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Benchmark Results on the Stability of an Uncontrolled Bicycle. Mechanics Seminar. Arend L. Schwab Google: Arend Schwab [I’m Feeling Lucky]. May 16, 2005 DAMTP, Cambridge University, UK. Laboratory for Engineering Mechanics Faculty of Mechanical Engineering. Acknowledgement.
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Benchmark Results on the Stability of an Uncontrolled Bicycle Mechanics Seminar Arend L. Schwab Google: Arend Schwab [I’m Feeling Lucky] May 16, 2005 DAMTP, Cambridge University, UK Laboratory for Engineering MechanicsFaculty of Mechanical Engineering
Acknowledgement Cornell University: Andy Ruina Jim Papadopoulos 2 Andrew Dressel TUdelft: Jaap Meijaard 1 • School of MMME, University of Nottingham, England, UK • PCMC , Green Bay, Wisconsin, USA
Motto Everbody knows how a bicycle is constructed … … yet nobody fully understands its operation!
Experiment Cornell University, Ithaca, NY, 1987: Yellow Bike in the Car Park
Experiment Cornell University, Ithaca, NY, 1987: Yellow Bike in the Car Park
Experiment Don’t try this at home !
Contents • Bicycle Model • Equations of Motion • Steady Motion and Stability • Benchmark Results • Myth and Folklore • Steering • Conclusions
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
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→ 2*2 on velocities Leaves:24-17 = 7independent Coordinates, and24-21 = 3independent Velocities (mobility) The system has: 3Degrees of Freedom, and4 (=7-3) Kinematic Coordinates
The Model 3 Degrees of Freedom: 4 Kinematic Coordinates: Input File with model definition:
Eqn’s of Motion For the degrees of freedom eqn’s of motion: and for kinematic coordinates nonholonomic constraints: State equations: with and
Steady Motion Steady motion: Stability of steady motion by linearized eqn’s of motion: and linearized nonholonomic constraints:
Linearized State State equations: Green:holonomic systems Linearized State equations: with and and
Straight Ahead Motion Upright, straight ahead motion : Turns out that the Linearized State eqn’s:
Straight Ahead Motion Linearized State eqn’s: Moreover, the lean angle j and the steer angle d are decoupled from the rear wheel rotation qr (forward speed ), resulting in: with
Stability of Straight Ahead Motion Linearized eqn’s of motion for lean and steering: with and the forward speed For a standard bicycle (Schwinn Crown) :
Root Loci Parameter: forward speed v v v Stable forward speed range4.1 < v < 5.7m/s
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 range4.1 < v < 5.7m/s
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 …
Comparison For a standard and distinct type of bicycle + rigid rider combination
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 < 1e-12 Perfect Match!
Myth & Folklore A Bicycle is self-stable because: - of the gyroscopic effect of the wheels !? - of the effect of the positive trail !? Not necessarily !
Myth & Folklore Forward speedv = 3 [m/s]:
Steering a Bike To turn right you have to steer … briefly to the LEFT and then let go of the handle bars.
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.
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.