Calculating Moments of Inertia for Arbitrary Sections

[cigarette]

Hi all,

Why are there so many moments of inertia for a given section? That is, i come across moments about the origin, x-axis, y-axis, centroid, xy-axis...etc... What are the differences? are there typical applications?

Please point me in the correct direction

Thanks in advance, any help is greatly appreciated

Cigarette

 

[Tide]

There are so many moments of inertia for the same reason there are so many vectors to describe the location of an object! Location and moment of inertia are always relative to something.

 

[cigarette]

Hi tide,

i don't quite get it. So let's say i have a planar region (2D) with vertices known, does it suffice if i find the moments of inertia about the origin for the section of interest? If i find the moments about the origin, does it make a difference if i find it about the centroid of the section? if i want to find the moments about the centroid, how can i do it (for an arbitrary polygon)? Derived from Green's theorem, the moments about the origin is

1/12 * sum { (y_{i+1} - y_{i} )(x_{i+1} + x_{i})(x_{i+1}^2 + x_{i}^2)
- (x_{i+1} - x_{i} )(y_{i+1} + y_{i})(y_{i+1}^2 + y_{i}^2)

(http://www.enel.ucalgary.ca/~shannon/v2/green/ ) I've tried deriving and it produces the same results

About the centroid the equation is
I_xx =
1/12 * sum { (x_{i+1} - x_{i} )(y_{i+1}^3 + y_{i}^2*y_{i+1} + y_{i}*y_{i+1}^2 + y_{i}^3)}

I can't tell the difference between the 2...

Since we are discussing 2D planar sections, are we talking about Area Moments? And area moments is expressed about the centroid of the area?

 

[Tide]

What exactly are you trying to do?

Moments of inertia are typically used when you need to analyze rotation and it's usually about some specific axis. Special shapes have "principle moments of inertia" based on their symmetry and you can use that to find the moments about an arbitrary axis using the translation rules for moments.

 

[cigarette]

What exactly are you trying to do?

Moments of inertia are typically used when you need to analyze rotation and it's usually about some specific axis. Special shapes have "principle moments of inertia" based on their symmetry and you can use that to find the moments about an arbitrary axis using the translation rules for moments.

Actually i am writing a program to provide the user with the geometrical properties of a section. The section is approximated by straight lines and all vertices are known. With regards to symmetry, i assume none, cos the section is "arbitrary". I aim to provide the user with the area, perimeter, centroid, and moments of inertia. these are all done except inertia, i figure it would be of more use to provide the user with the moments about the origin than the centroid. (i'm hoping to find the difference between the 2 so that i can my program can give more useful results.) And from here, maybe move on to radius of gyration and principle axes of inertia. I hope this clears some doubts.