Stretching a Torus - Maths Problem
Stretching a Torus - Maths Problem
(OP)
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..?
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..?





RE: Stretching a Torus - Maths Problem
You need to review Poisson's Ratio.
RE: Stretching a Torus - Maths Problem
A fast way to get a pretty good approximation is to assume the stretch on the i.d. causes a similar contraction of the diameter at a 1:2 ratio. But that's from a Taylor expansion, not Poisson. That Poisson was kinda a fishy guy, if you ask me.
RE: Stretching a Torus - Maths Problem
RE: Stretching a Torus - Maths Problem
RE: Stretching a Torus - Maths Problem
sorry, but expressed that way, i don't see any way to solve the problem without knowing the "unknown".
you say you know the ID of the shaft ... then either ...
1) guess the OD, assume a wall thikcness, or
2) ask whoever defined the ID what the OD is ??
RE: Stretching a Torus - Maths Problem
RE: Stretching a Torus - Maths Problem
Rob Stupplebeen
RE: Stretching a Torus - Maths Problem
Cheers
RE: Stretching a Torus - Maths Problem
You know original ring i.d., and cross-section diameter.
Assume the i.d. and mean diameter stretch the same amount. I.e. if 10% stretch is required to make the original i.d. become the new i.d., then assume the entire section stretches 10%. Calculate the reduction in cross section diameter (5%). Calculate the new o.d. and mean diameter.
At this point, you are close enough to see if you have a good seal. If you want to get picky, calculate the stretch in mean diameter from your calculation above, and recalculate the reduction in section diameter, and adjust other values accordingly. But, I'm willing to bet the deviations from 1st calc. to this second iteration calc. will be less than 1%.
Oh, and all of this is really only valid when the stretch is about 10% or less, beyond 20% stretch the Taylor expansion starts to show errors...
RE: Stretching a Torus - Maths Problem
If the O-ring has "any" stretch at all, then its INSIDE diameter = OD of the shaft. Period. the O-ring cannot "squish" the shaft, and if the two are inn contact, then the O-ring's ID has been determined.
But, to seal, the O-ring must flatten (at both inside and outside contact surfaces!), so the O ring OD (if - and only if! - its outside surface still touchs the ID of the device) will be the ID of the seal.
So, your problem consists of determining that the heated and cooled O-ring - with no restrains - overlaps both the OD of the shaft and the ID of the seal.
So heat it up and measure. Cool it down and measure.
RE: Stretching a Torus - Maths Problem
With what do you disagree?
RE: Stretching a Torus - Maths Problem
If thousandths matter, or tens of thousands, then you need to consider the expansion of the two constraining pieces of steel.
RE: Stretching a Torus - Maths Problem
The actual selection of an O-ring (material and geometry) and groove geometry to give good seal results, in various fluids, at various temperatures...would fill a book. Like the Parker O-ring Handbook for instance.
RE: Stretching a Torus - Maths Problem
RE: Stretching a Torus - Maths Problem
Generally you want the o-ring to have slight stretch in the installed condition. The opposite condition (compression of the O-ring required to seat it in its groove) can result in local buckling (wrinkling) of the ring surface, leading to leakage. The od of the ring, installed in the groove, should then be standing proud of the surrounding shaft surface (assuming the groove is on the shaft), and when installed in the mating sleeve, the ring will then undergo a slight squeeze, of somewhere between 10 and 30% of its section diameter. The actual value of squeeze can vary, and how much void is filled as the o-ring is squeezed will vary a bit too (you need to leave more room if the ring will see significant temperature variations, or if the compound used swells when it contacts the fluids to be sealed...).
But, we digress, I believe.
RE: Stretching a Torus - Maths Problem
http://mathworld.wolfram.com/Torus.html
Gives the Volume of a Torus as
V= (1/4)(pi)^2(R+r)(R-r)^2
r = (1/2)(ID)
R = (1/2)(OD)
If the volume is constant you can solve for the new R given a new r.
RE: Stretching a Torus - Maths Problem
http://mathworld.wolfram.com/CubicFormula.html
- Steve
RE: Stretching a Torus - Maths Problem
RE: Stretching a Torus - Maths Problem
You just take the logarithmic derivative of sreid's volume equation to get
dR/dr=(R+3r)/(3R+r)
RE: Stretching a Torus - Maths Problem
h
h
TTFN
FAQ731-376: Eng-Tips.com Forum Policies
RE: Stretching a Torus - Maths Problem
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.
RE: Stretching a Torus - Maths Problem
Zeke, you think cold fusion may occur? ;)
RE: Stretching a Torus - Maths Problem
Sorry, you know I facetiously meant "conservation of volume". Old habit. As soon as I write "conservation", the mass thing comes out.
RE: Stretching a Torus - Maths Problem
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".
RE: Stretching a Torus - Maths Problem
TTFN
FAQ731-376: Eng-Tips.com Forum Policies