Machine Kinematics
solved, then drawn
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The thread: Somebody had a problem

Every mechanism in the canon was invented to solve something specific — a piston that had to move straight, an indicator that had to trace, a film that had to stop thirty times a second. The problem explains the shape.
roller followerbase circle 30cycloidal, rise 20 over 120°, roller 8pressure angle 25.5° Prescribed motion

Prescribing motion

A linkage gives the motion its geometry allows. A cam gives the motion it was asked for, which sounds like an improvement and is a trade — the displacement becomes free and the derivatives stop being.

 ground 4, crank 1, coupler 3.5, rocker 35 attachment points, 150 solves each The paths points trace

What a coupler point draws

A point rigidly attached to the coupler of a four-bar traces a curve of degree six. Move the attachment a little and the curve changes a great deal. For most of the twentieth century the practical way to find the linkage that draws a wanted curve was to look it up in a book of printed atlases.

 10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹0%25%50%75%100%Watt, 1784Chebyshev, 1850sPeaucellier, 1864fraction of the available stroke useddeviation ÷ span120 solved positions per pointfifteen decades, and one flat line The paths points trace

The straight-line problem

Before 1800 a long true flat surface was harder to make than almost anything else, so guiding a piston straight without a slide was worth solving. Watt's answer was an approximation. Measuring how good an approximation, over how much of the stroke, turns out to be a more interesting question than whether it is exact.

 driver6 slotscentre distance 60.00 = crank ÷ sin(180°/6)index 60° per turn Prescribed motion

Stopping thirty times a second

A Geneva wheel turns continuous rotation into steps, and its one design requirement is that the pin enters the slot along the slot so the driven wheel starts and stops from rest. That fixes every dimension from the slot count. What it does not fix is the acceleration, which is why film sprocket holes tear.

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