Older days Elevators were run by DC motors, since their RPM was easy to control, and was much cheaper. But it had a worse torque characteristics, hence has a lot of jerks and bumps for the users.
Then as time passed Induction motors began to hit the market and most of the industries now use them just because they tough guys and withstand worse working conditions with almost no maintenance. But Wait!!, Induction motors are AC motors and controlling their speed was tough hence they were not recommended for Elevators until the VFD’s came in.
Hence most of them in modern days use a 3-phase induction motors with variable frequency drive electronic controls.
Because:
1.It has a long life and highly rigid structure.
2.No permanent magnet is involved hence no aging effect.
3.Since modern electronics have improved VFD’s (Variable frequency Drive), It is easy to control the Torque and therefore controlling the RPM of the motor.