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**n-coupled oscillator-- both series, and parallel.**

Hi all.

Ok, the title is a bit deceptive, as that is typically used to denote a string type oscillator system. In this case, I'm referring to multiple masses, and multiple springs.

To be specific, we're doing a system with 5 masses, and 13 springs. This means that there are 2 to 4 springs connected either in parallel, or in series to each of the masses.

From what I remember-- parallel springs are treated as:

1/k_eff = 1/k1 + 1/k2+ ... + 1/kn. Where n is the nth spring, and k is the spring constant.

And series springs are just k_eff = k1 + k2 + ... + kn. (the sum of all spring constants.)

Did I get this backwards, or is it correct?

Next, because we're doing a 5 mass system, we'll have a 5 x 5 matrix for the M matrix. Where the masses will be the diagonal elements of the M matrix. The K matrix however will be more complicated. Which is why I'm here.

It seems to me-- from previous lectures, and homework problems-- that the only elements of the K matrix that will have any non-zero values will be the diagonal, and the elements one space to the left, and one space to the right of the diagonal elements. Is this correct?

E.g. for a 2 x 2 matrix, the elements would be

k_11 -k_12

-k_21 k_22

In this case, the k_12 element = k_21. The reason would be due to a non-restoring force acting ON x_2, etc...

If we then expand this matrix out to n x n-- anything larger than a 2 x 2, all other elements--- e.g. sample 3 x 3 mat.

k_11 -k_12 0

-k_21 k_22 -k_23

0 -k_32 k_33

As you can see, only the diagonal, and "immediate" off-diagonal elements are non-zero.

My question-- does this play out in ALL oscillator matrices? I have asked my prof., but he has yet to answer.

In my specific 5 x 5 K matrix, this would in fact give me 13 elements, accounting for the 13 springs of the system.

The general equation to solve this system would be:

M^-1 K A = omega^2 A

where M, and K are n x n matrices, A is a column vector (and can be used to obtain our eigenvectors/normal modes), and omega^2 is the eigenvalue to the system-- can be multiple eignevalues.

A clarification of this aspect of a multiple oscillator system would be deeply appreciated.

Thanks.