Derivation of angular frequency of coupled Oscillator-vertical spring

[rshalloo]

Homework Statement

I'm trying to derive an expression for the angular frequency of a set of springs hanging vertically down with masses in between them. 3 cases:
2 equal masses and springs.
2 different masses and equal springs
3 equal masses and springs

Homework Equations

ω0=√(k/m)
F= -kx

The Attempt at a Solution

i know its a combination of 2 fundamental modes
and that the answer I'm aiming for for the first case is (ω1^2)=((3+√5)/2)(ω0)^2 and (ω2^2)=((3-√5)/2)(ω0)^2
But i can't get there.
I did out a force diagram for the general case of 2 uneven masses and 2 springs which i hoped would do for case 1 and 2 (case1 just being a special case of case2) and came up with 2 equations of motion
ma1=-2kx1 +kx2 +(m1+m2)g
ma2=-kx2 +kx1 + m2g

I really don't know where to go from here, can someone please point me in the right direction. I am looking for someone to give me the derivation but you know that feeling when you hit a brick wall? you :P

 

[diazona]

I'm not sure I understand the setup correctly. Are you saying you have 2 masses and 2 springs in the configuration (ceiling)-(spring1)-(mass1)-(spring2)-(mass2), such that the whole system is hanging vertically with mass 2 at the bottom? If that's the case, I don't think your equations are quite right.

In any case, once you get the equations, you can solve them by expressing them as a single differential equation for a 2-component vector. In other words, it goes into a form like this:
\begin{pmatrix}m_1 & 0 \\ 0 & m_2\end{pmatrix}\begin{pmatrix}\ddot{x}_1 \\ \ddot{x}_2\end{pmatrix} = \begin{pmatrix}k_{11} & k_{12} \\ k_{21} & k_{22}\end{pmatrix}\begin{pmatrix}x_1 \\ x_2\end{pmatrix} + \begin{pmatrix}C_1 \\ C_2\end{pmatrix}
or in more compact notation,
\tilde{M}\ddot{\vec{x}} = \tilde{K}\vec{x} + \vec{C}
and then you can find the normal modes and their frequencies by a procedure that basically boils down to computing the eigenvectors and eigenvalues of \tilde{M}^{-1}\tilde{K}.

 

[rshalloo]

Thanks Diazona, I've been at this all weekend but can't figure out where my equations are going wrong, could you possibly give me a hint or a worked example?


Thanks A million

Rob

 

[diazona]

I thought I gave you a hint - more than that, even, I told you how to solve the problem How about starting with the setup of your equations... what are the individual forces acting on each of the two masses? (If you can post a diagram of the configuration here that would be very helpful)

 

[rshalloo]

Heres my working out

 

[diazona]

Sorry for the delayed response, I was kind of busy. Thanks for posting the diagram though.

As far as setting up the diagram and equations, you just need to be consistent with the signs. That means once you've chosen a direction to be positive, make sure that x_1 > 0 actually does represent a displacement in that direction, and similarly for x_2 and the forces. (Technically, you don't need to do this, but it gets really confusing otherwise.)

I notice from your diagram that you're missing one force on the upper mass.