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Full-Text (PDF) Obtaining relations between the Magic Formula coefficients and tire physical properties B. Mashadi1*, H.Mousavi2, M.Montazeri3 1 Associate professor, 2 MSc graduate, School of Automotive Engineering, Iran University of Science and Technology, Tehran, Iran, 3 MSc graduate, Department of Mechanical Engineering, KNT University of Technology, Tehran, Iran. Abstract This paper introduces a technique that relates the coefficients of the Magic Formula tire model to the physical properties of the tire. For this purpose, the tire model is developed by ABAQUS commercial software. The output of this model for the lateral tire force is validated by available tire information and then used to identify the tire force properties. The Magic Formula coefficients are obtained from the validated model by using nonlinear least square curve fitting and Genetic Algorithm techniques. The loading and physical properties of the tire such as the internal pressure, vertical load and tire rim diameter are changed and tire lateral forces for each case are obtained. These values are then used to fit to the magic formula tire model and the coefficients for each case are derived. Results show the existing relationships between the Magic Formula coefficients and the loading and the physical properties of the tire. In order to investigate the effectiveness of the method, different parameter values are selected and the lateral force for each case are obtained by using the estimated coefficients as well as with the simulation and the results of the two methods are shown to be very close. This proves the effectiveness and the accuracy of the proposed method. Keywords: Magic formula, MF, Pacejka, tire model, Genetic algorithm, Least squares 1. Introduction based), empirical and semi-empirical models. Among the several useful quasi-static tire models, the semi- The pneumatic tires carry the vehicle weight, empirical Magic Formula (MF) tire model [1, 2] can reduce the vertical motions on irregular surfaces and be regarded as the best and most comprehensive supply sufficient forces to accelerate or decelerate the model for use in vehicle dynamics studies. In fact this vehicle. Tire cornering force holds the vehicle on a model has now been widely accepted and used corner and stabilizes the vehicle motion. Estimating throughout industry. This is due to its high level of the tire-road contact forces is, therefore, the first step correlation with tire test data and in addition, its in the vehicle handling and safety analysis. ability to be implemented relatively simply into a In the vehicle engineering, describing the tire vehicle simulation program. Moreover, its parameters forces is a complex problem. The reason is the have physical meanings that are useful for the complexity of the tire construction, geometry and characterization of tire shear force generation contact properties that make it hard to model these properties. However, difficulties remain in using the forces by using the fundamental and simple lows of model due to the volume of test data needed for a physics. The tire forces have non-linear and complex specific tire and also the process of finding the relation with quantities associated with the tire such coefficients of the formulae by curve fitting methods. as geometric measures and physical properties, tire All these techniques have been able to find relations pressure, temperature, friction, vertical forces, for the formula factors by using measured results for a longitudinal and lateral slips and more. specific tire. In addition, in most of these methods, Most of the literature concerning the tire behavior complex relations and calculations are employed to deals with the quasi-static part of the tire response in obtain the factors for the tire model. which steady forces and moments result from inputs Finite Element Models (FEM) are also useful such as slip or camber angle as the tire is rolling. A methods in modeling the behavior of tires. The tire large number of quasi-static tire models have been contact problem is one of important issues studied by proposed which include theoretical (or physically several researchers such as Noor and Tanner [3]. In their work in a NASA research program for the space International Journal of Automotive Engineering Vol. 5, Number 1, March 2015 912 Obtaining relations between the Magic Formula… shuttle tires, the status and developments of the paper aims at finding relations for the main computational models for tires are summarized. A coefficients of the Magic Formula tire model with the similar review has also been made again by Danielson physical properties so that one can express the tire [4]. ABAQUS is a commercial software that several forces without having to perform several expensive researchers have found it useful for tire contact force and time consuming tests. For this purpose a complex modeling. Examples are the works of Oden on FEM tire model will be used and validated in order to friction and rolling contact [5], Laursen and Simo in produce a large number of test results when the the field of contact problems with friction [6], physical properties of the tire are changed. In this Padovan on rolling visco-elastic materials [7] and paper an optimum method to have precise results and Wriggers on constitutive interface laws with friction less analyzing time is applied. [8]. Basically all early tire models were axi- 2. Methodology symmetric, with no tread pattern or only circumferential groves due to the limits of Among several methods to estimate the tire forces, computational resources. With the ever increasing the Magic formula model is widely employed in the computational power it is now possible to mesh a part analysis and modeling of the vehicle dynamics. This or even the whole tread with a detailed pattern (e.g. formula was introduced in 1987 by Pacejka and his Cho [9]) and perform all kinds of static analysis such colleagues [1], but it was improved over the years and as footprint shapes as functions of the axle load state various versions are available. This formula is a semi- and pressure. In an article written by Kabe and Koishi empirical model based on several expensive [10] a comparison between ABAQUS standard, experimental tests and software methods to fitting the explicit and experiments for steady state cornering formula to the test data. The formula contains a tires was made. The results of both implicit and number of coefficients that are functions of the explicit methods were closer to each other than to the effective parameters. The magic formula has been experiments. The implicit method, however, was able to express the tire behavior in such a way that the significantly faster. The prediction of tire cornering researchers used it extensively. It can be argued, forces is discussed by Tonuk and Unlusoy [11], where therefore, that possible relations between the magic a comparison with experiments is also presented. The formula coefficients and the physical properties of a application of a type of non-linear 3D finite element tire such as dimension, material, pressure and other tire model for simulating tire spindle force and factors are expected to exist. moment response during side slip is described by In this paper, it is intended to analyze the possible Darnell [12]. The simulation model is composed of relations between the Pacejka’s MF tire coefficients shell elements, which model the tread deformation, and its physical properties, by using a particular coupled to special purpose finite elements that model methodology. The proposed method is divided into the deformation of the sidewall and contact between two parts. At first, a model of the tire is constructed in the tread and the ground. Despite the simple model the Abacus software in order to obtain the tire-road the results corresponds quite well with experiments. contact forces. In the second part, by obtaining the A simulation with ABAQUS is studied by tire forces when the physical properties are varied, the Olatunbosun and Bolarinwa and the effect of tire data is fitted to the MF equation and the dependence design parameters on lateral forces and moments [13] of the coefficients on the physical parameters is is presented. Parametric studies are performed on a analyzed. simplified tire with no tread, negligible rim compliance and viscoelastic properties, shear forces 3. The Magic Formula model modeled with Coulomb friction. Explicit simulations to predict tire cornering forces are presented by This model represents uncoupled longitudinal and Koishi [14]. Besides a comparison with experiments, lateral forces, and self-aligning moment of the tire by parametric studies on the effect of inflation pressure, a unique formula. The model also provides a method belt angle and rubber modulus are performed. of dealing with the problem of combined cornering The empirical methods based on testing to obtain and braking (or acceleration) by making use of the the tire forces, have the disadvantage of repeating the uncoupled force/moment formula and some additional costly and time consuming tests, once the tire mathematical expressions. Although for the general physical properties need be changed. On the other non-linear vehicle models both the tractive and hand, in obtaining the factors of the Magic Formula cornering forces generated by tires are considered, for the idea of establishing relations based on the the sake of simplicity, only the uncoupled lateral physical properties of a tire has not been studied. This force model is used. International Journal of Automotive Engineering Vol. 5, Number 1, March 2015 B. Mashadi, H.Mousavi, M.Montazeri 913 The equations described here for the Magic = 2 + γ Sv (a10 Fz a11Fz ) (10) Formula representation of a tire are taken from the where a1 – a11 are constants and for a given tyre 1987 version [1]. This version was, however, later must be determined through a curve fitting procedure.
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