Help with coefficients matrix in spring system

[BiGyElLoWhAt]

Homework Statement

The system is a spring with constant 3k hanging from a ceiling with a mass m attached to it, then attached to that mass another spring with constant 2k and another mass m attached to that.
So spring -> mass -> spring ->mass.
Find the normal modes and characteristic system. I'm assuming it should be characteristic equation. Maybe not.

Homework Equations

dE/dt = 0

The Attempt at a Solution

So I have $T = 1/2 m \dot{x}_1^2 + 1/2 m (\dot{x}_1 + \dot{x}_2)^2$
but the real problem at this point is in V
$V = 1/2 (3k)x_1^2 +1/2 (2k)x_2^2 + mg(H - (2x_1 + x_2))$
I'm not sure how to get gravity into a coefficient matrix to solve this differential equation, since they are only dependent on 1 x term each. So x_1^2 goes in 1,1 and x_2^2 goes in 2,2 and x_1x_2/2 goes in both 1,2 and 2,1. I'm not sure what to do with x_1 and x_2 terms, though. Thanks.

 

[vela]

I don't think you meant to have time-derivatives in V.

One thing you can do is a change of variables: Complete the square first, e.g., $ax^2+bx$ becomes $a(x+\frac{b}{2a})^2+\text{constant}$. Then define new coordinates ($u=x+\frac{b}{2a}$) and rewrite the Lagrangian in terms of them

 

[BiGyElLoWhAt]

Yea I'm not sure why I have the dots there. Typos. So you're suggesting Lagrangian mechanics rather than applying conservation of energy?

 

[vela]

No, I just guessed you were doing Lagrangian mechanics.

 

[BiGyElLoWhAt]

We had a similar problem, on a HW and used $\frac{d}{dt} [\dot{q}^T A \dot{q} + q^T B q ]= 0$ with A and B the coefficient matrices that multiply to the equation. I was assuming that this would be a similar project, but the mgx term is giving me problems when I try to put it in said matrices.

 

[vela]

Did you try what I suggested to get rid of the linear terms?