How to know if the bearing of induction motor is healthy or not? there is any test for it ?

I was rather impressed by a good quality acoustic (expensive!) analyser I used. Portable so can use on many items, as well as steam traps etc. We know the ball race frequency at certain speeds so it can detect inner or outer race failures etc.very comprehensive. I am not a specialist in this instrument, but my opinion is it can be earlier detection than other methods such as vibe or thermal, sometimes.

The simplest answer is actually two things.

1) Observe the bearing (and lubricant) temperatures. Are they starting to trend away from "normal" ... particularly upward? If so - there may be a problem.

2) Observe the acoustical sound of the bearing (using a very simple probe, like a wooden or steel dowel .. or even a screwdriver). If it starts to sound different than normal, then things need to be examined more closely.

If it comes to having to take a more in-depth look at the bearings, the best thing is to use a vibration analyzer that will allow both waveform and harmonic amplitude/frequency measurement. You'll need to know the type of bearing, the bearing number (if not a sleeve bearing), and rotational speed of the shaft - as a minimum - to sort out what frequencies are of interest.

There are studies available about vibration in bearings indicating wear. Vibration can be picked up by an accelerometer but also by a microphone. A given bearing design at a given speed will show a spectral analysis (frequency component and magnitude) indicating a relative measurement of its health. Transfer the frequency (mathematically) to harmonics and speed is removed from the equation. Some of the studies draw a relationship between harmonic number and failure modes.
That said, many people use a simple stethoscope and a trained ear.

I have seen a wooden broom handle used. Simply cutting off the broom and filing edges to allow placing on a bearing housing, either motor bearing assembly or a bearing housing assembly. Using the handle place it on the assembly and then simply putting your ear to the other end of the wooden handle any noise is transmitted. The trick is to listen to a known good bearing and use your ear (hearing) to detect any changes, grinding, vibrations. This is a "very" simple process but can be used in a high noise industrial environment when equipment is running to detect a possible problem. Not very scientific or accurate but a good inexpensive way to find out if a problem may exist and then go back with sophisticated analyzers or vibration detection to determine causes. Typically good to determine if something will last until a PM (preventative maintenance) outage is scheduled.

Motaz - what Bob has listed about the broom handle is what I was thinking about when I mentioned an "acoustic" method.

My own personal "tool" is one I made myself several years ago, by welding a hollow ball bearing about 25 mm outside diameter welded to a solid steel shaft approximately 6 mm diameter. The whole thing - bearing and shaft together - is only about 450 mm length.

The end of the shaft away from the ball has been ground down to a taper (roughly 60 degrees or so) to achieve a bluntly-pointed end.

I would say that if you find a noisy bearing is already too late or very close to it. Unusual heat is many times related to excessive or lack of lubricant, bearing housing being out of tolerance, radial or axial load on shaft off limits, etc. thats the reason why temperature sensors are installed in large motors (too expensive for medium or small motors) but is a very good indicator of the condition of the bearing. regular monitoring is a very good practice.