Does eigenvalue analysis=modal analysis? Does eigenfrequency=natural frequency? What is eigenvalue?

Question

Does eigenvalue analysis=modal analysis? Does eigenfrequency=natural frequency? What is eigenvalue?

Does eigenvalue analysis=modal analysis? Does eigenfrequency=natural frequency? What is eigenvalue, eigenvectors, eigensystems?
Please explain what is the definition or what you understand by each term? List out their differences and similarities. Please bring in relevant concepts if needed!

Answer 1

Modal analysis is one application of Eigenvalue analysis but Eigenvalue analysis can be used for other things.

In the context of modal analysis (using FEM), the eigenvalue=eigenfrequency=natural frequency (mode).

The eigenvector is the diplacement vector or mode shape of the mode.

I haven't seen the term eigensystem, but it probably refers to the matrix and vector for the problem (i.e. everything as a whole).

Answer 2

The factor by which the magnitude is scaled is called the eigenvalue (help·info) of that vector. (See Fig. 1). Often, a transformation is completely described by its eigenvalues and eigenvectors. The eigenspace for a factor is the set of eigenvectors with that factor as eigenvalue.

These concepts play a major role in several branches of both pure and applied mathematics — appearing prominently in linear algebra, functional analysis, and to a lesser extent in nonlinear situations.

It is common to prefix any natural name for the vector with eigen instead of saying eigenvector. For example, eigenfunction if the eigenvector is a function, eigenmode if the eigenvector is a harmonic mode, eigenstate if the eigenvector is a quantum state, and so on (e.g. the eigenface example below). Similarly for the eigenvalue, e.g. eigenfrequency if the eigenvalue is (or determines) a frequency.

Answer 3

You're basically asking someone to explains a college semester's course in a few words. An eigenvalue is a vector which defines a non trivial solution to a set of equations. One subset of eigenvalues are eigenmodes which relate to modal analysis. They can define vibrations or wave function solutions. I guess you can roughly say that modal analysis is related to a subset of eigenvalues but eigenvalues are much more general. Take a look at wikipedia or mathworld.

Answer 4

Hello The eigen word is comun in quantum physics. I think is not possible to "mix" this word with modal analysis and natural frequency. In quantum physics people speak the eigen values refer to a solution of a Scrhodinger equation. In that equuation we have a density of prbabilitu wave function.. thats no represent a real word.

Answer 5

What Is Eigenvalue