1. 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]

2. Aug 9, 20052 Acknowledgement TUdelft: Jaap Meijaard 1 Jodi Kooiman 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. Aug 9, 20053 Motto Everybody knows how a bicycle is constructed … … yet nobody fully understands its operation!

4. Aug 9, 20054 Experiment Cornell University, Ithaca, NY, 1987: Yellow Bike in the Car Park

5. Aug 9, 20055 Experiment Cornell University, Ithaca, NY, 1987: Yellow Bike in the Car Park

6. Aug 9, 20056 Some Advice Don’t try this at home !

7. Aug 9, 20057 Contents Bicycle Model Equations of Motion Steady Motion and Stability Benchmark Results Experimental Validation Conclusions

8. Aug 9, 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. Aug 9, 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. Aug 9, 200510 The Model 3 Degrees of Freedom: 4 Kinematic Coordinates: Input File with model definition:

11. Aug 9, 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. Aug 9, 200512 Steady Motion Steady motion: Stability of steady motion by linearized eqn’s of motion: and linearized nonholonomic constraints:

13. Aug 9, 200513 Linearized State Linearized State equations: State equations: with and Green: holonomic systems

14. Aug 9, 200514 Straight Ahead Motion Turns out that the Linearized State eqn’s: Upright, straight ahead motion :

15. Aug 9, 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. Aug 9, 200516 Stability of Straight Ahead Motion with and a constant forward speed Linearized eqn’s of motion for lean and steering: For a standard bicycle (Schwinn Crown) :

17. Aug 9, 200517 Root Loci Parameter: forward speed v v v Stable forward speed range 4.1 < v < 5.7 m/s

18. Aug 9, 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. Aug 9, 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. Aug 9, 200520 Comparison For a standard and distinct type of bicycle + rigid rider combination

21. Aug 9, 200521 Compare Papadopoulos (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!

22. Aug 9, 200522 Experimental Validation Instrumented Bicycle, uncontrolled 2 rate gyros: -lean rate -yaw rate 1 speedometer: -forward speed 1 potentiometer -steering angle Laptop + Labview

23. Aug 9, 200523 Experimental Validation Linearized stability of the Uncontrolled Instrumented Bicycle Stable forward speed range: 4.0 < v < 7.8 [m/s]

24. Aug 9, 200524 An Experiment

25. Aug 9, 200525 Measured Data

26. Aug 9, 200526 Extract Eigenvalues Stable Weave motion is dominant Nonlinear fit function on the lean rate:

27. Aug 9, 200527 Extract Eigenvalues & Compare Nonlinear fit function on the lean rate: 2 = 5.52 [rad/s] 1 = -1.22 [rad/s] forward speed: 4.9 < v <5.4 [m/s]

28. Aug 9, 200528 Compare around critical weave speed

29. Aug 9, 200529 Just below critical weave speed

30. Aug 9, 200530 Compare at high and low speed

31. Aug 9, 200531 Conclusions - The Linearized Equations of Motion are Correct. Future Investigation: - Add a controller to the instrumented bicycle -> robot bike. - Investigate stability of steady cornering.

32. Aug 9, 200532 MATLAB GUI for Linearized Stability

33. Aug 9, 200533 Myth & Folklore A Bicycle is self-stable because: - of the gyroscopic effect of the wheels !? - of the effect of the positive trail !? Not necessarily !

34. Aug 9, 200534 Myth & Folklore Forward speed v = 3 [m/s]:

35. Aug 9, 200535 Steering a Bike To turn right you have to steer … briefly to the LEFT and then let go of the handle bars.

36. Aug 9, 200536 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.

37. Aug 9, 200537 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. - Add a human controller to the model. - Investigate stability of steady cornering.