Bifilar Pendulum w/ off-centred CM, need verification for derived eq.

[ChessGM]

Homework Statement

For an essay I'm looking at the MOI about the central diameter of a composite object (2 flush concentric cylinders, the external cylinder is shorter lengthwise and can slide along the longer one) with a centre of mass that is not on the central diameter. This is done through the external cylinder, and placing it some distance from the centre, and timing its period in a bifilar pendulum.

I have an equation but I'm not sure if its correct, if anyone could take the time to come up with it and cross-reference it that would be appreciated.

Relevant Equations

The "standard" equation is 2 x pi x r x sqrt(IL/mg), but now it does not work because it assumes a centred centre of mass.

Here are some diagrams:

The mass of the rod is M, and the external cylinder is m. This is what I came up with:

 

[berkeman]

Can you show your math work that led to that result? I'll send you a message with tips for posting math here at PF using LaTeX.