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Help with coefficients matrix in spring system

  1. Oct 16, 2016 #1

    BiGyElLoWhAt

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    1. The problem statement, all variables and given/known data
    The system is a spring with constant 3k hanging from a ceiling with a mass m attached to it, then attached to that mass another spring with constant 2k and another mass m attached to that.
    So spring -> mass -> spring ->mass.
    Find the normal modes and characteristic system. I'm assuming it should be characteristic equation. Maybe not.

    2. Relevant equations
    dE/dt = 0

    3. The attempt at a solution

    So I have ##T = 1/2 m \dot{x}_1^2 + 1/2 m (\dot{x}_1 + \dot{x}_2)^2##
    but the real problem at this point is in V
    ##V = 1/2 (3k)x_1^2 +1/2 (2k)x_2^2 + mg(H - (2x_1 + x_2))##
    I'm not sure how to get gravity into a coefficient matrix to solve this differential equation, since they are only dependent on 1 x term each. So x_1^2 goes in 1,1 and x_2^2 goes in 2,2 and x_1x_2/2 goes in both 1,2 and 2,1. I'm not sure what to do with x_1 and x_2 terms, though. Thanks.
     
    Last edited: Oct 16, 2016
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  3. Oct 16, 2016 #2

    vela

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    I don't think you meant to have time-derivatives in V.

    One thing you can do is a change of variables: Complete the square first, e.g., ##ax^2+bx## becomes ##a(x+\frac{b}{2a})^2+\text{constant}##. Then define new coordinates (##u=x+\frac{b}{2a}##) and rewrite the Lagrangian in terms of them
     
  4. Oct 16, 2016 #3

    BiGyElLoWhAt

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    Yea I'm not sure why I have the dots there. Typos. So you're suggesting Lagrangian mechanics rather than applying conservation of energy?
     
  5. Oct 16, 2016 #4

    vela

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    No, I just guessed you were doing Lagrangian mechanics.
     
  6. Oct 16, 2016 #5

    BiGyElLoWhAt

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    We had a similar problem, on a HW and used ##\frac{d}{dt} [\dot{q}^T A \dot{q} + q^T B q ]= 0## with A and B the coefficient matrices that multiply to the equation. I was assuming that this would be a similar project, but the mgx term is giving me problems when I try to put it in said matrices.
     
  7. Oct 17, 2016 #6

    vela

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    Did you try what I suggested to get rid of the linear terms?
     
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