# A mechano-electro-magnetical control circuit

I came up with the following mechano-electro-magnetical control circuit. The system as depicted here is in equilibrium, i.e. the upper spring s1 is relaxed , the upper circuit is closed (C1 producing a magnetic field), the lower spring s2 is slightly stretched, the lower circuit is open (C2 not producing a magnetic field). I wonder how this system behaves when m2 is driven into the direction of the magnet/coil C1 by an external force.

Eventually the magnetic force will exceed the force of spring s2, m2 will jump and close the contact. C2 will create a magnetic field that pushes away m1, opening the upper circuit. The magnetic field created by C1 will vanish, m2 will be drawn back by s2, and so forth.

Possibly, a limit cycle will be reached, possibly the stable ground state will be reached, depending on parameters like spring and coil constants, the voltages Vi and resistances Ri.

Can this system be solved analytically?

Has such a circuit been realized and is acutally used in any context? Any references?

• If you define assumptions pr specs of each transducer for a logic state {0,x,1} for x in between 0.1 states. Then if you assigned currents to produce forces with values for spring constant and mass , you might approach it in a logical sense or as an analog system with a transfer function for each object. But what are you trying to demonstrate logic? Or an analog widget? Jan 9, 2018 at 13:05
• Man, i invented something similar in third grade! Finally someone implements it, thank you! Be sure to tell how it ends up!
– user76844
Jan 9, 2018 at 13:14
• @Gregory: From Wikipedia I learn you must have been 7-8 years old when you invented this. Does this mean it's a trivial device or that you are a genius? Or just something inbetween? Jan 9, 2018 at 13:25
• Well i hope for the second option, but it's up to others to judge. And it was in USSR, back in 1990, so i was 9.
– user76844
Jan 9, 2018 at 13:26
• How old have you been? Jan 9, 2018 at 13:27

Can this system be solved analytically?

That depends on how you model it.

Generally, I'd say you could probably model this as a system with very limited state:

• position of the two small magnets

The spring force is a direct result of the position, so there's no need to consider that as state. Whether the switches are on or of depend on the position of the small magnets, too.

Then, you could set up equations describing the relationships like

"The speed of m1 is the mass of m1 times gravitational pull plus the magnetic pull, which depends on the switch2 state, i.e. whether m2 is at the 'closed' position."

Speed being the derivative of position over time, you then get a system of two differential equations. Let $p_i$ be the position of $m_i$, just set the mass of the magnets to 1, so that the gravitational pull becomes $g$, let the "closed height" be 0, and you'd get something like:

\begin{align} \dot p_1 &= &-g &+ \mathbf 1\left\{p_2 = 0\right\} \cdot \frac{V_1}{R_1}\\ \dot p_2 &= &g &- \mathbf 1\left\{p_1 = 0\right\} \cdot \frac{V_2}{R_2} \end{align}

with $\mathbf 1$ being the indicator function, ie. 1 when the condition in {} is true, 0 else.

To make this analytically easier, you'd go ahead and replace $\mathbf 1$ with something differentiable; I'd try an exponential function with negative exponent that departs from 1 very quickly (the larger const. $\gamma$, the more "noncontinuous" that will look from afar; maybe you'll find a solution that you can $\lim\limits_{\gamma\to\infty}$):

\begin{align} \dot p_1 &= &-g &+ e^{-\gamma{p_2}^{2}} \cdot \frac{V_1}{R_1}\\ \dot p_2 &= &g &- e^{-\gamma{p_1}^{2}} \cdot \frac{V_2}{R_2} \end{align}

Then, find a solution for $p_1$ and insert that into $\dot p_2$ (or vice versa).

There's various tricks that help you learn, which you usually learn during one of the control and systems theory lectures when studying engineering.

From your profile, I'd assume your more of an algebraian (is that a word?), but that's simply as much analysis as you need to stand :)

Since we're directly aiming for a system that both of us expect to behave periodically, I'd say: maybe apply the Fourier transform to above equations – it has nice properties w.r.t. derivations and exponential functions, and will end up delivering a discrete solution iff the system actually is periodic.

• (of course, if your figure is not meant to be interpreted as vertically aranged, but e.g. horizontally or in weightlessness, simply set $g=0$). Jan 9, 2018 at 13:05
• (and of course I omitted the Anfangswertproblem (initial value problem? Dunno) that's inherent to the question what $p_i(0)$ are.) Jan 9, 2018 at 13:10
• Are you serious with "help you hear"? Is this a standard approach of "understanding"? Jan 9, 2018 at 14:01
• We would have to reflect the fact that m1 and m2 are abruptly stopped when touching the contact. Jan 9, 2018 at 14:15
• @HansStricker "hear" was an unfortunate choice of words (probably was distracted when writing that sentence). The "abruptly stopped" is a) only true macroscopically and b) possible to approximate by increasing $\gamma$. But: My equations really are just an example, which, of course, is a simple model. If you wanted to complicate things: force on $m_i$ is, if you looked more closely physically, not only a function of the current through the electromagnet, but also a function of position relative to said magnet, magnet nonlinearities, speed at which the magnet traverses through the field… Jan 9, 2018 at 14:21