1 / 27

Stability of an Uncontrolled Bicycle

Delft University of Technology Laboratory for Engineering Mechanics Mechanical Engineering. Stability of an Uncontrolled Bicycle. Arend L. Schwab Laboratory for Engineering Mechanics Delft University of Technology The Netherlands.

andren
Download Presentation

Stability of an Uncontrolled Bicycle

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. Acknowledgement Cornell University Andy Ruina Jim Papadopoulos1 Andrew Dressel Delft University Jaap Meijaard2 • PCMC , Green Bay, Wisconsin, USA • School of 4M, University of Nottingham, England, UK

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

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

  5. The Model assumptions • 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

  6. The Model counting 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 = 7 Independent Coordinates, and24-21 = 3 Independent Velocities (mobility) The system has: 3 Degrees of Freedom, and4(=7-3) Kinematic Coordinates

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  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. Compare Papadopoulos & Hand (1988) MATLAB m-file for M, C1 K0 and K2 Papadopoulos & Schwab (2003): JBike6

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

  24. JBike6 MATLAB GUI

  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. Funny Bike ForwardSpeedv [m/s]: 3

  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.

More Related