From 2c28c25d952dd1bd790c645e92ca6a1900243780 Mon Sep 17 00:00:00 2001
From: "W. Trevor King" <wking@tremily.us>
Date: Sun, 5 May 2013 14:36:33 -0400
Subject: [PATCH] calibcant/discussion.tex: Cleanup peak-frequency discussion

Add $f_\text{max}$ for clarity, and link to this equation from the
vibration figure.
---
 src/calibcant/discussion.tex | 20 ++++++++++++--------
 src/calibcant/procedure.tex  | 10 ++++++----
 2 files changed, 18 insertions(+), 12 deletions(-)

diff --git a/src/calibcant/discussion.tex b/src/calibcant/discussion.tex
index 9173778..62d60b3 100644
--- a/src/calibcant/discussion.tex
+++ b/src/calibcant/discussion.tex
@@ -49,17 +49,21 @@ $\PSD_f$ in \cref{eq:model-psd-df}, it's useful to also calculate the
 frequency of the resonant peak.
 \begin{align}
   0 &= \deriv{f}{\PSD_f}
-    = \frac{2G_{1f}f}{\p({(f_0^2-f^2)^2 + \beta_f^2 f^2})^2}
-      \p({2(f_0^2-f^2) - \beta_f^2})
-    = 2(f_0^2-f^2) - \beta_f^2 \\
-  f^2 &= f_0^2 - \frac{\beta_f^2}{2} \\
-  f &= \sqrt{f_0^2 - \frac{\beta_f^2}{2}} \;,
+    = \frac{2G_{1f}f_\text{max}}
+           {\p({(f_0^2-f_\text{max}^2)^2 + \beta_f^2 f_\text{max}^2})^2}
+      \p({2(f_0^2-f_\text{max}^2) - \beta_f^2})
+    = 2(f_0^2-f_\text{max}^2) - \beta_f^2 \\
+  f_\text{max}^2 &= f_0^2 - \frac{\beta_f^2}{2} \\
+  f_\text{max} &= \sqrt{f_0^2 - \frac{\beta_f^2}{2}} \;,
   \label{eq:peak-frequency}
 \end{align}
 where we used $f\ne0$ during the large simplifying multiplication.  We
-see that the peak frequency is actually shifed from $f_0$ depending on
-the damping term $\beta_f$.  For overdamped cantilevers with large
-values of $\beta$, the peak frequency will not have a real solution.
+see that the peak frequency is shifted from $f_0$ depending on the
+damping term $\beta_f$.  For overdamped cantilevers with large values
+of $\beta$, the peak frequency will not have a real solution.%
+%
+\nomenclature{$f_\text{max}$}{The frequency of the peak power in
+  $\PSD_f$ (\cref{eq:eq:peak-frequency}).}
 
 \subsection{Propogation of errors}
 \label{sec:calibcant:discussion:errors}
diff --git a/src/calibcant/procedure.tex b/src/calibcant/procedure.tex
index cd012ec..e09a7fe 100644
--- a/src/calibcant/procedure.tex
+++ b/src/calibcant/procedure.tex
@@ -133,10 +133,12 @@ vibrations are configurable (with \hFconfig,
       panel shows the $\PSD_f(V_p,f)$ with a fit following
       \cref{eq:psd-Vp-offset}.  The constant offset $P_{0f}$, drawn as
       the horizontal line in the third panel, accounts for white noise
-      in the measurement circuit.  Only data in the blue region was
-      used when computing the best fit.  This is the first vibration
-      from the 2013-02-07T08-20-46 calibration, yielding a fitted
-      variance $\avg{V_p(t)^2}=96.90\pm0.99\U{mV$^2$}$.
+      in the measurement circuit.  The vertical line marks the peak
+      frequency $f_\text{max}$ (\cref{eq:peak-frequency}).  Only data
+      in the blue region was used when computing the best fit.  This
+      is the first vibration from the 2013-02-07T08-20-46 calibration,
+      yielding a fitted variance
+      $\avg{V_p(t)^2}=96.90\pm0.99\U{mV$^2$}$.
       \label{fig:calibcant:vibration}}
   \end{center}
 \end{figure}
-- 
2.26.2