𝑠

Sensitivity analysis of tire model micro-coefficients

S. Hamza1, A. Birouche1, M. Basset1, F. Anstett-Collin2 1MIPS-MIAM laboratory, Université de Haute-Alsace, Mulhouse, France

2CRAN laboratory , Université de Lorraine, France

4. Results

3. Polynomial chaos

Tires are a vehicle's most important safety

features. Indeed, tires are required to produce the

forces necessary to control the vehicle. Various

models have been proposed to describe the

behavior of the tire on the ground. These models

depend of numerous parameters which can be

distinguished in macro and micro parameters. One

semi-empirical model commonly used in vehicle

dynamics simulations, was developed by Pacejka

[1]. It is widely used to calculate steady-state tire

force and moment characteristics. This model

depends on various parameters.

In [2], it has been shown that the lateral stiffness 𝐾𝑦 and the slip angle are the parameters affecting

the lateral force variation. However, the lateral

stiffness 𝐾𝑦 depends on numerous parameters.

The tire lateral stiffness 𝐾𝑦 is obtained as a function of the vertical load 𝐹𝑧 and the camber angle

Introduction

Aim

1. The overview

• 𝐹𝑧 = 𝐹𝑧0: Situation without acceleration or braking.

• 𝐹𝑧 ≫ 𝐹𝑧0: Braking situation for tires of the front axle or acceleration for tires of the rear axle.

• 𝐹𝑧 ≪ 𝐹𝑧0: Acceleration situation for tires of the front axle or braking for tires of the rear axle.

Sensitivity index

2. Model

𝐾𝑦 = 𝑝1𝐹𝑧𝑠𝑖𝑛 2𝑎𝑟𝑐𝑡𝑎𝑛 𝐹𝑧𝑝2𝐹𝑍0 1 − 𝑝3 𝛾

5. References

[1] H. B. Pacejka. Tyre and Vehicle Dynamics. Elsevier, 2006.

[2] S. Hamza, A. Birouche, F. Anstett-Collin and M. Basset. Sensitivity analysis for the study of a tire model with correlated parameters and an arbitrary distribution.

International Conference on Sensitivity Analysis of Model Output, Nice, France, 1-4 July, 2013.

[3] G. Blatman, B. Sudret, Sparse polynomial chaos expansion and adaptive stochastic finite element using a regression approach, Comptes rendus de Mecanique 336 (6) (2008)

518-523.

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p1Fz

K y

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p2Fz

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p3FzK y

• To quantify the influence of micro-parameters

of Pacejka model on the lateral stiffness 𝐾𝑦 and, therefore, on lateral force.

Graphical representation of lateral stiffness 𝑲𝒚 as a function of vertical

load 𝑭𝒛 and micro-coefficients 𝒑𝟏, 𝒑𝟐 and 𝒑𝟑

𝐹𝑦

𝐹𝑥

𝑀𝑧

𝐹𝑧 𝜇 𝛼 𝜅 𝛾 ⋮ = ℳ(θ)

• 𝐹𝑧, 𝜇, 𝛼,𝜅 , 𝛾 input variables

• θ micro-coefficients

• 𝐹𝑦 the pure lateral force considered as the

output model and given by:

𝐹𝑦 = 𝜇𝐹𝑧 sin 𝐶 𝑎𝑟𝑐𝑡𝑎𝑛 𝐵 𝛼 + 𝑆ℎ −𝐸 𝐵 𝛼 + 𝑆ℎ −𝑎𝑟𝑐𝑡𝑎𝑛 𝐵 𝛼 + 𝑆ℎ + 𝑆𝑣

• The micro-coefficients are assumed to be

independent. The output can be approximated

by a sum of polynomial chaos as follows [3]:

𝑦≈ 𝑐𝑗𝜓𝑗𝑀𝑗=0 𝑢1, ⋯ , 𝑢𝑛

• The first order sensitivity index is computed

from polynomial chaos coefficient s and given

as:

𝑆𝑖 = 𝑐 𝑗2𝐸 𝜓𝑗2 𝑢𝑖𝑗∈Γ𝑖 𝑐 𝑗2𝐸 𝜓𝑗2 𝑢1, ⋯ , 𝑢𝑛𝑀𝑗=1

Where :

• 𝑀 is the number of PC coefficients.

• 𝑛 is the number of parameters.

• 𝜓𝑗 u is a multidimensional orthogonal

polynomials.

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Sp1

Sp2

Sp3

Sp1

Sp2

Sp3

Sp1

Sp2

Sp3

• Spi: Sensitivity index corresponding to parameter 𝜽𝒊