Mass moment of inertia of an irregular area about its centroid

[darkenergy]

Hi. Please help. I'm struggling to find the mass moment of inertia of an area.

Please see picture.

I already have the centroid of the area, and the exact area.

How would I find the mass moment of inertia of this area about its centroid?

I tried using AutoCAD, but it gives you the *area* moment of inertia which isn't helpful to me?

do you guys know any programs that find the mass moment of inertia of an irregular area for you?

 

[TheTick]

Most 3D CAD programs could give you this information.

Mathematically, one would divide the shape into discreet rectangles, triangles etc. Find the I of each parcel, and multiply by the distance^2 of the parcel centroid from the main centroid.

 

[rb1957]

isn't the mass moment = the area moment * density ?

 

[rb1957]

(of course) that assumes a 2D shape ...

otherwise you're left to do it from 1st principles,
by taking slices thru the body at different x-, y-, and z-

 

[darkenergy]

emm okay.

so let's say I have a 2D object in AutoCAD.

1) it works out the area moment of inertia for you
2) I multiply this by the depth of the area, giving you the 'volume moment of inertia'
3) multiply this by density
giving you the 'mass moment of inertia'?

would this work?

I'm a biologist, and I just always thought the area moment of inertia and mass moment of inertia are totally different things (mass moment of inertia = rotation, area moment of inertia = bending?)

 

[rb1957]

actually you need the polar moment of inertia ...
Ixx = integral (y^2*dA)
Iyy = integral (x^2*dA)

J = integral (r^2*dM) = (Ixx+Iyy)* (mass density)

you might have weight density (lbf/in^3), divide by g (386in/sec^2, or whatever units you're using)

 

[darkenergy]

hey; thanks
sorry - but I'm a dimension out:

what about the thickness of the area section?

 

[JStephen]

"Moment of Inertia" is used in different ways.

If you're rotating the blob about an axis going into the paper (ie, spinning it on the paper), you want the polar moment of inertia.

If you're flipping it about the line that is drawn across, then the area moment of inertia adjusted for mass instead of area would work.

Approaches to finding the moment of inertia: AutoCAD, as you're already using. Direct integration. Numerical integration by spreadsheet. Numerical solution using parallel axis theorem.

It's been ages, but seems like there was a Stoke's Theorem that relates integrals over the area to intergrals around the perimeter, and that might be of use in some circumstances.

 

[darkenergy]

I see. thanks a lot. Second option - I have a line horizontally and another vertically, so the area moment of inertia sounds good. nope the axis is not going into the paper...

thank yuou so much everybody!

 

[rb1957]

not just any line ... the axes need to be through the centroid.

you mentioned being one dimension out ...
Ixx = integral (y^2* dA)
J = intergral (r^2*dM) = integral(r^2*dA)*t*rho = [integral(x^2*dA) + integral(y^2*dA)]*rho*t
rho = mass density