Electrical equivalent of inverted pendulum

Question

Background

I am working on a summer project to design labs for undergraduate students. One of the topics is feedback and there is already quite a lot of stuff about passive feedback, so we might add some active feedback stuff. One really cool active feedback process is balancing the inverted pendulum (e.g. like in this video). I would like to create something like this, but purely electrical, because motors and mechanical parts are more expensive and cheap ones tend to jam quite easily (cheap enough for the labs is very cheap...).

What makes the inverted pendulum cool

So what we are looking for is an intrinsically unstable electrical configuration that needs to be actively adjusted (e.g. by a voltage from a programmable chip). Now one could e.g. think of a charged capacitor. It would discharge unless "balanced". But the balancing here is trivial, one just puts a voltage on and it becomes stable. In the mechanical analogy this would be just fixing the rotation axis in place. So another condition is that this "electrical pendulum" is unstable at all times like in the video and some parameter (e.g. a voltage) has to be adjusted constantly to keep it stable.

So the question is: Is there an electrical equivalent of this inverted pendulum balancing process?

Answer 1

While the paper linked to in sammy's answer does provide an equivalent circuit (page 5988), it seems of little practical use, since even in (unstable) equilibrium the current in L2 rises linearly with time. Within seconds (at most) it will reach a limit: saturation of the transformer core or maximum current (either I=U/R where R is the resistance of the transformer winding, or the power supply's maximum output current)

Apart from a NTC thermistor at some temperature above ambient (with the voltage constant, when temperature goes up resistance goes down, increasing the current and dissipated power, further increasing temperature; when temperature goes down, power drops, the NTC cools down further), I can't think of a solution with only passive components. With active components, an inherently unstable circuit is easy to make. The small angle approximation of a pendulum is equivalent to a mass on a spring, and an inverted one is equivalent to a spring with negative spring constant (F=-kx; with negative k, the force is in the direction of the displacement).

With that approximation, the equation for the pendulum becomes ${d^2\theta\over dt^2}={g\over \ell}\theta$

This can be simulated with two opamp integrator circuits:

The output of an integrator is $u_{\text{out}} = -\frac{1}{RC}\int_0^t u_{\text{in}}\, dt$

First integrator: With RC=1, if $u_{\text{in}}$ represents the angular acceleration, $-u_{\text{out}}$ will be the angular velocity.

Second integrator: With $RC={\ell\over g}$ , if $-u_{\text{in}}$ represents the angular velocity, $u_{\text{out}}$ will be $g\over \ell$ times the angular displacement.

With the output of the first integrator connected to the input of the second one and vice versa, the circuit will behave like the inverted pendulum (at small angles). It is inherently unstable and input offset voltage present in any real opamp will start it off. The voltage at the non-inverting input (+) of the first opamp (in a normal integrator circuit connected to the ground) represents the position of the pivot point, and balancing the pendulum can be done by adjusting that voltage. Note that dual supply opamps and a symmetric power supply are required to allow negative output voltage, since the pendulum must be able to swing both ways (no pun intended).

Answer 2

The following paper is devoted to electrical analogues to mechanical oscillators. Fig. 6 illustrates a simple coupled LC circuit which imitates the inverted pendulum on a cart given in Fig. 5. It does not include a circuit to balance the 'pendulum' as in the video.

https://core.ac.uk/download/files/508/12941830.pdf

I also found the following which may be useful. With much less mathematics than the above paper it shows how to construct electrical circuits which are analogs of mechanical systems.

http://lpsa.swarthmore.edu/Analogs/ElectricalMechanicalAnalogs.html