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..?

 

[MintJulep]

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

You need to review Poisson's Ratio.

 

[btrueblood]

Well, my first thought was a rubber O-ring, and for such materials a constant volume is a pretty good approximation. Of course, the inside surface of the O-ring will tend to flatten, so the "math" problem is more of a mental excersize than anything.

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.

 

[deangardner]

Yes it is to work out what the change in section of an o-ring would be when the environment goes cold and it's sealing onto a metal shaft. The o-ring guide lines also state that for approximation it can be assumed, in percentage, to be half the amount of stretch. An elongation of 1% corresponds to a reduction of the cross section (d2) of approx. 0.5% - "Taylor's Expansion". Thank you. That's a good aprox for what i'm doing.

 

[deangardner]

The problem is that i still can't work out what the new mean diameter will be... I only know the inside diameter that it has to stretch over.

 

[rb1957]

you need to determine the initial cross-section of an O-ring, so that when you stretch over a shaft of an unknown diameter you get a desired final cross-section ...

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 ??

 

[deangardner]

That is the question. What is the new OD?

 

[rstupplebeen]

With decent material properties this is trivial in FEA. I hope this helps.

Rob Stupplebeen

 

[sdra2]

Trelleborg have what I think you're after in their O-ring guide. Need to register with them to view it.

Cheers

 

[btrueblood]

Parker handbook also has that info.

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...

 

[racookpe1978]

No. I must politely disagree.

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.

 

[btrueblood]

racooke,

With what do you disagree?

 

[racookpe1978]

It appears you're overly relying on the theorectical (calculation of theorectical centers of expansion and diameters), rather than the simpler mechanics of a constrained flexible piece of rubber between two pieces of metal.

If thousandths matter, or tens of thousands, then you need to consider the expansion of the two constraining pieces of steel.

 

[btrueblood]

No, simply relying on the math to give an approximation of the contraction in section due to installed O-ring stretch, which was the OP question.

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.

 

[Ron]

For an o-ring, don't you generally assume that the O.D. of the ring is also the O.D. of the shaft, thus stretching the o-ring only by its thickness? I'm not sure where I heard that, but seems reasonable.

 

[btrueblood]

Ron,

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.

 

[sreid]

Wolfram

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.

 

[SomptingGuy]

This page may come in handy one you've got the equation into a cubic form.

http://mathworld.wolfram.com/CubicFormula.html

- Steve

 

[Ron]

BTB...thanks. I'm just a structural guy whose only experience with o-rings is changing them in my kitchen faucet and, many years ago, rebuilding the carburetors on my MGB and Pontiac LeMans.

 

[zekeman]

If you believe in conservation of mass and shape for this o-ring,( I don't) then you don't have to solve a messy cubic.
You just take the logarithmic derivative of sreid's volume equation to get
dR/dr=(R+3r)/(3R+r)