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]

Aug 9, 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

Aug 9, Motto Everybody knows how a bicycle is constructed … … yet nobody fully understands its operation!

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

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

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

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

Aug 9, 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

Aug 9, 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 = 3 independent Velocities (mobility) The system has: 3 Degrees of Freedom, and 4 (=7-3) Kinematic Coordinates

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

Aug 9, Eqn’s of Motion State equations: with and For the degrees of freedom eqn’s of motion: and for kinematic coordinates nonholonomic constraints:

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

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

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

Aug 9, 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

Aug 9, 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) :

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

Aug 9, 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]:

Aug 9, Comparison A Brief History of Bicycle Dynamics Equations Whipple Carvallo Sommerfeld & Klein Timoshenko, Den Hartog Döhring Neimark & Fufaev Robin Sharp Weir Kane Koenen Papadopoulos - and many more …

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

Aug 9, 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!

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

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

Aug 9, An Experiment

Aug 9, Measured Data

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

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

Aug 9, Compare around critical weave speed

Aug 9, Just below critical weave speed

Aug 9, Compare at high and low speed

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

Aug 9, MATLAB GUI for Linearized Stability

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

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

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

Aug 9, 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.

Aug 9, 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.