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# Flywheel Inertia with chamfer3

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## Flywheel Inertia with chamfer

(OP)
Hi

I am currently designing a flywheel to replicate the inertia of a true system in a test rig application.

I have designed a basic cylindrical flywheel to achieve the requested value of inertia. The problem is that the machining company have stipulated a chamfer to be added to either side of the outer radius of the flywheel.

It is important that the inertia of the flywheel is as accurate as possible, I was hoping to find a formula online that allows you to modify the standardised cylinder formula to account for a chamfer. I will be adjusting the value also when the true density of the metal being used has been communicated. the depth (h) is set at 0.1m only the radius will be manipulated.

Does anyone know what process I should go through to recalculate the flywheel radius accounting for the champfer?
Even point me in the right direction would be great.

Non champfer detail:

Inertia required : 5.87931658 kgm^2
rho, density : 7850kg/m^3
h, depth : 0.1m
r, radius : 0.262775m
m, 170.2896236 kg

Any help or advice would be appreciated.

5a

### RE: Flywheel Inertia with chamfer

CAD would do it for you. But won't this part need to be balanced after manufacture? Maybe as part of balancing you can also test and adjust the inertia. Is it so critical that you should consider the change in inertia due to growth under centripetal force?

### RE: Flywheel Inertia with chamfer

Look for the parallel axis theorem.

### RE: Flywheel Inertia with chamfer

(OP)
Thanks for the reply arbreen. Unfortunately I am not proficient on or don't have access to CAD software which I agree would be the quickest option. Which leaves me with the pen and paper option

The part will indeed require balancing but unfortunately after manufacture we are unable to then increase the radius to compensate for any differences so I need to find a way to calculate the radius incase of an increase. It's the inertia value that is the target value. I hope that made sense?

3DDave, I've had a quick look at that theorem but i'm struggling to find a valid example that will allow me to make the correct calculation for my problem. I will keep looking though.

### RE: Flywheel Inertia with chamfer

I was able to arrive at a solution through two methods. First, I used a combination of simple moment of inertia equations for cylinders and cones to get to the shape of a chamfer. Second, I integrated over the radius with a thin tube as my differential element. I was able to get identical results from both, but it would have been hard to catch my errors without the computer's numbers for confirmation. The formulas are a little too complicated to type out here, but you can check your work against this: according to my figures, machining a single 10mm X 45° chamfer on a 1000mm OD steel cylinder will remove a mass moment of inertia about the cylinder's axis of around 0.302 kg*m^2. On the other hand two 1mm chamfers on your example cylinder should only reduce your moment of inertia by about .001 kg*m^2, and if you're fretting over a change that small then you may have unrealistic tolerances for dimensions and material properties. I still think it would be better to measure the moment of inertia as manufactured and plug the value into a computer, or make it adjustable and dial it in.

3DDave, I also couldn't find a way to apply the parallel axis theorem. Could you elaborate or show an example?

### RE: Flywheel Inertia with chamfer

(OP)
Thanks for that arbreen

I will attempt to replicate your working. You said it's a lot to type in but is a picture of the working feasible?

From what you've calculated that certainly does provide a good snapshot into the effect on inertia of the chamfer. In the application I have the 3DOF model I have developed is certainly affected by the 10mm champfer but I agree that the 1mm shows very little effect.

I still hope to replicate your results so that I can make adjustment to the values and make a decision on adjustment of the radius to compensate the inertia loss.

Thanks again

### RE: Flywheel Inertia with chamfer

Two 10mm x 45deg chamfers on the outer corners of a 1000mm OD cylindrical flywheel will have a noticeable effect on inertia. And as others noted, depending on the dynamic balance requirements, the volume and location of material that must be removed to dynamically balance the flywheel will alter its inertia characteristics.

I've had some experience with dynamic balancing of high speed rotating components, and even though they were 100% machined from bar stock, they still required a surprising amount of material removed to achieve the dynamic balance condition needed.

### RE: Flywheel Inertia with chamfer

For a calc. model, I'd propose to put together the shape like this (if I understood the requirement correctly). The balancing will take it's toll, and if the material removal is not symmetrical to the regular shape the final inertia will need to be established by measurement. Depending on your tolerance requirements...
Regards

### RE: Flywheel Inertia with chamfer

Inserting your dimensions and density into solidworks I get the following moments of inertia:
No Chamfer - 5.87930122948
1mm x 1mm Chamfer - 5.87840967285 (-0.015%). Increase radius to 0.26279 gives Lxx = 5.87975
10mm x 10mm Chamfer - 5.79314711169 (-1.4%). Increase radius to 0.26374 gives Lxx = 5.87902

Most of those significant digits are useless unless your machinist can hit 10 micron tolerance and your material density is accurate to better than 0.001%.

The only way to get the inertia accurate to the last significant digit in your specification is to machine the flywheel slightly oversize, measure the inertia and remove material itteratively (finishing with fine emery) until the inertia is correct. Then you will need to maintain the flywheel at a constant temperature (+- 0.0001 degree C)

je suis charlie

### RE: Flywheel Inertia with chamfer

I am not trying to be silly but do you really think you can measure to the number of decimal places you are quoting.

If you start by sensibly rounding the numbers you may discover it all becomes negligible.

I spent years making rotating equipment test rigs and we never worried to the extent you seem top consider important and we used to make systems that ran to 200k rpm on a reasonably regular basis.

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