1. 1 Stability of an Uncontrolled Bicycle Delft University of Technology Laboratory for Engineering Mechanics Mechanical Engineering Dynamics Seminar, University of Nottingham, School of 4M, Oct 24, 2003 Arend L. Schwab Laboratory for Engineering Mechanics Delft University of Technology The Netherlands

2. 2 Acknowledgement Cornell University Andy Ruina Jim Papadopoulos 1 Andrew Dressel Delft University Jaap Meijaard 2 1)PCMC, Green Bay, Wisconsin, USA 2)School of 4M, University of Nottingham, England, UK

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

4. 4 Contents - The Model - FEM Modelling - Equations of Motion - Steady Motion and Stability - A Comparison - Myth and Folklore - Conclusions

5. 5 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 assumptions

6. 6 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 counting 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

7. 7 The SPACAR Model SPACAR Software for Kinematic and Dynamic Analysis of Flexible Multibody Systems; a Finite Element Approach. FEM-model : 2 Wheels, 2 Beams, 6 Hinges

8. 8 4 Nodal Coordinates: 2D Truss Element 3 Degrees of Freedom as a Rigid Body leaves: 1 Generalized Strain: Rigid Body Motion  this is the Constraint Equation FEM modelling (intermezzo)

9. 9 Generalized Nodes: Position Wheel Centre Contact Point Euler parameters Rotation Matrix: R(q) Wheel Element Rigid body pure rolling: 3 degrees of freedom In total 10 generalized coordinates Impose 7 Constraints Nodes (intermezzo)

10. 10 Radius vector: Rotated wheel axle: Normal on surface: Surface: Holonomic Constraints as zero generalized strains Strains Wheel Element Elongation: Lateral Bending: Contact point on the surface: Wheel perpendicular to the surface Normalization condition on Euler par: (intermezzo)

11. 11 Non-Holonomic Constraints as zero generalized slips Wheel Element Slips Generalized Slips: Velocity of material point of wheel at contact in c: Longitudinal slip Lateral slip Two tangent vectors in c: Radius vector: Angular velocity wheel: (intermezzo)

12. 12 The Model 3 Degrees of Freedom: 4 Kinematic Coordinates: Input File with model definition

13. 13 Eqn’s of Motion State equations: with and For the degrees of freedom eqn’s of motion: and for kinematic coordinates nonholonomic constraints:

14. 14 Steady Motion Steady motion: Stability of steady motion by linearized eqn’s of motion, and linearized nonholonomic constraints

15. 15 Linearized State Linearized State equations: State equations: with and

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

17. 17 Straight Ahead Motion in the Linearized State eqn’s: Moreover, the lean angle  and the steer angle  are decoupled from the rear wheel rotation  r (forward speed).

18. 18 Stability of the Motion with and the forward speed Linearized eqn’s of motion: For a standard bicycle (Schwinn Crown) we have:

19. 19 Root Loci Root Loci from the Linearized Equations of Motion, Parameter: forward speed Stable speed range 4.1 < v < 5.7 m/s v v v

20. 20 Check Stability Full Non-Linear Forward Dynamic Analysis with the same SPACAR model at different speeds. Forward Speed v [m/s]: 0 1.75 3.5 3.68 4.9 6.3 Stable speed range 4.1 < v < 5.7 m/s 4.5

21. 21 Compare 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 - 1987 Papadopoulos - and many more …

22. 22 Compare Papadopoulos & Hand (1988) Papadopoulos & Schwab (2003): JBike6 MATLAB m-file for M, C 1 K 0 and K 2

23. 23 Compare Papadopoulos (1987) with SPACAR (2003) Perfect Match, Relative Differences < 1e-12 !

24. 24 JBike6 MATLAB GUI

25. 25 Myth & Folklore A Bicycle is self-stable because: of the gyroscopic effect of the wheels !? of the effect of the positive trail !? Not necessarily !

26. 26 Funny Bike Forward Speed v [m/s]: 3

27. 27 Conclusions The Linearized Equations of Motion are Correct. A Bicycle can be Self-Stable even without Rotating Wheels and with Zero Trail. Further Investigation: Add a human controler to the model. Investigate stability of steady cornering.