Remove comment about Matlab code

figs/inkscape/convert_svg.sh

@@ -0,0 +1,18 @@
+#!/bin/bash
+
+# Directory containing SVG files
+INPUT_DIR="."
+
+# Loop through all SVG files in the directory
+for svg_file in "$INPUT_DIR"/*.svg; do
+    # Check if there are SVG files in the directory
+    if [ -f "$svg_file" ]; then
+        # Output PDF file name
+        pdf_file="../${svg_file%.svg}.pdf"
+        png_file="../${svg_file%.svg}"
+        
+        # Convert SVG to PDF using Inkscape
+        inkscape "$svg_file" --export-filename="$pdf_file" && \
+            pdftocairo -png -singlefile -cropbox "$pdf_file" "$png_file"
+    fi
+done

matlab/modal_1_meas_setup.m

@@ -10,98 +10,12 @@ addpath('./mat/'); % Path for data
 %% Colors for the figures
 colors = colororder;
 
-% Location of the Accelerometers
-% <<ssec:modal_accelerometers>>
-
-% The location of the accelerometers fixed to the micro-station is essential as it defines where the dynamics is measured.
-% A total of 23 accelerometers are fixed to the six key stages of the micro station: the lower and upper granites, the translation stage, the tilt stage, the spindle and the micro hexapod.
-% The position of the accelerometers are visually shown on a CAD model in Figure ref:fig:modal_location_accelerometers and their precise locations with respect to a frame located at the point of interest are summarized in Table ref:tab:modal_position_accelerometers.
-% Pictures of the accelerometers fixed to the translation stage and to the micro-hexapod are shown in Figure ref:fig:modal_accelerometer_pictures.
-
-% As all key stages of the micro-station are foreseen to behave as solid bodies, only 6 acrshort:dof can be considered per solid body.
-% However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured acrshort:dof) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section ref:ssec:modal_solid_body_assumption).
-
-% #+attr_latex: :options [t]{0.60\linewidth}
-% #+begin_minipage
-% #+name: fig:modal_location_accelerometers
-% #+caption: Position of the accelerometers
-% #+attr_latex: :width 0.99\linewidth :float nil
-% [[file:figs/modal_location_accelerometers.png]]
-% #+end_minipage
-% \hfill
-% #+attr_latex: :options [b]{0.38\linewidth}
-% #+begin_minipage
-% #+begin_scriptsize
-% #+name: tab:modal_position_accelerometers
-% #+caption: Positions in mm
-% #+attr_latex: :environment tabularx :width \linewidth :placement [b] :align Xccc
-% #+attr_latex: :booktabs t :float nil :center nil
-% #+RESULTS:
-% |              |  $x$ |  $y$ |  $z$ |
-% |--------------+------+------+------|
-% | Low. Granite | -730 | -526 | -951 |
-% | Low. Granite | -735 |  814 | -951 |
-% | Low. Granite |  875 |  799 | -951 |
-% | Low. Granite |  865 | -506 | -951 |
-% | Up. Granite  | -320 | -446 | -786 |
-% | Up. Granite  | -480 |  534 | -786 |
-% | Up. Granite  |  450 |  534 | -786 |
-% | Up. Granite  |  295 | -481 | -786 |
-% | Translation  | -475 | -414 | -427 |
-% | Translation  | -465 |  407 | -427 |
-% | Translation  |  475 |  424 | -427 |
-% | Translation  |  475 | -419 | -427 |
-% | Tilt         | -385 | -300 | -417 |
-% | Tilt         | -420 |  280 | -417 |
-% | Tilt         |  420 |  280 | -417 |
-% | Tilt         |  380 | -300 | -417 |
-% | Spindle      | -155 |  -90 | -594 |
-% | Spindle      |    0 |  180 | -594 |
-% | Spindle      |  155 |  -90 | -594 |
-% | Hexapod      |  -64 |  -64 | -270 |
-% | Hexapod      |  -64 |   64 | -270 |
-% | Hexapod      |   64 |   64 | -270 |
-% | Hexapod      |   64 |  -64 | -270 |
-% #+end_scriptsize
-% #+end_minipage
-
-% #+name: fig:modal_accelerometer_pictures
-% #+caption: Accelerometers fixed on the micro-station stages
-% #+attr_latex: :options [htbp]
-% #+begin_figure
-% #+attr_latex: :caption \subcaption{\label{fig:modal_accelerometers_ty} $T_y$ stage}
-% #+attr_latex: :options {0.49\textwidth}
-% #+begin_subfigure
-% #+attr_latex: :height 6cm
-% [[file:figs/modal_accelerometers_ty.jpg]]
-% #+end_subfigure
-% #+attr_latex: :caption \subcaption{\label{fig:modal_accelerometers_hexapod} Micro-Hexapod}
-% #+attr_latex: :options {0.49\textwidth}
-% #+begin_subfigure
-% #+attr_latex: :height 6cm
-% [[file:figs/modal_accelerometers_hexapod.jpg]]
-% #+end_subfigure
-% #+end_figure
-
-
 %% Load Accelerometer positions
 acc_pos = readtable('mat/acc_pos.txt', 'ReadVariableNames', false);
 acc_pos = table2array(acc_pos(:, 1:4));
 [~, i] = sort(acc_pos(:, 1));
 acc_pos = acc_pos(i, 2:4);
 
-% Force and Response signals
-% <<ssec:modal_measured_signals>>
-
-% The force sensor of the instrumented hammer and the accelerometers signals are shown in the time domain in Figure ref:fig:modal_raw_meas.
-% Sharp "impacts" can be seen for the force sensor, indicating wide frequency band excitation.
-% For the accelerometer, a much more complex signal can be observed, indicating complex dynamics.
-
-% The "normalized" acrfull:asd of the two signals are computed and shown in Figure ref:fig:modal_asd_acc_force.
-% Conclusions based on the time domain signals can be clearly seen in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer).
-% Similar results are obtained for all the measured frequency response functions.
-
-
 %% Load raw data
 meas1_raw = load('mat/meas_raw_1.mat');
 
@@ -117,12 +31,12 @@ time = linspace(0, meas1_raw.Track1_X_Resolution*length(meas1_raw.Track1), lengt
 %% Raw measurement of the Accelerometer
 figure;
 hold on;
-plot(time-22.2, meas1_raw.Track2, 'DisplayName', '$X_{j}$ [$m/s^2$]');
-plot(time-22.2, 1e-3*meas1_raw.Track1, 'DisplayName', '$F_{k}$ [kN]');
+plot(time-22.2, meas1_raw.Track2, 'DisplayName', '$X_{1,x}$ [$m/s^2$]');
+plot(time-22.2, 1e-3*meas1_raw.Track1, 'DisplayName', '$F_{z}$ [kN]');
 hold off;
 xlabel('Time [s]');
 ylabel('Amplitude');
-xlim([0, 0.2]);
+xlim([0, 0.2])
 ylim([-2, 2]);
 legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
 
@@ -138,8 +52,8 @@ Noverlap = floor(Nfft/2); % Overlap for frequency analysis
 %% Normalized Amplitude Spectral Density of the measured force and acceleration
 figure;
 hold on;
-plot(f, sqrt(pxx_acc./max(pxx_acc(f<200))), 'DisplayName', '$X_{j}$');
-plot(f, sqrt(pxx_force./max(pxx_force(f<200))), 'DisplayName', '$F_{k}$');
+plot(f, sqrt(pxx_acc./max(pxx_acc(f<200))), 'DisplayName', '$\Gamma_{X_{1,x}}$');
+plot(f, sqrt(pxx_force./max(pxx_force(f<200))), 'DisplayName', '$\Gamma_{F_{z}}$');
 hold off;
 set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
 xlabel('Frequency [Hz]'); ylabel('Normalized Spectral Density');
@@ -148,31 +62,6 @@ xticks([0:20:200]);
 ylim([0, 1])
 legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
 
-
-
-% #+name: fig:modal_raw_meas_asd
-% #+caption: Raw measurement of the acceleromter (blue) and of the force sensor at the Hammer tip (red) (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Density of the two signals (normalized) (\subref{fig:modal_asd_acc_force})
-% #+attr_latex: :options [htbp]
-% #+begin_figure
-% #+attr_latex: :caption \subcaption{\label{fig:modal_raw_meas}Time domain signals}
-% #+attr_latex: :options {0.49\textwidth}
-% #+begin_subfigure
-% #+attr_latex: :width 0.95\linewidth
-% [[file:figs/modal_raw_meas.png]]
-% #+end_subfigure
-% #+attr_latex: :caption \subcaption{\label{fig:modal_asd_acc_force}Amplitude Spectral Density (normalized)}
-% #+attr_latex: :options {0.49\textwidth}
-% #+begin_subfigure
-% #+attr_latex: :width 0.95\linewidth
-% [[file:figs/modal_asd_acc_force.png]]
-% #+end_subfigure
-% #+end_figure
-
-% The frequency response function $H_{jk}$ from the applied force $F_{k}$ to the measured acceleration $X_j$ is then computed and shown Figure ref:fig:modal_frf_acc_force.
-% The quality of the obtained data can be estimated using the /coherence/ function, which is shown in Figure ref:fig:modal_coh_acc_force.
-% Good coherence is obtained from $20\,\text{Hz}$ to $200\,\text{Hz}$ which corresponds to the frequency range of interest.
-
-
 %% Compute the transfer function and Coherence
 [G1,   f] = tfestimate(meas1_raw.Track1, meas1_raw.Track2, win, Noverlap, Nfft, Fs);
 [coh1, ~] = mscohere(  meas1_raw.Track1, meas1_raw.Track2, win, Noverlap, Nfft, Fs);

matlab/modal_2_frf_processing.m

@@ -10,75 +10,6 @@ addpath('./mat/'); % Path for data
 %% Colors for the figures
 colors = colororder;
 
-
-
-% #+name: fig:modal_local_to_global_coordinates
-% #+caption: Schematic of the measured motions of a solid body
-% #+RESULTS:
-% [[file:figs/modal_local_to_global_coordinates.png]]
-
-% The motion of the rigid body of figure ref:fig:modal_local_to_global_coordinates can be described by its displacement $\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]$ and (small) rotations $[\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]$ with respect to a reference frame $\{O\}$.
-
-% The motion $\vec{\delta} p_{i}$ of a point $p_i$ can be computed from $\vec{\delta} p$ and $\bm{\delta \Omega}$ using equation eqref:eq:modal_compute_point_response, with $\bm{\delta\Omega}$ defined in equation eqref:eq:modal_rotation_matrix.
-
-% \begin{equation}\label{eq:modal_compute_point_response}
-%   \vec{\delta} p_{i} &= \vec{\delta} p + \bm{\delta \Omega} \cdot \vec{p}_{i} \\
-% \end{equation}
-
-% \begin{equation}\label{eq:modal_rotation_matrix}
-%   \bm{\delta\Omega} = \begin{bmatrix}
-%    0              & -\delta\Omega_z &  \delta\Omega_y \\
-%    \delta\Omega_z &  0              & -\delta\Omega_x \\
-%   -\delta\Omega_y &  \delta\Omega_x &  0
-% \end{bmatrix}
-% \end{equation}
-
-% Writing this in a matrix form for the four points gives eqref:eq:modal_cart_to_acc.
-
-% \begin{equation}\label{eq:modal_cart_to_acc}
-% \left[\begin{array}{c}
-%   \delta p_{1x} \\ \delta p_{1y} \\ \delta p_{1z} \\\hline \vdots \\\hline \delta p_{4x} \\ \delta p_{4y} \\ \delta p_{4z}
-% \end{array}\right] =
-% \left[\begin{array}{ccc|ccc}
-%   1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\
-%   0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\
-%   0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline
-%     & \vdots &   &    & \vdots &   \\ \hline
-%   1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\
-%   0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\
-%   0 & 0 & 1 & p_{4y} & -p_{4x} & 0
-% \end{array}\right] \left[\begin{array}{c}
-%   \delta p_x \\ \delta p_y \\ \delta p_z \\ \hline \delta\Omega_x \\ \delta\Omega_y \\ \delta\Omega_z
-% \end{array}\right]
-% \end{equation}
-
-% Provided that the four sensors are properly located, the system of equation eqref:eq:modal_cart_to_acc can be solved by matrix inversion[fn:5].
-% The motion of the solid body expressed in a chosen frame $\{O\}$ can be determined using equation eqref:eq:modal_determine_global_disp.
-% Note that this matrix inversion is equivalent to resolving a mean square problem.
-% Therefore, having more accelerometers permits to have a better approximation of the motion of the solid body.
-
-% \begin{equation}
-% \left[\begin{array}{c}
-%   \delta p_x \\ \delta p_y \\ \delta p_z \\ \hline \delta\Omega_x \\ \delta\Omega_y \\ \delta\Omega_z
-% \end{array}\right] =
-% \left[\begin{array}{ccc|ccc}
-%   1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\
-%   0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\
-%   0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline
-%     & \vdots &   &    & \vdots &   \\ \hline
-%   1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\
-%   0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\
-%   0 & 0 & 1 & p_{4y} & -p_{4x} & 0
-% \end{array}\right]^{-1} \left[\begin{array}{c}
-%   \delta p_{1x} \\ \delta p_{1y} \\ \delta p_{1z} \\\hline \vdots \\\hline \delta p_{4x} \\ \delta p_{4y} \\ \delta p_{4z}
-% \end{array}\right] \label{eq:modal_determine_global_disp}
-% \end{equation}
-
-
-% From the CAD model, the position of the center of mass of each considered solid body is computed (see Table ref:tab:modal_com_solid_bodies).
-% Then, the position of each accelerometer with respect to the center of mass of the corresponding solid body can easily be derived.
-
-
 %% Load frequency response matrix
 load('frf_matrix.mat', 'freqs', 'frf');
 
@@ -100,45 +31,12 @@ solids.hexa = [1, 2, 3, 4]; % Hexapod
 % Names of the solid bodies
 solid_names = fields(solids);
 
-%% Save the acceleromter positions are well as the solid bodies
+%% Save the accelerometer positions are well as the solid bodies
 save('mat/geometry.mat', 'solids', 'solid_names', 'acc_pos');
 
 %% Extract the CoM of considered solid bodies
 model_com = reshape(table2array(readtable('mat/model_solidworks_com.txt', 'ReadVariableNames', false)), [3, 6]);
 
-
-
-% #+name: tab:modal_com_solid_bodies
-% #+caption: Center of mass of considered solid bodies with respect to the "point of interest"
-% #+attr_latex: :environment tabularx :width 0.6\linewidth :align lXXX
-% #+attr_latex: :center t :booktabs t
-% #+RESULTS:
-% |                   | $X$ [mm] | $Y$ [mm] | $Z$ [mm] |
-% |-------------------+----------+----------+----------|
-% | Bottom Granite    |       45 |      144 |    -1251 |
-% | Top granite       |       52 |      258 |     -778 |
-% | Translation stage |        0 |       14 |     -600 |
-% | Tilt Stage        |        0 |       -5 |     -628 |
-% | Spindle           |        0 |        0 |     -580 |
-% | Hexapod           |       -4 |        6 |     -319 |
-
-% Using eqref:eq:modal_determine_global_disp, the frequency response matrix $\mathbf{H}_\text{CoM}$ eqref:eq:modal_frf_matrix_com expressing the response at the center of mass of each solid body $D_i$ ($i$ from $1$ to $6$ for the $6$ considered solid bodies) can be computed from the initial acrshort:frf matrix $\mathbf{H}$.
-
-% \begin{equation}\label{eq:modal_frf_matrix_com}
-%   \mathbf{H}_\text{CoM}(\omega_i) = \begin{bmatrix}
-%   \frac{D_{1,T_x}}{F_x}(\omega_i) & \frac{D_{1,T_x}}{F_y}(\omega_i) & \frac{D_{1,T_x}}{F_z}(\omega_i) \\
-%   \frac{D_{1,T_y}}{F_x}(\omega_i) & \frac{D_{1,T_y}}{F_y}(\omega_i) & \frac{D_{1,T_y}}{F_z}(\omega_i) \\
-%   \frac{D_{1,T_z}}{F_x}(\omega_i) & \frac{D_{1,T_z}}{F_y}(\omega_i) & \frac{D_{1,T_z}}{F_z}(\omega_i) \\
-%   \frac{D_{1,R_x}}{F_x}(\omega_i) & \frac{D_{1,R_x}}{F_y}(\omega_i) & \frac{D_{1,R_x}}{F_z}(\omega_i) \\
-%   \frac{D_{1,R_y}}{F_x}(\omega_i) & \frac{D_{1,R_y}}{F_y}(\omega_i) & \frac{D_{1,R_y}}{F_z}(\omega_i) \\
-%   \frac{D_{1,R_z}}{F_x}(\omega_i) & \frac{D_{1,R_z}}{F_y}(\omega_i) & \frac{D_{1,R_z}}{F_z}(\omega_i) \\
-%   \frac{D_{2,T_x}}{F_x}(\omega_i) & \frac{D_{2,T_x}}{F_y}(\omega_i) & \frac{D_{2,T_x}}{F_z}(\omega_i) \\
-%   \vdots              & \vdots              & \vdots              \\
-%   \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}
-
-
 %% Frequency Response Matrix - Response expressed at the CoM of the solid bodies
 frfs_CoM = zeros(length(solid_names)*6, 3, 801);
 
@@ -167,14 +65,6 @@ end
 %% Save the computed FRF at the CoM
 save('mat/frf_com.mat', 'frfs_CoM');
 
-% Verification of solid body assumption
-% <<ssec:modal_solid_body_assumption>>
-
-% From the response of one solid body expressed by its 6 acrshortpl:dof (i.e. from $\mathbf{H}_{\text{CoM}}$), and using equation eqref:eq:modal_cart_to_acc, it is possible to compute the response of the same solid body at any considered position.
-% In particular, the response at the location of the four accelerometers can be computed and compared with the original measurements $\mathbf{H}$.
-% This is what is here done to check if solid body assumption is correct in the frequency band of interest.
-
-
 %% Compute the FRF at the accelerometer location from the CoM reponses
 frfs_A = zeros(size(frf));
 
@@ -197,15 +87,7 @@ for exc_dir = 1:3
   end
 end
 
-
-
-% The comparison is made for the 4 accelerometers fixed to the micro-hexapod (Figure ref:fig:modal_comp_acc_solid_body_frf).
-% The original frequency response functions and the ones computed from the CoM responses are well matching in the frequency range of interested.
-% Similar results are obtained for the other solid bodies, indicating that the solid body assumption is valid, and that a multi-body model can be used to represent the dynamics of the micro-station.
-% This also validates the reduction of the number of degrees of freedom from 69 (23 accelerometers with each 3 acrshort:dof) to 36 (6 solid bodies with 6 acrshort:dof).
-
-
-%% Comparaison of the original accelerometer response and reconstructed response from the solid body response
+%% Comparison of the original accelerometer response and reconstructed response from the solid body response
 exc_names = {'$F_x$', '$F_y$', '$F_z$'};
 DOFs = {'x', 'y', 'z', '\theta_x', '\theta_y', '\theta_z'};
 
@@ -223,11 +105,10 @@ for i = 1:length(accs_i)
 
   hold on;
   for dir_i = 1:3
-    plot(freqs, abs(squeeze(frf(3*(acc_i-1)+dir_i, exc_dir, :))), '-', 'DisplayName', sprintf('$a_{%i,%s}$', acc_i, DOFs{dir_i}));
+    plot(freqs, abs(squeeze(frf(3*(acc_i-1)+dir_i, exc_dir, :))), '-', 'color', [colors(dir_i,:), 0.5], 'linewidth', 2.5, 'DisplayName', sprintf('$a_{%i,%s}$ - meas', acc_i, DOFs{dir_i}));
   end
-  set(gca,'ColorOrderIndex',1)
   for dir_i = 1:3
-    plot(freqs, abs(squeeze(frfs_A(3*(acc_i-1)+dir_i, exc_dir, :))), '--', 'HandleVisibility', 'off');
+    plot(freqs, abs(squeeze(frfs_A(3*(acc_i-1)+dir_i, exc_dir, :))), '-', 'color', colors(dir_i, :), 'DisplayName', sprintf('$a_{%i,%s}$ - solid body', acc_i, DOFs{dir_i}));
   end
   hold off;
 
@@ -246,5 +127,6 @@ for i = 1:length(accs_i)
   set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'log');
   xlim([0, 200]); ylim([1e-6, 3e-2]);
   xticks([0:20:200]);
-  legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+  leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
+  leg.ItemTokenSize(1) = 15;
 end

matlab/modal_3_analysis.m

@@ -10,27 +10,6 @@ addpath('./mat/'); % Path for data
 %% Colors for the figures
 colors = colororder;
 
-% Number of modes determination
-% <<ssec:modal_number_of_modes>>
-% The acrshort:mif is here applied to the $n\times p$ acrshort:frf matrix where $n$ is a relatively large number of measurement DOFs (here $n=69$) and $p$ is the number of excitation DOFs (here $p=3$).
-
-% 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}
-% \end{equation}
-
-% \begin{equation} \label{eq:modal_svd}
-% [H(\omega)]_{n\times p} = [U(\omega)]_{n\times n} [\Sigma(\omega)]_{n\times p} [V(\omega)]_{p\times p}^H
-% \end{equation}
-
-% The acrshort:mif therefore yields to $p$ values that are also frequency dependent.
-% A peak in the acrshort:mif plot indicates the presence of a mode.
-% Repeated modes can also be detected by multiple singular values are having peaks at the same frequency.
-% The obtained acrshort:mif is shown on Figure ref:fig:modal_indication_function.
-% A total of 16 modes are 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.
-
-
 %% Load frequency response matrix
 load('frf_matrix.mat', 'freqs', 'frf');
 
@@ -46,7 +25,7 @@ end
 figure;
 hold on;
 for i = 1:size(MIF, 1)
-  plot(freqs, squeeze(MIF(i, i, :)));
+  plot(freqs, squeeze(MIF(i, i, :)), 'DisplayName', sprintf('MIF${}_%i$', i));
 end
 hold off;
 set(gca, 'Xscale', 'lin'); set(gca, 'Yscale', 'log');
@@ -54,33 +33,7 @@ xlabel('Frequency [Hz]'); ylabel('CMIF Amplitude');
 xticks([0:20:200]);
 xlim([0, 200]);
 ylim([1e-6, 2e-2]);
-
-% Verification of the modal model validity
-% <<ssec:modal_model_validity>>
-
-% In order to check the validity of the modal model, the complete $n \times n$ acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ is first synthesized from the modal parameters.
-% Then, the elements of this acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ that were already measured can be compared with the measured acrshort:frf matrix $\mathbf{H}$.
-
-% In order to synthesize the full acrshort:frf matrix, the eigenvectors $\phi_r$ are first organized in a matrix from as shown in equation eqref:eq:modal_eigvector_matrix.
-% \begin{equation}\label{eq:modal_eigvector_matrix}
-% \Phi = \begin{bmatrix}
-%   & & & & &\\
-%   \phi_1 & \dots & \phi_N & \phi_1^* & \dots & \phi_N^* \\
-%   & & & & &
-% \end{bmatrix}_{n \times 2m}
-% \end{equation}
-
-% The full acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ can be synthesize 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
-% \end{equation}
-
-% With $\mathbf{H}_{\text{mod}}(\omega)$ a diagonal matrix representing the response of the different modes eqref:eq:modal_modal_resp.
-% \begin{equation}\label{eq:modal_modal_resp}
-% \mathbf{H}_{\text{mod}}(\omega) = \text{diag}\left(\frac{1}{a_1 (j\omega - s_1)},\ \dots,\ \frac{1}{a_m (j\omega - s_m)}, \frac{1}{a_1^* (j\omega - s_1^*)},\ \dots,\ \frac{1}{a_m^* (j\omega - s_m^*)} \right)_{2m\times 2m}
-% \end{equation}
-
+ldg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
 
 %% Load modal parameters
 shapes_m = readtable('mat/mode_shapes.txt',              'ReadVariableNames', false);  % [Sign / Real / Imag]
@@ -138,19 +91,11 @@ for i = 1:length(freqs)
   Hsyn(:, :, i) = eigen_vec_ext_M*diag(1./(diag(modal_a_ext_M).*(j*2*pi*freqs(i) - diag(eigen_val_ext_M))))*eigen_vec_ext_M.';
 end
 
-%% Derivate two times to to have the acceleration response
+%% Derivate two times to have the acceleration response
 for i = 1:size(Hsyn, 1)
   Hsyn(i, :, :) = squeeze(Hsyn(i, :, :)).*(j*2*pi*freqs).^2;
 end
 
-
-
-% The comparison between the original measured frequency response functions and the synthesized ones from the modal model is done in Figure ref:fig:modal_comp_acc_frf_modal.
-% Whether the obtained match can be considered good or bad is quite arbitrary.
-% Yet, the modal model seems to be able to represent the coupling between different nodes and different direction which is quite important in a control point of view.
-% This can be seen in Figure ref:fig:modal_comp_acc_frf_modal_3 that shows the frequency response function between a force applied on node 11 (i.e. on the translation stage) in the $y$ direction to the measured acceleration at node $2$ (i.e. at the top of the micro-hexapod) in the $x$ direction.
-
-
 acc_o = 11; dir_o = 3;
 acc_i = 11; dir_i = 3;
 

modal-analysis.org

@@ -22,7 +22,7 @@
 #+BIND: org-latex-bib-compiler "biber"
 
 #+PROPERTY: header-args:matlab  :session *MATLAB*
-#+PROPERTY: header-args:matlab+ :comments org
+#+PROPERTY: header-args:matlab+ :comments no
 #+PROPERTY: header-args:matlab+ :exports none
 #+PROPERTY: header-args:matlab+ :results none
 #+PROPERTY: header-args:matlab+ :eval no-export
@@ -45,12 +45,6 @@
 #+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
 :END:
 
-#+begin_export html
-  <hr>
-  <p>This report is also available as a <a href="./modal-analysis.pdf">pdf</a>.</p>
-  <hr>
-#+end_export
-
 #+latex: \clearpage
 
 * Build                                                             :noexport:
@@ -143,12 +137,12 @@ To further improve the accuracy of the performance predictions, a model that bet
 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.
 
 Although the inertia of each solid body can easily be estimated from its geometry and material density, it is more difficult to properly estimate the stiffness and damping properties of the guiding elements connecting each solid body.
-Experimental modal analysis will be use to tune the model, and to verify that a multi-body model can accurately represent the dynamics of the micro-station.
+Experimental modal analysis will be used to tune the model, and to verify that a multi-body model can accurately represent the dynamics of the micro-station.
 
 The tuning approach for the multi-body model based on measurements is illustrated in Figure ref:fig:modal_vibration_analysis_procedure.
 First, a /response model/ is obtained, which corresponds to a set of frequency response functions computed from experimental measurements.
 From this response model, the modal model can be computed, which consists of two matrices: one containing the natural frequencies and damping factors of the considered modes, and another describing the mode shapes.
-This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is, to tune the mass of the considering solid bodies and the springs and dampers connecting the solid bodies.
+This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is, to tune the mass of the considered solid bodies and the springs and dampers connecting the solid bodies.
 
 #+begin_src latex :file modal_vibration_analysis_procedure.pdf
 \begin{tikzpicture}
@@ -184,16 +178,6 @@ 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.
 
-# #+name: tab:modal_section_matlab_code
-# #+caption: Report sections and corresponding Matlab files
-# #+attr_latex: :environment tabularx :width 0.5\linewidth :align lX
-# #+attr_latex: :center t :booktabs t
-# | *Sections*                           | *Matlab File*              |
-# |--------------------------------------+----------------------------|
-# | Section ref:sec:modal_meas_setup     | =modal_1_meas_setup.m=     |
-# | Section ref:sec:modal_frf_processing | =modal_2_frf_processing.m= |
-# | Section ref:sec:modal_analysis       | =modal_3_analysis.m=       |
-
 * Measurement Setup
 :PROPERTIES:
 :HEADER-ARGS:matlab+: :tangle matlab/modal_1_meas_setup.m
@@ -227,12 +211,12 @@ The obtained force and acceleration signals are described in Section ref:ssec:mo
 <<m-init-other>>
 #+end_src
 
-** Used Instrumentation
+** Instrumentation
 <<ssec:modal_instrumentation>>
 
-Three type of equipment are essential for a good modal analysis.
+Three types of equipment are essential for a good modal analysis.
 First, /accelerometers/ are used to measure the response of the structure.
-Here, 3-axis accelerometers[fn:1] shown in figure ref:fig:modal_accelero_M393B05 are used.
+Here, 3-axis accelerometers[fn:modal_1] shown in figure ref:fig:modal_accelero_M393B05 are used.
 These accelerometers were glued to the micro-station using a thin layer of wax for best results [[cite:&ewins00_modal chapt. 3.5.7]].
 
 #+name: fig:modal_analysis_instrumentation
@@ -259,16 +243,16 @@ These accelerometers were glued to the micro-station using a thin layer of wax f
 #+end_subfigure
 #+end_figure
 
-Then, an /instrumented hammer/[fn:2] (figure ref:fig:modal_instrumented_hammer) is used to apply forces to the structure in a controlled manner.
+Then, an /instrumented hammer/[fn:modal_2] (figure ref:fig:modal_instrumented_hammer) is used to apply forces to the structure in a controlled manner.
 Tests were conducted to determine the most suitable hammer tip (ranging from a metallic one to a soft plastic one).
 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 /acquisition system/[fn:3] (figure ref:fig:modal_oros) is used to acquire the injected force and response accelerations in a synchronized manner and with sufficiently low noise.
+Finally, an /acquisition system/[fn:modal_3] (figure ref:fig:modal_oros) is used to acquire the injected force and response accelerations in a synchronized manner and with sufficiently low noise.
 
 ** Structure Preparation and Test Planing
 <<ssec:modal_test_preparation>>
 
-To obtain meaningful results, the modal analysis of the micro-station in performed /in-situ/.
+To obtain meaningful results, the modal analysis of the micro-station is performed /in-situ/.
 To do so, all the micro-station stage controllers are turned "ON".
 This is especially important for stages for which the stiffness is provided by local feedback control, such as the air bearing spindle, and the translation stage.
 If these local feedback controls were turned OFF, this would have resulted in very low-frequency modes that were difficult to measure in practice, and it would also have led to decoupled dynamics, which would not be the case in practice.
@@ -280,7 +264,7 @@ The $H_{jk}$ element of this acrfull:frf matrix corresponds to the frequency res
 Measuring this acrshort:frf matrix is time consuming as it requires to make $n \times n$ measurements.
 However, due to the principle of reciprocity ($H_{jk} = H_{kj}$) and using the /point measurement/ ($H_{jj}$), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix $\mathbf{H}$ [[cite:&ewins00_modal chapt. 5.2]].
 Therefore, a minimum set of $n$ frequency response functions is required.
-This can be done either by measuring the response $X_{j}$ at a fixed acrshort:dof $j$ while applying forces $F_{i}$ for at all $n$ considered acrshort:dof, or by applying a force $F_{k}$ at a fixed acrshort:dof $k$ and measuring the response $X_{i}$ for all $n$ acrshort:dof.
+This can be done either by measuring the response $X_{j}$ at a fixed acrshort:dof $j$ while applying forces $F_{i}$ at all $n$ considered acrshort:dof, or by applying a force $F_{k}$ at a fixed acrshort:dof $k$ and measuring the response $X_{i}$ for all $n$ acrshort:dof.
 
 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.
@@ -296,47 +280,46 @@ Pictures of the accelerometers fixed to the translation stage and to the micro-h
 As all key stages of the micro-station are expected to behave as solid bodies, only 6 acrshort:dof can be considered for each solid body.
 However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured acrshort:dof) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section ref:ssec:modal_solid_body_assumption).
 
-#+attr_latex: :options [t]{0.60\linewidth}
+#+attr_latex: :options [b]{0.63\linewidth}
 #+begin_minipage
 #+name: fig:modal_location_accelerometers
 #+caption: Position of the accelerometers
-#+attr_latex: :width 0.99\linewidth :float nil
+#+attr_latex: :width 0.95\linewidth :float nil
 [[file:figs/modal_location_accelerometers.png]]
 #+end_minipage
 \hfill
-#+attr_latex: :options [b]{0.38\linewidth}
+#+attr_latex: :options [b]{0.36\linewidth}
 #+begin_minipage
 #+begin_scriptsize
-#+name: tab:modal_position_accelerometers
-#+caption: Positions in mm
+#+latex: \centering
 #+attr_latex: :environment tabularx :width \linewidth :placement [b] :align Xccc
 #+attr_latex: :booktabs t :float nil :center nil
-#+RESULTS:
-|              |  $x$ |  $y$ |  $z$ |
-|--------------+------+------+------|
-| Low. Granite | -730 | -526 | -951 |
-| Low. Granite | -735 |  814 | -951 |
-| Low. Granite |  875 |  799 | -951 |
-| Low. Granite |  865 | -506 | -951 |
-| Up. Granite  | -320 | -446 | -786 |
-| Up. Granite  | -480 |  534 | -786 |
-| Up. Granite  |  450 |  534 | -786 |
-| Up. Granite  |  295 | -481 | -786 |
-| Translation  | -475 | -414 | -427 |
-| Translation  | -465 |  407 | -427 |
-| Translation  |  475 |  424 | -427 |
-| Translation  |  475 | -419 | -427 |
-| Tilt         | -385 | -300 | -417 |
-| Tilt         | -420 |  280 | -417 |
-| Tilt         |  420 |  280 | -417 |
-| Tilt         |  380 | -300 | -417 |
-| Spindle      | -155 |  -90 | -594 |
-| Spindle      |    0 |  180 | -594 |
-| Spindle      |  155 |  -90 | -594 |
-| Hexapod      |  -64 |  -64 | -270 |
-| Hexapod      |  -64 |   64 | -270 |
-| Hexapod      |   64 |   64 | -270 |
-| Hexapod      |   64 |  -64 | -270 |
+|                   |  $x$ |  $y$ |  $z$ |
+|-------------------+------+------+------|
+| (17) Low. Granite | -730 | -526 | -951 |
+| (18) Low. Granite | -735 |  814 | -951 |
+| (19) Low. Granite |  875 |  799 | -951 |
+| (20) Low. Granite |  865 | -506 | -951 |
+| (13) Up. Granite  | -320 | -446 | -786 |
+| (14) Up. Granite  | -480 |  534 | -786 |
+| (15) Up. Granite  |  450 |  534 | -786 |
+| (16) Up. Granite  |  295 | -481 | -786 |
+| (9) Translation   | -475 | -414 | -427 |
+| (10) Translation  | -465 |  407 | -427 |
+| (11) Translation  |  475 |  424 | -427 |
+| (12) Translation  |  475 | -419 | -427 |
+| (5) Tilt          | -385 | -300 | -417 |
+| (6) Tilt          | -420 |  280 | -417 |
+| (7) Tilt          |  420 |  280 | -417 |
+| (8) Tilt          |  380 | -300 | -417 |
+| (21) Spindle      | -155 |  -90 | -594 |
+| (22) Spindle      |    0 |  180 | -594 |
+| (23) Spindle      |  155 |  -90 | -594 |
+| (1) Hexapod       |  -64 |  -64 | -270 |
+| (2) Hexapod       |  -64 |   64 | -270 |
+| (3) Hexapod       |   64 |   64 | -270 |
+| (4) Hexapod       |   64 |  -64 | -270 |
+#+latex: \captionof{table}{\label{tab:modal_position_accelerometers}Positions in mm}
 #+end_scriptsize
 #+end_minipage
 
@@ -411,6 +394,7 @@ For the accelerometer, a much more complex signal can be observed, indicating co
 
 The "normalized" acrfull:asd of the two signals were computed and shown in Figure ref:fig:modal_asd_acc_force.
 Conclusions based on the time domain signals can be clearly observed in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer).
+These data are corresponding to a hammer impact in the vertical direction and to the measured acceleration in the $x$ direction by accelerometer $1$ (fixed to the micro-hexapod).
 Similar results were obtained for all measured frequency response functions.
 
 #+begin_src matlab
@@ -431,12 +415,12 @@ time = linspace(0, meas1_raw.Track1_X_Resolution*length(meas1_raw.Track1), lengt
 %% Raw measurement of the Accelerometer
 figure;
 hold on;
-plot(time-22.2, meas1_raw.Track2, 'DisplayName', '$X_{j}$ [$m/s^2$]');
-plot(time-22.2, 1e-3*meas1_raw.Track1, 'DisplayName', '$F_{k}$ [kN]');
+plot(time-22.2, meas1_raw.Track2, 'DisplayName', '$X_{1,x}$ [$m/s^2$]');
+plot(time-22.2, 1e-3*meas1_raw.Track1, 'DisplayName', '$F_{z}$ [kN]');
 hold off;
 xlabel('Time [s]');
 ylabel('Amplitude');
-xlim([0, 0.2]);
+xlim([0, 0.2])
 ylim([-2, 2]);
 legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
 #+end_src
@@ -460,8 +444,8 @@ Noverlap = floor(Nfft/2); % Overlap for frequency analysis
 %% Normalized Amplitude Spectral Density of the measured force and acceleration
 figure;
 hold on;
-plot(f, sqrt(pxx_acc./max(pxx_acc(f<200))), 'DisplayName', '$X_{j}$');
-plot(f, sqrt(pxx_force./max(pxx_force(f<200))), 'DisplayName', '$F_{k}$');
+plot(f, sqrt(pxx_acc./max(pxx_acc(f<200))), 'DisplayName', '$\Gamma_{X_{1,x}}$');
+plot(f, sqrt(pxx_force./max(pxx_force(f<200))), 'DisplayName', '$\Gamma_{F_{z}}$');
 hold off;
 set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
 xlabel('Frequency [Hz]'); ylabel('Normalized Spectral Density');
@@ -476,7 +460,7 @@ exportFig('figs/modal_asd_acc_force.pdf', 'width', 'half', 'height', 'normal');
 #+end_src
 
 #+name: fig:modal_raw_meas_asd
-#+caption: Raw measurement of the accelerometer (blue) and of the force sensor at the Hammer tip (red) (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force})
+#+caption: Raw measurement of the accelerometer 1 in the $x$ direction (blue) and of the force sensor at the Hammer tip (red) for an impact in the $z$ direction (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force})
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:modal_raw_meas}Time domain signals}
@@ -493,7 +477,7 @@ exportFig('figs/modal_asd_acc_force.pdf', 'width', 'half', 'height', 'normal');
 #+end_subfigure
 #+end_figure
 
-The frequency response function $H_{jk}$ from the applied force $F_{k}$ to the measured acceleration $X_j$ is then computed and shown Figure ref:fig:modal_frf_acc_force.
+The frequency response function from the applied force to the measured acceleration is then computed and shown Figure ref:fig:modal_frf_acc_force.
 The quality of the obtained data can be estimated using the /coherence/ function (Figure ref:fig:modal_coh_acc_force).
 Good coherence is obtained from $20\,\text{Hz}$ to $200\,\text{Hz}$ which corresponds to the frequency range of interest.
 
@@ -532,7 +516,7 @@ exportFig('figs/modal_coh_acc_force.pdf', 'width', 'half', 'height', 'normal');
 #+end_src
 
 #+name: fig:modal_frf_coh_acc_force
-#+caption: Computed frequency response function from the applied force $F_{k}$ and the measured response $X_{j}$ (\subref{fig:modal_frf_acc_force}) as well as computed coherence (\subref{fig:modal_coh_acc_force})
+#+caption: 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})
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:modal_frf_acc_force} Frequency Response Function}
@@ -574,9 +558,9 @@ For each frequency point $\omega_{i}$, a 2D complex matrix is obtained that link
 \end{bmatrix}
 \end{equation}
 
-However, for the multi-body model being developed, only 6 solid bodies are considered, namely: the bottom granite, the top granite, the translation stage, the tilt stage, the spindle and the micro-hexapod.
+However, for the multi-body model, only 6 solid bodies are considered, namely: the bottom granite, the top granite, the translation stage, the tilt stage, the spindle and the micro-hexapod.
 Therefore, only $6 \times 6 = 36$ degrees of freedom are of interest.
-Therefore, the objective of this section is to to process the Frequency Response Matrix to reduce the number of measured acrshort:dof from 69 to 36.
+Therefore, the objective of this section is to process the Frequency Response Matrix to reduce the number of measured acrshort:dof from 69 to 36.
 
 The coordinate transformation from accelerometers acrshort:dof to the solid body 6 acrshortpl:dof (three translations and three rotations) is performed in Section ref:ssec:modal_acc_to_solid_dof.
 The $69 \times 3 \times 801$ frequency response matrix is then reduced to a $36 \times 3 \times 801$ frequency response matrix where the motion of each solid body is expressed with respect to its center of mass.
@@ -651,7 +635,7 @@ The goal here is to link these $4 \times 3 = 12$ measurements to the 6 acrshort:
 
 The motion of the rigid body of figure ref:fig:modal_local_to_global_coordinates can be described by its displacement $\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]$ and (small) rotations $[\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]$ with respect to the reference frame $\{O\}$.
 
-The motion $\vec{\delta} p_{i}$ of a point $p_i$ can be computed from $\vec{\delta} p$ and $\bm{\delta \Omega}$ using equation eqref:eq:modal_compute_point_response, with $\bm{\delta\Omega}$ defined in equation eqref:eq:modal_rotation_matrix.
+The motion $\vec{\delta} p_{i}$ of a point $p_i$ can be computed from $\vec{\delta} p$ and $\bm{\delta \Omega}$ using equation eqref:eq:modal_compute_point_response, with $\bm{\delta\Omega}$ defined in equation eqref:eq:modal_rotation_matrix [[cite:&ewins00_modal chapt. 4.3.2]].
 
 \begin{equation}\label{eq:modal_compute_point_response}
   \vec{\delta} p_{i} &= \vec{\delta} p + \bm{\delta \Omega} \cdot \vec{p}_{i} \\
@@ -684,30 +668,12 @@ Writing this in matrix form for the four points gives eqref:eq:modal_cart_to_acc
 \end{array}\right]
 \end{equation}
 
-Provided that the four sensors are properly located, the system of equation eqref:eq:modal_cart_to_acc can be solved by matrix inversion[fn:5].
-The motion of the solid body expressed in a chosen frame $\{O\}$ can be determined using equation eqref:eq:modal_determine_global_disp.
+Provided that the four sensors are properly located, the system of equation eqref:eq:modal_cart_to_acc can be solved by matrix inversion[fn:modal_5].
+The motion of the solid body expressed in a chosen frame $\{O\}$ can be determined by inverting equation eqref:eq:modal_cart_to_acc.
 Note that this matrix inversion is equivalent to resolving a mean square problem.
 Therefore, having more accelerometers permits better approximation of the motion of a solid body.
 
-\begin{equation}
-\left[\begin{array}{c}
-  \delta p_x \\ \delta p_y \\ \delta p_z \\ \hline \delta\Omega_x \\ \delta\Omega_y \\ \delta\Omega_z
-\end{array}\right] =
-\left[\begin{array}{ccc|ccc}
-  1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\
-  0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\
-  0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline
-    & \vdots &   &    & \vdots &   \\ \hline
-  1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\
-  0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\
-  0 & 0 & 1 & p_{4y} & -p_{4x} & 0
-\end{array}\right]^{-1} \left[\begin{array}{c}
-  \delta p_{1x} \\ \delta p_{1y} \\ \delta p_{1z} \\\hline \vdots \\\hline \delta p_{4x} \\ \delta p_{4y} \\ \delta p_{4z}
-\end{array}\right] \label{eq:modal_determine_global_disp}
-\end{equation}
-
-
-From the CAD model, the position of the center of mass of each considered solid body is computed (see Table ref:tab:modal_com_solid_bodies).
+From the CAD model, the position of the center of mass of each solid body is computed (see Table ref:tab:modal_com_solid_bodies).
 The position of each accelerometer with respect to the center of mass of the corresponding solid body can easily be determined.
 
 #+begin_src matlab
@@ -756,19 +722,19 @@ data2orgtable(1000*model_com', {'Bottom Granite', 'Top granite', 'Translation st
 
 #+name: tab:modal_com_solid_bodies
 #+caption: Center of mass of considered solid bodies with respect to the "point of interest"
-#+attr_latex: :environment tabularx :width 0.6\linewidth :align lXXX
+#+attr_latex: :environment tabularx :width 0.55\linewidth :align Xccc
 #+attr_latex: :center t :booktabs t
 #+RESULTS:
-|                   | $X$ [mm] | $Y$ [mm] | $Z$ [mm] |
-|-------------------+----------+----------+----------|
-| Bottom Granite    |       45 |      144 |    -1251 |
-| Top granite       |       52 |      258 |     -778 |
-| Translation stage |        0 |       14 |     -600 |
-| Tilt Stage        |        0 |       -5 |     -628 |
-| Spindle           |        0 |        0 |     -580 |
-| Hexapod           |       -4 |        6 |     -319 |
+|                   | $X$             | $Y$              | $Z$                |
+|-------------------+-----------------+------------------+--------------------|
+| Bottom Granite    | $45\,\text{mm}$ | $144\,\text{mm}$ | $-1251\,\text{mm}$ |
+| Top granite       | $52\,\text{mm}$ | $258\,\text{mm}$ | $-778\,\text{mm}$  |
+| Translation stage | $0$             | $14\,\text{mm}$  | $-600\,\text{mm}$  |
+| Tilt Stage        | $0$             | $-5\,\text{mm}$  | $-628\,\text{mm}$  |
+| Spindle           | $0$             | $0$              | $-580\,\text{mm}$  |
+| Hexapod           | $-4\,\text{mm}$ | $6\,\text{mm}$   | $-319\,\text{mm}$  |
 
-Using eqref:eq:modal_determine_global_disp, the frequency response matrix $\mathbf{H}_\text{CoM}$ eqref:eq:modal_frf_matrix_com expressing the response at the center of mass of each solid body $D_i$ ($i$ from $1$ to $6$ for the $6$ considered solid bodies) can be computed from the initial acrshort:frf matrix $\mathbf{H}$.
+Using eqref:eq:modal_cart_to_acc, the frequency response matrix $\mathbf{H}_\text{CoM}$ eqref:eq:modal_frf_matrix_com expressing the response at the center of mass of each solid body $D_i$ ($i$ from $1$ to $6$ for the $6$ considered solid bodies) can be computed from the initial acrshort:frf matrix $\mathbf{H}$.
 
 \begin{equation}\label{eq:modal_frf_matrix_com}
   \mathbf{H}_\text{CoM}(\omega_i) = \begin{bmatrix}
@@ -858,7 +824,7 @@ Similar results were obtained for the other solid bodies, indicating that the so
 This also validates the reduction in the number of degrees of freedom from 69 (23 accelerometers with each 3 acrshort:dof) to 36 (6 solid bodies with 6 acrshort:dof).
 
 #+begin_src matlab :exports none :results none
-%% Comparaison of the original accelerometer response and reconstructed response from the solid body response
+%% Comparison of the original accelerometer response and reconstructed response from the solid body response
 exc_names = {'$F_x$', '$F_y$', '$F_z$'};
 DOFs = {'x', 'y', 'z', '\theta_x', '\theta_y', '\theta_z'};
 
@@ -876,11 +842,10 @@ for i = 1:length(accs_i)
 
   hold on;
   for dir_i = 1:3
-    plot(freqs, abs(squeeze(frf(3*(acc_i-1)+dir_i, exc_dir, :))), '-', 'DisplayName', sprintf('$a_{%i,%s}$', acc_i, DOFs{dir_i}));
+    plot(freqs, abs(squeeze(frf(3*(acc_i-1)+dir_i, exc_dir, :))), '-', 'color', [colors(dir_i,:), 0.5], 'linewidth', 2.5, 'DisplayName', sprintf('$a_{%i,%s}$ - meas', acc_i, DOFs{dir_i}));
   end
-  set(gca,'ColorOrderIndex',1)
   for dir_i = 1:3
-    plot(freqs, abs(squeeze(frfs_A(3*(acc_i-1)+dir_i, exc_dir, :))), '--', 'HandleVisibility', 'off');
+    plot(freqs, abs(squeeze(frfs_A(3*(acc_i-1)+dir_i, exc_dir, :))), '-', 'color', colors(dir_i, :), 'DisplayName', sprintf('$a_{%i,%s}$ - solid body', acc_i, DOFs{dir_i}));
   end
   hold off;
 
@@ -899,7 +864,8 @@ for i = 1:length(accs_i)
   set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'log');
   xlim([0, 200]); ylim([1e-6, 3e-2]);
   xticks([0:20:200]);
-  legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+  leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
+  leg.ItemTokenSize(1) = 15;
 end
 #+end_src
 
@@ -908,7 +874,7 @@ exportFig('figs/modal_comp_acc_solid_body_frf.pdf', 'width', 'full', 'height', '
 #+end_src
 
 #+name: fig:modal_comp_acc_solid_body_frf
-#+caption: Comparaison of the original accelerometer response (solid curves) and the reconstructed response from the solid body response (dashed curves). Accelerometers 1 to 4 corresponding to the micro-hexapod are shown.
+#+caption: 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
 #+RESULTS:
 [[file:figs/modal_comp_acc_solid_body_frf.png]]
 
@@ -926,7 +892,7 @@ In order to perform the modal parameter extraction, the order of the modal model
 This is achived using the acrfull:mif in section ref:ssec:modal_number_of_modes.
 
 In section ref:ssec:modal_parameter_extraction, the modal parameter extraction is performed.
-The graphical display of the mode shapes can be computed from the modal model, which is quite quite useful for physical interpretation of the modes.
+The graphical display of the mode shapes can be computed from the modal model, which is quite useful for physical interpretation of the modes.
 
 To validate the quality of the modal model, the full acrshort:frf matrix is computed from the modal model and compared to the initial measured acrshort:frf (section ref:ssec:modal_model_validity).
 
@@ -953,11 +919,11 @@ To validate the quality of the modal model, the full acrshort:frf matrix is comp
 
 ** Number of modes determination
 <<ssec:modal_number_of_modes>>
-The acrshort:mif is here applied to the $n\times p$ acrshort:frf matrix where $n$ is a relatively large number of measurement DOFs (here $n=69$) and $p$ is the number of excitation DOFs (here $p=3$).
+The acrshort:mif is applied to the $n\times p$ acrshort:frf matrix where $n$ is a relatively large number of measurement DOFs (here $n=69$) and $p$ is the number of excitation DOFs (here $p=3$).
 
 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}
@@ -989,7 +955,7 @@ end
 figure;
 hold on;
 for i = 1:size(MIF, 1)
-  plot(freqs, squeeze(MIF(i, i, :)));
+  plot(freqs, squeeze(MIF(i, i, :)), 'DisplayName', sprintf('MIF${}_%i$', i));
 end
 hold off;
 set(gca, 'Xscale', 'lin'); set(gca, 'Yscale', 'log');
@@ -997,52 +963,53 @@ xlabel('Frequency [Hz]'); ylabel('CMIF Amplitude');
 xticks([0:20:200]);
 xlim([0, 200]);
 ylim([1e-6, 2e-2]);
+ldg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/modal_indication_function.pdf', 'width', 'wide', 'height', 'normal');
+exportFig('figs/modal_indication_function.pdf', 'width', 'normal', 'height', 'normal');
 #+end_src
 
-#+attr_latex: :options [t]{0.70\linewidth}
+#+attr_latex: :options [b]{0.70\linewidth}
 #+begin_minipage
 #+name: fig:modal_indication_function
 #+caption: Modal Indication Function
-#+attr_latex: :float nil :width 0.95\linewidth
+#+attr_latex: :float nil :scale 1
 [[file:figs/modal_indication_function.png]]
 #+end_minipage
 \hfill
 #+attr_latex: :options [b]{0.28\linewidth}
 #+begin_minipage
 #+begin_scriptsize
-#+name: tab:modal_obtained_modes_freqs_damps
-#+caption: Natural frequencies and modal damping
+#+latex: \centering
 #+attr_latex: :environment tabularx :width \linewidth :placement [b] :align ccc
 #+attr_latex: :booktabs t :float nil :center nil
-| Mode | Freq. [Hz] | Damp. [%] |
-|------+------------+-----------|
-|    1 |       11.9 |      12.2 |
-|    2 |       18.6 |      11.7 |
-|    3 |       37.8 |       6.2 |
-|    4 |       39.1 |       2.8 |
-|    5 |       56.3 |       2.8 |
-|    6 |       69.8 |       4.3 |
-|    7 |       72.5 |       1.3 |
-|    8 |       84.8 |       3.7 |
-|    9 |       91.3 |       2.9 |
-|   10 |      105.5 |       3.2 |
-|   11 |      106.6 |       1.6 |
-|   12 |      112.7 |       3.1 |
-|   13 |      124.2 |       2.8 |
-|   14 |      145.3 |       1.3 |
-|   15 |      150.5 |       2.4 |
-|   16 |      165.4 |       1.4 |
+| Mode | Frequency          | Damping    |
+|------+--------------------+------------|
+|    1 | $11.9\,\text{Hz}$  | $12.2\,\%$ |
+|    2 | $18.6\,\text{Hz}$  | $11.7\,\%$ |
+|    3 | $37.8\,\text{Hz}$  | $6.2\,\%$  |
+|    4 | $39.1\,\text{Hz}$  | $2.8\,\%$  |
+|    5 | $56.3\,\text{Hz}$  | $2.8\,\%$  |
+|    6 | $69.8\,\text{Hz}$  | $4.3\,\%$  |
+|    7 | $72.5\,\text{Hz}$  | $1.3\,\%$  |
+|    8 | $84.8\,\text{Hz}$  | $3.7\,\%$  |
+|    9 | $91.3\,\text{Hz}$  | $2.9\,\%$  |
+|   10 | $105.5\,\text{Hz}$ | $3.2\,\%$  |
+|   11 | $106.6\,\text{Hz}$ | $1.6\,\%$  |
+|   12 | $112.7\,\text{Hz}$ | $3.1\,\%$  |
+|   13 | $124.2\,\text{Hz}$ | $2.8\,\%$  |
+|   14 | $145.3\,\text{Hz}$ | $1.3\,\%$  |
+|   15 | $150.5\,\text{Hz}$ | $2.4\,\%$  |
+|   16 | $165.4\,\text{Hz}$ | $1.4\,\%$  |
+#+latex: \captionof{table}{\label{tab:modal_obtained_modes_freqs_damps}Identified modes}
 #+end_scriptsize
 #+end_minipage
 
 ** Modal parameter extraction
 <<ssec:modal_parameter_extraction>>
 
-Generally, modal identification consists of curve-fitting a theoretical expression to the actual measured acrshort:frf data.
+Generally, modal identification is using curve-fitting a theoretical expression to the actual measured acrshort:frf data.
 However, there are multiple levels of complexity, from fitting of a single resonance, to fitting a complete curve encompassing several resonances and working on a set of many acrshort:frf plots all obtained from the same structure.
 
 Here, the last method is used because it provides a unique and consistent model.
@@ -1052,6 +1019,7 @@ From the obtained modal parameters, the mode shapes are computed and can be disp
 
 #+name: fig:modal_mode_animations
 #+caption: Three obtained mode shape animations
+#+attr_latex: :options [hbtp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:modal_mode1_animation}$1^{st}$ mode at 11.9 Hz: tilt suspension mode of the granite}
 #+attr_latex: :options {\textwidth}
@@ -1084,11 +1052,11 @@ The levelers were then better adjusted.
 #+attr_latex: :width 0.6\linewidth
 [[file:figs/modal_airlock_picture.jpg]]
 
-The modal parameter extraction is made using a proprietary software[fn:4].
+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.
@@ -1112,10 +1080,10 @@ In order to synthesize the full acrshort:frf matrix, the eigenvectors $\phi_r$ a
 \end{bmatrix}_{n \times 2m}
 \end{equation}
 
-The full acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ can be synthesize using eqref:eq:modal_synthesized_frf.
+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.
@@ -1180,13 +1148,13 @@ for i = 1:length(freqs)
   Hsyn(:, :, i) = eigen_vec_ext_M*diag(1./(diag(modal_a_ext_M).*(j*2*pi*freqs(i) - diag(eigen_val_ext_M))))*eigen_vec_ext_M.';
 end
 
-%% Derivate two times to to have the acceleration response
+%% Derivate two times to have the acceleration response
 for i = 1:size(Hsyn, 1)
   Hsyn(i, :, :) = squeeze(Hsyn(i, :, :)).*(j*2*pi*freqs).^2;
 end
 #+end_src
 
-The comparison between the original measured frequency response functions and those synthesized ones from the modal model is presented in Figure ref:fig:modal_comp_acc_frf_modal.
+A comparison between original measured frequency response functions and synthesized ones from the modal model is presented in Figure ref:fig:modal_comp_acc_frf_modal.
 Whether the obtained match is good or bad is quite arbitrary.
 However, the modal model seems to be able to represent the coupling between different nodes and different directions, which is quite important from a control perspective.
 This can be seen in Figure ref:fig:modal_comp_acc_frf_modal_3 that shows the frequency response function from the force applied on node 11 (i.e. on the translation stage) in the $y$ direction to the measured acceleration at node $2$ (i.e. at the top of the micro-hexapod) in the $x$ direction.
@@ -1264,7 +1232,7 @@ exportFig('figs/modal_comp_acc_frf_modal_3.pdf', 'width', 'third', 'height', 'no
 #+end_src
 
 #+name: fig:modal_comp_acc_frf_modal
-#+caption: Comparison of the measured FRF with the synthesized FRF from the modal model.
+#+caption: Comparison of the measured FRF with the FRF synthesized from the modal model.
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:modal_comp_acc_frf_modal_1}From $F_{11,z}$ to $a_{11,z}$}
@@ -1292,11 +1260,11 @@ exportFig('figs/modal_comp_acc_frf_modal_3.pdf', 'width', 'third', 'height', 'no
 
 In this study, a modal analysis of the micro-station was performed.
 Thanks to an adequate choice of instrumentation and proper set of measurements, high quality frequency response functions can be obtained.
-The obtained frequency response functions indicate that the dynamics of the micro-station are complex, which is expected from a heavy stack stage architecture.
+The obtained frequency response functions indicate that the dynamics of the micro-station is complex, which is expected from a heavy stack stage architecture.
 It shows a lot of coupling between stages and different directions, and many modes.
 
 By measuring 12 degrees of freedom on each "stage", it could be verified that in the frequency range of interest, each stage behaved as a rigid body.
-This confirms that a solid body model can be used to properly model the micro-station.
+This confirms that a multi-body model can be used to properly model the micro-station.
 
 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.
@@ -1335,8 +1303,8 @@ colors = colororder;
 
 * Footnotes
 
-[fn:5]As this matrix is in general non-square, the Moore–Penrose inverse can be used instead.
-[fn:4]NVGate software from OROS company.
-[fn:3]OROS OR36. 24bits signal-delta ADC.
-[fn:2]Kistler 9722A2000. Sensitivity of $2.3\,mV/N$ and measurement range of $2\,kN$
-[fn:1]PCB 356B18. Sensitivity is $1\,V/g$, measurement range is $\pm 5\,g$ and bandwidth is $0.5$ to $5\,\text{kHz}$.
+[fn:modal_5]As this matrix is in general non-square, the Moore–Penrose inverse can be used instead.
+[fn:modal_4]NVGate software from OROS company.
+[fn:modal_3]OROS OR36. 24bits signal-delta ADC.
+[fn:modal_2]Kistler 9722A2000. Sensitivity of $2.3\,mV/N$ and measurement range of $2\,kN$
+[fn:modal_1]PCB 356B18. Sensitivity is $1\,V/g$, measurement range is $\pm 5\,g$ and bandwidth is $0.5$ to $5\,\text{kHz}$.

modal-analysis.tex

@@ -1,4 +1,4 @@
-% Created 2024-10-24 Thu 19:33
+% 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}
 
@@ -34,17 +34,16 @@
 \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.
 
 Although the inertia of each solid body can easily be estimated from its geometry and material density, it is more difficult to properly estimate the stiffness and damping properties of the guiding elements connecting each solid body.
-Experimental modal analysis will be use to tune the model, and to verify that a multi-body model can accurately represent the dynamics of the micro-station.
+Experimental modal analysis will be used to tune the model, and to verify that a multi-body model can accurately represent the dynamics of the micro-station.
 
 The tuning approach for the multi-body model based on measurements is illustrated in Figure \ref{fig:modal_vibration_analysis_procedure}.
 First, a \emph{response model} is obtained, which corresponds to a set of frequency response functions computed from experimental measurements.
 From this response model, the modal model can be computed, which consists of two matrices: one containing the natural frequencies and damping factors of the considered modes, and another describing the mode shapes.
-This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is, to tune the mass of the considering solid bodies and the springs and dampers connecting the solid bodies.
+This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is, to tune the mass of the considered solid bodies and the springs and dampers connecting the solid bodies.
 
 \begin{figure}[htbp]
 \centering
@@ -61,18 +60,16 @@ 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.
 This includes using appropriate instrumentation (presented in Section \ref{ssec:modal_instrumentation}) and properly preparing the structure to be measured (Section \ref{ssec:modal_test_preparation}).
 Then, the locations of the measured motions (Section \ref{ssec:modal_accelerometers}) and the locations of the hammer impacts (Section \ref{ssec:modal_hammer_impacts}) have to be chosen carefully.
 The obtained force and acceleration signals are described in Section \ref{ssec:modal_measured_signals}, and the quality of the measured data is assessed.
-\section{Used Instrumentation}
+\section{Instrumentation}
 \label{ssec:modal_instrumentation}
 
-Three type of equipment are essential for a good modal analysis.
+Three types of equipment are essential for a good modal analysis.
 First, \emph{accelerometers} are used to measure the response of the structure.
 Here, 3-axis accelerometers\footnote{PCB 356B18. Sensitivity is \(1\,V/g\), measurement range is \(\pm 5\,g\) and bandwidth is \(0.5\) to \(5\,\text{kHz}\).} shown in figure \ref{fig:modal_accelero_M393B05} are used.
 These accelerometers were glued to the micro-station using a thin layer of wax for best results \cite[chapt. 3.5.7]{ewins00_modal}.
@@ -104,11 +101,10 @@ 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}
 
-To obtain meaningful results, the modal analysis of the micro-station in performed \emph{in-situ}.
+To obtain meaningful results, the modal analysis of the micro-station is performed \emph{in-situ}.
 To do so, all the micro-station stage controllers are turned ``ON''.
 This is especially important for stages for which the stiffness is provided by local feedback control, such as the air bearing spindle, and the translation stage.
 If these local feedback controls were turned OFF, this would have resulted in very low-frequency modes that were difficult to measure in practice, and it would also have led to decoupled dynamics, which would not be the case in practice.
@@ -120,11 +116,10 @@ The \(H_{jk}\) element of this \acrfull{frf} matrix corresponds to the frequency
 Measuring this \acrshort{frf} matrix is time consuming as it requires to make \(n \times n\) measurements.
 However, due to the principle of reciprocity (\(H_{jk} = H_{kj}\)) and using the \emph{point measurement} (\(H_{jj}\)), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix \(\mathbf{H}\) \cite[chapt. 5.2]{ewins00_modal}.
 Therefore, a minimum set of \(n\) frequency response functions is required.
-This can be done either by measuring the response \(X_{j}\) at a fixed \acrshort{dof} \(j\) while applying forces \(F_{i}\) for at all \(n\) considered \acrshort{dof}, or by applying a force \(F_{k}\) at a fixed \acrshort{dof} \(k\) and measuring the response \(X_{i}\) for all \(n\) \acrshort{dof}.
+This can be done either by measuring the response \(X_{j}\) at a fixed \acrshort{dof} \(j\) while applying forces \(F_{i}\) at all \(n\) considered \acrshort{dof}, or by applying a force \(F_{k}\) at a fixed \acrshort{dof} \(k\) and measuring the response \(X_{i}\) for all \(n\) \acrshort{dof}.
 
 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}
 
@@ -136,45 +131,46 @@ Pictures of the accelerometers fixed to the translation stage and to the micro-h
 As all key stages of the micro-station are expected to behave as solid bodies, only 6 \acrshort{dof} can be considered for each solid body.
 However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured \acrshort{dof}) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section \ref{ssec:modal_solid_body_assumption}).
 
-\begin{minipage}[t]{0.60\linewidth}
+\begin{minipage}[b]{0.63\linewidth}
 \begin{center}
-\includegraphics[scale=1,width=0.99\linewidth]{figs/modal_location_accelerometers.png}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/modal_location_accelerometers.png}
 \captionof{figure}{\label{fig:modal_location_accelerometers}Position of the accelerometers}
 \end{center}
 \end{minipage}
 \hfill
-\begin{minipage}[b]{0.38\linewidth}
+\begin{minipage}[b]{0.36\linewidth}
 \begin{scriptsize}
-\captionof{table}{\label{tab:modal_position_accelerometers}Positions in mm}
+\centering
 \begin{tabularx}{\linewidth}{Xccc}
 \toprule
  & \(x\) & \(y\) & \(z\)\\
 \midrule
-Low. Granite & -730 & -526 & -951\\
-Low. Granite & -735 & 814 & -951\\
-Low. Granite & 875 & 799 & -951\\
-Low. Granite & 865 & -506 & -951\\
-Up. Granite & -320 & -446 & -786\\
-Up. Granite & -480 & 534 & -786\\
-Up. Granite & 450 & 534 & -786\\
-Up. Granite & 295 & -481 & -786\\
-Translation & -475 & -414 & -427\\
-Translation & -465 & 407 & -427\\
-Translation & 475 & 424 & -427\\
-Translation & 475 & -419 & -427\\
-Tilt & -385 & -300 & -417\\
-Tilt & -420 & 280 & -417\\
-Tilt & 420 & 280 & -417\\
-Tilt & 380 & -300 & -417\\
-Spindle & -155 & -90 & -594\\
-Spindle & 0 & 180 & -594\\
-Spindle & 155 & -90 & -594\\
-Hexapod & -64 & -64 & -270\\
-Hexapod & -64 & 64 & -270\\
-Hexapod & 64 & 64 & -270\\
-Hexapod & 64 & -64 & -270\\
+(17) Low. Granite & -730 & -526 & -951\\
+(18) Low. Granite & -735 & 814 & -951\\
+(19) Low. Granite & 875 & 799 & -951\\
+(20) Low. Granite & 865 & -506 & -951\\
+(13) Up. Granite & -320 & -446 & -786\\
+(14) Up. Granite & -480 & 534 & -786\\
+(15) Up. Granite & 450 & 534 & -786\\
+(16) Up. Granite & 295 & -481 & -786\\
+(9) Translation & -475 & -414 & -427\\
+(10) Translation & -465 & 407 & -427\\
+(11) Translation & 475 & 424 & -427\\
+(12) Translation & 475 & -419 & -427\\
+(5) Tilt & -385 & -300 & -417\\
+(6) Tilt & -420 & 280 & -417\\
+(7) Tilt & 420 & 280 & -417\\
+(8) Tilt & 380 & -300 & -417\\
+(21) Spindle & -155 & -90 & -594\\
+(22) Spindle & 0 & 180 & -594\\
+(23) Spindle & 155 & -90 & -594\\
+(1) Hexapod & -64 & -64 & -270\\
+(2) Hexapod & -64 & 64 & -270\\
+(3) Hexapod & 64 & 64 & -270\\
+(4) Hexapod & 64 & -64 & -270\\
 \bottomrule
 \end{tabularx}
+\captionof{table}{\label{tab:modal_position_accelerometers}Positions in mm}
 \end{scriptsize}
 \end{minipage}
 
@@ -193,7 +189,6 @@ Hexapod & 64 & -64 & -270\\
 \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}
 
@@ -223,7 +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}
 
@@ -233,6 +227,7 @@ For the accelerometer, a much more complex signal can be observed, indicating co
 
 The ``normalized'' \acrfull{asd} of the two signals were computed and shown in Figure \ref{fig:modal_asd_acc_force}.
 Conclusions based on the time domain signals can be clearly observed in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer).
+These data are corresponding to a hammer impact in the vertical direction and to the measured acceleration in the \(x\) direction by accelerometer \(1\) (fixed to the micro-hexapod).
 Similar results were obtained for all measured frequency response functions.
 
 \begin{figure}[htbp]
@@ -248,10 +243,10 @@ Similar results were obtained for all measured frequency response functions.
 \end{center}
 \subcaption{\label{fig:modal_asd_acc_force}Amplitude Spectral Density (normalized)}
 \end{subfigure}
-\caption{\label{fig:modal_raw_meas_asd}Raw measurement of the accelerometer (blue) and of the force sensor at the Hammer tip (red) (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force})}
+\caption{\label{fig:modal_raw_meas_asd}Raw measurement of the accelerometer 1 in the \(x\) direction (blue) and of the force sensor at the Hammer tip (red) for an impact in the \(z\) direction (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force})}
 \end{figure}
 
-The frequency response function \(H_{jk}\) from the applied force \(F_{k}\) to the measured acceleration \(X_j\) is then computed and shown Figure \ref{fig:modal_frf_acc_force}.
+The frequency response function from the applied force to the measured acceleration is then computed and shown Figure \ref{fig:modal_frf_acc_force}.
 The quality of the obtained data can be estimated using the \emph{coherence} function (Figure \ref{fig:modal_coh_acc_force}).
 Good coherence is obtained from \(20\,\text{Hz}\) to \(200\,\text{Hz}\) which corresponds to the frequency range of interest.
 
@@ -268,9 +263,8 @@ Good coherence is obtained from \(20\,\text{Hz}\) to \(200\,\text{Hz}\) which co
 \end{center}
 \subcaption{\label{fig:modal_coh_acc_force} Coherence}
 \end{subfigure}
-\caption{\label{fig:modal_frf_coh_acc_force}Computed frequency response function from the applied force \(F_{k}\) and the measured response \(X_{j}\) (\subref{fig:modal_frf_acc_force}) as well as computed coherence (\subref{fig:modal_coh_acc_force})}
+\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:
@@ -293,9 +287,9 @@ For each frequency point \(\omega_{i}\), a 2D complex matrix is obtained that li
 \end{bmatrix}
 \end{equation}
 
-However, for the multi-body model being developed, only 6 solid bodies are considered, namely: the bottom granite, the top granite, the translation stage, the tilt stage, the spindle and the micro-hexapod.
+However, for the multi-body model, only 6 solid bodies are considered, namely: the bottom granite, the top granite, the translation stage, the tilt stage, the spindle and the micro-hexapod.
 Therefore, only \(6 \times 6 = 36\) degrees of freedom are of interest.
-Therefore, the objective of this section is to to process the Frequency Response Matrix to reduce the number of measured \acrshort{dof} from 69 to 36.
+Therefore, the objective of this section is to process the Frequency Response Matrix to reduce the number of measured \acrshort{dof} from 69 to 36.
 
 The coordinate transformation from accelerometers \acrshort{dof} to the solid body 6 \acrshortpl{dof} (three translations and three rotations) is performed in Section \ref{ssec:modal_acc_to_solid_dof}.
 The \(69 \times 3 \times 801\) frequency response matrix is then reduced to a \(36 \times 3 \times 801\) frequency response matrix where the motion of each solid body is expressed with respect to its center of mass.
@@ -315,7 +309,7 @@ The goal here is to link these \(4 \times 3 = 12\) measurements to the 6 \acrsho
 
 The motion of the rigid body of figure \ref{fig:modal_local_to_global_coordinates} can be described by its displacement \(\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]\) and (small) rotations \([\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]\) with respect to the reference frame \(\{O\}\).
 
-The motion \(\vec{\delta} p_{i}\) of a point \(p_i\) can be computed from \(\vec{\delta} p\) and \(\bm{\delta \Omega}\) using equation \eqref{eq:modal_compute_point_response}, with \(\bm{\delta\Omega}\) defined in equation \eqref{eq:modal_rotation_matrix}.
+The motion \(\vec{\delta} p_{i}\) of a point \(p_i\) can be computed from \(\vec{\delta} p\) and \(\bm{\delta \Omega}\) using equation \eqref{eq:modal_compute_point_response}, with \(\bm{\delta\Omega}\) defined in equation \eqref{eq:modal_rotation_matrix} \cite[chapt. 4.3.2]{ewins00_modal}.
 
 \begin{equation}\label{eq:modal_compute_point_response}
   \vec{\delta} p_{i} &= \vec{\delta} p + \bm{\delta \Omega} \cdot \vec{p}_{i} \\
@@ -349,49 +343,31 @@ Writing this in matrix form for the four points gives \eqref{eq:modal_cart_to_ac
 \end{equation}
 
 Provided that the four sensors are properly located, the system of equation \eqref{eq:modal_cart_to_acc} can be solved by matrix inversion\footnote{As this matrix is in general non-square, the Moore–Penrose inverse can be used instead.}.
-The motion of the solid body expressed in a chosen frame \(\{O\}\) can be determined using equation \eqref{eq:modal_determine_global_disp}.
+The motion of the solid body expressed in a chosen frame \(\{O\}\) can be determined by inverting equation \eqref{eq:modal_cart_to_acc}.
 Note that this matrix inversion is equivalent to resolving a mean square problem.
 Therefore, having more accelerometers permits better approximation of the motion of a solid body.
 
-\begin{equation}
-\left[\begin{array}{c}
-  \delta p_x \\ \delta p_y \\ \delta p_z \\ \hline \delta\Omega_x \\ \delta\Omega_y \\ \delta\Omega_z
-\end{array}\right] =
-\left[\begin{array}{ccc|ccc}
-  1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\
-  0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\
-  0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline
-    & \vdots &   &    & \vdots &   \\ \hline
-  1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\
-  0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\
-  0 & 0 & 1 & p_{4y} & -p_{4x} & 0
-\end{array}\right]^{-1} \left[\begin{array}{c}
-  \delta p_{1x} \\ \delta p_{1y} \\ \delta p_{1z} \\\hline \vdots \\\hline \delta p_{4x} \\ \delta p_{4y} \\ \delta p_{4z}
-\end{array}\right] \label{eq:modal_determine_global_disp}
-\end{equation}
-
-
-From the CAD model, the position of the center of mass of each considered solid body is computed (see Table \ref{tab:modal_com_solid_bodies}).
+From the CAD model, the position of the center of mass of each solid body is computed (see Table \ref{tab:modal_com_solid_bodies}).
 The position of each accelerometer with respect to the center of mass of the corresponding solid body can easily be determined.
 
 \begin{table}[htbp]
 \caption{\label{tab:modal_com_solid_bodies}Center of mass of considered solid bodies with respect to the ``point of interest''}
 \centering
-\begin{tabularx}{0.6\linewidth}{lXXX}
+\begin{tabularx}{0.55\linewidth}{Xccc}
 \toprule
- & \(X\) [mm] & \(Y\) [mm] & \(Z\) [mm]\\
+ & \(X\) & \(Y\) & \(Z\)\\
 \midrule
-Bottom Granite & 45 & 144 & -1251\\
-Top granite & 52 & 258 & -778\\
-Translation stage & 0 & 14 & -600\\
-Tilt Stage & 0 & -5 & -628\\
-Spindle & 0 & 0 & -580\\
-Hexapod & -4 & 6 & -319\\
+Bottom Granite & \(45\,\text{mm}\) & \(144\,\text{mm}\) & \(-1251\,\text{mm}\)\\
+Top granite & \(52\,\text{mm}\) & \(258\,\text{mm}\) & \(-778\,\text{mm}\)\\
+Translation stage & \(0\) & \(14\,\text{mm}\) & \(-600\,\text{mm}\)\\
+Tilt Stage & \(0\) & \(-5\,\text{mm}\) & \(-628\,\text{mm}\)\\
+Spindle & \(0\) & \(0\) & \(-580\,\text{mm}\)\\
+Hexapod & \(-4\,\text{mm}\) & \(6\,\text{mm}\) & \(-319\,\text{mm}\)\\
 \bottomrule
 \end{tabularx}
 \end{table}
 
-Using \eqref{eq:modal_determine_global_disp}, the frequency response matrix \(\mathbf{H}_\text{CoM}\) \eqref{eq:modal_frf_matrix_com} expressing the response at the center of mass of each solid body \(D_i\) (\(i\) from \(1\) to \(6\) for the \(6\) considered solid bodies) can be computed from the initial \acrshort{frf} matrix \(\mathbf{H}\).
+Using \eqref{eq:modal_cart_to_acc}, the frequency response matrix \(\mathbf{H}_\text{CoM}\) \eqref{eq:modal_frf_matrix_com} expressing the response at the center of mass of each solid body \(D_i\) (\(i\) from \(1\) to \(6\) for the \(6\) considered solid bodies) can be computed from the initial \acrshort{frf} matrix \(\mathbf{H}\).
 
 \begin{equation}\label{eq:modal_frf_matrix_com}
   \mathbf{H}_\text{CoM}(\omega_i) = \begin{bmatrix}
@@ -406,7 +382,6 @@ Using \eqref{eq:modal_determine_global_disp}, the frequency response matrix \(\m
   \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}
 
@@ -422,9 +397,8 @@ This also validates the reduction in the number of degrees of freedom from 69 (2
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/modal_comp_acc_solid_body_frf.png}
-\caption{\label{fig:modal_comp_acc_solid_body_frf}Comparaison of the original accelerometer response (solid curves) and the reconstructed response from the solid body response (dashed curves). Accelerometers 1 to 4 corresponding to the micro-hexapod are shown.}
+\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).
@@ -434,16 +408,16 @@ In order to perform the modal parameter extraction, the order of the modal model
 This is achived using the \acrfull{mif} in section \ref{ssec:modal_number_of_modes}.
 
 In section \ref{ssec:modal_parameter_extraction}, the modal parameter extraction is performed.
-The graphical display of the mode shapes can be computed from the modal model, which is quite quite useful for physical interpretation of the modes.
+The graphical display of the mode shapes can be computed from the modal model, which is quite useful for physical interpretation of the modes.
 
 To validate the quality of the modal model, the full \acrshort{frf} matrix is computed from the modal model and compared to the initial measured \acrshort{frf} (section \ref{ssec:modal_model_validity}).
 \section{Number of modes determination}
 \label{ssec:modal_number_of_modes}
-The \acrshort{mif} is here applied to the \(n\times p\) \acrshort{frf} matrix where \(n\) is a relatively large number of measurement DOFs (here \(n=69\)) and \(p\) is the number of excitation DOFs (here \(p=3\)).
+The \acrshort{mif} is applied to the \(n\times p\) \acrshort{frf} matrix where \(n\) is a relatively large number of measurement DOFs (here \(n=69\)) and \(p\) is the number of excitation DOFs (here \(p=3\)).
 
 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}
@@ -457,45 +431,45 @@ 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{minipage}[t]{0.70\linewidth}
+\begin{minipage}[b]{0.70\linewidth}
 \begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/modal_indication_function.png}
+\includegraphics[scale=1,scale=1]{figs/modal_indication_function.png}
 \captionof{figure}{\label{fig:modal_indication_function}Modal Indication Function}
 \end{center}
 \end{minipage}
 \hfill
 \begin{minipage}[b]{0.28\linewidth}
 \begin{scriptsize}
-\captionof{table}{\label{tab:modal_obtained_modes_freqs_damps}Natural frequencies and modal damping}
+\centering
 \begin{tabularx}{\linewidth}{ccc}
 \toprule
-Mode & Freq. [Hz] & Damp. [\%]\\
+Mode & Frequency & Damping\\
 \midrule
-1 & 11.9 & 12.2\\
-2 & 18.6 & 11.7\\
-3 & 37.8 & 6.2\\
-4 & 39.1 & 2.8\\
-5 & 56.3 & 2.8\\
-6 & 69.8 & 4.3\\
-7 & 72.5 & 1.3\\
-8 & 84.8 & 3.7\\
-9 & 91.3 & 2.9\\
-10 & 105.5 & 3.2\\
-11 & 106.6 & 1.6\\
-12 & 112.7 & 3.1\\
-13 & 124.2 & 2.8\\
-14 & 145.3 & 1.3\\
-15 & 150.5 & 2.4\\
-16 & 165.4 & 1.4\\
+1 & \(11.9\,\text{Hz}\) & \(12.2\,\%\)\\
+2 & \(18.6\,\text{Hz}\) & \(11.7\,\%\)\\
+3 & \(37.8\,\text{Hz}\) & \(6.2\,\%\)\\
+4 & \(39.1\,\text{Hz}\) & \(2.8\,\%\)\\
+5 & \(56.3\,\text{Hz}\) & \(2.8\,\%\)\\
+6 & \(69.8\,\text{Hz}\) & \(4.3\,\%\)\\
+7 & \(72.5\,\text{Hz}\) & \(1.3\,\%\)\\
+8 & \(84.8\,\text{Hz}\) & \(3.7\,\%\)\\
+9 & \(91.3\,\text{Hz}\) & \(2.9\,\%\)\\
+10 & \(105.5\,\text{Hz}\) & \(3.2\,\%\)\\
+11 & \(106.6\,\text{Hz}\) & \(1.6\,\%\)\\
+12 & \(112.7\,\text{Hz}\) & \(3.1\,\%\)\\
+13 & \(124.2\,\text{Hz}\) & \(2.8\,\%\)\\
+14 & \(145.3\,\text{Hz}\) & \(1.3\,\%\)\\
+15 & \(150.5\,\text{Hz}\) & \(2.4\,\%\)\\
+16 & \(165.4\,\text{Hz}\) & \(1.4\,\%\)\\
 \bottomrule
 \end{tabularx}
+\captionof{table}{\label{tab:modal_obtained_modes_freqs_damps}Identified modes}
 \end{scriptsize}
 \end{minipage}
-
 \section{Modal parameter extraction}
 \label{ssec:modal_parameter_extraction}
 
-Generally, modal identification consists of curve-fitting a theoretical expression to the actual measured \acrshort{frf} data.
+Generally, modal identification is using curve-fitting a theoretical expression to the actual measured \acrshort{frf} data.
 However, there are multiple levels of complexity, from fitting of a single resonance, to fitting a complete curve encompassing several resonances and working on a set of many \acrshort{frf} plots all obtained from the same structure.
 
 Here, the last method is used because it provides a unique and consistent model.
@@ -503,7 +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}
@@ -541,7 +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}.
@@ -549,7 +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}
 
@@ -565,10 +538,10 @@ In order to synthesize the full \acrshort{frf} matrix, the eigenvectors \(\phi_r
 \end{bmatrix}_{n \times 2m}
 \end{equation}
 
-The full \acrshort{frf} matrix \(\mathbf{H}_{\text{syn}}\) can be synthesize using \eqref{eq:modal_synthesized_frf}.
+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}.
@@ -576,7 +549,7 @@ With \(\mathbf{H}_{\text{mod}}(\omega)\) a diagonal matrix representing the resp
 \mathbf{H}_{\text{mod}}(\omega) = \text{diag}\left(\frac{1}{a_1 (j\omega - s_1)},\ \dots,\ \frac{1}{a_m (j\omega - s_m)}, \frac{1}{a_1^* (j\omega - s_1^*)},\ \dots,\ \frac{1}{a_m^* (j\omega - s_m^*)} \right)_{2m\times 2m}
 \end{equation}
 
-The comparison between the original measured frequency response functions and those synthesized ones from the modal model is presented in Figure \ref{fig:modal_comp_acc_frf_modal}.
+A comparison between original measured frequency response functions and synthesized ones from the modal model is presented in Figure \ref{fig:modal_comp_acc_frf_modal}.
 Whether the obtained match is good or bad is quite arbitrary.
 However, the modal model seems to be able to represent the coupling between different nodes and different directions, which is quite important from a control perspective.
 This can be seen in Figure \ref{fig:modal_comp_acc_frf_modal_3} that shows the frequency response function from the force applied on node 11 (i.e. on the translation stage) in the \(y\) direction to the measured acceleration at node \(2\) (i.e. at the top of the micro-hexapod) in the \(x\) direction.
@@ -600,24 +573,21 @@ This can be seen in Figure \ref{fig:modal_comp_acc_frf_modal_3} that shows the f
 \end{center}
 \subcaption{\label{fig:modal_comp_acc_frf_modal_3}From $F_{11,y}$ to $a_{2,x}$}
 \end{subfigure}
-\caption{\label{fig:modal_comp_acc_frf_modal}Comparison of the measured FRF with the synthesized FRF from the modal model.}
+\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}
 
 In this study, a modal analysis of the micro-station was performed.
 Thanks to an adequate choice of instrumentation and proper set of measurements, high quality frequency response functions can be obtained.
-The obtained frequency response functions indicate that the dynamics of the micro-station are complex, which is expected from a heavy stack stage architecture.
+The obtained frequency response functions indicate that the dynamics of the micro-station is complex, which is expected from a heavy stack stage architecture.
 It shows a lot of coupling between stages and different directions, and many modes.
 
 By measuring 12 degrees of freedom on each ``stage'', it could be verified that in the frequency range of interest, each stage behaved as a rigid body.
-This confirms that a solid body model can be used to properly model the micro-station.
+This confirms that a multi-body model can be used to properly model the micro-station.
 
 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}