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The Double Elliptic Harmonograph is one of my favorites, and
for many years I had one set up in my basement. The basic design
of the double-elliptic harmonograph is shown at the left. The harmonograph
consists of two pendula hung end to end. The upper pendulum rod, F, terminates
in a universal joint which suspends it from the ceiling and allows it to
move freely in all directions. The lower pendulum, H, is hung from the
bottom of shelf K from a second universal joint. The figures are
drawn on a sheet of paper tacked to the shelf by a pen attached to a counterbalanced
arm. In my harmonograph the pendulum rods are oak two by fours about three
feet long, and four kilograms of slotted iron mass bolted to each pendulum
provide the motive energy.
The key to successful operation of any harmonograph which
relies on stored mechanical energy is low friction. Superimposed on all
of the motions is an overall damping factor which eventually brings the
pendula to a halt. If this damping is too large, the lines drawn by the
pen are too far apart, and the figure is uninteresting.
The mass on the upper pendulum is normally placed just
above the shelf. When the lower pendulum is removed, the harmonograph draws
the standard one-to-one Lissajous figures with the phasing controlled by
the initial motion of the system. That is, a phase difference of 90°
produces a circular Lissajous figure which gradually spirals in toward
the center as friction steals energy from the system. The placement of
the masses on the lower pendulum (often called the deflector) controls
the frequency ratio of the upper to lower pendulum. The system can be said
to be chaotic, as small changes in the initial conditions of the system
lead to greatly differing patterns. |
