Moments of inertia

Overview

In order to calculate an overall moment of inertia for the roller assembly, it helps to break it down into a series of simpler geometric shapes. For each of these shapes, the moment of inertia can be calculated independently, and once complete, the results for each section can be added (or subtracted) to produce a total for the entire roller assembly.

As an example, here is the process I followed for the brake disk assembly.

Brake disk

The brake assembly has a thinner (4mm) disk that ends up sandwiched between the electromagnets, and a larger central cylindrical section that fits onto the tapered roller, with both a truncated cone and a cylinder removed from the centre. In these pictures, this is turned onto it’s side, rotating around a central vertical axis. Apart from fitting the page layout better, this orientation makes more sense of the radius and height elements of the equations.

brake3_crop

The first step is to calculate the volume of each ‘feature’ from the measured dimensions. From this we can determine the mass, and then use these figures to find the moment of inertia.

Volume

There are a couple of equations needed for calculating the volume;

volume of a cylinder:

V = π r² h

volume of a truncated cone:

V = 1/3 π (r1² + r1+r2 + r2²) h

Starting with the thinner section of the disk – the first step is to calculate the volume of a cylinder representing the entire width of the disk (r = 66.5mm, h = 4mm).

We can then calculate the volume of a smaller cylinder, representing everything inside the central section (r = 25mm, h = 4mm). Then if we subtract one from the other, we are left with the volume of just the thin external section. Hopefully the image below makes sense of these three steps;

brake_disk_outer

Then we can perform a very similar process for the thicker central section. First we calculate the volume of the solid cylinder (r = 25mm, h = 26mm), then the volume of the features removed from the centre – in this instance a smaller cylinder (r = 10mm, h = 7mm) and a  truncated cone (r1 = 10mm, r2 = 12mm, h = 19mm). We then subtract the volume of both these features from the starting cylinder, leaving just the remaining material.

brake_disk_inner

Mass

Now we know the volume of all the features (and therefore the volume of the entire assembly), we can simply calculate the mass of each feature as a proportion of the overall weight of the brake assembly.

Moment of Inertia

The next step is to calculate the moment of inertia for each feature, just as we did for the volume calculations. The equations needed for this are;

moment of inertia of a solid cylinder:

I = 1/2 m r²

moment of inertia of a cone:

I = 3/10 m r²

You may notice this last equation is for a full cone, so in order to calculate the moment of inertia for our truncated cone, we need to perform one additional step. We calculate the volume, mass, and moment of inertia of both a full cone, and a smaller cone representing the portion that is removed. Subtracting one from the other will then leave us with the value we want for the remaining truncated portion. Again, hopefully the image below makes more sense of these steps;

brake_disk_cone

Once we have the moment of inertia for each individual feature, we can calculate the overall moment of inertia for the brake assembly. This is done by following exactly the same steps of adding and subtracting features that we used for the volume calculations above (i.e. large thin cylinder minus small thin cylinder, large central cylinder minus small central cylinder and small truncated cone).

Roller and flywheel

The same approach was then taken for both the main roller and the flywheel.

For the threaded sections of the roller, I cheated a little and treated them as cylinders with a diameter somewhere in-between the minor and major thread diameters.

Nuts and bearings

For the hex nuts, I cheated a lot and treated them as having an outer diameter  somewhere in-between the flats and the points. At this stage I’ve ignored the mass/inertia of the bearings completely.

I think all this cheating is an acceptable nod to reality, as these aspects only account for a tiny fraction of the overall moment of inertia.

Putting it all together

Finally we can sum the moment of inertia of each part (brake, roller, flywheel, nuts) to come up with an overall figure for the roller assembly,  which by my current reckoning is 0.005418 kg m² (allowing for howlers in my calculations/spreadsheet which I may yet uncover!).

There are elements outside of the roller assembly that will need to be considered, primarily the moment of inertia of the rear wheel. It is notable that my calculated moment of inertia for the roller assembly is considerably smaller than a very rough estimate of the moment of inertia for a rear wheel (~ 0.1 kg m²) .

Of course the roller is rotating a lot faster than the wheel, which greatly increases the angular momentum, but even so – my gut feel is that I ought to double check my figures. Either way, any gross errors should become apparent once I start performing some spin down and power testing…

Next steps

Next up will be measuring the moment of inertia of my powertap wheel, and then putting together a speed sensor to feed the roller data back to the PC for processing.

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