Stretching a Torus - Maths Problem 2

[deangardner]

I have a mathematical problem…

I’m trying to work out what the cross section of a Torus would be once it has been stretched over a shaft. So, for a given temperature the volume of the Torus would be the same no matter what amount of stretch you impart on the inside diameter. So the volume of a Torus is:

V = pi.r^2 x pi.Dmean (where r = Torus cross-section radius and Dmean equals the chord diameter of the Torus)

The problem is that I only know the inside diameter of the shaft that the torus has to be stretched onto and not what the chord diameter that the torus will end up being. If you rearrange for r, you get:

r^2 = v/pi^2.Dmean

The problem is that you only know what the inside diameter of Dmean is. Which is:

Dinside + ((Doutside – Dinside) / 2)

But I don’t know what Doutside is…

Can this problem be solved..?

 

[IRstuff]

Luckily, the cubic has a single real root.

http://files.engineering.com/getfil...-40c2-b2a4-706201a7737d&file=torus_volume.pdf

http://files.engineering.com/getfil...-49f7-b84f-1941dc86bed9&file=torus_volume.mcd

TTFN

FAQ731-376

 

[bcd]

For o-rings, you cna use the following formula to estimate the diameter of the cross section after stretching to a larger diameter:

CI = SQRT (CS^2 * (ID / IDi))

CI = cross section diameter installed
CS = cross section diameter in free state
ID = ID in free state
IDi = ID installed

Yes, there are many other factors in play such as poissons ratio, deflection where it contacts at the ID. Options are to build one and measure it, conduct an FEA, or use the math problem about to get an estimate.

 

[btrueblood]

"If you believe in conservation of mass ...( I don't)"

Zeke, you think cold fusion may occur?

 

[zekeman]

TRue,
Sorry, you know I facetiously meant "conservation of volume". Old habit. As soon as I write "conservation", the mass thing comes out.

 

[zekeman]

"Luckily, the cubic has a single real root"

What if that root is negative?

I guess instead of taking a vacation this summer one could obtain this closed form solution that could easily be gotten today on any of the plotting hand helds.Must admit that it is a fantastic soution.

BTW, where was this genius when I took algebra eons ago using the likes of DeCarte and Horner for "help".

 

[IRstuff]

Actually, the real root is always positive for plausible conditions. The equation is negative for r=0, and is positive for large r, so there must be a positive root. You can get some oddball answers, like an ID=8, r=134, but r will still be positive.

TTFN

FAQ731-376