bicycle tireroadcontact

MODELING TIRE-ROAD CONTACT OF A BICYCLE TIRE

JEROEN WIJLENS

Background

Introduction

Measurements

Tire model

Bicycle stability

Conclusions & recommendations

2

OVERVIEW

21 December 2012 Modelling tire-road contact of a bicycle tire

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BACKGROUND BICYCLE ACCIDENTS

Source: NOS


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BACKGROUND SOFIE PROJECT


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Why are tires important?

Tires are the only contact between bicycle and environment, they

influence the behavior of the whole bicycle.

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INTRODUCTION BICYCLE TIRES


Slip is defined as elongation of the spring:

αy = (Fy / Cy) if Fy < Fw

Slide is displacement of the contact point:

Fy > Fw

Friction coefficient μ and normal force Fn

determine transition between slip and slide.

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INTRODUCTION SLIP & SLIDE


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INTRODUCTION WHEEL PLANES

Ω

x

z

y

Plane through

wheel axis

Wheel plane

Road tangent plane

V

re

S

V = Vr = Ω·re


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INTRODUCTION FORCES & TORQUES

Fx

x

z

y

Fy

Tx

Ty

Tz

Ω

Fz


V

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INTRODUCTION LONGITUDINAL SLIP

x

z

y

Ty

V

Ω

Fz

Fx

Vr

Vs

κ = -Vsx / Vx


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INTRODUCTION

Fx

x

z

y

Fy

Tx

Ty

Tz

Ω

Fz


V

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INTRODUCTION SIDESLIP

V

x

z

y

α

Tz

Fz Fy

Vr

Ω

Vs

α = Vsy / Vx


V

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INTRODUCTION

Fx

x

z

y

Fy

Tx

Ty

Tz

Ω

Fz


V

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INTRODUCTION CAMBER & TURN SLIP

V

x

z

y

-

Tz

Fy

Tx Fz

Ω

ϕ = ( + Ω·sin( )) / Vx

𝝍

𝝍

𝜸

𝜸


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INTRODUCTION

Fx

x

z

y

Fy

Tx

Ty

Tz

Ω

Fz


V

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INTRODUCTION

Ω


PNEUMATIC TRAIL

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INTRODUCTION INPUT & OUTPUT OF TIRE-WHEEL SYSTEM


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INTRODUCTION BICYCLE STABILITY

Out-of-plane force and torques are important for bicycle stability:

• Lateral force Fy

• Aligning torque Tz

• Overturning torque Tx

These are measured using the rotating disk test machine


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GRAPHICAL OVERVIEW


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MEASUREMENTS ASSUMPTIONS

• Dry and clean conditions are assumed

• Texture on the tire tread is not taken into account

• Influences of temperature are neglected

• Only the tire deforms, rim and road do not deform

• Left and right behavior of the tire are the same


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MEASUREMENTS ROTATING DISK TEST MACHINE

Developed by department of Mechanical Engineering, University of Padova, Italy.


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MEASUREMENTS IMPOSED & MEASURED

Imposed:

• Sideslip angle αw

• Camber angle γw

• Vertical force Fz

• Inflation pressure Pi

Measured:

• Lateral force Fy

• Aligning torque Tz


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MEASUREMENTS ROTATING DISK TEST MACHINE


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MEASUREMENTS RAW VALUES


Vertical force Fz = 400 N

Inflation pressure Pi = 4 bar

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MEASUREMENTS RAW VALUES

• Curvature force due to circular path.

• Elimination of curvature effects


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MEASUREMENTS CORRECTED VALUES AS FUNCTION OF SIDESLIP

• Aligning torque due to asymmetric distribution of lateral force

and tends to realign the wheel.


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MEASUREMENTS CORRECTED VALUES AS FUNCTION OF CAMBER

• Aligning torque due to vertical component of rotational velocity

and tends to twist the wheel.


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GRAPHICAL OVERVIEW


Magic Formula tire model: curve fitting of measurement results

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TIRE MODEL MAGIC FORMULA

y is Fx, Fy or Tz B Stiffness factor

x is κ, α or γ C Shape factor (>0)

SH Horizontal shift D Peak Value

SV Vertical shift E Curvature factor (<1)

y(x) = D·sin[C·arctan{B·(x + SH) - E·(B·(x + SH) - arctan(B·(x + SH)))}] + SV


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B = 1.0

y(x) = D·sin[C·arctan{B·x - E·(B·x - arctan(B·x ))}]


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B = 1.5



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B = 2.0



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C = 1.0



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C = 1.5



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C = 2.0



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D = -1.0



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D = -0.75



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D = -0.5



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E = -5



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E = -10



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E = -20



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TIRE MODEL COMPOSITION

The PAC-MC tire model needs coefficients to describe:

• Vertical stiffness Kz and damping Cz.

• Longitudinal force Fx(κ)

• Lateral force Fy(α,γ)

• Aligning torque Tz(α,γ)

• Overturning torque Tx(γ)

• Rolling resistance torque Ty


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TIRE MODEL LATERAL FORCE

Fy = D·sin[f(α) + g(γ)]

f(α) = Cα·arctan{B α· α - E α·(B α· α - arctan(B α· α))}

g(γ) = Cγ·arctan{B γ· γ - E γ·(B γ· γ - arctan(B γ· γ))}

min(R(Fy(x), Fmy(x))); R = ∑(Fy(x) - Fmy(x))2


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TIRE MODEL LATERAL FORCE

Fy = D·sin[f(α) + g(γ)]

f(α) = Cα·arctan{B α· α - E α·(B α· α - arctan(B α· α))}

g(γ) = Cγ·arctan{B γ· γ - E γ·(B γ· γ - arctan(B γ· γ))}

min(R(Fy(x), Fmy(x))); R = ∑(Fy(x) - Fmy(x))2


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TIRE MODEL ALIGNING TORQUE

t(α) = Dt·cos[Ct·arctan{Bt·(α)}]·cos(α)

Tz = -t(α)·Fy(α) + Tzr(γ)


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TIRE MODEL OVERTURNING TORQUE

Due to camber of the wheel:

Tx = Fz · rc · γ


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TIRE MODEL SIMULATION ROTATING DISK TEST MACHINE

Camber γ is imposed


ϕ = ( + Ω·sin( )) / Vx 𝝍 𝜸

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TIRE MODEL SIMULATION ROTATING DISK TEST MACHINE

Sideslip α is imposed


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GRAPHICAL OVERVIEW


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BICYCLE STABILITY BENCHMARK BICYCLE

Stability of an uncontrolled bicycle


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Initial forward velocity V of 5 m/s


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CONCLUSIONS & RECOMMENDATIONS CONCLUSIONS


• Performance of a tire depends on the operating conditions.

• Magic Formula is suitable as model of a bicycle tire.

• Model is validated.

This tire model is suitable to use for investigation of bicycle stability

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CONCLUSIONS & RECOMMENDATIONS RECOMMENDATIONS


• Include turnslip in the tire model

• Accuracy of the measured aligning torque

• Magic Formula coefficient estimation

• Longitudinal slip

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Difference between Magic Formula tires and non-slipping rolling point-contact tires

Accelerating and braking


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QUESTIONS?

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FIGUREN: TRAIL ABOUT STEER AXIS


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