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Derivation of second order system transfer function

  1. Jan 17, 2011 #1

    I am trying to derive the general transfer function for a second order dynamic system, shown below:


    In order to do this I am considering a mass-spring-damper system, with an input force of f(t) that satisfies the following second-order differential equation:


    Using the following two relationships:



    I get this:





    Wheras my lecturer has the following in his notes:





    This obvisously gives the correct transfer function. So, from the two approaches, I have come to the conclusion that:


    But I do not understand the physical reasoning behind this. Can anyone offer any help with this?


  2. jcsd
  3. Jan 17, 2011 #2
    That is standard notation. The "trick" is to multiply the right hand side by [tex]\frac{k}{k}[/tex]. As for physical intuition. Perform a unit analysis. You should be able to draw a clear conclusion from that.
  4. Jan 17, 2011 #3
    Ah yes, I completely missed that. Although substituting [tex]\frac{k}{m}=\omega_n^2[/tex] leaves the gain of the system as [tex]\frac{1}{k}[/tex] which is then not dimensionless. I thought this transfer function was supposed to be dimensionless?
  5. Jan 17, 2011 #4
    No transfer functions are hardly dimensionless. Transfer functions are the ratio of system [tex]\frac{output}{input}[/tex]. Thus you can see that the transfer function can hold any units as long as it contains the output-input relationship you are looking for.
  6. Jan 18, 2011 #5
    Ok, thanks for your help viscousflow. It is very much appreciated.

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