Displacement of surfaces on a high speed rotating cylinder

[pineknotpineknot]

Hello, I'm new to Eng-tips forums. I'm looking for a very rare equation which will describe the radial displacement at any radial position on a thick walled open ended steel cylinder rotating at high speed (high centrifugal forces). There are many equations out there for thin walled cylinders (otherwise known as annular disks or rings). These thin walled equations appear in Roark's, Hearn's Mechanics of materials and other sources along with their associated equations for tangential and radial stress. I have yet to find a displacement equation for thick walled cylinders though. Can anyone help me on this? Thanks.

 

[vpl]

You might consider using the search function. I entered "thick walled cylinder" and pulled up a number of hits that seemed to be responsive to your situation.



Patricia Lougheed

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[metengr]

pineknotpineknot;
I would respectfully disagree with your statement ....

There are many equations out there for thin walled cylinders (otherwise known as annular disks or rings). These thin walled equations appear in Roark's, Hearn's Mechanics of materials and other sources along with their associated equations for tangential and radial stress.

In Roark's book (5th edition in my possession) there is a section on annular discs subject to dynamic and temperature stresses. Annular discs are not characterized as thin or thick-walled components for the determination of radial and tangential stresses. The characterization of annular discs as thin-walled is your assumption. There is no mention or reference that these equations are limited to thin-walled discs. In fact, Roarke mentions that if the annular disc is subjected to radial pressures from being mounted on a shaft, another table with formulas for thick-walled cylindrical components can be superimposed on the above inertia stresses.

Unless you want to go the way of complex FEM modeling to calculate strains based on locations in from the central axis, I would re-consider the formula's for calculating the change in outer radius and inner radius of an annular disc as first approximations in Roark’s book.

 

[prex]

Confirming what has been said by metengr, you find in Roark the equations for [σ]_r and [σ]_t for a rotating cylinder of any thickness.
If you want the expression of radial displacement [ξ] at any radius, it is easily obtained from Hooke's law as follows:
[ξ]=r[ε]_t=(r/E)([σ]_t-[ν][σ]_r)
You can cross check this expression with the displacements at inner and outer radii provided by Roark.

prex

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[pineknotpineknot]

Thanks for the responses, prex and metengr. The reason I am leary to use what is found in Roark's is because those same formulas for tangential and radial stress can be seen from Shigley and Hearn (I have attached a word file with those excerpts). In both Shigley and Hearn they are accompanied with warnings that they are only valid for thin walled cylinders. (r < 10*thickness) I would assume that if the stress equations are valid only for thin walls then the displacement equations in Roark (which appears side by side with those stress equations in other sources also) is also valid only for thin walls. Prex, could you give me a hint where I can confirm your statement that the radial displacement is always found by that Hooke's law equation for cylinders? Sorry I'm anal. I like to confirm everything on a few sources. Thanks again.

 

[prex]

If you read Hearn attentively, you'll see that the thin rotating disk limitation only holds for the first (very simplified) equation.
Concerning Shigley, I think that what they call thickness is in fact the thickness of the ring or the length of the cylinder (same terminology used by Hearn BTW). There is no limitation either on ring (axial) thickness or on cylinder (radial) thickness in the theoretical derivation that leads to the second order differential equation from which the expressions for radial and tangential stresses are derived. I think that the limitation set forth by Shigley is justified solely by practical considerations: a long rotating hollow cylinder would need to be supported very carefully on its shaft to not grossly violate the boundary conditions implied in the equations ([&sigma;]_r=0 at both R_i and R_o).
And the derivation of [&xi;] (whose expression BTW you have from University of Tennessee) is simply the application of Hooke's law, that you can find in any book on the theory of elasticity:
[&epsilon;]_t=([&sigma;]_t-[&nu;][&sigma;]_r)/E
[&epsilon;]_r=([&sigma;]_r-[&nu;][&sigma;]_t)/E

prex

http://www.xcalcs.com

: Online engineering calculations

http://www.megamag.it

: Magnetic brakes and launchers for fun rides

http://www.levitans.com

: Air bearing pads

 

[pineknotpineknot]

Guys, thanks for your help with this. I'll give it a go with this info.