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Finding DE for Two Springs and a Damper in Series

  1. Aug 29, 2011 #1
    Hi all! I'm having a bit of trouble writing the differential equation that governs this mechanical system of two springs and a hydraulic damper in series. Since there is no mass present I believe the resulting DE will be a first order equation of the form Ax'(t) + x(t) = f(t), where f(t) is the forcing function. However, I can not figure out how to manipulate the equations from the FBDs into this form. Once I have the DE I do not anticipate any trouble solving it. Any and all help you can provide to point me in the right direction is greatly appreciated.

    1. The problem statement, all variables and given/known data
    Find the transfer function, Xo(s)/Xi(s), for the mechanical system in the diagram below. The displacements xi and xo are measured from their respective equilibrium positions.

    Obtain the displacement, xo(t), when the input xi(t) is a step displacement of magnitude xi occurring at t=0.


    2. Relevant equations

    Components in series require the force to be constant throughout the chain. Lets call this force "P".

    The FBD at Junction I we will call Eq. 1:

    P = k1*(xi-y)

    FBD at Junction II we will call Eq. 2:

    k1*(xi-y) = k2*(y - xo)

    FBD at Junction III we will call Eq. 3:

    k2*(y - xo) = b1*x'o

    3. The attempt at a solution

    Since we are interested in the input and output, (xi and xo), my first thought is to eliminate the variable y by solving for it in Eq. 2 and substituting into Eq. 3.

    y =[itex]\frac{k_{1}x_{i}+k_{2}x_{o}}{k_{1}+k_{2}}[/itex]

    Subbing into Eq. 3:
    [itex]k_{2}(\frac{k_{1}x_{i}+k_{2}x_{o}}{k_{1}+k_{2}}-x_{o}) = b*x'_{o}[/itex]

    Dividing by [itex]k_{2}[/itex] and rearranging we get:

    [itex]\frac{b}{k_{2}}x'_{o} + x_{o} = \frac{k_{1}x_{i}+k_{2}x_{o}}{k_{1}+k_{2}}[/itex]

    Is this correct?

    Again, thanks for any guidance you can give me :)
    Last edited: Aug 29, 2011
  2. jcsd
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