An alternator has to be attached to some type of driver (diesel engine, steam turbine, windmill, etc.) to provide the torque to spin it. Some are direct coupled (the shaft ends are bolted together), some used a set of reducing gears. Some smaller units may used belts or chains with pulleys.
A single bearing alternator has no bearing on the driven end. It uses the bearing on the near end of the prime mover for support (through the coupling). This type of engine-alternator system cannot use a flexible coupling, because the weight of the alternator would place excessive shear force on it. With a hard coupling, both machines effectively create a single, long rotor. Assembling this type of system requires very small tolerances on alignment, and the foundation/support system must be very stiff to prevent excessive vibration from damaging the unit. In my experience, single-bearing units are also somewhat limited in size, as the hard coupling and the bearing on the driver would have to be pretty massive to support half the alternator.
A two bearing unit has support on both ends of each machine (driver and alternator), and can used in situation with higher vibration or lower quality foundations. They can use any type of coupling, since they have support on both ends of the shaft.
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