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Introductory Physics Homework Help
Finding DE for Two Springs and a Damper in Series
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[QUOTE="Paul9, post: 3475900, member: 318092"] 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. [h2]Homework Statement [/h2] Find the transfer function, X[SUB]o[/SUB](s)/X[SUB]i[/SUB](s), for the mechanical system in the diagram below. The displacements x[SUB]i[/SUB] and x[SUB]o[/SUB] are measured from their respective equilibrium positions. Obtain the displacement, x[SUB]o[/SUB](t), when the input x[SUB]i[/SUB](t) is a step displacement of magnitude x[SUB]i[/SUB] occurring at t=0. [ATTACH=full]145379[/ATTACH][h2]Homework Equations[/h2] Components in series require the force to be constant throughout the chain. Let's call this force "P". The FBD at Junction I we will call Eq. 1: P = k[SUB]1[/SUB]*(x[SUB]i[/SUB]-y) FBD at Junction II we will call Eq. 2: k[SUB]1[/SUB]*(x[SUB]i[/SUB]-y) = k[SUB]2[/SUB]*(y - x[SUB]o[/SUB]) FBD at Junction III we will call Eq. 3: k[SUB]2[/SUB]*(y - x[SUB]o[/SUB]) = b[SUB]1[/SUB]*x'[SUB]o[/SUB] [h2]The Attempt at a Solution[/h2] Since we are interested in the input and output, (x[SUB]i[/SUB] and x[SUB]o[/SUB]), 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 :) [/QUOTE]
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Introductory Physics Homework Help
Finding DE for Two Springs and a Damper in Series
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