Tangle matlab files

matlab/modal_1_meas_setup.m

@@ -13,12 +13,12 @@ colors = colororder;
 % Location of the Accelerometers
 % <<ssec:modal_accelerometers>>
 
-% The location of the accelerometers fixed to the micro-station is essential as it defines where the dynamics is measured.
-% A total of 23 accelerometers are fixed to the six key stages of the micro station: the lower and upper granites, the translation stage, the tilt stage, the spindle and the micro hexapod.
-% The position of the accelerometers are visually shown on a CAD model in Figure ref:fig:modal_location_accelerometers and their precise locations with respect to a frame located at the point of interest are summarized in Table ref:tab:modal_position_accelerometers.
+% 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 acrshort:dof can be considered per solid body.
+% As all key stages of the micro-station are expected to behave as solid bodies, only 6 acrshort:dof can be considered for each solid body.
 % However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured acrshort:dof) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section ref:ssec:modal_solid_body_assumption).
 
 % #+attr_latex: :options [t]{0.60\linewidth}
@@ -93,13 +93,13 @@ acc_pos = acc_pos(i, 2:4);
 % Force and Response signals
 % <<ssec:modal_measured_signals>>
 
-% The force sensor of the instrumented hammer and the accelerometers signals are shown in the time domain in Figure ref:fig:modal_raw_meas.
-% Sharp "impacts" can be seen for the force sensor, indicating wide frequency band excitation.
+% 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" 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.
+% 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.
 
 
 %% Load raw data
@@ -151,7 +151,7 @@ legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
 
 
 % #+name: fig:modal_raw_meas_asd
-% #+caption: Raw measurement of the acceleromter (blue) and of the force sensor at the Hammer tip (red) (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Density of the two signals (normalized) (\subref{fig:modal_asd_acc_force})
+% #+caption: Raw measurement of the accelerometer (blue) and of the force sensor at the Hammer tip (red) (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force})
 % #+attr_latex: :options [htbp]
 % #+begin_figure
 % #+attr_latex: :caption \subcaption{\label{fig:modal_raw_meas}Time domain signals}
@@ -169,7 +169,7 @@ legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
 % #+end_figure
 
 % The frequency response function $H_{jk}$ from the applied force $F_{k}$ to the measured acceleration $X_j$ is then computed and shown Figure ref:fig:modal_frf_acc_force.
-% The quality of the obtained data can be estimated using the /coherence/ function, which is shown in Figure ref:fig:modal_coh_acc_force.
+% The quality of the obtained data can be estimated using the /coherence/ function (Figure ref:fig:modal_coh_acc_force).
 % Good coherence is obtained from $20\,\text{Hz}$ to $200\,\text{Hz}$ which corresponds to the frequency range of interest.
 
 

matlab/modal_2_frf_processing.m

@@ -17,7 +17,7 @@ colors = colororder;
 % #+RESULTS:
 % [[file:figs/modal_local_to_global_coordinates.png]]
 
-% The motion of the rigid body of figure ref:fig:modal_local_to_global_coordinates can be described by its displacement $\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]$ and (small) rotations $[\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]$ with respect to a reference frame $\{O\}$.
+% The motion of the rigid body of figure ref:fig:modal_local_to_global_coordinates can be described by its displacement $\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]$ and (small) rotations $[\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]$ with respect to the reference frame $\{O\}$.
 
 % The motion $\vec{\delta} p_{i}$ of a point $p_i$ can be computed from $\vec{\delta} p$ and $\bm{\delta \Omega}$ using equation eqref:eq:modal_compute_point_response, with $\bm{\delta\Omega}$ defined in equation eqref:eq:modal_rotation_matrix.
 
@@ -33,7 +33,7 @@ colors = colororder;
 % \end{bmatrix}
 % \end{equation}
 
-% Writing this in a matrix form for the four points 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}
@@ -55,7 +55,7 @@ colors = colororder;
 % 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.
+% Therefore, having more accelerometers permits better approximation of the motion of a solid body.
 
 % \begin{equation}
 % \left[\begin{array}{c}
@@ -76,7 +76,7 @@ colors = colororder;
 
 
 % 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.
 
 
 %% Load frequency response matrix
@@ -100,7 +100,7 @@ solids.hexa = [1, 2, 3, 4]; % Hexapod
 % Names of the solid bodies
 solid_names = fields(solids);
 
-%% Save the acceleromter positions are well as the solid bodies
+%% Save the accelerometer positions are well as the solid bodies
 save('mat/geometry.mat', 'solids', 'solid_names', 'acc_pos');
 
 %% Extract the CoM of considered solid bodies
@@ -170,9 +170,9 @@ save('mat/frf_com.mat', 'frfs_CoM');
 % Verification of solid body assumption
 % <<ssec:modal_solid_body_assumption>>
 
-% From the response of one solid body expressed by its 6 acrshortpl:dof (i.e. from $\mathbf{H}_{\text{CoM}}$), and using equation eqref:eq:modal_cart_to_acc, it is possible to compute the response of the same solid body at any considered position.
-% In particular, the response at the location of the four accelerometers can be computed and compared with the original measurements $\mathbf{H}$.
-% This is what is here done to check if solid body assumption is correct in the frequency band of interest.
+% 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.
 
 
 %% Compute the FRF at the accelerometer location from the CoM reponses
@@ -199,10 +199,10 @@ end
 
 
 
-% The comparison is made for the 4 accelerometers fixed to the micro-hexapod (Figure ref:fig:modal_comp_acc_solid_body_frf).
-% The original frequency response functions and the ones computed from the CoM responses are well matching in the frequency range of interested.
-% Similar results are obtained for the other solid bodies, indicating that the solid body assumption is valid, and that a multi-body model can be used to represent the dynamics of the micro-station.
-% This also validates the reduction of the number of degrees of freedom from 69 (23 accelerometers with each 3 acrshort:dof) to 36 (6 solid bodies with 6 acrshort:dof).
+% 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).
 
 
 %% Comparaison of the original accelerometer response and reconstructed response from the solid body response

matlab/modal_3_analysis.m

@@ -25,9 +25,9 @@ colors = colororder;
 
 % 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.
+% 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 are found between 0 and $200\,\text{Hz}$.
+% 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.
 
 
@@ -58,10 +58,10 @@ ylim([1e-6, 2e-2]);
 % Verification of the modal model validity
 % <<ssec:modal_model_validity>>
 
-% In order to check the validity of the modal model, the complete $n \times n$ acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ is first synthesized from the modal parameters.
-% Then, the elements of this acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ that were already measured can be compared with the measured acrshort:frf matrix $\mathbf{H}$.
+% 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 $\mathbf{H}$.
 
-% 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.
+% 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}
 %   & & & & &\\
@@ -145,10 +145,10 @@ end
 
 
 
-% The comparison between the original measured frequency response functions and the synthesized ones from the modal model is done in Figure ref:fig:modal_comp_acc_frf_modal.
-% Whether the obtained match can be considered good or bad is quite arbitrary.
-% Yet, the modal model seems to be able to represent the coupling between different nodes and different direction which is quite important in a control point of view.
-% This can be seen in Figure ref:fig:modal_comp_acc_frf_modal_3 that shows the frequency response function between a force applied on node 11 (i.e. on the translation stage) in the $y$ direction to the measured acceleration at node $2$ (i.e. at the top of the micro-hexapod) in the $x$ direction.
+% 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.
 
 
 acc_o = 11; dir_o = 3;