A theory I have is that the oscillation is caused by weathercocking but maybe its the opposite and the rocket isn't stable enough.
If the rocket was not "stable enough", it would have veered way off vertical, or turned into a "sky writer":
But we are all guessing here based on video evidence.
The way to find out for sure is to build your rocket model in OpenRocket or RickSim, and compare the location of CG vs. CP. Have either you, or the team, modeled your rocket?
Also, did you actually fly an altimeter to measure the actual speed off the rail? In flight?
That data should give you answers to most of your questions.
Also, if you keep watching the video after the parachutes deploy, you can see that the section with the eggs and the 2 parachutes twirls and spins while coming down which reduced time and actually caused one of the eggs to crack. Why is this happening and how can we avoid this in the future. Any ideas or contributions are much appreciated.
Your rocket should not be spinning wildly under two fully opened chutes.
It looks like 1 of the 2 chutes did not fully inflate (time stamp 0.17-0.30), and the payload swung around wildly. Around 0.31 the second chute at least caught enough air to semi-inflate, but is still failing to provide equal coefficient of drag to that of the primary chute.
What was the delta between the modeled vs. observed descent speed under those 2 chutes?
As for the spinning during recovery, I believe that one of the parachutes has unequal-length shroud lines, causing it to "spill".
That is a good guess.
Could also be that the lines on one chute were tangled up.
Have both chutes been checked for construction abnormalities?
Did both land with lines relatively untangled?
Did you use a swivel on both chutes to keep the line tangling to the minimum?
My ultimate "correction", after equalizing the shroud line lengths, would be to run a line up through middle of one of the parachutes and attach the second parachute ABOVE the first one. By arranging them in a "linear" manner, the descent should have far less oscillation.
As anyone who has ever sailed will tell you, "taking" the wind away from the leeward sail / chute will reduce its effectiveness by a huge amount. This is great competitive advantage in sailing, but will also negate the effectiveness of the leeward (2nd level) chute.
You compensate for that by flying two larger chutes, but that simulating desired chute sizes unnecessarily difficult. Or just get two equal distant chutes to work properly, like NASA, SpaceX do with full size capsules (2-4 chutes in parallel), and these model Saturn V do as well:
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