English subtitles for clip: File:2 of 4 - Synthesis -A machine that uses gears, springs and levers to add sines and cosines.webm

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This astonishing machine uses just

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gears and springs

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to add together sinusoids.

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This synthesis, as its called,

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creates beautiful patterns

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that look nothing like sinusoids.

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The machine produces patterns like

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beats

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smooth peaks

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sharp corners,

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or even jagged edges.

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And these can all be made

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using this function.

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Here’s how this mechanical marvel

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implements that function.

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The machine has five key sections.

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First, a crank that varies x.

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Second, gears & cams

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that generate frequencies.

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Third, just a little bit above that

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rocker arms that

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create the sinusoids

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and amplitude bars to

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weigh those sinusoids.

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Fourth, at the top,

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springs and levers that

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add together these

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weighted sinusoids.

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And, lastly, on the front side

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a mechanism for recording

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this sum on a piece of paper.

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At the heart of the machine

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lies two sets of gears.

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This cone-shaped set has

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twenty gears that vary linearly

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in both size and number of teeth.

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It engages this cylindrical set of

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twenty gears all of the same size.

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Watch what happens as

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I turn the crank.

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The gears on the cone-shaped

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set all rotate together.

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The gears on the cylindrical set

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can rotate independently

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each at a rate determined

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by the corresponding

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gear on the cone.

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This gear rotates twice

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as fast as this gear

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this one three times as fast

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and so on.

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These gears generate

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the different frequencies.

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A cam on each gear

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drives a rocker arm

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up and down

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via a connecting rod.

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As each gear rotates,

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the corresponding rocker arm

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traces out an up-and-down

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or a sinusoidal motion

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with the same frequency.

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For example

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the first rocker arm

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oscillates to produce

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cosine one x,

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the tenth produces cosine 10x

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and the twentieth, cosine 20x.

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Of course, the arms in between

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produce cosine two x

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three x

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four x

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and so on.

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At the bottom a series of

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long vertical bars

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rest on each rocker arm.

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At the top these bars

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attach to levers that

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are spring loaded.

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The position of the rocker arm

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on the bar determines the coefficient.

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As I slide this bar to the edge

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of the rocker arm

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the lever at top rises

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that’s a coefficient of plus ten

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when I return it to

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the rocker arm’s pivot

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the lever returns to

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its original position

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a coefficient of zero.

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And as I continue to slide

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to the opposite edge

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the lever drops

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a coefficient of minus ten.

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Now, let me show you how the machine

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sums these sinusoidal motions

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weighted by their coefficients.

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I’ll set the machine up

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to produce beats,

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using sin(12x) and sin(10x).

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Each lever at the top of the machine

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has a spring that connects it

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to a pivoted bar.

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A large spring balances the pull of

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the series of springs on the levers.

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If we look at the top we see the spring

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extends more than

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a foot above the machine.

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As I turn the crank

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the amplitude bars

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drive the spring-loaded levers

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up and down

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this, in turn,

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moves the pivoted bar.

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Because the springs all pull on the

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same pivoted bar their

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displacements are summed.

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If we take a close look

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at the pivoted bar we see

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that it sweeps out a small arc

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we can barely see it go

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up and down.

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Let’s return to the front to see how

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the machine amplifies this motion

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and then records it.

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This lever magnifies the result

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from the summer

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but it’s still a very small motion.

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A wire connects the magnifying lever

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to a set of concentric wheels

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that amplify further the motion.

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This wire pulls on the hub of

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this inner wheel

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while a different wire is wrapped

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around the outer wheel.

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The wire attached to the larger wheel

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moves the pen vertically

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as a set of gears

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connected to the crank

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has been moving the paper

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horizontally under the pen.

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This is the same crank that

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rotates the gears to

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create the sinusoids

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and that also moves

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the pen up and down.

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Amazing as that synthesis is

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that adding together of sinusoids

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the machine does something

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even more fantastic to me.

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Given any periodic function

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it will calculate the correct mix

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of sines or cosines needed to

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synthesis that function.

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Understanding that process

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called analysis, will

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in the next video

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enrich our understand

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of the machine

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by looking at its operation

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from a different point of view.

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I’m Bill Hammack

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the engineer guy.

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Watch analysis

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the next video in the series.

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Click here if you want to

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learn more about the books and posters

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for sale.

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Or, watch this video

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where we go through every

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single page in the book.

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And, as a bonus,

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you can watch the machine

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spin around and around.