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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

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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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