(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Two equal masses (m) are constrained to move without friction, one on the positive x axis and one on the positive y axis. They are attached to two identical springs (force constant k) whose other ends are attached to the origin. In addition, the two masses are connected to each other by a third spring of force constant k'. The springs are chosen so that the system is in equilibrium with all three springs relaxed (length equal to unstretched length), What are the normal frequencies? Find and describe the normal modes. Consider only small displacements from equilibrium.

2. Relevant equations

Lagrangian Formalism: L = T (Kinetic Energy) - U (Potential Energy)

Taking T = (1/2)(summation across all j, k) M(j,k)*qdot(j)*qdot(k)

And U = (1/2)(summation across all j, k) K(j,k)*q(j)*q(k)

Where qdot is dq/dt, j and k represent subscripts, M(j,k) represents the (j,k) value in a jxk matrix and likewise for K(j,k). After this point there are some additional equations to find eigenvalue solutions (which represent the normal modes), but I'm not having trouble with that part so I'll leave them out for now.

3. The attempt at a solution

The problem I'm having is the set-up. The closest I've gotten to a reasonable solution took x and y as the general coordinates, where x represented the distance past equilibrium length for the x-mass, and likewise for the y-mass. This gave me the following equation:

T = (1/2)m*xdot^2+(1/2)m*ydot^2

Then, to find the distance the third spring (with constant k') is stretched, I used:

L represents the equilibrium distance for the two axial springs.

L'(at equilibrium)^2 = 2L^2

L'(stretched)^2 = (L+x)^2 + (L+y)^2

=2L^2 +2xL + 2yL + x^2 + y^2

So (delta)L^2 = 2xL + 2yL + x^2 + y^2

Thus, the potential energy of the system is:

U = mgL + mgy + (k/2)(x^2+y^2) + (k'/2)(2xL + 2yL + x^2 + y^2)

The first two terms represent the gravitational potential, with the first being a constant that can be thrown out of the Lagrangian.

This U needs to be rewritten as in the above formula to define the K matrix, but the formula I've got demands that each term contain either a cross-product between the general coordinates or a square of a single general coordinate. If what I'm doing is right, then how do I fit the 2xL and 2yL terms into my matrix, and if not, then what am I doing wrong?

It's occurred to me gravity might not be part of the problem as well - so that everything is symmetrical - but since the issue I'm having doesn't stem from that I haven't thought about it much further.

Thanks for any help you can give!

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# Homework Help: Finding Normal Modes of Oscillation with matrix representations

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