 This astonishing machine uses just gears and springs to add together sinusoids. This synthesis, as it's called, creates beautiful patterns that look nothing like sinusoids. The machine produces patterns like beets, 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, to crank the various Xs. Second, gears and 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 the sum on a piece of paper. At the heart of the machine lies two sets of gears. This cone-shaped set has 20 gears that vary linearly in both size and number of teeth. It engages the cylindrical set of 20 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 in up and down or a sinusoidal motion with the same frequency. For example, the first rocker arm oscillates to produce cosine 1x, the 10th produces cosine 10x, and the 20th cosine 20x. Of course, the arms in between produce cosine 2x, 3x, 4x, 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 arms 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 10. 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 10. 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 sine 12x and sine 10x. Each lever at the top of the machine has a spring that connects it to a pivoted bar. A large spring balances the pole 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 pin vertically, as a set of gears connected to the crank has been moving the paper horizontally under the pin. This is the same crank that rotates the gears to create the sinusoids and that also moves the pin 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 synthesize that function. Understanding that process, called analysis, will in the next video enrich our understanding of the machine by looking at its operation from a different point of view. I'm Bill Hammack, The Engineer Guy.