# Calculating inertia of motor

I'm currently working on a project, and for futher calculations down the road (the controller of the mechanical system) I need to calculate, as accurately as possible the interia of the system.

I'm running a BLDC motor, with a flywheel (with a relatively complicated geometry, and the appropriate connecting bits.

After reading the data sheet, and extracting from CAD the inertias of the individual parts along their axis of rotation, I get a total:

$$0.000046289\text{kg}\,*\text{m}^2$$

Which seems appropriate, (it's all very small btw).

But nothing of course is perfect, so I decided to approximate a closer to reality measurement by applying a current to the motor and measuring the rad/s over time, I did this for 0.3 amps, 0.4amps and 0.5amps.

In the current example, 0.5amps.

Here I used two different methods I'm aware of to calculate this, and used mathematica to approximate.

One, a very basic linear function developed from the first to the last point, and then taking the slope of the resulting function calculating it into the equation:

(this could be made slightly more accurate by calculating the median of the slope between each single point, but I digress)

$$J=\frac{I* \tau }{\alpha },\text{ }\alpha \to \frac{\omega _2-\omega _1}{t_2-t_1}$$

$$J$$ -> inertia $$[\text{kg}\,*\text{m}^2]$$

$$I$$ -> the motor amps [A]

$$\tau$$ -> the motors torque/amp [mNm/A]

$$\omega$$ -> omega [rad/s]

$$t$$ -> time [s]

$$R$$ -> Friction

This gave me an inertia of: $$0.000473021\text{kg}\,*\text{m}^2$$. This is an entire order of magnitude larger than the one I calculated from the datasheet and cad.

The second method, I used a nonlinear fit from the model of

$$\frac{d \left(J* \phi ''(t)=-R* \phi '(t)+I* \tau* \theta (t)\right)}{\text{dt}}$$

This gives me a very accurate fit:

And the resulting inertia of $$0.00037437\text{kg}\,*\text{m}^2$$

Depending on the amperage, the values vary slightly from amperage to amerage, this probably an artifact from the fact the system equation, when solved and taking the derivative from is:

$$\omega (t)\to \frac{I* \tau* \left(e^{\frac{R* t}{J}}-1\right)}{R}$$

R and J are a ratio, so this would explain away some of the issues being that the fitting function being used in Mathematica can't calculate such a ration accurately, of course, I know the friction coefficent from the datasheet, so this could be taken away...but I digress again:

My problem is that I'm unsure which of these values I can trust. Generally the cad calculated inertias are quite close to reality (these are consistently used in other engineering projects much more in value than my own), however my calculated and fit inertias match each other fairly well, but are an order of magnitude larger than the CAD values.

Is my methodology incorrect? Is there another way to calculate the inertias of a flywheel and electric motor?

Thanks for the help!

I can post the data points if requested.

• do you mean "inertia"? If so, you may want to check the spelling in the title of your question. And note, for a rotating part, you are implying that you are looking for "moment of inertia". – David White Feb 12 at 17:36
• Uff...thats a mistake. yes i meant inertia, thanks for the tip! – morbo Feb 12 at 17:42
• Please use MathJax to write equations. – exp ikx Feb 12 at 17:45
• @morbo, I wasn't being "picky" on your spelling. I wanted to make sure that I was understanding the question that you were asking. And, you're welcomed. – David White Feb 12 at 17:47
• updated the question with mathjax. – morbo Feb 12 at 20:30