Will have to make a video using proportional damping, but fundamentally this will just produce slight variations to the mass and stiffness matrices. The rest of the modeling will be the same.

In order to perform the diagonalization, we need only solve the eigenvalue problem (ie we solve for the mode shapes - we do NOT solve the response problem). Once diagonalized, the equations are decoupled and exist in the form of n-independent equations each of which represents a simple harmonic oscillator (SHO). Because we readily know the solution to the SHO equations, we can write them (in terms of the normal coordinates). The actual response of the system can then easily be written as a linear combination of these normal coordinates.

@@Freeball99 I'm still confused. Once we solve the eigenvalue problem, can't we just write x as a sum of the eigenvalues times a sinusoidal term of the corresponding frequency and be done with the problem? Also, all these matrix calculations seem like they would be computationally expensive. I still don't really see the point of this diagonalisation

This comes from my class notes from years ago. Not sure what book the teacher used, but it certainly could have been Meirovich. However, I think that "Mechanical Vibrations" by Rao has a pretty good explanation of normal modes.

You can do that. However, I find it easier (if doing it by hand) not to normalize the eigenvectors and then simply calculate the various omegas using the k's and m's from the diagonalized matrices...which is trivial.

Yes, certainly you could use other decompositions like SVD in vibrations problems. However, SVD is not as widely used as EVD in classical modal analysis, as EVD provides a more direct and straightforward way of obtaining the modal parameters of a system. However, SVD can still be a valuable tool in some vibration problems, especially when dealing with complex or non-square matrices, or when the goal is to extract the dominant patterns in the vibration data.

Generally, the introduction of aerodynamic loads will add damping to the system that is non-proportional damping. In this case, the normal mode method will not work as is. However, it is still possible to use a modified version of the normal mode method that takes into account the effects of non-proportional damping. One such method is the modal analysis method, which requires solving the equations of motion with the aerodynamic loads included and then transforming the results into modal coordinates.

I don't have the solution for this because it is a problem I made up. You should be able to get there from the information in the video. I will need to create a separate video to explain this because I'm not going to be able to answer it in the comments.

All the modes have been included. This was done in eqn 3 where I wrote the response in terms of the normal coordinates and mode shapes: {x} = [Φ]·{η}. The modal matrix, [Φ], includes all the modes.

Not really related. However, both allow one to extract the fundamental frequencies from the system, so they have that in common. In this particular video, we are dealing with ODEs and not a PDEs. We are using the normal mode method to decouple the equations of motions which allows us to easily solve each equation separately. We do this in the time domain. Fourier transform is used to transform a system from the time domain to the frequency domain.