How Do You Calculate Normal Modes in a Vertical Coupled Oscillator System?

[samjohnny]

Homework Statement

An object A with mass 3m is suspended from a fixed point O by a spring of constant k. A second object B with mass 2m is in turn suspended from A by an identical spring. The system moves along a vertical axis through O. Find the frequencies of the normal modes, and the normal coordinates.

Homework Equations

Hooke's law = -kx
F = md^2x/dt^2

The Attempt at a Solution

Note: I have the positive axis going vertically downwards. So for this question it seem clear to me that we have to account for the weight of each of the two objects as well as the forces due to the springs. For the equations of motion I got the following:

For object A: 5m*[d^2(xA)/dt^2] = 5*m*g - k*xA - k*(xA - xB)
For object B: 2m*[d^2(xB)/dt^2] = 2*m*g + k*(xA - xB)

Where xA and xB are the positions of objects A and B respectively.

What I'm not sure on firstly is whether I've got my signs right - I think I have but what do I know. for object A I put down a weight of 5mg since it's also carrying the mass of object B (I'm assuming the springs are of negligible mass). Now in order to calculate the frequencies of the normal modes and normal coordinates, I'm assuming the best way to do that and solve the coupled equations would be to arrange it in matrix form and then compute the Eigenvalues/vectors, but what's confusing me in how to set up the matrices are the mg terms as I've only worked on problems where the oscillators undergo horizontal movement and gravity isn't taken into account. Any assistance please?

Thanks

 

[TSny]

Hi, samjohnny.

It's important to be very clear on the notation. What is the meaning of xA and xB? Are you measuring both of these distances from the same fixed point? If so, what fixed point? If not, from where?

Do the springs have a nonzero natural length when they are not stretched? Do you need to include this in your equations?

For the equation of motion of mass A, you do not want to treat the mass as 5m. Mass B only affects mass A via the stretch of the lower spring and you are taking that into account in one of the other terms in the equation of motion.

To avoid the constant terms in the equation of motion, you should ultimately write the equations of motion in terms of displacements of the masses from their equilibrium positions (i.e., the positions of the masses when the system is hanging at rest).