04/19/23 Inverted Pendulum 1

Inverted Pendulum

2008 SummerDan Brunski

04/19/23 Inverted Pendulum 2

Theory

● A pendulum with an oscillating base can be described by the Mathieu equation

● Regions of stability exist in the Mathieu equation for δ < 0

● These regions correspond to an inverted pendulum position

0)cos(2

2

d

d

Mathieu equation

Inverted pendulum

Regions of stability of the Mathieu equation

Regular pendulum

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Equation Development

● The coordinates of a pendulum with mass m on an oscillating base b(t) are given by

0)cos(2

2

d

dt

20

l

g2

0

cos)(,sin, ltblyx

cossin22

1 222 lbmglblbmUTL

sin)cos(2

2

2

t

l

A

l

g

dt

d

b(t)

l

A sin

)cos()( tAtb

● The Lagrangian is then

● Using the Euler-Lagrange equation and substituting in b(t), the equation of motion is

● Making the following substitutions, the Mathieu equation is obtained

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What Didn’t Work

● Pendulums powered by either Black and Decker jigsaw or Dewalt reciprocating saw

● Pendulum dimensions: ─ 16-1/4” x 1-1/4” x 1/8”─ Length measured from pivot

center to end─ Lengths down to 6-3/4”

● Tried adding weights to various places

● Either the frequency wasn’t high enough, amplitude not great enough, modes of vibration in unintended directions, or a function of all three

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What Worked● Short piece of straw (44.5mm from

pivot to end) on a Pasco SF-9324 wave driver

● Foam pad under wave driver to keep noise down and stop it from bouncing around

● Function generator signal amplified to wave driver limit of 16Vpp

● Pivot made from 0.021” steel wire welded to a washer

● Confirmed stable settings:─ 41Hz – 77Hz─ 16Vpp ± 0.5V─ 5.39Vrms ± 0.1V─ 377mA ± 1mA─ 4mm ± 0.5mm amplitude

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Wiring Schematic

● Ammeter, Voltmeter, and oscilloscope are not necessary, but recommended to prevent damaging the wave driver

Function Generator

Frequency Counter

Amplifier

Ammeter Wave Driver Voltmeter

Oscilloscope