I think tomorrow ur have mechanical vibration exam...M.tech 2nd sem machine design

Thank you for your kind words. Comments like yours make all of this effort worth my while.

Thank you 👍

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

Yep, I agree. I'll be making a video on the normal mode method.

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.

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.

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

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.

kzfaq.info/get/bejne/mtekjdqL0rPMZKs.html

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.

Take a look at this. Hopefully it makes sense. dropbox.com/s/wdsoat8pew4fnn3/EigenvalueProblem.pdf

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.

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.

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.

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.

Correct with a slight modification...that the eigenvalues are equal to the square of the frequencies.

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

There you go: github.com/apf99/TwoDOF - also added it to the video description.

@@Freeball99 Thank you very much!

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!