Vermelding onderdeel organisatie 1 Benchmark Results on the Stability of an Uncontrolled Bicycle Mechanics Seminar May 16, 2005 DAMTP, Cambridge University, UK Laboratory for Engineering Mechanics Faculty of Mechanical Engineering Arend L. Schwab Google: Arend Schwab [I’m Feeling Lucky]
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Vermelding onderdeel organisatie 1 Benchmark Results on the Stability of an Uncontrolled Bicycle Mechanics Seminar May 16, 2005DAMTP, Cambridge University,
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Vermelding onderdeel organisatie
1
Benchmark Results on the Stability of an Uncontrolled BicycleMechanics Seminar
May 16, 2005 DAMTP, Cambridge University, UK
Laboratory for Engineering MechanicsFaculty of Mechanical Engineering
Arend L. SchwabGoogle: Arend Schwab [I’m Feeling Lucky]
May 16, 2005 2
Acknowledgement
TUdelft:Jaap Meijaard 1
Cornell University:Andy RuinaJim Papadopoulos 2
Andrew Dressel
1) School of MMME, University of Nottingham, England, UK2) PCMC , Green Bay, Wisconsin, USA
May 16, 2005 3
Motto
Everbody knows how a bicycle is constructed …
… yet nobody fully understands its operation!
May 16, 2005 4
Experiment
Cornell University, Ithaca, NY, 1987: Yellow Bike in the Car Park
May 16, 2005 5
Experiment
Cornell University, Ithaca, NY, 1987: Yellow Bike in the Car Park
May 16, 2005 6
Experiment
Don’t try this at home !
May 16, 2005 7
Contents
• Bicycle Model• Equations of Motion• Steady Motion and Stability• Benchmark Results• Myth and Folklore• Steering• Conclusions
May 16, 2005 8
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
May 16, 2005 9
The Model
4 Bodies → 4*6 coordinates(rear wheel, rear frame (+rider), front frame, front wheel)
Constraints:3 Hinges → 3*5 on coordinates2 Contact Pnts → 2*1 on coordinates
ComparisonFor a standard and distinct type of bicycle + rigid rider combination
May 16, 2005 21
ComparePapadopoulos (1987) with Schwab (2003) and Meijaard (2003)
pencil & paper SPACAR software AUTOSIM software
Relative errors in the entries in M, C and K are
< 1e-12
Perfect Match!
21 0 2( ) ( ) 0d d dv v Mq C q K K q
May 16, 2005 22
MATLAB GUI for Linearized Stability
May 16, 2005 23
Myth & Folklore
A Bicycle is self-stable because:
- of the gyroscopic effect of the wheels !?
- of the effect of the positive trail !?
Not necessarily !
May 16, 2005 24
Myth & Folklore
Forward speedv = 3 [m/s]:
May 16, 2005 25
Steering a Bike
To turn right you have to steer …
briefly to the LEFT
and then let go of the handle bars.
May 16, 2005 26
Steering a BikeStandard 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.
May 16, 2005 27
Conclusions
- The Linearized Equations of Motion are Correct.
- A Bicycle can be Self-Stable even without Rotating Wheels and with Zero Trail.