Stability of an Uncontrolled Bicycle

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

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

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

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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

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The Model
counting
4 Bodies ? 46 coordinates(rear wheel, rear
frame (rider), front frame, front
wheel) Constraints3 Hinges ? 35 on
coordinates2 Contact Pnts ? 21 on
coordinates ? 22 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
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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
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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)
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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)
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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)
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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)
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The Model
3 Degrees of Freedom
4 Kinematic Coordinates
Input File with model definition
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Eqns of Motion
For the degrees of freedom eqns of
motion
and for kinematic coordinates nonholonomic
constraints
State equations
and
with
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Steady Motion
Steady motion
Stability of steady motion by linearized eqns of
motion,
and linearized nonholonomic constraints
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Linearized State
State equations
Linearized State equations
with
and
and
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Straight Ahead Motion
Upright, straight ahead motion
Turns out that the Linearized State eqns
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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 eqns
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Stability of the Motion
Linearized eqns of motion
with and the forward
speed
For a standard bicycle (Schwinn Crown) we have
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Root Loci
Root l Loci from the Linearized Equations of
Motion, Parameter forward speed
v
v
v
Stable speed range 4.1 lt v lt 5.7 m/s
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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 lt v lt 5.7 m/s
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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

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Compare
Papadopoulos Hand (1988)
MATLAB m-file for M, C1 K0 and K2
Papadopoulos Schwab (2003) JBike6
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Compare
Papadopoulos (1987) with SPACAR (2003)
Perfect Match, Relative Differences lt 1e-12 !
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JBike6 MATLAB GUI
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Myth Folklore
A Bicycle is self-stable because
of the gyroscopic effect of the wheels !?
of the effect of the positive trail !?
Not necessarily !
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Funny Bike
ForwardSpeedv m/s
3
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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.