Article

A three-dimensional multibody tire model for research comfort and handling analysis as a structural framework for a multi-physical integrated system

Article

A three-dimensional multibody tire model for research comfort and handling analysis as a structural framework for a multi-physical integrated system

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Abstract

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.

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... Specifically, with increases in the operating speeds, an accurate understanding of tire kinematics and road-tire interaction [4] is important and necessary for the development of intelligent tires and intelligent vehicles. However, the multi-body simulation of the aforementioned features [5] remains a challenge in vehicle simulations. Finite element analysis (FEA)-based tire models, such as the nonlinear finite element tire model [6], absolute nodal coordinate formulation finite element tire [7], and wave finite element analysis (WFEA) tire model [8], were developed and introduced to assist with detailed material design and structural optimization of tires. ...
Article
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The study investigates the planar vibration characteristics of a radial tire with a large section ratio for a heavy load vehicle. A proposed tire model with a flexible belt on an elastic multi-stiffness foundation is investigated via theoretical modeling and an experimental modal analysis. Additionally, the parametric identification of the proposed tire model is discussed. When compared with the conventional sidewall stiffness model for general car tires, the analytical multi-stiffness function of a large section ratio sidewall is derived by combining the membrane feature caused by inflation pressure and the structural deformation caused by sidewall curvature. The Euler beam model specifically assesses the circumferential vibration of the flexible belt. Additionally, we combine the sidewall multi-stiffness function of the membrane pre-tension stiffness, and the structural deformation stiffness is integrated into the flexible belt model. The effect of inflation pressure on the circumferential vibration of the flexible belt is investigated via the experimental modal method and theoretical Euler beam model. The stretching, bending, shearing deformations, and stiffness characteristics of the curved sidewall arc are derived via the virtual work principle. The nonlinear stiffness characteristic of the tire sidewall is calculated relative to the radial deformation of the sidewall segment. The effect of the inhomogeneous section area on the multi-stiffness function is discussed. The experimental and theoretical results indicate the following: (1) The multi-stiffness function of the curved sidewall assesses the pre-tension membrane feature and structural deformation, which is related to the inflation pressure, structural curvature, and material properties; (2) the multi-stiffness function of the curved sidewall is nonlinear relative to the radial deformation of the sidewall arc, and the nonlinear characteristic is highlighted with high inflation pressure; (3) the inhomogeneous feature of the section area affects the multi-stiffness of the curved sidewall. Given the combined sidewall multi-stiffness function of the membrane feature and structural deformation, the elastic foundation model acts as an independent module to enrich the flexible belt beam model. Additionally, a flexible belt on an elastic multi-stiffness foundation tire model is suitable for a radial tire with a large section ratio of a heavy load vehicle or tires under impulsive loads and large deformations.
... Several and different simulation environments, each one characterized by peculiar solvers, features and physical subsystems like tire models (depending on the output required) are on the market. Indeed, literature is rich of tire models, the most used are the physical brush model [3] [4], Magic Formula [5], an empirical model characterized by low computational cost, TM-easy [6], characterized by a formulation simpler than the MF, however by a less accuracy, FEA models [7], adopted to evaluate static characteristics due to significant computational load, multibody models [8] [9], etc. Moreover, the need for higher model accuracy in certain conditions led the researchers to the development of other models like Bratch et al., that have developed a tire model for vehicle dynamics and accident reconstruction [10], R. Lot, who developed a motorcycle tire model [11], and Mancuson et al, that developed a model that considers also the comfort characteristics of the tire [12]. ...
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.
... 5 Furthermore, in vehicle dynamics, tires tread is made of different vulcanized polymers and fillers in order to ensure the vehicle performance and safe operation. [6][7][8] Actually, the friction phenomenon implicitly depends on the viscoelastic behavior of the tire tread compound, [9][10][11][12] where the hysteretic and adhesive phenomena respectively occur in the local contact patch due to tire block deformation and the chemical bound with road micro asperities. 13,14 In addition, the wear phenomenon of tire compound is mostly a result of strain energy loss due to viscoelastic friction mechanism as amount of an adhesive 15 and hysteretic ratio. ...
Article
Background The ultrasound technique, usually based on the transmission mode, is capable of providing the viscoelastic properties of polymers. Further techniques involving pulse-echo methods were also described in literature, but they still exhibit inaccuracies in the evaluation of the acoustic properties. Objective The manuscript focuses on an innovative approach for the characterization of the viscoelastic behavior of polymers employing the ultrasound methodology. The proposed procedure is based on the pulse-echo method in order to overcome possible inaccuracies in acoustic properties evaluation and in issues related to transmitter mode applications. Methods Starting from the pulse-echo method adopted for the acquisition, a novel formulation for data processing has been developed and described, allowing to determine the wave attenuation coefficient, in comparison to the commonly employed procedures involving ultrasound in polymers characterization, based on transmitter mode inspections. To carry out the study, a specifically designed ultrasound bench has been set up and three different polymers have been tested in the temperature range of interest. Results According to the proposed methodology, the loss factor towards the temperature is determined starting from the data acquired considering the identified attenuation coefficient and the measured sound velocities. The trustworthiness of the novel procedure has been proved comparing the obtained viscoelastic loss factor quantities to the reference master curves obtained by the standard Dynamic Mechanical Analysis characterizations carried out on the same polymer specimens. Conclusions A novel methodology involving ultrasound technology aiming to evaluate the viscoelasticity of the polymers using non-destructive approach has been developed. The results obtained are agreement with the standard viscoelastic master curves determined through the DMA.
... Indeed, the first step to achieve and the fundamental obstacle to overcome consists in tyre-road friction modelling and correct estimation for individual vehicles. Some very sophisticated FEM or Multibody models [2][3][4] are able to capture with great accuracy many phenomena related to the tyre dynamics, but they are characterised by extremely long calculation times because of their intrinsic complexity. The need for real-time tyre models, on the other hand, makes them unsuitable for braking and handling applications, and, as a consequence, they are mainly adopted to evaluate static characteristics, as stiffness, resonant frequencies and vibration modes. ...
Article
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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.
... Indeed, the first step to achieve and the fundamental obstacle to overcome consists in tyre-road friction modelling and correct estimation for individual vehicles. Some very sophisticated FEM or Multibody models [2][3][4] are able to capture with great accuracy many phenomena related to the tyre dynamics, but they are characterised by extremely long calculation times because of their intrinsic complexity. The need for real-time tyre models, on the other hand, makes them unsuitable for braking and handling applications, and, as a consequence, they are mainly adopted to evaluate static characteristics, as stiffness, resonant frequencies and vibration modes. ...
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.
... Simulations under various operational conditions were displayed, load changing from 0.5 to 4 KN and inflation pressure varied from 1.8 to 2.5 bar. Farroni [15] introduced a multibody tire model for normal contact applications. Tire structure is modelled as a three-dimensional array of interconnected nodes by means of spring-damper elements. ...
... Moreover, the tread block tangential slip is also not considered in this model. Farroni (2019) developed a 3-dimensional ride comfort and handling (RCH) tyre model, where the tyre was represented by an array of interconnected nodes through tension and rotational stiffness and damping. The temperature, tread viscoelastic compound characteristics and wear degradation can be incorporated in this model. ...
... In the present paper different tyre-road friction is considered for the each wheel which it is expected to have different slip based controllers for each one. The contact friction force is given by , where the adherence is a function of the slip [9]; and is the vertical force applied to the wheel which is transferred to the wheels during different car accelerations [10]. The th wheel dynamic is given by summing the rotational torque as . ...
... The tire characteristics estimated by this procedure can be used also to parametrize physical tire models like [8] [9] [10] without using bench tests. ...
Chapter
Measuring and modeling vehicle response delays is a fundamental prerequisite for dynamic simulations, both in real-time driving platforms and in offline systems. With a human driver the delays have a heavy influence on the driving feeling and for the achievement of the optimal lap times. If artificial intelligence is driving, lap times can be improved by using algorithms able to calculate vehicle states, whose reliable reproduction is deeply linked to the delay’s evaluation.
... The tires compound properties characterization is a key topic in the activities field concerning with the development of innovative methodologies of analysis of Vehicle Dynamics. Particularly, the vehicle performance and safety optimization require the knowledge of friction phenomena occurring within the contact area between the tire and road roughness [5,6,7,8,9]. ...
Chapter
Full-text available
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.
... Tire mechanics model is always used to describe the mathematical relationship between the tire mechanics outputs and the tire operating parameters [36][37][38]. In addition to the tire cornering characteristics, the tire mechanics models also include the tire longitudinal mechanical model, the tire vertical vibration model and the tire model in combined working conditions [39]. Different tire models are used to analyze the corresponding vehicle dynamics characteristics and establish the vehicle dynamics models for subsequent control design. ...
Article
Full-text available
Tire cornering characteristics have significant influence on vehicle lateral dynamics control. Unlike traditional tire mechanics models which are established based on the research experience or the mechanics mechanism, in this study, a novel experimental data-driven modeling approach is presented to model the tire cornering characteristics based on piecewise affine (PWA) identification method. In this approach, the highly nonlinear dynamic of the tire cornering characteristics is well approximated by several affine submodels acting on different regions. To obtain the experimental data which can accurately reflect the tire cornering characteristics, the tire tests are firstly carried out through a high-performance flat-plate test bench. On this basis, the PWA identification of the tire cornering characteristics is composed of the data clustering, the parameter estimation of the affine submodels and the calculation of the hyperplane coefficient matrices. The simulation results of the PWA identification model are finally compared with the experimental data to illustrate that the identified model has high accuracy in approximating the tire nonlinear cornering characteristics under wide range driving conditions.
... Moreover, the further discretization along lateral direction will allow to replicate the rib-by-rib adherence variations linked to footprint area modifications according to camber and dynamic stress acting during severe manoeuvres. Finally, the implementation of combined interactions, of self-alignment moment and of MF micro-parameters will lead to the achievement of a highly reliable tire simulation, useful for vehicle setup optimization and to feed lap time prediction algorithms [18]. ...
Chapter
The vehicle dynamic behaviour analysis is a crucial step for the evaluation of performance in terms of stability and safety. Tires play an important role by generating the interaction forces at each road-tire contact patch. The longitudinal and lateral dynamics are analysed by using instrumented vehicles with expensive high precision sensors to get a measurement of estimates of physical parameters of interest. This paper deals with the evaluation of vehicle under/oversteering behaviour and of braking performance using a Real-time (RT) simulator. The simulations were performed by using an efficient 15 Degrees of Freedoms (DOFs) Lumped-Parameter Full Vehicle Model (LPFVM), comprising a tire model with temperature-dependent properties. A virtual Driver-in-the-Loop (vDiL) scheme was used to perform test manouvers. The virtual driver is based on two PID regulators for speed and steering control. Finally, this paper reports the results of constant radius tests as defined by standard ISO4138 and of a braking manoeuvre. In both tests, a type-A road profile as defined by ISO 8608 standard was simulated.
... The procedure described is part of a larger research activity whose final aim, fixed with Prometeon Tire Group, is to integrate the algorithm with other tools that can predict tire thermal behavior and wear, identifying the rubber viscoelastic properties, the "strain energy loss" (SEL) and its thermal conductivities. These models and procedures can cooperate, providing a powerful instrument of analysis and simulation for the prediction of the real tire dynamics [19]. ...
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.
... In literature different tyre models have been proposed over the recent years. Some very advanced FEM or Multibody models [3][4][5] are capable of capturing many phenomena related to the tyre dynamics. Nevertheless, their intrinsic complexity makes them computationally demanding, and eventually unsuitable for real-time applications. ...
Article
Full-text available
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
Chapter
Real-time knowledge of the tire dynamic behaviour is a fundamental prerequisite for using electronic controls in vehicle dynamics, which are rapidly increasing their complexity and, consequently, the required accuracy of the measurement systems. Some examples of new advances from the automotive industry are driver assistance (ADAS) and roadside units (RSU), receivers installed on the road infrastructure capable of collecting vehicle data, with the purpose of monitoring the status of the road. Moreover, thanks to the low connection latency offered by the 5G network, interconnected vehicles will be able to passively communicate and apply autonomous driving strategies with the aim of avoiding road accidents [1]. Typically, the tire characterization is performed offline using the data acquired thanks to expensive test benches and instrumentation, subsequently, the parameters of specific models, analytic Pacejka [2] or Dugoff [3], formulations, multibody [4, 5] or FEA-based [6, 7] are identified. For the identification of tire models’ parameters directly using the signals acquired by means of on-board measurements, additional sensors not usually available in standard vehicles, such as the dynamometer hubs, are required. The UniNa vehicle dynamics research group has developed a tool called TRICK [8], able to estimate the physical quantities necessary to fully understand and characterize the tire dynamic behaviour from the CAN channels normally available in telemetry. The forces developed by the tires are derived from the equilibrium equations, while the tire slips are obtained from the kinematics equations [9]. The present paper describes the development stages which have been necessary to permit the TRICK tool working in real time, thus providing useful data regarding the vehicle and tire states to both the driver and the control systems. Issues related to the on-board instrument calibration procedure, noise filtering and interface development are specifically described.
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.
Chapter
Full-text available
Soft robots have been extensively studied for their ability to provide both good performance and safe human-robot interaction. In this paper, we present and compare the performance of two model-based control techniques with the common aim to independently and simultaneously control position and stiffness of a pneumatic soft robot’s joint. The dynamic system of a robot arm with flexible joints actuated by a pneumatic antagonistic pair of actuators, so-called McKibben artificial muscles, will be regarded, while its dynamic parameters will be considered imprecise. Simulation results are provided to verify the performance of the algorithms.
Article
In this work, a combined numerical/experimental analysis is performed for an automotive tyre. A preliminary experimental activity is realized on examined tyre to measure the temperatures of its layers under various operating conditions. In a second stage, a 3D CFD model of tyre is developed in a commercial code and steady RANS simulations are performed in the full range of angular velocity with the aim to refine the prediction of convective thermal power and heat transfer coefficient. CFD simulation results are passed to a user-defined 3D thermodynamic model to furnish a detailed and reliable tyre thermal output with the advantage of a low computational time. Tyre thermodynamic model, enhanced by CFD-related thermal characteristics, demonstrates the capability to properly forecast the measured temperature of tyre layers in a wide range of investigated operating conditions. The proposed numerical approach represents a valuable tool supporting the optimization of tyre behavior and the development of advanced control rules for optimal tyre management.
Chapter
In this paper the capability to reproduce the mechanical behaviour of viscoelastic materials is investigated comparing the response of a generalized Maxwell model and a relative fraction derivative model towards the experimental behavior of a selected viscoelastic material. In particular, the rheological models are mathematically described illustrating the advantages of the pole-zero formulation for a constrained parameters’ identification procedure. The effectiveness of the both models’ is then compared focusing on the ability of the models to adequately fit the experimental data with a minimum number of parameters, also addressing the possible computational issues.
Conference Paper
The use of computer simulations in motorcycle engineering makes it possible both to reduce designing time and costs and to avoid the risks and dangers associated with experiments and tests. The multi-body model for computer simulations can be built either by developing a mathematical model of the vehicle or by using commercial software for vehicle system dynamics. Even though the first method is more difficult and time-consuming than the second, maximum flexibility in the description of the features of the model can be obtained only by using an analytical model. Moreover, mathematical modelling has a high computation efficiency, whereas multi-body software requires a lot of time to carry out simulations. For the reasons above, the aim of this work was to develop a mathematical model of a motorcycle.
Conference Paper
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We study the use of kinematic and dynamic vehicle models for model-based control design used in autonomous driving. In particular, we analyze the statistics of the forecast error of these two models by using experimental data. In addition, we study the effect of discretization on forecast error. We use the results of the first part to motivate the design of a controller for an autonomous vehicle using model predictive control (MPC) and a simple kinematic bicycle model. The proposed approach is less computationally expensive than existing methods which use vehicle tire models. Moreover it can be implemented at low vehicle speeds where tire models become singular. Experimental results show the effectiveness of the proposed approach at various speeds on windy roads.
Article
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Owing to limitations of laboratory test rigs to reproduce some of the more extreme operating conditions (such as a tyre rolling over large obstacles), not all the tyre parameters and characteristics can be derived using tyre dynamic tests. However, most tyre dynamic responses and vibration properties can be virtually simulated using finite element (FE) analysis which has long been recognised as a significant simulation tool for tyre characteristics investigation. The objective of this study is to determine the rolling resistance of a tyre rolling on an uneven road by simulating the energy loss in the tyre and the longitudinal force. For simulating a tyre rolling on an uneven road, the road unevenness was generated using the inverse discrete Fourier transform (DFT) method, and the effects of different kinds of road unevenness and different travelling velocities on rolling resistance were investigated.
Article
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The importance of heavy vehicles traffic and the composition of bituminous mixes along with their mode of implementation influence the durability and the performance of wearing courses, especially resistance to polishing and the stripping of aggregates on the surface. This traffic-texture combination can be expressed through the distribution of pressures in the tyre-coating contact zone. The study presented in this article sets out an affordable methodology to measure the local pressures distribution and the real contact area between viscoelastic rubber and a surface road. The principle consists of intercalating a Prescale Fujifilm between a tyre and the road surface in specific temperature, humidity and load conditions in static mode. Then, using the analysis of images, we use the footprint left on the Fuji paper through the mechanical contact of the two surfaces to determine the distribution of local pressures and the real contact area with high precision and a good resolution. The effect caused by variations in the inflation pressure and intensity of the load is evaluated.
Book
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Book available at HALF PRICE (50% off), shipping included worldwide till MARCH 31, 2016 here http://www.springer.com/en/book/9789401785327 (Springer website) THE MORE YOU KNOW VEHICLE DYNAMICS, THE MORE YOU'LL BE SURPRISED Vehicle dynamics is often perceived as a quite intuitive subject. As a matter of fact, lots of people are able to drive a car. Nevertheless, without a rigorous mathematical formulation it is very difficult to truly understand the physical phenomena involved in the motion of a road vehicle. In this book, mathematical models of vehicles are developed, always paying attention to state the relevant assumptions and to provide explanations for each step. This approach allows for a deep, yet simple, analysis of the dynamics of vehicles, without having to resort to foggy concepts. The reader will soon achieve a clear understanding of the subject, which will be of great help both in dealing with the challenges of designing and testing new vehicles and in tackling new research topics. However, there is much more than that. Quite surprisingly, it is shown that several classical concepts, such as the understeer gradient or the roll axis, are either wrong or inadequate and need to be replaced. The book covers handling and performance of both road and race cars. A new approach, called MAP (Map of Achievable Performance), is presented and thoroughly discussed. It provides a global and intuitive picture of the handling features of a vehicle. Moreover, the book also deals with several relevant topics in vehicle dynamics that have never been discussed before. Even very experienced people should find the book interesting and stimulating. This new book is not a translation of the Italian Dinamica del Veicolo; by the same author. Actually, in some sense, this new book is totally different, with new topics and with new points of view for the topics covered in the Italian book as well.
Thesis
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The target of the activities described in the PhD thesis, fixed in collaboration with a motorsport racing team, with a high performance vehicle manufacturing company and with a tyre research and development technical centre is the development of a procedure able to estimate tyre interaction characteristics, reproducing them in simulation environments taking into account the fundamental friction and thermal phenomena concerning with tyre/road interaction. A first tool, called TRICK, has been developed with the aim to process data acquired from experimental test sessions, estimating tyre interaction forces and slip indices. Once characterized the vehicle, filtering and sensors output correction techniques have been employed on the available data, creating a robust procedure able to generate as an output a "virtual telemetry" and, following a specifically defined track driving routine, to provide tyre interaction experimental curves. TRICK virtual telemetry can be employed as an input for the second tool, TRIP-ID, developed with the aim to identify the parameters of a Pacejka Magic Formula tyre model. The advantage of this kind of procedure is the possibility to simulate the behaviour of a tyre without the bench characterizations provided by tyremakers, with the further benefit to reproduce the real interactions with road and the phenomena involved with it, commonly neglected in bench data. Among such phenomena, one of the most important is surely the effect that temperature induces on tyre performances, especially in racing applications. For this reason a specific model, called TRT, has been realized and characterized by means of proper thermodynamic tests, becoming a fundamental instrument for the simulation of a tyre behaviour as close to reality as possible. One of the most useful features provided by the model is the prediction of the so called "bulk temperature", recognized as directly linked with the tyre frictional performances. With the aim to analyse and understand the complex phenomena concerning with local contact between viscoelastic materials and rough surfaces, GrETA grip model has been developed. The main advantage to which the employment of the grip model conducts is constituted by the possibility to predict the variations induced by different tread compounds or soils on vehicle dynamics, leading to the definition of a setup able to optimise performances as a function of tyre the working conditions. The described models and procedures can cooperate, generating a many-sided and powerful instrument of analysis and simulation; the main features of the available employment solutions can be summarised as follows:  full geometric, thermodynamic, viscoelastic and structural characterization of tyres on which the analyses are focused;  estimation of the tyre interaction characteristic curves from experimental outdoor test data;  definition of a standard track driving procedure that employs tyres in multiple dynamic and thermal conditions;  identification of Pacejka Magic Formula tyre models parameters on the basis of the estimated tyre interaction characteristic curves;  estimation of surface, bulk and inner liner tyre temperatures for variable working conditions and real-time reproduction of tyre thermodynamic behaviour in simulation applications;  correlation of tyre thermal conditions with friction phenomena observable at the interface with road;  prediction of tyre frictional behaviour at tread compound and soil roughness variations;  modelling of tyre interaction by means of MF innovative formulations able to take into account grip and thermodynamic effects on vehicle dynamics;  definition of the optimal wheels and vehicle setup in order to provide the maximum possible performances improvement.
Book
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This application-oriented book introduces the associations between contact mechanics and friction and with it offers a deeper understanding of tribology. It deals with the associated phenomena of contact, adhesion, capillary forces, friction, lubrication, and wear from one consistent viewpoint. The author goes into (1) methods of rough estimation of tribological quantities, (2) methods for analytical calculations which attempt to minimize the necessary complexity, (3) the crossover into numerical simulation methods. With these methods the author conveys a consistent view of tribological processes in various scales of magnitude (from nanotribology to earthquake research). Also, system dynamic aspects of tribological systems, such as squeal and its suppression as well as other types of instabilities and spatial patterns are investigated. This book contains problems and worked solutions for individual chapters in which the reader can apply the theory to practical situations and deepen the understanding of the material. © Springer-Verlag Berlin Heidelberg 2010. All rights are reserved.
Article
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We have measured the surface topography and calculated the surface roughness power spectrum for an asphalt road surface. For the same surface we have measured the friction for a tire tread compound for velocities 10(-6) m s(-1) < v < 10(-3) m s(-1) at three different temperatures (at -8 °C, 20 °C and 48 °C). The friction data was shifted using the bulk viscoelasticity shift factor a(T) to form a master curve. We have measured the effective rubber viscoelastic modulus at large strain and calculated the rubber friction coefficient (and contact area) during stationary sliding and compared it to the measured friction coefficient. We find that for the low velocities and for the relatively smooth road surface we consider, the contribution to friction from the area of real contact is very important, and we interpret this contribution as being due to shearing of a very thin confined rubber smear film.
Article
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This study proposes a comprehensive analytical tire model for handling and ride comfort in low frequency ranges. A contact algorithm that is developed in this study provides a two-dimensional contact pressure distribution on even and uneven road surfaces with reasonable computational cost. Shear stresses and strains during cornering and braking are estimated by direct measurement of tread deformations. The model is validated against experimental force and moment data for general handling simulations. Cleat tests are also conducted and validated under different forward velocity and vertical load conditions for a tire vibration study.
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.
Book
The definitive book on tire mechanics by the acknowledged world expert. © 2012 Hans Pacejka Published by Elsevier Ltd All rights reserved.
Book
This application-oriented book introduces readers to the associations and relationships between contact mechanics and friction, providing them with a deeper understanding of tribology. It addresses the related phenomena of contacts, adhesion, capillary forces, friction, lubrication, and wear from a consistent point of view. The author presents (1) methods for rough estimates of tribological quantities, (2) simple and general methods for analytical calculations, and (3) the crossover into numerical simulation methods, the goal being to convey a consistent view of tribological processes at various scales of magnitude (from nanotribology to earthquake research). The book also explores the system dynamic aspects of tribological systems, such as squeal and its suppression, as well as other types of instabilities and spatial patterns. It includes problems and worked-out solutions for the respective chapters, giving readers ample opportunity to apply the theory to practical situations and to deepen their understanding of the material discussed. The second edition has been extended with a more detailed exposition of elastohydrodynamic lubrication, an updated chapter on numerical simulation methods in contact mechanics, a new section on fretting in the chapter on wear, as well as numerous new exercises and examples, which help to make the book an excellent reference guide. “Popov puts a reader equipped with good intuition into a position to tackle quite a few tribological problems in a semi-quantitative fashion. … The book is organized extremely well … The large number of solved problems at the end of each chapter is a definite advantage of the book. … great benefit to those who teach tribology. …Meaningful references are provided for those who want to refine the material as a teacher or to indulge in tribological research.” (Prof. Martin Müser, Tribology Letters, July, 2010)
Article
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.
Chapter
New structure elements have been developed and implemented in the TRT thermo-dynamic tyre model. The updated model aims to provide a complete tool to study and understand all the phenomena concerning the tyre behaviour in thermal transient conditions, since all the elements constituting its structure are modelled. The computational cost, connected to a more complex model to manage, has been decreased by simplifying the mesh of the previous model version and, thus, by reducing the state vector length so making it suitable for real time analyses.
Article
A three-dimensional prediction model has been developed in which the interaction between snow and a rolling tire with tread pattern is considered. An explicit finite element method (FEM) and a finite volume method (FVM) are used to model tire and snow respectively. Snow deformation is calculated by the Eulerian formulation to solve the complex interaction between snow and tire tread pattern. Coupling between a tire and snow is automatically computed by the coupling element. Numerical modeling of snow is essential to the tire performance prediction on snow. In this study, snow is assumed to be homogeneous and considered to be an elasto-plastic material. The Mohr-Coulomb yield model, in which the yield stress is a single function of pressure, is adopted. This function is investigated by tire traction tests under a wide range of tire contact pressures using several tires with different inflation pressures and patterns. The predicted results using the Mohr-Coulomb yield model are compared with those using the Capped Drucker-Pragger and the Cam-Clay yield models. Snow traction of a tire featuring different tread patterns is simulated by this technology. Results are shown to be in good qualitative agreement with experimental data.
Article
Accurate and efficient tire models for deformable terrain operations are essential for performing vehicle simulations. A direct application of an on-road tire model to simulate tire behavior on a deformable terrain such as soft soil is not possible. The methods for modeling and evaluation of performance of the wheeled vehicles on deformable terrains are influenced by different terrain properties in addition to design and operational parameters. These methods are ranged from very simple empirical methods to highly complex finite element methods. This survey covers the most used models that have been developed for wheeled vehicles in off-road applications. The emphasize is on the models that have made a significant contribution in advancement of techniques for characterizing soil, tire, soil–tire interaction, experimental analysis, model parameterization and model validation. A description is given for selected studies to familiarize the reader with the general terminologies, formulations and modeling approaches. More importantly, two summary tables are given for three groups of models in which the overall features of each model are reviewed and compared to other models. These tables can be used to understand the general picture of the available techniques, and facilitate selecting the appropriate model for future applications.
Article
The application of the rigid ring tyre model, in combination with a new enveloping model with elliptical cams, mounted in a quarter vehicle system is described for both specific obstacles, like potholes and bumps, and measured road profiles that fit the general category of ''broad-band random signals''. Validation results of both the tyre and vehicle models are presented. In addition, the models are analysed extensively to obtain insight into how the vehicle system behaves and to investigate how the enveloping model that generates an effective road surface contributes to this behaviour.
Article
A 3D simulation method was developed to investigate the tire–sand interactions, where the discrete element (DE) and finite element (FE) methods were coupled. An algorithm for the contact of FE and DE was introduced. The Hertz–Mindlin theory was applied to calculate the contact force among elements. A 3D numerical model was established based on the soil-bin experiment to validate the feasibility of the method in the analysis of the tire–sand interactions. The traveling process of the smooth tire on the sandy soil was simulated and the traction performance parameters of the tire under different slip values (0%, 10%, 20%, 30%, 40%, 50% and 60%) were obtained. The overall trend of the drawbar pull versus the slip ratio is qualitatively in agreement with current experimental results. The result also proved the feasibility of the 3D FE–DEM in the simulation of the tire–sand interactions.
Article
A Numerical–physical tyre model was developed . The whole model allows to obtain the road–tyre interactions so it can be used in vehicle dynamic simulations. In this article are presented its capabilities in normal interaction analysis. The normal interaction, i.e. the relationship between the normal load and the normal deflection, influences the tangential (longitudinal plus lateral) one, which determines the vehicle handling behaviour. The parameters used in this model depend on the structure of the tyre and they can be measured on the real tyre. The tyre has been schematized as composed by a flexible belt , the sidewalls and a rigid ring (Rim). The flexible belt is composed by a number of lumped masses linked by extensional and bending stiffnesses and dampers. The tyre model has been developed using the finite segment method. Using these method could be possible to include in the tyre simulations various non-linear structural effects due to large displacements and rotations. The model allows to simulate both steady state and transient conditions.
Article
When two solids are squeezed together they will in general not make atomic contact everywhere within the nominal (or apparent) contact area. This fact has huge practical implications and must be considered in many technological applications. In this paper I briefly review the basic theories of contact mechanics. I consider in detail a recently developed contact mechanics theory. I derive boundary conditions for the stress probability distribution function for elastic, elastoplastic and adhesive contact between solids and present numerical results illustrating some aspects of the theory. I analyze contact problems for very smooth polymer (PMMA) and Pyrex glass surfaces prepared by cooling liquids of glassy materials from above the glass transition temperature. I show that the surface roughness which results from the frozen capillary waves can have a large influence on the contact between the solids. The analysis suggests a new explanation for puzzling experimental results [L. Bureau, T. Baumberger, C. Caroli, arXiv:cond-mat/0510232 v1] about the dependence of the frictional shear stress on the load for contact between a glassy polymer lens and flat substrates. I discuss the possibility of testing the theory using numerical methods, e.g., finite element calculations.
Article
This paper focuses on the vehicle dynamic control system for a four in-wheel motor drive electric vehicle, aiming at improving vehicle stability under critical driving conditions. The vehicle dynamics controller is composed of three modules, i.e. motion following control, control allocation and vehicle state estimation. Considering the strong nonlinearity of the tyres under critical driving conditions, the yaw motion of the vehicle is regulated by gain scheduling control based on the linear quadratic regulator theory. The feed-forward and feedback gains of the controller are updated in real-time by online estimation of the tyre cornering stiffness, so as to ensure the control robustness against environmental disturbances as well as parameter uncertainty. The control allocation module allocates the calculated generalised force requirements to each in-wheel motor based on quadratic programming theory while taking the tyre longitudinal/lateral force coupling characteristic into consideration. Simulations under a variety of driving conditions are carried out to verify the control algorithm. Simulation results indicate that the proposed vehicle stability controller can effectively stabilise the vehicle motion under critical driving conditions.
Article
Owing to the advances in the simulation techniques of vehicle development and design, the modelling of the tyre is of special importance. Thereby, not only the reliability of quantitative results but also the extension to higher frequency ranges and the possibility to account for local road surface structures smaller than the tyre patch is becoming a necessity. A description of some tyre models and the required features for these last two aspects are presented. Reliable modelling of tyre characteristics based on measurements and efficient application with multi-body system (MBS) programs for vehicle dynamics simulation are checked with the ‘Tyre Model Performance Test (TMPT)’. With the TMPT handling and high frequency behaviour of tyre models are tested with a virtual test rig implemented in three different MBS software programs. Also, the validation of measurements is combined with specific capability tests to show the range of application of the tyre models. After an outline of the test conditions, the participating tyre models are introduced. A selection of results offers the possibility to compare the tyre models, the tyre model—MBS software combinations and simulation results with measurements. It becomes obvious that a thorough check of the tyre model and the interaction with the MBS software is essential to be aware of the range of application of simulations and the quality of the results.
Article
Free and forced vibrations of a tyre are predicted using a wave/finite element (WFE) approach. A short circumferential segment of the tyre is modelled using conventional finite element (FE) methods, a periodicity condition applied and the mass and stiffness matrices post-processed to yield wave properties. Since conventional FE methods are used, commercial FE packages and existing element libraries can be utilised. An eigenvalue problem is formulated in terms of the transfer matrix of the segment. Zhong's method is used to improve numerical conditioning. The eigenvalues and eigenvectors give the wavenumbers and wave mode shapes, which in turn define transformations between the physical and wave domains. A method is described by which the frequency dependent material properties of the rubber components of the tyre can be included without the need to remesh the structure. Expressions for the forced response are developed which are numerically well-conditioned. Numerical results for a smooth tyre are presented. Dispersion curves for real, imaginary and complex wavenumbers are shown. The propagating waves are associated with various forms of motion of the tread supported by the stiffness of the side wall. Various dispersion phenomena are observed, including curve veering, non-zero cut-off and waves for which the phase velocity and the group velocity have opposite signs. Results for the forced response are compared with experimental measurements and good agreement is seen. The forced response is numerically determined for both finite area and point excitations. It is seen that the size of area of the excitation is particularly important at high frequencies. When the size of the excitation area is small enough compared to the tread thickness, the response at high frequencies becomes stiffness-like (reactive) and the effect of shear stiffness becomes important.
Article
This paper describes extensions to the widely used TNO MF-Tyre 5.2 Magic Formula tyre model. The Magic Formula itself has been adapted to cope with large camber angles and inflation pressure changes. In addition, the description of the rolling resistance has been improved. Modelling of the tyre dynamics has been changed to allow a seamless and consistent switch from simple first-order relaxation behaviour to rigid ring dynamics. Finally, the effect of inflation pressure on the loaded radius and the tyre enveloping properties is discussed and some results are given to illustrate the capabilities of the model.
Article
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.
Article
This article is the second part of a two-part article looking at carcass deflections, contact pressure and shear stress distributions for a steady-rolling, slipping and cambered tyre. In the first part, a previously described and validated finite-element (FE) model of a racing-car tyre is developed further to extract detailed results which are not easily obtainable through measurements on an actual tyre. Generally, these results aid in the understanding of contact patch characteristics. In particular, they form a basis for the development of a simpler physical tyre model, which forms the focus of this part of the article. The created simpler tyre model has the following three purposes: (i) to reduce computational demand while retaining accuracy, (ii) to allow identification of tyre model features that are fundamental to an accurate representation of the contact stresses and (iii) to create a facility for better understanding of tyre wear mechanisms and thermal effects.Results generated agree well with the physically realistic rolling-tyre behaviour demonstrated by the FE model. Also, the model results indicate that an accurate simulation of the contact stresses requires a detailed understanding of carcass deformation behaviour.
Article
The tyre force and moment generating properties connected with the vehicle's horizontal motions are considered. Knowledge of tyre properties is necessary to properly design vehicle components and advanced control systems. For this purpose, mathematical models of the tyre are being used in vehicle simulation models. The steady-state empirical ‘Magic Formula tyre model’ is discussed. The aligning torque description is based on the concepts of pneumatic trail and residual torque. This facilitates its combined slip description. Following Michelin, weighting functions have been introduced to model the combined slip force generation. A full set of equations of the steady-state part of the model of the new version ‘Delft Tyre 97’ is presented. The non-steady state behaviour of the tyre is of importance in rapid transient maneuvres, when cornering on uneven roads and for the analysis of oscillatory braking and steering properties. A relatively simple model for longitudinal and lateral transient responses restricted to relatively low time and path frequencies is introduced.
Article
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.
Article
A physically based application-oriented tyre model, FTire, with detailed MBS and finite-element suspension models is discussed. The kernel of FTire has two parts, where the first one describes the tyre's structural stiffness, damping, and inertia properties. The second part consists of road evaluation, computation of the contact pressure distribution and distributed frictional forces. The FTire structure model consists of 80-200 lumped-masss nodes, replacing tire's steel cord, connected to the rim. The potential applications of FTire include tyre models for passenger cars, motercycles, tyre misuse, steering torque amount when parking, tread wear, and slow or sudden pressure loss.
Article
This paper examines an approach to model the vibrations of a deformed rolling tyre at low frequencies (below 500 Hz). The starting point for this approach is a finite element (FE) model of the tyre and the aim is to calculate the dynamic response of a rolling tyre including the details of its complex build up. This allows to relate the tyre design parameters to its vibro-acoustic properties. In this context, a modal approximation based on the eigenvalues and eigenvectors extracted from the detailed FE model of the tyre seems a computationally efficient possibility. In the proposed approach the natural frequencies and modeshapes of a deformed tyre are calculated in a standard FE package using the full (nonlinear) FE model. Subsequently, this modal base is transformed to determine the response of the rotating tyre in a fixed (Eulerian) reference frame. Furthermore, this approach makes it possible to define a receptance matrix for the rotating tyre. Results from relatively simple tyre models show that the effects of rotation are modelled correctly and are in accordance with results from literature.
Article
In the present paper, starting from equations of three-dimensional theory of elasticity, the system of the recurrent equations of the ray method is obtained allowing one to describe the dynamic behavior of thin solids. It is shown that the implementation of the recurrent equations of the ray method and the ray series derived from these equations is preferred over the hyperbolic equations of the Timoshenko type for finding the solution for the problems resulting in the generation and propagation of surfaces of discontinuity in thin bodies. This is due to the fact that in the theories of the Timoshenko type there occur two shear waves, on which shear disturbances along and transverse to the plate's middle surface propagate with different velocities, but within the framework of the proposed theory only one shear wave propagates, on which shear takes place both along the middle surface and transversely to the middle surface. The inconsistency of shear wave splitting into two waves is most pronounced in the problems wherein disturbances propagate along the rays possessing not only curvature but torsion as well, for example, along the helical lines located on the surface of a cylindrical shell.
Article
To investigate the interaction of tires on soft soils a lot of simulation method have been established. With the increasing capacity of numerical computers the finite element method (FEM) has gained rising importance as well in this field. But even though in recent time FEM modeling of tire–soil interaction has undergone a fast development, there are still improvements to be made for higher accuracy and reliability. The paper deals with a further development of tire–soil simulation with special view to an according two-dimensional (2-D) FEM model of an air-filled tire. In contrast to the existing 2-D tire models the new model does not try to find a mathematical description for the global reaction of tires, but is based on a mechanical reproduction of the basic components of a tire. Here the reproduction of the carcass as the most challenging part for a 2-D model is described in detail. Results from simulations with the new tire model on even and uneven roads that are compared to according test results prove the abilities of this model. Further results point out the influence of tire inflation reduction on soft soils with special view to the different effect on different soils.
Article
One of the fundamental problems in terramechanics is soil–tire system. Past achievements on this topic can be observed in various literatures. Fast development on CPU power of PC system enables us to apply numerical methods to this basic subject. Among others, finite element method (FEM) has been applied to simple problems of soil–tire system not only in 2D but also in 3D approach. However, it is noted that the current FEM technology cannot handle “singular” boundary conditions with sufficient accuracy of analysis. Typical example of this limitation can be seen in an application to traction tire–soil contact problems, where the contact point of tire lug tip behaves as the singular point of stress field. On the other hand, distinct or discrete element method (DEM) has in essence the capability of analyze microscopic deformation (or flow) of soil as many researchers have already been demonstrated. It is noted that DEM suffers large calculation time that is consumed not only at contact check between particulate elements but also at incremental time step. In our present study, we try to combine both merit of FEM and DEM together in order to analyze the soil–tire system interaction, where, for example, a tire and deep soil layer are modeled as FEM and soil surface layer as DEM. We propose simple algorithm of this FE–DE coupled method and sample program is developed that can solve some basic terramechanics problems in order to verify our idea. The obtained result shows qualitatively sufficient accuracy.
Article
The transient response of orthotropic, layered composite sandwich plates is investigated by using two new C0 four and nine node finite element formulations of a refined form of Reddy's higher-order theory. This refined third order theory accounts for parabolic variation of the transverse shear stresses, and requires no shear correction factors. The assumed strain approach is employed to model both thin and thick plates without any major defects like shear locking and parasitic spurious zero energy modes. A consistent mass matrix formulation is adopted. The Newmark direct integration scheme is used to solve the governing equilibrium equations. The parametric effects of plate aspect ratio, length to thickness ratio, boundary conditions and lamination scheme on the transient response are investigated. The present results are in very close agreement with earlier published results in the literature and can serve as a benchmark for future investigators.
Article
Nowadays virtual prototyping tools play an important part in the development of vehicles. For studying the dynamics of a vehicle, complex vehicle models are required that are composed of several accurately modelled components. As the tyre constitutes the only contact between the vehicle and the road surface, it is one of the most important components of a vehicle model. For performing ride comfort and durability analyses, there is a need for accurate tyre models that can predict the loads that are transmitted from the tyre to the wheel axle when driving over road irregularities. In this study, such a tyre simulation model is developed that can represent the dynamic response of a tyre when rolling over uneven road surfaces. The approach followed is the combination of the well-known rigid ring dynamic tyre model and a suitable enveloping model that generates a three-dimensional effective road surface, which is used as input for the rigid ring model. The thesis deals with the development of the enveloping models and with the extension of the rigid ring tyre model so that this model is capable of handling the effective road surface. It is shown that the combination of the rigid ring model and the enveloping model can be used successfully to describe the tyre dynamic response to uneven road surfaces. In this research project, numerous experiments have been carried out for model development, parameter identification and model validation. The results of many of these experiments are presented in the thesis.
Tyre models-desire and reality in respect of vehicle development
  • P Bosch
  • Ammon D Klempau
Bosch P, Ammon D and Klempau F. Tyre models-desire and reality in respect of vehicle development. Darmstadter Reifenkolloquium 2002; 17: 87-101.
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Oertel C and Fandre A. Das Reifenmodell System RMOD-K. ATZ 2001; 103: 1074-1079.
Tire modeling and friction estimation
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Svedenius J. Tire modeling and friction estimation. PhD Thesis, Lund University, Lund, 2007.
Ride comfort simulations and steps towards life time calculations: RMOD-K and ADAMS. International ADAMS user conference
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Oertel C and Fandre A. Ride comfort simulations and steps towards life time calculations: RMOD-K and ADAMS. International ADAMS user conference, Berlin, November 1999.
Multi-physical approach for tyre contact and wear mechanisms modelling
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Sakhnevych A. Multi-physical approach for tyre contact and wear mechanisms modelling. PhD Thesis, Universitad egli Studi di Napoli Federico II, Naples, 2017.