bicycle tireroadcontact

MODELING TIRE-ROAD CONTACT OF A BICYCLE TIRE

JEROEN WIJLENS

Background

Introduction

Measurements

Tire model

Bicycle stability

Conclusions & recommendations

OVERVIEW

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

BACKGROUND BICYCLE ACCIDENTS

Source: NOS

BACKGROUND SOFIE PROJECT

Why are tires important?

Tires are the only contact between bicycle and environment, they

influence the behavior of the whole bicycle.

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.

INTRODUCTION SLIP & SLIDE

INTRODUCTION WHEEL PLANES

Plane through

wheel axis

Wheel plane

Road tangent plane

V = Vr = Ω·re

INTRODUCTION FORCES & TORQUES

INTRODUCTION LONGITUDINAL SLIP

κ = -Vsx / Vx

INTRODUCTION

INTRODUCTION SIDESLIP

α = Vsy / Vx

INTRODUCTION

INTRODUCTION CAMBER & TURN SLIP

ϕ = ( + Ω·sin( )) / Vx

INTRODUCTION

PNEUMATIC TRAIL

INTRODUCTION INPUT & OUTPUT OF TIRE-WHEEL SYSTEM

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

GRAPHICAL OVERVIEW

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

MEASUREMENTS ROTATING DISK TEST MACHINE

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

MEASUREMENTS IMPOSED & MEASURED

Imposed:

• Sideslip angle αw

• Camber angle γw

• Vertical force Fz

• Inflation pressure Pi

Measured:

• Lateral force Fy

• Aligning torque Tz

MEASUREMENTS ROTATING DISK TEST MACHINE

MEASUREMENTS RAW VALUES

Vertical force Fz = 400 N

Inflation pressure Pi = 4 bar

MEASUREMENTS RAW VALUES

• Curvature force due to circular path.

• Elimination of curvature effects

MEASUREMENTS CORRECTED VALUES AS FUNCTION OF SIDESLIP

• Aligning torque due to asymmetric distribution of lateral force

and tends to realign the wheel.

MEASUREMENTS CORRECTED VALUES AS FUNCTION OF CAMBER

• Aligning torque due to vertical component of rotational velocity

and tends to twist the wheel.

GRAPHICAL OVERVIEW

Magic Formula tire model: curve fitting of measurement results

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

B = 1.0

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

B = 1.5

B = 2.0

C = 1.0

C = 1.5

C = 2.0

D = -1.0

D = -0.75

D = -0.5

E = -5

E = -10

E = -20

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

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

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

TIRE MODEL ALIGNING TORQUE

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

Tz = -t(α)·Fy(α) + Tzr(γ)

TIRE MODEL OVERTURNING TORQUE

Due to camber of the wheel:

Tx = Fz · rc · γ

TIRE MODEL SIMULATION ROTATING DISK TEST MACHINE

Camber γ is imposed

ϕ = ( + Ω·sin( )) / Vx 𝝍 𝜸

TIRE MODEL SIMULATION ROTATING DISK TEST MACHINE

Sideslip α is imposed

GRAPHICAL OVERVIEW

BICYCLE STABILITY BENCHMARK BICYCLE

Stability of an uncontrolled bicycle

Initial forward velocity V of 5 m/s

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

CONCLUSIONS & RECOMMENDATIONS RECOMMENDATIONS

• Include turnslip in the tire model

• Accuracy of the measured aligning torque

• Magic Formula coefficient estimation

• Longitudinal slip

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

Accelerating and braking

QUESTIONS?

FIGUREN

FIGUREN: TRAIL ABOUT STEER AXIS

bicycle tireroadcontact

tire tread

barmodelling tireroad

remodelling tireroad

modeling tireroad contactof

contact point

bicycle tirejeroen wijlens

lateral force fy

plane force

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