In the previous video,
I showed how turning the crank
on this machine generates
twenty different frequencies
also known as harmonics,
which are turned into cosines here
multiplied by coefficients
then summed
magnified
and the resulting function
plotted here on the front.
This process is called synthesis.
But this machine is called a
Harmonic Analyzer
which means it can be
used to solve the much more
difficult inverse problem called analysis.
For example
if I want to plot a
particular function here
how do I set the amplitude bars
to produce it.
To illustrate how the machine
does analysis let's start
with a square wave.
First, we use some unique properties
of the square wave.
Since it’s periodic
and even
all of the information that function
carries is contained
in half a period.
So we’ll take that half a period
and sample it at twenty
equally spaced points.
We use the values of these points
as the inputs for the machine.
When we turn the crank we produce
a new function that reveals
the correct coefficients.
To see how we get these coefficients
we’ll look at the side of the machine.
As I turn the crank the tips
of the rocker arms form a sinusoid.
The indices on the rocker arms run
from high on the left
to low on the right.
I’ll flip the video to
make it a little more intuitive.
When the rocker arm tips are
lined up in a straight line
we’ll call that crank number zero.
If the horizontal axis runs from
zero to pi
and the vertical axis from
minus one to plus one
then we can describe
the position of the rocker arms
with a cosine.
Every two turns of the crank
increases the frequency
of the cosine by one.
At these even numbered cranks
the values of the function
on the platen yield the coefficients
we’re looking for.
Let’s rewind and watch the plot
and the rocker arms simultaneously.
If we pause briefly at every
second crank a point is
marked on the function.
We create a total of 20 points.
If we look at the output
from the machine and
compare it to a sinc
calculated by a computer
we see that they are very similar.
Now, we’ll take the data points
from this sinc
scale them
and use these
values on the rocker arms.
Remember that our goal is to
program the analyzer so that
it will plot a square wave.
Now, as I turn the crank
the pen writes a horizontal line
then drops and writes
another flat section
which amazes me because
we’re adding only cosines
and then it rises to write
another horizontal line.
Of course, what we’re seeing is
a square wave.
What I’ve just shown
with the machine is
an essential feature
of fourier methods.
I can take a function
perform harmonic analysis
extract the coefficients
and then synthesize
that function to
approximate the original.
So, now that we see that
this machine can do harmonic synthesis
and analysis I’ll show you
in the next video
some details about how to
set up the analyzer to perform
these calculations.
I’m Bill Hammack
the Engineer Guy.
Next up in the series
is operation.
If you haven't seen
them already there's
also they intro
and synthesis videos.
You can learn more about
the book here.
And if you really want to
learn more about the book
watch the page by page.
If you're a fan of
oscillatory motion
you gotta watch the
bonus rocker arms video.