Machine Kinematics
solved, then drawn
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The thread: The singularities are the point

Dead centres, toggle positions and the configurations where a mechanism locks or changes branch are treated as the subject rather than as edge cases. They are where the mechanical advantage goes to infinity and where the machine stops working.
0153045600100200300crank angle (degrees)mechanical advantage (clipped at 60)toggle: advantage 1673worst μ = 46°four-bar 3.4/1.2/3/2.4, 720 solved positions222° apart Drawn wrongly

Two things called jamming

A four-bar's mechanical advantage peaks at 1,673 in one configuration and its transmission angle collapses in another, 222° away. Both get described as the mechanism jamming. One is enormous force output and the other is force disappearing into the bearings, and they are opposite situations.

 the redundant oneGrübler: 3(5−1) − 2(6) = 0Jacobian: 6 − rank 5 = 1 What can move

The mechanism Grübler says cannot move

Three parallel bars between two frames. Five links, six pins, and the criterion every engineering course teaches gives zero degrees of freedom — a structure. It is a mechanism, it is in drafting machines and locomotive coupling rods, and the formula cannot see why.

 0501001500100200300crank angle (degrees)transmission angle μ (degrees)40° design limitground 4, crank 1, coupler 3.5, rocker 3μ from 54.3° to 100.3° Linkages

The transmission angle

The angle at which the coupler meets the rocker decides how much of an applied force becomes useful output torque and how much goes into the bearings. It is pure geometry, it is computed here from every solved position rather than from a formula, and it is the number a linkage is judged by after Grashof has said it turns.

 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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