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I need help calculating the inertia tensor for a curved beam.

enter image description here

I found the formulas for this in the article https://hal.archives-ouvertes.fr/hal-01084693/document[page 20, Appendix A] and decided to check their correctness. And found the following:

  1. The formula for calculating the distance $\chi$ from point $O$ to the center of mass works correctly (tested in SolidWorks using mass analysis);

  2. The integration result does not correspond to the original integration formula for the inertia tensor components $I_{yy}$ and $I_{zz}$. $I_{xx}$ component works well when I align the $r$ vector (shown in the picture) with the $x$-axis. This corresponds to my tasks and there are no questions here yet.

enter image description here

  1. As far as I understood from the text, this tensor was searched for by placing the origin of coordinates in the center of mass.

My question is: How to find the inertia tensor during the rotation of a curved arc around the point O and, if possible, recalculate it for any point?

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  • $\begingroup$ Is the cross section of the beam always a rectangle (i.e. is the beam generated by sweeping a rectangle along a curved path)? $\endgroup$
    – Zegpi
    Commented Jan 1, 2023 at 1:27
  • $\begingroup$ @Zegpi The cross section can have any shape (square, round). $\endgroup$
    – ayr
    Commented Jan 1, 2023 at 4:19

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