Shape suggestions for known mass & inertia
Shape suggestions for known mass & inertia
(OP)
Hi all,
I am trying to develop a revolved shape that will have a known mass = 130 kg and known I (about the rotation axis) = 3.6e7 kg*mm^2 and must fit inside the cylinder with Do <= ~1200 mm, Di >= ~300 mm and Length < 450 mm.
Shape is completely arbitrary and density can be modified as well. I tried many obvious shapes and can not get the inertia high enough with the variables I have. Any suggestions? I am using Pro/ENGINEER's Behavioral Modeling functionality, but the problem solution is not dependant on this tool.
Thanks in advance.
I am trying to develop a revolved shape that will have a known mass = 130 kg and known I (about the rotation axis) = 3.6e7 kg*mm^2 and must fit inside the cylinder with Do <= ~1200 mm, Di >= ~300 mm and Length < 450 mm.
Shape is completely arbitrary and density can be modified as well. I tried many obvious shapes and can not get the inertia high enough with the variables I have. Any suggestions? I am using Pro/ENGINEER's Behavioral Modeling functionality, but the problem solution is not dependant on this tool.
Thanks in advance.
Best regards,
Matthew Ian Loew
"Luck is the residue of design."
Branch Rickey
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips Fora.





RE: Shape suggestions for known mass & inertia
RE: Shape suggestions for known mass & inertia
RE: Shape suggestions for known mass & inertia
In the limiting case, I=mass*R^2, for which I get 4.7E7 kg*mm*2, so it should be doable.
The problem, then, is that the density is too low.
Running a Mathcad solve block, I got 542.1 mm and 509.8 mm to match the moment and density of Aluminum.
TTFN
RE: Shape suggestions for known mass & inertia
RE: Shape suggestions for known mass & inertia
I suspect you looked at the Machinery's handbook where there is a typo error, it uses the "+" sign where it should be the "-" sign which is the correct one.
RE: Shape suggestions for known mass & inertia
TTFN
RE: Shape suggestions for known mass & inertia
I = 0.5*M*(Ro^2+Ri^2) = 3.6e7 kg*mm^2
for Ro = 600 mm, solving for Ri gives 440 mm
RE: Shape suggestions for known mass & inertia
http://230nsc1.phy-astr.gsu.edu/hbase/ihoop.html
TTFN
RE: Shape suggestions for known mass & inertia
Sorry for the confusion, I usually calculate the mass from the density multiply by the volume. Now I understand your point you try to keep M constant. You were correct I was mistaken.
I tried to calculate it from the basics using the material density "rou" and came to:
Mass=Pi*rou*(Ro^2-Ri^2)*Length
Inertia=0.5*Pi*rou(Ro^4-Ri^4)*Length
Now you have four unknown, Ro, Ri, Length and the density (rou).
From the Mass equation you can extract Ri^2 and insert it in the Inertia equation. This reduces to only three unknown Ro, the Length and the density (because the Inertia and the Mass are constant).
I assume that you are trying to design a flywheel. Therefore, to reduce stresses and unbalance to the minimum you need the smallest outside radius Ro. Therefore, I would choose the largest Length that you can allow and then draw the Ro against the density and choose a material that will give the minimum outside radius Ro. You also have to calculate the stress and the material has to able to withstand the stresses too.
RE: Shape suggestions for known mass & inertia
Interesting, my second edition of Fundamentals of Machine Component Design by Robert C. Juvinall & Kurt M. Marshek (www.amazon.com/exec/obidos/tg/detail/-/0471448443/qid=1...) has the equation with the “-“ sign. In another page that IRStuff references (http://230nsc1.phy-astr.gsu.edu/hbase/icyl.html#icyl3) the opposite equation is found.
In my analysis, however, I am not using the closed form solutions (because I did not want to be restricted to rectangular x-sections) for my general purpose feature. The optimizer could not create the geometry I wanted with the restrictions on max size. I will keep plugging away until I find a solution I like.
The maximum I value I can obtain is ~2.5e7 kg mm2 with
Do <= 914 mm and Length < 450 mm. (note: I was off with the earlier figure on the Do)
I can meet both goals with Do allowed to go to a value of 1083 mm.
Again, the solver in Pro/ENGINEER works great for me, but I can’t find a better geometry to fulfill the goal. I am still interested in additional commentary on the discrepancies on the equations.
Thanks all!
Best regards,
Matthew Ian Loew
"Luck is the residue of design."
Branch Rickey
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips Fora.
RE: Shape suggestions for known mass & inertia
However, your optimizer seems to be giving out goofy answers, but that may be because you've fixed Di at 300mm. That page I referenced has a moment calculator and the answers it gave matched my Mathcad calc. With all the mass at a diameter of 1200mm, you should get 4.7E7 kg*mm^2. That's the maximum mathematically possible for that diameter.
Is there some reason that Di can't go past 300mm? If there's some sort of limit, then the solution would be to have a hollow cylinder with a small number of flat annuli to get the required inner diameter, but put all remaining mass at the maximum radius. I guess the worst-case configuration would require a cylinder at the inner radius, a cylinder some maximum radius and a bunch of annuli to hold the two cylinders together.
TTFN
RE: Shape suggestions for known mass & inertia
Thanks for your continued interest. The optimizer seems spot on to me, the Do <= 914 mm and Length < 450 mm works out with the closed form equations, but I am short on the I. I am not forcing the Di to a value, the 300 mm figure was the minimum value that the value could take. In reality, I have the model as a thin revolved protrusion, so here are the actual variables:
Do is d335
Length is d330
Thickness is d323 so Di is derived from Do - 2 * Thickness
Here is the screenshot of the optimizer:
Again, my real limit on Di is 914 mm. It simply will not work with densities of earthly materials.
Best regards,
Matthew Ian Loew
"Luck is the residue of design."
Branch Rickey
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips Fora.
RE: Shape suggestions for known mass & inertia
The minimum mathematically possible radius is 526 mm.
Alternately, the mass must increase beyond 172 kg, or some combination thereof
TTFN
RE: Shape suggestions for known mass & inertia
RE: Shape suggestions for known mass & inertia
Here is where my model stands now:
Do = 1051.6 mm
Thickness = 0.82 mm
Length = 476.73 mm
Density = 0.0001 kg / mm3
Mass = 130 kg
I = 3.6 e7 kg * mm2
Best regards,
Matthew Ian Loew
"Luck is the residue of design."
Branch Rickey
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips Fora.
RE: Shape suggestions for known mass & inertia
Osmium is 22.6 gm/cm^3, which is more like 0.00002257 kg/mm^3 in your units.
With osmium, you can get Do = 1056.2, thickness = 3.4mm, and same length as you have
TTFN
RE: Shape suggestions for known mass & inertia
Sloppy job on subtracting in the previous post, thickness = 3.65 mm
TTFN
RE: Shape suggestions for known mass & inertia
I am just posting the current numbers from the model solution. The actual numbers in the optimization go off to about 15 significant figures. The density is not a problem, as I am trying to represent with simple geometry the properties of a far more complex system. The size restriction I was attempting to put on the analysis was to keep the solid geometry “hidden” within a certain cosmetic envelope. I use Pro/ENGINEER as an engineering tool, not just a modeling tool. So I am allowed certain freedoms in how certain engineering attributes get represented in the engineering models.
Best regards,
Matthew Ian Loew
"Luck is the residue of design."
Branch Rickey
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips Fora.
RE: Shape suggestions for known mass & inertia
If whatever you have does not occupy the full volume, then you need to have the envelope density be lower than the density of the material being used.
TTFN
RE: Shape suggestions for known mass & inertia
The envelope to package my parametric solid is smaller that the actual body that generates the properties. The actual body is also hollow, but that space can be used to “house” the representative solid. Think of a tire and an insert inside the tire representing the properties of the tire.
Best regards,
Matthew Ian Loew
"Luck is the residue of design."
Branch Rickey
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips Fora.
RE: Shape suggestions for known mass & inertia
TTFN
RE: Shape suggestions for known mass & inertia
No problem. Thanks for your continued interest!
Best regards,
Matthew Ian Loew
"Luck is the residue of design."
Branch Rickey
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips Fora.