Why single phase induction motor is vibrating?

The ball bearings in the motor probably need to be changed. If the motor does not have ball bearings then the bushings have worn out and the motor armature is dragging on the stator. Don't run this motor until the bearing have been repaired as excessive wear on the stator from armature rubbing will introduce metal filings into the stator windings which could short out and destroy the motor.

If it's the normal operating noise that you're asking about, a three phase motor has a smoothly spinning magnetic field driving it's rotor that has almost no uneven motion, and produces almost no vibration in the iron parts of the machine. A single phase motor runs with a pulsating field that swaps polarity twice per cycle, but DOESN'T actually rotate. This causes the iron in the motor to vibrate slightly whenever the AC is applied, and this vibration causes the noisier operation of a single phase motor compared to a three phase motor.

Single phase motor running noise can be reduced if the start winding li left partly energised during normal running. For instance, some room air conditioners leave a small capacitor connecting the starting winding so it delivers partial power during running to give a 'sort-of' rotating field in the single phase compressor motor, which helps to reduce the 'pulsating' component and enhance it with a 'rotating' field that quietens the compressor down while you're trying to sleep.....