Editorial Type:
Article Category: Research Article
 | 
Online Publication Date: 01 Mar 2008

Analytical Tire Forces and Moments with Physical Parameters2

Page Range: 3 – 42
DOI: 10.2346/1.2804130
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Abstract

The pneumatic tire behaves as a highly nonlinear system. Its complexity has limited the development of a complete and reasonable theory governing its mechanics. Practical tire models used in vehicle dynamics simulation and tire-related research rely basically on curve-fitted experimental data and empirical adjustments of theoretical models. This paper introduces a validated analytical model based on the physical properties of the tire by formulating the shear contact phenomena with elliptical normal pressure distribution and planar stress-strain laws. Adjustments are introduced to current methods for estimating distributed stiffness, the use of friction, and the forces saturation phenomena. The analytical model is formulated and normalized to accept tire physical parameters that are easily estimated from force and moments measurements. These parameters are universal to all tires: lateral and longitudinal stiffnesses, aligning pneumatic trail, overturning effective moment arm, and frictional properties. The method of using fundamental mechanics for modeling contact patch forces and moments with tire physical parameters constitutes a significant advancement. The tire model is validated with experimental data.

Keywords: tire mechanics; rubber friction; tire testing; tire forces and moments; tire modeling
Copyright: The Tire Society
FIG. 1.
FIG. 1.

Tire model and vehicle dynamics general interactions.

FIG. 2.
FIG. 2.

SAE tire axis system.

FIG. 3.
FIG. 3.

Tire normal pressure elliptical distribution.

FIG. 4.
FIG. 4.

Normalized pressure distribution—elliptical versus parabolic.

FIG. 5.
FIG. 5.

Tire footprint physics—lateral force.

FIG. 6.
FIG. 6.

Tire footprint physics—longitudinal force.

FIG. 7.
FIG. 7.

Tire contact footprint physics—composite longitudinal and lateral.

FIG. 8.
FIG. 8.

Tire element stiffness using consistent small angle approximation.

FIG. 9.
FIG. 9.

Footprint physics with tire physical properties.

FIG. 10.
FIG. 10.

Sliding position from leading edge versus composite slip function.

FIG. 11.
FIG. 11.

Adhesion or tire gearing.

FIG. 13.
FIG. 13.

Normalized footprint break points [(.) depends on tire friction values].

FIG. 12.
FIG. 12.

Adhesion and sliding fundamental properties—A: trivial zero slip, B: maximum adhesion, C: adhesion function is equal to sliding, D: adhesion potential rate is unity, E: adhesion only 20% of total contact patch, and F: complete sliding.

FIG. 14.
FIG. 14.

Peak coefficient of friction.

FIG. 15.
FIG. 15.

Logarithmic trend of peak friction.

FIG. 16.
FIG. 16.

Peak friction model results—lateral direction.

FIG. 17.
FIG. 17.

Peak friction model results—longitudinal results.

FIG. 18.
FIG. 18.

Decay of peak friction.

FIG. 19.
FIG. 19.

Decay of peak friction model—lateral direction.

FIG. 20.
FIG. 20.

Decay of peak friction model—longitudinal direction.

FIG. 21.
FIG. 21.

Tire lateral stiffness.

FIG. 22.
FIG. 22.

Tire lateral stiffness model.

FIG. 23.
FIG. 23.

Tire stiffness empirical model compared to polynomial fit.

FIG. 24.
FIG. 24.

Longitudinal stiffness normal load relation.

FIG. 25.
FIG. 25.

Longitudinal stiffness model.

FIG. 26.
FIG. 26.

Experimental data of aligning moment to lateral force.

FIG. 27.
FIG. 27.

Aligning moment pneumatic trail.

FIG. 28.
FIG. 28.

Overturning moment versus lateral force.

FIG. 29.
FIG. 29.

Overturning moment arm.

FIG. 30.
FIG. 30.

Lateral force versus inclination angle.

FIG. 31.
FIG. 31.

Inclination angle stiffness (camber thrust stiffness).

FIG. 32.
FIG. 32.

Lateral force validation.

FIG. 33.
FIG. 33.

Aligning moment validation.

FIG. 34.
FIG. 34.

Aligning moment validation (moment versus force).

FIG. 35.
FIG. 35.

Overturning moment validations.

FIG. 36.
FIG. 36.

Longitudinal force validations.

FIG. 37.
FIG. 37.

Combined forces validations.

FIG. 38.
FIG. 38.

Lateral force composite slip validations.

FIG. 39.
FIG. 39.

Longitudinal force composite force validations.

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