Benchmark Results on the Stability of an Uncontrolled Bicycle

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Benchmark Results on the Stability of an
Uncontrolled Bicycle

• Mechanics Seminar

Arend L. Schwab Google Arend Schwab Im
Feeling Lucky
May 16, 2005 DAMTP, Cambridge University, UK
Laboratory for Engineering MechanicsFaculty of
Mechanical Engineering
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Acknowledgement

• TUdelft
• Jaap Meijaard 1

Cornell University Andy Ruina Jim Papadopoulos
2 Andrew Dressel

• School of MMME, University of Nottingham,
  England, UK
• PCMC , Green Bay, Wisconsin, USA

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Motto
Everbody knows how a bicycle is constructed
yet nobody fully understands its operation!
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Experiment
Cornell University, Ithaca, NY, 1987 Yellow Bike
in the Car Park
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Experiment
Cornell University, Ithaca, NY, 1987 Yellow Bike
in the Car Park
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Experiment
Dont try this at home !
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Contents

• Bicycle Model
• Equations of Motion
• Steady Motion and Stability
• Benchmark Results
• Myth and Folklore
• Steering
• Conclusions

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The Model

• Modelling Assumptions
• rigid bodies
• fixed rigid rider
• hands-free
• symmetric about vertical plane
• point contact, no side slip
• flat level road
• no friction or propulsion

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The Model
4 Bodies ? 46 coordinates(rear wheel, rear
frame (rider), front frame, front
wheel) Constraints3 Hinges ? 35 on
coordinates2 Contact Pnts ? 21 on
coordinates ? 22 on velocities
Leaves 24-17 7 independent Coordinates,
and 24-21 3 independent Velocities (mobility)
The system has 3 Degrees of Freedom,
and 4 (7-3) Kinematic Coordinates
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The Model
3 Degrees of Freedom
4 Kinematic Coordinates
Input File with model definition
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Eqns of Motion
For the degrees of freedom eqns of motion
and for kinematic coordinates nonholonomic
constraints
State equations
with
and
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Steady Motion
Steady motion
Stability of steady motion by linearized eqns of
motion
and linearized nonholonomic constraints
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Linearized State
State equations
Green holonomic systems
Linearized State equations
with
and
and
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Straight Ahead Motion
Upright, straight ahead motion
Turns out that the Linearized State eqns
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Straight Ahead Motion
Linearized State eqns
Moreover, the lean angle j and the steer angle
d are decoupled from the rear wheel rotation qr
(forward speed ), resulting in
with
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Stability of Straight Ahead Motion
Linearized eqns of motion for lean and steering
with and the forward
speed
For a standard bicycle (Schwinn Crown)
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Root Loci
Parameter forward speed
v
v
v
Stable forward speed range 4.1 lt v lt 5.7 m/s
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Check Stability
by full non-linear forward dynamic analysis
forward speedv m/s
6.3
4.9
4.5
3.68
3.5
1.75
0
Stable forward speed range 4.1 lt v lt 5.7 m/s
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Comparison
A Brief History of Bicycle Dynamics Equations

• 1899 Whipple- 1901 Carvallo- 1903 Sommerfeld
  Klein- 1948 Timoshenko, Den Hartog- 1955
  Döhring- 1967 Neimark Fufaev- 1971 Robin
  Sharp- 1972 Weir- 1975 Kane- 1983 Koenen-
  1987 Papadopoulos
• and many more

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Comparison
For a standard and distinct type of bicycle
rigid rider combination
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Compare
Papadopoulos (1987) with Schwab (2003) and
Meijaard (2003) pencil paper
SPACAR software AUTOSIM software
Relative errors in the entries in M, C and K are
lt 1e-12 Perfect Match!
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MATLAB GUI for Linearized Stability
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Myth Folklore
A Bicycle is self-stable because
- of the gyroscopic effect of the wheels !?
- of the effect of the positive trail !?
Not necessarily !
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Myth Folklore
Forward speedv 3 m/s
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Steering a Bike
To turn right you have to steer
briefly to the LEFT and then let go of the
handle bars.
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Steering a Bike
Standard bike with rider at a stable forward
speed of 5 m/s, after 1 second we apply a steer
torque of 1 Nm for ½ a secondand then we let go
of the handle bars.
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Conclusions
- The Linearized Equations of Motion are
Correct. - A Bicycle can be Self-Stable even
without Rotating Wheels and with Zero Trail.
Future Investigation
- Add a human controler to the model. -
Investigate stability of steady cornering.