1. Inverted Pendulum
2008 Summer
Dan Brunski
11/24/2018
Inverted Pendulum

2. Theory
Mathieu equation
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
Regions of stability of the Mathieu equation
Regular pendulum
Inverted pendulum
11/24/2018
Inverted Pendulum

3. Equation Development
b(t)
The coordinates of a pendulum with mass m on an oscillating base b(t) are given by
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
11/24/2018
Inverted Pendulum

4. 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
11/24/2018
Inverted Pendulum

5. 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
11/24/2018
Inverted Pendulum

6. 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
11/24/2018
Inverted Pendulum