Date post: | 19-Dec-2015 |
Category: |
Documents |
View: | 221 times |
Download: | 2 times |
Vermelding onderdeel organisatie
1
A Multibody Dynamics Benchmark on the Equations of Motion of an Uncontrolled Bicycle
Fifth EUROMECH Nonlinear Dynamics ConferenceENOC-2005, Eindhoven, The Netherlands, 7-12 August 2005
Laboratory for Engineering MechanicsFaculty of Mechanical EngineeringDelft University of Technology The Netherlands
Arend L. SchwabGoogle: Arend Schwab [I’m Feeling Lucky]
Aug 9, 2005 2
Acknowledgement
TUdelft:Jaap Meijaard 1
Jodi Kooiman
Cornell University:Andy RuinaJim Papadopoulos 2
Andrew Dressel
1) School of MMME, University of Nottingham, England, UK2) PCMC , Green Bay, Wisconsin, USA
Aug 9, 2005 3
Motto
Everybody knows how a bicycle is constructed …
… yet nobody fully understands its operation!
Aug 9, 2005 5
Experiment
Cornell University, Ithaca, NY, 1987: Yellow Bike in the Car Park
Aug 9, 2005 6
Some Advice
Don’t try this at home !
Aug 9, 2005 7
Contents
• Bicycle Model• Equations of Motion• Steady Motion and Stability• Benchmark Results• Experimental Validation• Conclusions
Aug 9, 2005 8
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, 2005 9
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
Leaves: 24-17 = 7 independent Coordinates, and24-21 = 3 independent Velocities (mobility)
The system has: 3 Degrees of Freedom, and4 (=7-3) Kinematic Coordinates
Aug 9, 2005 10
The Model
3 Degrees of Freedom:
4 Kinematic Coordinates:
lean angle
steer angle
rear wheel rot.
d
r
q
r
r
front wheel rot.
yaw angle rear frame
rear contact pnt.
rear contact pnt.
f
k
x
y
q
Input File with model definition:
Aug 9, 2005 11
Eqn’s of Motion
1
dd
d
d d
k dt
q M f
q q
q Aq b
State equations:
with TM T MT and T f T f Mh
For the degrees of freedom eqn’s of motion:
and for kinematic coordinates nonholonomic constraints:
dq
kq
T d T T MTq T f Mh
k d q Aq b
Aug 9, 2005 12
Steady Motion
0d
constantd
constant
d
d
kt
q
q
q
Steady motion:
Stability of steady motion by linearized eqn’s of motion:
and linearized nonholonomic constraints:
d d d d k k M q C q K q K q 0
k d d d k k q A q B q B q
Aug 9, 2005 13
Linearized State
d d k d
d d
k d k k
M 0 0 q C K K q 0
0 I 0 q I 0 0 q 0
0 0 I q A B B q 0
1
dd
d
d d
k dt
q M f
q q
q Aq b
Linearized State equations:
State equations:
with, d
T T q
C T CT T Mh
, , , ,d k T T T q q q qK K K T KF T Mx f T Mh Cvand
and ,d k qB B B b
Green: holonomic systems
Aug 9, 2005 14
Straight Ahead Motion
d d k d
d d
k d k k
M 0 0 q C K K q 0
0 I 0 q I 0 0 q 0
0 0 I q A B B q 0
Turns out that the Linearized State eqn’s:
Upright, straight ahead motion :
lean angle 0
steer angle 0
rear wheel rot. speed / constantr v r
0
Aug 9, 2005 15
Straight Ahead Motion
d d k d
d d
k d k k
M 0 0 q C K K q 0
0 I 0 q I 0 0 q 0
0 0 I q A B B q 0
Linearized State eqn’s:
Moreover, the lean angleand the steer angle are decoupled from the rear wheel rotation r (forward speed ), resulting in:
0
rv r
x x 0 x x 0 x x 0
x x 0 , x x 0 , x x 0
0 0 x 0 0 0 0 0 0
d
M C K
lean angle
steer angle
rear wheel rot.
d
r
qwith
Aug 9, 2005 16
Stability of Straight Ahead Motion
with and a constant forward speed
Linearized eqn’s of motion for lean and steering:
1 0 2
130 3 0 40 1003 27 0 96, , ,
3 0.3 0.6 1.8 27 8.8 0 2.7
M C K K
21 0 2( ) ( ) 0d d dv v Mq C q K K q
lean
steer d
q rv r
For a standard bicycle (Schwinn Crown) :
Aug 9, 2005 17
Root Loci Parameter: forward speed
rv r
v
vv
Stable forward speed range 4.1 < v < 5.7 m/s
Aug 9, 2005 18
Check Stability by full non-linear forward dynamic analysis
Stable forward speed range 4.1 < v < 5.7 m/s
forward speedv [m/s]:
01.75
3.53.68
4.9
6.3
4.5
Aug 9, 2005 19
Comparison
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- 1983 Koenen- 1987 Papadopoulos
- and many more …
Aug 9, 2005 20
ComparisonFor a standard and distinct type of bicycle + rigid rider combination
Aug 9, 2005 21
ComparePapadopoulos (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!
21 0 2( ) ( ) 0d d dv v Mq C q K K q
Aug 9, 2005 22
Experimental Validation
Instrumented Bicycle, uncontrolled
2 rate gyros:
-lean rate
-yaw rate
1 speedometer:
-forward speed
1 potentiometer
-steering angle
Laptop + Labview
Aug 9, 2005 23
Experimental Validation
Linearized stability of the Uncontrolled Instrumented Bicycle
Stable forward speed range:
4.0 < v < 7.8 [m/s]
Aug 9, 2005 24
An Experiment
Aug 9, 2005 25
Measured Data
Aug 9, 2005 26
Extract EigenvaluesStable Weave motion is dominant
Nonlinear fit function on the lean rate:
11 2 2 3 2e [ cos( ) sin( )]tc c t c t
Aug 9, 2005 27
Extract Eigenvalues & Compare
Nonlinear fit function on the lean rate:
11 2 2 3 2e [ cos( ) sin( )]tc c t c t
2 = 5.52 [rad/s]
1 = -1.22 [rad/s]
forward speed:
4.9 < v <5.4 [m/s]
Aug 9, 2005 28
Compare around critical weave speed
Aug 9, 2005 29
Just below critical weave speed
Aug 9, 2005 30
Compare at high and low speed
Aug 9, 2005 31
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, 2005 32
MATLAB GUI for Linearized Stability
Aug 9, 2005 33
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, 2005 34
Myth & Folklore
Forward speedv = 3 [m/s]:
Aug 9, 2005 35
Steering a Bike
To turn right you have to steer …
briefly to the LEFT
and then let go of the handle bars.
Aug 9, 2005 36
Steering a BikeStandard 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 secondand then we let go of the handle bars.
Aug 9, 2005 37
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.