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.(adsbygoogle = window.adsbygoogle || []).push({});

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

Find the transfer function, X_{o}(s)/X_{i}(s), for the mechanical system in the diagram below. The displacements x_{i}and x_{o}are measured from their respective equilibrium positions.

Obtain the displacement, x_{o}(t), when the input x_{i}(t) is a step displacement of magnitude x_{i}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 = k_{1}*(x_{i}-y)

FBD at Junction II we will call Eq. 2:

k_{1}*(x_{i}-y) = k_{2}*(y - x_{o})

FBD at Junction III we will call Eq. 3:

k_{2}*(y - x_{o}) = b_{1}*x'_{o}

3. The attempt at a solution

Since we are interested in the input and output, (x_{i}and x_{o}), 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 :)

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# Finding DE for Two Springs and a Damper in Series

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