diff --git a/matlab/modal_1_meas_setup.m b/matlab/modal_1_meas_setup.m index 55a430d..c3b080d 100644 --- a/matlab/modal_1_meas_setup.m +++ b/matlab/modal_1_meas_setup.m @@ -13,12 +13,12 @@ colors = colororder; % Location of the 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 % <> -% 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. diff --git a/matlab/modal_2_frf_processing.m b/matlab/modal_2_frf_processing.m index a2f7e11..7771232 100644 --- a/matlab/modal_2_frf_processing.m +++ b/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 % <> -% 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 diff --git a/matlab/modal_3_analysis.m b/matlab/modal_3_analysis.m index 4939c84..4a27ad8 100644 --- a/matlab/modal_3_analysis.m +++ b/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 % <> -% 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;