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Displacement of a voigt model and a mass in series

  1. Feb 22, 2012 #1

    jk1

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

    A Voigt model in series with a mass is given (so Wall: (mass and damper in parallel) -- mass -- Force ---> ) with a force, F = Fo*cos(ωt) exciting/driving the system.
    Givens: Spring constant: K, Viscous constant: η, Mass element: m

    Find a solution for established harmonic oscillations in the form a*cos(ωt) + b*sin(ωt).


    2. Relevant equations

    F = mx''
    F = kx
    F = ηx'

    3. The attempt at a solution

    Let x1 be the displacement of the system (I know the displacement will be the same on the damper and the spring - I don't see how the mass can have it's own displacement, so I'm assuming it

    I labeled the forces on the spring and the damper both F1 and F2 respectively, and labeled the force on the mass F3.
    F1 + F2 = F3 = Fo*cos(ωt)

    I set kx + ηx' = Fo*cos(ωt) and solved the differential equation and got:

    x(t) = Ae^(-t*k/η) + Fo*((k * cos(ωt) + η * ω * sin(ωt)) / (η^2 * ω^2 + k^2))

    Then I solved using the initial condition x(0) = 0 because the voigt system shouldn't move instantaneously, and the mass shouldn't move unless the voigt model does.

    A = -Fo(k / (η^2 * ω^2 + k))

    But as you can see, there's an exponential term in my answer and that's not in the final form desired.

    I tried finding a different differential equation via mx'' = kx + ηx' = Fo*cos(ωt) but I'm not having much luck - I'm pretty sure I need to have the forcing function involved or it won't work (so I can't just solve mx'' - ηx' -kx = 0).

    Any suggestions?
     
    Last edited: Feb 22, 2012
  2. jcsd
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