How to calculate the inertia of gearbox?

[Sri_Vars]

Do I have to sum the inertia of all the gears and shafts? If so, how to do that?

 

[Baluncore]

Welcome to PF.
That depends on what type of gearbox it is.
The traditional automotive gearbox uses an input shaft that drives the lay (or counter) shaft. The output shaft is, in effect, a single unit. The ratio of the input shaft to the output shaft is gear dependent.

The moment of inertia of three subsystems must be measured or calculated.
1. The engine and flywheel should include the clutch pressure plate.
2. The gearbox input shaft and lay shaft should be grouped with the clutch friction disc.
3. The gearbox output shaft should be grouped with the drive shaft, differential, drive axles and all wheels.

 

[Sri_Vars]

Thank you Mr. Baluncore for your answer.
We are currently using the traditional gearbox only. But I am not sure how to calculate the inertia of the subsystems. It would be so helpful if you could brief it with formula.

 

[Baluncore]

You will need to dismantle the gearbox, then weigh and measure every component. Take a look at a gearbox parts diagram to see how many items are held on the main shaft.

The pitch diameter of a gear will be misleading due to the radius squared. You will have to approximate gear wheels as polygons, or find some other way.

A ball bearing race will probably have the inner race moving, the outer race static, and the balls spinning, somewhere in between.

https://en.wikipedia.org/wiki/Second_moment_of_area#Examples
https://en.wikipedia.org/wiki/List_of_moments_of_inertia

 

[Sri_Vars]

Thank you so much for your help!

 

[jack action]

Do I have to sum the inertia of all the gears and shafts? If so, how to do that?

No, because all gears and shafts don't necessarily have the same angular velocity. You can do a free body diagram for each rotating component which will give you a set of equations to solve (1 equation for 1 unknown).

This post shows how to do it with the equivalent kinetic energy method.

 

[jrmichler]

Do I have to sum the inertia of all the gears and shafts?

Yes. And you have to calculate the inertia of each component reflected to a common point in order to include the effects of gear ratios. Having done this calculation more than a few times, here's the procedure that I use. It's the same as the procedure in the post linked in Post #6 by @jack action above, just stated differently.

1) Calculate the inertia of each rotating assembly.
2) Calculate all gear ratios.
3) Define the starting point. It's normally the input shaft.
4) The input shaft is geared to the second shaft with a gear ratio. The inertia of the second shaft reflected to the input shaft is the inertia of the second shaft multiplied by the square of the gear ratio between the two shafts.
5) Proceed to the third shaft. The inertia of the third shaft reflected to the input shaft is the inertia of the third shaft multiplied by the square of the gear ratio between the input and third shafts.
6) The inertia of each shaft is the total inertia of the rotating components attached to that shaft.
7) When you have the inertia of all rotating components reflected to the input, then sum them. That's the inertia of the gearbox at the input shaft.

If you want the inertia at the output shaft, the procedure is similar, except that you start at the output shaft and work back to the input shaft.