Update content - 2026-10-14

content/book/hatch00_vibrat_matlab_ansys.md

@@ -622,19 +622,19 @@ And the damping matrix expressed in the principal coordinates is:
 We note:
 
 \begin{equation}
-  c\_{pi} = a + b \omega\_i^2 = 2 \xi\_i \omega\_i
+  c\_{pi} = a + b \omega\_i^2 = 2 \zeta\_i \omega\_i
 \end{equation}
 
-where \\(\xi\_i\\) is the percentage of critical damping for the i'th mode:
+where \\(\zeta\_i\\) is the percentage of critical damping for the i'th mode:
 
 \begin{equation}
-  \xi\_i = \frac{c\_i}{2 \sqrt{k\_{pi} m\_{pi}}} = \frac{c\_i}{2 m\_{pi} \sqrt{\omega\_i^2}} = \frac{a + b \omega\_i^2}{2 \omega\_i}
+  \zeta\_i = \frac{c\_i}{2 \sqrt{k\_{pi} m\_{pi}}} = \frac{c\_i}{2 m\_{pi} \sqrt{\omega\_i^2}} = \frac{a + b \omega\_i^2}{2 \omega\_i}
 \end{equation}
 
 And the equation in principal coordinates becomes:
 
 \begin{equation}
-  \ddot{z}\_{pi} + 2 \xi\_i \omega\_i \dot{z}\_{pi} + \omega\_i^2 z\_{pi} = F\_{pi}
+  \ddot{z}\_{pi} + 2 \zeta\_i \omega\_i \dot{z}\_{pi} + \omega\_i^2 z\_{pi} = F\_{pi}
 \end{equation}
 
 This type of damping is known as **proportional damping**, where the damping for each mode is proportional to the critical damping for that mode.
@@ -642,10 +642,10 @@ This type of damping is known as **proportional damping**, where the damping for
 
 #### Simple Proportional Damping {#simple-proportional-damping}
 
-Viscous damping in each mode is taken to be an arbitrary percentage \\(\xi\\) of the critical damping \\(c\_{cr}\\):
+Viscous damping in each mode is taken to be an arbitrary percentage \\(\zeta\\) of the critical damping \\(c\_{cr}\\):
 
 \begin{equation}
-  c = \frac{\xi}{2 \sqrt{k m}} = \xi \cdot c\_{cr}
+  c = \frac{\zeta}{2 \sqrt{k m}} = \zeta \cdot c\_{cr}
 \end{equation}
 
 
@@ -654,19 +654,19 @@ Viscous damping in each mode is taken to be an arbitrary percentage \\(\xi\\) of
 Recognizing that the higher modes of vibration damp out quickly, "relative" damping yields damping in proportion to frequencies in normal modes:
 
 \begin{equation}
-  \xi\_i = \frac{a + b\omega\_i^2}{2 \omega\_i}
+  \zeta\_i = \frac{a + b\omega\_i^2}{2 \omega\_i}
 \end{equation}
 
-If a value of \\(\xi\_1\\) for the first mode, is assumed:
+If a value of \\(\zeta\_1\\) for the first mode, is assumed:
 
 \begin{equation}
-  b = \frac{2 \xi\_1}{\omega\_1}
+  b = \frac{2 \zeta\_1}{\omega\_1}
 \end{equation}
 
 The value of the damping for any other mode is:
 
 \begin{equation}
-  \xi\_i = \xi\_1 \frac{\omega\_i}{\omega\_1}
+  \zeta\_i = \zeta\_1 \frac{\omega\_i}{\omega\_1}
 \end{equation}
 
 
@@ -861,7 +861,7 @@ It can be shown that for a general system with \\(m\\) undamped modes:
 If modes have some damping:
 
 \begin{equation}
-  \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}
+  \frac{z\_j}{F\_k} = \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + 2 \zeta\_i \omega\_i s + \omega\_i^2} \label{eq:general\_add\_tf\_damp}
 \end{equation}
 
 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\\).
@@ -923,13 +923,13 @@ The DC gain contributions of the eliminated modes are not included in the overal
 In order to eliminate this error, the Matlab function `modred` is introduced.
 Using `modred` is analogous to using Guyan reduction to reduce some less important degrees of freedom.
 
-We will discuss in this section two methods of sorting, one which is applicable for models with the same value of damping for all modes \\(\xi\_i = \xi\\) ("uniform" damping), and another which is applicable for models with different damping values for each mode ("non-uniform" damping).
+We will discuss in this section two methods of sorting, one which is applicable for models with the same value of damping for all modes \\(\zeta\_i = \zeta\\) ("uniform" damping), and another which is applicable for models with different damping values for each mode ("non-uniform" damping).
 
 The general equation for the overall transfer function of undamped and damped systems are:
 
 \begin{align}
   \frac{z\_j}{F\_k} &= \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + \omega\_i^2} \\\\
-  \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}
+  \frac{z\_j}{F\_k} &= \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + 2 \zeta\_i \omega\_i s + \omega\_i^2}
 \end{align}
 
 The **DC gain** of the i'th mode can be obtained by substituting \\(s = j\omega = 0\\):
@@ -943,12 +943,12 @@ where \\(z\_{nji} z\_{nki}\\) is the product of the j'th (output) row and k'th (
 At resonance, the **peak gain** amplitude of each mode is given by substituting \\(s = j \omega\_i\\):
 
 \begin{equation}
-  \text{Peak gain}\_i = \frac{z\_{ji}}{F\_{ki}} = \frac{-j}{2 \xi\_i} \frac{z\_{nji} z\_{nki}}{\omega\_i^2}
+  \text{Peak gain}\_i = \frac{z\_{ji}}{F\_{ki}} = \frac{-j}{2 \zeta\_i} \frac{z\_{nji} z\_{nki}}{\omega\_i^2}
 \end{equation}
 
-And we see that the peak gain for a mode is the DC gain of the same mode divided by \\(2 \xi\_i\\).
+And we see that the peak gain for a mode is the DC gain of the same mode divided by \\(2 \zeta\_i\\).
 
-If the same value of \\(\xi\\) is used for all modes, then all the DC gain terms are divided by the same \\(2 \xi\\) terms and the relative amplitude of the DC gains and peak gains are the same, so there is no difference between sorting a uniform damping model using DC gain or peak gain.
+If the same value of \\(\zeta\\) is used for all modes, then all the DC gain terms are divided by the same \\(2 \zeta\\) terms and the relative amplitude of the DC gains and peak gains are the same, so there is no difference between sorting a uniform damping model using DC gain or peak gain.
 
 However, if the modes have different damping the relationship between the DC gain and peak gain for all the modes is not a constant value and peak gain must be used to rank modes for importance.