The advantages of a flapwheel over a traditional disk are twofold: They appeared in the 1950s and their use was encouraged by developments in gas turbines.
The first flapwheels were made from cut sheets of abrasive, stacked into a multi-layered flat disk.
Portable flapwheels were made by bonding radial flaps of abrasive into a hub fitted with a spindle shank.
The extra lifetime this gave to the disk allowed their use for stock removal and metal shaping, not just finishing and polishing.
The flexibility of a flapwheel is an advantage for smoothing curved work and avoids the production of facets, as was a problem for rigid disks.