# Inertia tensor v.s pincipal axes moment of inertia

• kasoll
In summary, the conversation discusses calculating the inertia tensor from principal axes moment of inertia and how to reverse the calculation. It is explained that in the principal frame, the inertia tensor is diagonal and the off diagonal elements are zero. However, if there are non-zero off diagonal elements, the eigenvalue problem must be solved and the transformation matrix must be constructed to find the specific matrix.f

#### kasoll

Is there a method to calculate inertia tensor form principal axes moment of inertia?
Like now we have moment of inertia: (Ix,Iy,Iz)=(20,18,25), and hot to calculate the inertia tensor like
(Ixx,Ixy,Ixz
Iyx,Iyy,Iyz,
Izx,Izy,Izz)?

In the principal frame, the inertia tensor is diagonal meaning that Ixy = Iyz = Izx = 0. So in this case you would write
$$I=\begin{pmatrix} 20 & 0 & 0 \\ 0 & 18 & 0 \\ 0 & 0 & 25 \end{pmatrix}$$Does this answer your question?

• anorlunda and kasoll
Principle axis of moment of inertia is the axis passing through centroid or center of gravity of body.

The moment of inertia of a figure about a line is the sum of the products formed by multiplying the magnitude of each element (of area or of mass) by the square of its distance from the line. So the moment of inertia of a figure is the sum of moments of inertia of its parts.

Now we know that the moments of inertia of a figure about lines which intersect at a common point are generally unequal. The moment is greatest about one line and least about another line perpendicular to the first one. A set of three orthogonal lines consisting of these two and a line perpendicular to both are the principal axes of inertia of the figure relative to that point. If the point is the figure's centroid, the axes are the central principal axes of inertia. The moments of inertia about principal axes are principal moments of inertia.

• kasoll
In the principal frame, the inertia tensor is diagonal meaning that Ixy = Iyz = Izx = 0. So in this case you would write
$$I=\begin{pmatrix} 20 & 0 & 0 \\ 0 & 18 & 0 \\ 0 & 0 & 25 \end{pmatrix}$$Does this answer your question?
Thanks for the answer. But almost all the inertia tensor I have seen have non-zero valur as Ixy,Ixz,Iyz. I know to get principal axes moment of inertia from inertia tensor, which just looks like the one you mentioned.
But how to reverse the calculation? By multiply a matrix? And how to get the spercific matrix?

Thanks for the answer. But almost all the inertia tensor I have seen have non-zero valur as Ixy,Ixz,Iyz. I know to get principal axes moment of inertia from inertia tensor, which just looks like the one you mentioned.
But how to reverse the calculation? By multiply a matrix? And how to get the spercific matrix?
If there are non-zero off diagonal matrix elements, then you solve the eigenvalue problem, find the eigenvectors and construct the transformation matrix ##T## from the direction cosines as explained on pages 8-9 in the reference that you quoted. Read it carefully. Equation (15) says that ##I'=(T)~(I)~(T)^T##. If you want to go back the other way, then ##(I)=(T)^T~(I')~(T)##.

• kasoll