This astonishing machine uses just
gears and springs
to add together sinusoids.
This synthesis, as its called,
creates beautiful patterns
that look nothing like sinusoids.
The machine produces patterns like
beats
smooth peaks
sharp corners,
or even jagged edges.
And these can all be made
using this function.
Here’s how this mechanical marvel
implements that function.
The machine has five key sections.
First, a crank that varies x.
Second, gears & cams
that generate frequencies.
Third, just a little bit above that
rocker arms that
create the sinusoids
and amplitude bars to
weigh those sinusoids.
Fourth, at the top,
springs and levers that
add together these
weighted sinusoids.
And, lastly, on the front side
a mechanism for recording
this sum on a piece of paper.
At the heart of the machine
lies two sets of gears.
This cone-shaped set has
twenty gears that vary linearly
in both size and number of teeth.
It engages this cylindrical set of
twenty gears all of the same size.
Watch what happens as
I turn the crank.
The gears on the cone-shaped
set all rotate together.
The gears on the cylindrical set
can rotate independently
each at a rate determined
by the corresponding
gear on the cone.
This gear rotates twice
as fast as this gear
this one three times as fast
and so on.
These gears generate
the different frequencies.
A cam on each gear
drives a rocker arm
up and down
via a connecting rod.
As each gear rotates,
the corresponding rocker arm
traces out an up-and-down
or a sinusoidal motion
with the same frequency.
For example
the first rocker arm
oscillates to produce
cosine one x,
the tenth produces cosine 10x
and the twentieth, cosine 20x.
Of course, the arms in between
produce cosine two x
three x
four x
and so on.
At the bottom a series of
long vertical bars
rest on each rocker arm.
At the top these bars
attach to levers that
are spring loaded.
The position of the rocker arm
on the bar determines the coefficient.
As I slide this bar to the edge
of the rocker arm
the lever at top rises
that’s a coefficient of plus ten
when I return it to
the rocker arm’s pivot
the lever returns to
its original position
a coefficient of zero.
And as I continue to slide
to the opposite edge
the lever drops
a coefficient of minus ten.
Now, let me show you how the machine
sums these sinusoidal motions
weighted by their coefficients.
I’ll set the machine up
to produce beats,
using sin(12x) and sin(10x).
Each lever at the top of the machine
has a spring that connects it
to a pivoted bar.
A large spring balances the pull of
the series of springs on the levers.
If we look at the top we see the spring
extends more than
a foot above the machine.
As I turn the crank
the amplitude bars
drive the spring-loaded levers
up and down
this, in turn,
moves the pivoted bar.
Because the springs all pull on the
same pivoted bar their
displacements are summed.
If we take a close look
at the pivoted bar we see
that it sweeps out a small arc
we can barely see it go
up and down.
Let’s return to the front to see how
the machine amplifies this motion
and then records it.
This lever magnifies the result
from the summer
but it’s still a very small motion.
A wire connects the magnifying lever
to a set of concentric wheels
that amplify further the motion.
This wire pulls on the hub of
this inner wheel
while a different wire is wrapped
around the outer wheel.
The wire attached to the larger wheel
moves the pen vertically
as a set of gears
connected to the crank
has been moving the paper
horizontally under the pen.
This is the same crank that
rotates the gears to
create the sinusoids
and that also moves
the pen up and down.
Amazing as that synthesis is
that adding together of sinusoids
the machine does something
even more fantastic to me.
Given any periodic function
it will calculate the correct mix
of sines or cosines needed to
synthesis that function.
Understanding that process
called analysis, will
in the next video
enrich our understand
of the machine
by looking at its operation
from a different point of view.
I’m Bill Hammack
the engineer guy.
Watch analysis
the next video in the series.
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And, as a bonus,
you can watch the machine
spin around and around.