I Playing around on my flute I learned that it is just another way of playing Bb. In jazz band I used it a few times just for fun, because I found that it is a quicker than the other methods. Little did I know, it was there for a reason.
This key is used with your right first finger. Unlike all the other keys on the flute, this key is not meant to be played by the tip of your finger. Instead it is meant to be played at your middle knuckle. This is a much faster way of playing a passage like:
When playing this passage from the G#, A#, B, it is much quicker to use the Bb shake key, because you can hit it and continue on. You can use it anywhere you have this combination. It is also helpful from trilling A# to B.
Method #1 - When pressing the right hand, first finger button, you also press down 2 other keys. By having those other keys close down, this takes up more time.
Method #2 - The Thumb Bb key is a heavy and large button to press. This also takes more time to play.
Method #3 - This is the quickest way of playing Bb. When you press the shake key, only one padded key goes down. Try playing it quickly, or trilling from Bb to B, using all three methods.