Sun and Planet

By Spencer Connor

A bit of a curious arrangement, and an answer to the not-so-theoretical question of “how do I emulate a crank, but without using an actual crank because it’s patented”? In that effect it’s not the most direct solution, but it does have some unique qualities that make it more interesting. First is that it has an integrated gear reduction that can be tailored. which can be useful. Also fairly unusual (and mildly annoying for modelling) is that its equations of motion are transcendental so there is no algebraic solution. The reason for this is that while the planet gear doesn’t spin per se it does rotate back/forth based on the angle of the arm, which is determined from the meshing of the input gear with the planet gear, which is sort of like a mechanical circular reference. So you have to solve for the angle of the arm with a numerical solver, then solve for the height from that (thus the two equations below, the first of which being the annoyance). Like most signal generators of this type, as the amplitude of the signal gets smaller relative to the leg length it approaches being a sine wave. Additionally, the varying arm motion speed can be used as an output to create a pulsed rotation (lower animation).