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In rotational mechanical systems, there is a common ratio used when transferring quantities to different shafts.

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$$ \frac{N_D}{N_S} $$

Let's generalize number of teeth into number of a physical parameter associated with a connection.

The ratio then represents the number of the physical parameter of the destination over the source. This ratio is used in different manners depending on the type of quantity.

Rotational Mechanical System

  1. Gears link connections.

  2. The loop connection is the connection.

  3. The number of teeth is the physical parameter.

  4. For torque, you multiply by the ratio.

  5. For angle, you divide by the ratio.

  6. For impedance, you multiply by the square of the ratio.

Now get this, in the transformer this concept also applies. Now I know there are analogous quantities such as torque and voltage, angle and current and the two impedances for rotational mechanical and electrical networks. But I didn't notice at once that even this concept of transferring and referring quantities to connections it is not originally part of is also the same.

Transformer in Electrical System

  1. Windings link connections.

  2. The current connection is the connection.

  3. The number of turns is the physical parameter.

  4. For voltage, you multiply by the ratio.

  5. For current, you divide by the ratio.

  6. For impedance, you multiply by the square of the ratio.

I haven't seen any source say it blatantly as I did, but if you are familiar with the lessons you can verify if my concepts are correct. My question is, is there an overarching concept that covers why these two systems have a similar behavior? Are there formal terminologies and would such concept be extendable to other systems provided that they would also have analogs to the quantities mentioned above.

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  • $\begingroup$ Interesting question! Hydraulic systems also, with pressure and flow. One thing they all have in common is that there are two related variables, one of which is the time-derivative of the other e.g. force and momentum. There is also always an energy expression. This is where our engineering education dumbs it down a little bit... I think more 'serious' physics type classical mechanics courses are better set up to make generalizations $\endgroup$
    – Pete W
    Mar 27, 2021 at 1:45
  • $\begingroup$ Do you think a generalization doesn't exist yet? In this modern age, it feels weird that no one tried to establish one yet. I mean I get it because like in rotational mechanical systems there are gear trains and definitely we don't have winding trains in circuits, only a transformer. $\endgroup$
    – AndroidV11
    Mar 27, 2021 at 3:25
  • $\begingroup$ my guess is it exists but the mathematical vocabulary to talk about it is not that important for engineering, and is uninteresting for anyone else, so it didn't make the cut for inclusion in a 4-year or 5-year program. $\endgroup$
    – Pete W
    Mar 27, 2021 at 14:35
  • $\begingroup$ @AndroidV11 Electrical supply systems do have multi-level characteristics: transformers at 275kV/132kV, 132KV/11KV, 11KV/415V are common. That's equivalent to gear trains (with multiple take-off points). Resistance is friction, capacitance and reactance are inertia and springiness. $\endgroup$ Mar 27, 2021 at 16:42

3 Answers 3

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The common denominator is that both "machines"/devices are transformers of some kind. What they do is transform energy from one form to another. However the total energy (if you ignore losses) remains the same before and after the transformation.

In both of your examples there energy (or more precisely Power which is essentially energy in the unit of time) can be expressed in terms of a multiplication of two quantities.

Gear box Transformer
Power $M\omega$ $VI$

In both cases, because energy cannot be generated or destroyed, when one quantity increases there is a tradeoff in the other. So, I think of it as a balance.

For a given average Power you can't have Higher voltage and higher current. similarly, for a given average Power in a mechanical system you can't increase at the same time the torque and the rotational velocity. Only one at the time.

As to why there isn't yet a generalization, like for example the harmonic oscillator for dynamic systems (RLC circuit and mass-spring system), I think the only reason is that the mathematics are very simple, and there is no need to generalize. It would probably create more trouble explaining the concept from the mechanical and the electrical aspect, than the benefit of a unified approach. In any case, I am certain that gifted teachers when given the opportunity, provide hints to this insight.

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  • $\begingroup$ Yeah I get the conservation of power, is there some explanation why the number of physical quantity such as a teeth or turn is used in ratios when referring to different connections in a network? Yeah, with rotational mechanical networks the rotational mechanical energy is equal and with the transformer the copper loss is equal. It just seems odd that it works perfectly for both electrical and rotational mechanical networks. $\endgroup$
    – AndroidV11
    Mar 27, 2021 at 5:15
  • $\begingroup$ As I said there are other perfect analogies to mechanical and electrical systems. I've never tried to find any deeper connection than the energy. To me I guess it would have felt like searching for paranormal connections, although I guess I lean towards the practical side of things. $\endgroup$
    – NMech
    Mar 27, 2021 at 7:33
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    $\begingroup$ I would say the generalisation follows directly from the analogies. $\endgroup$
    – joojaa
    Mar 27, 2021 at 9:45
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There is no general concept except for proportionality

All you have discovered is that ratios are used throughout science and engineering. Simple machines (gears are only one example, you also have levers, pulleys, inclined planes, etc.), transformers, gas laws (PV/T=PV/T), optics...

The use of ratios is a fairly simple and ubiquitous concept in systems where variables are directly or inversely proportional to each other. But we also have relationships that are based on squares (gravity, kinetic energy, drag, E=MC^2...), cubes, exponential/logarythmic, geometric progressions and so on. Both electrical circuits and vibrating systems can be described using differential equations. Most (all?) engineering/science phenomenon are based on math, but not all are proportional relationships.

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This is well known feature called the Mechanical - Electrical analogies. As name implies there are in fact several of these spanning several domains, not only does it apply to transformers and gears it applies to spirngs, heat machines etc.

This was of great importance back in the day of analogue computers as this allowed you to simulate mechanical systems with electrical components. This is still used, the electrical engineers quite readily simulate heat loads with their circuit simulators. I have even seen the reverse done in a multibody simulator where a mechanical system simulated curcuitry, though it was more of a joke.

Simply, yes a gear is a transformer. Simply both are forms of a energy conservation used to exhange one linked property to another. Where the mechanical system is converting Force and Speed the electrical system is converting voltage and current. Similar systems exist also in the thermodynamic, acoustic and hydeaulic domains.

It isnt really suprising though, the technical requirements for all the systems are equivalent. If say one of the domains would come up with a super useful new basic use pattern all the others would seek to find the equivalent in case ot was useful.

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    $\begingroup$ +1 for the wikipedia link. $\endgroup$
    – NMech
    Mar 27, 2021 at 9:46

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