# Normal modes of three masses attached by two springs

I have the following system:

I've applied Newton's 2nd Law to the system and I have found the normal modes proceeding as an eigenvalues and eigenvectors problem. I obtained the frequencies $$\omega_1^2=0$$, $$\omega_2^2=\frac{k}{m}$$, $$\omega_3^2=\frac{2k}{M}+\frac{k}{m}$$.

My problem is I don't know how to interpret my results in a physical way.

I'll be thankful for any help.

• It might help you visualise these modes if you consider not just the frequencies of the oscillations but also the relative phases of the three masses. Dec 31, 2023 at 11:32
• to interpret you need the eigenvectors
– Eli
Jan 2 at 18:38

Hold your two hands up either side of your head, with fingers curled to make a fist, so that your head is located at the position of mass $$M$$ and your hands are at the locations of the two masses $$m$$. Now move your head and hands in such a way as to mimic the three normal modes, as follows:

mode of frequency zero: walk to the side holding hands at fixed distance from head.

next mode: head stays still, hand wobble in and out, both approaching head at same time

highest freq mode: hands move in same direction as each other, head moves in anti-phase. It's like a dance move, the one you see in quite a few cultures from generally eastern parts of the world (middle east and India for example).

To deduce the above from the maths, look carefully at your normal mode vectors, and plot graphs of $$x_1(t), x_2(t), x_3(t)$$ for each mode.

• Thanks!! I've also found the eigenvectors and everything's clear now :)
– aaa6
May 17 at 13:08