Moment of Inertia Matrix & Principal Axis Frame

Moment of Inertia Matrix & Principal Axis Frame | MATLAB Example Calculation

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Space Vehicle Dynamics 🚀 Lecture 19, part 1: Moment of inertia matrix (inertia tensor) and the principal axis frame. We discuss the calculation of the moment of inertia matrix for a rigid body and its properties. The eigenvalues of this matrix are the principal moments of inertia and the corresponding eigenvectors are the principal axes of rotation, special directions about which the rigid body can freely rotate in pure rotation (no wobbling). The transformation of the inertia matrix from one rigid body frame to another is given via the rotation matrix (direction cosine matrix).
We numerically demonstrate with a Matlab example. Starting with an inertia matrix, we find the principal moments and principal axes, and construct the principal axis frame.
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► Dr. Shane Ross 🌠 aerospace engineering professor, Virginia Tech
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► Space Vehicle Dynamics course videos (playlist)
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► Lecture notes (PDF)
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► References
Schaub & Junkins📘Analytical Mechanics of Space Systems, 4th edition, 2018
arc.aiaa.org/doi/book/10.2514...
► Software library (MATLAB, Python, Mathematica)
arc.aiaa.org/doi/suppl/10.251...
► Related Courses and Playlists by Dr. Ross
📚Three-Body Problem Orbital Mechanics
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📚Attitude Dynamics and Control
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📚Lagrangian and 3D Rigid Body Dynamics
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📚Center Manifolds, Normal Forms, and Bifurcations
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📚Nonlinear Dynamics and Chaos
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📚Hamiltonian Dynamics
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► Chapters
0:00 Rigid body dynamics, continuous mass distribution
5:03 Moment of inertia matrix entries, integral over body
12:00 Principal axis frame -- easiest, most natural, rigid body frame
16:38 Transforming moment of inertia matrix between frames
19:26 How long does a potato stay fresh?
19:37 Principal axis frame, how to find it
25:24 MATLAB example of finding principal axis frame
32:19 Principal axes physical significance
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Пікірлер: 12

@abrahamcabrera3849 Жыл бұрын

Thank you so much. Working through an online dynamics course and this really helped explain a concept my teacher only briefly reviewed

@ProfessorRoss 11 ай бұрын

You're very welcome! I'm glad I was able to provide some clarity on concepts touched on briefly. Thank *you* for watching.

@notheisenbear5686 2 жыл бұрын

Thanks much for posting this, most references focus on solving the eigenvalues/vector problem without connecting these quantities back to their physical significance. It seems to me that eig() produces the closest (small Phi) solution where eigs() sorts eigenvalues/vectors descending.

@noonesland2471 2 жыл бұрын

Thank you so much! It really helps me with these topics!

@ProfessorRoss 2 жыл бұрын

You're welcome! Glad to help.

@yasingarg8477 2 жыл бұрын

Thanks much, I have a question, how we can know we should rotate which axis?

@atharvabhomle3191 2 жыл бұрын

Sir, when we calculate principal axis fram from taken body frame we get f1 f2 f3. So does it mean that we shift our origin of body fixed frame to point (f1,f2,f3) to ge principal axis frame?

@KickenItOldSchool Жыл бұрын

thanks for uploading this

@satoyusei3075 2 жыл бұрын

Thank you for these lectures, they are helping me a lot. I am confused about 4:00. Are the B_Hc and B_W vectors the components of their vectors expressed in the B frame, so you will need to transpose those vectors and multiply them with a 3x1 matrix with components of the B frame's basis vectors to obtain the inertial vectors of B_Hc and B_W? Or are the bases of the B frame already included in B_Hc, B_Ic, and B_W in the equation in 4:00?

@ProfessorRoss 2 жыл бұрын

No multiplication is necessary. B_Hc, B_Ic, and B_W are the objects (vectors and matrix) already expressed in the B-frame.

@satoyusei3075 2 жыл бұрын

@@ProfessorRoss That makes a lot of sense now! Thank you!