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Improvement of steady-state PAC2002 under the sideslip and turn slip inputs based on the discrete theoretical tire model
Abstract
Semi-empirical tire model considering turn slip input is of great significance for the force feedback of vehicle at low speed or small turning radius conditions. In this paper a refined discrete model considering contact patch and the deformation of the belt/carcass which are from the finite element model is established. New force and moment calculation matrix equations are constructed and different iterative methods are compared and Richardson iteration has been chosen because of its best iteration speed. The tire dynamics characteristics under sideslip and turn slip inputs are analyzed based on the discrete model simulation results. It shows the PAC2002 influence coefficients of turn slip on peak value of side force and cornering stiffness can not be well expressed under different loads. And at small loads, the peak value and stiffness of aligning moment relative to turn slip under different sideslip angles can not well be expressed. According to above problems, the PAC2002 model is improved. After improvement, the average error of lateral force relative to sideslip angle under different turn slip is reduced from 3.7% to 1.4%, the average error of lateral force relative to turn slip under different sideslip angle is reduced from 9.9% to 3.6%. And the error of aligning moment relative to sideslip angle under different turn slip is reduced from 8.5% to 5.1% and the error of aligning moment relative to turn slip under different sideslip angle is reduced from 8.9% to 3.2% at small loads. The improvement proposed in this paper have a good result.
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This paper investigates the rolling dynamics of spherical wheels using the theoretical framework provided by the brush models. The analysis is mainly conducted under the assumption of vanishing sliding inside the contact patch. Different types of kinematics are considered: simply rolling wheels, rolling and tilting, and purely spinning. For the first two cases, a complete solution is derived concerning both the steady-state and transient behaviours. Some qualitative trends for the forces and moments generated inside the contact patch are then provided when accounting for limited friction. For the case of a purely spinning spherical wheel, it is shown that steady-state conditions are never possible owing to the assumption of vanishing sliding. Moreover, it is demonstrated that the shear stresses acting inside the contact patch grow unbounded if the additional contribution relating to the deflection of the bristle is not taken into account when calculating the total sliding velocity. In this case, a stationary solution may be eventually recovered as an asymptotic distribution only by assuming limited friction inside the contact patch.
Among load cases concerning vehicle suspension strength and durability, the misuse case is very special because of large obstacles. In this paper, an extension of the flexible belt model RMOD-K 7 concerning misuse situations is discussed. In normal rolling conditions, contact occurs between tire and road surface. To handle misuse deformations, the contact between the inner surface of the tire and the rim has to be dealt with. Results of the validation process are shown, leading to normal forces up to 70 [kN]. The model can be used together with mbs full vehicle models. One example is the determination of the critical test velocity, which generates the maximum of suspension stresses and can be found using RMOD-K 7.
The UniTire model is a nonlinear and non-steady-state tyre model for vehicle dynamics simulation and control under complex wheel motion inputs involving large lateral slip, longitudinal slip, turn slip and camber. The model is now installed in the driving simulator in the Automobile Dynamic Simulation Laboratory at Jilin University for studying vehicle dynamics and their control systems. Firstly, the nonlinear semiphysical steady-state tyre model, which complies with analytical boundary conditions up to the third order of the simplified physical model, is presented. Special attention has been paid to the expression for the dynamic friction coefficient between the tyre and the road surface and to the modification of the direction of the resultant force under combined-slip conditions. Based on the analytical non-steady-state tyre model, the effective slip ratios and quasi-steady-state concept are introduced to represent the non-steady-state nonlinear dynamic tyre properties in transient and large-slip-ratio cases. Non-steady-state tyre models of first-order approximation and of high-order approximation are developed on the basis of contact stress propagation processes. The UniTire model has been verified by different pure and combined test data and its simulation covered various complex wheel motion inputs, such as large lateral slip, longitudinal slip, turn slip and camber.
UniTire is a unified non-linear and non-steady tire model for vehicle dynamic simulation and control under complex wheel motion inputs, involving large lateral slip, longitudinal slip, turn-slip, and camber. The model is now installed in an ADSL driving simulator at Jilin University for studying vehicle dynamics and their control systems. In this paper, first, a brief history of UniTire development is introduced; then the application scope of UniTire and available interfaces to MBS software are presented; thirdly, a more detailed description of UniTire is given; fourthly, a tool aiming at parameterization of UniTire is also demonstrated; and finally, some comments on TMPT are made.
In radial tires, belt structure plays a role of minimizing the lateral deflection of carcass, which has a significant influence on the cornering and wear properties of a tire. The deflection of carcass affects the magnitude of tread block deformation when the tire is under the slip angle. As a result, it can change the cornering stiffness characteristics of the tire, especially when the vertical load is high.
During tire development, a tire design engineer tries to find the optimal belt ply angle that satisfies the various performance requirements simultaneously, but it is not an easy task because the effect of belt angle change is different depending on the size of the tire. There have been many attempts to construct a mathematical model that represents the structural properties of the belt package, including the string-based model and the beam on elastic foundation model. But, in many cases, only the in-plane bending of belt is considered and the shear deformation is not taken into consideration.
In this study, the effect of belt angle change on belt stiffness is analyzed using a mathematical model based on the Timoshenko beam theory. This model can account for the in-plane bending and shear deformation of the belt structure at the same time. The results of the analysis show how the contribution of bending and shear is changed depending on a tire design parameter, herein the belt cord angle. The effect of belt ply angle change on cornering stiffness is investigated by means of the brush model including belt flexibility. The prediction by the brush model is compared with the measurement using a Flat-trac machine, and the validity of the model is discussed.
This book highlights the mechanics of tire performance, offering detailed explanations of deriving basic equations for the fundamental properties of tires, and discussing ways to improve tire performance using these equations. It also compares the theory with practical measurements. The book commences with composite mechanics, which is the fundamental theory for belt and carcass tires, and covers classical, modified and discrete lamination theory. It then addresses the theory of tire shape and spring properties and the mechanics of tread pattern contact properties, as was well as the performance of various tires. This comprehensive book is a valuable resource for engineers involved in tire design and offers unique insights and examples of improvement of tire performances.
In striving for optimal comfort and safety conditions in road vehicles, today’s electronically controlled components provide a range of new options. These are developed and tested using computer simulations in software in the loop or hardware in the loop environments—an advancement that requires the modern automotive engineer to be able to build basic simulation models, handle higher level models, and operate simulation tools effectively.
Combining the fundamentals of vehicle dynamics with the basics of computer simulated modeling, Road Vehicle Dynamics: Fundamentals and Modeling Aspects draws on lecture notes from undergraduate and graduate courses given by the author, as well as industry seminars and symposiums, to provide practical insight on the subject. Requiring only a first course in dynamics and programming language as a prerequisite, this highly accessible book offers end-of-chapter exercises to reinforce concepts as well as programming examples and results using MATLAB®.
The book uses SI-units throughout, and begins with an introduction and overview of units and quantities, terminology and definitions, multibody dynamics, and equations of motion. It then discusses the road, highlighting both deterministic and stochastic road models; tire handling including contact calculation, longitudinal and lateral forces, vertical axis torques, and measurement and modeling techniques; and drive train components and concepts such as transmission, clutch, and power source.
Later chapters discuss suspension systems, including a dynamic model of rack-and-pinion steering as well as double-wishbone suspension systems; force elements such as springs, anti-roll bars, and hydro-mounts; and vehicle dynamics in vertical, longitudinal, and lateral directions using a simple model approach to examine the effects of nonlinear, dynamic, and active force elements. Highlighting useable knowledge, the book concludes with a three-dimension
Tire modeling is an ever-growing area of interest for vehicles as more efficient development processes are desired in terms of time and resources. Vehicle simulations offer an opportunity for development teams to predict tire and vehicle performance before the construction of a physical prototype. Michelin has identified the need for more robust and accurate tire models that can be used for such simulations to give an accurate description of the transient mechanical and thermal behavior of a tire. Rubber’s viscous and elastic properties are heavily dependent on their thermal state; when this effect is not modeled, it results in mathematical tire models that insufficiently predict tire performance. TameTire aims to fill this void for a broad range of maneuvers, track characteristics, and operating conditions based on the ability to predict tire forces and moments with physically based parameters. Some physical characteristics contained within a TameTire model include contact patch dimensions, tread, sidewall and belt stiffnesses, and rubber compound properties. Empirical tire models for handling have limited representation of tire physical properties due to the dependence on the measurement protocol and lack of identification of the thermal state of the tire. TameTire’s advance modeling techniques include capturing a tire’s thermal effects, thereby allowing for a more accurate and thorough analysis of tires behavior while being physically based (e.g., parameters for stiffness, rubber properties) and allowing the model to be grounded in the actual physics of a tire operating.
A vehicle parking manoeuvre is characterized by low or zero speed, small turning radius and large yaw velocity of the steered wheels. To predict the forces and moments generated by a wheel under these conditions, the Pacejka Magic Formula model has been extended to incorporate the effect of spin (turn slip model) in the past years. The extensions have been further developed and incorporated in the MFTyre/MF-Swift 6.2 model. This paper describes the development of a method for the identification of the turn slip parameters. Based on the operating conditions of a typical parking manoeuvre, the dominant parameters of the turn slip model are firstly defined. At an indoor test facility, the response of a tyre under the identified operating conditions is measured. An algorithm is developed to identify the dominant turn slip parameters from the measured responses. Wherever possible, the algorithm is based on direct analytical relationships between the turn slip model parameters and the measured signals, otherwise an iterative method is applied. By means of comparing vehicle simulation results to instrumented vehicle measurements, the turn slip model, including the identified parameters, is validated. The results show the effectiveness of the developed method.
By means of some instructive examples, the contribution shows how even complex phenomena and relations in tyre physics are better understood by using a physics-based tyre simulation model like FTire. In contrast to approximation-based phenomenological models, such an approach will give insight into, rather than requiring description of, the tyre's behaviour. Examples studied here comprise predicted influence of wheel load, inflation pressure, camber angle, and slow rolling speed on parking torque,predicted influence of inflation pressure on cornering stiffness and pneumatic trail,relaxation length: ramping up and down slip angle and wheel load,handling characteristic on very rough roads,a strange phenomenon: cleats that ‘attract’ a tyre. Related to these studies, user-friendly simulation tools on the basis of FTire are introduced, which help in understanding the above-mentioned complex tyre properties. One of these tools, being valuable both in teaching and for vehicle/tyre dynamics experts in industry and research, allows the user to interactively modify, during a running simulation, tyre geometry, material data, and operating conditions. The impact of these variations both on tyre forces and moments as well as on internal tyre states can be directly seen in a running animation, and later analysed with a large variety of post-processing tools. Animations for all case studies are available for download on www.cosin.eu/animationswww.cosin.eu/animations. All registered trademarks used here are properties of their respective owners.
TaMeTirE is a new thermo-mechanical tyre model for handling simulation developed by Michelin. Its main benefit compared to mathematical fit models is its capability to extrapolate tyre force and moment to the actual driving conditions of the vehicle by taking both speed and tyre temperature effects into account and showing their significant influence on the lateral force in pure cornering conditions. TaMeTirE can accurately predict the lateral force especially when the tyre is at the limit of friction.
An analytical tire model for cornering force (CF) and self-aligning torque (SAT) is described on the basis of the Fiala model. CF and SAT come mainly from the shear stress and sliding friction at the tread/road interface. The CF and SAT variations also change the kinetic conditions for their own generations, that is, the contact-pressure distribution, the tire-part deformation, and the relative position of the steering axis within the contact patch. The present model, the CF/SAT system model, which includes these conditional changes through the CF and SAT feedback loops, approximates the slip-angle dependence of CF and SAT withn high accuracy. The inverse data analysis, that is, least-squares fittings of the measured data to the model, give some experimental information about the tread friction coefficient (the adhesive and sliding friction coefficient) and the variation in contact-pressure distribution during cornering. The CF/SAT system model, as well as the CP/SATP model in Part I, may be useful for both the tire production at tire makers and the vehicle dynamics simulation at car makers.
A new analytical tire model for cornering power (CP) and self-aligning torque power (SATP) is proposed on the basis of the Fiala model. In a pneumatic tire, the self-deformation by the transmission of force and torque is so large to influence recursively the force and torque generation during cornering. Then CP and SATP have negative feedback loops: As CP and SATP increase, the steering transmission loss by the tire self-deformation also increases to depress further increments of CP and SATP. The present model, the CP/SATP system model, analytically describes the force and torque transmission with feedback loops by taking into account not only (i) the shear deformation of the tread rubber and (ii) the in-plane belt deflection, but also (iii) the out-of-plane sidewall rotation. As well as other practical systems with a negative feedback loop, the CP and SATP feedback stabilize the CP and SATP output level at a higher vertical load, and approximates the measured load dependence of CP and SATP with high accuracy. Although the sidewall rotation feedback by SATP has not been considered in conventional cornering studies, its contribution in recent radial tires is shown to be larger than that of belt deflection by CP. The model is applicable to not only the numerical simulation for tire design but also the inverse data analysis of the cornering test. Nonlinear least-squares fittings of the measured data to the model are excellent and give the dynamic estimates of tire-part stiffness.
In this paper, an overview of the TNO MF-SWIFT model is given. After an historical overview of the model development and the intended range of application, the model is described briefly. Next, the implementation in multibody and other simulation software is discussed, including also the available road models. After that, the parameterization of the tire model is described and the available identification tool is discussed. Finally, comments are given on the TMPT from the TNO point of view.
The history of ‘Magic Formula’ tire models is shortly described and the PAC2002 tire model is the latest generation of this type of tire models. PAC2002 is a semi-empirical tire model because it has both the empirical (mathematical formulas) and physical modeling components. The PAC2002 tire model can describe the behavior of tires traveling over relatively smooth road surfaces and its dynamical behavior is valid for frequencies up to 12 Hz. The structure of the model is explained and it is shown what the influences of the different Magic Formula-factors have on the characteristic. The classic tire model input quantities are extended with turn-slip, which is one of two components of tire spin. The functional expressions for steady-state cornering and braking/driving are given. In addition to the stretched-string (first-order relaxation effects) model to describe tire transient behavior, the newly added alternative contact-mass model is explained. The new formula set for tire spin and parking is given, which extends validity of the PAC2002 especially for low speed. Some typical advanced handling applications are given for which the tire model can be used. It is described what measurement program is needed to obtain the tire model parameters. A reduced measurement program can be used that is more cost-effective than and gives comparable accuracy to the full program. It is shown that the additions to PAC2002 (contact-mass transient model, tire spin and parking) result in considerable improvements compared to Magic Formula tire models with the traditional ‘first-order relaxation’ transient tire modeling. Next to the common applications of a Magic Formula tire model, the PAC2002 can be used for analyzing parking behavior, low- and zero-speed applications, changing friction properties as well as in more dynamic analysis such as ABS braking.
The first version of the tire simulation software Flexible ring Tire model (FTire) had been released in December 1998. Being subject to permanent improvement and several far-reaching model extensions since then, today it is one of the most widely used and generally accepted tire models for ride comfort, handling, and road load prediction.Strength of FTire is the strictly physical background, which perfectly fits both to Multi-Body Systems (MBS) and Finite Element Method (FEM) environments. Even though certain simplifications are unavoidable, this clean mechanical, thermo-dynamical, and tribological structure of the model guarantees a consistent and plausible model behavior even in situations that are not covered by respective measurements.The modelization takes into account most of the relevant excitation sources and non-linear transfer mechanisms, up to very high frequencies and extremely short wavelengths. The model's high level of detail is accompanied by a very comfortable program interface and a numerically robust and efficient solver. This allows the simulation of even extreme manoeuvres with moderate computation time. FTire can be used together with most of the important MBS packages and specialized vehicle dynamics programs.This contribution gives an overview on history, application, modelization, road models, parameterization, interfacing, availability, and future perspectives of FTire.
This paper describes the semi-physical tire model TMeasy for vehicle dynamics and handling analyses, as it was applied in the ‘low frequency tire models’ section of the research programme tire model performance test (TMPT). Despite more or less weak testing input data, the effort for the application of TMeasy remains limited due to its consequent ‘easy to use’ orientation. One particular feature of TMeasy is the wide physical meaning of its smart parameter set, which allows to sustain the identification process even under uncertain conditions. After a general introduction, the modelling concept of TMeasy is compactly described in this paper. Taking the standard tire interface (STI) to multibody simulation system (MBS) software into account, the way to apply TMeasy is briefly shown. This includes three selected examples of application. The final comments of the authors on TMPT describe the experiences and earnings received during the participation in that programme.
This paper discusses the tire model performance test (TMPT) participation of the LMS International comfort and durability tire model (LMS CDTire). CDTire is a family of three models based on a macroscopic physical description of tires, which is a compromise between scope of applicability and calculation time. A short overview of the CDTire model suite, including parameter identification (PI) software and road surface models, is complemented with application examples of full vehicle simulations utilizing different sub-models and different road surface models. The PI procedure for the TMPT tire, a non-commercial 205/55 R 16 passenger car tire, is reviewed and the results of the TMPT validation and capability tests are discussed.
A semiphysical model presented in this paper first includes the transient steer effect characterized by both the time delay of steering application and the stress relaxation of friction due to the tyre’s viscoelastic property. The model experimentally formulates the vertical force as a function of the tyre deflection, camber angle and lateral force. In order to describe the longitudinal and lateral forces as functions of the tyre’s characteristic parameters such as stiffness and friction, the basic formulations of the UA tyre model are reused. For self-aligning and overturning moments, first the application points of three forces are experimentally formulated, and then the moments are expressed as functions of the three forces and their application points. All characteristic parameters are experimentally determined as functions of the load, camber angle, slip angle, slip angle sweep rate and/or slip ratio for cornering and/or driving–braking. For the purpose of validation, the model is implemented into ADAMS and then its simulation results are compared with measurement data for various test conditions.
The accuracy of tyre models depends to a large extent on the measurement data used to assess model parameters. The MF-Swift tyre model parameters can be identified or estimated from various combinations of experimental data. The amount and required accuracy of the measurement data can be selected according to the tyre model application. This paper discusses a simulation study carried out using TNO’s MF-Swift implementation in Simulink (see www.delft-tyre.com) using sets of param-eters obtained from different combinations of measurement data that have been supplied for the tyre model performance test benchmark. The results show the relation between measurement effort and model accuracy, thus relating the most efficient test protocol to the tyre model application.
This paper discusses the parameter identification procedure for the LMS comfort and durability tyre (LMS CDTire) model. CDTire is a family of three models (namely CDTire 20, CDTire 30 and CDTire 40) based on a macroscopic physical description of tyres, which is a compromise between scope of applicability and speed. The ‘macroscopic description’ implies that the actual model contains elements representing, for example, the ‘belt’, but not individual cords of the belt layers. Therefore, model parameters such as the belt bending stiffness essentially have to be determined by testing the whole tyre in a number of different ways instead of assembling mechanical properties of all the individual elements of a tyre such as cord stiffness. The considered tests are standard tyre tests that can be performed by most tyre test facilities. The parameter identification process is explained in a generic way with exemplary focus on one specific identification subtask, as the complete process description would go beyond the scope of this article. Furthermore, some of the strategic decisions made in the build-up of the parameter identification process are highlighted. Together with a more involved discussion on some numerical aspects of the tyre test rig simulations, the choice of optimization algorithms is explained. The technical discussion is complemented with some results obtained with this parameter identification procedure on a 205/55 R16 non-commercial passenger car tyre.