Post on 18-Dec-2015
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THE INVERTED PENDULUMArthur EwenczykLeon FurchtgottWill SteinhardtPhilip SternAvi Ziskind
PHY 210Princeton University
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FORMULATING THE PROBLEM
Pendulum Mass (m) Length (l)
Oscillating Pivot Amplitude (A) Frequency (ω)
m
l
θ
A, ω
z
2
FORMULATING THE PROBLEM
Forces Acting on the Pendulum1. Gravity2. Force of the pivot3. [Friction]
m
l
θ
A, ω
z
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EQUATION OF MOTION
Newton’s Second Law
Linear motion:
Circular motion:
Equation of Motion
m
l
θ
A, ω
z2
2
d xF ma m
dt
2
2
dI I
dt
2
total 2
dI
dt
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TORQUES
Gravity:
Oscillating pivot: Force:
Torque:
m
l
θ
z
22
2
22
2
( ) cos( )
( )cos( )
( )cos( )
y t A t
d y ta A t
dt
d y tF ma m m A t
dt
grav sinr F
grav sinmgl
pivot sinr F
2pivot cos( )sinml A t 5
EQUATION OF MOTION2 2
2total grav pivot 2 2
d dI ml
dt dt
22 2
2sin cos( )sin
dml mgl mg A t
dt
2 2
2cos( ) sin 0
d g At
dt l l
Gravity term
Pivotterm
2
2
0
0
( ) sin
d g
dt lt t
m
θ
20
g
l
m
θ
2
2
0
0
( ) exp( )
d g
dt lt t
For example:
29.81m.s
19cm
g
l
10 7.19 rad.s
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PHYSICAL INTUITION2 2
202
cos( ) sin 0d A
tdt l
Pivot term (dominates for large values of
ω)
sinrF
2
2max
cos( )F m A t
F m A
1 2F F
2 1
2 1
Inertial (lab)frame
Non-inertialframe
(of pendulum)
Butikov, Eugene. On the dynamic stabilization of an inverted pendulum. Am. J. Phys., Vol. 69, No. 7, July 2001
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REPARAMETERIZATION
t Reparameterize:
220
2 2cos( ) sin 0
d A
d l
2
2cos( ) sin 0
d
d
A
l
2
0
Let:
Mathieu's Equation
2
2cos( ) 0
d
d
For small values of θ: sin
t
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STABILITY
unstable
stable
2
2cos( ) 0
d
d
Regular Pendulum:
Inverted Pendulum:
0
Smith, Blackburn, et. al.Stability and Hopf bifurcations in an inverted
pendulum. Am J. Phys. 60 (10), Oct 1992
Matlab Plot
0
stable
unstable
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PHYSICAL REGION
0
2A
l
Our values of ε and δ
Stability condition: (approximating A << l)
12 inch 0.013m
0.06719cm 0.19m
A
l (fixed)
2 202 2
7.19[2; 0.001]
[5 Hz; 200Hz]
(depends on ω)
3c
c
2.2 10
24.2 Hz
ε = 0.067
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SAMPLE PLOTS
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HOW WE BUILT IT
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HOW WE BUILT IT
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HOW WE BUILT IT
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WHY THE DRIVER DIDN’T WORK
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HOW WE BUILT IT
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HOW WE BUILT IT
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HOW WE BUILT IT
Some of the things we originally used to stabilize the pendulum
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HOW WE BUILT IT
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HOW WE BUILT IT
2020
HOW WE BUILT IT
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DATA AQUISITION
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DATA ACQUISITION
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DATA AQUISITION
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DATASpeed 1 (8 Hz) Speed 2 (13 Hz)
2525
Speed 3 (23 Hz) Speed ~2.8 (22 Hz)
147.8 147.9 148 148.1 148.2 148.3 148.4 148.5 148.6 148.7 148.8-10
-8
-6
-4
-2
0
2
Time
Ligh
t In
tens
ity
Speed 1
184.5 184.6 184.7 184.8 184.9 185 185.1 185.2 185.3 185.4 185.5-7
-6
-5
-4
-3
-2
-1
0
1
Time
Ligh
t In
tens
ity
Speed 2
206.8 206.9 207 207.1 207.2 207.3 207.4 207.5 207.6 207.7 207.8-2.5
-2
-1.5
-1
-0.5
0
0.5
1
Time
Ligh
t In
tens
ity
Speed 3
379.1 379.2 379.3 379.4 379.5 379.6 379.7 379.8 379.9 380 380.1-2.5
-2
-1.5
-1
-0.5
0
0.5
1
Time
Ligh
t In
tens
ity
Just Unstable
DATA AQUISITION
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1415 RPM = 23.58 Hz
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PROBLEMS WE HAD
Finding right amplitude and frequency Finding right equipment Stabilizing apparatus Getting apparatus to right frequency Soldering
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THANK YOU
Professor Page Wei Chen PHY 210 Colleagues Sam Cohen Mike Peloso
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