Presented by Markus Kunesch · IYPT 2009 – Tianjin, China Presented by Markus Kunesch F m g α....

Markus Kunesch, Michael Scherbela, Johannes Tiefnig, Angel UsunovBernhard Zatloukal

IYPT 2009 – Tianjin, China

Presented by Markus Kunesch

F

m g

α

Problem 7- Skateboarder:A skateboarder on a horizontal surface can accelerate from rest just by moving the body, without touching external support. Investigate the parameters that affect the motion of a skateboard propelled by this method.

IYPT 2009Problem No. 7 – SkateboarderMarkus Kunesch 2

Skateboarder

First experiments

Observations

Explanation Mathematical model

Advanced Experiments• Motion capturing

Comparison Conclusion


First Experiment - Observations


Aspect 1

• Body is turned

• Turning movement cannot bestopped totaly

• Velocity perpendicular to front wheels remains.

Front


Aspect 2

• Pivot is not in the middle

• Velocity perpendicular to front wheels remains.

?


Theoretical Consideration Aspect 1

• ConstraintsWheelpairs can not move sideways

γFront


Appell Equation

• Appell – equation[1]

γr

r

FpS=

∂∂

∑=i

ii pmS 2

21

pdirectioninforceFcoordinatep

functionAppellS

.........

Paul Émile Appell[1] Skateboard dynamics, Ipsolov, Smolnikov, Comput. Methods of applied mechanics and engineering Vol. 131, 1996, St. Petersburg

(1)

(2)


Appell Equation

• Appell - equation

rr

FpS=

∂∂

∑=i

ii pmS 2

21

22

21

21 αImaS +=

erskateboardofaccangularforcedragFonacceleratia

onacceleratiangularinertiaofmomentImassm

FunctionAppellS

r

...

.........

.........

δ

α

Front

(3)


Constraints

• Constraints:

lv

vvvv

y

x

2tan

sincos

⋅=

==

γχ

χχ

γ

χ

v

skateoardoflengthlaxisverticalvelocityv

axishorizontalvelocityvvelocityv

y

x

...

)(...)(...

...

Front

2l

(4)


Acceleration

va =

22

21

21 αImaS +=

δχα +=

erskateboardofaccangularonacceleratia


FunctionAppellS

.........

.........

δ

α

rr

FpS=

∂∂

γ

χ

v

Front

2l


Acceleration

rFlI

lIv

lImv

vS

=++

+=

∂∂ γδγ

γγγ tan2tan

cos4tan4

222

2

erskateboardofaccangularcoordinatepforcedragF

onacceleratiaonacceleratiangular

inertiaofmomentImassmfunctionAppellS

rr

...

............

.........

δ

α

γ

χ

rr

FpS=

∂∂

v

Front

2l

(5)


Acceleration

erskateboardofaccangularcoordinatepforcedragF



rr

...

............

.........

δ

α

γ

χ

v

Front

2l

(6)

+

−−=

γ

γδγγ

γ

22

22

tan4

tan2tancos4

lIm

lI

lIvF

vr


Theoretical consideration: Aspect 2

( )τχcos2

latotal =

skateboardofaccangularonacceleratitotala

onacceleratia

total

......

...

χ

( )

( ) dlaa total

⋅=⋅=

=⋅=

χτχ

τ

tan2

sin

d

τ

Front

2l

(6)


Advanced Experimental Setup


Cheap IMU – Motion capture


Experimental Setup


Controlling motions

Controlling motions:

• Angular acceleration ofskateboarder

• Turning of wheels

• Turning of skateboard

Parameters:

• Moment of inertia

• Mass of the skateboarder

• Skateboard length

• Drag Force

• Distance betweenwheels

Front


Calculation

-0,20

0,20,40,60,8

11,21,41,6

0,0 0,6 1,2 1,8 2,4 3,0 3,6 4,2 4,8 5,4 6,0

Velo

city

(m/s

)

Time (s)

Calculated velocity(with controlling motions)


Experiment

-0,5

0

0,5

1

1,5

2

0,0 0,6 1,2 1,8 2,4 3,0 3,6 4,2 4,8 5,4 6,0

Velo

city

(m/s

)

Time (s)

Meassured velocityError (±0.012 m/s max)


Comparison

-0,5

0

0,5

1

1,5

2

0,0 0,6 1,2 1,8 2,4 3,0 3,6 4,2 4,8 5,4 6,0

Velo

city

(m/s

)

Time (s)

ComparisonError (±0.012 m/s max)

calc

Exp1


• Performed in-depth experiments

• Full and detailed mathematical analysis

• The relevant parameters are• angle/angular velocity/angular acceleration ofthe body at the right time

• Moment of inertia of the skateboarder (z axis)• Mass of the skateboarder

• Length of the skateboard• Tilting at the right time• Raising at the right time• Turning of the skateboard at the right time

Conclusion


Conclusion

• Setting of the trucks• depending on the weight andstrength

• Surface• On a plain surface more speedis achieved, beginners shouldchoose rough surfaces

• Wheels•Hard wheels for speed, forbeginners difficult


References• Taschenbuch der Physik, Stöcker, Harri Deutsch, 2004

• A Treatise of the analytical dynamics of particles and rigid bodies E.T. Whittaker, M.A.1904 Cambridge, at the university press

• Mathematik für Physiker, Dr. rer. Nat. Helmut Fischer, Dr. rer. Nat. Helmut Kaul, B. G. Teubner, 2005

• Die Beschleunigung beim Slalomskateboarden, Ackermann Jürg, Strobel Maurus, Diplomarbeit ETH Zürich, 2000

• Skateboard dynamics, Ipsolov, Smolnikov, Comput. Methods of applied mechanicsand engineering Vol. 131, 1996, St. Petersburg

• Skateboarding Skills, Powell B, Firefly Books, Ontario

• http://www.geocities.com/CapeCanaveral/8341/bridge.htm, 3.05.09

• http://www.cruisin.de/cr_i/movies/pushen.pdf / 23.10.08

• http://www.cs.cmu.edu/afs/cs/academic/class/16741-s05/www/lecture5.pdf, 26.12.08

• Panasonic, Compact Quartz Angular Rate Sensors manual, 2005


First Experiments


Ad) Experimental Results

-2

-1,5

-1

-0,5

0

0,5

1

1,5

2

2,1

2,2

2,4

2,6

2,7

2,9

3,1

3,2

3,4

3,5

3,7

3,9

4,0

4,2

4,4

4,5

4,7

4,9

5,0

5,2

5,4

5,5

5,7

5,8

6,0

6,2

6,3

6,5

6,7

Pote

ntia

l (Vo

lts)

Time (sec)

Gyroscopic Sensor Output

roll


Ad) Experimental Results

-2,5

-2

-1,5

-1

-0,5

0

0,5

1

1,5

2

2,1

2,2

2,4

2,6

2,7

2,9

3,1

3,2

3,4

3,5

3,7

3,9

4,0

4,2

4,4

4,5

4,7

4,9

5,0

5,2

5,4

5,5

5,7

5,8

6,0

6,2

6,3

6,5

6,7

Pote

ntia

l (Vo

lts)

Time (sec)

Gyroscopic Sensor Output

pitch


Experimental Setup – Moment of Inertia

Laser to ensurethat body positiondoes not change


Ad) Calibration


Ad) Experimental Setup

Wheels86 1 Dur75 Dur – 90 Dur


Ad) Measurement of stiction


Ad) Calculation: Appell Equation

• Appell - equation

γr

r

FpS=

∂∂

∑=i

ii pmS 2

21

pdirectioninforcedragFcoordinatep

functionAppellS

.........


Ad) Constraints

• Constraints:

lv

vvvv

y

x

2tan

sincos

⋅=

==

γχ

χχ

γ

l χ

V

skateoardoflengthlVelocityv

......


Ad) Appell Function S

• Appell Function S forthe skateboarder:

22

21

21 αImaS +=

γ

lχ



......

.........

α


Ad) Acceleration

• a of skateboarder:va = γ

lχ

Assuming that centre ofmass is over yellow x-axis

22

21

21 αImaS +=

δχα +=


inertiaofmomentImassmFunctionAppellS

......

.........

α


Ad) Acceleration 1

γ

lχ

22

21

21 αJvmS +=

δχα +=

Constraint already included in V‘!

lv2tan ⋅= γχ

γγγχ 2cos

2tan2l

vlv

+=



......

.........

α

IYPT 2009Problem No. 7 – SkateboarderMarkus Kunesch 37pdirectioninforcedragP

coordinatepfunctionAppellS

...

...

...

Ad) Acceleration 2

γ

l χ

22

21

21 αIVmS +=

=++= δχδχα 2222

...cos

4tan442

222

2

22 ++=

γγγα

lV

lV

rr

PpS=

∂∂ rF

vS=

∂∂

⇒

δχα +=



......

.........

α




......

.........

α

Ad) Acceleration 3

γ

l χ

22

21

21 αIvmS +=

γδγγ

γγα tan4tancos8tan4

222

2

22

lv

lvv

lv

++=

=++= δχδχα 2222

γγγχ 2cos

2tan2l

vlv

+=


Ad) Acceleration 4

γδγγ

γγ tan2tancos4tan4

21

222

22

lIv

lIvv

lImvS

++

+=

22

21

21 αIvmS +=

FlI

lIv

lImv

vS

=++

+=

∂∂ γδγ

γγγ tan2tan

cos4tan4

222

2

forcedragPonacceleratia


functionAppellS

.........

.........

α


Ad) Additional experiment

00,20,40,60,8

11,21,41,61,8

2

0,0 0,6 1,2 1,8 2,4 3,0 3,6 4,2

Velo

city

[m/s

]

Time [s]

ComparisonSkateboarding problemsError (±0.012 m/s max)

cal

exp


Ad) Additional experiment

-0,5

0

0,5

1

1,5

2

2,5

0,0 0,8 1,6 2,4 3,2 4,0 4,8 5,6 6,4 7,2 8,0 8,8 9,6 10,4

Velo

city

[s]

Time [s]

ComparisonError (±0.012 m/s max)

calc

exp


Observations

• Skateboarder turns body(range: 90°)

• Deck is tilted wheelsturned

• Acceleration when front wheels touch the ground

• Pivot changes from onewheel to the other


Skateboarder

• First experiments

• Observations

• Explanation

• Mathematical model

• Advanced Experiments

– Motion capturing

• Comparison

• Conclusion

Presented by Markus Kunesch · IYPT 2009 – Tianjin, China Presented by Markus Kunesch F m g α....

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