Grammar check

figs/modal_local_to_global_coordinates.svg

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matlab/mat/acc_pos.txt

@@ -0,0 +1,23 @@
+    23  1.5500e-001 -9.0000e-002 -5.9400e-001     
+    22  0.0000e+000  1.8000e-001 -5.9400e-001     
+    21 -1.5500e-001 -9.0000e-002 -5.9400e-001     
+    20  8.6490e-001 -5.0600e-001 -9.5060e-001     
+    19  8.7500e-001  7.9900e-001 -9.5060e-001     
+    18 -7.3500e-001  8.1400e-001 -9.5060e-001     
+    17 -7.3000e-001 -5.2600e-001 -9.5060e-001     
+    16  2.9500e-001 -4.8100e-001 -7.8560e-001     
+    15  4.5000e-001  5.3400e-001 -7.8560e-001     
+    14 -4.8000e-001  5.3400e-001 -7.8560e-001     
+    13 -3.2000e-001 -4.4600e-001 -7.8560e-001     
+    12  4.7500e-001 -4.1900e-001 -4.2730e-001     
+    11  4.7500e-001  4.2400e-001 -4.2730e-001     
+    10 -4.6500e-001  4.0700e-001 -4.2730e-001     
+     9 -4.7500e-001 -4.1400e-001 -4.2730e-001     
+     8  3.8000e-001 -3.0000e-001 -4.1680e-001     
+     7  4.2000e-001  2.8000e-001 -4.1680e-001     
+     6 -4.2000e-001  2.8000e-001 -4.1680e-001     
+     5 -3.8500e-001 -3.0000e-001 -4.1680e-001     
+     4  6.4000e-002 -6.4000e-002 -2.7000e-001     
+     3  6.4000e-002  6.4000e-002 -2.7000e-001     
+     2 -6.4000e-002  6.4000e-002 -2.7000e-001     
+     1 -6.4000e-002 -6.4000e-002 -2.7000e-001     

matlab/mat/mode_damps.txt

@@ -0,0 +1,16 @@
+12.20318
+11.66888
+6.19561
+2.79104
+2.76253
+4.34928
+1.25546
+3.65470
+2.94088
+3.19084
+1.55526
+3.13166
+2.76141
+1.34304
+2.43201
+1.38400

matlab/mat/mode_freqs.txt

@@ -0,0 +1,16 @@
+11.86509
+18.55747
+37.82163
+39.07850
+56.31944
+69.78452
+72.49325
+84.83446
+91.26350
+105.47266
+106.57165
+112.67669
+124.20538
+145.30034
+150.52113
+165.42632

matlab/mat/mode_modal_a.txt

@@ -0,0 +1,16 @@
+4.13559e+003 +6.22828e+003
+2.76278e+002 +1.74197e+004
+-1.32270e+004 +2.17346e+004
+-2.48397e+005 -1.60998e+005
+-4.23967e+004 +7.06852e+004
+-7.36964e+003 +4.57024e+004
+1.37806e+005 +3.00336e+005
+-1.31109e+004 +2.81759e+004
+5.59259e+003 -4.27543e+004
+-5.28869e+004 +6.38436e+003
+3.71578e+004 +1.57745e+004
+-4.24659e+004 +7.90956e+003
+-3.57355e+004 +1.13161e+005
+5.24764e+004 -1.45211e+005
+1.97228e+005 +2.51758e+005
+-3.00273e+005 +3.27201e+005

matlab/mat/mode_modal_b.txt

@@ -0,0 +1,16 @@
+4.98475e+005 -2.49344e+005
+2.02102e+006 +2.05017e+005
+4.96035e+006 +3.45724e+006
+-4.12180e+007 +5.98638e+007
+2.45891e+007 +1.56880e+007
+1.98796e+007 +4.09986e+006
+1.37577e+008 -6.10466e+007
+1.47532e+007 +7.53272e+006
+-2.44115e+007 -3.92655e+006
+3.11045e+006 +3.51656e+007
+1.09485e+007 -2.47140e+007
+4.65546e+006 +3.02251e+007
+8.75076e+007 +3.03162e+007
+-1.31915e+008 -4.96844e+007
+2.42567e+008 -1.80683e+008
+3.35742e+008 +3.16782e+008

matlab/mat/model_solidworks_com.txt

@@ -0,0 +1,18 @@
+0.045
+0.144
+-1.251
+0.052
+0.258
+-0.778
+0.000
+0.014
+-0.600
+0.000
+-0.005
+-0.628
+0.000
+0.000
+-0.580
+-0.004
+0.006
+-0.319

matlab/modal_1_meas_setup.m

@@ -12,48 +12,77 @@ colors = colororder;
 
 % Location of the Accelerometers
 % <<ssec:modal_accelerometers>>
-% 4 tri-axis accelerometers are used for each solid body.
 
-% Only 2 could have been used as only 6DOF have to be measured, however, we have chosen to have some *redundancy*.
+% 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.
 
-% This could also help us identify measurement problems or flexible modes is present.
+% 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).
 
-% The position of the accelerometers are:
-% - 4 on the first granite
-% - 4 on the second granite
-% - 4 on top of the translation stage (figure ref:fig:accelerometers_ty)
-% - 4 on top of the tilt stage
-% - 3 on top of the spindle
-% - 4 on top of the hexapod (figure ref:fig:accelerometers_hexapod)
+% #+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
 
-% In total, 23 accelerometers are used: *69 DOFs are thus measured*.
-
-% The precise determination of the position of each accelerometer is done using the SolidWorks model (shown on figure ref:fig:location_accelerometers).
-
-% #+name: fig:accelerometer_pictures
-% #+caption: Accelerometers fixed on the micro-station
+% #+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:accelerometers_ty}$T_y$ stage}
+% #+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/accelerometers_ty.jpg]]
+% [[file:figs/modal_accelerometers_ty.jpg]]
 % #+end_subfigure
-% #+attr_latex: :caption \subcaption{\label{fig:accelerometers_hexapod}Micro-Hexapod}
+% #+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/accelerometers_hexapod.jpg]]
+% [[file:figs/modal_accelerometers_hexapod.jpg]]
 % #+end_subfigure
 % #+end_figure
 
-% #+name: fig:location_accelerometers
-% #+caption: Position of the accelerometers using SolidWorks
-% #+attr_latex: :width \linewidth
-% [[file:figs/location_accelerometers.png]]
-
-% The precise position of all the 23 accelerometer with respect to a frame located at the point of interest (located 270mm above the top platform of the hexapod) are shown in table ref:tab:position_accelerometers.
-
 
 %% Load Accelerometer positions
 acc_pos = readtable('mat/acc_pos.txt', 'ReadVariableNames', false);
@@ -61,59 +90,18 @@ acc_pos = table2array(acc_pos(:, 1:4));
 [~, i] = sort(acc_pos(:, 1));
 acc_pos = acc_pos(i, 2:4);
 
-% Signal Processing                                               :noexport:
-% <<ssec:modal_signal_processing>>
-% The measurements are averaged 10 times corresponding to 10 hammer impacts in order to reduce the effect of random noise.
-
-% Windowing is also used on the force and response signals.
-
-% A boxcar window (figure ref:fig:modal_windowing_force_signal) is used for the force signal as once the impact on the structure is done, the measured signal is meaningless.
-% The parameters are:
-% - *Start*: $3\%$
-% - *Stop*: $7\%$
-
-
-%% Boxcar window used for the force signal
-figure;
-plot(100*[0, 0.03, 0.03, 0.07, 0.07, 1], [0, 0, 1, 1, 0, 0]);
-xlabel('Time [\%]'); ylabel('Amplitude');
-xlim([0, 100]); ylim([0, 1]);
-
-
-
-% #+name: fig:modal_windowing_force_signal
-% #+caption: Boxcar window used for the force signal
-% #+RESULTS:
-% [[file:figs/modal_windowing_force_signal.png]]
-
-% An exponential window (figure ref:fig:modal_windowing_acc_signal) is used for the response signal as we are measuring transient signals and most of the information is located at the beginning of the signal.
-% The parameters are:
-% - FlatTop:
-%   - *Start*: $3\%$
-%   - *Stop*: $2.96\%$
-% - Decreasing point:
-%   - *X*: $60.4\%$
-%   - *Y*: $14.7\%$
-
-
-%% Exponential window used for acceleration signal
-x0 = 0.296;
-xd = 0.604;
-yd = 0.147;
-
-alpha = log(yd)/(x0 - xd);
-
-t = x0:0.01:1.01;
-y = exp(-alpha*(t-x0));
-
-figure;
-plot(100*[0, 0.03, 0.03, x0, t], [0, 0, 1, 1, y]);
-xlabel('Time [\%]'); ylabel('Amplitude');
-xlim([0, 100]); ylim([0, 1]);
-
 % 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');
 
@@ -126,46 +114,17 @@ impacts = [5.937, 11.228, 16.681, 22.205, 27.350, 32.714, 38.115, 43.888, 50.407
 % Time vector [s]
 time = linspace(0, meas1_raw.Track1_X_Resolution*length(meas1_raw.Track1), length(meas1_raw.Track1));
 
-
-
-% The force sensor 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, many resonances can be seen on the right, indicating complex dynamics
-
-
 %% Raw measurement of the Accelerometer
 figure;
-tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
-
-ax1 = nexttile([1,2]);
 hold on;
-plot(time, meas1_raw.Track2, 'DisplayName', 'Acceleration [$m/s^2$]');
-plot(time, 1e-3*meas1_raw.Track1, 'DisplayName', 'Force [kN]');
+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]');
 hold off;
 xlabel('Time [s]');
 ylabel('Amplitude');
-xlim([0, time(end)]);
-legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
-
-ax2 = nexttile();
-hold on;
-plot(time, meas1_raw.Track2);
-plot(time, 1e-3*meas1_raw.Track1);
-hold off;
-xlabel('Time [s]');
-set(gca, 'YTickLabel',[]);
-xlim([22.19, 22.4]);
-
-linkaxes([ax1,ax2],'y');
+xlim([0, 0.2]);
 ylim([-2, 2]);
-
-
-
-% #+name: fig:modal_raw_meas
-% #+caption: Raw measurement of the acceleromter (blue) and of the force sensor at the Hammer tip (red). Zoom on one impact is shown on the right.
-% #+RESULTS:
-% [[file:figs/modal_raw_meas.png]]
-
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
 
 %% Frequency Analysis
 Nfft = floor(5.0*Fs); % Number of frequency points
@@ -176,17 +135,11 @@ Noverlap = floor(Nfft/2); % Overlap for frequency analysis
 [pxx_force, f] = pwelch(meas1_raw.Track1, win, Noverlap, Nfft, Fs);
 [pxx_acc,   ~] = pwelch(meas1_raw.Track2, win, Noverlap, Nfft, Fs);
 
-
-
-% The "normalized" amplitude spectral density 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).
-
-
 %% Normalized Amplitude Spectral Density of the measured force and acceleration
 figure;
 hold on;
-plot(f, sqrt(pxx_force./max(pxx_force(f<200))), 'DisplayName', 'Force');
-plot(f, sqrt(pxx_acc./max(pxx_acc(f<200))), 'DisplayName', 'Acceleration');
+plot(f, sqrt(pxx_acc./max(pxx_acc(f<200))), 'DisplayName', '$X_{j}$');
+plot(f, sqrt(pxx_force./max(pxx_force(f<200))), 'DisplayName', '$F_{k}$');
 hold off;
 set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
 xlabel('Frequency [Hz]'); ylabel('Normalized Spectral Density');
@@ -197,12 +150,27 @@ legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
 
 
 
-% #+name: fig:modal_asd_acc_force
-% #+caption: Normalized Amplitude Spectral Density of the measured force and acceleration
-% #+RESULTS:
+% #+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 from the applied force to the measured acceleration can then be computed (Figure ref:fig:modal_frf_acc_force).
+% 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
@@ -217,16 +185,6 @@ set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'log');
 xlim([0, 200]);
 xticks([0:20:200]);
 
-
-
-% #+name: fig:modal_frf_acc_force
-% #+caption: Frequency Response Function between the measured force and acceleration
-% #+RESULTS:
-% [[file:figs/modal_frf_acc_force.png]]
-
-% The coherence between the input and output signals is also computed and found to be good between 20 and 200Hz (Figure ref:fig:modal_coh_acc_force).
-
-
 %% Frequency Response Function between the force and the acceleration
 figure;
 plot(f, coh1);

matlab/modal_2_frf_processing.m

@@ -10,33 +10,84 @@ addpath('./mat/'); % Path for data
 %% Colors for the figures
 colors = colororder;
 
-% Frequency Response Matrix
-% <<ssec:modal_frf_matrix>>
-% All the Frequency Response Functions measured are combined into one big array called the Frequency Response Matrix.
 
-% The frequency response matrix is an $n \times p \times q$ matrix with:
-% - $n$ the number of measurements: $23 \times 3$ (23 accelerometers measuring 3 directions each)
-% - $p$ the number of excitation inputs: $3$
-% - $q$ the number of frequency points $\omega_i$
 
-% Thus, the FRF matrix is an $69 \times 3 \times 801$ matrix.
+% #+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]]
 
-% For each frequency point $\omega_i$, a 2D matrix is obtained:
-% \begin{equation}
-%   \text{FRF}(\omega_i) = \begin{bmatrix}
-%   \frac{D_{1_x}}{F_x}(\omega_i) & \frac{D_{1_x}}{F_y}(\omega_i) & \frac{D_{1_x}}{F_z}(\omega_i) \\
-%   \frac{D_{1_y}}{F_x}(\omega_i) & \frac{D_{1_y}}{F_y}(\omega_i) & \frac{D_{1_y}}{F_z}(\omega_i) \\
-%   \frac{D_{1_z}}{F_x}(\omega_i) & \frac{D_{1_z}}{F_y}(\omega_i) & \frac{D_{1_z}}{F_z}(\omega_i) \\
-%   \frac{D_{2_x}}{F_x}(\omega_i) & \frac{D_{2_x}}{F_y}(\omega_i) & \frac{D_{2_x}}{F_z}(\omega_i) \\
-%   \vdots              & \vdots              & \vdots              \\
-%   \frac{D_{23_z}}{F_x}(\omega_i) & \frac{D_{23_z}}{F_y}(\omega_i) & \frac{D_{23_z}}{F_z}(\omega_i) \\
+% 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');
 
+%% 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);
+
 %% Accelerometers ID connected to each solid body
 solids = {};
 solids.gbot = [17, 18, 19, 20]; % bottom granite
@@ -52,189 +103,29 @@ solid_names = fields(solids);
 %% Save the acceleromter positions are well as the solid bodies
 save('mat/geometry.mat', 'solids', 'solid_names', 'acc_pos');
 
-
-
-% #+name: fig:aligned_accelerometers
-% #+caption: Aligned measurement of the motion of a solid body
-% #+RESULTS:
-% [[file:figs/aligned_accelerometers.png]]
-
-% The motion of the rigid body of figure ref:fig:aligned_accelerometers is defined by its displacement $\delta p$ and rotation $\vec{\Omega}$ with respect to the reference frame $\{O\}$.
-
-% The motions at points $1$ and $2$ are:
-% \begin{align*}
-%   \delta p_1 &= \delta p + \Omega \times p_1 \\
-%   \delta p_2 &= \delta p + \Omega \times p_2
-% \end{align*}
-
-% Taking only the $x$ direction:
-% \begin{align*}
-%   \delta p_{x1} &= \delta p_x + \Omega_y p_{z1} - \Omega_z p_{y1} \\
-%   \delta p_{x2} &= \delta p_x + \Omega_y p_{z2} - \Omega_z p_{y2}
-% \end{align*}
-
-% However, we have $p_{1y} = p_{2y}$ and $p_{1z} = p_{2z}$ because of the co-linearity of the two sensors in the $x$ direction, and thus we obtain
-% \begin{equation}
-%   \delta p_{x1} = \delta p_{x2}
-% \end{equation}
-
-% #+begin_important
-%   Two sensors that are measuring the motion of a rigid body in the direction of the line linking the two sensors should measure the same quantity.
-% #+end_important
-
-% We can verify that the rigid body assumption is correct by comparing the measurement of the sensors.
-
-% From the table ref:tab:position_accelerometers, we can guess which sensors will give the same results in the X and Y directions.
-
-% Comparison of such measurements in the X direction is shown on figure ref:fig:modal_solid_body_comp_x_dir.
-% Similar result is obtained for the Y direction.
-
-
-meas_dir = 1; % X
-exc_dir = 1; % X
-
-% Pair of accelerometers aligned in the X direction
-acc_i = [1 , 4 ;
-         2 , 3 ;
-         5 , 8 ;
-         6 , 7 ;
-         9 , 12;
-         10, 11;
-         14, 15;
-         18, 19;
-         21, 23];
-
-%% Comparaison of measured frequency response function for in the X directions for accelerometers aligned along X
-figure;
-tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
-for i = 1:size(acc_i, 1)
-  nexttile();
-  hold on;
-  plot(freqs, abs(squeeze(frf(meas_dir+3*(acc_i(i, 1)-1), exc_dir, :))), ...
-      'DisplayName', sprintf('%i', acc_i(i, 1)))
-  plot(freqs, abs(squeeze(frf(meas_dir+3*(acc_i(i, 2)-1), exc_dir, :))), ...
-      'DisplayName', sprintf('%i', acc_i(i, 2)))
-  hold off;
-  set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'log');
-  if i > 6
-    xlabel('Frequency [Hz]');
-    legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
-  else
-    set(gca, 'XTickLabel',[]);
-    legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
-  end
-
-  if rem(i, 3) == 1
-    ylabel('Amplitude');
-  else
-    set(gca, 'YTickLabel',[]);
-  end
-
-  xlim([0, 200]); ylim([1e-5, 2e-2]);
-end
-
-% TODO Verification of the principle of reciprocity               :noexport:
-% <<ssec:modal_reciprocity_principle>>
-% Because we expect our system to follow the principle of reciprocity.
-% That is to say the response $X_j$ at some degree of freedom $j$ due to a force $F_k$ applied on DOF $k$ should be the same as the response $X_k$ due to a force $F_j$:
-% \[ H_{jk} = \frac{X_j}{F_k} = \frac{X_k}{F_j} = H_{kj} \]
-
-% This comes from the fact that we expect to have symmetric mass, stiffness and damping matrices.
-
-% In order to access the quality of the data and the validity of the measured FRF, we then check that the reciprocity between $H_{jk}$ and $H_{kj}$ is of an acceptable level.
-
-% We can verify this reciprocity using 3 different pairs of response/force.
-
-
-  dir_names = {'X', 'Y', 'Z'};
-
-  figure;
-  for i = 1:3
-    subplot(3, 1, i)
-    a = mod(i, 3)+1;
-    b = mod(i-2, 3)+1;
-    hold on;
-    plot(freqs, abs(squeeze(frf(3*(11-1)+a, b, :))), 'DisplayName', sprintf('$\\frac{F_%s}{D_%s}$', dir_names{a}, dir_names{b}));
-    plot(freqs, abs(squeeze(frf(3*(11-1)+b, a, :))), 'DisplayName', sprintf('$\\frac{F_%s}{D_%s}$', dir_names{b}, dir_names{a}));
-    hold off;
-    set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'log');
-    if i == 3
-      xlabel('Frequency [Hz]');
-    else
-      set(gca, 'XTickLabel',[]);
-    end
-    if i == 2
-      ylabel('Amplitude [$\frac{m/s^2}{N}$]');
-    end
-    xlim([0, 200]);
-    legend('location', 'northwest');
-  end
-
-% From accelerometer DOFs to solid body DOFs - Matlab Implementation
-% First, we initialize a new FRF matrix =FRFs_O= which is an $n \times p \times q$ with:
-% - $n$ is the number of DOFs of the considered 6 solid-bodies: $6 \times 6 = 36$
-% - $p$ is the number of excitation inputs: $3$
-% - $q$ is the number of frequency points $\omega_i$
-
-% #+begin_important
-% For each frequency point $\omega_i$, the FRF matrix =FRFs_O= is a $n\times p$ matrix:
-% \begin{equation}
-%   \text{FRF}_\text{O}(\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}
-% where $D_i$ corresponds to the solid body number i.
-% #+end_important
-
-% Then, as we know the positions of the accelerometers on each solid body, and we have the response of those accelerometers, we can use the equations derived in the previous section to determine the response of each solid body expressed in the frame $\{O\}$.
-
-
-  FRFs_O = zeros(length(solid_names)*6, 3, 801);
-
-  for solid_i = 1:length(solid_names)
-    solids_i = solids.(solid_names{solid_i});
-
-    A = zeros(3*length(solids_i), 6);
-    for i = 1:length(solids_i)
-      acc_i = solids_i(i);
-
-      A(3*(i-1)+1:3*i, 1:3) = eye(3);
-      A(3*(i-1)+1:3*i, 4:6) = [ 0                  acc_pos(acc_i, 3) -acc_pos(acc_i, 2) ;
-                               -acc_pos(acc_i, 3)  0                  acc_pos(acc_i, 1) ;
-                                acc_pos(acc_i, 2) -acc_pos(acc_i, 1)  0];
-    end
-
-    for exc_dir = 1:3
-      FRFs_O((solid_i-1)*6+1:solid_i*6, exc_dir, :) = A\squeeze(FRFs((solids_i(1)-1)*3+1:solids_i(end)*3, exc_dir, :));
-    end
-  end
-
-% Center of Mass of each solid body
-% From solidworks, we can export the position of the center of mass of each solid body considered.
-% These are summarized in Table ref:tab:modal_com_solid_bodies
-
-
 %% Extract the CoM of considered solid bodies
 model_com = reshape(table2array(readtable('mat/model_solidworks_com.txt', 'ReadVariableNames', false)), [3, 6]);
 
-% From accelerometer DOFs to solid body DOFs - Expressed at the CoM
-% First, we initialize a new FRF matrix which is an $n \times p \times q$ with:
-% - $n$ is the number of DOFs of the considered 6 solid-bodies: $6 \times 6 = 36$
-% - $p$ is the number of excitation inputs: $3$
-% - $q$ is the number of frequency points $\omega_i$
 
-% #+begin_important
-% For each frequency point $\omega_i$, the FRF matrix is a $n\times p$ matrix:
-% \begin{equation}
-%   \text{FRF}_\text{CoM}(\omega_i) = \begin{bmatrix}
+
+% #+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) \\
@@ -246,14 +137,10 @@ model_com = reshape(table2array(readtable('mat/model_solidworks_com.txt', 'ReadV
 %   \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}
-% where 1, 2, ..., 6 corresponds to the 6 solid bodies.
-% #+end_important
-
-% Then, as we know the positions of the accelerometers on each solid body, and we have the response of those accelerometers, we can use the equations derived in the previous section to determine the response of each solid body expressed in their center of mass.
 
 
 %% Frequency Response Matrix - Response expressed at the CoM of the solid bodies
-FRFs_CoM = zeros(length(solid_names)*6, 3, 801);
+frfs_CoM = zeros(length(solid_names)*6, 3, 801);
 
 for solid_i = 1:length(solid_names)
   % Number of accelerometers fixed to this solid body
@@ -273,43 +160,30 @@ for solid_i = 1:length(solid_names)
   end
 
   for exc_dir = 1:3
-    FRFs_CoM((solid_i-1)*6+1:solid_i*6, exc_dir, :) = A\squeeze(frf((solids_i(1)-1)*3+1:solids_i(end)*3, exc_dir, :));
+    frfs_CoM((solid_i-1)*6+1:solid_i*6, exc_dir, :) = A\squeeze(frf((solids_i(1)-1)*3+1:solids_i(end)*3, exc_dir, :));
   end
 end
 
 %% Save the computed FRF at the CoM
-save('mat/frf_com.mat', 'FRFs_CoM');
+save('mat/frf_com.mat', 'frfs_CoM');
 
-% Verify that we find the original FRF from the FRF in the global coordinates
-% We have computed the Frequency Response Functions Matrix representing the response of the 6 solid bodies in their 6 DOFs with respect to their center of mass.
+% Verification of solid body assumption
+% <<ssec:modal_solid_body_assumption>>
 
-% From the response of one body in its 6 DOFs, we should be able to compute the FRF of each of its accelerometer fixed to it during the measurement, supposing that this stage is a solid body.
+% 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.
 
-% We can then compare the result with the original measurements.
-% This will help us to determine if:
-% - the previous inversion used is correct
-% - the solid body assumption is correct in the frequency band of interest
 
-% From the translation $\delta p$ and rotation $\delta \Omega$ of a solid body and the positions $p_i$ of the accelerometers attached to it, we can compute the response that would have been measured by the accelerometers using the following formula:
-% \begin{align*}
-%   \delta p_1 &= \delta p + \delta\Omega p_1\\
-%   \delta p_2 &= \delta p + \delta\Omega p_2\\
-%   \delta p_3 &= \delta p + \delta\Omega p_3\\
-%   \delta p_4 &= \delta p + \delta\Omega p_4
-% \end{align*}
-
-% Thus, we can obtain the FRF matrix =FRFs_A= that gives the responses of the accelerometers to the forces applied by the hammer.
-
-% It is implemented in matlab as follow:
-
-FRFs_A = zeros(size(frf));
+%% Compute the FRF at the accelerometer location from the CoM reponses
+frfs_A = zeros(size(frf));
 
 % For each excitation direction
 for exc_dir = 1:3
   % For each solid
   for solid_i = 1:length(solid_names)
-    v0 = squeeze(FRFs_CoM((solid_i-1)*6+1:(solid_i-1)*6+3, exc_dir, :));
-    W0 = squeeze(FRFs_CoM((solid_i-1)*6+4:(solid_i-1)*6+6, exc_dir, :));
+    v0 = squeeze(frfs_CoM((solid_i-1)*6+1:(solid_i-1)*6+3, exc_dir, :));
+    W0 = squeeze(frfs_CoM((solid_i-1)*6+4:(solid_i-1)*6+6, exc_dir, :));
 
     % For each accelerometer attached to the current solid
     for acc_i = solids.(solid_names{solid_i})
@@ -318,20 +192,17 @@ for exc_dir = 1:3
       % pos = acc_pos(acc_i, :).';
       posX = [0 pos(3) -pos(2); -pos(3) 0 pos(1) ; pos(2) -pos(1) 0];
 
-      FRFs_A(3*(acc_i-1)+1:3*(acc_i-1)+3, exc_dir, :) = v0 + posX*W0;
+      frfs_A(3*(acc_i-1)+1:3*(acc_i-1)+3, exc_dir, :) = v0 + posX*W0;
     end
   end
 end
 
 
 
-% We then compare the original FRF measured for each accelerometer =FRFs= with the "recovered" FRF =FRFs_A= from the global FRF matrix in the common frame.
-
-% The FRF for the 4 accelerometers on the Hexapod are compared on figure ref:fig:recovered_frf_comparison_hexa.
-% All the FRF are matching very well in all the frequency range displayed.
-
-% The FRF for accelerometers located on the translation stage are compared on figure ref:fig:recovered_frf_comparison_ty.
-% The FRF are matching well until 100Hz.
+% 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
@@ -344,7 +215,7 @@ exc_dir = 1; % Excited direction
 accs_i = solids.(solid_names{solid_i}); % Accelerometers fixed to this solid body
 
 figure;
-tiledlayout(2, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
+tiledlayout(2, 2, 'TileSpacing', 'Tight', 'Padding', 'None');
 
 for i = 1:length(accs_i)
   acc_i = accs_i(i);
@@ -356,7 +227,7 @@ for i = 1:length(accs_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, :))), '--', 'HandleVisibility', 'off');
   end
   hold off;
 
@@ -372,7 +243,8 @@ for i = 1:length(accs_i)
     set(gca, 'YTickLabel',[]);
   end
 
-  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  xlim([5, 200]); ylim([1e-6, 1e-1]);
-  legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
+  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);
 end

matlab/modal_3_analysis.m

@@ -10,32 +10,29 @@ addpath('./mat/'); % Path for data
 %% Colors for the figures
 colors = colororder;
 
-% Singular Value Decomposition - Modal Indication Function
-% The Mode Indicator Functions are usually used on $n\times p$ FRF matrix where $n$ is a relatively large number of measurement DOFs and $p$ is the number of excitation DOFs, typically 3 or 4.
+% 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 $p=3$).
 
-% In these methods, the frequency dependent FRF matrix is subjected to a singular value decomposition analysis which thus yields a small number (3 or 4) of singular values, these also being frequency dependent.
+% 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}
 
-% These methods are used to *determine the number of modes* present in a given frequency range, to *identify repeated natural frequencies* and to pre-process the FRF data prior to modal analysis.
+% \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}
 
-% From the documentation of the modal software:
-% #+begin_quote
-% The MIF consist of the singular values of the Frequency response function matrix. The number of MIFs equals the number of excitations.
-% By the powerful singular value decomposition, the real signal space is separated from the noise space. Therefore, the MIFs exhibit the modes effectively.
-% A peak in the MIFs plot usually indicate the existence of a structural mode, and two peaks at the same frequency point means the existence of two repeated modes.
-% Moreover, the magnitude of the MIFs implies the strength of the a mode.
-% #+end_quote
+% 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.
 
-% #+begin_important
-% The *Complex Mode Indicator Function* is defined simply by the SVD of the FRF (sub) matrix:
-% \begin{align*}
-%   [H(\omega)]_{n\times p} &= [U(\omega)]_{n\times n} [\Sigma(\omega)]_{n\times p} [V(\omega)]_{p\times p}^H\\
-%   [CMIF(\omega)]_{p\times p} &= [\Sigma(\omega)]_{p\times n}^T [\Sigma(\omega)]_{n\times p}
-% \end{align*}
-% #+end_important
-
-% We compute the Complex Mode Indicator Function.
-% The result is shown on Figure ref:fig:modal_indication_function.
 
+%% Load frequency response matrix
+load('frf_matrix.mat', 'freqs', 'frf');
 
 %% Computation of the modal indication function
 MIF = zeros(size(frf, 2), size(frf, 2), size(frf, 3));
@@ -58,35 +55,31 @@ xticks([0:20:200]);
 xlim([0, 200]);
 ylim([1e-6, 2e-2]);
 
-% Composite Response Function
-% An alternative is the Composite Response Function $HH(\omega)$ defined as the sum of all the measured FRF:
-% \begin{equation}
-%   HH(\omega) = \sum_j\sum_kH_{jk}(\omega)
+% 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}
 
-% Instead, we choose here to use the sum of the norms of the measured frf:
-% \begin{equation}
-%   HH(\omega) = \sum_j\sum_k \left|H_{jk}(\omega) \right|
+% 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}
 
-% The result is shown on figure ref:fig:modal_composite_reponse_function.
-
-
-%% Composite Response Function
-figure;
-hold on;
-plot(freqs, squeeze(sum(sum(abs(frf)))), '-k');
-hold off;
-xlabel('Frequency [Hz]'); ylabel('Amplitude');
-xlim([0, 200]);
-xticks([0:20:200]);
-
-% Importation of the modal parameters on Matlab
-% The obtained modal parameters are:
-% - Resonance frequencies in Hertz
-% - Modal damping ratio in percentage
-% - (complex) Modes shapes for each measured DoF
-% - Modal A and modal B which are parameters important for further normalization
+% 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}
 
 
 %% Load modal parameters
@@ -123,149 +116,91 @@ for mod_i = 1:mod_n
   end
 end
 
-% Modal Matrices
-% We would like to arrange the obtained modal parameters into two modal matrices:
-% \[ \Lambda = \begin{bmatrix}
-%   s_1 &        & 0 \\
-%       & \ddots &   \\
-%   0   &        & s_N
-% \end{bmatrix}_{N \times N}; \quad \Psi = \begin{bmatrix}
-%   & & \\
-%   \{\psi_1\} & \dots & \{\psi_N\} \\
-%   & &
-% \end{bmatrix}_{M \times N} \]
-% \[ \{\psi_i\} = \begin{Bmatrix} \psi_{i, 1_x} & \psi_{i, 1_y} & \psi_{i, 1_z} & \psi_{i, 2_x} & \dots & \psi_{i, 23_z} \end{Bmatrix}^T \]
-
-% $M$ is the number of DoF: here it is $23 \times 3 = 69$.
-% $N$ is the number of mode
-
-
-eigen_val_M = diag(2*pi*freqs_m.*(-damps_m/100 + j*sqrt(1 - (damps_m/100).^2)));
-eigen_vec_M = reshape(mode_shapes, [mod_n, acc_n*dir_n]).';
-
-
-
-% Each eigen vector is normalized: $\| \{\psi_i\} \|_2 = 1$
-
-
-% However, the eigen values and eigen vectors appears as complex conjugates:
-% \[ s_r, s_r^*, \{\psi\}_r, \{\psi\}_r^*, \quad r = 1, N \]
-
-% In the end, they are $2N$ eigen values.
-% We then build two extended eigen matrices as follow:
-% \[ \mathcal{S} = \begin{bmatrix}
-%   s_1 &        &     &       &        &      \\
-%       & \ddots &     &       & 0      &      \\
-%       &        & s_N &       &        &      \\
-%       &        &     & s_1^* &        &      \\
-%       & 0      &     &       & \ddots &      \\
-%       &        &     &       &        & s_N^*
-% \end{bmatrix}_{2N \times 2N}; \quad \Phi = \begin{bmatrix}
-%   & & & & &\\
-%   \{\psi_1\} & \dots & \{\psi_N\} & \{\psi_1^*\} & \dots & \{\psi_N^*\} \\
-%   & & & & &
-% \end{bmatrix}_{M \times 2N} \]
-
+%% Create the eigenvalue and eigenvector matrices
+eigen_val_M = diag(2*pi*freqs_m.*(-damps_m/100 + j*sqrt(1 - (damps_m/100).^2))); % Lambda = diagonal matrix
+eigen_vec_M = reshape(mode_shapes, [mod_n, acc_n*dir_n]).'; % Phi, vecnorm(eigen_vec_M) = 1
 
+% Add complex conjugate eigenvalues and eigenvectors
 eigen_val_ext_M = blkdiag(eigen_val_M, conj(eigen_val_M));
 eigen_vec_ext_M = [eigen_vec_M, conj(eigen_vec_M)];
 
-
-
-% We also build the Modal A and Modal B matrices:
-% \begin{equation}
-%   A = \begin{bmatrix}
-%     a_1 &        & 0   \\
-%         & \ddots &     \\
-%     0   &        & a_N
-%   \end{bmatrix}_{N \times N}; \quad B = \begin{bmatrix}
-%     b_1 &        & 0   \\
-%         & \ddots &     \\
-%     0   &        & b_N
-%   \end{bmatrix}_{N \times N}
-% \end{equation}
-% With $a_i$ is the "Modal A" parameter linked to mode i.
-
-
+%% "Modal A" and "Modal B" matrices
 modal_a_M = diag(complex(modal_a(:, 1), modal_a(:, 2)));
 modal_b_M = diag(complex(modal_b(:, 1), modal_b(:, 2)));
 
 modal_a_ext_M = blkdiag(modal_a_M, conj(modal_a_M));
 modal_b_ext_M = blkdiag(modal_b_M, conj(modal_b_M));
 
-% Matlab Implementation
-
+%% Synthesize the full FRF matrix from the modal model
 Hsyn = zeros(acc_n*dir_n, acc_n*dir_n, length(freqs));
 
 for i = 1:length(freqs)
-  Hsyn(:, :, i) = eigen_vec_ext_M*((j*2*pi*freqs(i)).^2*inv(modal_a_ext_M)/(diag(j*2*pi*freqs(i) - diag(eigen_val_ext_M))))*eigen_vec_ext_M.';
+  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
 
-
-
-% Because the synthesize frequency response functions are representing the displacement response in $[m/N]$, we multiply each element of the FRF matrix by $(j \omega)^2$ in order to obtain the acceleration response in $[m/s^2/N]$.
-
-
+%% Derivate two times to to have the acceleration response
 for i = 1:size(Hsyn, 1)
   Hsyn(i, :, :) = squeeze(Hsyn(i, :, :)).*(j*2*pi*freqs).^2;
 end
 
-% Original and Synthesize FRF matrix comparison
 
-acc_o = 1; dir_o = 1; dir_i = 1;
+
+% 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;
 
 figure;
-ax1 = subplot(2, 1, 1);
 hold on;
-plot(freqs, abs(squeeze(frf(3*(acc_o-1)+dir_o,          dir_i, :))), 'DisplayName', 'Original');
-plot(freqs, abs(squeeze(Hsyn(3*(acc_o-1)+dir_o, 3*(11-1)+dir_i, :))), 'DisplayName', 'Synthesize');
+plot(freqs, abs(squeeze(frf( 3*(acc_o-1)+dir_o,             dir_i, :))), 'DisplayName', 'Measured');
+plot(freqs, abs(squeeze(Hsyn(3*(acc_o-1)+dir_o, 3*(acc_i-1)+dir_i, :))), 'DisplayName', 'Synthesized');
 hold off;
+set(gca, 'xscale', 'lin');
 set(gca, 'yscale', 'log');
-set(gca, 'XTickLabel',[]);
+xlabel('Frequency [Hz]');
 ylabel('Magnitude [$\frac{m/s^2}{N}$]');
-title(sprintf('From acc %i %s to acc %i %s', 11, dirs(dir_i), acc_o, dirs(dir_o)))
-legend('location', 'northwest');
-
-ax2 = subplot(2, 1, 2);
-hold on;
-plot(freqs, mod(180/pi*phase(squeeze(frf(3*(acc_o-1)+dir_o,          dir_i, :)))+180, 360)-180);
-plot(freqs, mod(180/pi*phase(squeeze(Hsyn(3*(acc_o-1)+dir_o, 3*(11-1)+dir_i, :)))+180, 360)-180);
-hold off;
-yticks(-360:90:360); ylim([-180, 180]);
-xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
-
-linkaxes([ax1,ax2],'x');
+ldg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+ldg.ItemTokenSize = [10, 1];
+xticks([0:40:200]);
 xlim([1, 200]);
+ylim([1e-6, 1e-1]);
 
-% Synthesize FRF that has not yet been measured
-
-accs = [1]; dirs = [1:3];
+acc_o = 15; dir_o = 3;
+acc_i = 11; dir_i = 3;
 
 figure;
-ax1 = subplot(2, 1, 1);
 hold on;
-for acc_i = accs
-  for dir_i = dirs
-    plot(freqs, abs((1./(j*2*pi*freqs').^2).*squeeze(Hsyn(3*(acc_i-1)+dir_i, 3*(acc_i-1)+dir_i, :))), 'DisplayName', sprintf('Acc %i - %s', acc_i, dirs(dir_i)));
-  end
-end
+plot(freqs, abs(squeeze(frf( 3*(acc_o-1)+dir_o,             dir_i, :))), 'DisplayName', 'Measured');
+plot(freqs, abs(squeeze(Hsyn(3*(acc_o-1)+dir_o, 3*(acc_i-1)+dir_i, :))), 'DisplayName', 'Synthesized');
 hold off;
-set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
-set(gca, 'XTickLabel',[]);
-ylabel('Magnitude [$\frac{m}{N}$]');
-legend('location', 'southwest');
-
-ax2 = subplot(2, 1, 2);
-hold on;
-for acc_i = accs
-  for dir_i = dirs
-    plot(freqs, mod(180/pi*phase((1./(j*2*pi*freqs').^2).*squeeze(Hsyn(3*(acc_i-1)+dir_i, 3*(acc_i-1)+dir_i, :)))+180, 360)-180);
-  end
-end
-hold off;
-yticks(-360:90:360); ylim([-180, 180]);
-set(gca, 'xscale', 'log');
-xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
-
-linkaxes([ax1,ax2],'x');
+set(gca, 'xscale', 'lin');
+set(gca, 'yscale', 'log');
+xlabel('Frequency [Hz]');
+ylabel('Magnitude [$\frac{m/s^2}{N}$]');
+ldg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+ldg.ItemTokenSize = [10, 1];
+xticks([0:40:200]);
 xlim([1, 200]);
+ylim([1e-6, 1e-1]);
+
+acc_o = 2; dir_o = 1;
+acc_i = 11; dir_i = 2;
+
+figure;
+hold on;
+plot(freqs, abs(squeeze(frf( 3*(acc_o-1)+dir_o,             dir_i, :))), 'DisplayName', 'Measured');
+plot(freqs, abs(squeeze(Hsyn(3*(acc_o-1)+dir_o, 3*(acc_i-1)+dir_i, :))), 'DisplayName', 'Synthesized');
+hold off;
+set(gca, 'xscale', 'lin');
+set(gca, 'yscale', 'log');
+xlabel('Frequency [Hz]');
+ylabel('Magnitude [$\frac{m/s^2}{N}$]');
+ldg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+ldg.ItemTokenSize = [10, 1];
+xticks([0:40:200]);
+xlim([1, 200]);
+ylim([1e-6, 1e-1]);

modal-analysis.bib

@@ -1,12 +1,38 @@
+@article{wang11_extrac_real_modes_physic_matric,
+  author =       {Tong Wang and Lingmi Zhang and Kong Fah Tee},
+  title =        {Extraction of Real Modes and Physical Matrices From Modal
+                  Testing},
+  journal =      {Earthquake Engineering and Engineering Vibration},
+  volume =       10,
+  number =       2,
+  pages =        {219-227},
+  year =         2011,
+  doi =          {10.1007/s11803-011-0060-6},
+  url =          {https://doi.org/10.1007/s11803-011-0060-6},
+  DATE_ADDED =   {Tue Jul 9 15:51:21 2019},
+}
+
+@article{pastor12_modal_assur_criter,
+  author = {Miroslav Pastor and Michal Binda and Tom{\'a}{\v{s}} Har{\v{c}}arik},
+  title = {Modal Assurance Criterion},
+  journal = {Procedia Engineering},
+  volume = {48},
+  number = {nil},
+  pages = {543-548},
+  year = {2012},
+  doi = {10.1016/j.proeng.2012.09.551},
+  url = {https://doi.org/10.1016/j.proeng.2012.09.551},
+  DATE_ADDED = {Thu Jul 11 13:51:57 2019},
+}
+
 @book{ewins00_modal,
   author = {Ewins, DJ},
   title = {Modal testing: theory, practice and application},
-  year            = 2000,
+  year = {2000},
   publisher = {Wiley-Blackwell},
-  address         = {Baldock, Hertfordshire, England Philadelphia, PA},
-  isbn            = 0863802184,
   journal = {Research studies Pre, 2nd ed., ISBN-13},
-  keywords        = {favorite, identification},
   pages = {978--0863802188},
+  isbn = {0863802184},
+  address = {Baldock, Hertfordshire, England Philadelphia, PA},
+  keywords = {favorite, identification},
 }
-

modal-analysis.tex

@@ -1,8 +1,20 @@
-% Created 2024-06-26 Wed 17:58
+% Created 2024-10-24 Thu 19:33
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
 \input{preamble.tex}
+\newacronym{nass}{NASS}{Nano Active Stabilization System}
+\newacronym{asd}{ASD}{Amplitude Spectral Density}
+\newacronym{psd}{PSD}{Power Spectral Density}
+\newacronym{frf}{FRF}{Frequency Response Function}
+\newacronym{dof}{DoF}{Degree of freedom}
+\newacronym{svd}{SVD}{Singular Value Decomposition}
+\newacronym{mif}{MIF}{Mode Indicator Functions}
+\newglossaryentry{psdx}{name=\ensuremath{\Phi_{x}},description={{Power spectral density of signal $x$}}}
+\newglossaryentry{asdx}{name=\ensuremath{\Gamma_{x}},description={{Amplitude spectral density of signal $x$}}}
+\newglossaryentry{cpsx}{name=\ensuremath{\Phi_{x}},description={{Cumulative Power Spectrum of signal $x$}}}
+\newglossaryentry{casx}{name=\ensuremath{\Gamma_{x}},description={{Cumulative Amplitude Spectrum of signal $x$}}}
+\input{preamble_extra.tex}
 \bibliography{modal-analysis.bib}
 \author{Dehaeze Thomas}
 \date{\today}
@@ -12,7 +24,7 @@
  pdftitle={Micro-Station - Modal Analysis},
  pdfkeywords={},
  pdfsubject={},
- pdfcreator={Emacs 29.3 (Org mode 9.6)}, 
+ pdfcreator={Emacs 29.4 (Org mode 9.6)}, 
  pdflang={English}}
 \usepackage{biblatex}
 
@@ -23,47 +35,47 @@
 
 \clearpage
 
-In order 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) and springs and damper elements, is a good candidate to model the micro-station.
+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.
 
-Even though the inertia of each solid body can easily be estimated from its geometry and its material density, it is more difficult to properly estimate the stiffness and damping properties of the guiding elements connecting each solid body.
-The experimental modal analysis will be useful to verify that a multi-body model can represent accurately the dynamics of the micro-station and to help tuning the model.
+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.
 
-The approach of tuning the multi-body model from measurements is illustrated in Figure \ref{fig:modal_vibration_analysis_procedure}.
+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, and modal model can be computed, which consists of two matrices: one containing the natural frequencies and damping factors of the considered modes, and another one 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 say to tune the mass spring and damping properties of the considered solid bodies.
+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.
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/modal_vibration_analysis_procedure.png}
-\caption{\label{fig:modal_vibration_analysis_procedure}Figure caption}
+\caption{\label{fig:modal_vibration_analysis_procedure}Three models of the same structure. The goal is to tune a spatial model (i.e. mass, stiffness and damping properties) from a response model. The modal model can be used as an intermediate step.}
 \end{figure}
 
-The measurement setup used to obtain the response model is presented in Section \ref{sec:modal_meas_setup}.
-This includes the instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system), the test planing, and a first analysis of the obtained signals.
+The measurement setup used to obtain the response model is described in Section \ref{sec:modal_meas_setup}.
+This includes the instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system), test planing, and a first analysis of the obtained signals.
 
-In Section \ref{sec:modal_frf_processing}, the obtained frequency response functions between the forces applied using the instrumented hammer and the various accelerometers fixed to the structure are computed.
-These measurements are projected at the center of mass of each considered solid body to ease the further use of the results.
+In Section \ref{sec:modal_frf_processing}, the obtained frequency response functions between the forces applied by the instrumented hammer and the accelerometers fixed to the structure are computed.
+These measurements are projected at the center of mass of each considered solid body to facilitate the further use of the results.
 The solid body assumption is then verified, validating the use of the multi-body model.
 
 Finally, the modal analysis is performed in Section \ref{sec:modal_analysis}.
-It shows how complex the micro-station dynamics is, and the necessity of the developed more complex multi-body model.
+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 proper measurement setup is key.
-This include 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 location of the measured motion (Section \ref{ssec:modal_accelerometers}) and the location of the hammer impacts (Section \ref{ssec:modal_hammer_impacts}) have to be chosen carefully.
-Obtained force and acceleration signals are shown in Section \ref{ssec:modal_measured_signals}, and the quality of the measured data is verified.
+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}
 \label{ssec:modal_instrumentation}
 
-Three equipment are key to perform a good modal analysis.
+Three type 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 393B05. Sensitivity is \(10\,V/g\), measurement range is \(0.5\,g\) and bandwidth is \(0.7\) to \(450\,\text{Hz}\).} shown in figure \ref{fig:modal_accelero_M393B05} are used.
-These accelerometers are glued to the micro-station using a thin layer of wax for best results \cite[chapt. 3.5.7]{ewins00_modal}.
+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}.
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.33\textwidth}
@@ -87,56 +99,54 @@ These accelerometers are glued to the micro-station using a thin layer of wax fo
 \caption{\label{fig:modal_analysis_instrumentation}Instrumentation used for the modal analysis}
 \end{figure}
 
-Then, an \emph{instrumented hammer}\footnote{Kistler 9722A2000. Sensitivity of \(2.3\,mV/N\) and measurement range of \(2\,kN\)} (figure \ref{fig:modal_instrumented_hammer}) is used to apply forces to the structure in a controlled way.
-Tests have been conducted to determine the most suitable hammer tip (ranging from a metallic one to a soft plastic one).
-The softer tip has been 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.
+Then, an \emph{instrumented hammer}\footnote{Kistler 9722A2000. Sensitivity of \(2.3\,mV/N\) and measurement range of \(2\,kN\)} (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 \emph{acquisition system}\footnote{OROS OR36. 24bits signal-delta ADC.} (figure \ref{fig:modal_oros}) is used to acquire the injected force and the response accelerations in a synchronized way and with sufficiently low noise.
+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}
 
-In order to obtain meaningful results, the modal analysis of the micro-station in performed \emph{in-situ}.
-To do so, all the micro-station stages controllers are turned ``ON''.
-This is especially important for stages for which the stiffness is provided by local feedback control, which is case for the air bearing spindle, and the translation stage.
-If these local feedback control were turned OFF, this would have resulted in very low frequency modes difficult to measure in practice, and this would also have lead to decoupled dynamics which would not be the case in practice.
+To obtain meaningful results, the modal analysis of the micro-station in 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.
 
-The top part representing the active stabilization stage has been disassembled in order to reduce the complexity of the dynamics and also because the active stabilization stage and the sample will be added in the multi-body model afterwards.
+The top part representing the active stabilization stage was disassembled as the active stabilization stage will be added in the multi-body model afterwards.
 
+To perform the modal analysis from the measured responses, the \(n \times n\) frequency response function matrix \(\mathbf{H}\) needs to be measured, where \(n\) is the considered number of degrees of freedom.
+The \(H_{jk}\) element of this \acrfull{frf} matrix corresponds to the frequency response function from a force \(F_k\) applied at \acrfull{dof} \(k\) to the displacement of the structure \(X_j\) at \acrshort{dof} \(j\).
+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}.
 
-To perform the modal-analysis from the measured responses, the \(n \times n\) frequency response function matrix \(\mathbf{H}\) needs to be measured, where \(n\) is the considered number of degrees of freedom.
-The \(H_{jk}\) element of this FRF matrix corresponds to the frequency response function from a force \(F_k\) applied at DoF \(k\) to the displacement of the structure \(X_j\) at DoF \(j\).
-
-Measuring this FRF matrix is very time consuming as it requires to make \(n^2\) measurements.
-However thanks 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 needs to be measured.
-This can be done either by measuring the response \(X_{j}\) at a fixed DoF \(j\) while applying forces \(F_{i}\) for at all \(n\) considered DoF, or by applying a force \(F_{k}\) at a fixed DoF \(k\) and measuring the response \(X_{i}\) for all \(n\) 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 measured locations.
-In this modal-analysis, it is chosen to measure the response of the structure at all considered DoF, and to excite the structure at one location in three directions in order to have some redundancy and to make sure that all modes are properly excited.
+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}
 
-The location of the accelerometers fixed to the micro-station is essential as it defines where the dynamics is identified.
-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}.
+The location of the accelerometers fixed to the micro-station is essential because it defines where the dynamics is measured.
+A total of 23 accelerometers were 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 positions 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 DoF can be considered per solid body.
-However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured 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}).
+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}[b]{0.68\linewidth}
+\begin{minipage}[t]{0.60\linewidth}
 \begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/modal_location_accelerometers.png}
-\captionof{figure}{\label{fig:modal_location_accelerometers}Position of the accelerometers using SolidWorks}
+\includegraphics[scale=1,width=0.99\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]{0.31\linewidth}
-\begin{center}
-\scriptsize
-\begin{tabularx}{\linewidth}{Xcccc}
+\begin{minipage}[b]{0.38\linewidth}
+\begin{scriptsize}
+\captionof{table}{\label{tab:modal_position_accelerometers}Positions in mm}
+\begin{tabularx}{\linewidth}{Xccc}
 \toprule
  & \(x\) & \(y\) & \(z\)\\
 \midrule
@@ -165,13 +175,9 @@ Hexapod & 64 & 64 & -270\\
 Hexapod & 64 & -64 & -270\\
 \bottomrule
 \end{tabularx}
-\captionof{table}{\label{tab:modal_position_accelerometers}Accelerometer positions}
-
-\end{center}
+\end{scriptsize}
 \end{minipage}
 
-
-
 \begin{figure}[htbp]
 \begin{subfigure}{0.49\textwidth}
 \begin{center}
@@ -185,17 +191,16 @@ Hexapod & 64 & -64 & -270\\
 \end{center}
 \subcaption{\label{fig:modal_accelerometers_hexapod} Micro-Hexapod}
 \end{subfigure}
-\caption{\label{fig:modal_accelerometer_pictures}Accelerometers fixed on the micro-station}
+\caption{\label{fig:modal_accelerometer_pictures}Accelerometers fixed on the micro-station stages}
 \end{figure}
 
 \section{Hammer Impacts}
 \label{ssec:modal_hammer_impacts}
 
-The chosen location of the hammer impact corresponds to the location of accelerometer number \(11\) fixed to the translation stage.
-It was chosen to match the location of one accelerometer, because a \emph{point measurement} (i.e. a measurement of \(H_{kk}\)) is necessary to be able to reconstruct the full FRF matrix \cite{ewins00_modal}.
+The selected location of the hammer impact corresponds to the location of accelerometer number \(11\) fixed to the translation stage.
+It was chosen to match the location of one accelerometer, because a \emph{point measurement} (i.e. a measurement of \(H_{kk}\)) is necessary to be able to reconstruct the full \acrshort{frf} matrix \cite{ewins00_modal}.
 
-The impacts are performed in three directions, which are shown in figures \ref{fig:modal_impact_x}, \ref{fig:modal_impact_y} and \ref{fig:modal_impact_z}.
-This excitation point and the three considered directions allows to properly excite all the modes in the frequency band of interest and to provide good coherence for all the accelerometers as will be shown in the next section.
+The impacts were performed in three directions, as shown in figures \ref{fig:modal_impact_x}, \ref{fig:modal_impact_y} and \ref{fig:modal_impact_z}.
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.33\textwidth}
@@ -222,12 +227,13 @@ This excitation point and the three considered directions allows to properly exc
 \section{Force and Response signals}
 \label{ssec:modal_measured_signals}
 
-The force sensor 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, many resonances can be seen on the right, indicating complex dynamics.
+The force sensor of the instrumented hammer and the accelerometer signals are shown in the time domain in Figure \ref{fig:modal_raw_meas}.
+Sharp ``impacts'' can be observed 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'' amplitude spectral density 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).
+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).
+Similar results were obtained for all measured frequency response functions.
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.49\textwidth}
@@ -242,11 +248,11 @@ Conclusions based on the time domain signals can be clearly seen in the frequenc
 \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 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})}
+\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})}
 \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 \emph{coherence} function, which is shown in Figure \ref{fig:modal_coh_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.
 
 \begin{figure}[htbp]
@@ -267,11 +273,11 @@ Good coherence is obtained from \(20\,\text{Hz}\) to \(200\,\text{Hz}\) which co
 
 \chapter{Frequency Analysis}
 \label{sec:modal_frf_processing}
-All measurements where conducted and a \(n \times p \times q\) Frequency Response Functions Matrix were computed with:
+After all measurements are conducted, a \(n \times p \times q\) \acrlongpl{frf} matrix can be computed with:
 \begin{itemize}
-\item \(n = 69\): the number of output measured accelerations (23 3-axis accelerometers)
-\item \(p = 3\): the number of input force excitations
-\item \(q = 801\): the number of frequency points \(\omega_{i}\)
+\item \(n = 69\): number of output measured acceleration (23 3-axis accelerometers)
+\item \(p = 3\): number of input force excitation
+\item \(q = 801\): number of frequency points \(\omega_{i}\)
 \end{itemize}
 
 For each frequency point \(\omega_{i}\), a 2D complex matrix is obtained that links the 3 force inputs to the 69 output accelerations \eqref{eq:modal_frf_matrix_raw}.
@@ -287,39 +293,33 @@ 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 hexapod.
+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.
 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.
 
-The objective in this section is therefore to process the Frequency Response Matrix to reduce the number of measured DoFs from 69 to 36.
-
-In order to be able to perform this reduction of measured DoFs, the rigid body assumption first needs to be verified (Section \ref{ssec:modal_solid_body_first_check}).
-
-The coordinate transformation from accelerometers DoFs to the solid body 6 DoFs (three translations and three rotations) is performed in Section \ref{ssec:modal_acc_to_solid_dof}.
+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.
 
-To further validate this reduction of DoF and the solid body assumption, the frequency response function at the accelerometer location are synthesized from the reduced frequency response matrix and are compared with the initial measurements in Section \ref{ssec:modal_solid_body_assumption}.
-\section{First verification of the solid body assumption}
-\label{ssec:modal_solid_body_first_check}
+To validate this reduction of \acrshort{dof} and the solid body assumption, the frequency response function at the accelerometer location are ``reconstructed'' from the reduced frequency response matrix and are compared with the initial measurements in Section \ref{ssec:modal_solid_body_assumption}.
+\section{From accelerometer DOFs to solid body DOFs}
+\label{ssec:modal_acc_to_solid_dof}
 
-In this section, it is shown that two accelerometers fixed to a \emph{rigid body} at positions \(\vec{p}_1\) and \(\vec{p}_2\) such that \(\vec{p}_2 = \vec{p}_1 + \alpha \vec{x}\) will measure the same acceleration in the \(\vec{x}\) direction.
-Such situation is illustrated in Figure \ref{fig:modal_aligned_accelerometers}.
+Let us consider the schematic shown in Figure \ref{fig:modal_local_to_global_coordinates} where the motion of a solid body is measured at 4 distinct locations (in \(x\), \(y\) and \(z\) directions).
+The goal here is to link these \(4 \times 3 = 12\) measurements to the 6 \acrshort{dof} of the solid body expressed in the frame \(\{O\}\).
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/modal_aligned_accelerometers.png}
-\caption{\label{fig:modal_aligned_accelerometers}Aligned measurement of the motion of a solid body}
+\includegraphics[scale=1]{figs/modal_local_to_global_coordinates.png}
+\caption{\label{fig:modal_local_to_global_coordinates}Schematic of the measured motions of a solid body}
 \end{figure}
 
-The motion of the rigid body of figure \ref{fig:modal_aligned_accelerometers} is here 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 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 of points \(p_1\) and \(p_2\) can be computed from \(\vec{\delta} p\) and \(\bm{\delta \Omega}\) \eqref{eq:modal_p1_p2_motion}, with \(\bm{\delta\Omega}\) defined in \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}.
 
-\begin{subequations}\label{eq:modal_p1_p2_motion}
-\begin{align}
-  \vec{\delta} p_{1} &= \vec{\delta} p + \bm{\delta \Omega} \cdot \vec{p}_{1} \\
-  \vec{\delta} p_{2} &= \vec{\delta} p + \bm{\delta \Omega} \cdot \vec{p}_{2}
-\end{align}
-\end{subequations}
+\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}
@@ -329,52 +329,12 @@ The motion of points \(p_1\) and \(p_2\) can be computed from \(\vec{\delta} p\)
 \end{bmatrix}
 \end{equation}
 
-Considering only the \(x\) direction, equation \eqref{eq:modal_p1_p2_x_motion} is obtained.
-
-\begin{subequations}\label{eq:modal_p1_p2_x_motion}
-\begin{align}
-  \delta p_{x1} &= \delta p_x + \delta \Omega_y p_{z1} - \delta \Omega_z p_{y1} \\
-  \delta p_{x2} &= \delta p_x + \delta \Omega_y p_{z2} - \delta \Omega_z p_{y2}
-\end{align}
-\end{subequations}
-
-Because the two sensors are co-linearity in the \(x\) direction, \(p_{1y} = p_{2y}\) and \(p_{1z} = p_{2z}\), and \eqref{eq:modal_colinear_sensors_equal} is obtained.
-
-\begin{equation}\label{eq:modal_colinear_sensors_equal}
-  \boxed{\delta p_{x1} = \delta p_{x2}}
-\end{equation}
-
-It is therefore concluded that two position sensors fixed to a rigid body will measure the same quantity in the direction ``in line'' the two sensors.
-
-
-Such property can be used to verify that the considered stages are indeed behaving as rigid body in the frequency band of interest.
-From Table \ref{tab:modal_position_accelerometers}, the pairs of accelerometers that aligned in the X and Y directions can be identified.
-The response in the X direction of pairs of sensors aligned in the X direction are compared in Figure \ref{fig:modal_solid_body_comp_x_dir}.
-A good match is observed up to 200Hz.
-Similar result is obtained for the Y direction.
-This therefore indicates that the considered bodies are behaving as solid bodes in the frequency range of interest.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/modal_solid_body_comp_x_dir.png}
-\caption{\label{fig:modal_solid_body_comp_x_dir}Comparaison of measured frequency response function for in the X directions for accelerometers aligned along X. Amplitude is in \(\frac{m/s^2}{N}\). Accelerometer number is shown in the legend.}
-\end{figure}
-
-\section{From accelerometer DOFs to solid body DOFs}
-\label{ssec:modal_acc_to_solid_dof}
-
-Let's consider the schematic shown in Figure \ref{fig:modal_local_to_global_coordinates} where the motion of a solid body is measured at 4 distinct locations (in \(x\), \(y\) and \(z\) directions).
-The goal here is to link these \(4 \times 3 = 12\) measurements to the 6 DoFs of the solid body expressed in the frame \(\{O\}\).
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/modal_local_to_global_coordinates.png}
-\caption{\label{fig:modal_local_to_global_coordinates}Schematic of the measured motions of a solid body}
-\end{figure}
-
-Writing Eq. \eqref{eq:modal_p1_p2_motion} for the four displacement sensors in a matrix form gives \eqref{eq:modal_cart_to_acc}.
+Writing this in 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} \\
@@ -385,15 +345,13 @@ Writing Eq. \eqref{eq:modal_p1_p2_motion} for the four displacement sensors in a
   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] = \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]
 \end{equation}
 
-Supposing that the four sensors are properly located such that the system of equation \eqref{eq:modal_cart_to_acc} can be solved, the motion of the solid body expressed in a chosen frame \(\{O\}\) using the accelerometers attached to it can be determined using equation \eqref{eq:modal_determine_global_disp}.
-Note that this 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.
-
+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}.
+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}
@@ -414,9 +372,10 @@ Therefore, having more accelerometers permits to have a better approximation of
 
 
 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.
+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}
 \toprule
@@ -430,14 +389,12 @@ Spindle & 0 & 0 & -580\\
 Hexapod & -4 & 6 & -319\\
 \bottomrule
 \end{tabularx}
-\caption{\label{tab:modal_com_solid_bodies}Center of mass of considered solid bodies with respect to the ``point of interest''}
-
 \end{table}
 
-Using \eqref{eq:modal_determine_global_disp}, the frequency response matrix \(\bm{H}_\text{CoM}\) \eqref{eq:modal_frf_matrix_com} expressing the response \(D_i\) (\(i\) from \(1\) to \(6\) for the \(6\) considered solid bodies) at the center of mass of each solid body can be computed from the initial FRF matrix \(\bm{H}\).
+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}
-  \bm{H}_\text{CoM}(\omega_i) = \begin{bmatrix}
+  \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) \\
@@ -453,145 +410,66 @@ Using \eqref{eq:modal_determine_global_disp}, the frequency response matrix \(\b
 \section{Verification of solid body assumption}
 \label{ssec:modal_solid_body_assumption}
 
-From the response of one solid body along its 6 DoFs (from \(\bm{H}_{\text{CoM}}\)), and using \eqref{eq:modal_cart_to_acc}, it is possible to compute the response of the same solid body at any location, in particular at the location of the accelerometers fixed to this solid body.
+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 location.
+In particular, the responses at the locations of the four accelerometers can be computed and compared with the original measurements \(\mathbf{H}\).
+This is what is done here to check whether the solid body assumption is correct in the frequency band of interest.
 
-Comparing the computed response of a particular accelerometer from \(\bm{H}_{\text{CoM}}\) with the original measurements \(\bm{H}\) is useful to check if the change of coordinate \eqref{eq:modal_determine_global_disp} works as expected, and if the solid body assumption is correct in the frequency band of interest.
-
-The comparison is made for the 4 accelerometers fixed to the micro-hexapod in 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 3DoF) to 36 (6 solid bodies with 6 DoF).
+The comparison is made for the 4 accelerometers fixed on the micro-hexapod (Figure \ref{fig:modal_comp_acc_solid_body_frf}).
+The original frequency response functions and those computed from the CoM responses match well in the frequency range of interest.
+Similar results were 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 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{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 reconstructed response from the solid body response (dashed curves). For accelerometers 1 to 4 corresponding to the micro-hexapod.}
+\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.}
 \end{figure}
 
 \chapter{Modal Analysis}
 \label{sec:modal_analysis}
-The goal here is to extract the modal parameters describing the modes of station being studied, namely:
-\begin{itemize}
-\item the eigen frequencies and the modal damping (eigen values)
-\item the mode shapes (eigen vectors)
-\end{itemize}
-This is done from the FRF matrix previously extracted from the measurements.
+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).
+This is performed from the \acrshort{frf} matrix previously extracted from the measurements.
 
+In order to perform the modal parameter extraction, the order of the modal model has to be estimated (i.e. the number of modes in the frequency band of interest).
+This is achived using the \acrfull{mif} in section \ref{ssec:modal_number_of_modes}.
 
-In order to do the modal parameter extraction, we first have to estimate the order of the modal model we want to obtain.
-This corresponds to how many modes are present in the frequency band of interest.
-In section \ref{ssec:modal_number_of_modes}, we will use the Singular Value Decomposition and the Modal Indication Function to estimate the 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 modal parameter extraction methods generally consists of \textbf{curve-fitting a theoretical expression for an individual FRF to the actual measured data}.
-However, there are multiple level of complexity:
-\begin{itemize}
-\item works on a part of a single FRF curve
-\item works on a complete curve encompassing several resonances
-\item works on a set of many FRF plots all obtained from the same structure
-\end{itemize}
-The third method is the most complex but gives better results. This is the one we will use in section \ref{ssec:modal_parameter_extraction}.
-
-From the modal model, it is possible to obtain a graphic display of the mode shapes (section \ref{ssec:modal_mode_shapes}).
-
-
-In order to validate the quality of the modal model, we will synthesize the FRF matrix from the modal model and compare it with the FRF measured (section \ref{ssec:modal_model_validity}).
-
-
-The modes of the structure are expected to be complex, however real modes are easier to work with when it comes to obtain a spatial model from the modal parameters.
-\section{Determine the number of 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}
-\paragraph{Singular Value Decomposition - Modal Indication Function}
-The Mode Indicator Functions are usually used on \(n\times p\) FRF matrix where \(n\) is a relatively large number of measurement DOFs and \(p\) is the number of excitation DOFs, typically 3 or 4.
+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 \(p=3\)).
 
-In these methods, the frequency dependent FRF matrix is subjected to a singular value decomposition analysis which thus yields a small number (3 or 4) of singular values, these also being frequency dependent.
-
-These methods are used to \textbf{determine the number of modes} present in a given frequency range, to \textbf{identify repeated natural frequencies} and to pre-process the FRF data prior to modal analysis.
-
-From the documentation of the modal software:
-\begin{quote}
-The MIF consist of the singular values of the Frequency response function matrix. The number of MIFs equals the number of excitations.
-By the powerful singular value decomposition, the real signal space is separated from the noise space. Therefore, the MIFs exhibit the modes effectively.
-A peak in the MIFs plot usually indicate the existence of a structural mode, and two peaks at the same frequency point means the existence of two repeated modes.
-Moreover, the magnitude of the MIFs implies the strength of the a mode.
-\end{quote}
-
-\begin{important}
-The \textbf{Complex Mode Indicator Function} is defined simply by the SVD of the FRF (sub) matrix:
-\begin{align*}
-  [H(\omega)]_{n\times p} &= [U(\omega)]_{n\times n} [\Sigma(\omega)]_{n\times p} [V(\omega)]_{p\times p}^H\\
-  [CMIF(\omega)]_{p\times p} &= [\Sigma(\omega)]_{p\times n}^T [\Sigma(\omega)]_{n\times p}
-\end{align*}
-\end{important}
-
-We compute the Complex Mode Indicator Function.
-The result is shown on Figure \ref{fig:modal_indication_function}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/modal_indication_function.png}
-\caption{\label{fig:modal_indication_function}Modal Indication Function}
-\end{figure}
-
-\paragraph{Composite Response Function}
-An alternative is the Composite Response Function \(HH(\omega)\) defined as the sum of all the measured FRF:
-\begin{equation}
-  HH(\omega) = \sum_j\sum_kH_{jk}(\omega)
+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}
 
-Instead, we choose here to use the sum of the norms of the measured frf:
-\begin{equation}
-  HH(\omega) = \sum_j\sum_k \left|H_{jk}(\omega) \right|
+\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 result is shown on figure \ref{fig:modal_composite_reponse_function}.
+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 when multiple singular values have peaks at the same frequency.
+The obtained \acrshort{mif} is shown on Figure \ref{fig:modal_indication_function}.
+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}[htbp]
-\centering
-\includegraphics[scale=1]{figs/modal_composite_reponse_function.png}
-\caption{\label{fig:modal_composite_reponse_function}Composite Response Function}
-\end{figure}
-
-\section{Modal parameter extraction}
-\label{ssec:modal_parameter_extraction}
-\paragraph{OROS - Modal software}
-Modal identification are done within the Modal software of OROS.
-
-Several modal parameter extraction methods are available.
-We choose to use the ``broad band'' method as it permits to identify the modal parameters using all the FRF curves at the same time.
-It takes into account the fact the the properties of all the individual curves are related by being from the same structure: all FRF plots on a given structure should indicate the same values for the natural frequencies and damping factor of each mode.
-
-Such method also have the advantage of producing a \textbf{unique and consistent model} as direct output.
-
-In order to apply this method, we select the frequency range of interest and we give an estimate of how many modes are present.
-
-Then, it shows a stabilization charts, such as the one shown on figure \ref{fig:modal_stabilization_chart}, where we have to manually select which modes to take into account in the modal model.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=\linewidth]{img/modal_software/stabilisation_chart.jpg}
-\caption{\label{fig:modal_stabilization_chart}Stabilization Chart}
-\end{figure}
-
-We can then run the modal analysis, and the software will identify the modal damping and mode shapes at the selected frequency modes.
-
-\paragraph{Importation of the modal parameters on Matlab}
-The obtained modal parameters are:
-\begin{itemize}
-\item Resonance frequencies in Hertz
-\item Modal damping ratio in percentage
-\item (complex) Modes shapes for each measured DoF
-\item Modal A and modal B which are parameters important for further normalization
-\end{itemize}
-
-The obtained mode frequencies and damping are shown in Table \ref{tab:modal_obtained_modes_freqs_damps}.
-
-\begin{table}[htbp]
-\centering
-\scriptsize
-\begin{tabularx}{0.35\linewidth}{ccc}
+\begin{minipage}[t]{0.70\linewidth}
+\begin{center}
+\includegraphics[scale=1,width=0.95\linewidth]{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}
+\begin{tabularx}{\linewidth}{ccc}
 \toprule
-Mode & Frequency [Hz] & Damping [\%]\\
+Mode & Freq. [Hz] & Damp. [\%]\\
 \midrule
 1 & 11.9 & 12.2\\
 2 & 18.6 & 11.7\\
@@ -611,107 +489,47 @@ Mode & Frequency [Hz] & Damping [\%]\\
 16 & 165.4 & 1.4\\
 \bottomrule
 \end{tabularx}
-\caption{\label{tab:modal_obtained_modes_freqs_damps}Obtained eigen frequencies and modal damping}
+\end{scriptsize}
+\end{minipage}
 
-\end{table}
+\section{Modal parameter extraction}
+\label{ssec:modal_parameter_extraction}
 
-\paragraph{Theory}
-It seems that the modal analysis software makes the \textbf{assumption} of viscous damping for the model with which it tries to fit the FRF measurements.
+Generally, modal identification consists of 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.
 
-If we note \(N\) the number of modes identified, then there are \(2N\) eigenvalues and eigenvectors given by the software:
-\begin{align}
-  s_r &= \omega_r (-\xi_r + i \sqrt{1 - \xi_r^2}),\quad s_r^* \\
-  \{\psi_r\} &= \begin{Bmatrix} \psi_{1_x} & \psi_{2_x} & \dots & \psi_{23_x} & \psi_{1_y} & \dots & \psi_{1_z} & \dots & \psi_{23_z} \end{Bmatrix}^T, \quad \{\psi_r\}^*
-\end{align}
-for \(r = 1, \dots, N\) where \(\omega_r\) is the natural frequency and \(\xi_r\) is the critical damping ratio for that mode.
+Here, the last method is used because it provides a unique and consistent model.
+It takes into account the fact that the properties of all individual curves are related by being from the same structure: all \acrshort{frf} plots on a given structure should indicate the same values for the natural frequencies and damping factor of each mode.
 
-\paragraph{Modal Matrices}
-We would like to arrange the obtained modal parameters into two modal matrices:
-\[ \Lambda = \begin{bmatrix}
-  s_1 &        & 0 \\
-      & \ddots &   \\
-  0   &        & s_N
-\end{bmatrix}_{N \times N}; \quad \Psi = \begin{bmatrix}
-  & & \\
-  \{\psi_1\} & \dots & \{\psi_N\} \\
-  & &
-\end{bmatrix}_{M \times N} \]
-\[ \{\psi_i\} = \begin{Bmatrix} \psi_{i, 1_x} & \psi_{i, 1_y} & \psi_{i, 1_z} & \psi_{i, 2_x} & \dots & \psi_{i, 23_z} \end{Bmatrix}^T \]
-
-\(M\) is the number of DoF: here it is \(23 \times 3 = 69\).
-\(N\) is the number of mode
-
-Each eigen vector is normalized: \(\| \{\psi_i\} \|_2 = 1\)
-
-
-However, the eigen values and eigen vectors appears as complex conjugates:
-\[ s_r, s_r^*, \{\psi\}_r, \{\psi\}_r^*, \quad r = 1, N \]
-
-In the end, they are \(2N\) eigen values.
-We then build two extended eigen matrices as follow:
-\[ \mathcal{S} = \begin{bmatrix}
-  s_1 &        &     &       &        &      \\
-      & \ddots &     &       & 0      &      \\
-      &        & s_N &       &        &      \\
-      &        &     & s_1^* &        &      \\
-      & 0      &     &       & \ddots &      \\
-      &        &     &       &        & s_N^*
-\end{bmatrix}_{2N \times 2N}; \quad \Phi = \begin{bmatrix}
-  & & & & &\\
-  \{\psi_1\} & \dots & \{\psi_N\} & \{\psi_1^*\} & \dots & \{\psi_N^*\} \\
-  & & & & &
-\end{bmatrix}_{M \times 2N} \]
-
-We also build the Modal A and Modal B matrices:
-\begin{equation}
-  A = \begin{bmatrix}
-    a_1 &        & 0   \\
-        & \ddots &     \\
-    0   &        & a_N
-  \end{bmatrix}_{N \times N}; \quad B = \begin{bmatrix}
-    b_1 &        & 0   \\
-        & \ddots &     \\
-    0   &        & b_N
-  \end{bmatrix}_{N \times N}
-\end{equation}
-With \(a_i\) is the ``Modal A'' parameter linked to mode i.
-
-``Modal A'' and ``modal B'' are linked through the following formula:
-\[ B = - A \Lambda \]
-
-\section{Obtained Mode Shapes animations}
-\label{ssec:modal_mode_shapes}
-From the modal parameters, it is possible to show the modal shapes with an animation.
-
-Examples are shown in Figures \ref{fig:modal_mode_animations}.
+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{subfigure}{\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=\linewidth]{figs/modal_mode1_animation.jpg}
 \end{center}
-\subcaption{\label{fig:modal_mode1_animation}Mode 1}
+\subcaption{\label{fig:modal_mode1_animation}$1^{st}$ mode at 11.9 Hz: tilt suspension mode of the granite}
 \end{subfigure}
 \begin{subfigure}{\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=\linewidth]{figs/modal_mode6_animation.jpg}
 \end{center}
-\subcaption{\label{fig:modal_mode6_animation}Mode 6}
+\subcaption{\label{fig:modal_mode6_animation}$6^{th}$ mode at 69.8 Hz: vertical resonance of the spindle}
 \end{subfigure}
 \begin{subfigure}{\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=\linewidth]{figs/modal_mode13_animation.jpg}
 \end{center}
-\subcaption{\label{fig:modal_mode13_animation}Mode 13}
+\subcaption{\label{fig:modal_mode13_animation}$13^{th}$ mode at 124.2 Hz: lateral micro-hexapod resonance}
 \end{subfigure}
-\caption{\label{fig:modal_mode_animations}Instrumentation used for the modal analysis}
+\caption{\label{fig:modal_mode_animations}Three obtained mode shape animations}
 \end{figure}
 
-We can learn quite a lot from these mode shape animations.
-
-For instance, the mode shape of the first mode at 11Hz (figure \ref{fig:modal_mode1_animation}) seems to indicate that this corresponds to a suspension mode.
-
-This could be due to the 4 Airloc Levelers that are used for the granite (figure \ref{fig:modal_airloc}).
+These animations are useful for visually obtaining a better understanding of the system's dynamic behavior.
+For instance, the mode shape of the first mode at \(11\,\text{Hz}\) (figure \ref{fig:modal_mode1_animation}) indicates an issue with the lower granite.
+It turns out that four \emph{Airloc Levelers} are used to level the lower granite (figure \ref{fig:modal_airloc}).
+These are difficult to adjust and can lead to a situation in which the granite is only supported by two of them; therefore, it  has a low frequency ``tilt mode''.
+The levelers were then better adjusted.
 
 \begin{figure}[htbp]
 \centering
@@ -719,91 +537,87 @@ This could be due to the 4 Airloc Levelers that are used for the granite (figure
 \caption{\label{fig:modal_airloc}AirLoc used for the granite (2120-KSKC)}
 \end{figure}
 
-They are probably \textbf{not well leveled}, so the granite is supported only by two Airloc.
+The modal parameter extraction is made using a proprietary software\footnote{NVGate software from OROS company.}.
+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 \(a_r\).
 
-\section{Verify the validity of the Modal Model}
+\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
+\end{equation}
+
+The eigenvalues \(s_r\) and \(s_r^*\) can then be computed from equation \eqref{eq:modal_eigenvalues}.
+
+\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}
 
-There are two main ways to verify the validity of the modal model
-\begin{itemize}
-\item Synthesize FRF measurements that has been used to generate the modal model and compare
-\item Synthesize FRF that has not yet been measured. Then measure that FRF and compare
-\end{itemize}
+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 to the measured \acrshort{frf} matrix \(\mathbf{H}\).
 
-From the modal model, we want to synthesize the Frequency Response Functions that has been used to build the modal model.
-
-Let's recall that:
-\begin{itemize}
-\item \(M\) is the number of measured DOFs (\(3 \times n_\text{acc}\))
-\item \(N\) is the number of modes identified
-\end{itemize}
-
-We then have that the FRF matrix \([H_{\text{syn}}]\) can be synthesize using the following formula:
-\begin{important}
-\begin{equation}
-  [H_{\text{syn}}(\omega)]_{M\times M} = [\Phi]_{M\times2N} \left[\frac{Q_r}{j\omega - s_r}\right]_{2N\times2N} [\Phi]_{2N\times M}^T
+In order to synthesize the full \acrshort{frf} matrix, the eigenvectors \(\phi_r\) are first organized in 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}
-with \(Q_r = 1/M_{A_r}\)
-\end{important}
 
-An alternative formulation is:
-\[ H_{pq}(s_i) = \sum_{r=1}^N \frac{A_{pqr}}{s_i - \lambda_r} + \frac{A_{pqr}^*}{s_i - \lambda_r^*} \]
-with:
-\begin{itemize}
-\item \(A_{pqr} = \frac{\psi_{pr}\psi_{qr}}{M_{A_r}}\), \(M_{A_r}\) is called ``Modal A''
-\item \(\psi_{pr}\): scaled modal coefficient for output DOF \(p\), mode \(r\)
-\item \(\lambda_r\): complex modal frequency
-\end{itemize}
+The full \acrshort{frf} matrix \(\mathbf{H}_{\text{syn}}\) can be synthesize using \eqref{eq:modal_synthesized_frf}.
 
-From the modal software documentation:
-\begin{quote}
-\textbf{Modal A}
-Scaling constant for a complex mode. It has the same properties as modal mass for normal modes (undamped or proportionally damped cases). Assuming
-\begin{itemize}
-\item \(\psi_{pr}\) = Modal coefficient for measured degree of freedom p and mode r
-\item \(\psi_{qr}\) = Modal coefficient for measured degree of freedom q and mode r
-\item \(A_{pqr}\) = Residue for measured degree of freedom p, measured degree of q and mode r
-\item \(M_{Ar}\) = Modal A of mode r
-\end{itemize}
+\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}
 
-Then
-\[ A_{pqr} = \frac{\psi_{pr}\psi_{qr}}{M_{Ar}} \]
+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}
 
-
-\textbf{Modal B}
-Scaling constant for a complex mode. It has the same properties as modal stiffness for normal modes (undamped or proportionally damped cases). Assuming
-\begin{itemize}
-\item \(M_{Ar}\) = Modal A of mode r
-\item \(\lambda_r\) = System pole of mode r
-\end{itemize}
-
-Then
-\[ M_{Br} = - \lambda_r M_{Ar} \]
-\end{quote}
-
-The comparison between the original measured frequency response function and the synthesized one from the modal model is done in Figure \ref{fig:modal_comp_acc_frf_modal}.
+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}.
+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.
 
 \begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/modal_comp_acc_frf_modal.png}
-\caption{\label{fig:modal_comp_acc_frf_modal}description}
-\end{figure}
-
-Frequency response functions that has not been measured can be synthesized as shown in Figure \ref{fig:modal_synthesized_frf}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/modal_synthesized_frf.png}
-\caption{\label{fig:modal_synthesized_frf}description}
+\begin{subfigure}{0.33\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.99\linewidth]{figs/modal_comp_acc_frf_modal_1.png}
+\end{center}
+\subcaption{\label{fig:modal_comp_acc_frf_modal_1}From $F_{11,z}$ to $a_{11,z}$}
+\end{subfigure}
+\begin{subfigure}{0.33\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.99\linewidth]{figs/modal_comp_acc_frf_modal_2.png}
+\end{center}
+\subcaption{\label{fig:modal_comp_acc_frf_modal_2}From $F_{11,z}$ to $a_{15,z}$}
+\end{subfigure}
+\begin{subfigure}{0.33\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.99\linewidth]{figs/modal_comp_acc_frf_modal_3.png}
+\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.}
 \end{figure}
 
 \chapter{Conclusion}
 \label{sec:modal_conclusion}
 
-Validation of solid body model.
+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.
+It shows a lot of coupling between stages and different directions, and many modes.
 
-Further step: go from modal model to parameters of the solid body model.
+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.
 
+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

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ref.bib

@@ -1,38 +0,0 @@
-@article{wang11_extrac_real_modes_physic_matric,
-  author =       {Tong Wang and Lingmi Zhang and Kong Fah Tee},
-  title =        {Extraction of Real Modes and Physical Matrices From Modal
-                  Testing},
-  journal =      {Earthquake Engineering and Engineering Vibration},
-  volume =       10,
-  number =       2,
-  pages =        {219-227},
-  year =         2011,
-  doi =          {10.1007/s11803-011-0060-6},
-  url =          {https://doi.org/10.1007/s11803-011-0060-6},
-  DATE_ADDED =   {Tue Jul 9 15:51:21 2019},
-}
-
-@article{pastor12_modal_assur_criter,
-  author = {Miroslav Pastor and Michal Binda and Tom{\'a}{\v{s}} Har{\v{c}}arik},
-  title = {Modal Assurance Criterion},
-  journal = {Procedia Engineering},
-  volume = {48},
-  number = {nil},
-  pages = {543-548},
-  year = {2012},
-  doi = {10.1016/j.proeng.2012.09.551},
-  url = {https://doi.org/10.1016/j.proeng.2012.09.551},
-  DATE_ADDED = {Thu Jul 11 13:51:57 2019},
-}
-
-@book{ewins00_modal,
-  author = {Ewins, DJ},
-  title = {Modal testing: theory, practice and application},
-  year = {2000},
-  publisher = {Wiley-Blackwell},
-  journal = {Research studies Pre, 2nd ed., ISBN-13},
-  pages = {978--0863802188},
-  isbn = {0863802184},
-  address = {Baldock, Hertfordshire, England Philadelphia, PA},
-  keywords = {favorite, identification},
-}