Transformers and systems with gears have similar computation when having quantities refer to different parts. What is the general concept?

Question

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

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

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.