# Normal modes of the 3-mass-spring system

This question might seems naive, but I'm not a physics student and got puzzled here.

There are three masses placed on the plane and they form the 3 vertices of an equilateral triangle. Every two of them are connected by a spring with spring constant $k$. It's well known that this system has 6 normal modes:

1. translation in the x- or y- axis. These two form an 2d irre-rep of $D_3$ but with a zero frequency of $V$
2. a pure rotation. This corresponds to the sign rep of $D_3$.
3. breathing mode. This is the identity rep of $D_3$.
4. pumping mode. (this has two) These two correspond to the 2d irre-rep of $D_3$ but with a non-zero frequency of $V$.

My question arose when I was looking at the rotation mode: How does this mode look like? I was thinking it's like the follows:

There are no restoring forces in the springs: this is easy to see, but how can the masses move along the circle without any forces that holds them? (so they will not get pushed away by the centrifugal forces?)

The masses moving in a circle have to have a net force pulling towards the center. The force will be from the vector sum of the two springs that each mass is attached to. The force components tangential to the circular path will cancel but the components in the radial direction will add and point inwards. This will require that the springs stretch a little bit.

This is like a combination of the rotation mode and breathing mode. The question to answer is, "are the normal modes a mathematical decomposition or is it possible to physically realize every normal mode?" Your question highlights that the rotation mode, by itself, can't occur without some contribution from the breathing mode.

• @LaserMatter: Thanks for your answer, So rotation mode is only a mathematically derived mode and can not be perfectly realized in the real world? Commented Sep 24, 2016 at 6:09
• @LaserMatter On further consideration of your very good answer I think that the rotational mode has the stretched springs "built into it" so it is a realisable physically. What is not physically realisable is a rotation with unstretched springs? Commented Sep 24, 2016 at 7:57
• @Farcher yeah I agree. The breathing mode is an oscillation. So I was incorrect. Maybe that's what happens when I answer questions late at night. But the fact that the spring stretch to provide the needed centripetal force is still true. Commented Sep 24, 2016 at 13:40

There are no restoring forces in the springs: this is easy to see, ......

I do not think that this statement is correct.
The rotational mode has stretched springs but the masses do not vibrate whereas the breathing and two clapping modes do have the masses vibrating about the centre of mass. It is those stretched springs which provide the centripetal forces on each of the masses which allow the masses to rotate about their common centre of mass.

• I know. I just realized that rotational mode is a mathematically mode and in real world there can not be a pure rotational mode... Commented Sep 24, 2016 at 9:31
• @Zhao_L that's the right way to think about it, even though your conclusion is wrong. The usual mathematical concept of "normal modes" refers to infinitesimally small displacements from the starting condition - i.e. a very small motion along the tangent vectors to the circle through the three masses. But real life doesn't care about the mathematical approximations used to make a linear model of small displacements, and in real life the structure can rotate freely with constant angular velocity for any arbitrarily large angle. Commented Sep 24, 2016 at 14:38
• @alephzero Thanks! So strictly speaking, all modes are combinations of the normal modes only in a infinitesimally small neighborhood of the balanced state and the global picture showed in the animated gif is in fact mathematically incorrect? Commented Sep 24, 2016 at 15:38

Rotation will involve centripetal forces that have to be countered by combination of tension in the springs.

There are many other modes of vibration and their harmonics for this system.

1- two of the masses can be held apart vertically along y axis and the third mass can be pulled along x axis and released. System would vibrate by the 3rd mass going back and forth along x axis and the 2 masses can vibrate up and down along y axis, provided the displacements are small and no collision will happen.

2- Same configuration except this time the third mass is pulled to right and up, say by a tenth of the spring length then released. This time the masses will vibrate in 3 small circles which have a diameter of 2/3 tenths (approximately) of the spring length, while the hole system gently shimmies and turns around a randomly round center.
By changing the x and y of displacement of third mass we can force the system into some complex modes of vibration.

3- by trial and error we can find the suit spot where if we perturb the masses the system will go through many modes of vibration and resonate around the natural frequency of each mode then transit to a different mode. Similar to dance of a bunch of flys around a candle.

• But are not these other modes just linear combinations of the six modes cited by @LaserMatter? Commented Sep 24, 2016 at 8:01
• I think not. Modes I suggest have no symmetry with either axis or even polar symmetry. Like in example I gave by moving the third mass to x+1, y+1, the restoring force is going initially to be spring force between mass 3 and 2 as tension and mass 1 and 3 as compression, but gradually the inertia of J of rotation of axis of mass 1 and 2 will contribute to restoring force. Then again this J will change because of change of distance between mass one and 2. This will reduce the effect of the initial forces. Thus the system will pass through new phases of vibration configurations! Commented Sep 24, 2016 at 20:02