So What Is A Mode Shape Anyway? - The Eigenvalue Problem
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- เผยแพร่เมื่อ 6 ก.ย. 2024
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Structural Model for this Video: • Two Degree of Freedom ...
Code Repository: github.com/apf...
An explanation of the eigenvalue problem. What are natural frequencies and mode shapes anyway?
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I had such a massive mental blockage when it came to understand vibration theory. I have an exam in two days, and I have been covering 60-70% of my syllabus using your videos, you are absolutely amazing. Thank you for your ABSOLUTELY wonderful models that you built, they really help vizualize what multi-degree of freedom can be imagined. No matter your motive to do this, I feel like its genuinely philantropic, I hope God takes you to heaven.
Thank you for your kind words. Comments like yours make all of this effort worth my while.
hats off to your teaching! its really commendable how greatly you are explaining these topics! The resonance phenomenon, if I am not wrong, where the freq of forcing function equals the natural freq of structure was so well explained due to your animation. I wish all the good luck to you! And keep educating us!
I think tomorrow ur have mechanical vibration exam...M.tech 2nd sem machine design
Powerful stuff. How did all my teachers along the way fail to ever make all this so clear. So many applications. Thanks.
you are amazing mate!! if i pass my vibrations test tomorrow ill send you a pint from ireland :)
Your videos are absolut gold. I´m an engineer myself and stunned how clear and beautiful u present these ideas. Chapeau and cheers from germany
Great video. We show this to our students using an experimental 4 DOF tower. They enjoy it and it shows them the meaning of eigenvectors
This makes me smile! Thank you for sharing.
@@Freeball99 what programming language was that ??
@@freemanfreed1581 Python
@@Freeball99 i see. you have used pygame. good job man
Thank you for sharing your thoughts. It was really useful. Now, I have understood the mode shape and eigenvalue concept.
Cheers
Thank you for your valuable explanation.
Eigen value as well Eigen vector looks simple but powerful
Not sure why this guy doesn't have 18 million subscribers. Very good.
A remarkable video and creative way of explaining the problems with animation. It help me lot in understanding with visualization.
These videos are really great 👍 . I love the focus on Python (aka real world). I haven't been watching the Playlist in order, but about once a week I do a search for a particular type of problem and one of your videos is a perfect match!
thanks Sir,i understood the concept
Fantastic presentation, Clearly depicting the concept of Eigenvectors, Thank you so much for it. want a little , like it , for Modal Orthogonality property and its actual meaning in SRSS and CQC.
This was great. Really enjoyed the presentation and looking fwd to digging into your code.
This is classic thankyou make some more also good videos.
Really well explained and the code at the end really helped to visualise what was going on!
Thank you Sir! I'm here because of structural dynamics. ^_^
Very clear to understand .... Simple and complete. Great video recommendable for learning for young one to understand...
From 5:29 to 6:27, your statement that, because the determinant is zero, we can pick any of the equations, as they are dependent, we chose any! However the two doesn't yield same (X₁/X₂) values.
-2X₁ + (2-ω²)X₂ =0 yields
(X₁/X₂) = (2-ω²)/2
whereas (5-ω²)X₁ -2X₂ = 0 gives
(X₁/X₂) = 2/(5-ω²)!
Made same comment in the previous video as well!
You have a typo (5-ω²)X₁ should be (5-2ω²)X₁ if you plug in one of the eigenvalues for ω, you'll find that they are the same.
Explained brilliantly. Thanks
I think it would be a great idea to highlight the importance of eigenvector's orthogonality by showing how you can decouple the equations of the system
Yep, I agree. I'll be making a video on the normal mode method.
MINDBLOWING...
Thank you so much once more.
Such a brilliant video. Loved it. Thank you so much.
Siento que no expresa una formulación general, para sistemas TITO(two input two outputs), adicionalmente no explica el modo, la matriz modal, y su respectiva formulación, obvió no muestra las pruebas de Schmidt, para certificar que las respuestas obtenidas son adecuadas.
No muestra la fórmula de Ritz, para determinar frecuencia naturales para 2 niveles.
No explica nada generalizada, rigidez generalizada,...es un buen aporte para principiantes.
Great job 👍 especially the simulation .
Thank you 👍
I love this guy's voice
Nice! Awesome use of Python once again for the simulation at the end. And I didn't know that you could use pygame for those animations! I learned another thing ❤️
So sir that means, eigen values are the frequency of the vibration and the eigen vectors correspond to the pattern of vibration that the system will follow at a given frequency. Can we say this, if we want to make a statement in layman terms ?
Yes, this is correct with the slight modification that the eigenvalues are the frequencies squared (since these equations are 2nd order).
What a video. Thank you so much sir. God bless you
Hi can you do a video on complex mode shapes when considering un-proportional damping, with an explanation on real and imaginary components, what it’s physical significance. Couldn’t find a better person ask. Thank you heaps !!!
Yes, I can do this but will likely require a few primer videos first. I will add it to my list.
Freeball 🙏 yes that’s fine. couldn’t find anybody covers any material related to this topic . Thank you
Hello! Can you please explain the eigenvalues and eigenvectors if we have also 2 dampers in the system.
The most common way to treat damping is to assume proportional damping (which is a good assumption for many problems). In this case, the [c] matrix can be written as a linear combination of both the mass and stiffness matrices (i.e. [c] = α[m] + β[k]). In this case, adding damping can alter the effective stiffness matrix or the effective mass matrix. As a result of this, the eigenvectors and eigenvalues for the system will change depending on the values of α and β. I will have to make a video on this to explain it properly.
Question Professor: usually for standard eigen value/vector math problems you subtract identity matrix times lambda (or omega squared in this case) from your K matrix. However we are subtracting mass matrix (which is not the identity).
So how can the be related to the standard Indra of finding eigenvectors and values in the “classical” linear algebra math way?
Thanks!
Take a look at this. Hopefully it makes sense. dropbox.com/s/wdsoat8pew4fnn3/EigenvalueProblem.pdf
Great video! thank you
thanks
So correct me if I'm wrong, but your eigen values are your natural frequencies and your eigen vectors are your mode shapes. Correct?
Correct with a slight modification...that the eigenvalues are equal to the square of the frequencies.
Hi, great video i was just wondering how you would find the Eigen values if you incorporated a Damper matrix, C?
This is a good question, but difficult to answer without making a video...
Engineers will typically model viscous damping by using something called proportional damping. In which case, [c] = α[m] + β[k] where α and β are constants. So the damping is proportional to the stiffness and mass matrices (or said differently, the damping is proportional to the potential and kinetic energies of the system). This way, the system has the same eigenvalues as the undamped case and can be decoupled using the normal mode method. This is not the only technique used to decouple the equations of motion, but if has been shown to be effective in that it does not reduce the stability boundaries significantly.
It should also be mentioned that this technique is unnecessary if you are solving the equations of motion using numerical integration instead of the analytical approach. I will make a video on this in the future.
What are the mode shapes (and thus natural frequencies) for damped, MDOF systems, particularly when damping is not proportional? The Vibrations book by Rao says that modal analysis (and thus determination of mode shapes and natural frequencies) is only applicable/solvable for undamped systems while the one by Kelly has a procedure for general damping using augmented matrices.
But still, it is not clear to me how/what are the mode shapes for damped systems especially generally damped systems.
It's a good questions and beyond what I can explore in a text message.
On the one hand, from a numerical point-of-view, you could take the code from this video, add damping and run it for a forcing function of different frequencies until you hit resonance. Then you could examine the mode shapes.
On the other hand, from a mathematical point-of-view, we still like to refer to the natural frequencies even for a damped system. This is because the natural modes are orthogonal to one another and this is a very important property to be able to use when determining the response mathematically. As a result, in most cases, we assume the damping to be proportional damping. This means that we can write the damping matrix as a linear combination of the mass and stiffness matrices. Fundamentally this allows us to write the damped response in terms of the natural modes.
Damped modes look a lot like natural modes, but the peaks get squashed and shifted a little (for small to moderate amounts of damping).
@@Freeball99 hi Freeball. I only saw your reply today. Regarding the damped modes for a system, I actually just solved and plotted a 2dof system with general damping and whose initial displacements are that of the undamped modes. The vibrations are like what you said that are slightly offset. If the damping is proportional, the masses oscillate with no phase difference.
I would also like to express thanks and appreciation for your animating the 2dof system and sharing your code. It helped me tremendously in my vibrations class today. I would just like to ask how you learned/what resources you used to learn how to animate it? I'm interested to animate a 3dof system. I don't have any prior knowledge/experience with animation in Python although I am already familiar with it already. Plus there seems to be a dearth of learning resources on how to animate spring-mass systems step-by-step.
Lastly, in your animation code, what should I change in thr forcing function so I can transfer the applied force from mass 2 (the right mass)to mass 1 (the one attached on the wall)? I intend to make mass 2 as a vibration absorber to mass 1.
Again, thank you very much Freeball! Your videos are invaluable and deserves more views. Glad to have found your channel.
Great video, thanks. However, would you please comment on the orthogonality of the two eigenvectors. You said that their dot product is zero. However, if I take your example, one eigenvector is (0.593, 1) and the other is (-0.843, 1). Their dot product is (0.593 * -0.894 + 1 * 1) = 0.5. It appears that the eigenvectors are only orthogonal if the masses are identical. Am I doing anything wrong??
I believe those are the mode shapes which are not equal to the eigenvectors. Not 100% though.
@@UltimateBreloom Eigenvectors are mode shapes because they represent the deformed state of the structure as it vibrates at the respective eigenvalues (frequency).
I think I might have misspoken...slightly. The eigenvectors (which are the same thing as the mode shapes) are not orthogonal to one another, but rather are orthogonal with respect to the mass matrix (and also the stiffness matrix). This means that
transpose(v_i) · M · v_j = 0 if i ≠ j
where v_i, v_j are the eigenvectors and M is the mass matrix.
So, if you sandwich the mass matrix (or the stiffness matrix) in between then this relationship should hold.
@@Freeball99 Excellent! Thanks for the clarification. That works for both M and K matrices... transpose(v_i) · M · v_j = 0 and transpose(v_i) · K · v_j = 0 if i ≠ j
Nice video, thank you for sharing! I have a question, that is, what's the difference between mode shape and mode of vibration? Because i cannot find on the internet if the difference is that the the latter is the superposition of the former for many dofs, or perhaps one of the concepts contains also an info about the phase, or whatever... Thank you so much in advance!
I think what you have stated is fundamentally correct. The mode shapes describe the "shape" of the system when it vibrates in one of its natural modes. The mode of vibration refers to the superposition of these modes, but tend to focus on which mode/modes are being stimulated - i.e. which modes are contributing the most to the motion.
Eigenvectors are not necessarily orthogonal. They are linearly independent. Singular vectors are orthogonal. Am I wrong?
You are correct that, in general, eigenvectors are not necessarily orthogonal. However, for the case that the matrices are symmetric then this will always yield orthogonal eigenvectors. So for the case of vibrations problems, due to the symmetry of the [k] and [m] matrices - which is always true as far as I'm aware - the eigenvectors for the model will be orthogonal.
@@Freeball99 Thank you for the clarification Sir. Great video!
Thanks
thank you
great job
Could two mode shape of the two dof vibratory systems be negative or one must be positive and the other is negative?
No. One must be positive and one negative. You should always be able to write any response in terms of a linear combination of the mode shapes. In order to achieve this in 2DOF, you must have one mode where the masses are moving in the same direction and another mode where they are moving in opposite directions.
Hi. Can I get the link to the previous video showing the problem where the stiffness matrix is derived?
th-cam.com/video/dqtZwZmMh4w/w-d-xo.html
What device or software do you use for your lectures
The app is "Paper" by WeTransfer, running on an iPad Pro 13 inch and using an Apple Pencil. The voice is mine.
@@Freeball99 you voice is the best part
@@KakarotM99 That's kinda sus, my fam.
Anyone knows what todo if our one value of w is negative ?
You should not get negative values for ω. However in the over-damped case, you will get negative values for ω^2 in which case, ω will be an imaginary/complex number.
Are you willing to share the python codes you used in this video?
There you go: github.com/apf99/TwoDOF - also added it to the video description.
@@Freeball99 Thank you very much!
KCPD bro....
earthquakes are to a girl's guitar, they're just another good vibration
I am a civil engineer , Never have seen such a horrible video in my life ever.Totally hanged my brain. Actually i watched this movie to reading word Mode Shape & Eigenvalue in title which are used in seismic design. Now i thought it will be better to face EARTHQUAKE than watching this video. Bye i am going.........🧚