Calculating Principle Moments of Inertia

[bugatti79]

Hi Folks,

Is it possible to calculate the principle moments of inertia acting along the principle axes of inertia given the moments of inertia and their directional vectors. Ie , I have the following information

Moments of inertia ##J_1, J_2,J_3=18kgm2,15kgm2,6kgm2##

and the directional vectors

##\begin{bmatrix}J_{1x}& J_{1y}&J_{1z} \\ J_{2x} &J_{2y} &J_{2z} \\ J_{3x}&J_{3y} & J_{3z}\end{bmatrix}=\begin{bmatrix}0.4& 0.7&-0.2 \\ -0.8 &.1 &0.8 \\ 0.2&0.8 & 0.7\end{bmatrix}##

I also have the euler angles but I am not sure if there is some relationship between these and the directional vectors or indeed if I need them.
Any information will be appreciated.
Regards

 

[SteamKing]

Hi Folks,

Is it possible to calculate the principle moments of inertia acting along the principle axes of inertia given the moments of inertia and their directional vectors. Ie , I have the following information

Moments of inertia ##J_1, J_2,J_3=18kgm2,15kgm2,6kgm2##

and the directional vectors

##\begin{bmatrix}J_{1x}& J_{1y}&J_{1z} \\ J_{2x} &J_{2y} &J_{2z} \\ J_{3x}&J_{3y} & J_{3z}\end{bmatrix}=\begin{bmatrix}0.4& 0.7&-0.2 \\ -0.8 &.1 &0.8 \\ 0.2&0.8 & 0.7\end{bmatrix}##

I also have the euler angles but I am not sure if there is some relationship between these and the directional vectors or indeed if I need them.
Any information will be appreciated.
Regards

Here, we are talking about 'principal' axes and 'principal' moments of inertia.

You are using J for moments of inertia and direction cosines, so it's hard to know what information you have. There is a way to find the principal axes given a matrix of inertia values about some arbitrary coordinate system:

http://ocw.mit.edu/courses/aeronaut...fall-2009/lecture-notes/MIT16_07F09_Lec26.pdf

If you have only J1, J2, and J3, these may already be the principal moments of inertia.

 

[bugatti79]

Hi SteamKing,

Thanks for the reply. Actually, you are right. The J values are actually the principle moments of inertia but don't they act along the principle axes and not some set of direction cosines?

What I really want to do is calculate the moments of inertia for a new xyz coordinate system where the old and new coordinate system are related respectively by

-z=x, x=y and y=-z

So is it just a matter of rotating in steps of 90 deg and not consider the direction cosines given above...? Ie, what is the rotation matrix?

 

[SteamKing]

Hi SteamKing,

Thanks for the reply. Actually, you are right. The J values are actually the principle moments of inertia but don't they act along the principle axes and not some set of direction cosines?

The principal moments of inertia are found only about the principal axes. The inertia tensor has non-zero values only on the main diagonal when using the principal axes.

What I really want to do is calculate the moments of inertia for a new xyz coordinate system where the old and new coordinate system are related respectively by

-z=x, x=y and y=-z

So is it just a matter of rotating in steps of 90 deg and not consider the direction cosines given above...? Ie, what is the rotation matrix?

There are coordinate transformations for mass moments of inertia similar to those for area moments of inertia, which, of course, are defined only for 2-D planes.

The 3-D coordinate rotation matrix can be set up like in this article:

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

To calculate the moment of inertia of a body about an arbitrary axis, this article is recommended:

http://www.eng.auburn.edu/~marghitu/MECH2110/C_4.pdf

Eq. 4.8 on page 4 and the derivation above are what you are looking for, I believe.

 

[bugatti79]

The problem I have understanding is that I have the 3 J values which are the principle moments and so act along the principle axes as we have agreed. And we know the relationship between the old and new is just

##-z=x, x=y## and ##y=-z ##

So I can just use the 3D rotation matrix given in that wiki link you provided and rotate in steps of 90deg. I don't actually need the direction cosines or the euler angles, right?

 

[bugatti79]

There are 2 steps.
The first step is to use the euler angles to translate back to the original coordinate system then the second step is to translate this new tensor into the desired new co ordinate system. It works. Thanks