1

Stability of an Uncontrolled Bicycle

Delft University of TechnologyLaboratory for Engineering Mechanics

Mechanical Engineering

Dynamics Seminar, University of Nottingham, School of 4M, Oct 24, 2003

Arend L. SchwabLaboratory for Engineering Mechanics

Delft University of TechnologyThe Netherlands

2

Acknowledgement

Cornell University

Andy Ruina

Jim Papadopoulos1

Andrew Dressel

Delft University

Jaap Meijaard2

1) PCMC , Green Bay, Wisconsin, USA2) School of 4M, University of Nottingham, England, UK

3

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

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

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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

counting

Leaves: 24-17 = 7 Independent Coordinates, and24-21 = 3 Independent Velocities (mobility)

The system has: 3 Degrees of Freedom, and4 (=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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4 Nodal Coordinates:

2D Truss Element

),,,( 2211 yxyxx

3 Degrees of Freedom as a Rigid Body leaves:

1 Generalized Strain:

)( 02

122

12 xDε lyyxxl

Rigid Body Motion this is the Constraint Equation

0ε

0lll

FEM modelling

(intermezzo)

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Generalized Nodes:

Position Wheel Centre

Contact Point

Euler parameters

Rotation Matrix: R(q)

),,( zyx wwww

Wheel Element

),,,,( 3210 qqqqq

),,( zyx cccc

Rigid body pure rolling: 3 degrees of freedom

In total 10 generalized coordinates

Impose 7 Constraints

Nodes

(intermezzo)

10

)2/()( 02

01 rr rr

re w2

)(3 cg

ner )(4 wRadius vector:

Rotated wheel axle:

Normal on surface:

wcr

Surface:

ww eR(q)e

0)( xg

)(cn g

Holonomic Constraints as zero generalized strains

StrainsWheel Element

0xDε )(

Elongation:

Lateral Bending:

Contact point on the surface:

Wheel perpendicular to the surface

Normalization condition on Euler par: 12

05 qqq(intermezzo)

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Non-Holonomic Constraints as zero generalized slips

Wheel Element

0xxVs )(

Slips

Generalized Slips:

cs va1

cb 2s

Velocity of material point of wheel at contact in c:

rωwv c

Longitudinal slip

Lateral slip

Two tangent vectors in c:

)( , )( ww ernbera

Radius vector: wcr

Angular velocity wheel: ω(intermezzo)

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The Model

3 Degrees of Freedom:

4 Kinematic Coordinates:

r

d

rot. rear wheel

anglesteer

angle lean

q

r

r

pnt.contact rear

pnt.contact rear

framerear angleyaw

rot. lfront whee

y

x

f

k

q

Input File with model definition

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Eqn’s of Motion

bqA

q

fM

q

q

q

d

d

k

d

d

t

1

d

dState equations:

with MTTM T and MhfTf T

For the degrees of freedom eqn’s of motion:

and for kinematic coordinates nonholonomic constraints:

dq

kq

MhfTqMTT TdT

bqAq dk

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Steady Motion

constant

constant

0

d

d

k

d

d

tq

q

qSteady motion:

Stability of steady motion by linearized eqn’s of motion,

and linearized nonholonomic constraints

0qKqKqCqM kkdddd

kkdddk qBqBqAq

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Linearized State

0

0

0

q

q

q

BBA

00I

KKC

q

q

q

I00

0I0

00M

k

d

d

kd

kd

k

d

d

bqA

q

fM

q

q

q

d

d

k

d

d

t

1

d

d

Linearized State equations:

State equations:

with dTT

qMhTCTTC ,

qqqq CvMhTfxMTKFTKKK ,,,, TTTkd and

and qbBBB , kd

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Straight Ahead Motion

0

0

0

q

q

q

BBA

00I

KKC

q

q

q

I00

0I0

00M

k

d

d

kd

kd

k

d

d

Turns out that the Linearized State eqn’s:

Upright, straight ahead motion :

constant/ speed rot. rear wheel

0 anglesteer

0 angle lean

rvr

0

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Straight Ahead Motion

0

0

0

q

q

q

BBA

00I

KKC

q

q

q

I00

0I0

00M

k

d

d

kd

kd

k

d

d

in the Linearized State eqn’s:

Moreover, the lean angle and the steer angle are decoupled from the rear wheel rotation r (forward speed).

0

000

0xx

0xx

,

000

0xx

0xx

,

x00

0xx

0xxdKCM

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Stability of the Motion

with and the forward speed

Linearized eqn’s of motion:

7.20

960 ,

8.827

271003 ,

8.16.0

400 ,

3.03

3130201 KKCM

0)()( 2201 ddd vv qKKqCqM

steer

leandq rv r

For a standard bicycle (Schwinn Crown) we have:

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Root LociRoot Loci from the Linearized Equations of Motion,

Parameter: forward speed rv r

Stable speed range 4.1 < v < 5.7 m/s

v

vv

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Check StabilityFull Non-Linear Forward Dynamic Analysis

with the same SPACAR model at different speeds.

ForwardSpeedv [m/s]:

01.75

3.53.68

4.9

6.3

Stable speed range 4.1 < v < 5.7 m/s

4.5

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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)

0)()( 2201 ddd vv qKKqCqM

Papadopoulos & Schwab (2003): JBike6

MATLAB m-file for M, C1 K0 and K2

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Compare

Papadopoulos (1987) with SPACAR (2003)

0)()( 2201 ddd vv qKKqCqM

Perfect Match, Relative Differences < 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.