1. Vermelding onderdeel organisatie 1 Benchmark Results on the Stability of an Uncontrolled Bicycle Mechanics Seminar May 16, 2005DAMTP, Cambridge University, UK Laboratory for Engineering Mechanics Faculty of Mechanical Engineering Arend L. Schwab Google: Arend Schwab [I’m Feeling Lucky]

2. May 16, 20052 Acknowledgement TUdelft: Jaap Meijaard 1 Cornell University: Andy Ruina Jim Papadopoulos 2 Andrew Dressel 1)School of MMME, University of Nottingham, England, UK 2)PCMC, Green Bay, Wisconsin, USA

3. May 16, 20053 Motto Everbody knows how a bicycle is constructed … … yet nobody fully understands its operation!

4. May 16, 20054 Experiment Cornell University, Ithaca, NY, 1987: Yellow Bike in the Car Park

5. May 16, 20055 Experiment Cornell University, Ithaca, NY, 1987: Yellow Bike in the Car Park

6. May 16, 20056 Experiment Don’t try this at home !

7. May 16, 20057 Contents Bicycle Model Equations of Motion Steady Motion and Stability Benchmark Results Myth and Folklore Steering Conclusions

8. May 16, 20058 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

9. May 16, 20059 The Model 4 Bodies → 4*6 coordinates (rear wheel, rear frame (+rider), front frame, front wheel) Constraints: 3 Hinges → 3*5 on coordinates 2 Contact Pnts → 2*1 on coordinates → 2*2 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

10. May 16, 200510 The Model 3 Degrees of Freedom: 4 Kinematic Coordinates: Input File with model definition:

11. May 16, 200511 Eqn’s of Motion State equations: with and For the degrees of freedom eqn’s of motion: and for kinematic coordinates nonholonomic constraints:

12. May 16, 200512 Steady Motion Steady motion: Stability of steady motion by linearized eqn’s of motion: and linearized nonholonomic constraints:

13. May 16, 200513 Linearized State Linearized State equations: State equations: with and Green: holonomic systems

14. May 16, 200514 Straight Ahead Motion Turns out that the Linearized State eqn’s: Upright, straight ahead motion :

15. May 16, 200515 Straight Ahead Motion Linearized State eqn’s: Moreover, the lean angle  and the steer angle  are decoupled from the rear wheel rotation  r (forward speed ), resulting in: wit h

16. May 16, 200516 Stability of Straight Ahead Motion with and the forward speed Linearized eqn’s of motion for lean and steering: For a standard bicycle (Schwinn Crown) :

17. May 16, 200517 Root Loci Parameter: forward speed v v v Stable forward speed range 4.1 < v < 5.7 m/s

18. May 16, 200518 Check Stability by full non-linear forward dynamic analysis Stable forward speed range 4.1 < v < 5.7 m/s forward speed v [m/s]: 0 1.75 3.5 3.68 4.9 6.3 4.5

19. May 16, 200519 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 …

20. May 16, 200520 Comparison For a standard and distinct type of bicycle + rigid rider combination

21. May 16, 200521 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!

22. May 16, 200522 MATLAB GUI for Linearized Stability

23. May 16, 200523 Myth & Folklore A Bicycle is self-stable because: - of the gyroscopic effect of the wheels !? - of the effect of the positive trail !? Not necessarily !

24. May 16, 200524 Myth & Folklore Forward speed v = 3 [m/s]:

25. May 16, 200525 Steering a Bike To turn right you have to steer … briefly to the LEFT and then let go of the handle bars.

26. May 16, 200526 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 second and then we let go of the handle bars.

27. May 16, 200527 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.