Some questions about solving eigenproblems to do modal analysis

[blah900]

So for my graphics class I am currently reading "Stochastic Dynamics: Simulating the Effects of Turbulence on Flexible Structures" by Jos Stam

However, while starting to implement it, I've realized my understanding of math is not strong enough to fully understand what is going on so I have a couple of questions.

Given the mass matrix M and stiffness matrix K, how would I go about solving MQD=KQ where Q is the modal shapes matrix and D is a diagonal matrix of modal frequency involved with Q. I am not sure how Q and D can both be found from this one equation and exactly what part of this problem is a eigenproblem.

I am told that I should use ARPACK(which according to the paper uses Lanczos method) but without first understanding what is going on in that equation, I am not sure how to proceed. I would like any help or hint on understanding this.

 

[AlephZero]

Do you understand what the eigenvalues and vectors of a single matrix are? In other words do you understand why an equation like $Ax = \lambda x$ only has "interesting" solutions for some particular values of $\lambda$?

In principle, you can extend those ideas to your equation simply by taking $A = M^{-1}K$. In practice, you don't solve the problem that way numerically, because there are much more efficient ways to do it.

You shouldn't have to understand all the details of how the Lanczos algorithm in ARPACK actually works. All you need is the basic concept of what eigenvalues and vectors are, how to supply your matrices to the routine, and how the answers are returned from it.

 

[blah900]

So basically the D becomes the eigenvalues and Q ends up becoming the eigenvector of the
A = M^-1K
Ax=λx equation?

And I would just use ARPACK and find the part of the function that does this for me automatically given the stiffness matrix K and mass matrix M?

I think I am beginning to understand now, please correct me if I'm wrong.

 

[AlephZero]

That's right. You would usually write an eigenvalue $\lambda_i$ as $\omega^2_i$ where $\omega_i$ is the natural frequency (in radians per second), and the corresponding eigenvector is the vibratiion mode shape.

If you want more details on how the math relates to the physics, Google for "muiltiple degree of freedom dynamics" or MDOF.

I woudn't recommend trying to understand the Lanczos algorithm unless you already know quite a bit about linear algebra and computational methods, but it's widely used as a "general purpose" solution method for this type of eigenproblem with large matrices (up to order 100,000 or more).

 

[CellCoree]

First of all, it's great that you are reading and trying to implement a paper on stochastic dynamics for your graphics class! That is a very challenging and interesting topic. It is also completely normal to have questions and difficulties understanding the math involved. As a scientist, it is important to keep asking questions and seeking understanding, so let's try to break down the problem together.

The equation that you mentioned, MQD=KQ, is a simplified form of the modal analysis problem. In this equation, M represents the mass matrix, K represents the stiffness matrix, Q represents the modal shapes matrix, and D represents the diagonal matrix of modal frequencies. This equation is a representation of the dynamic behavior of a flexible structure, where M and K describe the physical properties of the structure and Q and D describe the modes of vibration.

Now, the goal of modal analysis is to find the modal shapes and frequencies for a given structure. This is where the eigenproblem comes in. An eigenproblem is a mathematical problem that involves finding the eigenvectors and eigenvalues of a given matrix. In this case, the eigenproblem is to find the modal shapes (Q) and frequencies (D) that satisfy the equation MQD=KQ.

To solve this eigenproblem, you can use the ARPACK library, which implements the Lanczos method. This method is a numerical algorithm that can efficiently compute a few eigenvectors and eigenvalues of a large matrix. In this case, the matrix that needs to be solved is MQ-K, and the ARPACK library will use the Lanczos method to find the modal shapes and frequencies.

In summary, the equation MQD=KQ is a simplified form of the modal analysis problem, and the eigenproblem involved is to find the modal shapes and frequencies that satisfy this equation. The ARPACK library with the Lanczos method can be used to efficiently solve this problem. I would recommend studying the theory behind the Lanczos method and the modal analysis process to better understand the math involved. Additionally, you can reach out to your instructor or classmates for further clarification and assistance. Keep up the good work and don't be afraid to ask questions!