I find this often happens in especially plastic shuttlecocks. Does it make an enormous impact on your badminton game if the plastic is slightly torn?
You probably are the only one having this problem. If you find the shuttlecock not up to standard. change it. In the event that it happens all the time, switch brand. Better yet, play feather.
Plastics are fine if you are new to the sport. Little tears to the skirt might not make any difference in playability. But if the thing wobbles in flight or becomes difficult to clear then it is time to make a change. Similarly there is a time when the cork can become too beat up to respond well. Everyone has to judge for himself. Welcome to this wonderful sport. And this wonderful site!
