Adding Inertia Tensors to 3D Shapes: Parallel Axis Theorem Explained

[daniel_i_l]

Lets say that I know the inertia tensors for a few different 3D shapes and I want to connect them together into one big composite shape. From what I understand, I first have to find the new center of mass, then using the parallel axis theorem find the new inertia tensors for each body along an axis going through the center, and then just add them up. My question in, what if the new inertia tensors around the center are around axes (plurel of axis?) that aren't pointing in the same direction. Is there a way to rotate the axis of an inertia tensor? Should I make a rotation matrix from each one of the bodies space to the new composite's space and then apply those to the inertia tensors before using the PAT? Can someone give me some details about the PAT in 3D?
Thanks.

 

[Ben Niehoff]

Yes, you can rotate the inertia tensor by what's called a "similarity transformation". If R is a rotation matrix and I is the inertia matrix, then I transforms as

I' = RIR^T

 

[daniel_i_l]

Thanks - then after I rotate all the tensors into the same coordinate system I can calculate new tensors around the new center of mass using the parallel axis theorem? How is it done in 3D?
Thanks.