I was surprised no one answered the question about the Player Piano Co.
drum beater pneumatic [MMD 980419].  It is covered in Larry Givens
book.

The part is a copy of a Coinola pneumatic and similar to a Link.
Inside the pneumatic is a little wooden flap valve.  As the pneumatic
closes, the flap valve closes off the supply and opens a flap valve on
the outside to let the air back in.  There is a helper spring in the
hinge end of the pneumatic to open the pneumatic.  The hook on the
outside of the pneumatic then closes the outside flap and opens the
inside flap valve to repeat the closing of pneumatic.

I would suggest taking the thing apart and checking to see if good ol'
PPCo used leather inside and outside for the flap valves.  I would also
check to make sure the flap valves were free to move and not bind.

These units work with the weight of the beater also pulling the
pneumatic open.  If it just closes and stays closed then the inside
flap is leaking.  There has to be some play between the two flap valves.
These are easy to make with a table saw.  In use they usually work
well.

Don Teach, Shreveport Music Co
1610 E. Bert Kouns, Shreveport,  LA  71105
dat-smc@juno.com