Finding the moment of inertia of a skeletal frame

[lzh]

Homework Statement

This isn't a homework problem but rather a problem I need to solve for a personal project of mine. Basically I need to find the moment of inertia of the following:
http://img860.imageshack.us/img860/3952/roboframe.jpg

Uploaded with ImageShack.us
The rotation is with respect to the top of the square frame

Homework Equations

I=\Sigmamr^2
Iend=mL^2/3
Icenter=mL^2/12

The Attempt at a Solution

What this problem comprises of essentially is just a bunch of rods with differing axis of rotation. I would simply need to sum the total. Now I've been only spoon fed the equations above for rotation on the end of a rod or center of a rod, never the equation for a rod with an axis of rotation off the rod. I would need to use the Iend equation for the rods in the middle, but the overall frame of this structure would need another equation. Could someone guide me to finding that equation?

thanks

 

[tiny-tim]

hi lzh!

(have a sigma: ∑ and try using the X² icon just above the Reply box )

use the parallel axis theorem …

moment of inertia about any point = moment of inertia about the centre of mass + md², where m is mass and d is distance from the centre of mass

(try it first for the end of a rod, to check it really works! )

 

[lzh]

hi,

thanks for the help!

if I were to have two motors spinning on this frame, one on the "top" of the frame, and one on the bottom. Should I be solving for the moment of inertia with respect to the center of mass, or should it be displaced?

 

[lzh]

sorry for the double post. But here are my calculations:
assume every rod has mass m. length of the orange/black rod is l and length of blue rod is r.
I=12rods*I(orange/black)+8rods*I(blue)
I=12*m(l^2/12+l^2/2)+8*(mr^2/3)

does this make sense?

 

[tiny-tim]

hi lzh!

if I were to have two motors spinning on this frame, one on the "top" of the frame, and one on the bottom.

i don't understand …

what are the motors doing?

are they, for example, making the frame spin about a fixed vertical axis?

Should I be solving for the moment of inertia with respect to the center of mass, or should it be displaced?

you should choose the centre of mass, or any point on the axis of rotation

assume every rod has mass m. length of the orange/black rod is l and length of blue rod is r.
I=12rods*I(orange/black)+8rods*I(blue)
I=12*m(l^2/12+l^2/2)+8*(mr^2/3)

does this make sense?

nooo …

about what axis are you measuring the moment of inertia ?

 

[lzh]

basically those two motors are attached to the frame, and the spinning end of the motor freely rotates. The frame is on a near frictionless spin table, so anything rotation from the motor will cause an opposite spin on the table. And so yea its making the frame spin about a fixed vertical axis.

I'm measuring the moment of inertial with respect to the vertical axis in the center of the frame. So i simply used the rod equation ml^2/12 and displaced it by how far it is from the center axis. Then I summed the total of all the rods that are not diagonal to the axis. For the blue rods, I usedI=ml^2/3.

Was this not right?

 

[tiny-tim]

I'm measuring the moment of inertial with respect to the vertical axis in the center of the frame. So i simply used the rod equation ml^2/12 and displaced it by how far it is from the center axis. Then I summed the total of all the rods that are not diagonal to the axis. For the blue rods, I usedI=ml^2/3.

for the blue rods, the axis goes though the centre of the rod, so use ml²/12 with nothing added, but with a longer l of course

for the other horizontal rods, your method is correct

for the vertical rods, starting with ml²/12 is wrong, you need instead the moment of inertia about the vertical axis through the centre of mass, which is … ?