Modeling a two-mass, spring, damper system I'm trying to model a system with two masses, two springs, two dampers, and one applied force using transfer functions. I'll then be inputting it into Simulink.
The system looks like this but there is a force applied to the right edge of ${ m }_{ 2 }$ pointing towards the right.
I already found the two differential equations of the system. Then I found one transfer function from each differential equation and ended up with these:
$\frac { { U }_{ 1 }(s) }{ { U }_{ 2 }(s) }$ and $\frac { { U }_{ 1 }(s){ U }_{ 2 }(s) }{ F(s) } $
I'm a little unsure of how to proceed from here. The end result should be the response of ${ U }_{ 1 }$ and ${ U }_{ 2 }$.
I haven't included the exact equations because I don't really want an exact answer (I'll never learn that way!). If I've left out any information please let me know. Thanks for any help!
EDIT: I guess the general question here is, how do you model this type of system using transfer functions?
EDIT 2: Here are my differential equations. Note that my variables differ slightly from the linked image.
${ \ddot { x }  }_{ 1 }=\frac { { k }_{ 12 } }{ { m }_{ 1 } } \left( { x }_{ 2 }-{ x }_{ 1 } \right) +\frac { { d }_{ 12 } }{ { m }_{ 1 } } \left( { \dot { x }  }_{ 2 }-{ \dot { x }  }_{ 1 } \right) -\frac { { k }_{ 1 } }{ { m }_{ 1 } } { x }_{ 1 }-\frac { { d }_{ 1 } }{ { m }_{ 1 } } { \dot { x }  }_{ 1 }$
${ \ddot { x }  }_{ 2 }=\frac { f(t) }{ { m }_{ 2 } } -\frac { { k }_{ 12 } }{ { m }_{ 2 } } \left( { x }_{ 2 }-{ x }_{ 1 } \right) -\frac { { d }_{ 12 } }{ { m }_{ 2 } } \left( { \dot { x }  }_{ 2 }-{ \dot { x }  }_{ 1 } \right) $
 A: I'm not sure, but from what you've said I
 don't think what you've got are the transfer functions. You probably wouldn't get a transfer function directly from each equation.
Rather, from two linear second-order differential equations you should get two s-domain equations. A time derivative becomes multiplication by s. So something like:
$$x''(t) = A x'(t) + B x(t) + Cy'(t) + D y(t) + f(t)$$
(where $A,B,C,D$ are constants) becomes:
$$s^2 X(s) = (As+B) X(s) + (Cs+D) Y(s) + F(s)$$
(assuming $x(0)=y(0)=0$ and $x'(0)=y'(0)=0$). Do the same for the other equation, and do some algebra on both equations to get something like:
$$X(s) = \frac{p(s)}{q(s)}F(s)$$
where $p$ and $q$ are polynomials. Then $\frac{p(s)}{q(s)}$ is the transfer function for $x$. From it you can tell how $x(t)$ reacts to the input $f(t)$. You get another transfer function for $y(t)$. 
Wikipedia's article has an example without dissipation. Here you have dissipation, you should be able to write each transfer function as a sum of terms $V/(s+\sigma)$, representing exponential decay with time constant $\sigma$, and/or terms $(Us + V)/((s + \sigma)^2 + \omega^2)$, representing a decaying sine wave with frequency $\omega$. Matlab/Simulink do a lot of the work for you.
