speed and temperature, but also on vehicle dynamics and characteristics of the infrastructure. It is not directly measurable with a physical sensor. The main objective of this thesis is to develop a system for the estimation of the tyre rolling resistance of a vehicle in real driving conditions. In order to achieve this objective, an indirect approach of estimation by using software sensor such observer. A multi-physical model of the tyre/road contact has been developed to integrate the various influencing parameters of rolling resistance and coupled with the full vehicle model. Indeed, the complexity of the model and the driving situations make this estimation difficult. Therefore, a variable gain (adaptive) unknown input observer is developed to ensure an accurate and robust estimation. This observation approach has been chosen for its robustness against modeling errors, parametric uncertainties and for its rapid convergence in finite time. An offline experimental validation is done on the test tracks of University Gustave Eiffel, Nantes in order to validate the estimation approach with an instrumented vehicle.
... Off-road ground vehicles face significant wear over their life cycle. Structural parameters like mass, suspension characteristics and tire performance will change over time [Farroni et al. (2017)]. Many of these changes are difficult to measure or quantify in real-time but can impact the overall dynamics of the vehicle. ...
Preprint
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Stabilizing vertical dynamics for on-road and off-road vehicles is an important research area that has been looked at mostly from the point of view of ride comfort. The advent of autonomous vehicles now shifts the focus more towards developing stabilizing techniques from the point of view of onboard proprioceptive and exteroceptive sensors whose real-time measurements influence the performance of an autonomous vehicle. The current solutions to this problem of managing the vertical oscillations usually limit themselves to the realm of active suspension systems without much consideration to modulating the vehicle velocity, which plays an important role by the virtue of the fact that vertical and longitudinal dynamics of a ground vehicle are coupled. The task of stabilizing vertical oscillations for military ground vehicles becomes even more challenging due lack of structured environments, like city roads or highways, in off-road scenarios. Moreover, changes in structural parameters of the vehicle, such as mass (due to changes in vehicle loading), suspension stiffness and damping values can have significant effect on the controller's performance. This demands the need for developing deep learning based control policies, that can take into account an extremely large number of input features and approximate a near optimal control action. In this work, these problems are addressed by training a deep reinforcement learning agent to minimize the vertical acceleration of a scaled vehicle travelling over bumps by controlling its velocity.
... This wear model was later further validated and investigated in a sensitivity analysis [18]. Only recently, Farroni et al. [19] developed a physical model of tyre wear to analyse the impact of thermal and frictional effects on vehicle performance. Similarly, Emami et al. [20] designed and developed a new portable test setup to study friction and wear, while Lepine et al. [21] presented a novel empirical tyre wear model for heavy vehicles that can be used to predict the wear for multi-axle vehicles based on route data and a vehicle model. ...
Article
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Effective emission control technologies and novel propulsion systems have been developed for road vehicles, decreasing exhaust particle emissions. However, work has to be done on non-exhaust traffic related sources such as tyre–road interaction and tyre wear. Given that both are inevitable in road vehicles, efforts for assessing and minimising tyre wear should be considered. The amount of tyre wear is because of internal (tyre structure, manufacturing, etc.) and external (suspension configuration, speed, road surface, etc.) factors. In this work, the emphasis is on the optimisation of such parameters for minimising tyre wear, but also enhancing occupant’s comfort and improving vehicle handling. In addition to the search for the optimum parameters, the optimisation is also used as a tool to identify and highlight potential trade-offs between the objectives and the various design parameters. Hence, initially, the tyre design (based on some chosen tyre parameters) is optimised with regards to the above-mentioned objectives, for a vehicle while cornering over both Class A and B road roughness profiles. Afterwards, an optimal solution is sought between the Pareto alternatives provided by the two road cases, in order for the tyre wear levels to be less affected under different road profiles. Therefore, it is required that the tyre parameters are as close possible and that they provide similar tyre wear in both road cases. Then, the identified tyre design is adopted and the optimum suspension design is sought for the two road cases for both passive and semi-active suspension types. From the results, significant conclusions regarding how tyre wear behaves with regards to passenger comfort and vehicle handling are extracted, while the results illustrate where the optimum suspension and tyre parameters have converged trying to compromise among the above objectives under different road types and how suspension types, passive and semi-active, could compromise among all of them more optimally.
... Tire strategy is not a trivial matter since it presents several complexities and challenges that must be taken into account: (i) There is a trade-off between tires of softer and harder compounds: the former allow faster laps, but softer tires tend to wear down more quickly, and will therefore cause the need for more pit stops. (ii) Racing performance does not depend only on the tire compound, but also on the wear of the tires being used (see Farroni et al. (2017)). Indeed, the term falling off the cliff is often used to refer to the point at which tires are so deteriorated that the lap time increases significantly compared to the lap time with low or medium tire degradation (see Terms (2019) for more on terminology). ...
Article
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Pit stops are a key element of racing strategy in several motor sports. Typically, these stops involve decisions such as in which laps to stop, and which type of tire, of three possible compounds, to set at each of these stops. There are several factors that increase the complexity of the task: the impact of lap times depending on the tire compound, the wear of the tires, unexpected events on the track such as safety cars and the weather, among others. This work presents a Dynamic Programming formulation that addresses the pit-stop strategy problem in order to optimize the laps in which to stop, and the tire changes that minimize the total race time. We show the relative performance of the optimal strategies for starting with tires of different compounds with different yellow-flag scenarios. Then, we extend the Dynamic Program (DP) to a Stochastic Dynamic Programming (SDP) formulation that incorporates random events such as yellow flags or rainy weather. We are able to visualize and compare these optimal pit-stop strategies obtained with these models in different scenarios. We show that the SDP solution, compared to the DP solution, tends to delay pit stops in order to benefit from a possible yellow flag. Finally, we show that the SDP outperforms the DP, especially in races in which yellow flags are likely to be waved more frequently.
Chapter
In steady-state conditions, explicit expressions for tyre characteristics may be derived using the theoretical framework provided by the brush theory. This chapter is, thus, dedicated to addressing the stationary problem from both the local and global perspectives. The fundamental concepts of critical slip and spin are introduced with respect to an isotropic tyre, and the deformation of the bristles inside the contact patch is investigated for different operating conditions of the tyre. Analytical functions describing the tyre forces and moment acting inside the contact patch are obtained for the particular case of a rectangular contact patch. The analysis is qualitative in nature.
Article
Vehicle dynamics is largely influenced by the phenomena occurring in the tire-road interface, and a great portion of these phenomena is mainly conditioned by the viscoelastic properties of the tire tread compound. It is not surprising that the possibility of obtaining the viscoelastic response of a compound by means of a nondestructive procedure is a growing research topic that affects application fields ranging from monitoring of the material performance during its entire life cycle to the quantitative analysis of product quality and repeatability of production processes. In this article, a novel nondestructive procedure for the viscoelastic characterization of tire tread compound is proposed. A portable instrument, based on instrumented indentation, was designed and prototyped with the aim to allow a real-time assessment of moduli directly on site. The testing procedure adopted to perform the test on three different compounds was described. A signal-processing procedure was developed for the identification of compound stiffness and damping parameters from which viscoelastic moduli were estimated. The results were also compared with the DMA characterization showing the same relative ranking between the compounds with a different trend in temperature due to the amount of the tests' indentation depth.
Chapter
One of the most diffused tire/road interaction models, widely employed in simulation applications, is the Pacejka’s Magic Formula (MF) [1, 2]. It is a semi-empirical model able to fit full scale test data, characterized by a large number of coefficients, often called micro-parameters, grouped basing on physical considerations in order to create specific functions, called macro-parameters. MF model coefficients provided by tire manufacturers are generally not fully representative of the behaviour of tires in contact with road. This is due to the testing conditions employed to identify model coefficients: tests are usually performed on a specific rolling bench or on a flat-trac (Tire testing system, commercialised by MTS Systems Corporation. It applies forces and motions to a tire running on a continuous flat belt.), that keep the tire in contact with a steel or an abrasive paper covered belt. The impossibility to test the tires under real working conditions causes unavoidable approximation errors, mainly due to differences in thermal exchanges and wear phenomena [3] between tire/belt and real tire/road contact. Therefore it is commonly necessary to modify the MF coefficients in order to improve the bench data correlation and to be able to validate vehicle models with data coming from experimental tests. The aim of the developed tool, called TRIP-ID (Tire/Road Interaction Parameters IDentification), is to provide an innovative procedure to identify the Pacejka coefficients basing on the experimental tests carried out measuring global vehicle data during outdoor track sessions. In the presented application, the procedure collects and processes the data provided by TRICK tool [4], allowing to eliminate the outlier points, to discriminate wear and thermal phenomena, taking into account the combined slip condition and the effects of vertical load and camber angle on the global grip. The innovative approach proposed can be useful to reproduce in real time simulation applications the feedback that high performances tires give to sport vehicle drivers, whose interest and skills are focused on keeping them in the optimal thermal range. The coupling of a properly modified MF model with a thermal and with a friction model can provide a reliable simulation and analysis instrument for drivers, carmakers and tire producers.
Chapter
Vehicle sideslip angle is an important signal related to lateral stability and essential for active safety control systems. Since direct measurement of sideslip angle requires expensive equipment, it should be estimated in a feasible way for implementation. This paper describes a novel cost-effective strategy of sideslip angle estimation using a disturbance observer. In this approach, modeling errors of a linear vehicle model are treated as unknown lumped disturbance, which is estimated by the disturbance observer. Simultaneously, a Luenberger observer estimates the sideslip angle and yaw rate. This method requires only simplified tire model and currently-available sensor measurements such as yaw rate, lateral acceleration, steering angle and longitudinal speed. The estimation performance of the proposed observers has been verified by comparing with an interacting multiple-models (IMM) approach via computer simulation studies using vehicle experimental data. The simulation results show effective and robust estimation performance of the proposed observer under various road surfaces.
Article
A new method to improve the passing performance of monorail train driving on a curve by adjusting the width of monorail beams is proposed. Equations for deformation of steering wheel and stabilizing wheel with respect to the contact relationship between monorail beams and trains are derived by geometric analysis. Then two kinds of adjustment schemes are proposed, equal width adjustment method and unequal width adjustment method. A curved guide way of radius 100 m is taken for the example to verify the validity of the improved method by comparing the tire deformation of monorail train driving on the beams before and after the adjustment of width. The results show that the improved adjustment method for the width of monorail beam is efficient to reduce tire deformation when the monorail train travels on a curve. For the guide way of radius 100 m, width of the beams should be reduced by 7 mm when the equal width adjustment method is adopted. When the unequal width adjustment method is adopted, the width of monorail beams at the position of steering wheel carriageway and stabilizing wheel carriageway should be reduced by 10 mm and 5 mm, respectively.
Article
Optimal control calculations are used to study the effect of tyre wear on race car performance. This is achieved by solving a minimum lap time optimal control problem over multiple laps using a dynamical model of a Formula One car. A previously developed thermodynamic model is enhanced by adding an additional state for the carcass temperature of the tyres. The tyre grip is modelled as a function of the tyres’ wear and temperature. Grip reduces when tyres get worn, or the tyres are not operated within their optimal temperature window. Overheating the tyres can accelerate wear, which in turn, degrades performance. The optimal control problem solver needs to ‘manage’ the state of tyres throughout a race (not just a single lap) to ensure that optimal race performance is achieved. At some point during a race a pit stop may be required to change worn tyres so that tyre grip can be restored. It is essential to understand the wear characteristics of various tyre compounds in order to determine the point when the time needed for a pit stop is justified in terms of subsequent racing performance.
Chapter
At a high level of description, the tyre may be thought of as a nonlinear dynamical system, which produces certain outputs, often referred to as tyre characteristics, when subjected to opportune inputs. This interpretation allows defining some fundamental quantities that contribute to determining both the steady-state and the transient response of the tyre. Amongst these, the slip variables play the most important role. In steady-state conditions, the tyre characteristics may be described as real analytic functions of the slips, which may be defined in three main different ways. The Jacobians of the steady-state characteristics with respect to the slip variables are often called matrices of generalised slip stiffnesses. The local properties of the steady-state tyre characteristics may be deduced from the entries of these matrices.
Article
Optimal control is used to study the management of tyre performance for a Formula One car. Tyre friction is compromised by accumulated wear and operation outside the tyre design temperature window – inappropriate thermal operation accelerates tyre wear. In this study tyre wear is modelled as a function of the tyre surface temperature and the power dissipated at the road contact. The tyre's frictional performance is modelled as a function of temperature and accumulated wear. A previously developed thermodynamic model is modified by the addition of a state representing the tyres' carcass temperature. Optimal control calculations are used to optimise (multi-lap) performance by improving grip and extending the life of the tyres. In combination, tyre wear and friction control can be used to schedule tyre changes and minimise race times.
Article
The purpose of this paper is to develop a multi-physical tyre model, which can take into account the influence of temperature and macro-texture of the road surface. The aim is to contribute to a better understanding and modelling of the tyre–road pavement interaction. This tyre–road interaction model can be used for friction estimation as well as rolling resistance estimation. Indeed, tyre–road friction has an undeniable influence on the rolling tyre; an accurate measurement of the tyre–road interaction is necessary for a better tyre design and simulations. The developed approach comprises mechanical and thermal models and gives dynamic forces and tyre temperatures as outputs. Comparison of simulations with experimental results to validate the model is presented and discussed. Special focus is given to the study of the impact of the macro-texture of road surface; numerical and experimental results are discussed. The experiments were done at IFSTTAR test tracks.
Conference Paper
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Designers and technicians involved in vehicle dynamics face during their daily activities with the need of reliable data regarding tyres and their physical behaviour. The solution is often provided by bench characterizations, rarely able to test tyres in real working conditions as concerns road surface and the consequential thermal and frictional phenomena. The aim of the developed procedure is the determination of the tyre/road interaction curves basing on the data acquired during experimental sessions performed employing the whole vehicle as a sort of moving lab, taking into account effects commonly neglected.
Conference Paper
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In the paper new structure elements have been developed and implemented in the already-existing TRT thermo-dynamic tyre model. The updated model aims to provide a complete tool to study and understand all the phenomena concerning the tyre in thermal transient conditions, since all the elements constituting its structure are finally modelled. The computational cost, connected to a more complex model to manage, was decreased by simplifying the mesh of the previous version of the model and, thus, by reducing the state vector length.
Article
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A theoretical tyre model was developed analytically using the proposed middle plane for predicting the three-dimensional global tyre wear qualitatively and quantitatively as a function of the road roughness and the dynamic characteristics of the vehicle. Diagrams of the tyre wear pattern across the tyre width and along the tyre circumference were drawn for different road roughnesses, camber angles and vehicle dynamic characteristics. The numerical results show how the three-dimensional tyre wear pattern varied when these factors changed. Numerical calculations also reveal that a unique wear pattern may be caused by the system’s inherent dynamic features, e.g. the modal shape. In addition, a sensitivity analysis shows that the slip ratio, the longitudinal stiffness, the tyre width, the velocity and the vertical force have the greatest effects on the frictional dissipation energy, in that order. With the established wear model, good tyre wear distribution predictions can be given qualitatively and quantitatively for the rolling surface of the tyre. Moreover, guidelines for obtaining less tyre wear are provided for tyre and car manufacturers.
Article
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In the paper a new physical tyre thermal model is presented. The model, called Thermo Racing Tyre (TRT) was developed in collaboration between the Department of Industrial Engineering of the University of Naples Federico II and a top ranking motorsport team. The model is three-dimensional and takes into account all the heat flows and the generative terms occurring in a tyre. The cooling to the track and to external air and the heat flows inside the system are modelled. Regarding the generative terms, in addition to the friction energy developed in the contact patch, the strain energy loss is evaluated. The model inputs come out from telemetry data, while its thermodynamic parameters come either from literature or from dedicated experimental tests. The model gives in output the temperature circumferential distribution in the different tyre layers (surface, bulk, inner liner), as well as all the heat flows. These information have been used also in interaction models in order to estimate local grip value.
Conference Paper
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In this paper the results of an experimental activity carried out with the aim to investigate on the frictional behaviour of visco-elastic materials in sliding contact with road asperities is presented. Experiments are carried out using a prototype of pin on disk machine whose pin is constituted by a specimen of rubber coming from a commercial tyre while the disk may be in glass, marble or abrasive paper. Tests are performed both in dry and wet conditions. Roughness of the disk materials is evaluated by a tester and by a laser scan device. Temperature in proximity of the contact patch is measured by pyrometer pointed on the disk surface in the pin trailing edge, while room temperature is measured by a thermocouple. Sliding velocity is imposed by an inverter controlled motor driving the disk and measured by an incremental encoder. Vertical load is imposed applying calibrated weights on the pin and friction coefficients are measured acquiring the longitudinal forces signal by means of a load cell. As regards to the road roughness, the experimental results show a marked dependence with road Ra index. Dry and wet tests performed on different micro-roughness profiles (i.e. glass and marble) highlighted that friction coefficient in dry conditions is greater on smoother surfaces, while an opposite tendency is shown in wet conditions. Although affected by uncertainties the results confirm the dependence of friction on temperature, vertical load and track conditions.
Article
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This paper presents novel aspects regarding the physically motivated modelling of rubber stationary sliding friction on rough surfaces. The description of dynamic contact is treated within the framework of a generalized Greenwood–Williamson theory for rigid/soft frictional pairings. Due to the self-affinity of rough surfaces, both hysteresis and adhesion friction components arise from a multi-scale excitation of surface roughness. Beside a complete analytical formulation of contact parameters, the morphology of macrotexture is considered via the introduction of a second scaling range at large length scales which mostly contribute to hysteresis friction. Moreover, adhesion friction is related to the real area of contact combined with the kinetics of interfacial peeling effects. Friction experiments carried out with different rubbers on rough granite and asphalt point out the relevance of hysteresis and adhesion friction concepts on rough surfaces. The two scaling ranges approach significantly improves the description of wet and dry friction behaviour within the range of low sliding velocity. In addition, material and surface effects are predicted and understood on a physical basis. The applicability of such modelling is of high interest for materials developers and road constructors regarding the prediction of wet grip performance of tyres on road tracks.
Conference Paper
The TRT model, developed to accurately reproduce the tire thermal dynamics in all the vehicle working conditions, has to be physically characterized [1][2]. An appropriate non-destructive procedure, that allows to obtain the thermal diffusivity of completely different tire layers, is described. The heat is directly supplied on the tire tread surface trough a specifically powered laser, while two thermal cameras acquire temperatures reached on both the outer and the inner layers. Using the above instrumentation layout to acquire the tire radial and circumferential temperature gradients and a specifically developed mathematical TRTLab model based on the use of Fourier's equation of diffusion applied to a three dimensional domain, allows to estimate the tire thermal diffusivity.
Conference Paper
The evaluation of the tire tread viscoelastic characteristics, especially by means of non-destructive procedures, is a particularly interesting topic for motorsport teams and companies, used to work with unknown and confidential compounds. The availability of such information would define new scenarios in vehicle analysis field, as the possibility to provide physical inputs to tire grip models or the study of the suspensions setup able to make tires work inside their optimal thermal working range. The employment of commercial devices allows to select by means of specific indices the optimal combination of tires to be installed on a vehicle, but it does not provide any information physically correlated with the tread polymers characteristics. The aim of the presented activity is the modelling of one of the cited devices, a dynamic dial indicator, interacting with a viscoelastic half-space. The obtained results allow, analyzing the signals acquired by the device, to identify the tread equivalent stiffness and damping as a function of tire working temperature, providing the basic guidelines for the development of an innovative procedure for a full non-destructive viscoelastic characterization of the tire compounds. Index Terms-Material non-destructive characterization, temperature effect, tire tread compound behavior, TSD, viscoelastic characteristics.
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The Nature of Viscoelastic Behavior. Illustrations of Viscoelastic Behavior of Polymeric Systems. Exact Interrelations among the Viscoelastic Functions. Approximate Interrelations among the Linear Viscoelastic Functions. Experimental Methods for Viscoelastic Liquids. Experimental Methods for Soft Viscoelastic Solids and Liquids of High Viscosity. Experimental Methods for Hard Viscoelastic Solids. Experimental Methods for Bulk Measurements. Dilute Solutions: Molecular Theory and Comparisons with Experiments. Molecular Theory for Undiluted Amorphous Polymers and Concentrated Solutions Networks and Entanglements. Dependence of Viscoelastic Behavior on Temperature and Pressure. The Transition Zone from Rubberlike to Glasslike Behavior. The Plateau and Terminal Zones in Uncross-Linked Polymers. Cross-Linked Polymers and Composite Systems. The Glassy State. Crystalline Polymers. Concentrated Solutions, Plasticized Polymers, and Gels. Viscoelastic Behavior in Bulk (Volume) Deformation. Applications to Practical Problems. Appendices. Author & Subject Indexes.
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This study investigates, experimentally and analytically, the polygonal wear of truck/bus and car tires and elucidates the generation mechanism of polygonal wear caused by the first natural vibration mode of the tire in the vertical direction. In the analysis, the phenomenon is modelled as a time delay system accompanied by wear in which the amount of wear of the tire is fed back as forced displacement in the vertical direction after the time period of tire rotation. The progress of the polygonal wear of the tire is very slow and is caused by unstable vibration in the steady-state wear process generated in the limited regions where the products of polygonal numbers and the rotational speed of the tire are less than the natural frequencies of the tire system. The tire is deformed almost to the shape of a regular polygon when polygonal wear occurs. Good agreement between experimental and analytical results concerning the occurrence of polygonal wear of the tire was confirmed.
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This paper deals with the frictional behaviour of a tyre tread elementary volume in sliding contact with road asperities. Friction is supposed as composed by two main components: adhesion and deforming hysteresis. The target, fixed in collaboration with a motorsport racing team and with a tyre manufacturing company, is to provide an estimation of local grip for on-line analyses and real time simulations and to evaluate and predict adhesive and hysteretic frictional contributions arising at the interface between tyre tread and road. A way to approximate asperities, based on rugosimetric analyses on macro and micro scale, has been introduced. The adhesive component of friction has been estimated by means of a new approach based on two different models found in literature, whose parameters have been identified thanks to a wide experimental campaign previously carried out. The hysteretic component of friction has been estimated by means of an energy balance taking into account rubber viscoelastic behaviour, characterized by means of proper DMA tests, and internal stress / strain distribution due to indentation with road. Model results are finally shown and discussed and the validation experimental procedure is described. The correct reproduction of friction phenomenology and the model prediction capabilities are highlighted making particular reference to grip variability due to changes in working conditions.
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The mechanisms of heat generation and heat dissipation of tire are first discussed, and the heat generation capacity and the heat dissipation capacity per unit time are obtained through the analysis of both tire contact pressure distribution and heat dissipation method. Then,according to heat balance condition of tire, the theory model of the steady-state surface temperature of tire is established based on tire temperature field tests. Finally, the effects of speed, load, and tire pressure on the steady-state surface temperature of tire are analysed. The results show that the established model of the steady-state surface temperature of rolling tire is feasible, and the results can reflect the thermal conditions of tire, and therefore provides a theoretical foundation to predict the steady-state surface temperature for various types of tires under different driving conditions.
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Firstly, the formula of tire wear is established considering temperature effect and the dynamic characteristics of vehicles. In addition, the effects of speed, ambient temperatures, tire pressure and sprung mass for tire wear are analyzed. Finally, the main impact factors of tire tread wear are obtained through the parameters sensitivity analysis. The results show that: the established model of tire wear is feasible, and the results can reflect the wear conditions of tires, which provides a theoretic foundation to predict tire wear for different types of tire under different running conditions.
Article
A wheel is said to slip if its travelling velocity differs in absolute value from the circumferential velocity, or if it has a component in the direction of the axle. Slip produces sliding which is in general confined to the rear part of the area of contact, the sliding range increasing with increasing slip. Assuming proportionality between abrasion and frictional energy dissipation, wear has been calculated for a simple model as function of slip and physical properties of the wheel. At moderate slip, wear is found to increase as the square of the slip; at constant slip, wear increases with increasing stiffness, and decreases with increasing elastic hysteresis of the wheel. The results allow an estimate to be made of the dependence of wear on speed of travel under practical conditions. Experimental evidence is adduced in support of the theory.ZusammenfassungMan sagt von einem Rade, dass es schlüpft, wenn die Fahrgeschwindigkeit sich dem Absolutwert nach von der Umfangsgeschwindigkeit unterscheidet, oder wenn sie eine Komponente in Richtung der Achse hat. Schlupf verursacht Rutschen, das sich im allgemeinen auf den rückwärtigen Teil der Berührungsfläche beschränkt. Dabei wächst der Rutschbereich mit zunehmendem Schlupf an. Der Verschleiss ist für ein einfaches Modell mit der Annahme berechnet worden, dass Abrieb und verlorene Reibungsenergie einander proportional sind. Es stellt sich heraus, dass bei mässigem Schlupf der Verschleiss mit dem Quadrate des Schlupfes ansteigt. Bei konstantem Schlupf nimmt der Verschleiss mit zunehmender Steifheit des Rades zu, während er mit wachsender elastischer Hysterese abnimmt. Aus den Resultaten lässt sich die Abhängigkeit des Verschleisses von der Fahrgeschwindigkeit abschätzen. Experimentelle Ergebnisse unterstützen die Theorie.
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The report compares wear formulas in the literature, and examines the fatigue phenomenon as well as the temperature effect on wear rate. Finally it looks into rubber deformation as related to basic thermodynamic concepts and examines the feasibility of an energy balance approach for the wear process.
Chapter
Thermal Analysis techniques are used in a wide range of disciplines, from pharmacy and foods to polymer science, materials and glasses; in fact any field where changes in sample behaviour are observed under controlled heating or controlled cooling conditions. The wide range of measurements possible provide fundamental information on the material properties of the system under test, so thermal analysis has found increasing use both in basic characterisation of materials and in a wide range of applications in research, development and quality control in industry and academia. Principles and Applications of Thermal Analysis is written by manufacturers and experienced users of thermal techniques. It provides the reader with sound practical instruction on how to use the techniques and gives an up to date account of the principle industrial applications. By covering basic thermogravimetric analysis (TGA), differential scanning calorimetry (DSC) including the new approach of Fast Scanning DSC, together with dynamic mechanical analysis (DMA /TMA) methods, then developing the discussion to encompass industrial applications, the book serves as an ideal introduction to the technology for new users. With a strong focus on practical issues and relating the measurements to the physical behaviour of the materials under test, the book will also serve as an important reference for experienced analysts.
Article
Due to their many economic and ecological implications the possibility to predict tyre wear is of major importance to tyre manufacturers, fleet owners and governments. Based on these observations, in 2000 a three-year project named Tyre and Road Wear and Slip assessment (TROWS) was started. One of the TROWS objectives was to provide a tool able to numerically predict tyre global wear as well as to qualitatively determine the wear distribution. The proposed methodology combines a mathematical model of the tyre with an experimentally determined local friction and wear law. Thus, tyre abrasion due to each single manoeuvre can be determined. Full-scale experimental tests were carried out with two Peugeot 406 cars on a public road course in Italy. Each car was equipped with a different set of tyres: one car was equipped with four all-season tyres (from now on called A tyres) and the other car was equipped with four winter tyres (from now on called B tyres). Both sets of tyres had a 195/65 R15 size. The collected data was used to validate the model. The methodology proved to give qualitatively good tyre wear predictions.
Article
The interpretation of certain phenomena occuring at nominally flat surfaces in stationary or sliding contact is dependent on the assumed distribution of the real area of contact between the surfaces. Since there is little direct evidence on which to base an estimate of this distribution, the approach used is to set up a simple model and compare the deduced theory (e.g., the deduced dependence of the experimental observables on the load) with the experimental evidence. The main conclusions are as follows. (a) The electrical contact resistance depends on the model used to represent the surfaces; the most realistic model is one in which increasing the load increases both the number and size of the contact areas. (b) In general, mechanical wear should also depend on the model. However, in wear experiments showing the simplest behavior, the wear rate is proportional to the load, and these results can be explained by assuming removal of lumps at contact areas formed by plastic deformation; moreover, this particular deduction is independent of the assumed model. This suggests that a basic assumption of previous theories, that increasing the load increases the number of contacts without affecting their average size, is redundant.
Article
A method for estimating the three-dimensional (3D) footprint of a 16.9R38 pneumatic tyre was developed. The method was based on measured values of contact pressure at the soil–tyre interface and wheel contact length determined from the contact pressures and the depths and widths of ruts formed in the soil. The 3D footprint was investigated in an area of the field where the pressure sensors of the tyre passed in a soft clay soil. The tyre was instrumented with six miniature pressure sensors, three on the lug face and the remaining three on the under-tread region between two lugs. The instrumented tyre was run at a constant forward speed of 0.27 m/s and 23% slip on a soft soil, 0.48 MPa cone index, 25.6% d.b. moisture content for four wheel load and tyre pressure combination treatments. The 3D footprint assessment derived from soil–tyre interface stress used in this research is a unique methodology, which could precisely relate the trend profile of the 3D footprint to the measured rut depth. The tyre–soil interface contact pressure distributions results showed that as inflation pressure increased the soil strength increased significantly near the centre of the tyre as a compaction increase sensed with the cone penetrometer.
Article
A theoretical model for the adhesive friction between elastomers and rough solid surfaces is proposed on the basis of opening crack propagation processes at the boundary of the contact interfaces and the rate processes of formation of molecular bonds on the solid surface. This model, which is expressed as a product of the terms related to the two abovementioned processes, requires some measurable and fitted parameters such as the frictional shear strength expressed as a function of viscoelastic dissipation, rate-dependent elasticity, density of bonded molecular chains at a contact junction, critical velocity related to viscoelastic relaxation, and critical velocity related to the rate process of formation of molecular bonds on the solid surface. The friction-velocity relationship exhibits a remarkable fit to previously obtained experimental results for polymers such as engineering rubber, gels, and plastics (glassy polymers), and all fitting parameters are physically reasonable. The viscoelastic index "n" is also related to the "glass-to-rubber transition" of a nanometer-thick polymer layer for frictional behavior. Thus, from a practical viewpoint, this model can be used effectively for fitting the adhesive friction behavior of polymers.
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