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The UniTire model: A nonlinear and non-steady-state tyre model for vehicle dynamics simulation
Abstract
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.
... Therefore, the nonlinear tire model is essential in this system to describe the force condition at each tire. The UniTire tire model can express tire forces more accurately not only in single working conditions but also in complex working conditions [26], which means it is ideally suited to the part of the vehicle drifting motion. If the UniTire tire model is applied directly to the narrow tilting vehicle drifting controller, there will be a large amount of computation in the controller design. ...
The narrow tilting vehicle receives extensive public attention because of traffic congestion and environmental pollution, and the active rolling motion control is a traffic safety precaution that reduces the rollover risk caused by the structure size of the narrow vehicle. The drifting motion control reflects the relatively updated attentive research of the regular-size vehicle, which can take full advantage of the vehicle’s dynamic performance and improve driving safety, especially when tires reach their limits. The narrow tilting vehicle drifting control is worthy of research to improve the driving safety of the narrow tilting vehicle, especially when tires reach the limit. The nonlinear narrow tilting vehicle dynamic model is established with the UniTire model to describe the vehicle motion characteristics and is simplified to reduce the computation of the drifting controller design. The narrow tilting vehicle drifting controller is designed based on the robust theory with uncertain external disturbances. The controller has a wide application, validity, and robustness and whose performance is verified by realizing different drifting motions with different initial driving motions. The narrow tilting vehicle drifting robust control has some practical and theoretical significance for more research.
... Afterwards, the authors performed another analytical study to overcome the shortcoming of the previous approach [12,13]. For the purpose of describing the complicated nonlinearity characteristics of a modern tire, several improved approaches of tire models with new conceptual parameters are still being developed and enhanced [14][15][16][17]. ...
... After nearly 30 years of development, the UniTire model has gradually developed and matured. It has a good expression ability for various working conditions and a simple model with outstanding prediction and extrapolation ability [9]. ...
Tires are the only link between vehicles and roads and provide all forces for vehicle movement, including driving, steering, and braking. Therefore, the tire model is fundamental for vehicle dynamics, and an efficient tire model is necessary for further technology development of vehicle dynamics and control. In this article, a simplified physical tire model featuring tire hysteresis properties is presented. Firstly, an elastic hysteresis system is introduced to the tire physical model. Then, combining with resistor–capacitor operator, force–deformation characteristics of the elastic hysteresis system are established. Correspondingly, the force–deformation characteristics are applied to describe tire nonlinear dynamics, and a unified expression of HysTire, i.e., a semi-empirical tire model featuring hysteresis characteristics, is established. Finally, the nonlinear dynamics of two tires are modeled to verify the effectiveness and superiority of HysTire. Comparison results with the magic formula model (MF) show that the HysTire provides accuracy like that of the MF, while it is much better in the profiles of predictive and extrapolative capabilities.
... For tire and vehicle dynamics simulations and analyses, different types of tire models have been developed and categorized as empirical and theoretical models [1,2]. The empirical model, or combined theoretical and empirical model, is used for vehicle dynamics, and its model parameters are obtained by fitting measurements [2], including the MF/PAC2002 [2][3][4][5], TMeasy [6][7][8], UniTire [9,10], Hankook-Tire [11], TameTire [12,13], MF-Swift [14,15], FTire [16,17], CDTire [18,19] and RMOD-K [20,21] models, etc. ...
Theoretical tire models are often used in tire dynamics analysis and tire design. In the past, scholars have carried out a lot of research on theoretical model modeling; however, little progress has been made on its solution. This paper focuses on the numerical solution of the theoretical model. New force and moment calculation matrix equations are constructed, and different iterative methods are compared. The results show that the modified Richardson iteration method proposed in this paper has the best convergence-stability in the steady and unsteady state calculation, which mathematically solves the problem of nonconvergence of discrete theoretical models in the published reference. A novel discrete method for solving the total deformation of tires is established based on the Euler method. The unsteady characteristics of tire models are only related to the path frequency without changing its parameters, so the unsteady state ability of the tire model can be judged based on this condition. It shows that the method in the references have significant differences at different speeds with the same path frequency under turn slip or load variations input, but the method proposed in this paper has good results.
... Researchers have proposed various theoretical tire models, including but not limited to the Magic Formula [17,18], Kamm circle [19], Nicolas-Comstock [20], average lumped LuGre [21,22], UniTire [23], and Dugoff [24]. Meanwhile, finite element (FE) tire-pavement interaction models were also created to study the tire mechanics. ...
Overloaded vehicles have a variety of adverse effects; they not only damage pavements, bridges, and other infrastructure but also threaten the safety of human life. Therefore, it is necessary to address the problem of overloading, and this requires the accurate identification of the vehicle weight. Many methods have been used to identify vehicle weights. Most of them use contact methods that require sensors attached to or embedded in the road or bridge, which have disadvantages such as high cost, low accuracy, and poor durability. The authors have developed a vehicle weight identification method based on computer vision. The methodology identifies the tire–road contact force by establishing the relationship using the tire vertical deflection, which is extracted using computer vision techniques from the tire image. The focus of the present paper is to study the tire–road contact mechanism and develop tire contact force equations. Theoretical derivations and numerical simulations were conducted first to establish the tire force–deformation equations. The proposed vision-based vehicle weight identification method was then validated with field experiments using two passenger cars and two trucks. The effects of different tire specifications, loads, and inflation pressures were studied as well. The experiment showed that the results predicted by the proposed method agreed well with the measured results. Compared with the traditional method, the developed method based on tire mechanics and computer vision has the advantages of high accuracy and efficiency, easy operation, low cost, and there is no need to lay out sensors; thus, it provides a new approach to vehicle weighing.
This paper presents an investigation into high-performance tire characteristics at handling limits. Analysis of the tire characteristics has been conducted using the data obtained from indoor test rigs and vehicle tests. The test rig utilised a flat track test platform to obtain tire forces with precise controllability and measurability. As for the vehicle tests, driving data of a professional driver was obtained for repeatability through a high-performance test vehicle equipped with a wheel force transducer (WFT) at each wheel and a Differential Global Positioning System (DGPS). This study is primarily focused on obtaining a deeper insight into vehicle control through an analysis of tire force characteristics at handling limits. The analysis of indoor test data utilised the Magic Formula to obtain an appropriate model capable of describing real-world tire characteristics. An optimisation-based friction ellipse has been proposed to describe the tire force limits in particular. The solution of the proposed optimisation problem allows for an envelope to be generated that accurately reflects the tire force limits. Additionally, a normalised friction circle model has been proposed to quantify tire friction usage based on the proposed friction ellipse, providing a methodology to monitor the friction use of each wheel.
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.
Tire is the only component of a vehicle that makes contact with the road; thus, much important vehicle performance relies on the complex interaction between the tire and road. Tire models are used to describe tire behaviors in terms of mechanical, thermal and wear properties. These models are considered critical in various areas, such as vehicle dynamics and advanced vehicle control, ride comfort, and energy efficiency.
The empirical or semi-empirical model is widely used for vehicle simulation because of its high accuracy but relies on massive experimental data of tire force and moment. Therefore, tire mechanical prediction is of great significance to improve tire modeling efficiency and to reduce costs. Typical prediction methods or models based on normalization presented by Pacejka and Radt assume that different loads could be close to one normalized curve, but it is not always accurate enough to predict tire force under pure slip conditions and there is no mention of friction prediction methods. In the paper, a theoretical model considering the deformation of belt/carcass is established, which lays the foundation for a normalization model. A method for separation of friction from test data and the proportional assumption of longitudinal and lateral peak friction coefficient between target tire and reference tire are proposed, and the experimental results show that this assumption is acceptable. Finally, according to the separation of friction method and assumption, a new prediction method for tire force under pure slip conditions is presented and validated by comparison with the experimental data. It shows that the proposed method has good prediction capability with satisfactory accuracy.
This article introduces an in-plane dynamic tire model with real-time simulation capabilities. The tire belt is discretized into uniformly distributed mass points, and numerous massless contact elements are added between adjacent mass points. The efficient bending force calculation method and Savkoor dynamic friction coefficient formula are included in the model. Newton’s method and the Newmark-beta method are combined to solve the dynamic equations of the tire model, and the simulation results show that a reasonable model structure could meet the real-time calculation requirements. Particle swarm algorithm was used to identify the model parameters under different conditions. Finally, the model is validated and compared with the test data, showing that the model can express the static, steady-state, and high-frequency dynamic mechanical characteristics of the tire with high accuracy and efficiency.
This study presents an analytical approach for determining tyre dynamic properties in a sequence of three papers. Most of the necessary parameters are determined using tyre geometry, orientation and some experimental data. Explicit formulations are derived analytically for the tyre dynamic properties as functions of the slip ratio, slip angle, camber angle and other tyre dynamic parameters. These formulations can be used for the general vehicle simulations in braking/traction and steering manoeuvres with a varying camber angle on regular or irregular terrains. In the first paper a brief review of some previous studies is given. Tyre modelling, contact, slip and friction properties are also studied. Tyre forces and moments due to pure slips are investigated. An algorithmic procedure is provided, in the summary for computational purposes. In the second paper, tyre forces and moments due to comprehensive slips are studied. Finally, in the third paper these analytical models are validated against some experimental data.
Equations of motion applicable to a “running-band” model of the pneumatic tire, as developed by von Schlippe, are transformed to the frequency domain, and side-force and aligning-moment responses are determined for a tire undergoing sinusoidal (a) lateral oscillations of the wheel center plane; (b) steering oscillations with no lateral motion of the wheel center plane; and (c) lateral and yawing motions phased so that the wheel center plane is always aligned with the direction of motion, i.e., the slip angle is zero. The analytical results obtained for the postulated model show that the side force and aligning moment caused by a steering oscillation (namely, the motion described by (b) above) is equal to that produced by pure side-slipping (i.e., (a) above) plus that caused by pure yawing (i.e., (c) above). Numerical evaluation of the derived frequency-response expressions yields an aligning-moment response to steering judged to be extremely significant in view of the unstable behavior often exhibited by wheels with a steering degree of freedom. Comparison of theoretical predictions with experimental data obtained recently by Saito indicates that the “running-band” model is a good first-order approximation of the lateral nonstationary characteristics of pneumatic tires.
Tire cornering properties in non-steady state conditions (NSSC) are analyzed and reviewed in the aspects of theory, test and simulation. Based on the established tire models of cornering properties in steady state conditions, theoretical tire models of cornering properties in NSSC are established with complex deformations of carcass under consideration. Tire cornering tests in NSSC are carried out in a platform-type tire test rig. Then four structural parameters in the derived tire models are identified. Numerical simulations in spatial domain can be realized under different kinds of operation conditions according to the simulation algorithm derived from the initial integral expressions of the derived tire models. The theoretical results show good agreements with test data both in frequency domain and in spatial domain. The presented models and simulation data can be applied to analysis and simulation of vehicle dynamics.
The determination of the crystal and the molecular structure of a five-coordinate palladium(II) complex and its 1H and 31P NMR studies are described. The crystals of [Pd(PMe2Ph)3(2,9-Me2-phen)][BF4]2 (C38H45,N2P3PdB2F8) are monoclinic, space group P21/n, with Z = 4 and unit cell dimensions: a = 23.346(6), b = 13.429(2), c = 12.929(2) Å, β = 91.96(2)°. The structure was determined by a heavy-atom method from diffractometer data and refined to an R value of 0.048 for 4601 unique observed reflections. Three phosphorus atoms from the three dimethylphenylphosphines, and a nitrogen atom from the 2,9-dimethyl-1,10-phenanthroline (2,9-Me2-phen), form an approximately square-planar configuration about the palladium atom; PdP(1) 2.283(1), PdP(2) 2.395(2), PdP(3) 2.407(2) and PdN(1) 2.164(5) Å. The remaining nitrogen atom is at the apex of a distorted square pyramid, PdN(10) 2.588(5) Å. The 1H NMR spectrum shows fluxional motion which averages the two halves of the 2,9-Me2-phen.
This research has been divided into four parts for the purpose of serial publication in the International Journal of Vehicle Design. For the sake of completeness the abstracts for all four parts are given here. Part 1: Review of theories of rubber friction. Previous theoretical work on the dynamic properties of tyres is reviewed. Extensions of a theory relating tyre cornering properties to conditions of braking and traction are considered, along with factors relevant to a further extension to the dynamic properties of an actual pneumatic tyre. A theoretical derivation of the six components of force and moment generated in a tyre is given which involves integrating the vertical and friction forces acting on a rubber block and the resultant moments over the area of contact. Part 2: Experimental investigations of rubber friction and deformation of a tyre. Experiments designed to determine the basic properties of a tyre needed for the calculation of the six components of force and moment are described. Similar consideration is then given to experiments aimed at determining the frictional properties needed for the calculation of the six components. Part 3: Calculation of the six components of force and moment of a tyre. The method of calculating the six components of force and moment generated in a tyre is presented, and then applied. Partial results of the calculation are presented. Part 4: Influence of running conditions on the calculations and experiments. The results of calculations performed in previous sections of the paper are applied to the study of the influence of running conditions on the six components of force and moment in the case of real tyres. An experiment to determine the six components is described, and the experimental values compared with the calculated values.
This research has been divided into four parts for the purpose of serial publication in the International Journal of Vehicle Design. For the sake of completeness the abstracts for all four parts are given here. Part 1: Review of theories of rubber friction. Previous theoretical work on the dynamic properties of tyres is reviewed. Extensions of a theory relating tyre cornering properties to conditions of braking and traction are considered, along with factors relevant to a further extension to the dynamic properties of an actual pneumatic tyre. A theoretical derivation of the six components of force and moment generated in a tyre is given which involves integrating the vertical and friction forces acting on a rubber block and the resultant moments over the area of contact. Part 2: Experimental investigations of rubber friction and deformation of a tyre. Experiments designed to determine the basic properties of a tyre needed for the calculation of the six components of force and moment are described. Similar consideration is then given to experiments aimed at determining the frictional properties needed for the calculation of the six components. Part 3: Calculation of the six components of force and moment of a tyre. The method of calculating the six components of force and moment generated in a tyre is presented, and then applied. Partial results of the calculation are presented. Part 4: Influence of running conditions on the calculations and experiments. The results of calculations performed in previous sections of the paper are applied to the study of the influence of running conditions on the six components of force and moment in the case of real tyres. An experiment to determine the six components is described, and the experimental values compared with the calculated values.