postheadericon How to Make a Three Pendulum Rotary Harmonograph

A harmonograph is a mechanical device that uses swinging pendulums to draw pictures. This particular harmonograph was designed by Karl Sims[1] but harmonographs in general are believed to be originally invented in 1844 by Scottish mathematician Hugh Blackburn. There are other types of harmonographs, but this 3-pendulum rotary type gives a wide variety of pleasant results, and is fairly easy to build once you’ve settled on a design and have acquired the appropriate materials and tools. This is a great project to do with kids and can result in endless experiments creating new types of geometric designs.

Two lateral pendulums swing back and forth at right angles to each other with arms connecting to a pen. One moves the pen from side to side, and the second moves it from front to back on the paper. A third “rotary” pendulum moves the paper by swinging on any axis or in circular motions, while the pen is drawing on it. The combined motions of all three pendulums generate the resulting drawing.

How to Make a Three Pendulum Rotary Harmonograph
How to Make a Three Pendulum Rotary Harmonograph
How to Make a Three Pendulum Rotary Harmonograph
How to Make a Three Pendulum Rotary Harmonograph

1 Start by building a sturdy table. This table top is a 3’x3′ square of 3/4″ thick plywood. The legs are 1?” x 1?” square and about 37″ long, with triangular braces cut from 1?” x 8″x 12″ wooden pieces. The legs are splayed out slightly to give the table strength, and to allow the rotary pendulum to swing without hitting a leg. Screw and/or glue the braces to the legs first, and then cut their tops together at a slight angle with a table or circular saw.

How to Make a Three Pendulum Rotary Harmonograph

Adjust the leg lengths to give a table-top height of about 37″.

How to Make a Three Pendulum Rotary Harmonograph

If you want to be able to fit the table through doorways without taking off the legs, you might need slightly shorter legs and pendulums.

How to Make a Three Pendulum Rotary Harmonograph
How to Make a Three Pendulum Rotary Harmonograph

2 Drill holes for the pendulums. Drill 3 holes of 3″ diameter through the table surface for the pendulums to hang through. The hole for the rotary pendulum should be centered in a corner about 8″ from each side just clear of the leg brace underneath. The other two holes should be aligned near the opposite edges, about 8″ from the common side, and 3″ from the other. You’ll need a special large circular drill bit for this. Alternatively you can first drill a smaller hole, and then cut a wider opening with a jigsaw.

How to Make a Three Pendulum Rotary Harmonograph

3 Mount two metal plates (about 1?” x 4″) on the sides of the two lateral pendulum holes, and drill a small indentation in the center of each plate. Start the indentation in the metal plate with a small drill bit (such as 1/8″) and then continue with a larger bit (such as 1/4″). Be careful not to drill all the way through.

How to Make a Three Pendulum Rotary Harmonograph

Unless you have a good drill press, it may be easier to position the indentations of the plates on the table after you create the fulcrum blocks with protruding screws below, because it can be harder to accurately position the screws in the blocks later to align with the indentations.

How to Make a Three Pendulum Rotary Harmonograph

4 Build the pendulums. The pendulum shafts are made from 4′ long 3/4″ diameter wooden dowels. Hardwood blocks, 3/4″ x 1?” oak, are used as a fulcrums so the pendulums can rest and rock on the table with minimal friction. For the two lateral pendulums, use 5″ long blocks, and for the rotary pendulum, use a 2?” long block. Drill a 3/4″ hole through the center of each block, and screw 1?” #10 screws through each end. The tips of the screws will rest in the indentations of the metal plates on the table to keep them in place while rocking.

How to Make a Three Pendulum Rotary Harmonograph

5 Insert a dowel through the 3/4″ hole in each fulcrum block. Glue and/or screw them such that the screw tips are 12″ from the top end of the dowel and facing downwards. The pendulum should hang down 36″ below the top surface of the table with about a 1″ clearance from the floor.

How to Make a Three Pendulum Rotary Harmonograph

6 Build the gimbal. The rotary pendulum needs a gimbal mechanism that allows it to swing in any direction. This is made from a large metal washer with 2?” outer and 1″ inner diameter. The washer rests on screw tips protruding from under the table, and then the pendulum rests on the washer. Drill pairs of indentations into the washer on each side, off 90 degrees between the sides, to allow rocking on two perpendicular axes. Here are views of the rotary pendulum gimbal from above and below the table. In the view from below, note the oak blocks (3/4″ x 1?” x 5″) supporting the two screws (1?” #10) protruding upwards and diagonally.

How to Make a Three Pendulum Rotary Harmonograph
How to Make a Three Pendulum Rotary Harmonograph

7 File down the bottom edges of these blocks as needed so the pendulum doesn’t hit them when it swings. Likewise file down the corners of the fulcrum block if necessary so it doesn’t hit the table when swinging.

How to Make a Three Pendulum Rotary Harmonograph

8 Attach the weights. 2? lb lifting weights from a sporting goods store work well, but typically have a 1″ interior hole. You can stack several weights and slide them together onto the 3/4″ pendulum dowel by using a 5″ long 3/4″ metal pipe nipple, with a 3/4″ to 1″ bushing screwed onto the lower end. A 1″ steel clamp attached to the dowel fixes the weights from sliding off, and allows easy adjustment of the weight’s height to give different swinging frequencies.

How to Make a Three Pendulum Rotary Harmonograph

9 Create the paper platform. Cut about 1″ off the top of the rotary pendulum dowel, so it is slightly lower than the other two. Then mount an 11″x11″ square of thin 1/8″ board to the top of this pendulum, using a small oak block glued to it for support, with a 3/4″ hole for the dowel. Wrap some tape around the top of the dowel to get a tight fit, or just glue it on.

How to Make a Three Pendulum Rotary Harmonograph

Use two rubber bands, or some clips, to hold the paper in place on the platform.

How to Make a Three Pendulum Rotary Harmonograph
How to Make a Three Pendulum Rotary Harmonograph

10 Create the arms. Connect a 30″ long balsa stick to the top of each lateral pendulum using a thin nail. Bend the nail back and forth a little in the balsa to allow the the arm to rotate smoothly and move up and down slightly. The nail hole will slowly loosen further during use.

How to Make a Three Pendulum Rotary Harmonograph

11 Finally attach the two arms together with a doubled-over rubber band as shown. Note that if you plan to use your harmonograph regularly, such as in a museum setting, a more robust solution for these arms that won’t wear as quickly might be necessary.

How to Make a Three Pendulum Rotary Harmonograph

12 Fashion a pen holder. To make a simple pen-holder, drill a 1/2″ hole on the end of one arm, and cut about 4″ down the center of the arm to make a clothes-pin like device. Alternatively, just glue a real clothes-pin to the end of one of the arms. Pictures of both versions are shown.

How to Make a Three Pendulum Rotary Harmonograph

13 Make the pen lifter. It is convenient to be able to raise and lower the pen gently without disturbing the motion of the pendulums. To do this, insert a 30″ pole into a hole near the center of the table, just far enough from the paper platform that it won’t be hit by it (about 12″ from the rotary pendulum hole).

How to Make a Three Pendulum Rotary Harmonograph

14 Attach an oak block under the table for a deeper hole and better support.

How to Make a Three Pendulum Rotary Harmonograph

15 Tie a string to the balsa arms where they are connected together, lead it through a screw-eye on top of the pole, and back down to a small jam cleat or groove where it can hold the pen in place above the paper until you are ready to lower it.

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