Principal axes of a triangular plate
I’ve been working on understanding the rotational behavior of rigid bodies, and I recently tackled the problem of finding the principal axes of inertia for a triangular plate. This post summarizes my approach and the key results.
I considered a right-angled triangle with uniform density \rho and base vertices at (0, 0, 0), (h,0,0), and (h, h, 0). I assumed the vertical component of the element is negligible. The moment of inertia tensor for this triangle, as I calculated, is:
\mathbf{I} = \frac{mh^2}{36} \begin{bmatrix} 2 & -1 & 0 \\ -1 & 2 & 0 \\ 0 & 0 & 4 \end{bmatrix}
where m is the mass of the triangle and h is the length of the base.
To find the principal axes, I needed to solve the eigenvalue problem for the matrix:
\mathbf{M} = \begin{bmatrix} 2 & -1 & 0 \\ -1 & 2 & 0 \\ 0 & 0 & 4 \end{bmatrix}
This involved finding the eigenvalues \lambda and eigenvectors \mathbf{v} that satisfy the equation \mathbf{M}\mathbf{v} = \lambda\mathbf{v}. After some calculations, I determined the eigenvalues to be \lambda_1 = 1, \lambda_2 = 3, and \lambda_3 = 4. These eigenvalues represent the principal moments of inertia.
Next, I computed the corresponding eigenvectors. These eigenvectors point in the direction of the principal axes. I normalized these vectors to obtain:
\mathbf{u}_1 = \begin{bmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{bmatrix}, \quad \mathbf{u}_2 = \begin{bmatrix} -1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{bmatrix}, \quad \mathbf{u}_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}
These eigenvectors define the principal axes of the triangular plate. The axis corresponding to the smallest eigenvalue (\lambda_1 = 1) is the axis about which the plate is easiest to rotate. Conversely, the axis associated with the largest eigenvalue (\lambda_3 = 4) is the axis about which the plate is hardest to rotate.
I then constructed the diagonalized inertia tensor:
\mathbf{I}' = \frac{mh^2}{36} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{bmatrix}
This diagonal form simplifies the analysis of rotational motion.
Finally, I visualized the principal axes with respect to the center of mass of the triangle. This helped me better understand the orientation of these axes relative to the geometry of the plate.
For more insights into this topic, you can find the details here.