Update content - 2026-10-14

content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md

@@ -345,21 +345,21 @@ If we left multiply the equation by \\(\Phi^T\\) and we use the orthogonalily re
 \\[ diag(\mu\_i) \ddot{z} + \Phi^T C \Phi + diag(\mu\_i \omega\_i^2) z = \Phi^T f \\]
 
 If \\(\Phi^T C \Phi\\) is diagonal, the **damping is said classical or normal**. In this case:
-\\[ \Phi^T C \Phi = diag(2 \xi\_i \mu\_i \omega\_i) \\]
+\\[ \Phi^T C \Phi = diag(2 \zeta\_i \mu\_i \omega\_i) \\]
 
-One can verify that the Rayleigh damping \eqref{eq:rayleigh\_damping} complies with this condition with modal damping ratios \\(\xi\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\).
+One can verify that the Rayleigh damping \eqref{eq:rayleigh\_damping} complies with this condition with modal damping ratios \\(\zeta\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\).
 
 And we obtain decoupled modal equations \eqref{eq:modal\_eom}.
 
 <div class="cbox">
 
 \begin{equation}\label{eq:modal\_eom}
-  \ddot{z} + 2 \xi \Omega \dot{z} + \Omega^2 z = z^{-1} \Phi^T f
+  \ddot{z} + 2 \zeta \Omega \dot{z} + \Omega^2 z = z^{-1} \Phi^T f
 \end{equation}
 
 with:
 
--   \\(\xi = diag(\xi\_i)\\)
+-   \\(\zeta = diag(\zeta\_i)\\)
 -   \\(\Omega = diag(\omega\_i)\\)
 -   \\(\mu = diag(\mu\_i)\\)
 
@@ -373,12 +373,12 @@ Typical values of the modal damping ratio are summarized on [Table 2](#table--ta
   Typical Damping ratio
 </div>
 
-| **Damping Ratio**              | **Application**         |
-|--------------------------------|-------------------------|
-| \\(\xi \simeq 0.001 - 0.005\\) | Space structures        |
-| \\(\xi \simeq 0.01 - 0.02\\)   | Mechanical engineering  |
-| \\(\xi \simeq 0.05\\)          | Civil engineering       |
-| \\(\xi \simeq 0.2\\)           | When ground is involved |
+| **Damping Ratio**                | **Application**         |
+|----------------------------------|-------------------------|
+| \\(\zeta \simeq 0.001 - 0.005\\) | Space structures        |
+| \\(\zeta \simeq 0.01 - 0.02\\)   | Mechanical engineering  |
+| \\(\zeta \simeq 0.05\\)          | Civil engineering       |
+| \\(\zeta \simeq 0.2\\)           | When ground is involved |
 
 The assumption of classical damping is often justified for light damping, but it is questionable when the damping is large.
 
@@ -407,7 +407,7 @@ With:
 -   \\(D\_i(\omega)\\) is the dynamic amplification factor of mode \\(i\\) given by
 
 \begin{equation}\label{eq:dynamic\_amplification}
-  D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i}
+  D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \zeta\_i \omega/\omega\_i}
 \end{equation}
 
 <a id="figure--fig:neglected-modes"></a>
@@ -489,7 +489,7 @@ By looking at [Figure 7](#figure--fig:collocated-control-frf), we see that negle
 The open-loop transfer function of a lighly damped structure with a collocated actuator/sensor pair can be written:
 
 \begin{equation}\label{eq:frf\_lighly\_damped\_collocated}
-  G(s) = G\_0 \frac{\Pi\_i(s^2/z\_i^2 + 2 \xi\_i s/z\_i + 1)}{\Pi\_j(s^2/\omega\_j^2 + 2 \xi\_j s /\omega\_j + 1)}
+  G(s) = G\_0 \frac{\Pi\_i(s^2/z\_i^2 + 2 \zeta\_i s/z\_i + 1)}{\Pi\_j(s^2/\omega\_j^2 + 2 \zeta\_j s /\omega\_j + 1)}
 \end{equation}
 
 The corresponding Bode plot is represented in [Figure 9](#figure--fig:alternating-p-z). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
@@ -576,14 +576,14 @@ The transfer function between the total force and the current \\(i\\) applied to
 <div class="cbox">
 
 \begin{equation}\label{eq:proof\_mass\_eq}
-  \frac{F}{i} = \frac{-s^2 T}{s^2 + 2\xi\_p \omega\_p s + \omega\_p^2}
+  \frac{F}{i} = \frac{-s^2 T}{s^2 + 2\zeta\_p \omega\_p s + \omega\_p^2}
 \end{equation}
 
 with:
 
 -   \\(T\\) is the transducer constant
 -   \\(\omega\_p = \frac{k}{m}\\) is the natural frequency of the spring-mass system
--   \\(\xi\_p\\) is the damping ratio
+-   \\(\zeta\_p\\) is the damping ratio
 
 </div>
 
@@ -609,7 +609,7 @@ If \\(x\_0\\) is the displacement of the support and if the voice coil is open (
 By using the two equations, we obtain:
 
 \begin{equation}\label{eq:geophone}
-  \frac{e}{\dot{x\_0}} = \frac{-s^2 T}{s^2 + 2\xi\_p\omega\_p s + \omega\_p^2}
+  \frac{e}{\dot{x\_0}} = \frac{-s^2 T}{s^2 + 2\zeta\_p\omega\_p s + \omega\_p^2}
 \end{equation}
 
 Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\).