• Name: Tom DeFayette and Gene Clark
• Address: Trinity University, Department of Physics
• Phone 210-999-7380 Fax is 210-999-7423
• email: tdefayet@trinity.edu eclark@trinity.edu

A Double Pendulum Constructed From Inline Roller Skate Wheels

Abstract

The double pendulum is a popular device for demonstration of deterministic chaos. It is mechanically simple and has visually impressive dynamics. However, it must have low friction in order to sustain undriven oscillations long enough to make the chaotic character of the motion obvious. A design for a very low friction double pendulum that is large enough for class demonstrations has been published. However, this design requires machinist skills and access to a machine shop. This note presents a design that has the properties of low friction and large size required for a successful classroom demonstration and the advantage that it can be constructed entirely from in-line roller skate wheels and common wooden and metal rod stock. No machining or skill is required for construction.

Parts List for One Inline Roller Skate Double Pendulum

Quantity Item (all units in inches, unless noted otherwise)

Note. All items except inline roller skate wheels can be purchased at a hardware store. Dowels and threaded rods are normally sold in 2 or 3 foot lengths. Use the wood saw to trim dowels and a hacksaw with a 24 tooth blade to cut the threaded rod.

Inline Roller Skate Double Pendulum Tool List

• Use a 11/32 drill bit to drill holes into four wheels for the wooden dowel --
• Use a 27/64 drill bit to drill hole into the wheel for the 5/16-18 x 6 1/2 long threaded rod (lower pendulum arm)
• Vise to hold wheels when drilling holes for dowels and threaded rod
• Wood crosscut saw.
• Hacksaw with 24 tooth blade
• Tape measure or ruler.
• 1/4"Flat tip screwdriver

Safety Concerns

Assembly

• 1. Assemble all materials from the Parts List before proceeding.
• 2. If using used skate wheels, remove bearings and unthreaded axles from wheels by pressing the bearing out by hand or with a screw driver. These bearings are easily removed and inserted. New bearings and axles are sold separately from the wheels.
• 3. Clean bearings, old or new, by soaking in mineral spirits. If the bearing has a dust/dirt cap, remove cap with a small screwdriver. New and used inline roller skate bearings are packed with white lithium grease. This grease hinders rotation of the bearing and therefore movement of the pendulum. Soak the bearings in mineral spirits for approximately one hour. You may need to use a cotton swap to remove the grease after soaking. If compressed air is available it can be used to blow out the grease. It is especially important to wear safety glasses during this procedure since the bearing will spin rapidly in the race while you are holding onto it. It will not be necessary to replace the dust/dirt cap. Place the bearings and the unthreaded axles off to the side. You will use them latter.
• 4. Drill holes in wheels. Before inserting or reinserting the bearings and axles, drill required holes in the five wheels. Four wheels will be drilled with the 11/32" drill bit. One wheel will be drilled with a 27/64" drill bit. Drill holes as deep as possible, but do not drill into the plastic wheel hub where the bearing will be located. Ensure you drill your hole perpendicular to the hub and not at an angle. If you do not do this correctly, turn the wheel over and try drilling a hole correctly on the other side of the wheel.
• 5. Insert bearing and unthreaded axle into each wheel. This is a simple process and done without tools. Reverse procedure you used for removing the bearings from old wheels or follow the instructions which came with new wheels.
• 6. Insert wooden dowels. Insert hardwood dowels into the holes of the four wheels having a 11/32" hole. This is a simple press fit. Using only your hands, press the dowel into the hole as deep as possible. Two wheels will be connected together by a dowel. Glue is not required to hold the dowel in the wheel. Do not use any lubricant on the dowel or in the wheel hole. You can add a small amount of rubber glue or a poly-acrylic resin (sold as a "clear sealant" at hardware stores) in each hole before inserting the dowel. However, you may not be able to latter remove the dowels from the wheels if you use an adhesive. Using an adhesive may interfere with your ability to conduct such experiments as the affect of pendulum arm length on oscillations.
• 7. Insert the 5/16 -18 threaded rod. Insert the 5/16 - 18 x 61/2 threaded rod into the wheel having the 27/64" hole. This rod should be hand screwed into the wheel hole. Screw the rod into the hole as far as possible. No abrasives or lubricants are necessary. The threaded rod will tap threads in the wheel hole. This wheel and threaded rod comprises the bottom arm of your double pendulum.
• 8. Assemble the top axle. The 1/4 -20 x 10" threaded rod is the axle for the top pendulum arm. Insert this rod through two of the skate wheels having a 11/32" hole. Use four of the hex nuts and four nylon washers to position the wheels on this axle. Hand-tighten the hex nuts. The wheels should be separated approximately 2 inches as measured from the center of each wheel; not the side edges of the wheels. These dimensions are not critical. Experiment and see what happens. After completing this step, you have two skate wheels skewed on a threaded rod. Fasten the top axle with wheels to a sturdy object using the c-clamps and wood pieces. The wood pieces grip the threaded rod between the c-clamps and the sturdy object.
• 9. Assemble the lower axle. The 1/4 - 20 x 4" threaded rod is the axle for the lower pendulum arm. This lower axle connects the two wheels hanging on wooden dowels from the top pendulum. Placed in between these two wheels is the wheel having the 5/16 - 18 x 61/2 " threaded rod. Place two nylon washers between each wheel to separate the wheels on the axle. Use the hex nuts to secure all the wheels by placing one hex nut on each end of the threaded rod lower axle. Hand-tighten only. The axle should protrude from the hex nut by approximately 1/2 inch. Do not use thread locker at this time on your hex nuts.
• 10. Test your double pendulum. Carefully swing the pendulum. The pendulum should move smoothly, not make noises, and not come apart or loosen at the joints. Look for any unstable joint or connection and any erratic movements during the swings. Check the c-clamps holding the top axle to ensure they stay tight. When you are satisfied that the movement of the double pendulum is smooth and that the apparatus is safe, loosen each hex nut and place one drop of the thread locker on the threaded rod where the hex nut will be located. Hand tighten the hex nut to where it was located before adding the thread locker. The thread locker will keep the hex nut from loosening on the threaded rod. The thread locker will "lock" the hex nut in position in about 6 hours. As an alternative to using thread locker, you can use teflon pipe thread tape. Wrap a small piece of teflon pipe thread tape around the threaded rod and hand tighten the hex nut over the tape.

Optimizing the design

There are several ways to optimize the oscillation of your double pendulum. The double pendulum shown in our picture has sections cut from the lower wheels in order to reduce the mass at the lower axis of rotation. Use a hacksaw blade to trim pieces of the wheel. The oscillations are affected by many variables which impact on the pendulum's movement. We attempted to increase the amplitude, frequency, and period of the oscillation by screwing nuts of different masses onto the lower pendulum arm. Since the wooden dowels are easily disconnected from the wheels, you can experiment with various lengths of dowels. As with all pendulums, your double pendulum's oscillations are affected by friction and the moment of inertia of the components. The design presented here has the advantage of producing many chaotic oscillations of the lower pendulum. If the lower pendulum has a smaller moment of inertia it will swing at a much larger angular velocity and so will lose energy to friction at a faster rate.

How long should the lower pendulum continue to swing over the top? When constructed using these plans the lower pendulum will produce about 30 chaotic oscillations when dropped from the point of unstable equilibrium above the support point for the upper pendulum.

How can you record quantitative data from this pendulum? You can record quantitative data from the double pendulum by attaching a magnet to any point whose positions you want to record. Use a Hall effect sensor to detect the position. A time series of the Hall probe output with a magnet on the end of the lower pendulum is usually sufficient to diagnose chaos.

Reference: "Chaos in a double pendulum," Troy Shinbrot, Celso Grebogi, Jack Wisdom, and James A. Yorke, Am.J.Phys. 60, 491-499 (June 1992).