Double Elliptical Pendulum Harmonograph

The following is an except from a letter published in the column from a father-daughter team of Concord, Massachusetts:

"Our most recent machine is a version of the elegantly simple double-elliptic pendulum first described in 1907 by British experimenters. It consists of a small wooden platform weighted with 50 pounds of iron [see illustration at lower left]. The platform is suspended through a yoke and a length of iron pipe by ball-bearing gimbals fixed to the ceiling. The figure is drawn on paper that is taped to the platform; the drawing instrument is a pen attached to one end of a counterweight level arm. This simple arrangement generates only circles, ellipses, and straight lines. The interest of these figures can be enhanced by making identical drawings and superposing one on the other with a slight displacement; moiré patterns appear."

"The most astonishing increase in the versatility of the machine can be made, however, by suspending a second, less massive pendulum from the bottom of the platform, again using a gimbal arrangement. When it vibrates its own natural period, the shorter pendulum perturbs the motion of the swinging platform in an infinite number of ways. The handsomest patterns are generated when the frequency of the upper pendulum bears a whole-number ratio to that of the lower pendulum - a ratio such as 3:2 or 2:1. The adjustable factors that describe any given figure are then the two amplitudes of the upper pendulum, the phase angle between them, the phase between upper and lower pendulum, the phase and amplitudes of the lower pendulum and the ratios of the frequencies of the upper and lower pendulums. We have scarcely begun to investigate the variety of patterns that can be generated with the double-elliptic pendulum machine."

I wrote to C. L. Strong in 1966 with questions before I built mine, and again later with samples of my patterns.

I adapted my own design from the information and illustrations in the column, and additional feedback from Mr. Strong. I designed it to just clear the concrete basement floor in our house and to be suspended from the floor joists overhead as shown in the column.

Finally, in March, 1967, I had iron pipe cut and threaded. I spent $40 on eight ball-bearing "pillow blocks" to construct the needed U-joints. I scavenged the 50 pounds of sash weights from Inland Metals in Spokane.

I must have drawn a scale drawing and somehow figured out what lengths of pipe I needed. I'm surprised that I don't seem to have my original "plan". It must have been an interesting design and construction project because of the way all the components related to one another. I probably built it from the top down.

This could be an expensive project today depending on your access to pipe-cutting and threading tools, etc. For example, I had the ends of eight of the pcs of pipe milled rather than threaded so that they would fit into the bushings of the pillow blocks. I was lucky to have the old-fashioned West Valley hardware store a block way where the man often had time on his hands to do some of these things for me.

After getting it built, I found that the pen apparatus was just as important as the pendulum. I spent quite some time perfecting my control over that design - otherwise, tracing would be botched, or the level of friction would be too high, etc. You must be able to lower and raise the pen gently and cleanly, the pen can't be too massive, or too light.

Because I had a lightweight counter-balance pen arm, it turned out that I needed to add pressure by means of a delicate spring to keep the pen from continuing upwards and leaving the paper at times, resulting in 'skips'. (As the DEPH swings, the pen must be able to move up and down. -- the paper is flat, and the table is swinging in a spherical section.) The lowering, pressure-setting, and raising were all controlled by a single lever.

To store the DEPH out of the way, I would: remove the lower pendulum; open the top of the writing platform and remove the sash weights; unscrew the device from the flange at the top; put the whole thing out of the way.

The DEPH is currently on loan to the Physics Department at Lewis & Clark College in Portland, Oregon.

Before building the DEPH, I experimented with some other setups described in the column. I started by taking time-lapse exposures in a darkened room -- shooting straight up at a point of light swinging freely in "circles". Since it's virtually impossible to start a freely-suspended object swinging in a perfect circle, the ellipse will rotate and also spiral down due to friction. The locus of this orbit can be quiet beautiful. The suspended flashlight might also twirl or wobble adding variation. The 'point' of light can be a slit, making the width of the light line vary as the flashlist orbits. I had a darkroom, so we could shoot the film, then take it right into the developing process all in the same evening.

I also made a crude Twin Pendulum Harmonograph, also shown in the column, but it didn't produce very interesting results.

It's always a good idea to keep a detailed, dated, lab notebook for such projects. I didn't.

Bibliographic refs from Sci Am, not too useful:

Harmonic Curves, William F. Rigge, SJ, The Creighton University, 1926.

Harmonic Vibrations and Vibration Figures, Joseph Goold, Charles E. Benham, Richard Kerr and L.R. Wilberforce, edited by Herbert C. Newton, Newton & Co., 1909.