Predicting Motion of a Swing on a Non-Horizontal Branch

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TL;DR Summary

Do you understand swings?

A swing is suspended from a non-horizontal tree branch. Points C and D are fixed in space. All 4 line segments in the diagram have constant distance. After some initial "kick" imparts energy to the system the only force acting externally on the system is gravity.

Is it possible to predict the motion of the swing?

 

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Yes, if we have enough information to completely specify the problem. What are the four points and what is attached between them? Where is the “initial kick” applied and what force is it, applied for how long? How is the mass of the swing distributed?

 

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AB is the seat. AC and BD are the ropes. CD is the branch. I assume the system remains under tension. The initial kick could be positioning the swing away from the minimum energy position then releasing. Mass is centered on the swing seat with some non zero moment of inertia.

 

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I define $\theta$ to be the angle a rope makes relative to z and $\phi$ is the angle relative to x. The branch is in the xz plane. From conservation of energy I got
$$ \frac{r_A^2}{2} \left[\left(\frac{\partial\theta_A}{\partial t}\right)^2 + \left(\frac{\partial\phi_A}{\partial t}\right)^2\right]+ \frac{r_B^2}{2}\left[\left(\frac{\partial\theta_B}{\partial t}\right)^2 + \left(\frac{\partial\phi_B}{\partial t}\right)^2\right] - r_A\cos \theta_A - r_B\cos\theta_B = 0$$
Am I on the right track?