Glad it was helpful! This was for an undergraduate attitude dynamics course. You might appreciate a more advanced lecture on Poisson Brackets and Non-Canonical Hamiltonian Systems, where the Euler's equations are just one example, ruclips.net/video/mRCsBC1fQCo/видео.html
I've only skimmed through this and most of it hasn't gone in yet but I have a question related to 38:11. I am a mechanical engineer and have designed a system where two rotating bodies intersect around the center of a sphere similar to the diagram shown at 38:11. The two axis of rotation are almost perpendicular with each other. However I am struggling to find any use that applies to it apart from a neat piece of art. Any insight appreciated.
I'm not quite sure I understand the question, and fear there might be a misunderstanding about what this sphere represents. Correct me if I'm wrong: The sphere shown at 38:11 doesn't represent a physical object -- it's a sphere in the 3-dimensional abstract space of angular momentum vector components, (H1, H2, H3). The sphere represents the physical constraint that momentum is conserved (so doesn't grow or increase in magnitude, it has a constant 'radius' in this space). The lines on it represent the paths of possible motion of the angular momentum in the absence of any external torque on the rigid body. In any case, I'm curious about this situation you've described. Do you have a sketch of a video of it?
@@ProfessorRoss My question is related to angular momentum as I'm trying to figure out how the forces would react in my model. If you had equal angular momentum acting at 3 perpendicular axis but around the same point, what would happen? I have uploaded a short of one 3d model showing 2 axis - ruclips.net/user/shortsKPNqZf3bpW8?si=qI9f9MIeRdYqH8Pb I haven't shown the mechanism that is used to enable this motion. I have printed off a model that uses 3 axis but in a different way to the one in the video. My theory is that it could create a gyroscope without any drift as it has 3 axis. Unfortunately due to the limitations of 3d printing I couldn't spin the one I've made fast enough to create gyroscopic forces. I've been experimenting with magnetic bearings that will allow for higher speeds.
@@angus433 That's pretty fascinating! To properly model this object, I wouldn't treat it as a rigid body, but as a quasi-rigid body, or more specifically a variable-geometry body. There hasn't been much work on this, as far as I can find out. My group did some modeling of one for a study of flying snakes, which change their shape cyclically while airborne and thus change their mass distribution (ruclips.net/video/0PQ1_XpjBso/видео.html). The idea is you have Euler's equations, but with a modification since the mass distribution (and thus, the moment of inertia) is changing in time, so in addition to some 'rigid-body terms', you also get 'variable-geometry terms'. Even the study of these objects with no external torque (the 'free body' problem) hasn't been properly documented, as far as I know.
@@ProfessorRoss What a great study very interesting. I wonder if the snake shifting it's mass provides any additional forward or upward movement. I haven't found anything on the type of mechanism I've designed but I'm sure it has some useful applications. Do you know of any source for studying quasi-rigid bodies?
I am taking this course again throughout the vacation and it helps me a lot to remind me of the overall Dynamics. Thank you. I just have one question. 15:11 I am curious about how were you able to take this dot product without knowing H_c and b_3 in N-frame. Is it because you took the dot product for those two vectors in the B-frame and used the fact that the answer of dot product in any frame is the same?
Yes, that's correct. Since I wrote H_c in the B-frame, I can easily write the unit vector H_c/|H_c| in the B-frame, and then take the dot product with b_3 direction, which just means keep the third component of the unit vector H_c/|H_c|. And this dot product is the same in any frame, including the N-frame. Note that there's an ERROR at 12:57; there should be an I3 in front of ω3 in the third entry of the H_c vector
Excellent explanation! Thank you I was wondering what the maximum angle can be between the angular velocity vector and the angular momentum vector of an axisymmetric body
Very nice visualization of energy level sets on S2. This helped me better understand Hamiltonian reduction in my symplectic geometry course.
Glad it was helpful! This was for an undergraduate attitude dynamics course. You might appreciate a more advanced lecture on Poisson Brackets and Non-Canonical Hamiltonian Systems, where the Euler's equations are just one example, ruclips.net/video/mRCsBC1fQCo/видео.html
@@ProfessorRoss these lectures look very interesting! Definitely going on my reading/watch list when I get some free time. Thanks
It was really helpful. Thank you so much
Absolutely useful. Thank you very much for this excellent Lecture¡¡¡
Glad it was helpful!
I've only skimmed through this and most of it hasn't gone in yet but I have a question related to 38:11. I am a mechanical engineer and have designed a system where two rotating bodies intersect around the center of a sphere similar to the diagram shown at 38:11. The two axis of rotation are almost perpendicular with each other. However I am struggling to find any use that applies to it apart from a neat piece of art. Any insight appreciated.
I'm not quite sure I understand the question, and fear there might be a misunderstanding about what this sphere represents. Correct me if I'm wrong: The sphere shown at 38:11 doesn't represent a physical object -- it's a sphere in the 3-dimensional abstract space of angular momentum vector components, (H1, H2, H3). The sphere represents the physical constraint that momentum is conserved (so doesn't grow or increase in magnitude, it has a constant 'radius' in this space). The lines on it represent the paths of possible motion of the angular momentum in the absence of any external torque on the rigid body.
In any case, I'm curious about this situation you've described. Do you have a sketch of a video of it?
@@ProfessorRoss My question is related to angular momentum as I'm trying to figure out how the forces would react in my model. If you had equal angular momentum acting at 3 perpendicular axis but around the same point, what would happen? I have uploaded a short of one 3d model showing 2 axis - ruclips.net/user/shortsKPNqZf3bpW8?si=qI9f9MIeRdYqH8Pb
I haven't shown the mechanism that is used to enable this motion. I have printed off a model that uses 3 axis but in a different way to the one in the video. My theory is that it could create a gyroscope without any drift as it has 3 axis. Unfortunately due to the limitations of 3d printing I couldn't spin the one I've made fast enough to create gyroscopic forces. I've been experimenting with magnetic bearings that will allow for higher speeds.
@@angus433 That's pretty fascinating! To properly model this object, I wouldn't treat it as a rigid body, but as a quasi-rigid body, or more specifically a variable-geometry body. There hasn't been much work on this, as far as I can find out. My group did some modeling of one for a study of flying snakes, which change their shape cyclically while airborne and thus change their mass distribution (ruclips.net/video/0PQ1_XpjBso/видео.html). The idea is you have Euler's equations, but with a modification since the mass distribution (and thus, the moment of inertia) is changing in time, so in addition to some 'rigid-body terms', you also get 'variable-geometry terms'. Even the study of these objects with no external torque (the 'free body' problem) hasn't been properly documented, as far as I know.
@@ProfessorRoss What a great study very interesting. I wonder if the snake shifting it's mass provides any additional forward or upward movement. I haven't found anything on the type of mechanism I've designed but I'm sure it has some useful applications. Do you know of any source for studying quasi-rigid bodies?
I am taking this course again throughout the vacation and it helps me a lot to remind me of the overall Dynamics. Thank you. I just have one question.
15:11
I am curious about how were you able to take this dot product without knowing H_c and b_3 in N-frame. Is it because you took the dot product for those two vectors in the B-frame and used the fact that the answer of dot product in any frame is the same?
Yes, that's correct. Since I wrote H_c in the B-frame, I can easily write the unit vector H_c/|H_c| in the B-frame, and then take the dot product with b_3 direction, which just means keep the third component of the unit vector H_c/|H_c|. And this dot product is the same in any frame, including the N-frame. Note that there's an ERROR at 12:57; there should be an I3 in front of ω3 in the third entry of the H_c vector
Excellent, thank you!
Wil, Thank you for watching! I'm glad it was helpful.
Thank you from Korea
Thank you too!
Excellent explanation!
Thank you
I was wondering what the maximum angle can be between the angular velocity vector and the angular momentum vector of an axisymmetric body
Thank you!!
You're welcome!
Is your background Valencia?
Indeed, it is! I got to visit Valencia and really loved it. It was my children's first experience swimming in the Mediterranean at the beach.
I been thinking about it. It’s like 9024 and 9025 level it where it can’t roll once on 3125. I don’t know? Interesting
Thankyou
👍👍👍👍