Work suspension table part

test-bench-nano-hexapod.org

@@ -105,6 +105,9 @@ Add these documents:
 - [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/test-bench-vibration-table/vibration-table.org][test-bench-vibration-table]]
 - [ ] *Use corrected APA parameters in the initialization script*
 
+
+Suspended table EPDM: ID00/test_bench/table_dyn
+
 *Goal of this test bench*:
 - Properly mount the nano-hexapod
 - Verify all is working
@@ -141,6 +144,8 @@ Maybe the rest is not so interesting here as it will be presented again in the n
 - [ ] Modes of the encoder supports
 - [ ] ...
 
+** TODO [#C] Remove un-used matlab scripts and src files
+
 ** TODO [#B] Make nice subfigures for identified modes
 SCHEDULED: <2024-10-26 Sat>
 
@@ -4111,94 +4116,24 @@ After all six struts are mounted, the mounting tool (Figure ref:fig:test_nhexa_c
 <<sec:test_nhexa_table>>
 ** Introduction
 
-# In EPDM: ID00/test_bench/table_dyn
+When a dynamical system is fixed to a support (such as a granite or an optical table), its dynamics will couple to the support dynamics.
+This may results in additional modes appearing in the system dynamics, which are difficult to predict and model.
+Two prevent this issue, strategy adopted here is to mount the nano-hexapod on top a suspended table with low frequency suspension modes.
 
-This document is divided as follows:
+In such a case, the modes of the suspended table are chosen to be at much lower frequency than those of the nano-hexapod such that they are well decoupled.
-- Section ref:ssec:test_nhexa_table_setup: the experimental setup and all the instrumentation are described
+An other key advantage is that the suspension modes of the suspended table can be easily modelled using Simscape.
-- Section ref:ssec:test_nhexa_table_identification: the table dynamics is identified
+Therefore, the measured dynamics of the nano-hexapod on top of the suspended table can be compared to a simscape model representing the same experimental conditions.
-- Section ref:ssec:test_nhexa_table_model: a Simscape model of the vibration table is developed and tuned from the measurements
+The model of the Nano-Hexapod can thus be precisely tuned to match the measured dynamics.
 
-** Experimental Setup
+The developed suspended table is presented in Section ref:ssec:test_nhexa_table_setup.
-<<ssec:test_nhexa_table_setup>>
+The modal analysis of the table is done in ref:ssec:test_nhexa_table_identification.
-*** Introduction                                                    :ignore:
+Finally, the Simscape model representing the suspended table is tuned to match the measured modes (Section ref:ssec:test_nhexa_table_model).
 
-- [ ] Redo the CAD view
+** Matlab Init                                              :noexport:ignore:
+#+begin_src matlab
+%% test_nhexa_table.m
+#+end_src
 
-#+name: fig:vibration-table-cad-view
-#+caption: CAD View of the vibration table
-#+attr_latex: :width 0.8\linewidth
-[[file:figs/vibration-table-cad-view.png]]
-
-*** Suspended table
-
-- Dimensions :: 450 mm x 450 mm x 60 mm
-- Mass :: 21.3 kg (bot=7.8, top=7.6, mid=5.9kg)
-- Interface plate :: 3.2kg
-
-#+name: fig:compliance_optical_table
-#+caption: Compliance of the B4545A optical table
-#+attr_latex: :width 0.8\linewidth
-[[file:figs/test_nhexa_compliance_table.png]]
-
-If we include including the bottom interface plate:
-- Total mass: 24.5 kg
-- CoM: 42mm below Center of optical table
-- Ix = 0.54, Iy = 0.54, Iz = 1.07 (with respect to CoM)
-
-*** Springs
-
-Helical compression spring
-make of steel wire (52SiCrNi5) with rectangular cross section
-SZ8005 20 x 044 from Steinel
-L0 = 44mm
-Spring rate = 17.8 N/mm
-
-[[file:figs/test_nhexa_table_springs.jpg]]
-
-** Identification of the table's response
-<<ssec:test_nhexa_table_identification>>
-
-(4x) 3D accelerometer [[https://www.pcbpiezotronics.fr/produit/accelerometres/356b18/][PCB 356B18]]
-
-#+name: tab:list_modes
-#+caption: List of the identified modes
-#+attr_latex: :environment tabularx :width 0.5\linewidth :align ccX
-#+attr_latex: :center t :booktabs t :float t
-|   | Freq. [Hz] | Description   |
-|---+------------+---------------|
-| 1 |        1.3 | X-translation |
-| 2 |        1.3 | Y-translation |
-| 3 |       1.95 | Z-rotation    |
-| 4 |       6.85 | Z-translation |
-| 5 |        8.9 | Tilt          |
-| 6 |        8.9 | Tilt          |
-| 7 |        700 | Flexible Mode |
-
-#+name: fig:test_nhexa_mode_shapes_rigid_table
-#+caption: Mode shapes of the 6 suspension modes (from 1Hz to 9Hz)
-#+attr_latex: :width \linewidth
-[[file:figs/test_nhexa_mode_shapes_rigid_table.png]]
-
-#+name: fig:ModeShapeHF1_crop
-#+caption: First flexible mode of the table at 700Hz
-#+attr_latex: :width 0.3\linewidth
-[[file:figs/ModeShapeHF1_crop.gif]]
-
-
-** Simscape Model of the suspended table
-:PROPERTIES:
-:header-args:matlab+: :tangle matlab/simscape_model.m
-:END:
-<<ssec:test_nhexa_table_model>>
-*** Introduction                                                    :ignore:
-In this section, the Simscape model of the vibration table is described.
-
-#+name: fig:simscape_vibration_table
-#+caption: 3D representation of the simscape model
-#+attr_latex: :width 0.8\linewidth
-[[file:figs/simscape_vibration_table.png]]
-
-*** Matlab Init                                            :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
 <<matlab-dir>>
 #+end_src
@@ -4223,79 +4158,101 @@ In this section, the Simscape model of the vibration table is described.
 <<m-init-other>>
 #+end_src
 
-*** Simscape Sub-systems
+** Experimental Setup
-<<sec:simscape_parameters>>
+<<ssec:test_nhexa_table_setup>>
 
-Parameters for sub-components of the simscape model are defined below.
+The design of the suspended table is quite straightforward.
+First, an optical table with high frequency flexible mode was selected[fn:5].
+Then, four springs[fn:6] were selected with low enough spring rate such that the suspension modes are below 10Hz.
+Finally, some interface elements were designed, and mechanical lateral mechanical stops were added (Figure ref:fig:test_nhexa_suspended_table_cad).
 
-**** Springs
+#+name: fig:test_nhexa_suspended_table_cad
-<<sec:simscape_springs>>
+#+caption: CAD View of the vibration table. Purple cylinders are representing the soft springs.
+#+attr_latex: :width 0.7\linewidth
+[[file:figs/test_nhexa_suspended_table_cad.jpg]]
 
-The 4 springs supporting the suspended optical table are modelled with "bushing joints" having stiffness and damping in the x-y-z directions:
+** Modal analysis of the suspended table
+<<ssec:test_nhexa_table_identification>>
 
-#+begin_src matlab
+In order to perform a modal analysis of the suspended table, a total of 15 3-axis accelerometers[fn:7] were fixed to the breadboard.
-%% This parameters are defined in the Mask of the suspended table
+Using an instrumented hammer, the first 9 modes could be identified and are summarized in Table ref:tab:test_nhexa_suspended_table_modes.
-% spring.kx = 0.5e3; % X- Stiffness [N/m]
+The first 6 modes are suspension modes (i.e. rigid body mode of the breadboard) and are located below 10Hz.
-% spring.cx = 15; % X- Damping [N/(m/s)]
+The next modes are flexible modes of the breadboard as shown in Figure ref:fig:test_nhexa_table_flexible_modes, and located above 700Hz.
 
-% spring.ky = 0.5e3; % Y- Stiffness [N/m]
+#+attr_latex: :options [t]{0.45\linewidth}
-% spring.cy = 15; % Y- Damping [N/(m/s)]
+#+begin_minipage
+#+name: fig:test_nhexa_suspended_table
+#+caption: Mounted suspended table. Only 1 or the 15 accelerometer is mounted on top
+#+attr_latex: :width 0.99\linewidth :float nil
+[[file:figs/test_nhexa_suspended_table.jpg]]
+#+end_minipage
+\hfill
+#+attr_latex: :options [b]{0.45\linewidth}
+#+begin_minipage
+#+begin_scriptsize
+#+name: tab:test_nhexa_suspended_table_modes
+#+caption: Obtained modes of the suspended table
+#+attr_latex: :environment tabularx :width 0.9\linewidth :placement [b] :align clX
+#+attr_latex: :booktabs t :float nil :center t
+#+RESULTS:
+| *Modes* | *Frequency* | *Description*    |
+|---------+-------------+------------------|
+|     1,2 | 1.3 Hz      | X-Y translations |
+|       3 | 2.0 Hz      | Z rotation       |
+|       4 | 6.9 Hz      | Z translation    |
+|     5,6 | 9.5 Hz      | X-Y rotations    |
+|---------+-------------+------------------|
+|       7 | 701 Hz      | "Membrane" Mode  |
+|       8 | 989 Hz      | Complex mode     |
+|       9 | 1025 Hz     | Complex mode     |
+#+end_scriptsize
+#+end_minipage
 
-% spring.kz = 1e3; % Z- Stiffness [N/m]
-% spring.cz = 50; % Z- Damping [N/(m/s)]
 
-% spring.z0 = 32e-3; % Equilibrium z-length [m]
+#+name: fig:test_nhexa_table_flexible_modes
-#+end_src
+#+caption: Three identified flexible modes of the suspended table
+#+attr_latex: :options [htbp]
+#+begin_figure
+#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_table_flexible_mode_1}Flexible mode at 701Hz}
+#+attr_latex: :options {\textwidth}
+#+begin_subfigure
+#+attr_latex: :width \linewidth
+[[file:figs/test_nhexa_table_flexible_mode_1.jpg]]
+#+end_subfigure
+#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_table_flexible_mode_2}Flexible mode at 989Hz}
+#+attr_latex: :options {\textwidth}
+#+begin_subfigure
+#+attr_latex: :width \linewidth
+[[file:figs/test_nhexa_table_flexible_mode_2.jpg]]
+#+end_subfigure
+#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_table_flexible_mode_3}Flexible mode at 1025Hz}
+#+attr_latex: :options {\textwidth}
+#+begin_subfigure
+#+attr_latex: :width \linewidth
+[[file:figs/test_nhexa_table_flexible_mode_3.jpg]]
+#+end_subfigure
+#+end_figure
 
-**** Inertial Shaker (IS20)
+** Simscape Model of the suspended table
-<<sec:simscape_inertial_shaker>>
+:PROPERTIES:
+:header-args:matlab+: :tangle matlab/simscape_model.m
+:END:
+<<ssec:test_nhexa_table_model>>
 
-The inertial shaker is defined as two solid bodies:
+The Simscape model of the suspended table simply consists of two solid bodies connected by 4 springs.
-- the "housing" that is fixed to the element that we want to excite
+The 4 springs are here modelled with "bushing joints" that have stiffness and damping properties in x, y and z directions.
-- the "inertial mass" that is suspended inside the housing
+The 3D representation of the model is displayed in Figure ref:fig:test_nhexa_suspended_table_simscape where the 4 "bushing joints" are represented by the blue cylinders.
 
-The inertial mass is guided inside the housing and an actuator (coil and magnet) can be used to apply a force between the inertial mass and the support.
+#+name: fig:test_nhexa_suspended_table_simscape
-The "reacting" force on the support is then used as an excitation.
+#+caption: 3D representation of the simscape model
+#+attr_latex: :width 0.8\linewidth
+[[file:figs/test_nhexa_suspended_table_simscape.png]]
 
-#+name: tab:is20_characteristics
+The model order is 12, and it represents the 6 suspension modes.
-#+caption: Summary of the IS20 datasheet
+The inertia properties of the parts are set from the geometry and material densities.
-#+attr_latex: :environment tabularx :width 0.4\linewidth :align lX
+The stiffness of the springs was initially set from the datasheet nominal value of $17.8\,N/mm$ and then reduced down to $14\,N/mm$ to better match the measured suspension modes.
-#+attr_latex: :center t :booktabs t :float t
+The stiffness of the springs in the horizontal plane is set at $0.5\,N/mm$.
-| Characteristic  | Value      |
+The obtained suspension modes of the simscape model are compared with the measured ones in Table ref:tab:test_nhexa_suspended_table_simscape_modes.
-|-----------------+------------|
-| Output Force    | 20 N       |
-| Frequency Range | 10-3000 Hz |
-| Moving Mass     | 0.1 kg     |
-| Total Mass      | 0.3 kg     |
-
-From the datasheet in Table ref:tab:is20_characteristics, we can estimate the parameters of the physical shaker.
-
-These parameters are defined below
-**** 3D accelerometer (356B18)
-<<sec:simscape_accelerometers>>
-
-An accelerometer consists of 2 solids:
-- a "housing" rigidly fixed to the measured body
-- an "inertial mass" suspended inside the housing by springs and guided in the measured direction
-
-The relative motion between the housing and the inertial mass gives a measurement of the acceleration of the measured body (up to the suspension mode of the inertial mass).
-
-#+name: tab:356b18_characteristics
-#+caption: Summary of the 356B18 datasheet
-#+attr_latex: :environment tabularx :width 0.5\linewidth :align lX
-#+attr_latex: :center t :booktabs t :float t
-| Characteristic      | Value               |
-|---------------------+---------------------|
-| Sensitivity         | 0.102 V/(m/s2)      |
-| Frequency Range     | 0.5 to 3000 Hz      |
-| Resonance Frequency | > 20 kHz            |
-| Resolution          | 0.0005 m/s2 rms     |
-| Weight              | 0.025 kg            |
-| Size                | 20.3x26.1x20.3 [mm] |
-
-*** Identification
-<<sec:simscape_parameters>>
-Let's now identify the resonance frequency and mode shapes associated with the suspension modes of the optical table.
 
 #+begin_src matlab
 %% Configure Simscape Model
@@ -4312,79 +4269,20 @@ io(io_i) = linio([mdl, '/F_v'],  1, 'openoutput'); io_i = io_i + 1;
 G = linearize(mdl, io);
 G.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
 G.OutputName  = {'Vdx', 'Vdy', 'Vdz', 'Vrx', 'Vry', 'Vrz'};
-#+end_src
 
-#+begin_src matlab
+%% Compute the resonance frequencies
-freqs = logspace(0, 2, 1000);
-figure;
-hold on;
-plot(freqs, abs(squeeze(freqresp(G(1,1), freqs, 'Hz'))), 'DisplayName', '$x$');
-plot(freqs, abs(squeeze(freqresp(G(2,2), freqs, 'Hz'))), 'DisplayName', '$y$');
-plot(freqs, abs(squeeze(freqresp(G(3,3), freqs, 'Hz'))), 'DisplayName', '$z$');
-plot(freqs, abs(squeeze(freqresp(G(4,4), freqs, 'Hz'))), 'DisplayName', '$R_x$');
-plot(freqs, abs(squeeze(freqresp(G(5,5), freqs, 'Hz'))), 'DisplayName', '$R_y$');
-plot(freqs, abs(squeeze(freqresp(G(6,6), freqs, 'Hz'))), 'DisplayName', '$R_z$');
-hold off;
-set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-xlabel('Frequency [Hz]'); ylabel('Amplitude [$m/s/N$]');
-legend('location', 'northeast');
-#+end_src
-
-#+begin_src matlab :results output replace :exports results :tangle no
-size(G)
-#+end_src
-
-#+RESULTS:
-: size(G)
-: State-space model with 6 outputs, 6 inputs, and 12 states.
-
-Compute the resonance frequencies
-#+begin_src matlab
 ws = eig(G.A);
 ws = ws(imag(ws) > 0);
 #+end_src
 
-#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+#+name: tab:test_nhexa_suspended_table_simscape_modes
-data2orgtable([sort(imag(ws))'/2/pi; 1.3, 1.3, 1.95, 6.85, 8.9, 9.5], {'Simscape', 'Experimental'}, {'x', 'y', 'Rz', 'Dz', 'Rx', 'Ry'}, ' %.2f ');
+#+caption: Comparison of the identified suspension modes with the Simscape model and measured experimentally
-#+end_src
+#+attr_latex: :environment tabularx :width 0.6\linewidth :align Xcccc
-
+#+attr_latex: :center t :booktabs t
-#+RESULTS:
+| Directions   | $D_x$, $D_y$ | $R_z$  | $D_z$  | $R_x$, $R_y$ |
-|              |    x |    y |   Rz |   Dz |   Rx |   Ry |
+|--------------+--------------+--------+--------+--------------|
-|--------------+------+------+------+------+------+------|
+| Experimental | 1.3 Hz       | 2.0 Hz | 6.9 Hz | 9.5 Hz       |
-| Simscape     | 1.28 | 1.28 | 1.82 | 6.78 | 9.47 | 9.47 |
+| Simscape     | 1.3 Hz       | 1.8 Hz | 6.8 Hz | 9.5 Hz       |
-| Experimental |  1.3 |  1.3 | 1.95 | 6.85 |  8.9 |  9.5 |
-
-
-And the associated response of the optical table
-#+begin_src matlab
-x_mod = zeros(6, 6); % 6 modes, 6 outputs
-
-for i = 1:length(ws)
-    xi = evalfr(G(1,:), ws(i));
-    x_mod(:,i) = xi./norm(xi);
-end
-#+end_src
-
-The results are shown in Table ref:tab:mode_shapes.
-The motion associated to the mode shapes are just indicative.
-
-#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
-data2orgtable([flip(imag(ws)/2/pi)'; flip(abs(x_mod),2)], {'$\omega_0$ [Hz]', 'x', 'y', 'z', 'Rx', 'Ry', 'Rz'}, {}, ' %.1f ');
-#+end_src
-
-#+name: tab:mode_shapes
-#+caption: Resonance frequency and approximation of the mode shapes
-#+attr_latex: :environment tabularx :width 0.4\linewidth :align Xcccccc
-#+attr_latex: :center t :booktabs t :float t
-#+RESULTS:
-| $\omega_0$ [Hz] | 8.2 | 8.2 | 8.2 | 5.8 | 5.6 | 5.6 |
-|-----------------+-----+-----+-----+-----+-----+-----|
-| x               | 0.0 | 0.0 | 0.0 | 0.0 | 0.1 | 0.5 |
-| y               | 0.0 | 0.0 | 0.0 | 0.0 | 0.5 | 0.0 |
-| z               | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 |
-| Rx              | 1.0 | 0.0 | 0.0 | 0.0 | 0.8 | 0.0 |
-| Ry              | 0.0 | 1.0 | 0.0 | 0.0 | 0.2 | 0.9 |
-| Rz              | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 |
 
 * Nano-Hexapod Dynamics
 <<sec:test_nhexa_dynamics>>
@@ -5899,6 +5797,10 @@ freqs = 2*logspace(1, 3, 1000);
 #+END_SRC
 
 * Footnotes
+
+[fn:7]PCB 356B18. Sensitivity is $1\,V/g$, measurement range is $\pm 5\,g$ and bandwidth is $0.5$ to $5\,\text{kHz}$.
+[fn:6]"SZ8005 20 x 044" from Steinel. The spring rate is specified at $17.8\,N/mm$
+[fn:5]The 450 mm x 450 mm x 60 mm Nexus B4545A from Thorlabs.
 [fn:4]As the accuracy of the FARO arm is $\pm 13\,\mu m$, the true straightness is probably better than the values indicated. The limitation of the instrument is here reached.
 [fn:3]The height dimension is better than $40\,\mu m$. The diameter fit of 182g6 and 24g6 with the two plates is verified.
 [fn:2]Location of all the interface surfaces with the flexible joints are checked. The fits (182H7 and 24H8) with the interface element are checked.

test-bench-nano-hexapod.tex

@@ -1,4 +1,4 @@
-% Created 2024-10-27 Sun 11:12
+% Created 2024-10-27 Sun 14:49
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -133,214 +133,124 @@ After all six struts are mounted, the mounting tool (Figure \ref{fig:test_nhexa_
 \label{sec:test_nhexa_table}
 \section{Introduction}
 
-This document is divided as follows:
+When a dynamical system is fixed to a support (such as a granite or an optical table), its dynamics will couple to the support dynamics.
-\begin{itemize}
+This may results in additional modes appearing in the system dynamics, which are difficult to predict and model.
-\item Section \ref{ssec:test_nhexa_table_setup}: the experimental setup and all the instrumentation are described
+Two prevent this issue, strategy adopted here is to mount the nano-hexapod on top a suspended table with low frequency suspension modes.
-\item Section \ref{ssec:test_nhexa_table_identification}: the table dynamics is identified
+
-\item Section \ref{ssec:test_nhexa_table_model}: a Simscape model of the vibration table is developed and tuned from the measurements
+In such a case, the modes of the suspended table are chosen to be at much lower frequency than those of the nano-hexapod such that they are well decoupled.
-\end{itemize}
+An other key advantage is that the suspension modes of the suspended table can be easily modelled using Simscape.
+Therefore, the measured dynamics of the nano-hexapod on top of the suspended table can be compared to a simscape model representing the same experimental conditions.
+The model of the Nano-Hexapod can thus be precisely tuned to match the measured dynamics.
+
+The developed suspended table is presented in Section \ref{ssec:test_nhexa_table_setup}.
+The modal analysis of the table is done in \ref{ssec:test_nhexa_table_identification}.
+Finally, the Simscape model representing the suspended table is tuned to match the measured modes (Section \ref{ssec:test_nhexa_table_model}).
 
 \section{Experimental Setup}
 \label{ssec:test_nhexa_table_setup}
-\begin{itemize}
+
-\item[{$\square$}] Redo the CAD view
+The design of the suspended table is quite straightforward.
-\end{itemize}
+First, an optical table with high frequency flexible mode was selected\footnote{The 450 mm x 450 mm x 60 mm Nexus B4545A from Thorlabs.}.
+Then, four springs\footnote{``SZ8005 20 x 044'' from Steinel. The spring rate is specified at \(17.8\,N/mm\)} were selected with low enough spring rate such that the suspension modes are below 10Hz.
+Finally, some interface elements were designed, and mechanical lateral mechanical stops were added (Figure \ref{fig:test_nhexa_suspended_table_cad}).
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=0.8\linewidth]{figs/vibration-table-cad-view.png}
+\includegraphics[scale=1,width=0.7\linewidth]{figs/test_nhexa_suspended_table_cad.jpg}
-\caption{\label{fig:vibration-table-cad-view}CAD View of the vibration table}
+\caption{\label{fig:test_nhexa_suspended_table_cad}CAD View of the vibration table. Purple cylinders are representing the soft springs.}
-\end{figure}
-\subsection{Suspended table}
-
-\begin{description}
-\item[{Dimensions}] 450 mm x 450 mm x 60 mm
-\item[{Mass}] 21.3 kg (bot=7.8, top=7.6, mid=5.9kg)
-\item[{Interface plate}] 3.2kg
-\end{description}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=0.8\linewidth]{figs/test_nhexa_compliance_table.png}
-\caption{\label{fig:compliance_optical_table}Compliance of the B4545A optical table}
 \end{figure}
 
-If we include including the bottom interface plate:
+\section{Modal analysis of the suspended table}
-\begin{itemize}
-\item Total mass: 24.5 kg
-\item CoM: 42mm below Center of optical table
-\item Ix = 0.54, Iy = 0.54, Iz = 1.07 (with respect to CoM)
-\end{itemize}
-
-\subsection{Springs}
-
-Helical compression spring
-make of steel wire (52SiCrNi5) with rectangular cross section
-SZ8005 20 x 044 from Steinel
-L0 = 44mm
-Spring rate = 17.8 N/mm
-
-\begin{center}
-\includegraphics[scale=1]{figs/test_nhexa_table_springs.jpg}
-\end{center}
-
-\section{Identification of the table's response}
 \label{ssec:test_nhexa_table_identification}
 
-(4x) 3D accelerometer \href{https://www.pcbpiezotronics.fr/produit/accelerometres/356b18/}{PCB 356B18}
+In order to perform a modal analysis of the suspended table, a total of 15 3-axis accelerometers\footnote{PCB 356B18. Sensitivity is \(1\,V/g\), measurement range is \(\pm 5\,g\) and bandwidth is \(0.5\) to \(5\,\text{kHz}\).} were fixed to the breadboard.
+Using an instrumented hammer, the first 9 modes could be identified and are summarized in Table \ref{tab:test_nhexa_suspended_table_modes}.
+The first 6 modes are suspension modes (i.e. rigid body mode of the breadboard) and are located below 10Hz.
+The next modes are flexible modes of the breadboard as shown in Figure \ref{fig:test_nhexa_table_flexible_modes}, and located above 700Hz.
 
-\begin{table}[htbp]
+\begin{minipage}[t]{0.45\linewidth}
-\centering
+\begin{center}
-\begin{tabularx}{0.5\linewidth}{ccX}
+\includegraphics[scale=1,width=0.99\linewidth]{figs/test_nhexa_suspended_table.jpg}
+\captionof{figure}{\label{fig:test_nhexa_suspended_table}Mounted suspended table. Only 1 or the 15 accelerometer is mounted on top}
+\end{center}
+\end{minipage}
+\hfill
+\begin{minipage}[b]{0.45\linewidth}
+\begin{scriptsize}
+\begin{center}
+\begin{tabularx}{0.9\linewidth}{clX}
 \toprule
- & Freq. [Hz] & Description\\
+\textbf{Modes} & \textbf{Frequency} & \textbf{Description}\\
 \midrule
-1 & 1.3 & X-translation\\
+1,2 & 1.3 Hz & X-Y translations\\
-2 & 1.3 & Y-translation\\
+3 & 2.0 Hz & Z rotation\\
-3 & 1.95 & Z-rotation\\
+4 & 6.9 Hz & Z translation\\
-4 & 6.85 & Z-translation\\
+5,6 & 9.5 Hz & X-Y rotations\\
-5 & 8.9 & Tilt\\
+\midrule
-6 & 8.9 & Tilt\\
+7 & 701 Hz & ``Membrane'' Mode\\
-7 & 700 & Flexible Mode\\
+8 & 989 Hz & Complex mode\\
+9 & 1025 Hz & Complex mode\\
 \bottomrule
 \end{tabularx}
-\caption{\label{tab:list_modes}List of the identified modes}
+\captionof{table}{\label{tab:test_nhexa_suspended_table_modes}Obtained modes of the suspended table}
+
+\end{center}
+\end{scriptsize}
+\end{minipage}
 
-\end{table}
 
 \begin{figure}[htbp]
-\centering
+\begin{subfigure}{\textwidth}
-\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_mode_shapes_rigid_table.png}
+\begin{center}
-\caption{\label{fig:test_nhexa_mode_shapes_rigid_table}Mode shapes of the 6 suspension modes (from 1Hz to 9Hz)}
+\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_table_flexible_mode_1.jpg}
+\end{center}
+\subcaption{\label{fig:test_nhexa_table_flexible_mode_1}Flexible mode at 701Hz}
+\end{subfigure}
+\begin{subfigure}{\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_table_flexible_mode_2.jpg}
+\end{center}
+\subcaption{\label{fig:test_nhexa_table_flexible_mode_2}Flexible mode at 989Hz}
+\end{subfigure}
+\begin{subfigure}{\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_table_flexible_mode_3.jpg}
+\end{center}
+\subcaption{\label{fig:test_nhexa_table_flexible_mode_3}Flexible mode at 1025Hz}
+\end{subfigure}
+\caption{\label{fig:test_nhexa_table_flexible_modes}Three identified flexible modes of the suspended table}
 \end{figure}
 
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=0.3\linewidth]{figs/ModeShapeHF1_crop.gif}
-\caption{\label{fig:ModeShapeHF1_crop}First flexible mode of the table at 700Hz}
-\end{figure}
-
-
 \section{Simscape Model of the suspended table}
 \label{ssec:test_nhexa_table_model}
-In this section, the Simscape model of the vibration table is described.
+
+The Simscape model of the suspended table simply consists of two solid bodies connected by 4 springs.
+The 4 springs are here modelled with ``bushing joints'' that have stiffness and damping properties in x, y and z directions.
+The 3D representation of the model is displayed in Figure \ref{fig:test_nhexa_suspended_table_simscape} where the 4 ``bushing joints'' are represented by the blue cylinders.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=0.8\linewidth]{figs/simscape_vibration_table.png}
+\includegraphics[scale=1,width=0.8\linewidth]{figs/test_nhexa_suspended_table_simscape.png}
-\caption{\label{fig:simscape_vibration_table}3D representation of the simscape model}
+\caption{\label{fig:test_nhexa_suspended_table_simscape}3D representation of the simscape model}
 \end{figure}
-\subsection{Simscape Sub-systems}
-\label{sec:simscape_parameters}
 
-Parameters for sub-components of the simscape model are defined below.
+The model order is 12, and it represents the 6 suspension modes.
-
+The inertia properties of the parts are set from the geometry and material densities.
-\paragraph{Springs}
+The stiffness of the springs was initially set from the datasheet nominal value of \(17.8\,N/mm\) and then reduced down to \(14\,N/mm\) to better match the measured suspension modes.
-\label{sec:simscape_springs}
+The stiffness of the springs in the horizontal plane is set at \(0.5\,N/mm\).
-
+The obtained suspension modes of the simscape model are compared with the measured ones in Table \ref{tab:test_nhexa_suspended_table_simscape_modes}.
-The 4 springs supporting the suspended optical table are modelled with ``bushing joints'' having stiffness and damping in the x-y-z directions:
-
-\paragraph{Inertial Shaker (IS20)}
-\label{sec:simscape_inertial_shaker}
-
-The inertial shaker is defined as two solid bodies:
-\begin{itemize}
-\item the ``housing'' that is fixed to the element that we want to excite
-\item the ``inertial mass'' that is suspended inside the housing
-\end{itemize}
-
-The inertial mass is guided inside the housing and an actuator (coil and magnet) can be used to apply a force between the inertial mass and the support.
-The ``reacting'' force on the support is then used as an excitation.
 
 \begin{table}[htbp]
 \centering
-\begin{tabularx}{0.4\linewidth}{lX}
+\begin{tabularx}{0.6\linewidth}{Xcccc}
 \toprule
-Characteristic & Value\\
+Directions & \(D_x\), \(D_y\) & \(R_z\) & \(D_z\) & \(R_x\), \(R_y\)\\
 \midrule
-Output Force & 20 N\\
+Experimental & 1.3 Hz & 2.0 Hz & 6.9 Hz & 9.5 Hz\\
-Frequency Range & 10-3000 Hz\\
+Simscape & 1.3 Hz & 1.8 Hz & 6.8 Hz & 9.5 Hz\\
-Moving Mass & 0.1 kg\\
-Total Mass & 0.3 kg\\
 \bottomrule
 \end{tabularx}
-\caption{\label{tab:is20_characteristics}Summary of the IS20 datasheet}
+\caption{\label{tab:test_nhexa_suspended_table_simscape_modes}Comparison of the identified suspension modes with the Simscape model and measured experimentally}
-
-\end{table}
-
-From the datasheet in Table \ref{tab:is20_characteristics}, we can estimate the parameters of the physical shaker.
-
-These parameters are defined below
-\paragraph{3D accelerometer (356B18)}
-\label{sec:simscape_accelerometers}
-
-An accelerometer consists of 2 solids:
-\begin{itemize}
-\item a ``housing'' rigidly fixed to the measured body
-\item an ``inertial mass'' suspended inside the housing by springs and guided in the measured direction
-\end{itemize}
-
-The relative motion between the housing and the inertial mass gives a measurement of the acceleration of the measured body (up to the suspension mode of the inertial mass).
-
-\begin{table}[htbp]
-\centering
-\begin{tabularx}{0.5\linewidth}{lX}
-\toprule
-Characteristic & Value\\
-\midrule
-Sensitivity & 0.102 V/(m/s2)\\
-Frequency Range & 0.5 to 3000 Hz\\
-Resonance Frequency & > 20 kHz\\
-Resolution & 0.0005 m/s2 rms\\
-Weight & 0.025 kg\\
-Size & 20.3x26.1x20.3 [mm]\\
-\bottomrule
-\end{tabularx}
-\caption{\label{tab:356b18_characteristics}Summary of the 356B18 datasheet}
-
-\end{table}
-
-\subsection{Identification}
-\label{sec:simscape_parameters}
-Let's now identify the resonance frequency and mode shapes associated with the suspension modes of the optical table.
-
-\begin{verbatim}
-size(G)
-State-space model with 6 outputs, 6 inputs, and 12 states.
-\end{verbatim}
-
-
-Compute the resonance frequencies
-\begin{center}
-\begin{tabular}{lrrrrrr}
- & x & y & Rz & Dz & Rx & Ry\\
-\hline
-Simscape & 1.28 & 1.28 & 1.82 & 6.78 & 9.47 & 9.47\\
-Experimental & 1.3 & 1.3 & 1.95 & 6.85 & 8.9 & 9.5\\
-\end{tabular}
-
-\end{center}
-
-
-And the associated response of the optical table
-The results are shown in Table \ref{tab:mode_shapes}.
-The motion associated to the mode shapes are just indicative.
-
-\begin{table}[htbp]
-\centering
-\begin{tabularx}{0.4\linewidth}{Xcccccc}
-\toprule
-\(\omega_0\) [Hz] & 8.2 & 8.2 & 8.2 & 5.8 & 5.6 & 5.6\\
-\midrule
-x & 0.0 & 0.0 & 0.0 & 0.0 & 0.1 & 0.5\\
-y & 0.0 & 0.0 & 0.0 & 0.0 & 0.5 & 0.0\\
-z & 0.0 & 0.0 & 0.0 & 1.0 & 0.0 & 0.0\\
-Rx & 1.0 & 0.0 & 0.0 & 0.0 & 0.8 & 0.0\\
-Ry & 0.0 & 1.0 & 0.0 & 0.0 & 0.2 & 0.9\\
-Rz & 0.0 & 0.0 & 1.0 & 0.0 & 0.0 & 0.0\\
-\bottomrule
-\end{tabularx}
-\caption{\label{tab:mode_shapes}Resonance frequency and approximation of the mode shapes}
 
 \end{table}