Zephyr writes:

My question is, is that normal? should the "drive shaft" on an
electic motor have some play in any direction?
Is the the play in the shaft there by design?

How large a motor is this?

If the motor has ball bearings, there will be very little end play,
probably too little for you to feel. But it's pretty common for
fractional-horsepower electric motors to use bronze sleeve bearings
instead of ball bearings, and sleeve bearings allow for axial movement
of the shaft through the bearings. There will be thrust washers of some
sort to prevent the shaft from moving *too* far, but there may be
substantial end play between the two points where the thrust washers
operate.

But why is it noisy? Even with sleeve bearings, the motor is usually
put together so that the magnetic field (when operating) pulls the
rotor to a point about half-way between its limits of play, and the
thrust washers don't need to do anything. If this motor drives a fan,
and the fan blows air parallel to the shaft, then it will put some end
thrust on the shaft and might even pull the shaft against one end of its
travel. Mounting the motor with the shaft vertical may do that too,
courtesy of gravity. Even then, the thrust bearing is steel running on
bronze, and should be quiet unless worn or unlubricated.

Dave