I don't think that there is any scientific justification for Harmonographs, unless one remembers that they are used to draw the fancy repeating scrollwork on paper currencies. But physicists and civilians continue to be drawn to them, fascinated by the patterns they produce.

   Harmonographs are mechanical devices that trace out the resultant of two or more simple harmonic motions, typically along mutually-perpendicular axes. The device at the left is a modification, ca. 1875, by Herbert Newton of an earlier design by S.C. Tisley 

   I first saw and used this type of harmonograph as a junior physics major at Amherst College in the fall of 1957. It had been built by Robert Romer of the physics faculty.

   Since then I have built several similar devices, often quite crude, but always including ball-bearing joints instead of the knife-edge bearings used in this example, which is in the Garland Collection of Classic Physics Apparatus at Vanderbilt University.


   This more complicated harmonograph is in the collection of historical apparatus at Harvard University. Like the apparatus above, it has two pendula swinging at right angles to each other. However, the left-hand pendulum's suspended mass can be drawn up by the clockwork mechanism in the center. The circular stage also has a rotating top, thus adding two more simple harmonic motions at right angles to each other with the same frequency but a phase difference of ninety degrees.

   The design is by S.C. Tisley, 1878.

   The firm of Tisley and Spiller of London manufactured this double-pendulum harmonograph, first described by S.C. Tisley in 1873, for use in the lecture hall. The two pendulums drive a stylus that scrapes the carbon from the surface of a smoked glass plate placed on the built-in overheat projector. Consequently, the audience saw a black screen on which a white line traced out the harmonic figure. 
      The 1889 catalogue of acoustic apparatus published by Rudolph Koenig of Paris lists this apparatus as "Wheatstone's apparatus for mechanically compounding two rectangular vibratory motions" at a price of 300 francs. A small bead at the top of the pivoted, upright rod on the left-hand side is driven by two simple harmonic motions at right angles to each other. In a fashion similar to the kaleidophone , also invented by Wheatstone, the apparatus is operated in a dark room, and the image of a single bright light source is observed by reflection from the bead. 

  This apparatus is at Harvard University..

  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. 

REFERENCES: 1. Robert H. Romer, "A Double Pendulum 'Art Machine'", Am. J. Phys., 38, 116 (1970)
                          2. Robert. J. Whitaker, "Harmonographs. I. Pendulum Design", Am. J. Phys., 69, 162-173 (2001)
                          3. Thomas B. Greenslade, Jr., “The Double-Elliptic Harmonograph”, Phys. Teach., 36, 90-91 (1998)
                          4. Thomas B. Greenslade, Jr., “Nineteenth Century Textbook Illustrations XXVII, Harmonographs",
                                                                                                               Phys. Teach. 17, 256-8 (1979)

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