Consider the following system consisting of 3 masses and 4 springs :
I have learned that this system posseses three normal modes, corresponding to its three natural frequencies, say $\omega_0$, $\omega_1$ and $\omega_2$.
I'm interested in the movement of the masses that each of the three normal modes represents.
I know that one normal mode correponds to the middle mass $m_2$ being fixed and the other outer masses $m_1$ and $m_3$ moving in opposite directions with the same frequency … But what about the two other normal modes ?
For a system of 2 masses and 3 springs, I've learned that the first normal mode represents the motion of the two masses, with the same frequency and with same phase; while the second normal represents the motion of the two masses, with some other common frequency but with phase difference of ninety degrees ( one is moving in the opposite direction of the other ).
But what about a system of 3 masses and 4 springs? Or say, $N$ masses and $(N+1)$ springs ?
I've found many mathematical demonstrations (with eigenvectors and eigenvalues) of the solution of this system, representing the motion of the masses, but I can't see the physical interpretation.
Best Answer
If you let $x_i$ be the position of the $m_i$, you can write a set of coupled equations $$\begin {pmatrix} m_1\ddot{x_1}\\ m_2\ddot{x_2}\\ m_3\ddot{x_3} \end {pmatrix}=A\begin {pmatrix} x_1\\ x_2\\ x_3 \end {pmatrix}$$ where $A$ gives the forces from the springs. With more masses and springs you have more lines in the equation. If all the masses are the same, you can divide them into $A$. You find the frequencies by finding the eigenvalues of $A$ and the modes by finding the corresponding eigenvectors. Roughly speaking, the modes will be like the modes of a string. The lowest mode will have all the masses moving in the same direction, as this produces the least stretching on the springs. The next mode will have one change of direction, so the left hand batch of masses will oscillate opposite to the right hand batch of masses. Each higher mode will have another sign change.