Vermelding onderdeel organisatie 1 A Multibody Dynamics Benchmark on the Equations of Motion of an Uncontrolled Bicycle Fifth EUROMECH Nonlinear Dynamics Conference ENOC-2005, Eindhoven, The Netherlands, 7-12 August 2005 Laboratory for Engineering Mechanics Faculty of Mechanical Engineering Delft University of Technology The Netherlands Arend L. Schwab Google: Arend Schwab [I’m Feeling Lucky]
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Vermelding onderdeel organisatie 1 A Multibody Dynamics Benchmark on the Equations of Motion of an Uncontrolled Bicycle Fifth EUROMECH Nonlinear Dynamics.
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Vermelding onderdeel organisatie
1
A Multibody Dynamics Benchmark on the Equations of Motion of an Uncontrolled Bicycle
Fifth EUROMECH Nonlinear Dynamics ConferenceENOC-2005, Eindhoven, The Netherlands, 7-12 August 2005
Laboratory for Engineering MechanicsFaculty of Mechanical EngineeringDelft University of Technology The Netherlands
Arend L. SchwabGoogle: Arend Schwab [I’m Feeling Lucky]
Aug 9, 2005 2
Acknowledgement
TUdelft:Jaap Meijaard 1
Jodi Kooiman
Cornell University:Andy RuinaJim Papadopoulos 2
Andrew Dressel
1) School of MMME, University of Nottingham, England, UK2) PCMC , Green Bay, Wisconsin, USA
Aug 9, 2005 3
Motto
Everybody knows how a bicycle is constructed …
… yet nobody fully understands its operation!
Aug 9, 2005 5
Experiment
Cornell University, Ithaca, NY, 1987: Yellow Bike in the Car Park
Aug 9, 2005 6
Some Advice
Don’t try this at home !
Aug 9, 2005 7
Contents
• Bicycle Model• Equations of Motion• Steady Motion and Stability• Benchmark Results• Experimental Validation• Conclusions
Aug 9, 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
Aug 9, 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
Aug 9, 2005 21
ComparePapadopoulos (1987) with Schwab (2003) and Meijaard (2003)
1: Pencil & Paper 2: SPACAR software 3: 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
Aug 9, 2005 22
Experimental Validation
Instrumented Bicycle, uncontrolled
2 rate gyros:
-lean rate
-yaw rate
1 speedometer:
-forward speed
1 potentiometer
-steering angle
Laptop + Labview
Aug 9, 2005 23
Experimental Validation
Linearized stability of the Uncontrolled Instrumented Bicycle
Stable forward speed range:
4.0 < v < 7.8 [m/s]
Aug 9, 2005 24
An Experiment
Aug 9, 2005 25
Measured Data
Aug 9, 2005 26
Extract EigenvaluesStable Weave motion is dominant
Nonlinear fit function on the lean rate:
11 2 2 3 2e [ cos( ) sin( )]tc c t c t
Aug 9, 2005 27
Extract Eigenvalues & Compare
Nonlinear fit function on the lean rate:
11 2 2 3 2e [ cos( ) sin( )]tc c t c t
2 = 5.52 [rad/s]
1 = -1.22 [rad/s]
forward speed:
4.9 < v <5.4 [m/s]
Aug 9, 2005 28
Compare around critical weave speed
Aug 9, 2005 29
Just below critical weave speed
Aug 9, 2005 30
Compare at high and low speed
Aug 9, 2005 31
Conclusions
- The Linearized Equations of Motion are Correct.
Future Investigation:
- Add a controller to the instrumented bicycle -> robot bike.
- Investigate stability of steady cornering.
Aug 9, 2005 32
MATLAB GUI for Linearized Stability
Aug 9, 2005 33
Myth & Folklore
A Bicycle is self-stable because:
- of the gyroscopic effect of the wheels !?
- of the effect of the positive trail !?
Not necessarily !
Aug 9, 2005 34
Myth & Folklore
Forward speedv = 3 [m/s]:
Aug 9, 2005 35
Steering a Bike
To turn right you have to steer …
briefly to the LEFT
and then let go of the handle bars.
Aug 9, 2005 36
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.
Aug 9, 2005 37
Conclusions
- The Linearized Equations of Motion are Correct.
- A Bicycle can be Self-Stable even without Rotating Wheels and with Zero Trail.
Future Investigation:
- Validate the modelling assumptions by means of experiments.