Analogous electric circuit for mass-spring-damper system

[STEMucator]

Homework Statement

Find the analogous electrical circuit for the following mass spring damper system.

Homework Equations

The Attempt at a Solution

I am rusty with writing equations of motion. I wanted to see if someone could check my work.

Looking at the diagram, there are three equations to write. Also there should be a third displacement variable, call it $x_3$, between $k_3$ and $b$. Assume down is positive.

For mass $m_1$:

$$m_1x'' = -k_1x_1 + k_2(x_2 - x_1) + k_3(x_3 - x_1) + b(x_2' - x_3') + p(t)$$

For mass $m_2$:

$$m_2x'' = -k_2(x_2 - x_1) - b(x_2' - x_3') - k_3(x_3 - x_1)$$

At the node in between the damper and spring:

$$0 = -k_3(x_3 - x_1) + b(x_2' - x_3')$$

Do these look okay?

 

[gneill]

You do realize that you can draw an analogous electrical circuit for the mechanical system without writing and solving the differential equations, right?

 

[STEMucator]

You do realize that you can draw an analogous electrical circuit for the mechanical system without writing and solving the differential equations, right?

Yes this is possible, but I was hoping to understand how to write the equations of motion anyway. It would be nice to know how to write them for a more complicated system, so I would still like to know if I've done that properly.

I'll give your idea a try though. Here is my attempt:

The battery on the far right corresponds to $p(t)$.

 

[gneill]

Yes, your figure looks okay to me. You've chosen the Force ⇒ Voltage paradigm. You could also have used the Force ⇒ Current paradigm where masses become capacitors rather than inductors.

For your equations, at a glance they look fine except I don't see where you've accounted for gravity acting on the masses.

 

[STEMucator]

Yes, your figure looks okay to me. You've chosen the Force ⇒ Voltage paradigm. You could also have used the Force ⇒ Current paradigm where masses become capacitors rather than inductors.

Okay.

For your equations, at a glance they look fine except I don't see where you've accounted for gravity acting on the masses.

Yeah I find this weird because in my textbook they never seem to account for the force of gravity on a mass.

They compensate for this by making a substitution like so:

 

[gneill]

Ah, I see. Yes, that substitution works and makes the math simpler. Of course, to match the model's predicted position to a real-world position one would need to know the equilibrium position's offset in real-world coordinates.