Twin Elliptic Pendulum Harmonograph Drawings

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Ratios Larger Than 1:1


The Twin Elliptic Pendulum Harmonograph combines two separate conical motions around a single axis by suspending a “deflector” pendulum below the main pendulum. There are several different ways to configure a Twin Elliptic Pendulum, but they are usually comprised of a moving drawing table with a pen arm that floats up and down vertically from a separate fixed support.

Interesting Properties of the Twin Elliptic Pendulum 

With the Twin Elliptic Pendulum, a 1:1 ratio cannot be achieved. The nicest drawings are produced when the ratios are higher, such as with a 3:1 or 5:3 ratio. It can be demonstrated that when both pendulums are rotated in the same direction (concurrent motion) at a ratio of X:Y,  the drawings produce X-Y cusps (or loops); and if the pendulums are rotated in opposite directions (counter-current motion), then a ratio of X+Y cusps is produced. When the X and Y are both odd, symmetrical drawings are produced in both concurrent and counter-current pendulum motion (examples: 3:1 or 5:3 ratios). The 2:1 motion produces a deltoid figure and the 3:1 produces the asteroid; with more complex hypocycloids created at other ratios. 

Motion and Ratio Adjustments  

It takes a bit of practice to achieve counter-current motion with the Twin Elliptic Pendulum. It also takes some practice (and a bit of trial and error) to produce a perfect “tuning” of the pendulum periods so that precise ratio’s of motion can be achieved.  As a simple rule of thumb:  You increase the ratio by either raising (or lightening) the upper pendulum; or lowering (or adding weight) to the lower deflector pendulum.  Note that frictional dampening becomes more apparent as the x and y are increased. 

Click on the drawings below to see them full sized.


3:1 Ratio Examples (corresponds to a musical 5th):

Concurrent Motion, 3:1 Ratio (3-1=2 cusps in the example above)

Counter-current Motion, 3:1 Ratio (3+1=4 cusps in the example above)

Introducing a more linear (back and forth) pendulum motion, rather than circular, dramatically changes the design, but the ratio is still 3:1

3:1 Motion Ratio (with linear aspect)

X:Y Unknown