Eigenvalues and Eigenvectors - How do I work out which eigenvalue corresponds to which eigenvector?

I have a 7-DOF vibrating car model and would like to work out the eigenvalues (or natural frequencies) of each individual mode of the car, where the body has 3-DOF (pitch, roll and vertical displacement) and then each wheel has 1-DOF for its vertical displacement. I have pre-defined the mass and stiffness matrices so that I can find the eigenvectors and eigenvalues using "[Evecs,Evals]=eig(inv(M)*K)", which gives me matrices containing the eigenvectors and eigenvalues.

The problem I have is that the Evals matrix is mixed up, such that I do not know for definite which eigenvalue corresponds to which eigenvector. Because the car is assumed symmetric across its width, I can easily work out the three body modes but I do not know how to distinguish between the four wheel modes.

This issue only seems to occur with high-DOF systems, as I have successfully performed the exact same method on a 4-DOF half-car model. Has anyone else encountered this problem? Is there an easy way for me to resolve it?

[Adam]

The eigenvalues and eigenvectors will always be in the same order as each other.

[Jan]

Note: Use the more stable M\K instead of inv(M) * K.

[Steven Lord]

Or solve the generalized eigenvalue problem directly using eig(K, M).

[Benjamin Harrington]

Steven - using eig(K,M) presents the eigenvalues in the correct order (relative to how I defined the mass and stiffness matrices), thanks for your suggestion!

Answer 1

[Jan]

No, the order of eigen-values and eigen-vectors is not mixed. (As Adam has said already)

[Benjamin Harrington]

Hi Jan,

I think I phrased my question slightly wrong - yes, the order of the eigenvalues and eigenvectors are the same. I have attached the code that I am using in case that helps illustrate my problem better. As you should be able to see, the way I have defined my mass and stiffness matrices implies that the eigenvalues should be arranged in this order: body vertical mode, body roll mode, body pitch mode, front left wheel mode, front right wheel mode, rear right wheel mode, rear left wheel mode. Furthermore, I know that the eigenvalues of the four wheel modes should all be similar and that the eigenvalues of the three body modes should also be similar. This suggests that the 1st, 2nd and 3rd eigenvalues/eigenvectors should represent the three body modes and then the remaining four represent each wheel mode.

When I run the script, the 3rd eigenvalue is not similar to the 1st and 2nd eigenvalues. Likewise, the 5th eigenvalue is not similar to the 4th, 6th and 7th eigenvalues. Thankfully, as I mentioned in my original question, the car is symmetric across its width and so I can work out which the three body modes by looking at the eigenvectors (1st is body vertical, 2nd is body pitch and 5th is body roll). As you should be able to see from the code, this leaves the 3rd, 4th, 6th and 7th eigenvalues for the four wheel modes - this is where my problem lies. When I look at the eigenvectors, the magnitudes and polarities of the values do not obviously indicate which mode corresponds to which wheel.

Also, when you say M/K I assume you mean K/M? Having said that, due to matrix algebra rules K/M does not equal inv(M)*K so I do not see how they are interchangeable.

Thanks, and here is the code:

[Benjamin Harrington]

I have just followed this up by using Steven's advice in the comments and the eigenvalues are presented in the correct order, thus solving my problem! Having said that, the method I originally used does do something strange to the order of the eigenvalues when compared with the way I defined the mass and stiffness matrices, so for curiosity's sake I would still appreciate a solution to my question (if there is one)

Answer 2

[John D'Errico]

I see from your comments to Jan that you still want an answer on how to know or fix the order of the eigenvalues. In your words:

"As you should be able to see, the way I have defined my mass and stiffness matrices implies that the eigenvalues should be arranged in this order: body vertical mode, body roll mode, body pitch mode, front left wheel mode, front right wheel mode, rear right wheel mode, rear left wheel mode. "

NO. You cannot know that the eigenvalues & eigenvectors are in any order that corresponds to specific things about your system. If that happens SOMETIMES, then you got lucky! It need not happen in general, nor should you presume that it will ever happen.

The eigenvalues are essentially the roots of a polynomial. At least, you can think of them that way, regardless of how they end up being computed. They need not correspond to any physical features of your system, and there is no presumption of order.

[Benjamin Harrington]

Hi John,

Point taken! In that case then, how would you suggest I go about working out which eigenvalue/eigenvector pair corresponds to which mode of the system? I have been trying to look at the magnitudes and polarities of the values within each eigenvector to influence my decision, but I have found that it is very easy to get confused.

[Benjamin Harrington]

This is the matrix of eigenvectors for the system:

[-0.5238 0.6427 0.0128 0.0168 0 0 0;

0 0 0 0 0.9305 0.0809 -0.0444;

0.7941 0.6527 -0.0157 0.02 0 0 0;

-0.2168 0.0234 -0.707 -0.0018 0.2162 -0.703 -0.061;

-0.2168 0.0234 -0.707 -0.0018 -0.2162 0.703 0.061;

0.0232 0.2827 0.002 -0.7069 -0.1428 0.0501 -0.7038;

0.0232 0.2827 0.002 -0.7069 0.1428 -0.0501 0.7038]

and this is the vector of eigenvalues that accompanies it:

10000*

[0.0297;

0.0379;

1.0621;

1.1434;

0.085;

1.3903;

1.1413].

Correct me if I am wrong, but I think that the 1st eigenvector/eigenvalue pair is the body vertical displacement, the 2nd is the body pitch and the 5th is the body roll. As for the individual wheels, I have no idea.

[John D'Errico]

Eigenvectors rarely have simple meanings that you can attribute to them. If you get lucky, you may see something that makes sense for some eigenvectors, when you think about the linear combination of the parameters that they imply.