Hi SethO,
The forces working on a helicopter are quite a few. If we just ignore most of the “secondary” influences, I would sketch the main parts as follow. The point where the blades of the rotor meet, are nearly equal as if one would have a hook there. The set of of the blades and their functions are also ignored here for a moment. So, given that the main weight is the helicopter body with the engine inside, we have a hook point above the center of gravity, which results in the movements you describe. Again, ignoring secondary forces, we can build this easily in Dynamics with a connector. The size of the sphere.1 stands here for the inertia, so increasing this size increases the “pendulum” movement. (File …DYhc_01) IF you place an object way below the Sphere.1 , as a child of the Sphere, you could place a spring between this object and Sphere.1 to create a little bit of secondary influences.
A friction field might help as well.
Just to share an alternative, I have set up one with MoGraph as well, but no Dynamics here, just Delay Effector work. (File …MGhc_01). It uses the movement like a Matrix Object and the Delay Effector Combination Spring and Even to fake the helicopter behavior.
Scene files:
https://www.amazon.com/clouddrive/share/JA4UsIpssDrxOlgPwwK9T5yT4HuDwdRs9qScWADT3QH
However, given the flight path in your scene will maybe not be possible. I would work more manually.
All the best