Have Inertia Tensor for Object. Find It for Object after Rotation

In summary, the conversation discusses finding the new inertia tensor for an object that has been rotated around its center of mass in a three-dimensional coordinate system. It is possible to use a matrix transform to compute the new tensor, which can save time compared to recomputing it for each orientation. The conversation also includes a solution for this problem and clarifies the difference between rotation matrices used in different programming languages.
  • #1
RoshanBBQ
280
0

Homework Statement


Let's say I have a coordinate system that has (0,0,0) at the CM of an object, and I know the object's inertia tensor for that coordinate, T. (T is a 3x3 inertia tensor where (1,1) is moment of inertia about x-axis, (2,2) is moment of inertia about y-axis, and (3,3) is moment of inertia about z-axis).

I then rotate the object about its CM by a degrees about x-axis, b degrees about y-axis, and c degrees about the z-axis. I am using the convention where a positive angle rotates the object clockwise if the positive portion of the axis about which rotation is happening is approaching you from your viewing spot.

Is there a simple matrix transform I can do to T to find the new inertia tensor, T'? This is for a coding project. As of now, I am recomputing T' for each orientation. I would like to speed things up if possible. I am searching online with little success. If one of you gentlemen or ladies knows the relation off the top of your head, it would save me much time!EDIT: I found something about finding the inertia tensor for a new, rotated coordinate. Is it the case that rotating an object a b and c would be the tensor found about the coordinates rotated -a -b and -c?
 
Last edited:
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  • #2
Here is the answer for future people with this question:
Let us say you know the rotation matrix to rotate a set of x,y, and z coordinates like so:
[tex]\begin{pmatrix}
x_{new}\\
y_{new}\\
z_{new}
\end{pmatrix}
= R
\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}
[/tex]
And you rotated your object using R. If you have T, the old inertia tensor before rotation, the new one is found by
[tex]T_{new} = RTR^T[/tex]

However, and this is an annoying lack of complete resource from places I searched (I had to derive this myself), if you have (LIKE IN MATLAB) a rotation matrix used like so:
[tex]\begin{pmatrix}
x_{new} & y_{new} & z_{new}
\end{pmatrix}
=
\begin{pmatrix}
x & y & z
\end{pmatrix}
R
[/tex]

Then, we have
[tex]T_{new} = R^T T R[/tex]

R is a 3x3 rotation matrix. In MATLAB, your rotation matrix will be 4x4. Simply discard 4th row and column.
 
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1. What is an inertia tensor?

An inertia tensor is a mathematical representation of an object's mass distribution with respect to its rotation. It is a 3x3 matrix that describes how an object's mass is distributed around its center of mass.

2. How is the inertia tensor calculated?

The inertia tensor is calculated by integrating the mass of each infinitesimal element of the object over its entire volume. This integration takes into account the object's shape, size, and mass distribution.

3. How does the inertia tensor change after rotation?

After rotation, the inertia tensor changes because the object's mass distribution with respect to its new axis of rotation is different than its previous axis. This leads to a change in the values of the inertia tensor, reflecting the object's new moment of inertia.

4. How is the inertia tensor useful in physics?

The inertia tensor is useful in physics because it allows for the calculation of an object's rotational motion. It is used in various equations, such as the angular momentum equation and the equations of motion, to analyze the rotational behavior of objects.

5. Can the inertia tensor be used for any type of object?

The inertia tensor can be used for any rigid body object, regardless of its shape or size. However, it is most commonly used for objects with a symmetrical shape, as it simplifies the calculations.

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