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 UniversityAndy Ruina

Jim Papadopoulos1

Andrew Dressel

Delft UniversityJaap 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.

4

Contents

- The Model

- FEM Modelling

- Equations of Motion

- Steady Motion and Stability

- A Comparison

- Myth and Folklore

- Conclusions

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

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

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

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)

9

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)

11

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)

12

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

yx

f

k

q

Input File with model definition

13

Eqn’s of Motion

bqAq

fM

qqq

d

d

k

d

d

t

1

dd

State 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

14

Steady Motion

constantconstant

0

dd

k

d

d

tqqqSteady motion:

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

and linearized nonholonomic constraints

0qKqKqCqM kkdddd

kkdddk qBqBqAq

15

Linearized State

000

qqq

BBA00I

KKC

qqq

I000I000M

k

d

d

kd

kd

k

d

d

bqAq

fM

qqq

d

d

k

d

d

t

1

dd

Linearized State equations:

State equations:

with dTT

qMhTCTTC ,

qqqq CvMhTfxMTKFTKKK ,,,, TTTkd and

and qbBBB , kd

16

Straight Ahead Motion

000

qqq

BBA00I

KKC

qqq

I000I000M

k

d

d

kd

kd

k

d

d

Turns out that the Linearized State eqn’s:

Upright, straight ahead motion :

constant/ speed rot. rear wheel0 anglesteer 0 angle lean

rvr

0

17

Straight Ahead Motion

000

qqq

BBA00I

KKC

qqq

I000I000M

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

0000xx0xx

,0000xx0xx

,x000xx0xx

dKCM

18

Stability of the Motion

with and the forward speed

Linearized eqn’s of motion:

7.20960

,8.827

271003 ,

8.16.0400

,3.033130

201 KKCM

0)()( 2201 ddd vv qKKqCqM

steer leandq rv r

For a standard bicycle (Schwinn Crown) we have:

19

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

20

Check StabilityFull Non-Linear Forward Dynamic Analysis

with the same SPACAR model at different speeds.

ForwardSpeedv [m/s]:

01.75

3.53.68

4.96.3

Stable speed range 4.1 < v < 5.7 m/s

4.5

21

CompareA 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

ComparePapadopoulos & Hand (1988)

0)()( 2201 ddd vv qKKqCqM

Papadopoulos & Schwab (2003): JBike6

MATLAB m-file for M, C1 K0 and K2

23

ComparePapadopoulos (1987) with SPACAR (2003)

0)()( 2201 ddd vv qKKqCqM

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.