Principal Axes: Understanding the Confusion

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In summary, there is disagreement on whether the directions of the principal axes are the eigenvectors of the tensor of inertia wrt the Oxyz axes or obtained by applying the matrix R that diagonalizes the tensor of inertia. However, it is agreed that the principal axes are the coordinate axes of the rotation R that diagonalizes the tensor of inertia, and that this rotation can be found by choosing the axes to study the object along its principal axes. There is also a concrete problem involving a square of uniform mass in the Oxy plane that reveals an error in the book's construction of R from the eigenvectors. The principal axes are not only the eigenvalues of the tensor of inertia, but also the axis of greater moment of inertia,
  • #1
quasar987
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The book was vague on this subject and as a result, everyone is in disagreement on this.

All my friends think that the directions of the principal axes are the eigenvectors of the tensor of inertia wrt some Oxyz axes.

I think that the directions of the principal axes are obtained by applying on the Oxyz axes the matrix R that diagonalizes the tensor of inertia. (i.e. the matrix whose lines are the eigenvectors of I)

The T.A. thinks we're both right.
 
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  • #2
Principal Axes are the coordinate Axes of the rotation R that diagonalizes I. (defined in Fowles and Cassiday)

"the matrix whose lines are the eigenvectors of I" are just the standard axes.
 
  • #3
This is pretty much the definition I have in my book. But does that mean that the principal axes are the eigenvectors of I or the standard axes acted on by R?
 
  • #5
I've read all your source robphy and none seem to support my position.

Why do you say both are correct?

And how could both be correct? I have before me a concrete problem and clearly the eigenvectors and the Oxyz system acted on by R give different directions.
 
  • #6
The moment of inertia is a 3-dimensional positive-definite symmetric matrix.

So, its eigenvalues are real and its eigenvectors are mutually orthogonal [or can be chosen to be]. (These eigenvectors are the principal axes, and the eigenvalues are the principal moments.)

So, there is a rotation that will orient the xyz-triad along the triad of mutually-orthogonal eigenvectors. If I am not mistaken, that rotation will diagonalize the moment of inertia matrix. It amounts to having chosen the axes to study the object along its principal axes.


Maybe you should post the concrete problem.
 
  • #7
I will, because this is too mysterious.

There is a square of uniform mass M and sides 'a' in the Oxy plane with corners at (0,0), (a,0), (0,a) and (a,a).

Everyone agrees that the inertia tensor is

[tex]I=M\left(\begin{array}{ccc}a^2/3&-a^2/4&0\\-a^2/4&a^2/3&0\\0&0&2a^2/3\end{array}\right)[/tex]

Everyone also agrees that the eigenvectors are

[tex]\omega^{(1)}=\left(\begin{array}{c}0 \\ 0\\ 1 \end{array}\right)[/tex]
[tex]\omega^{(2)}=\left(\begin{array}{c}-1 \\ 1\\ 0 \end{array}\right)[/tex]
[tex]\omega^{(3)}=\left(\begin{array}{c}1 \\ 1\\ 0 \end{array}\right)[/tex]

However, this means that the matrix R which diagonalizes I is

[tex]R=\left(\begin{array}{ccc}0&0&1\\-1&1&0\\1&1&0\end{array}\right)[/tex]

And now if I apply R on, say, [tex]\hat{x}[/tex], I get

[tex]R\hat{x}=\left(\begin{array}{ccc}0&0&1\\-1&1&0\\1&1&0\end{array}\right)\left(\begin{array}{c}1 \\ 0\\ 0 \end{array}\right)=\left(\begin{array}{c}0 \\ -1\\ 1 \end{array}\right)[/tex]

which is not equal to any of the eigenvectors.
 
  • #8
quasar987 said:
I will, because this is too mysterious.

There is a square of uniform mass M and sides 'a' in the Oxy plane with corners at (0,0), (a,0), (0,a) and (a,a).

Everyone agrees that the inertia tensor is

[tex]I=M\left(\begin{array}{ccc}a^2/3&-a^2/4&0\\-a^2/4&a^2/3&0\\0&0&2a^2/3\end{array}\right)[/tex]

Everyone also agrees that the eigenvectors are

[tex]\omega^{(1)}=\left(\begin{array}{c}0 \\ 0\\ 1 \end{array}\right)[/tex]
[tex]\omega^{(2)}=\left(\begin{array}{c}-1 \\ 1\\ 0 \end{array}\right)[/tex]
[tex]\omega^{(3)}=\left(\begin{array}{c}1 \\ 1\\ 0 \end{array}\right)[/tex]

However, this means that the matrix R which diagonalizes I is

[tex]R=\left(\begin{array}{ccc}0&0&1\\-1&1&0\\1&1&0\end{array}\right)[/tex]

And now if I apply R on, say, [tex]\hat{x}[/tex], I get

[tex]R\hat{x}=\left(\begin{array}{ccc}0&0&1\\-1&1&0\\1&1&0\end{array}\right)\left(\begin{array}{c}1 \\ 0\\ 0 \end{array}\right)=\left(\begin{array}{c}0 \\ -1\\ 1 \end{array}\right)[/tex]

which is not equal to any of the eigenvectors.
No, you wrote down the transpose of R. The eigenvectors are the columns of R, not the rows.
 
  • #9
Ok, it's an error in the book then because it is clearly indicated how to construct R from the eigenvectors and he lines up the eigenvectors (as opposed to "rows up").

Thanks for pointing that out marcusl.
 
  • #10
I apologize. I just want to add the principal axes are not only eingevalues of something. They are also the axis of bigger moment of inertia, the axis of least moment of inertia and the third, perpendicular to previous ones.
 
  • #11
Glad to help :biggrin:
 
  • #12
Yeah, me too... I guess. :devil:
 
  • #13
Rob, you gave all the answers. I just caught the error...
 

FAQ: Principal Axes: Understanding the Confusion

What are the principal axes?

The principal axes, also known as principal axes of inertia, are the three mutually perpendicular axes that pass through the center of mass of an object and define its orientation in space.

Why are the principal axes important?

The principal axes are important because they help us understand how an object rotates and moves in space. They also provide a convenient reference frame for describing an object's motion and understanding its stability.

How are the principal axes determined?

The principal axes are determined by calculating the moments of inertia of an object about three mutually perpendicular axes. The axes with the largest and smallest moments of inertia are the principal axes, with the axis of largest moment of inertia referred to as the principal axis of greatest inertia.

What is the relationship between the principal axes and an object's symmetry?

The principal axes are closely related to an object's symmetry. If an object has a high degree of symmetry, its principal axes will coincide with its symmetry axes. This means that the object will have equal moments of inertia about each principal axis, resulting in simpler equations of motion.

Can an object have more than three principal axes?

No, an object can only have three principal axes. This is because an object's orientation in space can be fully described by three mutually perpendicular axes. Any additional axes would not provide any new information about the object's orientation.

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