Feedback Control: Modeling Mechanical System with Circuit

[Bluestribute]

Homework Statement

Find the transfer function for the following mechanical system with force input fin and output x2.

Homework Equations

The Attempt at a Solution

The reason I left the equations blank is because I'm not sure how to appropriately model this scenario. I've seen two different lectures give two different ways of modeling the same thing (one says to model mass as a capacitor, the other an inductor). I also have conflicting information about modeling force: one says current, one says voltage.

Right now, I have two nodes (x1 and x2) connected with a capacitor in between (1/k, with again, conflicting information). The two nodes are connected to a reference ground with two inductors of value m. The current source (fin) is going to node x2.

 

[gneill]

There are two common electrical - mechanical analogous modelling paradigms. They are equally valid. As you say, one models mass as an inductor while the other models it as a capacitor.

Have a look at this web page: Analogous Electrical and Mechanical Systems from Swarthmore College.

 

[Bluestribute]

That's actually one of the website I used haha. So I guess now it's just making sure the circuit is correct. I have the force (current) in one loop that has the x2 node and m2 inductor. Then another loop which has the x2 and m2, a 1/k capacitor, and ends with an m1 inductor from the x1 node.

But my answer doesn't match the solution of m1s^2+k/m1m2s^4+k(m1+m2)s^2

 

[gneill]

What does that solution represent? It doesn't look like a transfer function to me. Maybe it's the grouping of the terms; Are there enough parentheses in the expression to make the order of operations unambiguous? Should it perhaps be:

(m1s^2+k) / ( m1m2s^4+k(m1+m2)s^2 )

And is it the transfer function X2(s)/f or V2(s)/f ?

 

[Bluestribute]

X2/Fin = (m1s^2 + k) / (m1m2s^4 + k(m1 + m2)s^2) is the transfer function they got

 

[gneill]

Okay. That works.

I should tell you that I've always been more comfortable with the mass == capacitance paradigm for these sorts of problems. It's probably just me, but I always seem to trip myself up with how to "terminate" the inductors properly. The capacitor version is easy: Capacitors always have one leg grounded.

Anyways, I think your model needs to be changed slightly. You have a mass-spring oscillator that's anchored to another mass. So m1 and the spring become an LC "tank circuit", and m2 becomes another L that connects it to the force (voltage source). Something like this:

 

[Bluestribute]

So I'm going to do it the capacitance way just 'cause that's what we have in our lecture notes (the brief brief notes).

I have the spring and mass 1 in series, and that series is in parallel with the second mass, which is in series with the current supply. So the laplace equations are something like:

(1/m₁s² + 1/k) = seriesA
(1/seriesA + 1/m₂s²)^-1 = seriesB = answer. I get something similar actually, but I have extra terms in there. I end up getting:

m₂s²(m₁s²+k) / (m₁s²(m₂s²k + 1) + k

So close . . .

EDIT: Oh . . . I see . . . I didn't inverse my second mass . . . Couldn't see underneath all my erase marks haha! So when done correctly, I get the same answer. Sweet, thanks for clearing that up