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Update Content - 2024-12-17

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Thomas Dehaeze
2024-12-17 16:38:55 +01:00
parent 59f9b8ae81
commit 91ca6068b5
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content/book/hatch00_vibrat_matlab_ansys.md
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@@ -252,7 +252,7 @@ where:
#### Eigenvalues / Characteristic Equation {#eigenvalues-characteristic-equation}
Re-injecting normal modes <eq:principal_mode> into the equation of motion <eq:tdof_eom> gives the eigenvalue problem:
Re-injecting normal modes \eqref{eq:principal\_mode} into the equation of motion \eqref{eq:tdof\_eom} gives the eigenvalue problem:
\begin{equation}
(\bm{k} - \omega\_i^2 \bm{m}) \bm{z}\_{mi} = 0
@@ -310,7 +310,7 @@ Virtual interpretation of the eigenvectors are shown in [Figure 5](#figure--fig:
#### Modal Matrix {#modal-matrix}
The modal matrix is an \\(n \times m\\) matrix with columns corresponding to the \\(m\\) system eigenvectors as shown in Eq. <eq:modal_matrix>
The modal matrix is an \\(n \times m\\) matrix with columns corresponding to the \\(m\\) system eigenvectors as shown in Eq. \eqref{eq:modal\_matrix}
\begin{equation}
\bm{z}\_m = \begin{bmatrix}
@@ -873,7 +873,7 @@ If modes have some damping:
\frac{z\_j}{F\_k} = \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \label{eq:general\_add\_tf\_damp}
\end{equation}
Equations <eq:general_add_tf> and <eq:general_add_tf_damp> shows that in general every transfer function is made up of **additive combinations of single degree of freedom systems**, with each system having its DC gain determined by the appropriate eigenvector entry product divided by the square of the eigenvalue, \\(z\_{nji} z\_{nki}/\omega\_i^2\\), and with resonant frequency defined by the eigenvalue \\(\omega\_i\\).
Equations \eqref{eq:general\_add\_tf} and \eqref{eq:general\_add\_tf\_damp} shows that in general every transfer function is made up of **additive combinations of single degree of freedom systems**, with each system having its DC gain determined by the appropriate eigenvector entry product divided by the square of the eigenvalue, \\(z\_{nji} z\_{nki}/\omega\_i^2\\), and with resonant frequency defined by the eigenvalue \\(\omega\_i\\).
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