This content is not included in
your SAE MOBILUS subscription, or you are not logged in.
Stochastic Analysis of Tire-Force Equations
Technical Paper
2007-01-4259
ISSN: 0148-7191, e-ISSN: 2688-3627
Annotation ability available
Sector:
Language:
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.
Recommended Content
Authors
Topic
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. O'Brien, J. “Terramechanics. Land Locomotion Mechanics” A.A. Balkema Publishers 2004
- Pacejka, H.B. “Tyre and Vehicle Dynamics” Butterworth-Heinemann 1st 2002
- Wong, J. Y. “Theory of Ground Vehicles” John Wiley & Sons, Inc. Third 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. Spanos, P.D. “Stochastic Finite Elements” Dover Publications Inc Mineola, New York 2003
- Ghanem, R.G. Spanos, P.D. “Polynomial Chaos in Stochastic Finite Element” Journal of Applied Mechanics 1990 57 197 202
- Ghanem, R.G. Spanos, P.D. “Spectral Stochastic Finite-Element Formulation for Reliability Analysis” ASCE Journal of Engineering Mechanics 1991 117 10 2351 2372
- Ghanem, R.G. Spanos, P.D. “A Stochastic Galerkin Expansion for Nonlinear Random Vibration Analysis” Probabilistic Engineering Mechanics 1993 8 3 255 264
- Xiu, D. Lucor, D. Su, C.-H. Karniadakis, G.E. “Stochastic Modeling of Flow-Structure Interactions using Generalized Polynomial Chaos” J. Fluids Engineering 124 51 59 2002
- Xiu, D. Karniadakis, G. E. “The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations” Journal of Sci Comput 2002 24 2 619 644
- Xiu, D. Karniadakis, G.E. “Modeling Uncertainty in Flow Simulations via Generalized Polynomial Chaos” Journal of Computational Physics 2003 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 191 4927 4928
- Sandu, C. Sandu, A. Chan, B.J. Ahmadian, M. “Treating Uncertainties in Multibody Dynamic Systems using a Polynomial Chaos Spectral Decomposition” Proc. of the ASME IMECE 2004, 6 th Annual Symposium on “Advanced Vehicle Technology”, Paper number IMECE2004-60482 Nov. 14-19 2004 Anaheim, CA
- Sandu, C. Sandu, A. Chan, B.J. Ahmadian, M. “Treatment of Constrained Multibody Dynamic Systems with Uncertainties” Proc. of the SAE Congress 2005, Paper number 2005-01-0936 April 11-14 2005 Detroit, MI.
- Sandu, A. Sandu, C. Ahmadian, M. “Modeling Multibody Dynamic Systems With Uncertainties. Part I: Theoretical and Computational Aspects” Multibody System Dynamics Springer Netherlands 1384-5640 10.1007/s11044-006-9007-5 2006 June 29 1 23 23
- Sandu, C. Sandu, A. Ahmadian, M. “Modeling Multibody Dynamic Systems With Uncertainties. Part II: Numerical Applications” Multibody System Dynamics Springer Netherlands 1384-5640 10.1007/s11044-006-9008-4 2006 15 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. 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 June 23-26 2007 Fairbanks, AK.