Moment of inertia of system in 3D

[PJani]

Hey i am working on something and i need to know how to calculate moment of inertia of a 3D system of objects.

I know these variables:
Mass of whole system
Center of mass of whole system

Center of mass of each object
Offset of each object
Mass of each object
Moment of Inertia of each individual object(its precalculated)

I think i need to use parallel axis theorem but i am not sure how to actually calculate the "sum".

Is this right:

\vec{I_i}= \vec{I_{com_i}} + mass_i * (\vec{com_{system}}-\vec{offset_i} + \vec{com_i})^2
\vec{I_{system}} = \sum^{N}_{i=1}{\vec{I_i}}

?

 

[haruspex]

Hey i am working on something and i need to know how to calculate moment of inertia of a 3D system of objects.

I know these variables:
Mass of whole system
Center of mass of whole system

Center of mass of each object
Offset of each object
Mass of each object
Moment of Inertia of each individual object(its precalculated)

I think i need to use parallel axis theorem but i am not sure how to actually calculate the "sum".

Is this right:

\vec{I_i}= \vec{I_{com_i}} + mass_i * (\vec{com_{system}}-\vec{offset_i} + \vec{com_i})^2
\vec{I_{system}} = \sum^{N}_{i=1}{\vec{I_i}}

?

How are you defining an object's "offset"? I expected it to mean offset of object's c.o.m. from system's c.o.m., which would be \vec{com_i}-\vec{com_{system}}.
Anyway, assuming you want the MI about the system's c.o.m, I make the answer
\vec{I_i}= \vec{I_{com_i}} + mass_i * |\vec{com_i}-\vec{com_{system}}|^2

 

[PJani]

How are you defining an object's "offset"? I expected it to mean offset of object's c.o.m. from system's c.o.m., which would be \vec{com_i}-\vec{com_{system}}.
Anyway, assuming you want the MI about the system's c.o.m, I make the answer
\vec{I_i}= \vec{I_{com_i}} + mass_i * |\vec{com_i}-\vec{com_{system}}|^2

offset is position of object from center of system(not the com_{system}) the com_i is "local" center of mass.

 

[haruspex]

offset is position of object from center of system(not the com_{system}) the com_i is "local" center of mass.

OK, so is the MI required about the system c.o.m. or about the system centre (= origin?).
Anyway, my equation was wrong because I forgot to say that the vectors to use are only the components orthogonal to the axis of rotation.

 

[PJani]

Actually is the system centre. The system com is not "known" till the end of calculation/iteration

How do you mean by orthogonal. Because everything is axis aligned...

 

[haruspex]

The full expression of moment of inertia of a 3D object is a matrix. If you know the specific axis you care about then you can take moments about that, but things can tricky. If that is not a principal axis of the object then rotation about it will not be stable. And in general it's not even stable about all of the principal axes.