How to Find Normal Mode Frequencies for Three Masses in an Equilateral Triangle?

[eventhorizonof]

Three equal masses arranged in a equilateral triangle are connected by 'springs' with force constants 'k'

the coordinates of the masses are:

mass 1 at [0, $$\sqrt{3}$$/2*L]
mass 2 at [L/2, 0]
mass 3 at [-L/2,0]

find the normal mode frequencies.

The only part i am having trouble with is setting up the potential energy i know what to do after I have the potential energy.

so far i have

U = 1/2 * k (x_2 - x_3)^2


i know there are more terms in the potential but i am having trouble projecting the deviations from equilibrium onto the diagonals.

 

[olgranpappy]

well, there's 6 degrees of freedom, so there are 6 normal modes. 2 are translations as a whole and 1 is rotation as a whole. there are 3 vibrational mode, one of which is the "breathing" mode, and then there are 2 others.

 

[eventhorizonof]

neglecting all rotation what will the other terms of the potential be?

 

[olgranpappy]

$$U=\frac{k}{2}((\vec x^{(1)}-\vec x^{(2)})^2+(\vec x^{(1)}-\vec x ^{(3)})^2+(\vec x^{(2)}-\vec x^{(3)})^2)$$

 

[eventhorizonof]

in your notation vector x 1 is the location of the first mass [x1,y1] , x 2 second mass [x2,y2],...

right?

 

[olgranpappy]

yeah