1. Definition of the eigenvalue problem#
1.1. Generalities#
Consider the following eigenvalue problem:
Find
\((\lambda ,\Phi )\mathrm{\in }\mathrm{ℂ}x{\mathrm{ℂ}}^{n}\mathrm{/}({\lambda }^{2}\mathrm{B}+\lambda \mathrm{C}+\mathrm{A})\Phi \mathrm{=}0\) eq 1.1-1
where \(A,C,B\) are positive symmetric real matrices of order \(n\).
A distinction is made between two cases:
quadratic problem: \(C\ne 0\),
generalized problem: \(C=0\).
\(\lambda\) is called eigenvalue and \(\Phi\) eigenvector. In the following, we will talk about clean mode for \(\Phi\) and we will introduce the concept of natural frequency.
To solve this problem, several methods are available in Code_Aster and the reader is referred to the [R5.01.01] and [R5.01.02] documents.
1.2. Generalized problem#
The generalized problem can be written in the form:
Find
\((\lambda ,\Phi )\in ℝx{ℝ}^{n}/(-{\lambda }^{2}B+A)\Phi =0\) eq 1.2-1
Two other quantities are introduced which make it possible to characterize the natural mode:
\(\lambda =\omega =(2\pi f)\) eq 1.2-2
where
\(\omega\): natural pulsation associated with the natural mode \(\Phi\),
\(f\): natural frequency associated with natural mode \(\Phi\).
It is also shown that the eigenmodes are \(A\) and \(B\) orthogonal, i.e.:
\(\mathrm{\{}\begin{array}{c}{\Phi }^{\text{iT}}\mathrm{A}{\Phi }^{\mathrm{j}}\mathrm{=}{\delta }_{\text{ij}}{\Phi }^{\text{iT}}\mathrm{A}{\Phi }^{\mathrm{i}}\\ {\Phi }^{\text{iT}}\mathrm{B}{\Phi }^{\mathrm{j}}\mathrm{=}{\delta }_{\text{ij}}{\Phi }^{\text{iT}}\mathrm{B}{\Phi }^{\mathrm{i}}\end{array}\) eq 1.2-3
where \(\left({\Phi }^{\mathrm{i}},{\Phi }^{\mathrm{j}}\right)\) are two clean modes.
1.3. Quadratic problem#
The quadratic problem [éq 1.1-1] can be put in another form of double size (we speak of linear reduction [R5.01.02]):
Find
\((\lambda ,F)\in ℂx{ℂ}^{n}/(\lambda \left[\begin{array}{cc}0& B\\ B& C\end{array}\right]+\left[\begin{array}{cc}-B& 0\\ 0& A\end{array}\right])(\begin{array}{c}\lambda \Phi \\ \Phi \end{array})=0\) eq 1.3-1
We pose next: \(\stackrel{ˆ}{B}=\left[\begin{array}{cc}0& B\\ B& C\end{array}\right]\stackrel{ˆ}{A}=\left[\begin{array}{cc}-B& 0\\ 0& A\end{array}\right]\).
Since matrices \(A,C,B\) are real, the eigenvalues and modes are imaginary conjugated two by two.
Three other quantities are introduced which make it possible to characterize the natural mode:
\(\lambda \mathrm{=}a+\mathit{ib}\text{=}\text{-}\frac{\xi \omega }{\sqrt{1\mathrm{-}{\xi }^{2}}}+i\omega \text{=}\text{-}\frac{\xi (2\pi f)}{\sqrt{1\mathrm{-}{\xi }^{2}}}+i(2\pi f)\) eq 1.3-2
where |
\(\omega\): natural pulsation associated with the natural mode \(\Phi\), |
\(f\): natural frequency associated with the natural mode \(\Phi\), |
|
\(\xi\): reduced amortization. |
It is also shown that the eigenmodes are \(\left[\begin{array}{cc}0& B\\ B& C\end{array}\right]\) and \(\left[\begin{array}{cc}-B& 0\\ 0& A\end{array}\right]\) orthogonal, i.e.:
\(\{\begin{array}{}-{\lambda }_{i}{\lambda }_{j}{\Phi }^{\text{iT}}B{\Phi }^{j}+{\Phi }^{\text{iT}}A{\Phi }^{j}={\delta }_{\text{ij}}(-{\lambda }_{i}^{2}{\Phi }^{\text{iT}}B{\Phi }^{i}+{\Phi }^{\text{iT}}A{\Phi }^{i})\\ ({\lambda }_{i}+{\lambda }_{j}){\Phi }^{\text{iT}}B{\Phi }^{j}+{\Phi }^{\text{iT}}C{\Phi }^{j}={\delta }_{\text{ij}}(2{\lambda }_{i}{\Phi }^{\text{iT}}B{\Phi }^{i}+{\Phi }^{\text{iT}}C{\Phi }^{i})\end{array}\) eq 1.3-3
where \(({\mathrm{\lambda }}_{i},{\mathrm{\lambda }}_{j})\) are the eigenvalues associated respectively with the \(({\Phi }^{i},{\Phi }^{j})\) eigenmodes.
Note:
the proper modes are therefore not \(\mathrm{A},\mathrm{B}\text{ou}\mathrm{C}\) orthogonal.