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I'm modeling a shape that is best described as a partial, hollow torus. Here's what it looks like: https://i.stack.imgur.com/rZopR.png

In my application, the angle can vary from 0 to 85 degrees. I'm writing a program that iterates through this shape and requires the x and y coordinates of the center of gravity for any angle. I don't know where to start deriving this expression. I've tried using this page as a guide: How to find the center of mass of half a torus?

The results I got from that page didn't seem reasonable. The hollow, non-symmetrical aspects of my shape make it distinct from the previous question. Where should I start with this?

(If it's still not clear what I'm helping, this is what I'm after: https://i.stack.imgur.com/r8Ruf.png)

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MattM
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Let $R$ be the distance from the center of the torus to the center of a circular section, $r_1$ and $r_2$ be the inner and outer radii of each circular section (according to your picture that should correspond to $r_2=D_t/2$, $r_1=D_t/2-t$ and $R=(3/4)D_t$). If you consider a thin slice of your torus, corresponding to a small angle $\Delta\theta$, then you can find the distance of its center of mass from the torus axis by summing over thin strips, corresponding to an angle $\phi$ around the center of the circular section, and over $r$ running from $r_1$ to $r_2$: $$ r_{slice}= {\int_{r_1}^{r_2}\int_0^{2\pi}(R-r\cos\phi)^2\Delta\theta\,r\,d\phi\,dr \over \int_{r_1}^{r_2}\int_0^{2\pi}(R-r\cos\phi)\Delta\theta\,r\,d\phi\,dr} =R+{r_1^2+r_2^2\over4R}. $$ The center of mass of the torus lies along the bisector of angle $\theta$. We can get its distance from the torus axis by averaging the projection of $r_{slice}$ on the bisector: $$ r_{torus}={1\over\theta}\int_{-\theta/2}^{\theta/2}r_{slice}\cos\alpha\,d\alpha= \left(R+{r_1^2+r_2^2\over4R}\right){2\over\theta}\sin{\theta\over2}. $$ In the above formula, of course, $\theta$ must be expressed in radians.

enter image description here

Intelligenti pauca
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  • I don't see the hollow portion comes to play in your derivation. Is this for a solid torus?

    If so can I simply apply the same equation to find the CM of the hollow torus and use composite body equations?

     – MattM Jul 19 '16 at 15:06
  • Additionally I am getting R=(3/4)Dt: 1/4th from the axis to the edge of the torus (which is slightly vague in my figure) and an additional 1/2 from the edge of the torus to the center. – MattM Jul 19 '16 at 15:15
  • My formula refers to a hollow torus: I've added a picture for a better understanding. Please ask if single passages are not clear. Of course you are right about $R$: I'll correct immediately. – Intelligenti pauca Jul 19 '16 at 17:32
  • I've also written in full the integrals giving $r_{slice}$, hope it is clearer now. Once computed $r_{slice}$, getting the position of the center of mass is straightforward. – Intelligenti pauca Jul 19 '16 at 17:41
  • Thank you so much for your help - this is fantastic. Your image helps a great deal. My torus is hollow, but to approximate it as a shell may be too large of an assumption. The thickness (t in the image in my original post) can grow as large as 25% of the torus radius r. Does this invalidate the derivation? This is why I asked about using composite bodies. – MattM Jul 19 '16 at 18:35
  • Yes, if thickness is not negligible there may be a difference. I'll think about that. – Intelligenti pauca Jul 19 '16 at 19:30
  • I've updated my answer to take thickness into account: fortunately there wasn't much to change. – Intelligenti pauca Jul 19 '16 at 20:20
  • Fantastic. Thank you so much! – MattM Jul 19 '16 at 21:39