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Stochastic Analysis of Tire-Force Equations
Technical Paper
2007-01-4259
ISSN: 0148-7191, e-ISSN: 2688-3627
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English
Abstract
The most popular semi-empirical models for predicting different aspects of the pneumatic tires performance under steady-state conditions are the Friction Ellipse Model (FEM) and the Magic Formula Model (MFM). The Friction Ellipse Model calculates the longitudinal and the lateral forces in the tire contact patch based on the slip ratio, the slip angle, the normal forces at the tire, and the friction coefficients between the tire and the road surface. The Magic Formula Model describes the cornering forces, the braking forces, and the aligning moment as functions of the slip ratio, the slip angle, and the normal forces at the tire. In the real operational environment, key parameters at the interface of the vehicle with the road, such as the slip ratio, the slip angle, the friction coefficients, and the normal force do not have constant values, but always change in time; thus, it is not possible to capture the effect of such uncertainties on the tire behavior (resultant force and moments) using a deterministic model. In addition, current measuring techniques have certain limitations and sometime non-negligible measurement errors could be a source of relatively rough approximations in estimating some important parameters involved in vehicle dynamics simulations and control algorithms.
In this study we treat the uncertainty in key parameters associated with the tire-road interface using a polynomial chaos approach. The approach has been proved to be more computationally efficient than traditional stochastic methods such as Monte Carlo (MC), while it can nicely accommodate nonlinear systems with large uncertainties. In this paper, FEM and the MFM have been extended from deterministic to stochastic models, to account for the uncertainties in the tire-road friction coefficient, the slip ratio, the slip angle, and the normal forces in the contact patch. Although a uniform distribution has been considered for each of the stochastic parameters of interest, the approach presented is not limited to this type of distribution. In addition to the analysis of the impact that the uncertainty in one parameter has on the dynamics of the tire for the specific tire model considered, we also studied the tire response under the scenario where multiple parameters behave in a stochastic manner simultaneously. The modeling approach presented in this paper is able to capture the stochastic nature of parameters of interest and to predict the response of the system under those uncertainties, in an effort to provide a better understanding and a more realistic prediction of the tire-road interaction than a deterministic formulation. This is possible since the stochastic models give the response as a range of possible values and the analysis can further benefit from the corresponding probability density function.
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Citation
Li, L. and Sandu, C., "Stochastic Analysis of Tire-Force Equations," SAE Technical Paper 2007-01-4259, 2007, https://doi.org/10.4271/2007-01-4259.Also In
Commercial Vehicle Chassis & Suspension Systems and Effect of Tire, Suspension & Chassis Failure on Vehicle Dynamics and Control
Number: SP-2146 ; Published: 2007-10-30
Number: SP-2146 ; Published: 2007-10-30
References
- Wong, J. Y. - “Terramechanics and Off-road Vehicles”, Elsevier, 1989.
- Muro, T. and O'Brien, J. - “Terramechanics. Land Locomotion Mechanics”, A.A. Balkema Publishers, 2004.
- Pacejka, H.B. - “Tyre and Vehicle Dynamics”, Butterworth-Heinemann, 1st Edition, 2002.
- Wong, J. Y. - “Theory of Ground Vehicles”, John Wiley & Sons, Inc., Third Edition, 2001.
- Pacejka, H.B., Sharp, R.S. - “Shear Force Development by Pneumatic Tyres in Steady State Conditions: A Review of Modeling Aspects”, Vehicle System Dynamics, 20, 121-176,1991.
- Gim, G. - “Vehicle Dynamic Simulation with A Comprehensive Model for Pneumatic Tires”, Ph.D Dissertation, University of Arizona, 1988.
- Gim, G., Nikravesh, P. E. - “An Analytical Model of Pneumatic Tyre for Vehicle Dynamics Simulations. Part 2: Comprehensive Slips”, International Journal of Vehicle Design, 12(1), 19-39, 1991.
- Eichhorn, U., Roth, J. - “Prediction and Monitoring of Tyre/Road Friction”, Proceedings of FISITA, London, June, 1992.
- Kiencke, U. - “Realtime Estimation of Adhesion Characteristic Between Tyres and Road”, Proceedings of IFAC Congress, 15-18, Sydney, 1993.
- Miller, S. L., Youngber, B., Millie, A., Schweizer, P., Gerdes, J. C. - “Calculating Longitudinal Wheel Slip and Tire Parameters Using GPS Velocity”, Proceedings of the American Control Conference, 1800-1805, Arlington, VA, June 25-27, 2001
- Cherouat, H., Braci, M., Diop, S. - “Vehicle Velocity, Side Slip Angles and Yaw Rate Estimation”, IEEE ISIE, 349-354, Dubrovnik, Croatia, June 20-23, 2005.
- Ghanem, R.G., and Spanos, P.D. - “Stochastic Finite Elements”, Dover Publications Inc, Mineola, New York, 2003.
- Ghanem, R.G. and Spanos, P.D. - “Polynomial Chaos in Stochastic Finite Element”, Journal of Applied Mechanics, 1990, Vol. 57, 197-202
- Ghanem, R.G. and Spanos, P.D. - “Spectral Stochastic Finite-Element Formulation for Reliability Analysis”, ASCE Journal of Engineering Mechanics, 1991, Vol. 117, No. 10, 2351-2372
- Ghanem, R.G., Spanos, P.D. - “A Stochastic Galerkin Expansion for Nonlinear Random Vibration Analysis”, Probabilistic Engineering Mechanics, 1993, Vol. 8, No. 3, 255-264
- Xiu, D., Lucor, D., Su, C.-H. and Karniadakis, G.E. - “Stochastic Modeling of Flow-Structure Interactions using Generalized Polynomial Chaos”, J. Fluids Engineering, Vol. 124, 51-59, 2002.
- Xiu, D., Karniadakis, G. E. - “The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations”, Journal of Sci Comput, 2002: Vol. 24, No. 2: 619-644.
- Xiu, D., Karniadakis, G.E. - “Modeling Uncertainty in Flow Simulations via Generalized Polynomial Chaos”, Journal of Computational Physics, 2003: Vol. 187: 137-167.
- Xiu, D., Karniadakis, G.E. - “Modeling Uncertainty in Steady-state Diffusion problems via Generalized Polynomial Chaos”, Computer Methods in Applied Mechanics and Engineering, 2002: Vol. 191: 4927-4928.
- Sandu, C., Sandu, A., Chan, B.J., and Ahmadian, M. - “Treating Uncertainties in Multibody Dynamic Systems using a Polynomial Chaos Spectral Decomposition”, Proc. of the ASME IMECE 2004, 6th Annual Symposium on “Advanced Vehicle Technology”, Paper number IMECE2004-60482, 9 pages, Nov. 14-19, 2004 Anaheim, CA.
- Sandu, C., Sandu, A., Chan, B.J., and Ahmadian, M. - “Treatment of Constrained Multibody Dynamic Systems with Uncertainties”, Proc. of the SAE Congress 2005, Paper number 2005-01-0936, 11 pages, April 11-14, 2005, Detroit, MI.
- Sandu, A., Sandu, C., and Ahmadian, M. - “Modeling Multibody Dynamic Systems With Uncertainties. Part I: Theoretical and Computational Aspects”, Multibody System Dynamics, Publisher: Springer Netherlands, ISSN: 1384-5640 (Paper) 1573-272X (Online), DOI 10.1007/s11044-006-9007-5, 2006: June 29: 1-23 (23).
- Sandu, C., Sandu, A., and Ahmadian, M. - “Modeling Multibody Dynamic Systems With Uncertainties. Part II: Numerical Applications”, Multibody System Dynamics, Publisher: Springer Netherlands, ISSN: 1384-5640 (Paper) 1573-272X (Online), DOI: 10.1007/s11044-006-9008-4, 2006: Vol. 15, No. 3: 241-262 (22).
- Li, L., Sandu, C., Sandu, A. - “Modeling and Simulation of a Full Vehicle with Parametric and External Uncertainties”, Proc. of the 2005 ASME Int. Mechanical Engineering Congress and Exposition, 7th VDC Annual Symposium on “Advanced Vehicle Technologies”, Session 4: Advances in Vehicle Systems Modeling and Simulation, Paper number IMECE2005-82101, Nov. 6-11, 2005, Orlando, FL.
- Sandu, C., Sandu, A., Li, L. - “Stochastic Modeling of Terrain Profiles and Soil Parameters”, SAE 2005 Transactions Journal of Commercial Vehicles, V114-2, 2005-01-3559, 211-220, Feb, 2006.
- Li, L, Sandu, C. - “Algorithm for the Prediction of Traction Performance of Terrain Vehicles”, Proc. of the 2006 ASME Int. Mechanical Engineering Congress and Exposition, Paper number IMECE2006-13968, Nov. 6-10, 2006 Chicago, IL.
- Li, L., Sandu, C., Lee, J., and Liu, Q. - “Development of Tire-on-Stochastic Snow Models Using a Polynomial Chaos Approach”, Proc. of the 2007 Asia-Pacific and North America 2007 Regional ISTVS Conference, Paper number PI-074, 16 pages, June 23-26, 2007, Fairbanks, AK.