What is the Relationship Between a Sand Pendulum and Infinite Series?

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I'm working on a pendulum that drops a stream of sand as it swings, similar to the one in this video. How could this pendulum be modeled or represented by infinite series?

My idea so far is to use parametric equations for harmonic motion, but I'm not sure how that would work for this type of curve.

Could anyone point me to a resource for this type of pendulum?

(This is my first time posting on stack exchange, sorry if I didn't follow the protocol in some way.)