Finding Normal Mode Frequencies for Two Hanging Masses with Springs

[Pacopag]

Homework Statement

A mass M_1 is suspended by a spring with constant k_1. A second mass M_2 is suspended from the first by a spring with constant k_2. The equilibrium height of the first is y_1 and the equilibrium height of the second is y_2.

I just want to write down the Lagrangian for now.

Homework Equations

L=T-V (i.e. Lagrangian is Kinetic energy minus Potential energy).

The Attempt at a Solution

I'm using as generalized coordinates the displacements of the two masses from equilibrium \delta_1 and \delta_2
I'm pretty sure that the kinetic energy parts are easy
T_1 = {1\over 2}M_1 \dot\delta_1^2 and T_2 = {1\over 2}M_2 \dot\delta_2^2
Now for the potential energy for the second mass I think is
V_2 = -M_2 g (y_2+\delta_1+\delta_2) + {1\over 2}k_2(\delta_2-\delta_1)^2.
The first term is the gravitational part (taking the zero to be at the ceiling where the first spring is anchored) and the second term is the spring part.
But for the first mass, I'm tempted to write
V_1 = -M_1 g (y_1+\delta_1) + {1\over 2}k_1 \delta_1^2.
I think the spring part is wrong because M_1 is attached to spring 2, so there should be some contribution involving k_2. This leads me to want to add another term like {1\over 2}k_2(\delta_2-\delta_1)^2, but I don't know if it should be positive or negative, and also I feel like I am 'double counting' the spring term in V_2. For example, if spring 1 is streched and spring 2 is compressed, I expect V_1 to be increased by the compression of spring 2. I hope this is clear.

 

[Pacopag]

I've almost convinced myself that I should have
V1=-M_1 g(y_1+\delta_1)+{1\over 2}k_1\delta_1^2\pm{1\over 2}k_2(\delta_2-\delta_1)^2.
The sign is fuzzy to me, but thinking about the force (saying that down is negative), I think I should keep the minus. If this is correct, then I can get the equations of motion, which brings me to my real question: How do I find the normal mode frequencies? From similar examples, I've tried putting \delta_j = A_j e^{i\omega t} (i.e. same frequency for both displacements (gen. coords)), into the equations of motion, then finding \omega such that the solution exists. But, the gravitational terms don't allow us to cancel off the exponentials, which usually happens. I hope you can follow.

 

[Pacopag]

Sorry. It should just be
V_1=-M_1 g(y_1+\delta_1)+{1\over 2}k_1\delta_1^2.
But I still don't know how to find the normal mode frequencies.