coupled pendulum oscillation

1. The problem statement, all variables and given/known data

Consider 2 pendulums with the same length L, but 2 different masses Ma and Mb. They are coupled by a spring of spring constant k which is attached to the bobs (the masses).

a) find the equations of motion

b) find the frequencies and configurations of the normal modes

2. Relevant equations

θ”(t) = -(g/L)sinθ ≈ -(g/L)θ (using small angle approximation to keep the diffeq linear)

3. The attempt at a solution

a) I came up with:
θa”(t) = -(g/L)θa – (k/Ma)(θa – θb)

θb”(t) = -(g/L)θb – (k/Mb)(θb – θa)

b) I tried using θa(t) = θAcos(ωt – ø) and θb(t) = θBcos(ωt – ø) to solve the differential equation but I could not obtain a symmetrical matrix since the masses of the bobs are different.

Attached Images
File Type: jpg pendulum.jpg (64.0 KB)

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