bicycle tireroadcontact
TRANSCRIPT
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MODELING TIRE-ROAD CONTACT OF A BICYCLE TIRE
JEROEN WIJLENS
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Background
Introduction
Measurements
Tire model
Bicycle stability
Conclusions & recommendations
2
OVERVIEW
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BACKGROUND BICYCLE ACCIDENTS
Source: NOS
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BACKGROUND SOFIE PROJECT
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BACKGROUND SOFIE PROJECT
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BACKGROUND SOFIE PROJECT
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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.
8
INTRODUCTION BICYCLE TIRES
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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.
9
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
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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
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V
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INTRODUCTION SIDESLIP
V
x
z
y
α
Tz
Fz Fy
Vr
Ω
Vs
α = Vsy / Vx
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V
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INTRODUCTION
Fx
x
z
y
Fy
Tx
Ty
Tz
Ω
Fz
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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
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V
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INTRODUCTION
Ω
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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
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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
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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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TIRE MODEL MAGIC FORMULA
B = 1.0
y(x) = D·sin[C·arctan{B·x - E·(B·x - arctan(B·x ))}]
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TIRE MODEL MAGIC FORMULA
B = 1.5
y(x) = D·sin[C·arctan{B·x - E·(B·x - arctan(B·x ))}]
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TIRE MODEL MAGIC FORMULA
B = 2.0
y(x) = D·sin[C·arctan{B·x - E·(B·x - arctan(B·x ))}]
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TIRE MODEL MAGIC FORMULA
C = 1.0
y(x) = D·sin[C·arctan{B·x - E·(B·x - arctan(B·x ))}]
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TIRE MODEL MAGIC FORMULA
C = 1.5
y(x) = D·sin[C·arctan{B·x - E·(B·x - arctan(B·x ))}]
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TIRE MODEL MAGIC FORMULA
C = 2.0
y(x) = D·sin[C·arctan{B·x - E·(B·x - arctan(B·x ))}]
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TIRE MODEL MAGIC FORMULA
D = -1.0
y(x) = D·sin[C·arctan{B·x - E·(B·x - arctan(B·x ))}]
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TIRE MODEL MAGIC FORMULA
D = -0.75
y(x) = D·sin[C·arctan{B·x - E·(B·x - arctan(B·x ))}]
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TIRE MODEL MAGIC FORMULA
D = -0.5
y(x) = D·sin[C·arctan{B·x - E·(B·x - arctan(B·x ))}]
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TIRE MODEL MAGIC FORMULA
E = -5
y(x) = D·sin[C·arctan{B·x - E·(B·x - arctan(B·x ))}]
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TIRE MODEL MAGIC FORMULA
E = -10
y(x) = D·sin[C·arctan{B·x - E·(B·x - arctan(B·x ))}]
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TIRE MODEL MAGIC FORMULA
E = -20
y(x) = D·sin[C·arctan{B·x - E·(B·x - arctan(B·x ))}]
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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
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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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BICYCLE STABILITY BENCHMARK BICYCLE
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BICYCLE STABILITY BENCHMARK BICYCLE
Stability of an uncontrolled bicycle
Initial forward velocity V of 5 m/s
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BICYCLE STABILITY BENCHMARK BICYCLE
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BICYCLE STABILITY BENCHMARK BICYCLE
Stability of an uncontrolled bicycle
Initial forward velocity V of 3 m/s
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CONCLUSIONS & RECOMMENDATIONS CONCLUSIONS
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• 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
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• Include turnslip in the tire model
• Accuracy of the measured aligning torque
• Magic Formula coefficient estimation
• Longitudinal slip
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BICYCLE STABILITY BENCHMARK BICYCLE
Difference between Magic Formula tires and non-slipping rolling point-contact tires
Accelerating and braking
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BICYCLE STABILITY BENCHMARK BICYCLE
Stability of an uncontrolled bicycle
Initial forward velocity V of 5 m/s
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QUESTIONS?
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FIGUREN
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FIGUREN
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FIGUREN
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FIGUREN
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FIGUREN
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FIGUREN
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FIGUREN
21 December 2012 Modelling tire-road contact of a bicycle tire
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FIGUREN
21 December 2012 Modelling tire-road contact of a bicycle tire
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FIGUREN
21 December 2012 Modelling tire-road contact of a bicycle tire
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FIGUREN
21 December 2012 Modelling tire-road contact of a bicycle tire
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FIGUREN
21 December 2012 Modelling tire-road contact of a bicycle tire
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FIGUREN
21 December 2012 Modelling tire-road contact of a bicycle tire
![Page 74: Bicycle tireroadcontact](https://reader033.vdocuments.mx/reader033/viewer/2022051212/55872fe0d8b42a16218b46a9/html5/thumbnails/74.jpg)
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FIGUREN
21 December 2012 Modelling tire-road contact of a bicycle tire
![Page 75: Bicycle tireroadcontact](https://reader033.vdocuments.mx/reader033/viewer/2022051212/55872fe0d8b42a16218b46a9/html5/thumbnails/75.jpg)
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FIGUREN: TRAIL ABOUT STEER AXIS
21 December 2012 Modelling tire-road contact of a bicycle tire