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Equations of Motion

  1. Nov 19, 2015 #1

    Zondrina

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    Homework Helper

    1. The problem statement, all variables and given/known data

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

    Screen Shot 2015-11-19 at 7.47.55 PM.png

    2. Relevant equations


    3. 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?
     
  2. jcsd
  3. Nov 19, 2015 #2

    gneill

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    Staff: Mentor

    You do realize that you can draw an analogous electrical circuit for the mechanical system without writing and solving the differential equations, right?
     
  4. Nov 20, 2015 #3

    Zondrina

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    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:

    724aff635db084cf0a586208cb8528bc.png

    The battery on the far right corresponds to ##p(t)##.
     
  5. Nov 20, 2015 #4

    gneill

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    Staff: Mentor

    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.
     
  6. Nov 20, 2015 #5

    Zondrina

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    Okay.

    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:

    Screen Shot 2015-11-20 at 10.01.05 AM.png
    Screen Shot 2015-11-20 at 10.01.13 AM.png
     
  7. Nov 20, 2015 #6

    gneill

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    Staff: Mentor

    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.
     
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