Relay burning after some comutations

matbks:
I believe my relay got damaged because it worked without a ground wired on it (for less than one minute) wich made it oscilate very fast, making a loud noise.

Which could easily have damaged your arduino.

When discussing a circuit is always best to show the schematic you are working to.

Photo of a hand drawn circuit is fine.
You appear to have mastered eagle though which you may find easier to post.