Henri P. Gavin ("Mathematical Properties of Stiffness Matrices") gave some information about eigenvalues of stiffness matrices. "http://people.duke.edu/~hpgavin/cee421/matrix.pdf"
I need a simpler explanation. Consider a scenario of two surfaces in contact (not rigid contact, but unglued close-fit).
If the eigenvalues of 6 x 6 stiffness matrix resulting from this model are 100, 1200, 1250, 1300, 1320, 1330,
1. What does these values mean?
2. How can I locate the points of applications of each of these eigenvalues on the model?
3. What would be their influence on the model?
4. What would be the influence of a change in any of the values of the eigenvalues on the model?
http://mathworld.wolfram.com/Eigenvalue.html might be a better source for you, but all it says is that each eigenvalue replaces the value of the transformation for a specific vector that we call then eigenvector. Eigen, from memory, means true, so that eigenvalue means true value. In this case, they do the job of an entire matrix. You would have to use the rule they give you: AX=lambda X, to find out all Xs for which that can happen. Lambda must be a scalar, as you see. Each X will be one point of application, if I understand well your personal meanings. They should have no influence on the model, it is just another way of expressing it or something, like at most. If you change the eigenvalues, you have to change the eigenvectors, right? If your model mentions them, then you should have a different model.
Hello Samuel,
The eigenvalues of any matrix show how the elements are distributed in the matrix or if their values are not in the same range i.e if the matrix is well-conditioned or ill-conditionned. For example if you define the volume of a polyhedron by a (nxn) matrix, the computation of the eigenvalues gives you the information if your volume is regular or bad-shaped (volume=area or line) using the definition of the conditioning number with the 2-norm. The definition of the spectral radius shows you the greater radius of the ellipsoide drawn with these eigenvalues.
For spring and mass systems.
The eigenvalues and eigenvectors give the normal modes of vibration of a spring mass system. All other vibrations are a linear combination of these normal modes of vibration. The eigenvalues are related to the frequency of vibration of a particular normal mode.
When you have an operator acting on a something and the same something comes back out but multiplied by a constant value then the something is called an eigenfunction (if it is functions you are dealing with) or an eigenvector (if it is vectors you are dealing with).The multiplying constant (usually called lambda) is the eigenvalue.
"Eigen" means "same" as
OPERATOR@( function)=lamda*(same function as on left hand side)
or
OPERATOR@(vector)=lamda*(same vector as on left hand side )
@ means "acting on"
OPERATOR can be a matrix or it could be a first derivative or a second derivative or whatever, depending on your problem.
Now, take the OPERATOR first derivative (call it D) and OPERATOR second derivative (call it D2) and consider the exponential function exp(k*x)
Then
D@(exp(k*x)=k*exp(k*x)
exp(k*x) is an eigenfunction under this scheme and k is the eigenvalue
similarily
D2@(exp(k*x)=k*k*exp(k*x)
exp(k*x) is an eigenfunction under this scheme and k squared is the eigenvalue.
These properties of the exponential function under differentiation allows coupled linear differential equations to be set up and solved as eigen system problems.
In coupled spring mass systems the eigenvalues relate to the frequency of vibrations of the linearly independent solutions. The eigenvectors are related to each linearly independent solution. Physically these are the normal modes of vibration of the system.
By the principle of superposition you can have other solutions which are linear combinations of the independent solutions. Physically this means that all other modes of vibration (of which there are an infinite number of) are a linear combination of the normal modes of vibration.
For your situation you need to know the governing equations and how they can be set up as an eigenvalue problem, Then you will know what the eigenvalues mean.
see
https://www.youtube.com/watch?v=9NlGuQ26y80
https://www.youtube.com/watch?v=chh2XybEUjg
https://www.youtube.com/watch?v=aJlSjNAB-2Y
https://www.youtube.com/watch?v=eD4WbTKKZpU
In my opinion :: any matrix corresponding to a physical system can be easily understood by making a plot of pseudo-spectra .For a literature on it ;go through the work of E.B.Davies on Harmonic oscillator .Then you can apply to your problem.
B.Rath
Hi Samuel, In the context of stiffness matrices, the eigenvectors are called "principal screws", an idea introduced by Robert Ball in his treatise on screws. The stiffness matrix maps infinitesimal motions - combining translation and rotation of a rigid body, called twists - to generalised forces (force + torque) or wrenches acting on it. Wrenches are dual to twists. The principal screws, similar to the way that eigenvectors of distinct eigenvalues for any symmetric matrix are orthogonal, are mutually "reciprocal", that is (s_j)^T Q_0 (s_i) = 0. Here Q_0 is the Klein form on the space of twists. This is an indefinite form so it is possible for twists to be self-reciprocal.
In your example, there is one principal screw where the wrench induced by the corresponding twist is smaller in magnitude than for the others and this will be a minimum response for the system.
You can find out more in Jon Selig's papers such as
http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=845218
Peter
Dear Samuel,
A detailed and simple explanation is in these remarks:
if you want for example to compute [K-wi^2*M]{phi}={0} where K:is the stiffness matrix, M: the mass matrix, wi: the frequencies, wi^2=lambdai: the eigenvalues and phi: the eigenvectors such that C=0: the damping matrix and I: the identity matrix then:
determinant([M^(-1)*K -lambdai*I])=0 gives you the eigenvalues of the M^(-1)*K matrix i.e: of your analyzed structure which the dynamic motion can be explained as a combination (real dynamic response) of several ideal modes showing its corresponding displacements (eigenvectors). Each eigenvector is calculated with each eigenvalue. An eigenvalue shows the square system vibrating frequency in the corresponding mode. Any change on these values will obviously influence the model vibrating response as a consequence of the change on the displacement mode.
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