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Thomas Dehaeze 2024-11-14 10:45:19 +01:00
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@ -45,12 +45,6 @@
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png") #+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END: :END:
#+begin_export html
<hr>
<p>This report is also available as a <a href="./modal-analysis.pdf">pdf</a>.</p>
<hr>
#+end_export
#+latex: \clearpage #+latex: \clearpage
* Build :noexport: * Build :noexport:
@ -143,7 +137,7 @@ To further improve the accuracy of the performance predictions, a model that bet
A multi-body model consisting of several rigid bodies connected by kinematic constraints (i.e. joints), springs and damper elements is a good candidate to model the micro-station. A multi-body model consisting of several rigid bodies connected by kinematic constraints (i.e. joints), springs and damper elements is a good candidate to model the micro-station.
Although the inertia of each solid body can easily be estimated from its geometry and material density, it is more difficult to properly estimate the stiffness and damping properties of the guiding elements connecting each solid body. Although the inertia of each solid body can easily be estimated from its geometry and material density, it is more difficult to properly estimate the stiffness and damping properties of the guiding elements connecting each solid body.
Experimental modal analysis will be use to tune the model, and to verify that a multi-body model can accurately represent the dynamics of the micro-station. Experimental modal analysis will be used to tune the model, and to verify that a multi-body model can accurately represent the dynamics of the micro-station.
The tuning approach for the multi-body model based on measurements is illustrated in Figure ref:fig:modal_vibration_analysis_procedure. The tuning approach for the multi-body model based on measurements is illustrated in Figure ref:fig:modal_vibration_analysis_procedure.
First, a /response model/ is obtained, which corresponds to a set of frequency response functions computed from experimental measurements. First, a /response model/ is obtained, which corresponds to a set of frequency response functions computed from experimental measurements.
@ -296,47 +290,46 @@ Pictures of the accelerometers fixed to the translation stage and to the micro-h
As all key stages of the micro-station are expected to behave as solid bodies, only 6 acrshort:dof can be considered for each solid body. 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). However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured acrshort:dof) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section ref:ssec:modal_solid_body_assumption).
#+attr_latex: :options [t]{0.60\linewidth} #+attr_latex: :options [b]{0.63\linewidth}
#+begin_minipage #+begin_minipage
#+name: fig:modal_location_accelerometers #+name: fig:modal_location_accelerometers
#+caption: Position of the accelerometers #+caption: Position of the accelerometers
#+attr_latex: :width 0.99\linewidth :float nil #+attr_latex: :width 0.95\linewidth :float nil
[[file:figs/modal_location_accelerometers.png]] [[file:figs/modal_location_accelerometers.png]]
#+end_minipage #+end_minipage
\hfill \hfill
#+attr_latex: :options [b]{0.38\linewidth} #+attr_latex: :options [b]{0.36\linewidth}
#+begin_minipage #+begin_minipage
#+begin_scriptsize #+begin_scriptsize
#+name: tab:modal_position_accelerometers #+latex: \centering
#+caption: Positions in mm
#+attr_latex: :environment tabularx :width \linewidth :placement [b] :align Xccc #+attr_latex: :environment tabularx :width \linewidth :placement [b] :align Xccc
#+attr_latex: :booktabs t :float nil :center nil #+attr_latex: :booktabs t :float nil :center nil
#+RESULTS:
| | $x$ | $y$ | $z$ | | | $x$ | $y$ | $z$ |
|--------------+------+------+------| |-------------------+------+------+------|
| Low. Granite | -730 | -526 | -951 | | (17) Low. Granite | -730 | -526 | -951 |
| Low. Granite | -735 | 814 | -951 | | (18) Low. Granite | -735 | 814 | -951 |
| Low. Granite | 875 | 799 | -951 | | (19) Low. Granite | 875 | 799 | -951 |
| Low. Granite | 865 | -506 | -951 | | (20) Low. Granite | 865 | -506 | -951 |
| Up. Granite | -320 | -446 | -786 | | (13) Up. Granite | -320 | -446 | -786 |
| Up. Granite | -480 | 534 | -786 | | (14) Up. Granite | -480 | 534 | -786 |
| Up. Granite | 450 | 534 | -786 | | (15) Up. Granite | 450 | 534 | -786 |
| Up. Granite | 295 | -481 | -786 | | (16) Up. Granite | 295 | -481 | -786 |
| Translation | -475 | -414 | -427 | | (9) Translation | -475 | -414 | -427 |
| Translation | -465 | 407 | -427 | | (10) Translation | -465 | 407 | -427 |
| Translation | 475 | 424 | -427 | | (11) Translation | 475 | 424 | -427 |
| Translation | 475 | -419 | -427 | | (12) Translation | 475 | -419 | -427 |
| Tilt | -385 | -300 | -417 | | (5) Tilt | -385 | -300 | -417 |
| Tilt | -420 | 280 | -417 | | (6) Tilt | -420 | 280 | -417 |
| Tilt | 420 | 280 | -417 | | (7) Tilt | 420 | 280 | -417 |
| Tilt | 380 | -300 | -417 | | (8) Tilt | 380 | -300 | -417 |
| Spindle | -155 | -90 | -594 | | (21) Spindle | -155 | -90 | -594 |
| Spindle | 0 | 180 | -594 | | (22) Spindle | 0 | 180 | -594 |
| Spindle | 155 | -90 | -594 | | (23) Spindle | 155 | -90 | -594 |
| Hexapod | -64 | -64 | -270 | | (1) Hexapod | -64 | -64 | -270 |
| Hexapod | -64 | 64 | -270 | | (2) Hexapod | -64 | 64 | -270 |
| Hexapod | 64 | 64 | -270 | | (3) Hexapod | 64 | 64 | -270 |
| Hexapod | 64 | -64 | -270 | | (4) Hexapod | 64 | -64 | -270 |
#+latex: \captionof{table}{\label{tab:modal_position_accelerometers}Positions in mm}
#+end_scriptsize #+end_scriptsize
#+end_minipage #+end_minipage
@ -411,6 +404,7 @@ For the accelerometer, a much more complex signal can be observed, indicating co
The "normalized" acrfull:asd of the two signals were computed and shown in Figure ref:fig:modal_asd_acc_force. 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). Conclusions based on the time domain signals can be clearly observed in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer).
These data are corresponding to an hammer impact in the vertical direction and to the measured acceleration in the $x$ direction by accelerometer $1$ (fixed to the micro-hexapod).
Similar results were obtained for all measured frequency response functions. Similar results were obtained for all measured frequency response functions.
#+begin_src matlab #+begin_src matlab
@ -431,12 +425,12 @@ time = linspace(0, meas1_raw.Track1_X_Resolution*length(meas1_raw.Track1), lengt
%% Raw measurement of the Accelerometer %% Raw measurement of the Accelerometer
figure; figure;
hold on; hold on;
plot(time-22.2, meas1_raw.Track2, 'DisplayName', '$X_{j}$ [$m/s^2$]'); plot(time-22.2, meas1_raw.Track2, 'DisplayName', '$X_{1,x}$ [$m/s^2$]');
plot(time-22.2, 1e-3*meas1_raw.Track1, 'DisplayName', '$F_{k}$ [kN]'); plot(time-22.2, 1e-3*meas1_raw.Track1, 'DisplayName', '$F_{z}$ [kN]');
hold off; hold off;
xlabel('Time [s]'); xlabel('Time [s]');
ylabel('Amplitude'); ylabel('Amplitude');
xlim([0, 0.2]); xlim([0, 0.2])
ylim([-2, 2]); ylim([-2, 2]);
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1); legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
#+end_src #+end_src
@ -460,8 +454,8 @@ Noverlap = floor(Nfft/2); % Overlap for frequency analysis
%% Normalized Amplitude Spectral Density of the measured force and acceleration %% Normalized Amplitude Spectral Density of the measured force and acceleration
figure; figure;
hold on; hold on;
plot(f, sqrt(pxx_acc./max(pxx_acc(f<200))), 'DisplayName', '$X_{j}$'); plot(f, sqrt(pxx_acc./max(pxx_acc(f<200))), 'DisplayName', '$\Gamma_{X_{1,x}}$');
plot(f, sqrt(pxx_force./max(pxx_force(f<200))), 'DisplayName', '$F_{k}$'); plot(f, sqrt(pxx_force./max(pxx_force(f<200))), 'DisplayName', '$\Gamma_{F_{z}}$');
hold off; hold off;
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Normalized Spectral Density'); xlabel('Frequency [Hz]'); ylabel('Normalized Spectral Density');
@ -476,7 +470,7 @@ exportFig('figs/modal_asd_acc_force.pdf', 'width', 'half', 'height', 'normal');
#+end_src #+end_src
#+name: fig:modal_raw_meas_asd #+name: fig:modal_raw_meas_asd
#+caption: Raw measurement of the accelerometer (blue) and of the force sensor at the Hammer tip (red) (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force}) #+caption: Raw measurement of the accelerometer 1 in the $x$ direction (blue) and of the force sensor at the Hammer tip (red) for an impact in the $z$ direction (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force})
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:modal_raw_meas}Time domain signals} #+attr_latex: :caption \subcaption{\label{fig:modal_raw_meas}Time domain signals}
@ -493,7 +487,7 @@ exportFig('figs/modal_asd_acc_force.pdf', 'width', 'half', 'height', 'normal');
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
The frequency response function $H_{jk}$ from the applied force $F_{k}$ to the measured acceleration $X_j$ is then computed and shown Figure ref:fig:modal_frf_acc_force. The frequency response function from the applied force to the measured acceleration is then computed and shown Figure ref:fig:modal_frf_acc_force.
The quality of the obtained data can be estimated using the /coherence/ function (Figure ref:fig:modal_coh_acc_force). 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. Good coherence is obtained from $20\,\text{Hz}$ to $200\,\text{Hz}$ which corresponds to the frequency range of interest.
@ -532,7 +526,7 @@ exportFig('figs/modal_coh_acc_force.pdf', 'width', 'half', 'height', 'normal');
#+end_src #+end_src
#+name: fig:modal_frf_coh_acc_force #+name: fig:modal_frf_coh_acc_force
#+caption: Computed frequency response function from the applied force $F_{k}$ and the measured response $X_{j}$ (\subref{fig:modal_frf_acc_force}) as well as computed coherence (\subref{fig:modal_coh_acc_force}) #+caption: Computed frequency response function from the applied force $F_{z}$ to the measured response $X_{1,x}$ (\subref{fig:modal_frf_acc_force}) as well as computed coherence (\subref{fig:modal_coh_acc_force})
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:modal_frf_acc_force} Frequency Response Function} #+attr_latex: :caption \subcaption{\label{fig:modal_frf_acc_force} Frequency Response Function}
@ -651,7 +645,7 @@ The goal here is to link these $4 \times 3 = 12$ measurements to the 6 acrshort:
The motion of the rigid body of figure ref:fig:modal_local_to_global_coordinates can be described by its displacement $\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]$ and (small) rotations $[\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]$ with respect to the reference frame $\{O\}$. The motion of the rigid body of figure ref:fig:modal_local_to_global_coordinates can be described by its displacement $\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]$ and (small) rotations $[\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]$ with respect to the reference frame $\{O\}$.
The motion $\vec{\delta} p_{i}$ of a point $p_i$ can be computed from $\vec{\delta} p$ and $\bm{\delta \Omega}$ using equation eqref:eq:modal_compute_point_response, with $\bm{\delta\Omega}$ defined in equation eqref:eq:modal_rotation_matrix. The motion $\vec{\delta} p_{i}$ of a point $p_i$ can be computed from $\vec{\delta} p$ and $\bm{\delta \Omega}$ using equation eqref:eq:modal_compute_point_response, with $\bm{\delta\Omega}$ defined in equation eqref:eq:modal_rotation_matrix [[cite:&ewins00_modal chapt. 4.3.2]].
\begin{equation}\label{eq:modal_compute_point_response} \begin{equation}\label{eq:modal_compute_point_response}
\vec{\delta} p_{i} &= \vec{\delta} p + \bm{\delta \Omega} \cdot \vec{p}_{i} \\ \vec{\delta} p_{i} &= \vec{\delta} p + \bm{\delta \Omega} \cdot \vec{p}_{i} \\
@ -756,17 +750,17 @@ data2orgtable(1000*model_com', {'Bottom Granite', 'Top granite', 'Translation st
#+name: tab:modal_com_solid_bodies #+name: tab:modal_com_solid_bodies
#+caption: Center of mass of considered solid bodies with respect to the "point of interest" #+caption: Center of mass of considered solid bodies with respect to the "point of interest"
#+attr_latex: :environment tabularx :width 0.6\linewidth :align lXXX #+attr_latex: :environment tabularx :width 0.55\linewidth :align Xccc
#+attr_latex: :center t :booktabs t #+attr_latex: :center t :booktabs t
#+RESULTS: #+RESULTS:
| | $X$ [mm] | $Y$ [mm] | $Z$ [mm] | | | $X$ | $Y$ | $Z$ |
|-------------------+----------+----------+----------| |-------------------+-----------------+------------------+--------------------|
| Bottom Granite | 45 | 144 | -1251 | | Bottom Granite | $45\,\text{mm}$ | $144\,\text{mm}$ | $-1251\,\text{mm}$ |
| Top granite | 52 | 258 | -778 | | Top granite | $52\,\text{mm}$ | $258\,\text{mm}$ | $-778\,\text{mm}$ |
| Translation stage | 0 | 14 | -600 | | Translation stage | $0$ | $14\,\text{mm}$ | $-600\,\text{mm}$ |
| Tilt Stage | 0 | -5 | -628 | | Tilt Stage | $0$ | $-5\,\text{mm}$ | $-628\,\text{mm}$ |
| Spindle | 0 | 0 | -580 | | Spindle | $0$ | $0$ | $-580\,\text{mm}$ |
| Hexapod | -4 | 6 | -319 | | Hexapod | $-4\,\text{mm}$ | $6\,\text{mm}$ | $-319\,\text{mm}$ |
Using eqref:eq:modal_determine_global_disp, the frequency response matrix $\mathbf{H}_\text{CoM}$ eqref:eq:modal_frf_matrix_com expressing the response at the center of mass of each solid body $D_i$ ($i$ from $1$ to $6$ for the $6$ considered solid bodies) can be computed from the initial acrshort:frf matrix $\mathbf{H}$. Using eqref:eq:modal_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 $\mathbf{H}$.
@ -989,7 +983,7 @@ end
figure; figure;
hold on; hold on;
for i = 1:size(MIF, 1) for i = 1:size(MIF, 1)
plot(freqs, squeeze(MIF(i, i, :))); plot(freqs, squeeze(MIF(i, i, :)), 'DisplayName', sprintf('MIF${}_%i$', i));
end end
hold off; hold off;
set(gca, 'Xscale', 'lin'); set(gca, 'Yscale', 'log'); set(gca, 'Xscale', 'lin'); set(gca, 'Yscale', 'log');
@ -997,45 +991,46 @@ xlabel('Frequency [Hz]'); ylabel('CMIF Amplitude');
xticks([0:20:200]); xticks([0:20:200]);
xlim([0, 200]); xlim([0, 200]);
ylim([1e-6, 2e-2]); ylim([1e-6, 2e-2]);
ldg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file none #+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/modal_indication_function.pdf', 'width', 'wide', 'height', 'normal'); exportFig('figs/modal_indication_function.pdf', 'width', 'normal', 'height', 'normal');
#+end_src #+end_src
#+attr_latex: :options [t]{0.70\linewidth} #+attr_latex: :options [b]{0.70\linewidth}
#+begin_minipage #+begin_minipage
#+name: fig:modal_indication_function #+name: fig:modal_indication_function
#+caption: Modal Indication Function #+caption: Modal Indication Function
#+attr_latex: :float nil :width 0.95\linewidth #+attr_latex: :float nil :scale 1
[[file:figs/modal_indication_function.png]] [[file:figs/modal_indication_function.png]]
#+end_minipage #+end_minipage
\hfill \hfill
#+attr_latex: :options [b]{0.28\linewidth} #+attr_latex: :options [b]{0.28\linewidth}
#+begin_minipage #+begin_minipage
#+begin_scriptsize #+begin_scriptsize
#+name: tab:modal_obtained_modes_freqs_damps #+latex: \centering
#+caption: Natural frequencies and modal damping
#+attr_latex: :environment tabularx :width \linewidth :placement [b] :align ccc #+attr_latex: :environment tabularx :width \linewidth :placement [b] :align ccc
#+attr_latex: :booktabs t :float nil :center nil #+attr_latex: :booktabs t :float nil :center nil
| Mode | Freq. [Hz] | Damp. [%] | | Mode | Frequency | Damping |
|------+------------+-----------| |------+--------------------+------------|
| 1 | 11.9 | 12.2 | | 1 | $11.9\,\text{Hz}$ | $12.2\,\%$ |
| 2 | 18.6 | 11.7 | | 2 | $18.6\,\text{Hz}$ | $11.7\,\%$ |
| 3 | 37.8 | 6.2 | | 3 | $37.8\,\text{Hz}$ | $6.2\,\%$ |
| 4 | 39.1 | 2.8 | | 4 | $39.1\,\text{Hz}$ | $2.8\,\%$ |
| 5 | 56.3 | 2.8 | | 5 | $56.3\,\text{Hz}$ | $2.8\,\%$ |
| 6 | 69.8 | 4.3 | | 6 | $69.8\,\text{Hz}$ | $4.3\,\%$ |
| 7 | 72.5 | 1.3 | | 7 | $72.5\,\text{Hz}$ | $1.3\,\%$ |
| 8 | 84.8 | 3.7 | | 8 | $84.8\,\text{Hz}$ | $3.7\,\%$ |
| 9 | 91.3 | 2.9 | | 9 | $91.3\,\text{Hz}$ | $2.9\,\%$ |
| 10 | 105.5 | 3.2 | | 10 | $105.5\,\text{Hz}$ | $3.2\,\%$ |
| 11 | 106.6 | 1.6 | | 11 | $106.6\,\text{Hz}$ | $1.6\,\%$ |
| 12 | 112.7 | 3.1 | | 12 | $112.7\,\text{Hz}$ | $3.1\,\%$ |
| 13 | 124.2 | 2.8 | | 13 | $124.2\,\text{Hz}$ | $2.8\,\%$ |
| 14 | 145.3 | 1.3 | | 14 | $145.3\,\text{Hz}$ | $1.3\,\%$ |
| 15 | 150.5 | 2.4 | | 15 | $150.5\,\text{Hz}$ | $2.4\,\%$ |
| 16 | 165.4 | 1.4 | | 16 | $165.4\,\text{Hz}$ | $1.4\,\%$ |
#+latex: \captionof{table}{\label{tab:modal_obtained_modes_freqs_damps}Identified modes}
#+end_scriptsize #+end_scriptsize
#+end_minipage #+end_minipage
@ -1186,7 +1181,7 @@ for i = 1:size(Hsyn, 1)
end end
#+end_src #+end_src
The comparison between the original measured frequency response functions and those synthesized ones from the modal model is presented in Figure ref:fig:modal_comp_acc_frf_modal. A comparison between original measured frequency response functions and synthesized ones from the modal model is presented in Figure ref:fig:modal_comp_acc_frf_modal.
Whether the obtained match is good or bad is quite arbitrary. 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. 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. 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.
@ -1296,7 +1291,7 @@ The obtained frequency response functions indicate that the dynamics of the micr
It shows a lot of coupling between stages and different directions, and many modes. It shows a lot of coupling between stages and different directions, and many modes.
By measuring 12 degrees of freedom on each "stage", it could be verified that in the frequency range of interest, each stage behaved as a rigid body. By measuring 12 degrees of freedom on each "stage", it could be verified that in the frequency range of interest, each stage behaved as a rigid body.
This confirms that a solid body model can be used to properly model the micro-station. This confirms that a multi-body model can be used to properly model the micro-station.
Although a lot of effort was put into this experimental modal analysis of the micro-station, it was difficult to obtain an accurate modal model. 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. However, the measurements are useful for tuning the parameters of the micro-station multi-body model.

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@ -1,4 +1,4 @@
% Created 2024-10-24 Thu 19:33 % Created 2024-11-14 Thu 10:42
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -39,7 +39,7 @@ To further improve the accuracy of the performance predictions, a model that bet
A multi-body model consisting of several rigid bodies connected by kinematic constraints (i.e. joints), springs and damper elements is a good candidate to model the micro-station. A multi-body model consisting of several rigid bodies connected by kinematic constraints (i.e. joints), springs and damper elements is a good candidate to model the micro-station.
Although the inertia of each solid body can easily be estimated from its geometry and material density, it is more difficult to properly estimate the stiffness and damping properties of the guiding elements connecting each solid body. Although the inertia of each solid body can easily be estimated from its geometry and material density, it is more difficult to properly estimate the stiffness and damping properties of the guiding elements connecting each solid body.
Experimental modal analysis will be use to tune the model, and to verify that a multi-body model can accurately represent the dynamics of the micro-station. Experimental modal analysis will be used to tune the model, and to verify that a multi-body model can accurately represent the dynamics of the micro-station.
The tuning approach for the multi-body model based on measurements is illustrated in Figure \ref{fig:modal_vibration_analysis_procedure}. 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. First, a \emph{response model} is obtained, which corresponds to a set of frequency response functions computed from experimental measurements.
@ -136,45 +136,46 @@ Pictures of the accelerometers fixed to the translation stage and to the micro-h
As all key stages of the micro-station are expected to behave as solid bodies, only 6 \acrshort{dof} can be considered for each solid body. 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}). However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured \acrshort{dof}) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section \ref{ssec:modal_solid_body_assumption}).
\begin{minipage}[t]{0.60\linewidth} \begin{minipage}[b]{0.63\linewidth}
\begin{center} \begin{center}
\includegraphics[scale=1,width=0.99\linewidth]{figs/modal_location_accelerometers.png} \includegraphics[scale=1,width=0.95\linewidth]{figs/modal_location_accelerometers.png}
\captionof{figure}{\label{fig:modal_location_accelerometers}Position of the accelerometers} \captionof{figure}{\label{fig:modal_location_accelerometers}Position of the accelerometers}
\end{center} \end{center}
\end{minipage} \end{minipage}
\hfill \hfill
\begin{minipage}[b]{0.38\linewidth} \begin{minipage}[b]{0.36\linewidth}
\begin{scriptsize} \begin{scriptsize}
\captionof{table}{\label{tab:modal_position_accelerometers}Positions in mm} \centering
\begin{tabularx}{\linewidth}{Xccc} \begin{tabularx}{\linewidth}{Xccc}
\toprule \toprule
& \(x\) & \(y\) & \(z\)\\ & \(x\) & \(y\) & \(z\)\\
\midrule \midrule
Low. Granite & -730 & -526 & -951\\ (17) Low. Granite & -730 & -526 & -951\\
Low. Granite & -735 & 814 & -951\\ (18) Low. Granite & -735 & 814 & -951\\
Low. Granite & 875 & 799 & -951\\ (19) Low. Granite & 875 & 799 & -951\\
Low. Granite & 865 & -506 & -951\\ (20) Low. Granite & 865 & -506 & -951\\
Up. Granite & -320 & -446 & -786\\ (13) Up. Granite & -320 & -446 & -786\\
Up. Granite & -480 & 534 & -786\\ (14) Up. Granite & -480 & 534 & -786\\
Up. Granite & 450 & 534 & -786\\ (15) Up. Granite & 450 & 534 & -786\\
Up. Granite & 295 & -481 & -786\\ (16) Up. Granite & 295 & -481 & -786\\
Translation & -475 & -414 & -427\\ (9) Translation & -475 & -414 & -427\\
Translation & -465 & 407 & -427\\ (10) Translation & -465 & 407 & -427\\
Translation & 475 & 424 & -427\\ (11) Translation & 475 & 424 & -427\\
Translation & 475 & -419 & -427\\ (12) Translation & 475 & -419 & -427\\
Tilt & -385 & -300 & -417\\ (5) Tilt & -385 & -300 & -417\\
Tilt & -420 & 280 & -417\\ (6) Tilt & -420 & 280 & -417\\
Tilt & 420 & 280 & -417\\ (7) Tilt & 420 & 280 & -417\\
Tilt & 380 & -300 & -417\\ (8) Tilt & 380 & -300 & -417\\
Spindle & -155 & -90 & -594\\ (21) Spindle & -155 & -90 & -594\\
Spindle & 0 & 180 & -594\\ (22) Spindle & 0 & 180 & -594\\
Spindle & 155 & -90 & -594\\ (23) Spindle & 155 & -90 & -594\\
Hexapod & -64 & -64 & -270\\ (1) Hexapod & -64 & -64 & -270\\
Hexapod & -64 & 64 & -270\\ (2) Hexapod & -64 & 64 & -270\\
Hexapod & 64 & 64 & -270\\ (3) Hexapod & 64 & 64 & -270\\
Hexapod & 64 & -64 & -270\\ (4) Hexapod & 64 & -64 & -270\\
\bottomrule \bottomrule
\end{tabularx} \end{tabularx}
\captionof{table}{\label{tab:modal_position_accelerometers}Positions in mm}
\end{scriptsize} \end{scriptsize}
\end{minipage} \end{minipage}
@ -233,6 +234,7 @@ For the accelerometer, a much more complex signal can be observed, indicating co
The ``normalized'' \acrfull{asd} of the two signals were computed and shown in Figure \ref{fig:modal_asd_acc_force}. 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). Conclusions based on the time domain signals can be clearly observed in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer).
These data are corresponding to an hammer impact in the vertical direction and to the measured acceleration in the \(x\) direction by accelerometer \(1\) (fixed to the micro-hexapod).
Similar results were obtained for all measured frequency response functions. Similar results were obtained for all measured frequency response functions.
\begin{figure}[htbp] \begin{figure}[htbp]
@ -248,10 +250,10 @@ Similar results were obtained for all measured frequency response functions.
\end{center} \end{center}
\subcaption{\label{fig:modal_asd_acc_force}Amplitude Spectral Density (normalized)} \subcaption{\label{fig:modal_asd_acc_force}Amplitude Spectral Density (normalized)}
\end{subfigure} \end{subfigure}
\caption{\label{fig:modal_raw_meas_asd}Raw measurement of the accelerometer (blue) and of the force sensor at the Hammer tip (red) (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force})} \caption{\label{fig:modal_raw_meas_asd}Raw measurement of the accelerometer 1 in the \(x\) direction (blue) and of the force sensor at the Hammer tip (red) for an impact in the \(z\) direction (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force})}
\end{figure} \end{figure}
The frequency response function \(H_{jk}\) from the applied force \(F_{k}\) to the measured acceleration \(X_j\) is then computed and shown Figure \ref{fig:modal_frf_acc_force}. The frequency response function from the applied force to the measured acceleration is then computed and shown Figure \ref{fig:modal_frf_acc_force}.
The quality of the obtained data can be estimated using the \emph{coherence} function (Figure \ref{fig:modal_coh_acc_force}). 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. Good coherence is obtained from \(20\,\text{Hz}\) to \(200\,\text{Hz}\) which corresponds to the frequency range of interest.
@ -268,7 +270,7 @@ Good coherence is obtained from \(20\,\text{Hz}\) to \(200\,\text{Hz}\) which co
\end{center} \end{center}
\subcaption{\label{fig:modal_coh_acc_force} Coherence} \subcaption{\label{fig:modal_coh_acc_force} Coherence}
\end{subfigure} \end{subfigure}
\caption{\label{fig:modal_frf_coh_acc_force}Computed frequency response function from the applied force \(F_{k}\) and the measured response \(X_{j}\) (\subref{fig:modal_frf_acc_force}) as well as computed coherence (\subref{fig:modal_coh_acc_force})} \caption{\label{fig:modal_frf_coh_acc_force}Computed frequency response function from the applied force \(F_{z}\) to the measured response \(X_{1,x}\) (\subref{fig:modal_frf_acc_force}) as well as computed coherence (\subref{fig:modal_coh_acc_force})}
\end{figure} \end{figure}
\chapter{Frequency Analysis} \chapter{Frequency Analysis}
@ -315,7 +317,7 @@ The goal here is to link these \(4 \times 3 = 12\) measurements to the 6 \acrsho
The motion of the rigid body of figure \ref{fig:modal_local_to_global_coordinates} can be described by its displacement \(\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]\) and (small) rotations \([\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]\) with respect to the reference frame \(\{O\}\). The motion of the rigid body of figure \ref{fig:modal_local_to_global_coordinates} can be described by its displacement \(\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]\) and (small) rotations \([\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]\) with respect to the reference frame \(\{O\}\).
The motion \(\vec{\delta} p_{i}\) of a point \(p_i\) can be computed from \(\vec{\delta} p\) and \(\bm{\delta \Omega}\) using equation \eqref{eq:modal_compute_point_response}, with \(\bm{\delta\Omega}\) defined in equation \eqref{eq:modal_rotation_matrix}. The motion \(\vec{\delta} p_{i}\) of a point \(p_i\) can be computed from \(\vec{\delta} p\) and \(\bm{\delta \Omega}\) using equation \eqref{eq:modal_compute_point_response}, with \(\bm{\delta\Omega}\) defined in equation \eqref{eq:modal_rotation_matrix} \cite[chapt. 4.3.2]{ewins00_modal}.
\begin{equation}\label{eq:modal_compute_point_response} \begin{equation}\label{eq:modal_compute_point_response}
\vec{\delta} p_{i} &= \vec{\delta} p + \bm{\delta \Omega} \cdot \vec{p}_{i} \\ \vec{\delta} p_{i} &= \vec{\delta} p + \bm{\delta \Omega} \cdot \vec{p}_{i} \\
@ -377,16 +379,16 @@ The position of each accelerometer with respect to the center of mass of the cor
\begin{table}[htbp] \begin{table}[htbp]
\caption{\label{tab:modal_com_solid_bodies}Center of mass of considered solid bodies with respect to the ``point of interest''} \caption{\label{tab:modal_com_solid_bodies}Center of mass of considered solid bodies with respect to the ``point of interest''}
\centering \centering
\begin{tabularx}{0.6\linewidth}{lXXX} \begin{tabularx}{0.55\linewidth}{Xccc}
\toprule \toprule
& \(X\) [mm] & \(Y\) [mm] & \(Z\) [mm]\\ & \(X\) & \(Y\) & \(Z\)\\
\midrule \midrule
Bottom Granite & 45 & 144 & -1251\\ Bottom Granite & \(45\,\text{mm}\) & \(144\,\text{mm}\) & \(-1251\,\text{mm}\)\\
Top granite & 52 & 258 & -778\\ Top granite & \(52\,\text{mm}\) & \(258\,\text{mm}\) & \(-778\,\text{mm}\)\\
Translation stage & 0 & 14 & -600\\ Translation stage & \(0\) & \(14\,\text{mm}\) & \(-600\,\text{mm}\)\\
Tilt Stage & 0 & -5 & -628\\ Tilt Stage & \(0\) & \(-5\,\text{mm}\) & \(-628\,\text{mm}\)\\
Spindle & 0 & 0 & -580\\ Spindle & \(0\) & \(0\) & \(-580\,\text{mm}\)\\
Hexapod & -4 & 6 & -319\\ Hexapod & \(-4\,\text{mm}\) & \(6\,\text{mm}\) & \(-319\,\text{mm}\)\\
\bottomrule \bottomrule
\end{tabularx} \end{tabularx}
\end{table} \end{table}
@ -457,38 +459,39 @@ The obtained \acrshort{mif} is shown on Figure \ref{fig:modal_indication_functio
A total of 16 modes were found between 0 and \(200\,\text{Hz}\). 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}. The obtained natural frequencies and associated modal damping are summarized in Table \ref{tab:modal_obtained_modes_freqs_damps}.
\begin{minipage}[t]{0.70\linewidth} \begin{minipage}[b]{0.70\linewidth}
\begin{center} \begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/modal_indication_function.png} \includegraphics[scale=1,scale=1]{figs/modal_indication_function.png}
\captionof{figure}{\label{fig:modal_indication_function}Modal Indication Function} \captionof{figure}{\label{fig:modal_indication_function}Modal Indication Function}
\end{center} \end{center}
\end{minipage} \end{minipage}
\hfill \hfill
\begin{minipage}[b]{0.28\linewidth} \begin{minipage}[b]{0.28\linewidth}
\begin{scriptsize} \begin{scriptsize}
\captionof{table}{\label{tab:modal_obtained_modes_freqs_damps}Natural frequencies and modal damping} \centering
\begin{tabularx}{\linewidth}{ccc} \begin{tabularx}{\linewidth}{ccc}
\toprule \toprule
Mode & Freq. [Hz] & Damp. [\%]\\ Mode & Frequency & Damping\\
\midrule \midrule
1 & 11.9 & 12.2\\ 1 & \(11.9\,\text{Hz}\) & \(12.2\,\%\)\\
2 & 18.6 & 11.7\\ 2 & \(18.6\,\text{Hz}\) & \(11.7\,\%\)\\
3 & 37.8 & 6.2\\ 3 & \(37.8\,\text{Hz}\) & \(6.2\,\%\)\\
4 & 39.1 & 2.8\\ 4 & \(39.1\,\text{Hz}\) & \(2.8\,\%\)\\
5 & 56.3 & 2.8\\ 5 & \(56.3\,\text{Hz}\) & \(2.8\,\%\)\\
6 & 69.8 & 4.3\\ 6 & \(69.8\,\text{Hz}\) & \(4.3\,\%\)\\
7 & 72.5 & 1.3\\ 7 & \(72.5\,\text{Hz}\) & \(1.3\,\%\)\\
8 & 84.8 & 3.7\\ 8 & \(84.8\,\text{Hz}\) & \(3.7\,\%\)\\
9 & 91.3 & 2.9\\ 9 & \(91.3\,\text{Hz}\) & \(2.9\,\%\)\\
10 & 105.5 & 3.2\\ 10 & \(105.5\,\text{Hz}\) & \(3.2\,\%\)\\
11 & 106.6 & 1.6\\ 11 & \(106.6\,\text{Hz}\) & \(1.6\,\%\)\\
12 & 112.7 & 3.1\\ 12 & \(112.7\,\text{Hz}\) & \(3.1\,\%\)\\
13 & 124.2 & 2.8\\ 13 & \(124.2\,\text{Hz}\) & \(2.8\,\%\)\\
14 & 145.3 & 1.3\\ 14 & \(145.3\,\text{Hz}\) & \(1.3\,\%\)\\
15 & 150.5 & 2.4\\ 15 & \(150.5\,\text{Hz}\) & \(2.4\,\%\)\\
16 & 165.4 & 1.4\\ 16 & \(165.4\,\text{Hz}\) & \(1.4\,\%\)\\
\bottomrule \bottomrule
\end{tabularx} \end{tabularx}
\captionof{table}{\label{tab:modal_obtained_modes_freqs_damps}Identified modes}
\end{scriptsize} \end{scriptsize}
\end{minipage} \end{minipage}
@ -576,7 +579,7 @@ With \(\mathbf{H}_{\text{mod}}(\omega)\) a diagonal matrix representing the resp
\mathbf{H}_{\text{mod}}(\omega) = \text{diag}\left(\frac{1}{a_1 (j\omega - s_1)},\ \dots,\ \frac{1}{a_m (j\omega - s_m)}, \frac{1}{a_1^* (j\omega - s_1^*)},\ \dots,\ \frac{1}{a_m^* (j\omega - s_m^*)} \right)_{2m\times 2m} \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} \end{equation}
The comparison between the original measured frequency response functions and those synthesized ones from the modal model is presented in Figure \ref{fig:modal_comp_acc_frf_modal}. A comparison between original measured frequency response functions and synthesized ones from the modal model is presented in Figure \ref{fig:modal_comp_acc_frf_modal}.
Whether the obtained match is good or bad is quite arbitrary. 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. 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. 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.
@ -612,7 +615,7 @@ The obtained frequency response functions indicate that the dynamics of the micr
It shows a lot of coupling between stages and different directions, and many modes. It shows a lot of coupling between stages and different directions, and many modes.
By measuring 12 degrees of freedom on each ``stage'', it could be verified that in the frequency range of interest, each stage behaved as a rigid body. By measuring 12 degrees of freedom on each ``stage'', it could be verified that in the frequency range of interest, each stage behaved as a rigid body.
This confirms that a solid body model can be used to properly model the micro-station. This confirms that a multi-body model can be used to properly model the micro-station.
Although a lot of effort was put into this experimental modal analysis of the micro-station, it was difficult to obtain an accurate modal model. 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. However, the measurements are useful for tuning the parameters of the micro-station multi-body model.