Saddles - iypt solutions

14
13. ROTATING SADDLE
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Task
A ball is placed in the middle of a rotating
saddle.
Investigate it's dynamics and explain
the conditions under which the ball
does not fall off the saddle.
How can the rotation
help the stability?
• Static saddle:
just rolls off
• Rotating saddle:
rolls around the saddle
 effect of slopes
cancels
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14
(laboratory ref. frame)
1. Goes around with the
saddle
Centripetal force needed
 unstable
2. Remains stationary
(Rolls back quickly enough)
stable
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1. Sufficient saddle
rotation
– Cancels the effect of
slopes
2. Rolling backwards
– Avoids centrifugal
force
How can these
be achieved?
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Existing theory
• Thompson: The rotating-saddle trap: a mechanical
analogy to RF-electricquadrupoleion trapping?
 Canadian journal of Physics, Vol. 80, 2002
• Koch: Konzeption und Aufbau einer mobilen
Experimentiereinheit fur Schuleräprsentationen zum
Thema Teilchenfallen
 Universität Stuttgard, 2004
• Point mass in gravitational potential
– Constrained to saddle’s surface

mgh0 2
U  x' , y'  = 2 x'  y' 2
r0

F = U
• Mathematical trick:
– coordinates in complex plane z  x  iy
z  k (x2  y2 )
14
b
POSITION
z τ  = ( Ae
Solution
+β+ t
+ Be
 β+ t
+ Ce
+β-t
+ De
 β-t
Ω: angular velocity (saddle rotation)
The only requirement
for stability:
f
>
f
c
Condition of stability (z does not diverge):
A,B,C,D: determined by initial conditions
β± = ± 2
gh0
1
2 2
r0 
gh0
 0 .5
2 2
r0 
f ≥
2gh 0
2πr0
 f CRITICAL
)e
i
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EXPERIMENTAL VERIFICATION
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Apparatus: Saddles
h0 = 6,5cm
r0 = 8cm
h0 = 1,5cm
r0 = 8cm
• fc = 1,08 Hz
material = plastic
• fc = 2,25 Hz
material = nylons
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Radius range:
1,88 cm – 5,0 cm
Mass range:
2,46 g 26,56 g
hollow
Radius range:
0,63 cm – 3,26 cm
Mass range:
8,39 g 35,79 g
solid
Apparatus: Balls
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14
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Stability vs. Frequency
YELLOW saddle
big ping pong ball
WHY?
20
4,5
fc
18
16
3,5
Lifetime (s)
Lifetime (s)
fc
4
14
12
10
8
6
3
2,5
2
1,5
drag forces
1
4
0,5
2
0
BLUE saddle
big ping pong ball
0
0,4
0,6
0,8
1
1,2
1,4
frequency (Hz)
f  fC
1,6
1,8
2
0,7
0,9
1,1
1,3
1,5
1,7
1,9
2,1
frequency (Hz)
Significant increase in lifetime
but clearly not infinite
2,3
2,5
2,7
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Second condition:
Must roll back fast enough
Friction/rolling resistance:
Drags the ball
to rotate with the saddle
Ball becomes unstable
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Literature: Effect of friction


• Thompson: FFriction   kv
– Analytical solution; always diverges

• Koch: FFriction

v
 k 
v
– Numerical solution; no record of stability
Stability
Maximal
lifetime
14
Parametres affecting the lifetime
1. Drag forces & Friction
2. Frequency
3. Ball
4. Initial position
?
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1. DRAG FORCES & FRICTION
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Effect of friction
Thompson:
1
 r0 
TL =
ln 
σ Ω R




TL = trapping lifetime
σ ~ friction coefficient
R = initial distance from
the center
r0= trap's radius
Higher friction  lower lifetime
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1. Lifetime vs. Friction: Experiment
DRY
NYLON SADDLE
NYLON SADDLE
SOAKED WITH WATER
μ=0,25
>
μ=0,09
PREDICTION CONFIRMED
MAXIMUM LIFETIME:
MAXIMUM LIFETIME:
2,25s
<
6,33s
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Not so simple
Koch’s article:
Teflonspray
(lower friction)
Lifetime
Clean saddle
(higher friction)
10,1s
HIGHER FRICTION
HIGHER LIFETIME
?
54,7s
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What if the ball does
NOT slip?
EXPECTATION:
FRICTION
HighSUFFICIENT
friction 
Slipping 
Short lifetime
Avoids slipping
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Sufficient friction: no slipping
Similar to zero friction (no slip)
Dragging effect:
only rolling resistance
(much lower than
dynamic friction)
Relatively stable:
• Zero friction
• Sufficient friction
Measurement:
Slipping vs. Rolling
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SUFFICIENT
FRICTION
=
AVERAGE LIFETIME:
AVERAGE LIFETIME:
HIGHER LIFETIME
Slipping (dynamic friction)
Rolling (static friction)
(30 measurements)
(30 measurements)
2,7 s
0,4 s
8,8 s
2,6 s
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Parameters affecting lifetime
1. Friction
– dynamic: the lower, the longer lifetime
– static: fulfills
condition
?
2. Frequency
3. Moment of inertia
4. Initial position
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2. FREQUENCY
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JUMPING
14
1. Ball free to move upwards
• Very fast rotation:
• Height changes harmonically
height
UP
UP
DOWN
DOWN
Harmonic
frequency = 2Ω
time
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Condition of jumping
h  kr cos(2t )
• Saddle shape in polar coordinates
2
• Vertical acceleration:
a  2  kr 2 cos(2t )
2
ag
ag
Constrained
to surface
Jumps
Critical frequency
for jumping:
r
h
f  f jump
1

4r
g
k
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Estimation vs. reality
TOO
HIGH
FREQUENCY
• We measured the distance in which jump occurred
and estimated the frequency
1
g
f 
NO MORE CONSTRAINED
TO 0,14 Hz
f  3,43Hz
4r k
• Measured frequency:
f LOWER
 3,69 Hz
LIFETIME
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Parameters affecting lifetime
1. Friction
2. Frequency
– Lower limit: rise of lifetime
– Upper limit: jumping
?
3. Moment of inertia
4. Initial position
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3. MOMENT OF INERTIA
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Hollow VS. Solid Ball
Greater moment of inertia
More energy needed for rolling
M  J 
Lower speed
Longer lifetime
2
2
J = mR
3
Should have
longer lifetime than
2
2
J = mR
5
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EXPERIMENT
Greater
moment
of
2
2
J = mR
J = mR
3
inertia
5
=
AVERAGE LIFETIME:
AVERAGE LIFETIME:
longer lifetime
2
2
(30 measurements)
(30 measurements)
8,96 s ± 1,74 s
2,11 s ± 0,34 s
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Parameters affecting lifetime
1. Friction
2. Frequency
3. Moment of inertia
– Higher moment of inertia = longer
lifetime
?
4. Initial position
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4. INITIAL POSITION
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INITIAL POSITION
Lifetime VS Initial distance
2,5
Lifetime [s]
2
1,5
1
0,5
saddle
0
0,10
0,30
0,50
R [cm]
0,70
The further from the center we place the ball,
the sooner it falls off
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Conclusions
• Conditions under which the ball should be
stable
20
18
16
Lifetime (s)
– Sufficient saddle rotation
14
 Theory: Critical frequency fc
 Experiment never stable (rise od lifetime)
– Avoiding centripetal force
 Theory: No or low drag force
 Our contribution: by backward rotation
12
10
8
6
4
2
0
0,4 0,6 0,8
1
1,2 1,4 1,6 1,8
frequency (Hz)
2
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Conclusions
• Examined
– Friction:
 Theory: solution only for specific case
 Our contribution: sufficient friction = more stable
– Jumping (not mentioned in theory)
 upper limit for frequency exists + estimation
– Rotation of the ball (not mentioned)
 Dependence on the moment of inertia
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Thank you for your attention
Ball's trajectory (Yellow saddle)
small ping pong ball
10
8
6
Y (cm)
4
2
0
-15
-13
-11
-9
-7
-5
-3
-1
1
-2
-4
-6
X (cm)
3
5
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APPENDICES
14
Prediction: Stability vs. Frequency
f  f critical
f  f critical
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Similar to
predicted
behaviour
f  fC
Ball's trajectory (Yellow saddle)
small ping pong ball
10
But always
limited
lifetime
8
6
Y (cm)
4
2
0
-15
-13
-11
-9
-7
-5
-3
-1
1
-2
f  fC
-4
-6
X (cm)
3
5
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Apparatus: Rotation
• Rotation: driller
– Frequency range:
0.6Hz 3.7Hz
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Small ping pong ball
YELLOW saddle
BLUE saddle
small ping pong ball
small ping pong ball
12
2,5
10
8
Trapping lifetime (s)
Trapping Lifetime (s)
2
6
4
1
0,5
2
0
0
0,4
1,5
0,6
0,8
1
1,2
frequency (Hz)
1,4
1,6
1,8
2
1
1,5
2
2,5
frequency (Hz)
3
3,5
4
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INITIAL POSITION
Lifetime VS Initial distance
2,5
Lifetime [s]
2
1,5
1
saddle
0,5
0
0,10
0,30
0,50
R [cm]
0,70
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Hollow balls' parameters
white ball:
r=2,9cm
m=46,79g
big red-yellow ball:
r=5,0cm
m=26,56g
ping-pong ball:
d= 3,94 cm
m= 2,46 g
α (friction coef.) = 0,19
small ping pong ball:
d= 3,76 cm
m= 3,16 g
big orange ball:
r= 3,26 cm
m= 6,93g
α (friction coef.) = 0,25
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Hollow balls' parameters
Big green ball:
r= 3,26 cm
m= 13,30g
α (friction coef.) = sufficient
small green ball:
r= 1,77 cm
m= 17,72g
Small metal ball:
r=0,63cm
m=8,39g
Big metal ball:
r= 1,01 cm
m= 35,79g
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Sufficient friction (no slipping)
No friction
• 2.

  
ma  mg  N  I
Sufficient friction

   
ma  mg  N  I  F
 

J  RF  n
 
 nv

R
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Saddle
• Convex in one direction;
concave in the other
• Various saddle types:
z  k(x  y )
2
2
• Convenient for mathematical description
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POINT MASS vs BALL
DEEPER SADDLE
M = Fr= J ϵ
TRAP'S
CENTER
THE BIGGER
COMPONENT OF FG
CAUSES THE TORQUE
THE LESS IT RESEMBLES
POINT MASS
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'DEEP' vs 'SHALLOW' saddle
MAXIMUM LIFETIME:
4,07 s
MAXIMUM LIFETIME:
23,91 s
THE SMALLER THE SADDLE'S HEIGHT
IS, THE MORE STABLE THE BALL IS
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FRICTION
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Friction
–blue saddle
f 
F friction
FN
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Friction
–blue saddle+ WATER
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TRAPPING LIFETIME
THOMPSON'S
ARTICLE
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THEORY
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Theory
Gravitational potential:
- assigned to the rotating
frame (fixed to U)

mgh0 2
U  x' , y'  = 2 x'  y' 2
r0

- converted to the laboratory
frame:
mgh0 2 2
U ( x , y)=
[( x − y )cos(2Ω t )+2xy sin (2Ω t )]
2
r0
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using the following formula
yields
F=− ∇ U
2
∂ x 2mgh 0
[− x cos(2Ω t )− y sin (2 Ω t )]
2 = 2
∂t
r0
∂ 2 y 2mgh 0
= 2
[ y cos(2Ω t )− x sin (2Ω t )]
2
∂t
r0
using dimensionless parameters
converting to the complex plane
τ = Ω t and q= gh0
r 20 Ω 2
( z= x+iy) ,
the 2 equations are reduced into:
∂2 z
i2 τ
+2q∗
e
=0
2
∂τ
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Applying another substitution
yields the solution:
+β+ τ
f (τ )= Ae
− β+ τ
+Be
+β- τ
+Ce
z (τ )= f (τ )ei τ
− β- τ
+De
where A,B,C,D are real parameters depending on initial conditions and
β ±= √± 2∣q∣− 1
β± ∈ R - {0} => result will diverge in any case => particle is
trapped only if β± ∈ I, thus
2∣q∣≤ 1 => q ≤ 0,5
q=
gh0
2
Ω r
2
0
≤ 0,5
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• The condition for stability is:
•
2gh 0
√
Ω≥
r0
regardless of initial position of the ball
q≤0,5
2gh 0
√
f≥
2 πr 0
q>0,5
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Limited trapping lifetime
1) unstable trapping parameters (q>0,5)
r0
1
T L=
ln ( )
β+ Ω
R
2) friction (q
r0
1
T L=
ln ( )
√2q− 1 R
r0
1
T L=
ln ( )
βΩ
R

β ~ friction coefficient
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14
BALL'S PATH
– short trapping lifetimes
video
14
BALL'S MOTION
– longer trapping lifetimes
OUR DIAGRAM
THEORETICAL DIAGRAM
Ball's trajectory (Yellow saddle)
small ping pong ball
10
8
6
Y (cm)
4
2
0
-15
-13
-11
-9
-7
-5
-3
-1
1
3
-2
-4
-6
X (cm)
KOCH'S ARTICLE
video
11:35
5
Ball's trajectory (Yellow saddle)
STABLE TRAPPING
PARAMETERS
14
small ping pong ball
10
8
6
4
Y (cm)
•
2
0
-15
-13
-11
-9
-7
-5
-3
-1
1
-2
-4
-6
X (cm)
stable
unstable
3
5
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BALL'S MOTION
– longer trapping lifetimes
Ball's trajectory (Yellow saddle)
Ball's Trajectory (BLUE saddle)
small ping pong ball
big ping pong ball
10
1
8
0,8
0,6
6
0,4
y (cm)
Y (cm)
4
2
0,2
0
-2,6
-2,1
-1,6
-1,1
-0,6
-0,1
-0,2
0
-15
-13
-11
-9
-7
-5
-3
-1
1
3
-0,4
5
-2
-0,6
-0,8
-4
-1
-6
X (cm)
x (cm)
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BALL'S PATH
– longer trapping lifetimes
OUR DIAGRAM
THEORETICAL DIAGRAM
Ball's distance form the center in time
small ping pong ball
5
4,5
4
3,5
r (cm)
3
2,5
2
1,5
1
0,5
0
4,4
4,6
4,8
5
5,2
5,4
t (s)
KOCH'S ARTICLE
5,6
5,8
6
6,2
6,4
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ANGLE
14
β ± = √± 2∣q∣− 1
β ± ∈ R− { 0}=> result will diverge in any case =>
particle is trapped only if
2∣q∣≤ 1
β ±∈ I
, thus
=> q ≤ 0,5
gh0
q = 2 2 ≤0,5
 r0
The condition for stability is:
2gh 0
√
Ω≥
r0
f ≥
2 gh0
2πr0
14
Theory (Thompson's article)
Gravitational potential:
- assigned to the rotating
frame (fixed to U)
U ( x ' , y ' )=
mgh0
r0
2
( x ' 2− y ' 2 )
- converted to the laboratory
frame:
mgh0 2 2
U ( x , y)=
[( x − y )cos(2Ω t )+2xy sin (2Ω t )]
2
r0
14
β ± = √± 2∣q∣− 1
β ± ∈ R− { 0}=> result will diverge in any case =>
particle is trapped only if
β ±∈ I
, thus
gh0
q = 2 2 ≤0,5
 r0
The condition for stability is:
2gh 0
√
Ω≥
r0
2gh 0
f≥
2 r0
14
SOURCES
R.I. Thompson, T.J. Harmon, and M.G. Ball:
The rotating-saddle trap: a mechanical analogy to RFelectricquadrupoleion trapping?
(Can. J. Phys. Vol. 80, 2002)
Wolfgang Rueckner, Justin Georgi, Douglass Goodale, Daniel
Rosenberg, David Tavilla:
Rotating saddle Paul trap
(American Journal of Physics 63, 186 (1995); doi: 10.1119/1.17983)
A. K. Johnson and J. A. Rabchuk:
A bead on a hoop rotating about a horizontal axis: A one-dimensional
ponderomotive trap
(Citation: American Journal of Physics 77, 1039 (2009); doi:
10.1119/1.3167216)
Tobias Koch:
Konzeption und Aufbau einer mobilen Experimentiereinheit fur
Schuleräprsentationen zum Thema Teilchenfallen
14
Stara prezentacia
14
Friction & Initial conditions
friction in equations
exponential growth
limited trapping lifetimetime TL
2 reasons for lifetime limitation:
1) unstable trapping parameters
2) friction
f < f critical
14
Conclusion
2gh 0
2πr0
 f CRITICAL
• Conditions under which the ball should be stable
– Critical frequency
• We found its limitations and examined the effects
– Jumping
– Friction, initial position
– Rotation of the ball
+VERIFICATION
• We constructed 2 saddle traps with
parameters, used different types of balls
different
• Analysis of ball
and motion while being trapped
+comparison with theory
14
Conclusions
Our contribution
1. Stability I.
Calculation of
critical frequency
II. Infinite trapping
lifetime for
f  fc
I.
Experimental
verification
II. Experimentally
confuted
•
Lifetime rise for
f  fc
14
Conclusions
1. Friction
Our contribution
i.
Dynamic I.
Solved for special
case only
II. not mentioned
ii.
Static
I.
Experimental
verification
II. Correlates best with
point-mass theoryoptimal case
2. Jumping
2) not mentioned
III. upper limit for frequency
+ estimation
3. Rotation
3) not mentioned
IV. Dependence on
moment of inertia +
verification

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