Vector Derivatives (the Equation of Coriolis) and the Angular Velocity Vector

[@ChristopherLum]

In case it is helpful, here are all the Flight Mechanics videos in a single playlist kzbin.info/aero/PLxdnSsBqCrrEx3A6W94sQGClk6Q4YCg-h. You can support this channel via Patreon at www.patreon.com/christopherwlum. Please let me know what you think in the comments. Thanks for watching!

[@Oxcentric]

I've not even got halfway through and I have to say this is a fantastically executed explanation - so much effort and thought has gone into making this as clear as possible for a beginner! Great work!

[@AlexandraSurprise]

AA 516 - Allie S

[@thomasstiglich3484]

Another great video, as always. I believe that there is a slight notational mistake at the end of the video (1:09:59). Property ii) should read w_b/a = -w_a/b. I believe that you have it as w_b/a = -w_b/a..

[@ChristopherLum]

Thomas, thanks for catching this, you are absolutely correct, I wrote it down incorrectly at 1:09:59 (I showed the correct form at 1:06:00 but switched the indices here). Thanks for pointing this out and thanks for watching!

[@davidtelgen8114]

AE 512: Holy notation, batman!

[@timproby7624]

[AE 512]: I got a little lost on what $\bar{V}$ and $\bar{omega}_{b/r}$ were? * $\bar{omega}_{b/r}$ is the vector where we are rotating about? * $\bar{V}$ is the vector being rotated about $\bar{omega}_{b/r}$? And the time derivatives of both are how each vector is changing with time. So the axis of rotation (omega) is changing with time which then means V is also changing?

[@ChristopherLum]

Tim, do you have a timestamp of the video or page number in the notes that I can refer to?

[@timproby7624]

@@ChristopherLum I rewatched the video which clarified my original question

[@Jenicek109]

Man, love your practical demonstrations. :D This is the good stuff!

[@morganwalker940]

dang son you got so many frames this class should be called Aaron Brothers

[@ChristopherLum]

Haha, Morgan for the win!

[@bsgove]

AE512: I always found Coriolis very unintuitive because it isn't as straightforward as other affects on objects. The math doesn't lie though.

[@scientifically5812]

Thank you for your video… It is amazing content. at 1:08:16 there is a equation Vdot (in a) = Vdot (in b) + omega b/a x V(in b) … My question is, in which coordinate system should you express omega b/a? I hope it is also expressed in b frame?

[@김금하-w2e]

You are amazing! Thank you very much for the video.

[@ChristopherLum]

You're very welcome, there are several other similar videos on the channel. Please feel free to check them out and let me know what you think. Thanks for watching!

[@WalkingDeaDJ]

Jason-AE512: Thanks for the explanation. Well structured and I appreciate it.

[@ToanLuu158]

AE512:Your effort on this topic is commendable. Thank you for providing us with a real visualization through your diagram.

[@Kumky605]

AA516: Forgot to comment on this when I watched it the first time but loved the visual approach to working through these equations!

[@markweisa8328]

Your videos are really super valuable and have excellent explanations. I have a question here that seems confusing a little bit to me. From the intuition perspective not through the mathematical proof, how is the angular velocity vector rate of change independent of frames in the 3D world, where it's supposed to be a vector like any other vector and has its own projections on different frames? Besides, I guess this vector can be changed in two ways, one is its magnitude and the other is its orientation. So in the case of changing the orientation, I think the rate of change would be different between frames. Thanks in advance.

[@tonymcdaniel6878]

AA516: I still cannot wrap my head around that the angular acceleration vector is the same from different reference frames

[@shavykashyap]

AA 516: The visual approach to explain these concepts is very helpful. Thank you, Professor.

[@jackpascho2561]

AA516: I found the demonstration and explanation of what r_b/r and w_b/r are (7-8 min) to be very insightful, I think it's easy (at least for me) to get lost in all the sub and super scripts so seeing a simple physical depiction of what it was as well as putting it into words is a great way to help remember what each variable/subscript means later in the derivation

[@nathanlipshutz3140]

AE512: Interested to see how we will use the Coriolis equation in the future.

[@AndersonYuan-j7g]

AA 516: Celeste Yuan (Late due to name discrepancies discussed)

[@Colin_Baxter_UW]

AA516: Seeing all of the steps written out was very helpful

[@BoyRod]

Pour les français, la vidéo est à propos de la formule de Bour

[@edwardmau5877]

AE 512 The camera point of view was neat.

[@reesetaylor3506]

AE 512: Very confusing topic made clear by the demonstration, thank you for making this relatable to real world applications. I am a little confused by the switch between frame notation between the two methods @43:00. Is writing for example r_{p/b} the same as writing r_{p}^b? I interpret r_{p/b} as the position vector from the body frame's origin to point P, and r_{p}^b as the position vector of point P numercally expressed in F_b. These seem like they are describing the same thing, so why the change in notation?

[@ChristopherLum]

Good question. The superscript is used to describe which frame the vector is expressed in. The subscript is used to describe start/end points of the vector. Mostly pay attention to the superscipt as this is what is used in future discussions.

[@kutdem6321]

AA516: I've watched Flight Mech 01,02,03,04 videos. Thank you for simplifying complex concepts and making them easy to understand, I enjoyed first weeks lectures. (Kutay Demiralay)

[@ChristopherLum]

Great, it looks like you are ahead of the curve. Keep me posted on how things go as the weeks progress!

[@chayweaver.2995]

AE512: @1:05:50 you say that w_a/b + w_b/a has to equal 0 in order for that cross product to equal zero. I don't think this is entirely true. What about if w_a/b + w_b/a was equal to V? or -V? If this were the case then that cross product would still equal zero even though w_a/b +w_b/a is non-zero.

[@ChristopherLum]

This needs to be true for any aribtrary V, not just the special case that you mention. As such, the only way this is true for any V is if the property mentioned applies

[@BryanLiberman]

AE512: Thanks for the Video Dr. Lum. I liked in the beginning how you referenced real airplanes when describing these mechanics. I think it would be very cool to see a video going through some of the most popular airplanes in culture and history, for those aerospace engineers that like airplanes, but can't name specific models/species of aircraft due to lack of exposure!

[@ChristopherLum]

Great idea, let me put that on the list of videos to make in the future

[@9700784176]

self note, not a comment: skip this video and go to 7/29

[@milesrobertroane955]

AA516: I love all the models you make! I am not great with linear algebra, so thanks for saying explicitly how you can swap the terms in a cross-product for the cost of a negative sign. I'm sure remembering that will be helpful in the future! And another sidenote, I'm glad we set mu = -phi, because remembering the left-hand rotation would've been hard.

[@ChristopherLum]

Yeah, left-handed and other non-standard notation is no fun at all :)

[@rowellcastro2683]

AA516: Another very informative lecture video. Thank you for going more in depth than the book explaining the derivation of the equations and the visualizations really do help a lot. Thanks Professor!

[@ChristopherLum]

Glad it was helpful!

[@PatrickGalvin519]

AA 516: It was nice to see the derivation for the equation - I had seen the equation itself before in a Kinematics & Dynamics class but not the derivation so it's nice to see where it comes from. I also appreciated being able to see the derivation from two different perspectives.

[@ChristopherLum]

Glad it was helpful!

[@SayedTorak]

AE512: That pilot is not have a fun time. Love the different camera angles. appreciate the effort.

[@ChristopherLum]

Awesome, I'm glad it was helpful, this took a surprisingly long time to film and edit together!

[@ravisaripalli6735]

At the conclusion of Coriolis eqn. derivation, why the reference frane super scruot for Vector V was dropped. I would have thought that every vector reference frame need specified ..Would very much appreciate your response to this

[@ChristopherLum]

Hi Ravi, Thanks for reaching out, I'm glad you enjoyed the video. Unfortunately I'm unable to respond to questions on KZbin due to the sheer volume of inquiries that I receive. That being said, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum or via the 'Thanks' button underneath the video as I'll be able to answer questions there. Given your interest in the topic, I'd love to have you as a Patron as I'm able to talk/interact personally with Patrons. Thanks for watching! -Chris

[@Elclaapo]

You were born to teach Cristopher! Many many many thanks for the work you do for the community

[@ChristopherLum]

I'm glad it was helpful. There are several related videos on the channel. Please feel free to check them out and I would love to hear what you think in the comments. Thanks for watching!

[@triskitman124]

Liked because of the Michael Bay hate!

[@ChristopherLum]

Haha, nice!

[@jernejvolk3678]

You are doing an amazing thing by sharing your vast knowledge about the topics you are lecturing. Keep up the good work!

[@ChristopherLum]

Hi Jernej, Thanks for the kind words, I'm glad you enjoyed the video. If you find these videos helpful, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum or via the 'Thanks' button underneath the video. Given your interest in this topic, I'd love to have you a as a Patron as I'm able to talk/interact personally with all Patrons. I can also answer any questions and provide code/downloads on Patreon. Thanks for watching! -Chris

[@Po-ChihHuang]

AA516 Very importance topic and very easy explanation

[@ChristopherLum]

Hi, don't worry about leaving comments on the older videos, just add notes on the current videos as I just want to make sure people are keeping up with the lecture schedule. You'll easily get full participation points if you are on schedule. I don't count late comments as it isn't clear that you watched the video at the correct time.

[@sethwhittington28]

AE512: I really appreciate the effort and production value that you put into these videos. The props and use of drone cam footage help me a lot more than pages upon pages of mathematic symbols.

[@ChristopherLum]

Yeah, I'm a visual learner and also prefer to look at pictures rather than symbols.

[@jacobcarroll3933]

Good stuff! I love that we get that same result using different methods.

[@akk2766]

Loving these videos. I was following this through and trying to understand but I think I'm missing something fundamental, unless there's an error in the Small Angle Analysis section. I've been thrown off by the 3x1 matrices at 56:19 and 56:36 - are they swapped per chance? And should the one at 56:36 have a -1 instead of 1? I'm 4 years late but hopefully not too late... 🙏

[@akk2766]

I think I understand why this changed as you explain at ~1:04:54 - am I in the ball park?

[@ChristopherLum]

Hi, Thanks for reaching out, I'm glad you enjoyed the video. Unfortunately I'm unable to respond to questions on KZbin due to the sheer volume of inquiries that I receive. That being said, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum or via the 'Thanks' button underneath the video as I'll be able to answer questions there. Given your interest in the topic, I'd love to have you as a Patron as I'm able to talk/interact personally with Patrons. Thanks for watching! -Chris

[@wcottee]

Thank you for your efforts. I have always had a problem understanding describing the rate of change of a vector in an inertial frame using a rotating frame's coordinate basis (around 1:07:00). The RHS of the equation ( V + w x V) are all in the body (rotating) frame, correct? Which would mean V in the (an) inertial frame is also in the body frame...but frame is not inertial? What am I missing?

[@thomasstiglich3484]

Yes. RHS is all in body frame. The equals sign does not set the two FRAMES equal. The RHS describes the rotation of the vector V in Fb as perceived in the fixed inertial frame.

[@wcottee]

@@thomasstiglich3484 Thank you .

[@wcottee]

@@thomasstiglich3484 Thank you for the clarification, Thomas.

[@ShivaShakerii]

AA516: Your approach to visualizing concepts before diving into the details is absolutely rare, and I believe it's the most effective way to teach these subjects. As someone aspiring to become a professor, I find your teaching style incredibly inspiring and helpful. Great work!

[@ChristopherLum]

I'm glad it is helpful. Please let me know what you think of future videos as well. I'm constantly trying to update the delivery format, especially since I can put more details into these offline discussions.

[@ChristopherLum]

I forgot to mention, but we should chat at some point if you're interested in pursuing a career in academica. Please remind me at one of our office hours and I'd love to hear what you're interested in.

[@patrickfollis9417]

AA 516: when you say that w_a/b = -w_b/a, is this true within a single reference frame? (such as the inertial frame for example) Thanks! -Patrick F

[@ChristopherLum]

Hi Patrick, I think I understand the question but in short, yes, that should hold true within a frame. If you were to express this in different frames then it is possible that the values in the individual vector elements would change but the vectors are the same.

[@patrickfollis9417]

@@ChristopherLum makes sense, thanks

[@edengebretsadik7793]

AE512

[@golardachhetri6152]

I wonder how ur videos are not seen by many...... Really helped me to get the image in my head

[@ChristopherLum]

I'm glad it was helpful. There are several related videos on the channel. Please feel free to check them out and I would love to hear what you think in the comments. Thanks for watching!

[@yaffetbedru6612]

AA516

[@Pluto4Planet2024]

Between the camera equipment, whiteboard, expo markers, etc. I think this KZbin video had a larger budget than most Gen 1 Transformers cartoons.

[@leonardonovais2513]

I can't believe I spent the best part of the last two days trying to prove a property you just needed 4 lines to prove (at 1:03:26). Thank you I guess, but I feel frustrated as heck.

[@BahadirTubay]

Amazing demonstration!

[@antonete125]

49:28 I am confused becouse if you sit on an aircraft and you see the aircraft, the aircraft is not moving, it does not have velocity or acceleration, so the acceleration with respect to the body fixed referene frame must be zero, isn’t it?

[@xinlunzhao9803]

Is there an assumption that the direction of the angular velocity is always the same throughout the derivations? If not, the whole stuff will become super lengthy due to the product rule.

[@inderbhangal5419]

AA516: Amazing lecture with lots of great descriptions.

[@jamesdavey3101]

Hey! What's wrong with the Michael Bay Transformers! :C

[@giannimorandi4018]

Auguri Mirko

[@huiwan9065]

Best demo I've seen regarding the body-reference-frame and global-reference-frame

[@clickle23]

Great teaching Chris, I did enjoy it the example is so vivid only poor Starscream though :-P However I have a quick question though. For Equation of Coriolis, (when there is no body self motion, only rotation w.r.t. R frame), R_dot(p/b) seen in the R frame = w(b/r) x r(p/b). I understood there is no difference for vector w(b/r) represented either in R frame or B frame as both shares the common axis of rotation, however, there will be a difference for vector r(p/b) represented in R frame or B frame , it seems in the equation the vector r(p/b) is represented in the body frame, however the left of equation needs to be represented in R frame, do we have a mismatch here? looks like the vector r(p/b) represented in B frame needs to be rotated into R frame before cross w(b/r)(can be represented also in R frame in this case)?

[@boeing797screamliner]

AA516: These lectures are fantastic, no trouble understanding the material so far

[@ChristopherLum]

Gary, thanks, please keep me posted as you move through the material and let me know if anything needs clarification or further discussion.

[@boeing797screamliner]

@@ChristopherLum Will do

[@kenrickchan1165]

Turns out all I really wanted for Christmas was u̇.

[@karlbalitaan]

AE512: Another great video Chris! I really enjoyed the visual demo at the start. You might have to update your plane model to the F-18 once Top Gun Maverick comes out in May

[@ChristopherLum]

Haha, I know, the f14 is getting a little dated. Luckily the movie has been delayed for years so my references are still ok 😁

[@zacharywilliams2436]

great use of visual aids to demonstrate rotation in multiple reference frames

[@yoloswag6242]

funny guy haha thanks

[@ChristopherLum]

I'm glad it was helpful. There are more similar videos on the channel. Please feel free to check them out and let me know what you think in the comments. Thanks for watching!

[@milesbridges3547]

AA 516: I appreciate the use of visual aids in your explanations/derivations as it helps make the discussion clear and easier to follow. The derivation of the equation of coriolis was very well done!

[@ChristopherLum]

I find one benefit of videos on KZbin is that you can do post processing and add these types of perspectives. I'm glad it was helpful!

[@marshawnlynch6310]

I'm just here so I won't get fined.

[@aryarajendra1088]

This is good stuff!

[@ChristopherLum]

Hi Arya, Thanks for the kind words, I'm glad you enjoyed the video. If you find these videos helpful, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum or via the 'Thanks' button underneath the video. Given your interest in this topic, I'd love to have you a as a Patron as I'm able to talk/interact personally with all Patrons. Thanks for watching! -Chris

[@marilynjasmer8365]

Nice to see the two derivations of the coriolis equation through two different methods

[@abhijeetdhillon9591]

You are the teacher i never had ,sir ! :(

[@ChristopherLum]

I'm glad it was helpful. This is part of a series of videos (see the pinned comment). Please feel free to check them out and let me know what you think, thanks for watching.

[@abhijeetdhillon9591]

@@ChristopherLum Dear sir , in the 6dof model can we attach motors and checkout the how the battery power reduces with time , i am doing a project on vtol with electric propulsion ,your help would be really appreciated!

[@khaliddandago3436]

Wow. Well done for the immense effort you put for us to understand your explanations.

[@ChristopherLum]

Hi Khalid, Thanks for the kind words, I'm glad you enjoyed the video. If you find these videos helpful, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum or via the 'Thanks' button underneath the video. Given your interest in this topic, I'd love to have you a as a Patron as I'm able to talk/interact personally with all Patrons. I can also answer any questions, provide code, notes, downloads, etc. on Patreon. Thanks for watching! -Chris

[@jamesferrese2350]

lots of math but cool stuff

[@Elamin008]

you are my hero

[@cjyoo1410]

AE512: Great video Chris! I've never seen the Coriolis equation derived using Rodrigues’ Rotation Formula. Also just for clarification, the 4th property of angular velocity (where the angular acceleration is the same in either frame) is only true when the vector is parallel to the rotation axis, correct?

[@ChristopherLum]

Hi CJ, great to hear from you again! Is this similar to what hw1, p5 is asking about?

[@cjyoo1410]

@@ChristopherLum Yes, I guess it's the same question problem 5 is asking. The cross product term is zero because we chose the vector to be equal to the rotation vector.

[@mikeandrewputerakurnia5572]

NICEEE

[@suryatejag6463]

I love your lectures sir...thank you

[@ChristopherLum]

Hi Surya, Thanks for the kind words, I'm glad you enjoyed the video. If the find the these videos to be helpful, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum. Given your interest in this topic, I'd love to have you a as a Patron as I'm able to talk/interact personally with all Patrons. Thanks for watching! -Chris

[@Hannah08LV]

thanks for the lecture!

[@helenkuni9039]

Great demos!

[@sidraman6950]

great use of visual aids to demonstrate rotation in multiple reference frames