These pieces are bushings, or "collets" to hold the auger
in older large mortising machines (for woodworking)such as 
Greenlee, Oliver,Fay & Egan, etc.  3 styles.  The nut is to 
prevent the augerfrom being rammed up in the spindle on 
automatic machines where there is a lot of force and no 
opportunity for operator sensing of the loads.