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I'm making a double pendulum for an art installation, and I would like to add energy to the system to compensate for friction and air drag losses so it can run more or less indefinitely.

The design so far has the top pendulum connecting to an axle at the top which is parallel to the ground and can rotate freely. So far I've sketched out a design which uses a rotational position sensor and a microcontroller to determine the pendulum shaft's current speed and position. A small, reversible electric motor will connect to the axle with a somewhat "limp" drive belt around the consistency of a rubber band (to lessen the impact of rapid chaotic changes mismatching with power input direction). As the axle rotates unpredictably, the control software will reverse or advance the motor to add a slight force in the current direction of rotation. Once tuned, the hope is that it will not perceptibly interfere with the natural motion of the pendulum while adding enough energy to keep the pendulum swinging with the same total energy as it started at.

Questions: 1) Are there mechanically simpler ways of adding energy to such a chaotic system? 2) Will energy added to the top pendulum distribute to the bottom pendulum in a way that would leave a casual observer unaware, or would the system develop obviously unnatural motion? 3) How can I roughly estimate (+-50%, to size the drive motor) the average energy input required to overcome friction and drag losses?

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  • $\begingroup$ I would suggest some more research into double pendulums. It is entirely plausible that artificially driving this is theoretically impossible as you'd be deterministically kicking a chaotic system. The accepted answer is not a good example as that toy has additional degrees of freedom compared to a double pendulum. $\endgroup$ – Paul Uszak Apr 13 '17 at 12:11
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Perpetual motion pendulum desk toys have a magnetic coil in the base that is pulsed at the correct pendulum displacement. This applies a small assisting force to the permanent magnet in the pendulum each pass. A question on Electrical Engineering SE explains the driving circuit.

If visually you can not accept a magnet passing near a coil, a DC motor would be an option too. I would probably not interface with a rubber band as that would be a very sloppy system with backlash to address. I would connect it direct drive or with a toothed belt. You could develop a circuit or micro-controller program that would provide positive feedback to the motor. Basically reading the voltage from the motor and adding a percentage to that. The polarity would change clockwise vs counter clockwise as would the assisting current.

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  • $\begingroup$ Excellent info, thanks! I'm curious about the motor approach - would this entail a loop of read voltage -> set voltage -> read voltage, or is there a way to do this continuously using a single motor? Are there any commonly available types of motor that have 2+ isolated circuits (read from one, set another)? $\endgroup$ – crypticsymbols Apr 13 '17 at 5:23
  • $\begingroup$ All dc motors generate "back emf" when turning. A motor that is not moving has no back emf and will draw lots of current (locked-rotor amps). A motor that is turning at a constant rpm with zero friction or load will have a back emf equal to the applied voltage. Back emf can be measured intermittent or continuously as explained in this EE SE answer: electronics.stackexchange.com/questions/54997/… $\endgroup$ – ericnutsch Apr 13 '17 at 6:35
  • $\begingroup$ @ericnutsch How would you know the "correct time" as by definition the lower AND upper pendulum swings are totally unpredictable? Don't forget that adding energy in a regular fashion to a chaotic system might dampen it or bring it into synchronisation. $\endgroup$ – Paul Uszak Apr 13 '17 at 12:06
  • $\begingroup$ @PaulUszak, I was referring to a single pulse being engaged ahead of the displacement as opposed to a continuously applied torque. But good catch that could be confusing; I will edit it. $\endgroup$ – ericnutsch Apr 14 '17 at 0:47
  • $\begingroup$ Ah no. That's exactly what I meant not to do. The system is chaotic. Are you sure that it will remain so if you keep kicking it regularly, and when would you know when to kick it? It's not a clock and the swing isn't periodic - it's random. Have a look at YouTube and you'll see that... $\endgroup$ – Paul Uszak Apr 14 '17 at 2:07
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To show a complete free falling double pendulum, a clutch separating your motor would work sufficiently.

I'd still recommend a position and angular velocity sensor - when the magnitude of rotation slows down below a specified set point, have the motor begin speeding up the clutch to a speed beyond the angular rotation of the motor. When the angular rotation is in the correct direction, engage the clutch. Then have a random number generator dictate how long the clutch stays engaged, to ensure every second of the demonstration is new. During this energization, this will be noticeable to the eye, but it will make for a perpetual double pendulum that acts truly and completely random.

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