Moment of Inertia | Solve a Problem with Centroid & Steiner's Theorem

[HERiTAGE]

Hey ppl it's my first post wohooo. :)

I've been trying o solve this problem for ages and seriously i don't know what I'm doing wrong... It's not a difficult problem but i can't seem to solve it.
I have to calculate the MoI of this figure:

http://mypicfordc.com.sapo.pt/moi.JPG

Where a=89.9 [cm], b=57.8 [cm], c=44.8 [cm]

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I've been able to accuraely calculate the centroid of the figure (xG=50.261 ; yG= 44.607) and I've reached a MoI value that is close to the solution but it's not close enough to be considered correct.

My MoI values are:
Ix= 5305338 ; Iy= 4131811 ; Ixy= 1580647

Correct values are surrounding:
Ix= 5800000 ; Iy= 4700000 ; Ixy= 1500000 (not completely accurate, just an aprox.) The main values are all correct (Areas, xGi, yGi, Sxi, Syi) because i need the to find the Centroid, which is correct.
I'm using Steiner's theorem to find the moment of inertia.

Please help me, I'm driving nutts...

I can scan my calculus so you can see what i am doing wrong but it won't do you much because it's kind of messy. (if you still would like to see it, just ask)

Thanks in advance. :)

 

[J77]

Does Steiner's formula compute the inertia using the double integral?

(I'm not familiar with it at all!)

Assuming it does, I don't see how, if you computed the centroid using double integration, that your moments of inertia can be wrong.

Perhaps you could put up extended workings?

 

[HERiTAGE]

No, no integration is required since the shapes that compose the figure are pretty common. (quarter circle, rectangle and triangle)

It's kind of hard for me to explain how Steiner Theorem works through a keyboard and speaking in another language than my natural one... :)

What Steiner theorem does is, calculate the Moment of Inertia of a figure considering a given point.

The formula for the I'x (i.e.) is:

I'x= Ix+dy*A

Where Ix is the Moment of Inertia of your figure (there are tables that give you formulas for Moments of Inertia on squares, circles, etc) dy is the distance between your figure's centroid and the main figure's centroid and A is the Area of your figure.

example using my figure:

The moment of inertia on a quarter circle is giver by:

Ix=(pi*r^4)/16

So using Steiner's theorem to calculate the MoI of the Quarter circle on the main figure's centroid we get:

Ix'=(pi*r^4)/16 + dy * (pi*r^2)/4

where dy is the difference between yG of the quarter circle and yG of the main figure.

 

[J77]

Ahhh, I see. Never heard it called Steiner's...

http://en.wikipedia.org/wiki/Parallel_axes_rule

 

[HERiTAGE]

Well can anyone help? please?

 

[Doc Al]

If you have the MOI of each piece (quarter circle, triangle, square) about its COM, use the parallel axis theorem to find the MOI of each about the centroid. Then just add them up. Of course, the masses of the missing pieces of the square will be negative.

 

[HERiTAGE]

I've done that.
That's how i got that result.

The thing is, i know how to solve the problem, but i keep getting a wrong result.
I even checked the problem step by step with a friend of mine and we both got to the conclusion that the methodology was correct. Only the results weren't. :\

I really need to solve this problem coz it's the final of a series of 20 problems that i must solve in order to go to the final exam and it would suck to flunk a discipline just because i couldn't solve one problem.

 

[Doc Al]

What Steiner theorem does is, calculate the Moment of Inertia of a figure considering a given point.

The formula for the I'x (i.e.) is:

I'x= Ix+dy*A

Where Ix is the Moment of Inertia of your figure (there are tables that give you formulas for Moments of Inertia on squares, circles, etc) dy is the distance between your figure's centroid and the main figure's centroid and A is the Area of your figure.

Not clear to me what you are doing here. As has been suggested, use the parallel axis theorem. Using your notation, that would be:
I'x= Ix+ M(dy)^2, where M is the mass of the piece. (See the link that J77 provided.)

 

[HERiTAGE]

It's hard for me to explain what I'm trying to do.
I am using the parallel axis theorem (a.k.a. Steiner's Theorem).

Hmm maybe i should upload my scanned calculus...

http://mypicfordc.com.sapo.pt/moical.jpg"

*Forget about the lower right values for a, b and c. The correct ones are near the figure.

*Edit*

Never mind... I've already solved the problem. I had a mistake on on of the tables where i calculate the centroid of each composing figure.
Thanks for the help anyway.