foggyengineer
New member
- Joined
- Dec 29, 2025
- Messages
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I'm an engineer. I know this is for kid's homework, but I feel silly posting this to a technical forum. It's essentially an algebra problem. I'm hoping someone will find this interesting and want to help.
I have a ring with an odd cross section, and I want to simplify it to a rectangular cross section while maintaining mass moment of inertia and mass. I have a set of equations, and need help with getting the formulae I need for my spreadsheet.
Given:
[imath]I_x[/imath] - mass moment of inertia about the x-axis
m - mass
t - thickness of annulus
A - effective area of cross section
[imath]\rho[/imath] - density
V - volume
Other Variables - Find any 2:
[imath]r_1[/imath] - radius of inner circle
[imath]r_2[/imath] - radius of outer circle
h - width of annulus
[imath]r_e[/imath] - effective radius, centroid of cross section of annulus such that re = (r1 + h/2) or (r2 - h/2)
Formulae:
[math]I_x = (m/2)(r_1^2+r_2^2)[/math][math]A=\pi(r_2^2-r_1^2)[/math][math]r_e=r_1+h/2=r_2-h/2[/math][math]V=A \cdot t[/math]
The Other Variables are flexible; they do not need a rigid definition so long as the Given variables are constant. I think that I have too few knowns for the number of variables, and some assumptions can be made here. That is why I have defined r1 & r2 in terms of re and h. I can assume one or two and have the others. And that's where I need the help. My brain tells me that these should be fixed values, since mass and moment of inertia must be constant, which is why I think this is just an algebra problem, but I just can't see the solution for the fog in my brain.
⇒ I need a formulae for any 2 of the Other Variables which maintain given mass and moment of inertia.
I have a ring with an odd cross section, and I want to simplify it to a rectangular cross section while maintaining mass moment of inertia and mass. I have a set of equations, and need help with getting the formulae I need for my spreadsheet.
Given:
[imath]I_x[/imath] - mass moment of inertia about the x-axis
m - mass
t - thickness of annulus
A - effective area of cross section
[imath]\rho[/imath] - density
V - volume
Other Variables - Find any 2:
[imath]r_1[/imath] - radius of inner circle
[imath]r_2[/imath] - radius of outer circle
h - width of annulus
[imath]r_e[/imath] - effective radius, centroid of cross section of annulus such that re = (r1 + h/2) or (r2 - h/2)
Formulae:
[math]I_x = (m/2)(r_1^2+r_2^2)[/math][math]A=\pi(r_2^2-r_1^2)[/math][math]r_e=r_1+h/2=r_2-h/2[/math][math]V=A \cdot t[/math]
The Other Variables are flexible; they do not need a rigid definition so long as the Given variables are constant. I think that I have too few knowns for the number of variables, and some assumptions can be made here. That is why I have defined r1 & r2 in terms of re and h. I can assume one or two and have the others. And that's where I need the help. My brain tells me that these should be fixed values, since mass and moment of inertia must be constant, which is why I think this is just an algebra problem, but I just can't see the solution for the fog in my brain.
⇒ I need a formulae for any 2 of the Other Variables which maintain given mass and moment of inertia.
