Rework the full document once

modal-analysis.tex

@@ -1,4 +1,4 @@
-% Created 2024-10-24 Thu 09:36
+% Created 2024-10-24 Thu 17:41
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -8,12 +8,14 @@
 \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{nass-uniaxial-model.bib}
+\bibliography{modal-analysis.bib}
 \author{Dehaeze Thomas}
 \date{\today}
 \title{Micro-Station - Modal Analysis}
@@ -134,18 +136,17 @@ Pictures of the accelerometers fixed to the translation stage and to the micro-h
 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}).
 
-\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}
+\includegraphics[scale=1,width=0.99\linewidth]{figs/modal_location_accelerometers.png}
-\captionof{figure}{\label{fig:modal_location_accelerometers}Position of the accelerometers using SolidWorks in mm}
+\captionof{figure}{\label{fig:modal_location_accelerometers}Position of the accelerometers}
 \end{center}
 \end{minipage}
 \hfill
-\begin{minipage}[b]{0.31\linewidth}
+\begin{minipage}[b]{0.38\linewidth}
-\begin{center}
+\begin{scriptsize}
-\captionof{table}{\label{tab:modal_position_accelerometers}Accelerometer positions}
+\captionof{table}{\label{tab:modal_position_accelerometers}Positions in mm}
-\scriptsize
+\begin{tabularx}{\linewidth}{Xccc}
-\begin{tabularx}{\linewidth}{Xcccc}
 \toprule
  & \(x\) & \(y\) & \(z\)\\
 \midrule
@@ -174,11 +175,9 @@ Hexapod & 64 & 64 & -270\\
 Hexapod & 64 & -64 & -270\\
 \bottomrule
 \end{tabularx}
-\end{center}
+\end{scriptsize}
 \end{minipage}
 
-
-
 \begin{figure}[htbp]
 \begin{subfigure}{0.49\textwidth}
 \begin{center}
@@ -298,79 +297,15 @@ However, for the multi-body model being developed, only 6 solid bodies are consi
 Therefore, only \(6 \times 6 = 36\) degrees of freedom are of interest.
 The objective in this section is therefore to process the Frequency Response Matrix to reduce the number of measured \acrshort{dof} from 69 to 36.
 
-In order to be able to perform this reduction of measured \acrshort{dof}, the rigid body assumption first needs to be verified (Section \ref{ssec:modal_solid_body_first_check}).
-
 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 \acrshort{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}.
+To validate this reduction of \acrshort{dof} and the solid body assumption, the frequency response function at the accelerometer location are recomputed 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}
-
-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}.
-
-\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}
-\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 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}.
-
-\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_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}
-
-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 \acrshort{dofs} of the solid body expressed in the frame \(\{O\}\).
+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
@@ -396,8 +331,9 @@ Writing Eq. \eqref{eq:modal_p1_p2_motion} for the four displacement sensors in a
 \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}.
+Provided that the four sensors are properly located, the system of equation \eqref{eq:modal_cart_to_acc} can be solved by matrix inversion.
-Note that this inversion is equivalent to resolving a mean square problem.
+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.
 
 
@@ -439,10 +375,10 @@ Hexapod & -4 & 6 & -319\\
 \end{tabularx}
 \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 \acrshort{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) \\
@@ -458,14 +394,14 @@ 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 \acrshort{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 along 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.
+Comparing the computed response of a particular accelerometer from \(\mathbf{H}_{\text{CoM}}\) with the original measurements \(\mathbf{H}\) is use to check if 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 (Figure \ref{fig:modal_comp_acc_solid_body_frf}).
-
-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 3acrshort:dof) to 36 (6 solid bodies with 6 \acrshort{dof}).
+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}).
 
 \begin{figure}[htbp]
 \centering
@@ -475,129 +411,49 @@ This also validates the reduction of the number of degrees of freedom from 69 (2
 
 \chapter{Modal Analysis}
 \label{sec:modal_analysis}
-The goal here is to extract the modal parameters describing the modes of station being studied, namely:
+The goal here is to extract the modal parameters describing the modes of station being studied, namely the natural frequencies and the modal damping (i.e. the eigenvalues) and the mode shapes (.i.e. the eigenvectors).
-\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 \acrshort{frf} matrix previously extracted from the measurements.
 
+In order to perform the modal parameter extraction, the order of the modal model needs to be estimated (i.e. the number of modes in the frequency band of interest).
+This is done 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.
+In section \ref{ssec:modal_parameter_extraction}, the modal parameter extraction is performed.
-This corresponds to how many modes are present in the frequency band of interest.
+Graphical display of the mode shapes can be computed from the modal model, which is quite quite useful to have a physical interpretation of the modes.
-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.
 
-
+To validate the quality of the modal model, the full \acrshort{frf} matrix is computed from the modal model and compared with the initial measured \acrshort{frf} (section \ref{ssec:modal_model_validity}).
-The modal parameter extraction methods generally consists of \textbf{curve-fitting a theoretical expression for an individual \acrshort{frf} to the actual measured data}.
-However, there are multiple level of complexity:
-\begin{itemize}
-\item works on a part of a single \acrshort{frf} curve
-\item works on a complete curve encompassing several resonances
-\item works on a set of many \acrshort{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 \acrshort{frf} matrix from the modal model and compare it with the \acrshort{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}
 \label{ssec:modal_number_of_modes}
-\paragraph{Singular Value Decomposition - Modal Indication Function}
+The \acrshort{mif} is here applied to the \(n\times p\) \acrshort{frf} matrix where \(n\) is a relatively large number of measurement DOFs (here \(n=69\)) and \(p\) is the number of excitation DOFs (here \(p=3\)).
-The Mode Indicator Functions are usually used on \(n\times p\) \acrshort{frf} matrix where \(n\) is a relatively large number of measurement DOFs and \(p\) is the number of excitation DOFs, typically 3 or 4.
 
-In these methods, the frequency dependent \acrshort{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 \(\Sigma\) is obtained from a \acrshort{svd} of the \acrshort{frf} matrix \eqref{eq:modal_svd}.
-
+\begin{equation} \label{eq:modal_cmif}
-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 \acrshort{frf} data prior to modal analysis.
+  [CMIF(\omega)]_{p\times p} = [\Sigma(\omega)]_{p\times n}^T [\Sigma(\omega)]_{n\times p}
-
-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 \acrshort{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 \acrshort{frf}:
-\begin{equation}
-  HH(\omega) = \sum_j\sum_kH_{jk}(\omega)
 \end{equation}
 
-Instead, we choose here to use the sum of the norms of the measured frf:
+\begin{equation} \label{eq:modal_svd}
-\begin{equation}
+[H(\omega)]_{n\times p} = [U(\omega)]_{n\times n} [\Sigma(\omega)]_{n\times p} [V(\omega)]_{p\times p}^H
-  HH(\omega) = \sum_j\sum_k \left|H_{jk}(\omega) \right|
 \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 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{figure}[htbp]
+\begin{minipage}[t]{0.70\linewidth}
-\centering
+\begin{center}
-\includegraphics[scale=1]{figs/modal_composite_reponse_function.png}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/modal_indication_function.png}
-\caption{\label{fig:modal_composite_reponse_function}Composite Response Function}
+\captionof{figure}{\label{fig:modal_indication_function}Modal Indication Function}
-\end{figure}
+\end{center}
-
+\end{minipage}
-\section{Modal parameter extraction}
+\hfill
-\label{ssec:modal_parameter_extraction}
+\begin{minipage}[b]{0.28\linewidth}
-\paragraph{OROS - Modal software}
+\begin{scriptsize}
-Modal identification are done within the Modal software of OROS.
+\captionof{table}{\label{tab:modal_obtained_modes_freqs_damps}Natural frequencies and modal damping}
-
+\begin{tabularx}{\linewidth}{ccc}
-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 \acrshort{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 \acrshort{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 \acrshort{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]
-\caption{\label{tab:modal_obtained_modes_freqs_damps}Obtained eigen frequencies and modal damping}
-\centering
-\scriptsize
-\begin{tabularx}{0.35\linewidth}{ccc}
 \toprule
-Mode & Frequency [Hz] & Damping [\%]\\
+Mode & Freq. [Hz] & Damp. [\%]\\
 \midrule
 1 & 11.9 & 12.2\\
 2 & 18.6 & 11.7\\
@@ -617,197 +473,134 @@ Mode & Frequency [Hz] & Damping [\%]\\
 16 & 165.4 & 1.4\\
 \bottomrule
 \end{tabularx}
-\end{table}
+\end{scriptsize}
+\end{minipage}
 
-\paragraph{Theory}
+\section{Modal parameter extraction}
-It seems that the modal analysis software makes the \textbf{assumption} of viscous damping for the model with which it tries to fit the \acrshort{frf} measurements.
+\label{ssec:modal_parameter_extraction}
 
-If we note \(N\) the number of modes identified, then there are \(2N\) eigenvalues and eigenvectors given by the software:
+The modal identification generally consists of curve-fitting a theoretical expression for an individual \acrshort{frf} to the actual measured data.
-\begin{align}
+However, there are multiple level of complexity, from fitting of a single resonance, a complete curve encompassing several resonances and working on a set of many \acrshort{frf} plots all obtained from the same structure.
-  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.
 
-\paragraph{Modal Matrices}
+Here, the last method is used as it gives a unique and consistent model as direct output.
-We would like to arrange the obtained modal parameters into two modal matrices:
+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.
-\[ \Lambda = \begin{bmatrix}
-  s_1 &        & 0 \\
-      & \ddots &   \\
-  0   &        & s_N
-\end{bmatrix}_{N \times N}; \quad \Psi = \begin{bmatrix}
-  & & \\
-  \{\psi_1\} & \dots & \{\psi_N\} \\
-  & &
-\end{bmatrix}_{M \times N} \]
-\[ \{\psi_i\} = \begin{Bmatrix} \psi_{i, 1_x} & \psi_{i, 1_y} & \psi_{i, 1_z} & \psi_{i, 2_x} & \dots & \psi_{i, 23_z} \end{Bmatrix}^T \]
 
-\(M\) is the number of \acrshort{dof}: here it is \(23 \times 3 = 69\).
+From the obtained modal parameters, the mode shapes are computed and can be displayed in the form of animations.
-\(N\) is the number of mode
+Three mode shapes are shown in Figure \ref{fig:modal_mode_animations}.
 
-Each eigen vector is normalized: \(\| \{\psi_i\} \|_2 = 1\)
+These animations are quite useful to easily get a better understanding of the system.
-
+For instance, the mode shape of the first mode at 11Hz (figure \ref{fig:modal_mode1_animation}) indicates that there is 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}).
-However, the eigen values and eigen vectors appears as complex conjugates:
+These are difficult to tune so that the granite is well supported by four of them and not ``wobbly'' on just two of them.
-\[ s_r, s_r^*, \{\psi\}_r, \{\psi\}_r^*, \quad r = 1, N \]
-
-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}.
 
 \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}).
-
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1,width=0.6\linewidth]{figs/modal_airlock_picture.jpg}
 \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 \(a_r\).
+
+\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 \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{Verify the validity of the Modal Model}
 \label{ssec:modal_model_validity}
 
-There are two main ways to verify the validity of the modal model
+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.
-\begin{itemize}
+Then, the elements of this \acrshort{frf} matrix \(\mathbf{H}_{\text{syn}}\) that were already measured can be compared with the measured elements \(\mathbf{H}\).
-\item Synthesize \acrshort{frf} measurements that has been used to generate the modal model and compare
+New measurements may be performed to compare with elements of the synthesized \acrshort{frf} matrix that were not initialized measured to build the modal model.
-\item Synthesize \acrshort{frf} that has not yet been measured. Then measure that \acrshort{frf} and compare
-\end{itemize}
 
-From the modal model, we want to synthesize the Frequency Response Functions that has been used to build the modal model.
+In order to synthesize the full \acrshort{frf} matrix, the eigenvectors \(\phi_r\) are first reorganized in a matrix from as shown in equation \eqref{eq:modal_eigvector_matrix}.
-
+\begin{equation}\label{eq:modal_eigvector_matrix}
-Let's recall that:
+\Phi = \begin{bmatrix}
-\begin{itemize}
+  & & & & &\\
-\item \(M\) is the number of measured DOFs (\(3 \times n_\text{acc}\))
+  \phi_1 & \dots & \phi_N & \phi_1^* & \dots & \phi_N^* \\
-\item \(N\) is the number of modes identified
+  & & & & &
-\end{itemize}
+\end{bmatrix}_{n \times 2m}
-
-We then have that the \acrshort{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
 \end{equation}
-with \(Q_r = 1/M_{A_r}\)
-\end{important}
 
-An alternative formulation is:
+The full \acrshort{frf} matrix \(\mathbf{H}_{\text{syn}}\) can be synthesize using \eqref{eq:modal_synthesized_frf}.
-\[ H_{pq}(s_i) = \sum_{r=1}^N \frac{A_{pqr}}{s_i - \lambda_r} + \frac{A_{pqr}^*}{s_i - \lambda_r^*} \]
-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}
 
-From the modal software documentation:
+\begin{equation}\label{eq:modal_synthesized_frf}
-\begin{quote}
+  [\mathbf{H}_{\text{syn}}(\omega)]_{n\times n} = [\Phi]_{n\times2m} [\mathbf{H}_{\text{mod}}(\omega)]_{2m\times2m} [\Phi]_{2m\times n}^T
-\textbf{Modal A}
+\end{equation}
-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}
 
-Then
+With \(\mathbf{H}_{\text{mod}}(\omega)\) a diagonal matrix representing the response of the different modes \eqref{eq:modal_modal_resp}.
-\[ A_{pqr} = \frac{\psi_{pr}\psi_{qr}}{M_{Ar}} \]
+\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 match is rather good considering the complex dynamics and the different directions considered.
 
 \begin{figure}[htbp]
-\centering
+\begin{subfigure}{0.33\textwidth}
-\includegraphics[scale=1]{figs/modal_comp_acc_frf_modal.png}
+\begin{center}
-\caption{\label{fig:modal_comp_acc_frf_modal}description}
+\includegraphics[scale=1,width=0.99\linewidth]{figs/modal_comp_acc_frf_modal_1.png}
-\end{figure}
+\end{center}
-
+\subcaption{\label{fig:modal_comp_acc_frf_modal_1}From $F_{11,z}$ to $a_{11,z}$}
-Frequency response functions that has not been measured can be synthesized as shown in Figure \ref{fig:modal_synthesized_frf}.
+\end{subfigure}
-
+\begin{subfigure}{0.33\textwidth}
-\begin{figure}[htbp]
+\begin{center}
-\centering
+\includegraphics[scale=1,width=0.99\linewidth]{figs/modal_comp_acc_frf_modal_2.png}
-\includegraphics[scale=1]{figs/modal_synthesized_frf.png}
+\end{center}
-\caption{\label{fig:modal_synthesized_frf}description}
+\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 adequate choice of instrumentation and proper set of measurements, high quality frequency response functions could be obtained.
+As could be expected from a heavy stacked stages architecture, the obtained frequency response functions indicate that the dynamics of the micro-station is complex.
+It shows lots of coupling between stages and different directions, as well as many modes with various damping properties.
 
-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 is behaving as a rigid body.
+This confirms that a solid-body model can be used to properly model the micro-station.
 
+Even though lots of efforts were put in the proper modal analysis of the micro-station, it was stiff very difficult to obtain an accurate modal model.
+Yet, the measurements will be quite useful when tuning the parameters of the multi-body model.
 
 \printbibliography[heading=bibintoc,title={Bibliography}]
+
+\printglossaries
 \end{document}