I HAVE A GEARBOX WITH 24 TEETH AT INPUT SHAFT , 87 TEETH AND 24 AT FIRST STAGE, 22 &84 TEETH AT SECOND STAGE AND 81 TEETH AT THIRD STAGE. I SUGGEST THE GEARBOX HAVE A PROBLEM.HOW CAN I KNOW THE NATURAL FREQUENCY OF THE GEARMESH FREQUENCY?

First thing first, how you know that there is some gearbox problem.
If there is an increase in vibration? At how much of rotative speed? Type of bearings and gearbox?

Gear mesh is simply the no.of teeth X rtotative speed of shaft.

The first thing we need to know is the input RPM to the first shaft. Then you can easily calculate your "fault" frequencies.

Am I right in thinking that this is a four shaft speed reducer box ie the following is the gear combination:

Input shaft 24 teeth meshing with second shaft 87 teeth.
Second shaft 24 teeth meshing with 84 teeth third shaft.
Third shaft 22 teethe meshing with 81 teeth on the output shaft.
This gives a total reduction of about 10.8:1 ratio.
As Yahoo says, your natural or fault frequencies are simply a multiple of each shaft speed X number of teeth.

Sorry to say it

But natural frequency = eigen value, has nothing to do with a gear mesh freqeuncy.

Let's take it in the simple way: Are you running lets say 1000 rpm (imput shaft speed) and you have a bull gear with 75 teeths and a pinion with 25 teeths => gear ratio of 3.0, output speed is 3000rpm and you will have a gear mesh freq. of 1000 rpm x 75 teeths = 75000 rpm (or 1250Hz), or 3000rpm x 25 teeths = 75000 rpm.
Do you have the number of teets and the speed of the gear shaft then you easy can calculate the gear mesh freq. It is a complete different story when you are looking for the the natural frequency of a rotor bearing system.

The above frequency could of course coinside with a natural freqeuncy of one of the shafts, and then we talk about a resonance situation. Important here is that the location of the natural frequendy of a shaft supported by two bearings depends on the shaft properties (stiffness) and the support properties (effective dynamic stiffness and damping terms)

I hope this clarifies it a little.

As several have pointed out you need to be more detailed in your description of this gearbox. But a quick look at the gearsets as you've described them says that the probility of a tooth repeat frequency situation is high. If indeed you have a 24 x 87 in mesh then they share a common prime factor of 3. You also mention you have a 22 x 84 at the second stage; this set has a common prime factor of 2.

If you ever take this gearset apart make sure all gearsets are marked and go back together with the same index of the teeth, otherwise you could create problems. By index the same teeth must be in mesh; mark two teeth of the gear with "X's" and the pinion tooth between them with an "O". Then when you reassemble make sure they go back together with the same "index." This should be done to every meshing pair, especially until you are sure what meshes with what and you can positively rule out the potential of a tooth repeat frequency situation.

I suggest you go to http://www.bently.com/articles/691winter.asp and download an article that I wrote on gear generated frequencies.

John from PA

My question is:
Is this something new that has shown up in this gearbox or has it been there for some time and getting worse or is this the first time the gearbox has been checked for vibration?

If the natural frequency of a gear/shaft is a possible problem but this is the first time the problem has surfaced, I would think that something other than a natural frequency is the problem.

quote:

24 TEETH AT INPUT SHAFT , 87 TEETH AND 24 AT FIRST STAGE, 22 &84 TEETH AT SECOND STAGE AND 81 TEETH AT THIRD STAGE

The "22 &84" in the above quote makes me wonder if the setup of the gearbox is different from a straight forward reduction.

Is the gear setup anywhere close to the attached drawing?

Seems like if the one on the left is it, then if 1000 rpm were the input speed then the output would be about 24 rpm, but if the one on the right is it then the output would be about 312 rpm.
But then again I might be totally wrong all the way around.

Can you show us a plot or more info?