Deriving Resonant Frequency of Multiple Mass Mass-Spring Systems

[verd]

How does one begin to derive the resonant frequencies of a multiple mass mass-spring system? (3-mass system)
OK, so deriving this isn't a homework problem. I'd just like to know how. It's been a couple of years since I've taken a physics course. I was asked what the resonant frequencies of each mode of a 3-mass mass-spring system was. I found the formulas on the internet, but am not sure how these are derived. Does anyone know how to derive these? ...If anyone could get me started on one, that would be great.

Relevant equations can be found in this PDF document for each mode of oscillation, page 13 & 14!
"academic.reed.edu/physics/courses/phys100/Lab%20Manuals/Vibrational%20Modes/normalmodes.pdf"[/URL]
(There are accompanying images in the PDF that I can't format the same way on a forum)Does anyone have any suggestions on where to begin?Thanks!

 

[Barkan]

it's been 3 years since this is asked.
but, in any case, i just try to answer this question. (Correct me if I'm wrong)
(also, considering the fact that above given link is dead)

suppose we have 2dof mass spring system

GND -- Spring 1 (k1) -- Mass 1 (m1) -- Spring 2 (k2) -- Mass 2 (m2)

there are two resonance frequencies (w1, w2) but they are NOT equal to following

w1 (NOT EQUAL TO) sqrt(k1/m1)
w2 (NOT EQUAL TO) sqrt(k2/m2)All you need is to write the motion equations. I do it for 2 Dof system but anyone can modify it for 3 DoF system as well.

x1 and x2 are the position functions of m1 and m2 respectively.
ddx1 and ddx2 are accelerations, dx1 and dx2 are velocities.

m1 ddx1 + (k1+k2) x1 - k2 x2 = 0 (If a force applied to m1, put this force to right hand side instead of zero)
m2 ddx2 + k2 x2 - k2 x1 = 0 (the same here)

Mass and stiffness matrix take the following form (in Matlab notation)

M = [m1, 0; 0, m2]
K = [k1+k2, -k2; -k2, k2]

Eigenvalues of the following matrix gives you 2 distinct resonance frequencies. Since the matrix is 2x2 there are two eigen values.

K-Lamda*M

In case of 3 DoF system the above matrix becomes 3x3 and its 3 eigenvalues give you 3 distinct resonant frequencies.

Best