# What's the physical interpretation of an arbitrary normal mode for masses and springs?

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.

• The physical interpretation is that any free movement of this system can be understood as a superposition of its eigenmodes moving at its eigenfrequencies (beware of the complications in case of degeneracies). – CuriousOne Dec 13 '14 at 15:54
• answer coming, but note that the normal modes will depend on the values of the masses and strength of the springs - would be possible to make some useful predictions if, for example, all masses equal and all springs have same k. – tom Dec 13 '14 at 15:57

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.

• Man, i was thinking exactly like that. Can you tell me if the masses will be moving with a common frequency ( but perharps distinct amplitude ) in all these normal modes ? – nerdy Dec 13 '14 at 16:27
• Yes, the common frequency is where the eigenstuff comes from. You assume that each mass moves with a common frequency, but its own amplitude and phase. Then $\ddot x_i= \omega^2 x_i$ , you get $(A-\omega^2)\vec{x}=0$ and you want nontrivial solutions. – Ross Millikan Dec 13 '14 at 16:34
• This answer was fantastic.The simple comparison with the string makes the physical interpretation of normal modes of any coupled oscillatory system crystal clear.I'm amazed i could not find such helpful insight in the literature. Thanks a bunch. – nerdy Dec 13 '14 at 16:36
• I wish i could upvote more :( – nerdy Dec 13 '14 at 18:35

I hope this is useful as an answer to your question, because although this is not about your exact system it gives a helpful interpretation of what normal modes are.

For the water molecule we can consider it as three masses linked by two identical springs. Similar to your system there are three normal modes, which can be represented as the following motions shown in the diagram (taken from http://www4.ncsu.edu/~franzen/public_html/CH795N/lecture/XIV/image964.gif)

So there are three distinct normal modes of motion of the water molecule, which can be understood as the following motions of the water molecule.

In the same way, normal modes that you calculate for your system will correspond to different cooperative motions of the three masses. Hope this is useful.

I suggest that your modes may look something like the modes below - Note the difference between 2 and 3 is that in 3 all motions are 'in phase' and in 2 the molecules at the ends are moving $\pi \over 2$ or 90$\deg$ out of phase with each other

please note that these are for masses all equal and springs all equal and even then I may have got it wrong.

• Yes, but could you tell me which motions ??? I know one normal mode correponds to the middle mass $m_2$ being fixed while $m_1$ and $m_3$ move to opposite directions. What about the other two normal modes ? – nerdy Dec 13 '14 at 16:04
• ok - will post suggestions – tom Dec 13 '14 at 16:05
• Are you sure the second mode doesn't correponds to all masses moving in the same direction, like Ross Milikan suggested ? – nerdy Dec 13 '14 at 16:28
• @nerdy - no Ross is probably correct - but with all of these the actual motion will depend on the values of the masses and the spring constants. – tom Dec 13 '14 at 16:35