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Rework measurement setup section

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Thomas Dehaeze 2024-06-26 16:17:00 +02:00
parent 1fe38c4ba3
commit f03d226c09
21 changed files with 948 additions and 436 deletions
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mat/
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ltximg/
slprj/
matlab/slprj/

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@ -94,35 +94,66 @@
org-ref-acronyms-before-parsing))
#+END_SRC
* Notes :noexport:
Prefix is =modal=
* Introduction :ignore:
In order to properly make a multi-body model of the micro-station, an experimental modal-analysis is performed.
In order to further improve the accuracy of the performance predictions, a model that better represents the micro-station dynamics is required.
A multi-body model, consisting of several rigid bodies connected by kinematic constraints (i.e. joints) and springs and damper elements, is a good candidate to model the micro-station.
In fact, even though it is easy to estimate the inertia of each solid body from its geometry and its material density, it is much more difficult to properly estimate the stiffness and damping properties of the guiding elements connecting each solid body.
Even though the inertia of each solid body can easily be estimated from its geometry and its material density, it is more difficult to properly estimate the stiffness and damping properties of the guiding elements connecting each solid body.
The experimental modal analysis will be useful to verify that a multi-body model can represent accurately the dynamics of the micro-station and to help tuning the model.
In this report, an experimental modal analysis is perform in order to ease the development of the multi-body model.
The approach of tuning the multi-body model from 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.
From this response model, and modal model can be computed, which consists of two matrices: one containing the natural frequencies and damping factors of the considered modes, and another one describing the mode shapes.
This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is to say to tune the mass spring and damping properties of the considered solid bodies.
The measurement setup used for modal analysis is presented in Section ref:sec:modal_meas_setup.
This includes the instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system), the test planing, and the first analysis of the obtained signals.
#+begin_src latex :file modal_vibration_analysis_procedure.pdf
\begin{tikzpicture}
\node[block, inner sep = 8pt, align=center] (1) {Description\\of structure};
\node[block, inner sep = 8pt, align=center, left=1.0 of 1] (2) {Vibration\\Modes};
\node[block, inner sep = 8pt, align=center, left=1.0 of 2] (3) {Response\\Levels};
\draw[<->] (1) -- (2);
\draw[<->] (2) -- (3);
\node[above] (labelt) at (1.north) {Spatial Model};
\node[] at (2|-labelt) {Modal Model};
\node[] at (3|-labelt) {Response Model};
\node[align = center, font=\tiny, below] (labelb) at (1.south) {Mass\\Stiffness\\Damping};
\node[align = center, font=\tiny] at (2|-labelb) {Natural Frequencies\\Mode Shapes\\};
\node[align = center, font=\tiny] at (3|-labelb) {Time Responses\\Frequency Responses\\};
\end{tikzpicture}
#+end_src
#+name: fig:modal_vibration_analysis_procedure
#+caption: Figure caption
#+RESULTS:
[[file:figs/modal_vibration_analysis_procedure.png]]
The measurement setup used to obtain the response model is presented in Section ref:sec:modal_meas_setup.
This includes the instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system), the test planing, and a first analysis of the obtained signals.
In Section ref:sec:modal_frf_processing, the obtained frequency response functions between the forces applied using the instrumented hammer and the various accelerometers fixed to the structure are computed.
These measurements are then projected at the center of mass of each solid body to ease the further use of the results.
The solid body assumption is also validated.
These measurements are projected at the center of mass of each considered solid body to ease the further use of the results.
The solid body assumption is then verified, validating the use of the multi-body model.
Finally, the modal analysis is performed in Section ref:sec:modal_analysis.
It shows how complex the micro-station dynamics is, and the necessity of the developed more complex multi-body model.
[[cite:&ewins00_modal]]
#+name: tab:modal_section_matlab_code
#+caption: Report sections and corresponding Matlab files
#+attr_latex: :environment tabularx :width 0.5\linewidth :align lX
#+attr_latex: :center t :booktabs t
| *Sections* | *Matlab File* |
|--------------------------------------+----------------------------|
| Section ref:sec:modal_meas_setup | =modal_1_meas_setup.m= |
| Section ref:sec:modal_frf_processing | =modal_2_frf_processing.m= |
| Section ref:sec:modal_analysis | =modal_3_analysis.m= |
# #+name: tab:modal_section_matlab_code
# #+caption: Report sections and corresponding Matlab files
# #+attr_latex: :environment tabularx :width 0.5\linewidth :align lX
# #+attr_latex: :center t :booktabs t
# | *Sections* | *Matlab File* |
# |--------------------------------------+----------------------------|
# | Section ref:sec:modal_meas_setup | =modal_1_meas_setup.m= |
# | Section ref:sec:modal_frf_processing | =modal_2_frf_processing.m= |
# | Section ref:sec:modal_analysis | =modal_3_analysis.m= |
* Measurement Setup
:PROPERTIES:
@ -131,11 +162,10 @@ Finally, the modal analysis is performed in Section ref:sec:modal_analysis.
<<sec:modal_meas_setup>>
** Introduction :ignore:
ref:ssec:modal_instrumentation
ref:ssec:modal_test_preparation
ref:ssec:modal_accelerometers
ref:ssec:modal_hammer_impacts
ref:ssec:modal_measured_signals
In order to perform an experimental modal analysis, a proper measurement setup is key.
This include using appropriate instrumentation (presented in Section ref:ssec:modal_instrumentation) and properly preparing the structure to be measured (Section ref:ssec:modal_test_preparation).
Then, the location of the measured motion (Section ref:ssec:modal_accelerometers) and the location of the hammer impacts (Section ref:ssec:modal_hammer_impacts) have to be chosen carefully.
Obtained force and acceleration signals are shown in Section ref:ssec:modal_measured_signals, and the quality of the measured data is verified.
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -160,120 +190,122 @@ ref:ssec:modal_measured_signals
** Used Instrumentation
<<ssec:modal_instrumentation>>
In order to perform to *Modal Analysis* and to obtain first a *Response Model*, the following devices are used:
- An *acquisition system* (OROS) with 24bits ADCs (figure ref:fig:modal_oros)
- 3 tri-axis *Accelerometers* (figure ref:fig:modal_accelero_M393B05) with parameters shown on table ref:tab:modal_accelero_M393B05
- An *Instrumented Hammer* with various Tips (figure ref:fig:modal_instrumented_hammer)
Three equipment are key to perform a good modal analysis.
First, /accelerometers/ are used to measure the response of the structure.
Here, 3-axis accelerometers[fn:1] shown in figure ref:fig:modal_accelero_M393B05 are used.
These accelerometers are glued to the micro-station using a thin layer of wax for best results [[cite:&ewins00_modal chapt. 3.5.7]].
#+name: fig:modal_analysis_instrumentation
#+caption: Instrumentation used for the modal analysis
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:modal_accelero_M393B05}3-axis accelerometer}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :height 6cm
[[file:figs/modal_accelero_M393B05.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_instrumented_hammer}Instrumented hammer}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :height 6cm
[[file:figs/modal_instrumented_hammer.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_oros}OROS acquisition system}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :height 6cm
[[file:figs/modal_oros.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_accelero_M393B05}Accelerometer (M393B05)}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :height 6cm
[[file:figs/modal_accelero_M393B05.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_instrumented_hammer}Instrumented Hammer}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :height 6cm
[[file:figs/modal_instrumented_hammer.jpg]]
#+end_subfigure
#+end_figure
The acquisition system permits to auto-range the inputs (probably using variable gain amplifiers) the obtain the maximum dynamic range.
This is done before each measurement.
Anti-aliasing filters are also included in the system.
Then, an /instrumented hammer/[fn:2] (figure ref:fig:modal_instrumented_hammer) is used to apply forces to the structure in a controlled way.
Tests have been conducted to determine the most suitable hammer tip (ranging from a metallic one to a soft plastic one).
The softer tip has been found to give best results as it injects more energy in the low frequency range where the coherence was low, such that the overall coherence was improved.
#+name: tab:modal_accelero_M393B05
#+caption: 393B05 Accelerometer Data Sheet
#+attr_latex: :environment tabularx :width 0.4\linewidth :align lX
#+attr_latex: :center t :booktabs t
| Sensitivity | $10\,V/g$ |
| Measurement Range | 0.5 g pk |
| Broadband Resolution | $\SI{4}{\micro\g\rms}$ |
| Frequency Range | $0.7$ to $\SI{450}{Hz}$ |
| Resonance Frequency | $> \SI{2.5}{kHz}$ |
Tests have been conducted to determine the most suitable Hammer tip.
This has been found that the softer tip gives the best results.
It excites more the low frequency range where the coherence is low, the overall coherence was improved.
The accelerometers are glued on the structure.
Finally, an /acquisition system/[fn:3] (figure ref:fig:modal_oros) is used to acquire the injected force and the response accelerations in a synchronized way and with sufficiently low noise.
** Structure Preparation and Test Planing
<<ssec:modal_test_preparation>>
All the stages are turned ON.
This is done for two reasons:
- Be closer to the real dynamic of the station in used
- If the control system of stages are turned OFF, this would results in very low frequency modes un-identifiable with the current setup, and this will also decouple the dynamics which would not be the case in practice
This is critical for the translation stage and the spindle as their is no stiffness in the free DOF (air-bearing for the spindle for instance).
The alternative would have been to mechanically block the stages with screws, but this may result in changing the modes.
In order to obtain meaningful results, the modal analysis of the micro-station in performed /in-situ/.
To do so, all the micro-station stages controllers are turned "ON".
This is especially important for stages for which the stiffness is provided by local feedback control, which is case for the air bearing spindle, and the translation stage.
If these local feedback control were turned OFF, this would have resulted in very low frequency modes difficult to measure in practice, and this would also have lead to decoupled dynamics which would not be the case in practice.
The stages turned ON are:
- Translation Stage
- Tilt Stage
- Spindle and Slip-Ring
- Hexapod
The top part representing the active stabilization stage has been disassembled in order to reduce the complexity of the dynamics and also because the active stabilization stage and the sample will be added in the multi-body model afterwards.
The top part representing the NASS and the sample platform have been removed in order to reduce the complexity of the dynamics and also because this will be further added in the model inside Simscape.
To perform the modal-analysis from the measured responses, the $n \times n$ frequency response function matrix $\mathbf{H}$ needs to be measured, where $n$ is the considered number of degrees of freedom.
The $H_{jk}$ element of this FRF matrix corresponds to the frequency response function from a force $F_k$ applied at DoF $k$ to the displacement of the structure $X_j$ at DoF $j$.
All the stages are moved to their zero position (Ty, Ry, Rz, Slip-Ring, Hexapod).
Measuring this FRF matrix is very time consuming as it requires to make $n^2$ measurements.
However thanks to the principle of reciprocity ($H_{jk} = H_{kj}$) and using the /point measurement/ ($H_{jj}$), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix $\mathbf{H}$ [[cite:&ewins00_modal chapt. 5.2]].
Therefore, a minimum set of $n$ frequency response functions needs to be measured.
This can be done either by measuring the response $X_{j}$ at a fixed DoF $j$ while applying forces $F_{i}$ for at all $n$ considered DoF, or by applying a force $F_{k}$ at a fixed DoF $k$ and measuring the response $X_{i}$ for all $n$ DoF.
All other elements have been remove from the granite such as another heavy positioning system.
The goal is to identify the full $N \times N$ FRF matrix $H$ (where $N$ is the number of degree of freedom of the system):
\begin{equation}
H_{jk} = \frac{X_j}{F_k}
\end{equation}
However, from only one column or one line of the matrix, we can compute the other terms thanks to the principle of reciprocity.
Either we choose to identify $\frac{X_k}{F_i}$ or $\frac{X_i}{F_k}$ for any chosen $k$ and for $i = 1,\ ...,\ N$.
We here choose to identify $\frac{X_i}{F_k}$ for practical reasons:
- it is easier to glue the accelerometers on all the stages and excite only a one particular point than doing the opposite
The measurement thus consists of:
- always excite the structure at the same location with the Hammer
- Move the accelerometers to measure all the DOF of the structure
We will measured 3 columns (3 impacts location) in order to have some redundancy and to make sure that all modes are excited.
It is however not advised to measure only one row or one column as one or more modes may be missed by an unfortunate choice of force or acceleration measured locations.
In this modal-analysis, it is chosen to measure the response of the structure at all considered DoF, and to excite the structure at one location in three directions in order to have some redundancy and to make sure that all modes are properly excited.
** Location of the Accelerometers
<<ssec:modal_accelerometers>>
4 tri-axis accelerometers are used for each solid body.
Only 2 could have been used as only 6DOF have to be measured, however, we have chosen to have some *redundancy*.
The location of the accelerometers fixed to the micro-station is essential as it defines where the dynamics is identified.
A total of 23 accelerometers are fixed to the six key stages of the micro station: the lower and upper granites, the translation stage, the tilt stage, the spindle and the micro hexapod.
The position of the accelerometers are visually shown on a CAD model in Figure ref:fig:modal_location_accelerometers and their precise locations with respect to a frame located at the point of interest are summarized in Table ref:tab:modal_position_accelerometers.
Pictures of the accelerometers fixed to the translation stage and to the micro-hexapod are shown in Figure ref:fig:modal_accelerometer_pictures.
# In reality 3 are needed as if only two are used, the rotation along the line connecting the two accelerometers will not be measured.
As all key stages of the micro-station are foreseen to behave as solid bodies, only 6 DoF can be considered per solid body.
However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured DoF) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section ref:ssec:modal_solid_body_assumption).
This could also help us identify measurement problems or flexible modes is present.
#+attr_latex: :options [b]{0.68\linewidth}
#+begin_minipage
#+name: fig:modal_location_accelerometers
#+caption: Position of the accelerometers using SolidWorks
#+attr_latex: :width 0.9\linewidth :float nil
[[file:figs/modal_location_accelerometers.png]]
#+end_minipage
\hfill
#+attr_latex: :options [b]{0.31\linewidth}
#+begin_minipage
#+name: tab:modal_position_accelerometers
#+caption: Accelerometer positions
#+attr_latex: :environment tabularx :width \linewidth :align Xcccc
#+attr_latex: :center t :booktabs t :font \scriptsize :float nil
#+RESULTS:
| | $x$ | $y$ | $z$ |
|--------------+------+------+------|
| Low. Granite | -730 | -526 | -951 |
| Low. Granite | -735 | 814 | -951 |
| Low. Granite | 875 | 799 | -951 |
| Low. Granite | 865 | -506 | -951 |
| Up. Granite | -320 | -446 | -786 |
| Up. Granite | -480 | 534 | -786 |
| Up. Granite | 450 | 534 | -786 |
| Up. Granite | 295 | -481 | -786 |
| Translation | -475 | -414 | -427 |
| Translation | -465 | 407 | -427 |
| Translation | 475 | 424 | -427 |
| Translation | 475 | -419 | -427 |
| Tilt | -385 | -300 | -417 |
| Tilt | -420 | 280 | -417 |
| Tilt | 420 | 280 | -417 |
| Tilt | 380 | -300 | -417 |
| Spindle | -155 | -90 | -594 |
| Spindle | 0 | 180 | -594 |
| Spindle | 155 | -90 | -594 |
| Hexapod | -64 | -64 | -270 |
| Hexapod | -64 | 64 | -270 |
| Hexapod | 64 | 64 | -270 |
| Hexapod | 64 | -64 | -270 |
#+end_minipage
The position of the accelerometers are:
- 4 on the first granite
- 4 on the second granite
- 4 on top of the translation stage (figure ref:fig:modal_accelerometers_ty)
- 4 on top of the tilt stage
- 3 on top of the spindle
- 4 on top of the hexapod (figure ref:fig:modal_accelerometers_hexapod)
In total, 23 accelerometers are used: *69 DOFs are thus measured*.
The precise determination of the position of each accelerometer is done using the SolidWorks model (shown on figure ref:fig:modal_location_accelerometers).
#+name: fig:modal_accelerometer_pictures
#+caption: Accelerometers fixed on the micro-station
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:modal_accelerometers_ty} $T_y$ stage}
#+attr_latex: :options {0.49\textwidth}
@ -289,13 +321,6 @@ The precise determination of the position of each accelerometer is done using th
#+end_subfigure
#+end_figure
#+name: fig:modal_location_accelerometers
#+caption: Position of the accelerometers using SolidWorks
#+attr_latex: :width \linewidth
[[file:figs/modal_location_accelerometers.png]]
The precise position of all the 23 accelerometer with respect to a frame located at the point of interest (located 270mm above the top platform of the hexapod) are shown in table ref:tab:modal_position_accelerometers.
#+begin_src matlab
%% Load Accelerometer positions
acc_pos = readtable('mat/acc_pos.txt', 'ReadVariableNames', false);
@ -308,69 +333,35 @@ acc_pos = acc_pos(i, 2:4);
data2orgtable([[1:23]', 1000*acc_pos], {'Hexapod', 'Hexapod', 'Hexapod', 'Hexapod', 'Tilt', 'Tilt', 'Tilt', 'Tilt', 'Translation', 'Translation', 'Translation', 'Translation', 'Upper Granite', 'Upper Granite', 'Upper Granite', 'Upper Granite', 'Lower Granite', 'Lower Granite', 'Lower Granite', 'Lower Granite', 'Spindle', 'Spindle', 'Spindle'}, {'Stage', 'ID', '$x$ [mm]', '$y$ [mm]', '$z$ [mm]'}, ' %.0f ');
#+end_src
#+name: tab:modal_position_accelerometers
#+caption: Position of the accelerometers
#+attr_latex: :environment tabularx :width 0.5\linewidth :align Xcccc
#+attr_latex: :center t :booktabs t :font \scriptsize
#+RESULTS:
| | ID | $x$ [mm] | $y$ [mm] | $z$ [mm] |
|---------------+----+----------+----------+----------|
| Hexapod | 1 | -64 | -64 | -270 |
| Hexapod | 2 | -64 | 64 | -270 |
| Hexapod | 3 | 64 | 64 | -270 |
| Hexapod | 4 | 64 | -64 | -270 |
| Tilt | 5 | -385 | -300 | -417 |
| Tilt | 6 | -420 | 280 | -417 |
| Tilt | 7 | 420 | 280 | -417 |
| Tilt | 8 | 380 | -300 | -417 |
| Translation | 9 | -475 | -414 | -427 |
| Translation | 10 | -465 | 407 | -427 |
| Translation | 11 | 475 | 424 | -427 |
| Translation | 12 | 475 | -419 | -427 |
| Upper Granite | 13 | -320 | -446 | -786 |
| Upper Granite | 14 | -480 | 534 | -786 |
| Upper Granite | 15 | 450 | 534 | -786 |
| Upper Granite | 16 | 295 | -481 | -786 |
| Lower Granite | 17 | -730 | -526 | -951 |
| Lower Granite | 18 | -735 | 814 | -951 |
| Lower Granite | 19 | 875 | 799 | -951 |
| Lower Granite | 20 | 865 | -506 | -951 |
| Spindle | 21 | -155 | -90 | -594 |
| Spindle | 22 | 0 | 180 | -594 |
| Spindle | 23 | 155 | -90 | -594 |
** Hammer Impacts
<<ssec:modal_hammer_impacts>>
Only 3 impact points are used.
The impact points are shown on figures ref:fig:modal_impact_x, ref:fig:modal_impact_y and ref:fig:modal_impact_z.
We chose this excitation point as it seems to excite all the modes in the frequency band of interest and because it provides good coherence for all the accelerometers.
The chosen location of the hammer impact corresponds to the location of accelerometer number $11$ fixed to the translation stage.
It was chosen to match the location of one accelerometer, because a /point measurement/ (i.e. a measurement of $H_{kk}$) is necessary to be able to reconstruct the full FRF matrix [[cite:ewins00_modal]].
# TODO - Explain that the excitation is at the location of acceleromter 11
# This is very important to measure the excitation point
From [[cite:ewins00_modal]]:
Most modal test require a point mobility measurement as one of the measured FRF.
This is hard to achieve as both force and response transducer should be at the same point on the structure.
The impacts are performed in three directions, which are shown in figures ref:fig:modal_impact_x, ref:fig:modal_impact_y and ref:fig:modal_impact_z.
This excitation point and the three considered directions allows to properly excite all the modes in the frequency band of interest and to provide good coherence for all the accelerometers as will be shown in the next section.
#+name: fig:modal_hammer_impacts
#+caption: The three hammer impacts used for the modal analysis
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:modal_impact_x} $X$ impact}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :width \linewidth
#+attr_latex: :width 0.8\linewidth
[[file:figs/modal_impact_x.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_impact_y} $Y$ impact}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :width \linewidth
#+attr_latex: :width 0.8\linewidth
[[file:figs/modal_impact_y.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_impact_z} $Z$ impact}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :width \linewidth
#+attr_latex: :width 0.8\linewidth
[[file:figs/modal_impact_z.jpg]]
#+end_subfigure
#+end_figure
@ -440,6 +431,14 @@ exportFig('figs/modal_windowing_acc_signal.pdf', 'width', 'normal', 'height', 'n
** Force and Response signals
<<ssec:modal_measured_signals>>
The force sensor and the accelerometers signals are shown in the time domain in Figure ref:fig:modal_raw_meas.
Sharp "impacts" can be seen for the force sensor, indicating wide frequency band excitation.
For the accelerometer, many resonances can be seen on the right, indicating complex dynamics.
The "normalized" amplitude spectral density of the two signals are computed and shown in Figure ref:fig:modal_asd_acc_force.
Conclusions based on the time domain signals can be clearly seen in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer).
#+begin_src matlab
%% Load raw data
meas1_raw = load('mat/meas_raw_1.mat');
@ -454,47 +453,24 @@ impacts = [5.937, 11.228, 16.681, 22.205, 27.350, 32.714, 38.115, 43.888, 50.407
time = linspace(0, meas1_raw.Track1_X_Resolution*length(meas1_raw.Track1), length(meas1_raw.Track1));
#+end_src
The force sensor and the accelerometers signals are shown in the time domain in Figure ref:fig:modal_raw_meas.
Sharp "impacts" can be seen for the force sensor, indicating wide frequency band excitation.
For the accelerometer, many resonances can be seen on the right, indicating complex dynamics
#+begin_src matlab :exports none :results none
%% Raw measurement of the Accelerometer
figure;
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([1,2]);
hold on;
plot(time, meas1_raw.Track2, 'DisplayName', 'Acceleration [$m/s^2$]');
plot(time, 1e-3*meas1_raw.Track1, 'DisplayName', 'Force [kN]');
plot(time-22.2, meas1_raw.Track2, 'DisplayName', '$X_{j}$ [$m/s^2$]');
plot(time-22.2, 1e-3*meas1_raw.Track1, 'DisplayName', '$F_{k}$ [kN]');
hold off;
xlabel('Time [s]');
ylabel('Amplitude');
xlim([0, time(end)]);
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
ax2 = nexttile();
hold on;
plot(time, meas1_raw.Track2);
plot(time, 1e-3*meas1_raw.Track1);
hold off;
xlabel('Time [s]');
set(gca, 'YTickLabel',[]);
xlim([22.19, 22.4]);
linkaxes([ax1,ax2],'y');
xlim([0, 0.2]);
ylim([-2, 2]);
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_raw_meas.pdf', 'width', 'full', 'height', 'normal');
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/modal_raw_meas.pdf', 'width', 'half', 'height', 'normal');
#+end_src
#+name: fig:modal_raw_meas
#+caption: Raw measurement of the acceleromter (blue) and of the force sensor at the Hammer tip (red). Zoom on one impact is shown on the right.
#+RESULTS:
[[file:figs/modal_raw_meas.png]]
#+begin_src matlab
%% Frequency Analysis
Nfft = floor(5.0*Fs); % Number of frequency points
@ -506,15 +482,12 @@ Noverlap = floor(Nfft/2); % Overlap for frequency analysis
[pxx_acc, ~] = pwelch(meas1_raw.Track2, win, Noverlap, Nfft, Fs);
#+end_src
The "normalized" amplitude spectral density of the two signals are computed and shown in Figure ref:fig:modal_asd_acc_force.
Conclusions based on the time domain signals can be clearly seen in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer).
#+begin_src matlab :exports none :results none
%% Normalized Amplitude Spectral Density of the measured force and acceleration
figure;
hold on;
plot(f, sqrt(pxx_force./max(pxx_force(f<200))), 'DisplayName', 'Force');
plot(f, sqrt(pxx_acc./max(pxx_acc(f<200))), 'DisplayName', 'Acceleration');
plot(f, sqrt(pxx_acc./max(pxx_acc(f<200))), 'DisplayName', '$X_{j}$');
plot(f, sqrt(pxx_force./max(pxx_force(f<200))), 'DisplayName', '$F_{k}$');
hold off;
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Normalized Spectral Density');
@ -524,16 +497,31 @@ ylim([0, 1])
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_asd_acc_force.pdf', 'width', 'wide', 'height', 'normal');
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/modal_asd_acc_force.pdf', 'width', 'half', 'height', 'normal');
#+end_src
#+name: fig:modal_asd_acc_force
#+caption: Normalized Amplitude Spectral Density of the measured force and acceleration
#+RESULTS:
#+name: fig:modal_raw_meas_asd
#+caption: Raw measurement of the acceleromter (blue) and of the force sensor at the Hammer tip (red) (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Density of the two signals (normalized) (\subref{fig:modal_asd_acc_force})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:modal_raw_meas}Time domain signals}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/modal_raw_meas.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_asd_acc_force}Amplitude Spectral Density (normalized)}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/modal_asd_acc_force.png]]
#+end_subfigure
#+end_figure
The frequency response function from the applied force to the measured acceleration can then be computed (Figure ref:fig:modal_frf_acc_force).
The frequency response function from the applied force $F_{k}$ to the measured acceleration $X_j$ is then computed and shown Figure ref:fig:modal_frf_acc_force.
The quality of the obtained data can be estimated using the /coherence/ function, which is shown in Figure ref:fig:modal_coh_acc_force.
Good coherence is obtained from $20\,\text{Hz}$ to $200\,\text{Hz}$ which corresponds to the frequency range of interest.
#+begin_src matlab
%% Compute the transfer function and Coherence
@ -551,17 +539,10 @@ xlim([0, 200]);
xticks([0:20:200]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_frf_acc_force.pdf', 'width', 'wide', 'height', 'normal');
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/modal_frf_acc_force.pdf', 'width', 'half', 'height', 'normal');
#+end_src
#+name: fig:modal_frf_acc_force
#+caption: Frequency Response Function between the measured force and acceleration
#+RESULTS:
[[file:figs/modal_frf_acc_force.png]]
The coherence between the input and output signals is also computed and found to be good between 20 and 200Hz (Figure ref:fig:modal_coh_acc_force).
#+begin_src matlab :exports none :results none
%% Frequency Response Function between the force and the acceleration
figure;
@ -573,13 +554,26 @@ xticks([0:20:200]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_coh_acc_force.pdf', 'width', 'wide', 'height', 'normal');
exportFig('figs/modal_coh_acc_force.pdf', 'width', 'half', 'height', 'normal');
#+end_src
#+name: fig:modal_coh_acc_force
#+caption: Coherence between the measured force and acceleration
#+RESULTS:
#+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})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:modal_frf_acc_force} Frequency Response Function}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/modal_frf_acc_force.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_coh_acc_force} Coherence}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/modal_coh_acc_force.png]]
#+end_subfigure
#+end_figure
* Frequency Analysis
:PROPERTIES:
@ -1677,6 +1671,11 @@ Validation of solid body model.
Further step: go from modal model to parameters of the solid body model.
# Conclusion:
# - Validation of rigid body assumption
# - Explain how this helps tuning the multi-body model
# - complex dynamics: need multi-body model of the micro-station to represent the limited compliance...
* Bibliography :ignore:
#+latex: \printbibliography[heading=bibintoc,title={Bibliography}]
@ -1701,3 +1700,9 @@ addpath('./mat/'); % Path for data
%% Colors for the figures
colors = colororder;
#+END_SRC
* Footnotes
[fn:3]OROS OR36. 24bits signal-delta ADC.
[fn:2]Kistler 9722A2000. Sensitivity of $2.3\,mV/N$ and measurement range of $2\,kN$
[fn:1]PCB 393B05. Sensitivity is $10\,V/g$, measurement range is $0.5\,g$ and bandwidth is $0.7$ to $450\,\text{Hz}$.

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@ -1,4 +1,4 @@
% Created 2023-10-07 Sat 15:11
% Created 2024-06-26 Wed 16:06
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -12,7 +12,7 @@
pdftitle={Micro-Station - Modal Analysis},
pdfkeywords={},
pdfsubject={},
pdfcreator={Emacs 29.1 (Org mode 9.5.2)},
pdfcreator={Emacs 29.3 (Org mode 9.6)},
pdflang={English}}
\usepackage{biblatex}
@ -23,173 +23,154 @@
\clearpage
In order to properly make a multi-body model of the micro-station, an experimental modal-analysis is performed.
In order to further improve the accuracy of the performance predictions, a model that better represents the micro-station dynamics is required.
A multi-body model, consisting of several rigid bodies connected by kinematic constraints (i.e. joints) and springs and damper elements, is a good candidate to model the micro-station.
In fact, even though it is easy to estimate the inertia of each solid body from its geometry and its material density, it is much more difficult to properly estimate the stiffness and damping properties of the guiding elements connecting each solid body.
Even though the inertia of each solid body can easily be estimated from its geometry and its material density, it is more difficult to properly estimate the stiffness and damping properties of the guiding elements connecting each solid body.
The experimental modal analysis will be useful to verify that a multi-body model can represent accurately the dynamics of the micro-station and to help tuning the model.
In this report, an experimental modal analysis is perform in order to ease the development of the multi-body model.
The approach of tuning the multi-body model from measurements is illustrated in Figure \ref{fig:modal_vibration_analysis_procedure}.
First, a \emph{response model} is obtained, which corresponds to a set of frequency response functions computed from experimental measurements.
From this response model, and modal model can be computed, which consists of two matrices: one containing the natural frequencies and damping factors of the considered modes, and another one describing the mode shapes.
This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is to say to tune the mass spring and damping properties of the considered solid bodies.
In Section \ref{sec:modal_meas_setup} the measurement setup is presented.
The instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system) is presented, and obtained signals
Obtained frequency response functions between the forces applied using the instrumented hammer and the various accelerometers fixed to the structure are
Section \ref{sec:modal_frf_processing}
Section \ref{sec:modal_analysis}
\cite{ewins00_modal}
\begin{table}[htbp]
\caption{\label{tab:modal_section_matlab_code}Report sections and corresponding Matlab files}
\begin{figure}[htbp]
\centering
\begin{tabularx}{0.5\linewidth}{lX}
\toprule
\textbf{Sections} & \textbf{Matlab File}\\
\midrule
Section \ref{sec:modal_meas_setup} & \texttt{modal\_1\_meas\_setup.m}\\
Section \ref{sec:modal_frf_processing} & \texttt{modal\_2\_frf\_processing.m}\\
Section \ref{sec:modal_analysis} & \texttt{modal\_3\_analysis.m}\\
\bottomrule
\end{tabularx}
\end{table}
\includegraphics[scale=1]{figs/modal_vibration_analysis_procedure.png}
\caption{\label{fig:modal_vibration_analysis_procedure}Figure caption}
\end{figure}
The measurement setup used to obtain the response model is presented in Section \ref{sec:modal_meas_setup}.
This includes the instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system), the test planing, and a first analysis of the obtained signals.
In Section \ref{sec:modal_frf_processing}, the obtained frequency response functions between the forces applied using the instrumented hammer and the various accelerometers fixed to the structure are computed.
These measurements are projected at the center of mass of each considered solid body to ease the further use of the results.
The solid body assumption is then verified, validating the use of the multi-body model.
Finally, the modal analysis is performed in Section \ref{sec:modal_analysis}.
It shows how complex the micro-station dynamics is, and the necessity of the developed more complex multi-body model.
\chapter{Measurement Setup}
\label{sec:modal_meas_setup}
\ref{ssec:modal_instrumentation}
\ref{ssec:modal_test_preparation}
\ref{ssec:modal_accelerometers}
\ref{ssec:modal_hammer_impacts}
\ref{ssec:modal_measured_signals}
In order to perform an experimental modal analysis, a proper measurement setup is key.
This include using appropriate instrumentation (presented in Section \ref{ssec:modal_instrumentation}) and properly preparing the structure to be measured (Section \ref{ssec:modal_test_preparation}).
Then, the location of the measured motion (Section \ref{ssec:modal_accelerometers}) and the location of the hammer impacts (Section \ref{ssec:modal_hammer_impacts}) have to be chosen carefully.
Obtained force and acceleration signals are shown in Section \ref{ssec:modal_measured_signals}, and the quality of the measured data is verified.
\section{Used Instrumentation}
\label{ssec:modal_instrumentation}
In order to perform to \textbf{Modal Analysis} and to obtain first a \textbf{Response Model}, the following devices are used:
\begin{itemize}
\item An \textbf{acquisition system} (OROS) with 24bits ADCs (figure \ref{fig:modal_oros})
\item 3 tri-axis \textbf{Accelerometers} (figure \ref{fig:modal_accelero_M393B05}) with parameters shown on table \ref{tab:modal_accelero_M393B05}
\item An \textbf{Instrumented Hammer} with various Tips (figure \ref{fig:modal_instrumented_hammer})
\end{itemize}
\begin{figure}
Three equipment are key to perform a good modal analysis.
First, \emph{accelerometers} are used to measure the response of the structure.
Here, 3-axis accelerometers\footnote{PCB 393B05. Sensitivity is \(10\,V/g\), measurement range is \(0.5\,g\) and bandwidth is \(0.7\) to \(450\,\text{Hz}\).} shown in figure \ref{fig:modal_accelero_M393B05} are used.
These accelerometers are glued to the micro-station using a thin layer of wax for best results (see \cite[chapt. 3.5.7]{ewins00_modal}).
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,height=6cm]{figs/modal_accelero_M393B05.jpg}
\end{center}
\subcaption{\label{fig:modal_accelero_M393B05}3-axis accelerometer}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,height=6cm]{figs/modal_instrumented_hammer.jpg}
\end{center}
\subcaption{\label{fig:modal_instrumented_hammer}Instrumented hammer}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,height=6cm]{figs/modal_oros.jpg}
\end{center}
\subcaption{\label{fig:modal_oros}OROS acquisition system}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,height=6cm]{figs/modal_accelero_M393B05.jpg}
\end{center}
\subcaption{\label{fig:modal_accelero_M393B05}Accelerometer (M393B05)}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,height=6cm]{figs/modal_instrumented_hammer.jpg}
\end{center}
\subcaption{\label{fig:modal_instrumented_hammer}Instrumented Hammer}
\end{subfigure}
\caption{\label{fig:modal_analysis_instrumentation}Instrumentation used for the modal analysis}
\end{figure}
The acquisition system permits to auto-range the inputs (probably using variable gain amplifiers) the obtain the maximum dynamic range.
This is done before each measurement.
Anti-aliasing filters are also included in the system.
Then, an \emph{instrumented hammer}\footnote{Kistler 9722A2000. Sensitivity of \(2.3\,mV/N\) and measurement range of \(2\,kN\)} (figure \ref{fig:modal_instrumented_hammer}) is used to apply forces to the structure in a controlled way.
Tests have been conducted to determine the most suitable hammer tip (ranging from hard metallic one to soft plastic).
The softer tip has been found to give best results as it permits to inject more energy in the low frequency range where the coherence was low, such that the overall coherence was improved.
\begin{table}[htbp]
\caption{\label{tab:modal_accelero_M393B05}393B05 Accelerometer Data Sheet}
\centering
\begin{tabularx}{0.4\linewidth}{lX}
\toprule
Sensitivity & \(10\,V/g\)\\
Measurement Range & 0.5 g pk\\
Broadband Resolution & \(\SI{4}{\micro\g\rms}\)\\
Frequency Range & \(0.7\) to \(\SI{450}{Hz}\)\\
Resonance Frequency & \(> \SI{2.5}{kHz}\)\\
\bottomrule
\end{tabularx}
\end{table}
Tests have been conducted to determine the most suitable Hammer tip.
This has been found that the softer tip gives the best results.
It excites more the low frequency range where the coherence is low, the overall coherence was improved.
The accelerometers are glued on the structure.
Finally, an \emph{acquisition system}\footnote{OROS OR36. 24bits signal-delta ADC.} (figure \ref{fig:modal_oros}) is used to acquire the injected force and the response accelerations in a synchronized way and with sufficiently low noise.
\section{Structure Preparation and Test Planing}
\label{ssec:modal_test_preparation}
All the stages are turned ON.
This is done for two reasons:
\begin{itemize}
\item Be closer to the real dynamic of the station in used
\item If the control system of stages are turned OFF, this would results in very low frequency modes un-identifiable with the current setup, and this will also decouple the dynamics which would not be the case in practice
\end{itemize}
This is critical for the translation stage and the spindle as their is no stiffness in the free DOF (air-bearing for the spindle for instance).
The alternative would have been to mechanically block the stages with screws, but this may result in changing the modes.
In order to obtain meaningful results, the modal analysis of the micro-station in performed \emph{in-situ}.
To do so, the local position control of all the micro-station stages are turned ON.
This is especially important for stages for which the stiffness is provided by local feedback control, which is case for the air bearing spindle, and the \(T_y\) stage.
If these local feedback control were turned OFF, this would have resulted in very low frequency modes difficult to measure in practice, and this would also have lead to decoupled dynamics which would not be the case in practice.
The stages turned ON are:
\begin{itemize}
\item Translation Stage
\item Tilt Stage
\item Spindle and Slip-Ring
\item Hexapod
\end{itemize}
The top part representing the active stabilization stage and the sample platform have been removed in order to reduce the complexity of the dynamics and also because the active stabilization stage and the sample will be added in the multi-body model afterwards.
The top part representing the NASS and the sample platform have been removed in order to reduce the complexity of the dynamics and also because this will be further added in the model inside Simscape.
To perform the modal-analysis from the measured responses, the \(n \times n\) frequency response function matrix \(\mathbf{H}\) needs to be measured, with \(n\) is the considered number of degrees of freedom.
The \(H_{jk}\) element of this FRF matrix corresponds to the frequency response function from a force \(F_k\) applied at DoF \(k\) to the displacement of the structure \(X_j\) at DoF \(j\).
All the stages are moved to their zero position (Ty, Ry, Rz, Slip-Ring, Hexapod).
Measuring this FRF matrix is very time consuming as it requires to make \(n^2\) measurements.
However thanks to the principle of reciprocity (\(H_{jk} = H_{kj}\)) and using the \emph{point measurement} (\(H_{jj}\)), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix \(\mathbf{H}\) \cite[chapt. 5.2]{ewins00_modal}.
Therefore, a minimum set of \(n\) frequency response functions needs to be measured.
This can be done either by measuring the response \(X_{j}\) at a fixed DoF \(j\) while applying forces \(F_{i}\) for at all \(n\) considered DoF, or by applying a force \(F_{k}\) at a fixed DoF and measuring the response \(X_{i}\) for all \(n\) DoF.
All other elements have been remove from the granite such as another heavy positioning system.
The goal is to identify the full \(N \times N\) FRF matrix \(H\) (where \(N\) is the number of degree of freedom of the system):
\begin{equation}
H_{jk} = \frac{X_j}{F_k}
\end{equation}
However, from only one column or one line of the matrix, we can compute the other terms thanks to the principle of reciprocity.
Either we choose to identify \(\frac{X_k}{F_i}\) or \(\frac{X_i}{F_k}\) for any chosen \(k\) and for \(i = 1,\ ...,\ N\).
We here choose to identify \(\frac{X_i}{F_k}\) for practical reasons:
\begin{itemize}
\item it is easier to glue the accelerometers on all the stages and excite only a one particular point than doing the opposite
\end{itemize}
The measurement thus consists of:
\begin{itemize}
\item always excite the structure at the same location with the Hammer
\item Move the accelerometers to measure all the DOF of the structure
\end{itemize}
We will measured 3 columns (3 impacts location) in order to have some redundancy and to make sure that all modes are excited.
It is however not advice to measure only one row or one column as one or more modes may be missed by an unfortunate choice of force or acceleration measured locations.
In this modal-analysis, it is chosen to measure the response of the structure at all considered DoF, and to excite the structure at one location in three directions in order to have some redundancy and to make sure that all modes are excited.
\section{Location of the Accelerometers}
\label{ssec:modal_accelerometers}
4 tri-axis accelerometers are used for each solid body.
Only 2 could have been used as only 6DOF have to be measured, however, we have chosen to have some \textbf{redundancy}.
The location of the accelerometers fixed to the micro-station is essential as it defines at which points the dynamics is identified.
A total of 23 accelerometers are fixed to the six key stages of the micro station: the lower and upper granites, the translation stage, the tilt stage, the spindle and the micro hexapod.
The position of the accelerometers are visually shown on a CAD model in Figure \ref{fig:modal_location_accelerometers} and their precise locations with respect to a frame located at the point of interest are summarized in Table \ref{tab:modal_position_accelerometers}.
Pictures of the accelerometers fixed to the translation stage and to the micro-hexapod are shown in Figure \ref{fig:modal_accelerometer_pictures}.
This could also help us identify measurement problems or flexible modes is present.
As all key stages of the micro-station are foreseen to behave as solid bodies, only 6 DoF can be considered per solid body.
However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured DoF) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section \ref{ssec:modal_solid_body_assumption}).
The position of the accelerometers are:
\begin{itemize}
\item 4 on the first granite
\item 4 on the second granite
\item 4 on top of the translation stage (figure \ref{fig:modal_accelerometers_ty})
\item 4 on top of the tilt stage
\item 3 on top of the spindle
\item 4 on top of the hexapod (figure \ref{fig:modal_accelerometers_hexapod})
\end{itemize}
\begin{minipage}[b]{0.68\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/modal_location_accelerometers.png}
\captionof{figure}{\label{fig:modal_location_accelerometers}Position of the accelerometers using SolidWorks}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.31\linewidth}
\begin{center}
\scriptsize
\begin{tabularx}{\linewidth}{Xcccc}
\toprule
& \(x\) & \(y\) & \(z\)\\
\midrule
Low. Granite & -730 & -526 & -951\\
Low. Granite & -735 & 814 & -951\\
Low. Granite & 875 & 799 & -951\\
Low. Granite & 865 & -506 & -951\\
Up. Granite & -320 & -446 & -786\\
Up. Granite & -480 & 534 & -786\\
Up. Granite & 450 & 534 & -786\\
Up. Granite & 295 & -481 & -786\\
Translation & -475 & -414 & -427\\
Translation & -465 & 407 & -427\\
Translation & 475 & 424 & -427\\
Translation & 475 & -419 & -427\\
Tilt & -385 & -300 & -417\\
Tilt & -420 & 280 & -417\\
Tilt & 420 & 280 & -417\\
Tilt & 380 & -300 & -417\\
Spindle & -155 & -90 & -594\\
Spindle & 0 & 180 & -594\\
Spindle & 155 & -90 & -594\\
Hexapod & -64 & -64 & -270\\
Hexapod & -64 & 64 & -270\\
Hexapod & 64 & 64 & -270\\
Hexapod & 64 & -64 & -270\\
\bottomrule
\end{tabularx}
\captionof{table}{\label{tab:modal_position_accelerometers}Accelerometer positions}
\end{center}
\end{minipage}
In total, 23 accelerometers are used: \textbf{69 DOFs are thus measured}.
The precise determination of the position of each accelerometer is done using the SolidWorks model (shown on figure \ref{fig:modal_location_accelerometers}).
\begin{figure}
\begin{subfigure}{0.49\textwidth}
@ -207,76 +188,31 @@ The precise determination of the position of each accelerometer is done using th
\caption{\label{fig:modal_accelerometer_pictures}Accelerometers fixed on the micro-station}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=\linewidth]{figs/modal_location_accelerometers.png}
\caption{\label{fig:modal_location_accelerometers}Position of the accelerometers using SolidWorks}
\end{figure}
The precise position of all the 23 accelerometer with respect to a frame located at the point of interest (located 270mm above the top platform of the hexapod) are shown in table \ref{tab:modal_position_accelerometers}.
\begin{table}[htbp]
\caption{\label{tab:modal_position_accelerometers}Position of the accelerometers}
\centering
\scriptsize
\begin{tabularx}{0.5\linewidth}{Xcccc}
\toprule
& ID & \(x\) [mm] & \(y\) [mm] & \(z\) [mm]\\
\midrule
Hexapod & 1 & -64 & -64 & -270\\
Hexapod & 2 & -64 & 64 & -270\\
Hexapod & 3 & 64 & 64 & -270\\
Hexapod & 4 & 64 & -64 & -270\\
Tilt & 5 & -385 & -300 & -417\\
Tilt & 6 & -420 & 280 & -417\\
Tilt & 7 & 420 & 280 & -417\\
Tilt & 8 & 380 & -300 & -417\\
Translation & 9 & -475 & -414 & -427\\
Translation & 10 & -465 & 407 & -427\\
Translation & 11 & 475 & 424 & -427\\
Translation & 12 & 475 & -419 & -427\\
Upper Granite & 13 & -320 & -446 & -786\\
Upper Granite & 14 & -480 & 534 & -786\\
Upper Granite & 15 & 450 & 534 & -786\\
Upper Granite & 16 & 295 & -481 & -786\\
Lower Granite & 17 & -730 & -526 & -951\\
Lower Granite & 18 & -735 & 814 & -951\\
Lower Granite & 19 & 875 & 799 & -951\\
Lower Granite & 20 & 865 & -506 & -951\\
Spindle & 21 & -155 & -90 & -594\\
Spindle & 22 & 0 & 180 & -594\\
Spindle & 23 & 155 & -90 & -594\\
\bottomrule
\end{tabularx}
\end{table}
\section{Hammer Impacts}
\label{ssec:modal_hammer_impacts}
Only 3 impact points are used.
The impact points are shown on figures \ref{fig:modal_impact_x}, \ref{fig:modal_impact_y} and \ref{fig:modal_impact_z}.
We chose this excitation point as it seems to excite all the modes in the frequency band of interest and because it provides good coherence for all the accelerometers.
The impact location corresponds to the location of accelerometer \(11\) fixed to the translation stage.
The impacts are performed in three directions, which are shown in figures \ref{fig:modal_impact_x}, \ref{fig:modal_impact_y} and \ref{fig:modal_impact_z}.
This excitation point and the three considered directions were chosen in order to properly excite all the modes in the frequency band of interest and because it provides good coherence for all the accelerometers.
From \cite{ewins00_modal}:
Most modal test require a point mobility measurement as one of the measured FRF.
This is hard to achieve as both force and response transducer should be at the same point on the structure.
This excitation point corresponds to the location of one accelerometer, because a \emph{point measurement} is necessary to be able to reconstruct the full FRF matrix \cite{ewins00_modal}.
\begin{figure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/modal_impact_x.jpg}
\includegraphics[scale=1,width=0.8\linewidth]{figs/modal_impact_x.jpg}
\end{center}
\subcaption{\label{fig:modal_impact_x} $X$ impact}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/modal_impact_y.jpg}
\includegraphics[scale=1,width=0.8\linewidth]{figs/modal_impact_y.jpg}
\end{center}
\subcaption{\label{fig:modal_impact_y} $Y$ impact}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/modal_impact_z.jpg}
\includegraphics[scale=1,width=0.8\linewidth]{figs/modal_impact_z.jpg}
\end{center}
\subcaption{\label{fig:modal_impact_z} $Z$ impact}
\end{subfigure}
@ -285,39 +221,48 @@ This is hard to achieve as both force and response transducer should be at the s
\section{Force and Response signals}
\label{ssec:modal_measured_signals}
The force sensor and the accelerometers signals are shown in the time domain in Figure \ref{fig:modal_raw_meas}.
Sharp ``impacts'' can be seen for the force sensor, indicating wide frequency band excitation.
For the accelerometer, many resonances can be seen on the right, indicating complex dynamics
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/modal_raw_meas.png}
\caption{\label{fig:modal_raw_meas}Raw measurement of the acceleromter (blue) and of the force sensor at the Hammer tip (red). Zoom on one impact is shown on the right.}
\end{figure}
For the accelerometer, many resonances can be seen on the right, indicating complex dynamics.
The ``normalized'' amplitude spectral density of the two signals are computed and shown in Figure \ref{fig:modal_asd_acc_force}.
Conclusions based on the time domain signals can be clearly seen in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer).
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/modal_asd_acc_force.png}
\caption{\label{fig:modal_asd_acc_force}Normalized Amplitude Spectral Density of the measured force and acceleration}
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/modal_raw_meas.png}
\end{center}
\subcaption{\label{fig:modal_raw_meas}Time domain signals}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/modal_asd_acc_force.png}
\end{center}
\subcaption{\label{fig:modal_asd_acc_force}Amplitude Spectral Density (normalized)}
\end{subfigure}
\caption{\label{fig:modal_raw_meas_asd}Raw measurement of the acceleromter (blue) and of the force sensor at the Hammer tip (red) (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Density of the two signals (normalized) (\subref{fig:modal_asd_acc_force})}
\end{figure}
The frequency response function from the applied force to the measured acceleration can then be computed (Figure \ref{fig:modal_frf_acc_force}).
The frequency response function from the applied force \(F_{k}\) to the measured acceleration \(X_j\) is then computed and shown Figure \ref{fig:modal_frf_acc_force}.
The quality of the obtained data can be estimated using the \emph{coherence} function, which is shown in Figure \ref{fig:modal_coh_acc_force}.
Good coherence is obtained from \(20\,\text{Hz}\) to \(200\,\text{Hz}\) which corresponds to the frequency range of interest.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/modal_frf_acc_force.png}
\caption{\label{fig:modal_frf_acc_force}Frequency Response Function between the measured force and acceleration}
\end{figure}
The coherence between the input and output signals is also computed and found to be good between 20 and 200Hz (Figure \ref{fig:modal_coh_acc_force}).
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/modal_coh_acc_force.png}
\caption{\label{fig:modal_coh_acc_force}Coherence between the measured force and acceleration}
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/modal_frf_acc_force.png}
\end{center}
\subcaption{\label{fig:modal_frf_acc_force} Frequency Response Function}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/modal_coh_acc_force.png}
\end{center}
\subcaption{\label{fig:modal_coh_acc_force} Coherence}
\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})}
\end{figure}
\chapter{Frequency Analysis}
@ -489,7 +434,6 @@ The possibles choices are:
\end{itemize}
\begin{table}[htbp]
\caption{\label{tab:modal_frame_comparison}Advantages and disadvantages for the choice of reference frame}
\centering
\scriptsize
\begin{tabularx}{\linewidth}{XXX}
@ -501,6 +445,8 @@ Common Frame & We can compare the motion of each solid body & Small \(\theta_{x,
Frames at joint position & Directly gives which joint direction can be blocked & How to choose the joint position?\\
\bottomrule
\end{tabularx}
\caption{\label{tab:modal_frame_comparison}Advantages and disadvantages for the choice of reference frame}
\end{table}
The choice of the frame depends of what we want to do with the data.
@ -515,7 +461,6 @@ From solidworks, we can export the position of the center of mass of each solid
These are summarized in Table \ref{tab:modal_com_solid_bodies}
\begin{table}[htbp]
\caption{\label{tab:modal_com_solid_bodies}Center of mass of considered solid bodies}
\centering
\begin{tabularx}{0.6\linewidth}{lXXX}
\toprule
@ -529,6 +474,8 @@ Spindle & 0 & 0 & -580\\
Hexapod & -4 & 6 & -319\\
\bottomrule
\end{tabularx}
\caption{\label{tab:modal_com_solid_bodies}Center of mass of considered solid bodies}
\end{table}
@ -722,7 +669,6 @@ The obtained modal parameters are:
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}
@ -747,6 +693,8 @@ Mode & Frequency [Hz] & Damping [\%]\\
16 & 165.4 & 1.4\\
\bottomrule
\end{tabularx}
\caption{\label{tab:modal_obtained_modes_freqs_damps}Obtained eigen frequencies and modal damping}
\end{table}
\paragraph{Theory}
@ -938,5 +886,6 @@ Validation of solid body model.
Further step: go from modal model to parameters of the solid body model.
\printbibliography[heading=bibintoc,title={Bibliography}]
\end{document}