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combining inertial tensors.

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I was curious if its possible to combine two inertial tensors? two arbitaries shaped objects have inertial tensors Ia and Ib I want to join these objects rigidly together, any thoughts on how I go about this, and how I combine the two matrices into one so that I get the correct tensors out... ...or do I have to abandon both matricies and do some over the voluems again?

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Of course you can!
Inertial matrices are additive, and it come from their definition: its the integral on the volume of the object of r*r*dm (r being the distance beetween the point that wheighs dm and the point where we calculate I).
So, you can do I(G)=Ia(G)+Ib(G).
However, it is really impossible to do I(G)=Ia(Ga)+Ib(Gb), or you''ll have to get Ia(G) from Ia(G) with the Huygens formula.

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Guest Anonymous Poster
As I recall, if you have two inertial tensors about the centers of mass of their respective objects, call them Ia and Ib for object A and object B respectively, and you want a new inertial tensor for the combination of the two, spaced a distance of r units apart, about the center of mass of object A, this would be:

Ia + r^2 * Ib

It should be fairly easy to see how to extend this if you don''t want it to be about the center of mass of either of the two objects.

So, mikamikaze is right if you want both objects to be superimposed on top of each other such that their centers of mass coincide. The above is just a little more general.

Isn''t this right?


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The general moment of inertia of a body about the origin is:

I = Ic + M(R^2 I - RR)

R is the vector displacement of the body''s centre of mass from the origin, so this formula can be used to give the moment of inertia of a body not centred at the origin.

Ic is the moment of inertia of the body about it''s centre of mass, which is what people usually understand by the moment of inertia.

M is the mass of the body
R^2 is R.R or |R| ^2, the square of the length. I is the identity matrix.

RR is a dyad. This is way of multiplying two vectors to generate a matrix, giving in this case the 3x3 matrix

RxRx RxRy RxRz
RxRy RyRy RyRz
RxRz RyRz RzRz

where R = {Rx, Ry, Rz}

If you are cacluating a new MOI for two joined bodies first work out the position of the centre of mass of the combined body. Then use the above formula twice, getting something like

I = Ia + Ma(Ra^2 I - RaRa) + Ib + Mb(Rb^2 I - RbRb)

Where Ra, Rb are the offsets of the two bodies from their combined centre of mass, Ma, Mb are their masses. If their centre of masses are at the same point then Ra and Rb are both zero vectors and the above formula becomes

I = Ia + Ib

as expected.

(for those interested this is from Goldstein''s Classical Mechanics, pages 197 & 198 - I knew the basic principles but had to look up the formula)

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