Change transpose symbol

modal-analysis.org

@@ -933,7 +933,7 @@ The acrshort:mif is applied to the $n\times p$ acrshort:frf matrix where $n$ is
 
 The complex modal indication function is defined in equation eqref:eq:modal_cmif where the diagonal matrix $\Sigma$ is obtained from a acrlong:svd of the acrshort:frf matrix as shown in equation eqref:eq:modal_svd.
 \begin{equation} \label{eq:modal_cmif}
-  [CMIF(\omega)]_{p\times p} = [\Sigma(\omega)]_{p\times n}^T [\Sigma(\omega)]_{n\times p}
+  [CMIF(\omega)]_{p\times p} = [\Sigma(\omega)]_{p\times n}^{\intercal} [\Sigma(\omega)]_{n\times p}
 \end{equation}
 
 \begin{equation} \label{eq:modal_svd}
@@ -1066,7 +1066,7 @@ The modal parameter extraction is made using a proprietary software[fn:modal_4].
 For each mode $r$ (from $1$ to the number of considered modes $m=16$), it outputs the frequency $\omega_r$, the damping ratio $\xi_r$, the eigenvectors $\{\phi_{r}\}$ (vector of complex numbers with a size equal to the number of measured acrshort:dof $n=69$, see equation eqref:eq:modal_eigenvector) and a scaling factor $a_r$.
 
 \begin{equation}\label{eq:modal_eigenvector}
-\{\phi_i\} = \begin{Bmatrix} \phi_{i, 1_x} & \phi_{i, 1_y} & \phi_{i, 1_z} & \phi_{i, 2_x} & \dots & \phi_{i, 23_z} \end{Bmatrix}^T
+\{\phi_i\} = \begin{Bmatrix} \phi_{i, 1_x} & \phi_{i, 1_y} & \phi_{i, 1_z} & \phi_{i, 2_x} & \dots & \phi_{i, 23_z} \end{Bmatrix}^{\intercal}
 \end{equation}
 
 The eigenvalues $s_r$ and $s_r^*$ can then be computed from equation eqref:eq:modal_eigenvalues.
@@ -1093,7 +1093,7 @@ In order to synthesize the full acrshort:frf matrix, the eigenvectors $\phi_r$ a
 The full acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ can be obtained using eqref:eq:modal_synthesized_frf.
 
 \begin{equation}\label{eq:modal_synthesized_frf}
-  [\mathbf{H}_{\text{syn}}(\omega)]_{n\times n} = [\Phi]_{n\times2m} [\mathbf{H}_{\text{mod}}(\omega)]_{2m\times2m} [\Phi]_{2m\times n}^T
+  [\mathbf{H}_{\text{syn}}(\omega)]_{n\times n} = [\Phi]_{n\times2m} [\mathbf{H}_{\text{mod}}(\omega)]_{2m\times2m} [\Phi]_{2m\times n}^{\intercal}
 \end{equation}
 
 With $\mathbf{H}_{\text{mod}}(\omega)$ a diagonal matrix representing the response of the different modes eqref:eq:modal_modal_resp.

modal-analysis.tex

@@ -1,4 +1,4 @@
-% Created 2025-03-25 Tue 21:57
+% Created 2025-04-07 Mon 14:25
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -24,7 +24,7 @@
  pdftitle={Micro-Station - Modal Analysis},
  pdfkeywords={},
  pdfsubject={},
- pdfcreator={Emacs 29.4 (Org mode 9.6)}, 
+ pdfcreator={Emacs 30.1 (Org mode 9.7.26)}, 
  pdflang={English}}
 \usepackage{biblatex}
 
@@ -34,7 +34,6 @@
 \tableofcontents
 
 \clearpage
-
 To further improve the accuracy of the performance predictions, a model that better represents the micro-station dynamics is required.
 A multi-body model consisting of several rigid bodies connected by kinematic constraints (i.e. joints), springs and damper elements is a good candidate to model the micro-station.
 
@@ -61,8 +60,6 @@ The solid body assumption is then verified, validating the use of the multi-body
 
 Finally, the modal analysis is performed in Section \ref{sec:modal_analysis}.
 This shows how complex the micro-station dynamics is, and the necessity of having a model representing its complex dynamics.
-
-
 \chapter{Measurement Setup}
 \label{sec:modal_meas_setup}
 In order to perform an experimental modal analysis, a suitable measurement setup is essential.
@@ -104,7 +101,6 @@ Tests were conducted to determine the most suitable hammer tip (ranging from a m
 The softer tip was found to give best results as it injects more energy in the low-frequency range where the coherence was low, such that the overall coherence was improved.
 
 Finally, an \emph{acquisition system}\footnote{OROS OR36. 24bits signal-delta ADC.} (figure \ref{fig:modal_oros}) is used to acquire the injected force and response accelerations in a synchronized manner and with sufficiently low noise.
-
 \section{Structure Preparation and Test Planing}
 \label{ssec:modal_test_preparation}
 
@@ -124,7 +120,6 @@ This can be done either by measuring the response \(X_{j}\) at a fixed \acrshort
 
 It is however not advised to measure only one row or one column, as one or more modes may be missed by an unfortunate choice of force or acceleration measurement location (for instance if the force is applied at a vibration node of a particular mode).
 In this modal analysis, it is chosen to measure the response of the structure at all considered \acrshort{dof}, and to excite the structure at one location in three directions in order to have some redundancy, and to ensure that all modes are properly identified.
-
 \section{Location of the Accelerometers}
 \label{ssec:modal_accelerometers}
 
@@ -194,7 +189,6 @@ However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured \acrs
 \end{subfigure}
 \caption{\label{fig:modal_accelerometer_pictures}Accelerometers fixed on the micro-station stages}
 \end{figure}
-
 \section{Hammer Impacts}
 \label{ssec:modal_hammer_impacts}
 
@@ -224,7 +218,6 @@ The impacts were performed in three directions, as shown in figures \ref{fig:mod
 \end{subfigure}
 \caption{\label{fig:modal_hammer_impacts}The three hammer impacts used for the modal analysis}
 \end{figure}
-
 \section{Force and Response signals}
 \label{ssec:modal_measured_signals}
 
@@ -272,7 +265,6 @@ Good coherence is obtained from \(20\,\text{Hz}\) to \(200\,\text{Hz}\) which co
 \end{subfigure}
 \caption{\label{fig:modal_frf_coh_acc_force}Computed frequency response function from the applied force \(F_{z}\) to the measured response \(X_{1,x}\) (\subref{fig:modal_frf_acc_force}) as well as computed coherence (\subref{fig:modal_coh_acc_force})}
 \end{figure}
-
 \chapter{Frequency Analysis}
 \label{sec:modal_frf_processing}
 After all measurements are conducted, a \(n \times p \times q\) \acrlongpl{frf} matrix can be computed with:
@@ -390,7 +382,6 @@ Using \eqref{eq:modal_cart_to_acc}, the frequency response matrix \(\mathbf{H}_\
   \frac{D_{6,R_z}}{F_x}(\omega_i) & \frac{D_{6,R_z}}{F_y}(\omega_i) & \frac{D_{6,R_z}}{F_z}(\omega_i)
 \end{bmatrix}
 \end{equation}
-
 \section{Verification of solid body assumption}
 \label{ssec:modal_solid_body_assumption}
 
@@ -408,7 +399,6 @@ This also validates the reduction in the number of degrees of freedom from 69 (2
 \includegraphics[scale=1]{figs/modal_comp_acc_solid_body_frf.png}
 \caption{\label{fig:modal_comp_acc_solid_body_frf}Comparison of the original accelerometer responses and the reconstructed responses from the solid body response. Accelerometers 1 to 4 corresponding to the micro-hexapod are shown. Input is a hammer force applied on the micro-hexapod in the \(x\) direction}
 \end{figure}
-
 \chapter{Modal Analysis}
 \label{sec:modal_analysis}
 The goal here is to extract the modal parameters describing the modes of the micro station being studied, namely, the natural frequencies and the modal damping (i.e. the eigenvalues) as well as the mode shapes (.i.e. the eigenvectors).
@@ -427,7 +417,7 @@ The \acrshort{mif} is applied to the \(n\times p\) \acrshort{frf} matrix where \
 
 The complex modal indication function is defined in equation \eqref{eq:modal_cmif} where the diagonal matrix \(\Sigma\) is obtained from a \acrlong{svd} of the \acrshort{frf} matrix as shown in equation \eqref{eq:modal_svd}.
 \begin{equation} \label{eq:modal_cmif}
-  [CMIF(\omega)]_{p\times p} = [\Sigma(\omega)]_{p\times n}^T [\Sigma(\omega)]_{n\times p}
+  [CMIF(\omega)]_{p\times p} = [\Sigma(\omega)]_{p\times n}^{\intercal} [\Sigma(\omega)]_{n\times p}
 \end{equation}
 
 \begin{equation} \label{eq:modal_svd}
@@ -441,7 +431,6 @@ The obtained \acrshort{mif} is shown on Figure \ref{fig:modal_indication_functio
 A total of 16 modes were found between 0 and \(200\,\text{Hz}\).
 The obtained natural frequencies and associated modal damping are summarized in Table \ref{tab:modal_obtained_modes_freqs_damps}.
 
-\begin{figure}
 \begin{minipage}[b]{0.70\linewidth}
 \begin{center}
 \includegraphics[scale=1,scale=1]{figs/modal_indication_function.png}
@@ -477,8 +466,6 @@ Mode & Frequency & Damping\\
 \captionof{table}{\label{tab:modal_obtained_modes_freqs_damps}Identified modes}
 \end{scriptsize}
 \end{minipage}
-\end{figure}
-
 \section{Modal parameter extraction}
 \label{ssec:modal_parameter_extraction}
 
@@ -490,7 +477,7 @@ It takes into account the fact that the properties of all individual curves are
 
 From the obtained modal parameters, the mode shapes are computed and can be displayed in the form of animations (three mode shapes are shown in Figure \ref{fig:modal_mode_animations}).
 
-\begin{figure}
+\begin{figure}[hbtp]
 \begin{subfigure}{\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=\linewidth]{figs/modal_mode1_animation.jpg}
@@ -528,7 +515,7 @@ The modal parameter extraction is made using a proprietary software\footnote{NVG
 For each mode \(r\) (from \(1\) to the number of considered modes \(m=16\)), it outputs the frequency \(\omega_r\), the damping ratio \(\xi_r\), the eigenvectors \(\{\phi_{r}\}\) (vector of complex numbers with a size equal to the number of measured \acrshort{dof} \(n=69\), see equation \eqref{eq:modal_eigenvector}) and a scaling factor \(a_r\).
 
 \begin{equation}\label{eq:modal_eigenvector}
-\{\phi_i\} = \begin{Bmatrix} \phi_{i, 1_x} & \phi_{i, 1_y} & \phi_{i, 1_z} & \phi_{i, 2_x} & \dots & \phi_{i, 23_z} \end{Bmatrix}^T
+\{\phi_i\} = \begin{Bmatrix} \phi_{i, 1_x} & \phi_{i, 1_y} & \phi_{i, 1_z} & \phi_{i, 2_x} & \dots & \phi_{i, 23_z} \end{Bmatrix}^{\intercal}
 \end{equation}
 
 The eigenvalues \(s_r\) and \(s_r^*\) can then be computed from equation \eqref{eq:modal_eigenvalues}.
@@ -536,7 +523,6 @@ The eigenvalues \(s_r\) and \(s_r^*\) can then be computed from equation \eqref{
 \begin{equation}\label{eq:modal_eigenvalues}
   s_r = \omega_r (-\xi_r + i \sqrt{1 - \xi_r^2}), \quad s_r^* = \omega_r (-\xi_r - i \sqrt{1 - \xi_r^2})
 \end{equation}
-
 \section{Verification of the modal model validity}
 \label{ssec:modal_model_validity}
 
@@ -555,7 +541,7 @@ In order to synthesize the full \acrshort{frf} matrix, the eigenvectors \(\phi_r
 The full \acrshort{frf} matrix \(\mathbf{H}_{\text{syn}}\) can be obtained using \eqref{eq:modal_synthesized_frf}.
 
 \begin{equation}\label{eq:modal_synthesized_frf}
-  [\mathbf{H}_{\text{syn}}(\omega)]_{n\times n} = [\Phi]_{n\times2m} [\mathbf{H}_{\text{mod}}(\omega)]_{2m\times2m} [\Phi]_{2m\times n}^T
+  [\mathbf{H}_{\text{syn}}(\omega)]_{n\times n} = [\Phi]_{n\times2m} [\mathbf{H}_{\text{mod}}(\omega)]_{2m\times2m} [\Phi]_{2m\times n}^{\intercal}
 \end{equation}
 
 With \(\mathbf{H}_{\text{mod}}(\omega)\) a diagonal matrix representing the response of the different modes \eqref{eq:modal_modal_resp}.
@@ -589,7 +575,6 @@ This can be seen in Figure \ref{fig:modal_comp_acc_frf_modal_3} that shows the f
 \end{subfigure}
 \caption{\label{fig:modal_comp_acc_frf_modal}Comparison of the measured FRF with the FRF synthesized from the modal model.}
 \end{figure}
-
 \chapter{Conclusion}
 \label{sec:modal_conclusion}
 
@@ -603,8 +588,6 @@ This confirms that a multi-body model can be used to properly model the micro-st
 
 Although a lot of effort was put into this experimental modal analysis of the micro-station, it was difficult to obtain an accurate modal model.
 However, the measurements are useful for tuning the parameters of the micro-station multi-body model.
-
 \printbibliography[heading=bibintoc,title={Bibliography}]
-
 \printglossaries
 \end{document}

preamble.tex

@@ -9,6 +9,13 @@
 
 \usepackage[stylemods=longextra]{glossaries-extra}
 
+\usepackage{amssymb}
+\usepackage{amsmath}
+\usepackage{amsthm}
+\usepackage{bm}
+\usepackage{dsfont}
+\usepackage{empheq} % for drawing boxes around equations
+
 \setabbreviationstyle[acronym]{long-short}
 \setglossarystyle{long-name-desc}