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Derivation of second order system transfer function

by ryan88
Tags: derivation, function, order, transfer
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Jan17-11, 11:02 AM
P: 39

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:

[tex]\frac{d^2y}{dt^2}+2\zeta\omega_n\frac{dy}{dt}+\omega_n^2y=\frac{f(t)}{m }[/tex]

[tex]\mathcal{L}\left\{\frac{d^2y}{dt^2}\right\}+2\zeta\omega_n\mathcal{L}\l eft\{\frac{dy}{dt}\right\}+\omega_n\mathcal{L}\left\{y\right\}=\frac{1} {m}\mathcal{L}\left\{f(t)\right\}[/tex]



Wheras my lecturer has the following in his notes:

[tex]\frac{d^2y}{dt^2}+2\zeta\omega_n\frac{dy}{dt}+\omega_n^2y=K\omega_n^2x( t)[/tex]

[tex]\mathcal{L}\left\{\frac{d^2y}{dt^2}\right\}+2\zeta\omega_n\mathcal{L}\l eft\{\frac{dy}{dt}\right\}+\omega_n\mathcal{L}\left\{y\right\}=K\omega_ n^2\mathcal{L}\{x(t)\}[/tex]



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?


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Jan17-11, 06:13 PM
P: 273
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.
Jan17-11, 06:48 PM
P: 39
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?

Jan17-11, 08:30 PM
P: 273
Derivation of second order system transfer function

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.
Jan18-11, 02:33 AM
P: 39
Ok, thanks for your help viscousflow. It is very much appreciated.


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