A Multibody Dynamics Benchmark on the Equations of Motion of an Uncontrolled Bicycle

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 Arend L. Schwab Google: Arend Schwab [I’m Feeling Lucky] Laboratory for Engineering MechanicsFaculty of Mechanical Engineering Delft University of Technology The Netherlands

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

3. Motto Everybody knows how a bicycle is constructed … … yet nobody fully understands its operation!

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

5. Some Advice Don’t try this at home !

6. Contents • Bicycle Model • Equations of Motion • Steady Motion and Stability • Benchmark Results • Experimental Validation • Conclusions

7. 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

8. 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

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

10. Eqn’s of Motion For the degrees of freedom eqn’s of motion: and for kinematic coordinates nonholonomic constraints: State equations: with and

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

12. Linearized State State equations: Green:holonomic systems Linearized State equations: with and and

13. Straight Ahead Motion Upright, straight ahead motion : Turns out that the Linearized State eqn’s:

14. 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

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

16. Root Loci Parameter: forward speed v v v Stable forward speed range4.1 < v < 5.7m/s

17. 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

18. 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 …

19. Comparison For a standard and distinct type of bicycle + rigid rider combination

20. 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!

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

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

23. An Experiment

24. Measured Data

25. Extract Eigenvalues Stable Weave motion is dominant Nonlinear fit function on the lean rate:

26. Extract Eigenvalues & Compare l2 = 5.52 [rad/s] l1 = -1.22 [rad/s] forward speed: 4.9 < v <5.4 [m/s] Nonlinear fit function on the lean rate:

27. Compare around critical weave speed

28. Just below critical weave speed

29. Compare at high and low speed

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

31. MATLAB GUI for Linearized Stability

32. Myth & Folklore A Bicycle is self-stable because: - of the gyroscopic effect of the wheels !? - of the effect of the positive trail !? Not necessarily !

33. Myth & Folklore Forward speedv = 3 [m/s]:

34. Steering a Bike To turn right you have to steer … briefly to the LEFT and then let go of the handle bars.

35. 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.

36. 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.