# Transform an Inertia Tensor

I am trying to provide colleagues with a spreadsheet method of transforming the inertia properties of a complex shaped body to a different coordinate system, involving only rotation.

I've read that this can be achieved by multiplying the inertia tensor by the transform of the matrix of direction cosines and then multiplying the matrix of direction cosines by the previous result:

[I'] = [T][I][T]^T

The body in question has been drawn in CAD so that the principal axes and inertias, and the inertia relative to a global axis system, can be obtained. In addition I created a second global coordinate system with a common origin to the original system and with a simple rotation about a single axis. This gives me a further set of values relative to this second global coordinate system. I then put all the numbers into Excel expecting to use the above matrix maths to transform one of the inertia tensors into any of the others using the appropriate direction cosines. Unfortunately, that isn't working, so I assume I am misunderstanding something.

As an example:

T (the direction cosines for the principal axes) =

0.32819818 -0.0000209 0.94460889

-0.00012885 -0.99999999 0.00002264

0.94460888 -0.00012914 -0.32819818

T^T (i.e. the inverse) using MINVERSE(Array) =

0.32819818 -0.000128846 0.944608882

-2.09024E-05 -0.999999993 -0.000129143

0.944608889 2.26412E-05 -0.32819818

I (the principal inertias) =

148478195.6 0 0

0 271583441.8 0

0 0 281696001

So, finding [I][T]^T using MMult(Array1,Array2) =

48730273.62 -19130.84153 140253822.4

-5676.743151 -271583439.8 -35073.17172

266092546.6 6377.931954 -92452114.86

Then finding [T]{[I][T]^T} =

267346572.3 5422.037743 -41300039.62

5422.182592 271583439.7 14908.35047

-41300039.57 14907.89697 162827626.4

The CAD gives the inertia relative to the global system as:

267346571.9 389.2392844 41300040.2

389.2392844 271583441.9 -2858.85771

41300040.2 -2858.85771 162827624.7

The diagonal terms are pretty well correct, but the others are not. The second example using a simple rotation about a single axis also produced incorrect results, although this time only two of the three products were correct and none of the moments. I won't post that here unless requested.

Can anyone shed any light on what I'm doing wrong?

Many thanks,

Simon.