Modal Analysis Using The Normal Mode Method

Decoupling of the equations of motion by transforming them into the system's normal coordinates. We derive the theory behind the transformation step-by-step and work an example to demonstrate the method.

LINKS:
Two Degree of Freedom Problem Without Damping

So What is a Mode Shape Anyway - Solving the Eigenvalue Problem

Forced Vibration of a SDOF System:

Differential Equations Primer - Finding the Homogeneous (Transient) Solution:

Differential Equations Primer - Finding the Particular (Steady-State) Solution:

CHAPTERS:
0:00 Intro
0:52 Step-By-Step
10:30 Recap
11:30 Orthogonality of Normal Modes
17:50 Example Problem
22:46 Putting it All Together
26:00 Outro

[SLCMuralha]

Much appreciated! I only wish I had your videos when going through undergrad ME courses. It would have helped tremendously.

[abhishekdixit]

Thanx for such informative videos... Shows ur indepth knowledge about the subject...
Binge watching ur content🎉

[ibrahimhawraasafaaibrahimu]

you absolutely deserve a subscribe and a like and a share

[ignatiusjacquesreilly]

Great video! If you have time I'd be interested in seeing this problem solved with proportional damping.

[rk]

Thank you for the wonderfully clear step by step explanation. When I studied linear algebra, the inverse of the eigenvalue matrix was not it's transpose, so I have no Idea why the diagonalization worked. I'm glad it did, but I miss the point of the diagonalization, because you already have had to solve the system to do the diagonalization? I was hoping decoupling would lead to a way to construct arbitrary motion as a linear combination of the decoupled components.

[tryfonasthemas]

Is there any relationship of what you illustrated with the fourier transform? The steps we used remind me a lot of how one solves pdes in fourier space by converting them to odes.

[Enggoat]

Please may you post the answer to the example problem in the comments or elsewhere, just to be able to check understanding of solving the problems?

[nickgenin]

professor, why not normalize it by the mass and get the [m] as diagonal 1 and the stiffness as omegai^2?

[KapilanKillewalavan]

What book are you referencing with this? Leonard meirovitch?

[sdleakey]

Was just curious how this method still works when you don't include all of the eigenmodes of the system (as would be typical in FEA). I'm struggling to connect those math dots.

[Itsgallon]

Does the projection to modal coordinates still work to diagonalize mass and stiffness matrices in the solution of an aeroelastic system where aerodynamics are now introduced to the structural dynamics?

[christosgeorgiadis]

Could we use singular value decompostion (SVD) instead of eigenvalue decomposition that was used in this video? Do you know if this method is used in vibration problems. I'm curius. Thank you.

[tryfonasthemas]

Is there any relationship of what you illustrated with the fourier transform? The steps we used remind me a lot of how one solves pdes in fourier space by converting them to odes.

[tryfonasthemas]

Is there any relationship of what you illustrated with the fourier transform? The steps we used remind me a lot of how one solves pdes in fourier space by converting them to odes.