I learned about the analogousness between mechanical and electrical systems a few months back (with the help of Feynman). Yesterday, my professor was lecturing about this topic, when she told that electrical systems can be written in two ways – either by using the voltage source, or by using the current source (drawing the equivalent analogs, and writing differential equations).
I've always thought current sources to be abstract constructs for the electrical engineers to solve problems in their boring way, node analysis and stuff, since it's voltage that drives the free electrons (hence, current).
Anyways, my confusion was about something called "Force-Current" analogous system, where the situation has gone crazy. Here's my (rather ugly) drawing of her example. We were asked to draw the equivalent electrical analogs. (It's a convention that whenever we make use of current source, we analyze them using nodes)
I can understand the $F\to V$ system alright. Because, it makes use of the analog:
- $x\to q$
- $m\to L$
- $c\to R$
- $k\to 1/C$
But, it becomes hard to perceive, when current source $F\to I$ comes into play. Now, the analogs are crazy… (Note: This analog wasn't discussed by Feynman)
- $x\to \phi$
- $m\to C$
- $c\to 1/R$
- $k\to 1/L$
Is it right, actually? I'm skeptical about this kind of mapping. In this case, $G=1/R$ is conductance (which is the thing analogous to the damping factor $c$). Would this mean that conductance is the actual resistance here? And, more weirdo – capacitance is analogous to mass.
The DEs for this system are somewhat long. I'll choose an RLC circuit to express my point.
In terms of $F\to V$, (using $x\to q$)
$$V_R=R \frac{dq}{dt},\ \ V_L=L\frac{d^2 q}{dt^2},\ \ V_C=\frac{q}C$$
Now, using $F\to I$, (using $x\to \phi$)
$$I_R=\frac{1}R\frac{d\phi}{dt},\ \ I_L=\frac{\phi}{L},\ \ I_C=C\frac{d^2 \phi}{dt^2}$$
The $x\to q$ was fine, because motion of charge is current, and that's the basis for all electrical systems. But, the $x\to \phi$ is inconceivable. The magnetic flux, how $\phi$ changes, induced emf, etc. are needed for inductors. Yeah, but how can they be applied to resistors or capacitors?
With this framework for writing the differential equations, it seems as if magnetic flux is the clockwork behind the working of such a system.
This horror made me think that the $F\to I$ analog (using $\phi$) is just another abstract mathematical construct for writing differential equations. Am I right? Is it reasonable at all?
Best Answer
I think there is something wrong with your mapping.
Looking at http://lpsa.swarthmore.edu/Analogs/ElectricalMechanicalAnalogs.html , I see the following table:
This is inconsistent with the mapping you are showing.
I can understand this table - I can't understand yours. I think an error crept in - which would reasonably explain your confusion.
Looking forward to comments!