From fb2bbb5ee341cb1e415864cd041c61e3d2f6a49b Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 2 Feb 2021 19:29:51 +0100 Subject: [PATCH] Rename file and export to pdf --- figs/apa300ML.pdf | Bin 0 -> 1314466 bytes figs/expected_hysteresis.pdf | Bin 0 -> 55491 bytes figs/expected_hysteresis.png | Bin 0 -> 79435 bytes index.html | 326 +--------- index.org | 210 ------ ref.bib | 24 + test-bench-apa300ml.html | 358 +++++++++++ test-bench-apa300ml.org | 1157 ++++++++++++++++++++++++++++++++++ test-bench-apa300ml.pdf | Bin 0 -> 1713805 bytes 9 files changed, 1540 insertions(+), 535 deletions(-) create mode 100644 figs/apa300ML.pdf create mode 100644 figs/expected_hysteresis.pdf create mode 100644 figs/expected_hysteresis.png mode change 100644 => 120000 index.html delete mode 100644 index.org create mode 100644 ref.bib create mode 100644 test-bench-apa300ml.html create mode 100644 test-bench-apa300ml.org create mode 100644 test-bench-apa300ml.pdf diff --git a/figs/apa300ML.pdf 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zNJQ%Ca~CgS4AP}SIUBI*B(C>f9ovF^O^GiyIkK)QTY*#*)|Azk!qr?+wANy zXZ=4%QF^J#>EA+}m4%p_fj@I{_jN!5NLlIqY+Xy9N_RC2g#MGaZ=V2mW1J(Vevv5C zs-U3YP>iCQFMO#`m>`qCxvvUYw>|J9k>j?+sL0g_O^CMV@tt@TbC zMOSR!@g?t*2pAA^s=)3duWY10 z<0wBjUrwxgsoAgRrDmnJa&jnL_7P?z8IkvaU+QP_5XBR~5n0QVyhn{&8CbQ)hN6sJ zJSZX}0GXeVE756EtocA6AU~gaag7%~Rz-Md;Be>M)Y3lj#q9&9?Lx*Tr`k%qnCk2< z#met^NFKe>r7jJ<=&13y5KgU*r^Ak71RssjiWR5dj;-!8fA;K7Pu4V+A3CkXqrJj? z`#Htc;u+_cbnjBF{C|Jm^7jn!_r40wYJ-C{_}g~ - - - - - -Amplifier Piezoelectric Actuator APA300ML - Test Bench - - - - - - - - -

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Amplifier Piezoelectric Actuator APA300ML - Test Bench

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-The goal of this test bench is to extract all the important parameters of the Amplified Piezoelectric Actuator APA300ML. -

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-This include: -

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  • Stroke
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  • Stiffness
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  • Hysteresis
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  • Gain from the applied voltage \(V_a\) to the generated Force \(F_a\)
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  • Gain from the sensor stack strain \(\delta L\) to the generated voltage \(V_s\)
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  • Dynamical behavior
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apa300ML.png -

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Figure 1: Picture of the APA300ML

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1 Model of an Amplified Piezoelectric Actuator and Sensor

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-Consider a schematic of the Amplified Piezoelectric Actuator in Figure 2. -

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apa_model_schematic.png -

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Figure 2: Amplified Piezoelectric Actuator Schematic

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-A voltage \(V_a\) applied to the actuator stacks will induce an actuator force \(F_a\): -

-\begin{equation} - F_a = g_a \cdot V_a -\end{equation} - -

-A change of length \(dl\) of the sensor stack will induce a voltage \(V_s\): -

-\begin{equation} - V_s = g_s \cdot dl -\end{equation} - -

-We wish here to experimental measure \(g_a\) and \(g_s\). -

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-The block-diagram model of the piezoelectric actuator is then as shown in Figure 3. -

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apa-model-simscape-schematic.png -

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Figure 3: Model of the APA with Simscape/Simulink

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2 Test-Bench Description

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-Here are the documentation of the equipment used for this test bench: -

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test_bench_apa_alone.png -

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Figure 4: Schematic of the Test Bench

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3 Measurement Procedure

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3.1 Stroke Measurement

-
-

-Using the PD200 amplifier, output a voltage: -\[ V_a = 65 + 85 \sin(2\pi \cdot t) \] -To have a quasi-static excitation between -80 and 150V. -

- -

-As the gain of the PD200 amplifier is 20, the DAC output voltage should be: -\[ V_{dac}(t) = 3.25 + 4.25\sin(2\pi \cdot t) \] -

- -

-Verify that the voltage offset is zero! -

- -

-Measure the output vertical displacement \(d\) using the interferometer. -

- -

-Then, plot \(d\) as a function of \(V_a\), and perform a linear regression. -Conclude on the obtained stroke. -

-
-
- -
-

3.2 Stiffness Measurement

-
-

-Add some (known) weight \(\delta m g\) on the suspended mass and measure the deflection \(\delta d\). -This can be tested when the piezoelectric stacks are open-circuit. -

- -

-As the stiffness will be around \(k \approx 10^6 N/m\), an added mass of \(m \approx 100g\) will induce a static deflection of \(\approx 1\mu m\) which should be large enough for a precise measurement using the interferometer. -

- -

-Then the obtained stiffness is: -

-\begin{equation} - k = \frac{\delta m g}{\delta d} -\end{equation} -
-
- -
-

3.3 Hysteresis measurement

-
-

-Supply a quasi static sinusoidal excitation \(V_a\) at different voltages. -

- -

-The offset should be 65V, and the sin amplitude can range from 1V up to 85V. -

- -

-For each excitation amplitude, the vertical displacement \(d\) of the mass is measured. -

- -

-Then, \(d\) is plotted as a function of \(V_a\) for all the amplitudes. -

-
-
- -
-

3.4 Piezoelectric Actuator Constant

-
-

-Using the measurement test-bench, it is rather easy the determine the static gain between the applied voltage \(V_a\) to the induced displacement \(d\). -Use a quasi static (1Hz) excitation signal \(V_a\) on the piezoelectric stack and measure the vertical displacement \(d\). -Perform a linear regression to obtain: -

-\begin{equation} - d = g_{d/V_a} \cdot V_a -\end{equation} - -

-Using the Simscape model of the APA, it is possible to determine the static gain between the actuator force \(F_a\) to the induced displacement \(d\): -

-\begin{equation} - d = g_{d/F_a} \cdot F_a -\end{equation} - -

-From the two gains, it is then easy to determine \(g_a\): -

-\begin{equation} - g_a = \frac{F_a}{V_a} = \frac{F_a}{d} \cdot \frac{d}{V_a} = \frac{g_{d/V_a}}{g_{d/F_a}} -\end{equation} -
-
- -
-

3.5 Piezoelectric Sensor Constant

-
-

-From a quasi static (1Hz) excitation of the piezoelectric stack, measure the gain from \(V_a\) to \(V_s\): -

-\begin{equation} - V_s = g_{V_s/V_a} V_a -\end{equation} - -

-Using the simscape model, compute the static gain from the actuator force \(F_a\) to the strain of the sensor stack \(dl\): -

-\begin{equation} - dl = g_{dl/F_a} F_a -\end{equation} - -

-Then, the static gain from the sensor stack strain \(dl\) to the general voltage \(V_s\) is: -

-\begin{equation} - g_s = \frac{V_s}{dl} = \frac{V_s}{V_a} \cdot \frac{V_a}{F_a} \cdot \frac{F_a}{dl} = \frac{g_{V_s/V_a}}{g_a \cdot g_{dl/F_a}} -\end{equation} - -

-Alternatively, we could impose an external force to add strain in the APA that should be equally present in all the 3 stacks and equal to 1/5 of the vertical strain. -This external force can be some weight added, or a piezo in parallel. -

-
-
- -
-

3.6 Capacitance Measurement

-
-

-Measure the capacitance of the 3 stacks individually using a precise multi-meter. -

-
-
- -
-

3.7 Dynamical Behavior

-
-

-Perform a system identification from \(V_a\) to the measured displacement \(d\) by the interferometer and by the encoder, and to the general voltage \(V_s\). -

- -

-This can be performed using different excitation signals. -

- -

-This can also be performed with and without the encoder fixed to the APA. -

-
-
- -
-

3.8 Compare the results obtained for all 7 APA300ML

-
-

-Compare all the obtained parameters for all the test APA. -

-
-
-
- -
-

4 Measurement Results

-
-
-
-

Author: Dehaeze Thomas

-

Created: 2021-01-04 lun. 14:44

-
- - diff --git a/index.html b/index.html new file mode 120000 index 0000000..911cda8 --- /dev/null +++ b/index.html @@ -0,0 +1 @@ +test-bench-apa300ml.html \ No newline at end of file diff --git a/index.org b/index.org deleted file mode 100644 index c22e3e4..0000000 --- a/index.org +++ /dev/null @@ -1,210 +0,0 @@ -#+TITLE: Amplifier Piezoelectric Actuator APA300ML - Test Bench -:DRAWER: -#+LANGUAGE: en -#+EMAIL: dehaeze.thomas@gmail.com -#+AUTHOR: Dehaeze Thomas - -#+HTML_LINK_HOME: ../index.html -#+HTML_LINK_UP: ../index.html - -#+HTML_HEAD: -#+HTML_HEAD: - -#+BIND: org-latex-image-default-option "scale=1" -#+BIND: org-latex-image-default-width "" - -#+LaTeX_CLASS: scrreprt -#+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full] -#+LaTeX_HEADER_EXTRA: \input{preamble.tex} - -#+PROPERTY: header-args:matlab :session *MATLAB* -#+PROPERTY: header-args:matlab+ :comments org -#+PROPERTY: header-args:matlab+ :exports both -#+PROPERTY: header-args:matlab+ :results none -#+PROPERTY: header-args:matlab+ :eval no-export -#+PROPERTY: header-args:matlab+ :noweb yes -#+PROPERTY: header-args:matlab+ :mkdirp yes -#+PROPERTY: header-args:matlab+ :output-dir figs - -#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}") -#+PROPERTY: header-args:latex+ :imagemagick t :fit yes -#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150 -#+PROPERTY: header-args:latex+ :imoutoptions -quality 100 -#+PROPERTY: header-args:latex+ :results file raw replace -#+PROPERTY: header-args:latex+ :buffer no -#+PROPERTY: header-args:latex+ :tangle no -#+PROPERTY: header-args:latex+ :eval no-export -#+PROPERTY: header-args:latex+ :exports results -#+PROPERTY: header-args:latex+ :mkdirp yes -#+PROPERTY: header-args:latex+ :output-dir figs -#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png") -:END: - -* Introduction :ignore: - -The goal of this test bench is to extract all the important parameters of the Amplified Piezoelectric Actuator APA300ML. - -This include: -- Stroke -- Stiffness -- Hysteresis -- Gain from the applied voltage $V_a$ to the generated Force $F_a$ -- Gain from the sensor stack strain $\delta L$ to the generated voltage $V_s$ -- Dynamical behavior - -#+name: fig:apa300ML -#+caption: Picture of the APA300ML -[[file:figs/apa300ML.png]] - -* Model of an Amplified Piezoelectric Actuator and Sensor - -Consider a schematic of the Amplified Piezoelectric Actuator in Figure [[fig:apa_model_schematic]]. - -#+name: fig:apa_model_schematic -#+caption: Amplified Piezoelectric Actuator Schematic -[[file:figs/apa_model_schematic.png]] - -A voltage $V_a$ applied to the actuator stacks will induce an actuator force $F_a$: -\begin{equation} - F_a = g_a \cdot V_a -\end{equation} - -A change of length $dl$ of the sensor stack will induce a voltage $V_s$: -\begin{equation} - V_s = g_s \cdot dl -\end{equation} - -We wish here to experimental measure $g_a$ and $g_s$. - -The block-diagram model of the piezoelectric actuator is then as shown in Figure [[fig:apa-model-simscape-schematic]]. - -#+begin_src latex :file apa-model-simscape-schematic.pdf - \begin{tikzpicture} - \node[block={2.0cm}{2.0cm}, align=center] (model) at (0,0){Simscape\\Model}; - \node[block, left=1.0 of model] (ga){$g_a(s)$}; - \node[block, right=1.0 of model] (gs){$g_s(s)$}; - - \draw[<-] (ga.west) -- node[midway, above]{$V_a$} node[midway, below]{$[V]$} ++(-1.0, 0); - \draw[->] (ga.east) --node[midway, above]{$F_a$} node[midway, below]{$[N]$} (model.west); - \draw[->] (model.east) --node[midway, above]{$dl$} node[midway, below]{$[m]$} (gs.west); - \draw[->] (gs.east) -- node[midway, above]{$V_s$} node[midway, below]{$[V]$} ++(1.0, 0); - \end{tikzpicture} -#+end_src - -#+name: fig:apa-model-simscape-schematic -#+caption: Model of the APA with Simscape/Simulink -#+RESULTS: -[[file:figs/apa-model-simscape-schematic.png]] - -* Test-Bench Description - -#+begin_note -Here are the documentation of the equipment used for this test bench: -- Voltage Amplifier: [[file:doc/PD200-V7-R1.pdf][PD200]] -- Amplified Piezoelectric Actuator: [[file:doc/APA300ML.pdf][APA300ML]] -- DAC/ADC: Speedgoat [[file:doc/IO131-OEM-Datasheet.pdf][IO313]] -- Encoder: [[file:doc/L-9517-9678-05-A_Data_sheet_VIONiC_series_en.pdf][Renishaw Vionic]] and used [[file:doc/L-9517-9862-01-C_Data_sheet_RKLC_EN.pdf][Ruler]] -- Interferometer: [[https://www.attocube.com/en/products/laser-displacement-sensor/displacement-measuring-interferometer][Attocube IDS3010]] -#+end_note - -#+name: fig:test_bench_apa_alone -#+caption: Schematic of the Test Bench -[[file:figs/test_bench_apa_alone.png]] - -* Measurement Procedure -** Introduction :ignore: - -** Stroke Measurement - -Using the PD200 amplifier, output a voltage: -\[ V_a = 65 + 85 \sin(2\pi \cdot t) \] -To have a quasi-static excitation between -80 and 150V. - -As the gain of the PD200 amplifier is 20, the DAC output voltage should be: -\[ V_{dac}(t) = 3.25 + 4.25\sin(2\pi \cdot t) \] - -Verify that the voltage offset is zero! - -Measure the output vertical displacement $d$ using the interferometer. - -Then, plot $d$ as a function of $V_a$, and perform a linear regression. -Conclude on the obtained stroke. - -** Stiffness Measurement - -Add some (known) weight $\delta m g$ on the suspended mass and measure the deflection $\delta d$. -This can be tested when the piezoelectric stacks are open-circuit. - -As the stiffness will be around $k \approx 10^6 N/m$, an added mass of $m \approx 100g$ will induce a static deflection of $\approx 1\mu m$ which should be large enough for a precise measurement using the interferometer. - -Then the obtained stiffness is: -\begin{equation} - k = \frac{\delta m g}{\delta d} -\end{equation} - -** Hysteresis measurement - -Supply a quasi static sinusoidal excitation $V_a$ at different voltages. - -The offset should be 65V, and the sin amplitude can range from 1V up to 85V. - -For each excitation amplitude, the vertical displacement $d$ of the mass is measured. - -Then, $d$ is plotted as a function of $V_a$ for all the amplitudes. - -** Piezoelectric Actuator Constant - -Using the measurement test-bench, it is rather easy the determine the static gain between the applied voltage $V_a$ to the induced displacement $d$. -Use a quasi static (1Hz) excitation signal $V_a$ on the piezoelectric stack and measure the vertical displacement $d$. -Perform a linear regression to obtain: -\begin{equation} - d = g_{d/V_a} \cdot V_a -\end{equation} - -Using the Simscape model of the APA, it is possible to determine the static gain between the actuator force $F_a$ to the induced displacement $d$: -\begin{equation} - d = g_{d/F_a} \cdot F_a -\end{equation} - -From the two gains, it is then easy to determine $g_a$: -\begin{equation} - g_a = \frac{F_a}{V_a} = \frac{F_a}{d} \cdot \frac{d}{V_a} = \frac{g_{d/V_a}}{g_{d/F_a}} -\end{equation} - -** Piezoelectric Sensor Constant - -From a quasi static (1Hz) excitation of the piezoelectric stack, measure the gain from $V_a$ to $V_s$: -\begin{equation} - V_s = g_{V_s/V_a} V_a -\end{equation} - -Using the simscape model, compute the static gain from the actuator force $F_a$ to the strain of the sensor stack $dl$: -\begin{equation} - dl = g_{dl/F_a} F_a -\end{equation} - -Then, the static gain from the sensor stack strain $dl$ to the general voltage $V_s$ is: -\begin{equation} - g_s = \frac{V_s}{dl} = \frac{V_s}{V_a} \cdot \frac{V_a}{F_a} \cdot \frac{F_a}{dl} = \frac{g_{V_s/V_a}}{g_a \cdot g_{dl/F_a}} -\end{equation} - -Alternatively, we could impose an external force to add strain in the APA that should be equally present in all the 3 stacks and equal to 1/5 of the vertical strain. -This external force can be some weight added, or a piezo in parallel. - -** Capacitance Measurement - -Measure the capacitance of the 3 stacks individually using a precise multi-meter. - -** Dynamical Behavior - -Perform a system identification from $V_a$ to the measured displacement $d$ by the interferometer and by the encoder, and to the general voltage $V_s$. - -This can be performed using different excitation signals. - -This can also be performed with and without the encoder fixed to the APA. - -** Compare the results obtained for all 7 APA300ML - -Compare all the obtained parameters for all the test APA. - -* Measurement Results diff --git a/ref.bib b/ref.bib new file mode 100644 index 0000000..0bd32bc --- /dev/null +++ b/ref.bib @@ -0,0 +1,24 @@ +@article{souleille18_concep_activ_mount_space_applic, + author = {Souleille, Adrien and Lampert, Thibault and Lafarga, V and + Hellegouarch, Sylvain and Rondineau, Alan and Rodrigues, + Gon{\c{c}}alo and Collette, Christophe}, + title = {A Concept of Active Mount for Space Applications}, + journal = {CEAS Space Journal}, + volume = 10, + number = 2, + pages = {157--165}, + year = 2018, + publisher = {Springer}, +} + +@phdthesis{poel10_explor_activ_hard_mount_vibrat, + author = {van der Poel, Gerrit Wijnand}, + doi = {10.3990/1.9789036530163}, + isbn = {978-90-365-3016-3}, + keywords = {parallel robot}, + school = {University of Twente}, + title = {An Exploration of Active Hard Mount Vibration Isolation for + Precision Equipment}, + url = {https://doi.org/10.3990/1.9789036530163}, + year = 2010, +} diff --git a/test-bench-apa300ml.html b/test-bench-apa300ml.html new file mode 100644 index 0000000..1b7a0fb --- /dev/null +++ b/test-bench-apa300ml.html @@ -0,0 +1,358 @@ + + + + + + +Amplifier Piezoelectric Actuator APA300ML - Test Bench + + + + + + + + +
+ UP + | + HOME +
+

Amplifier Piezoelectric Actuator APA300ML - Test Bench

+ +
+

This report is also available as a pdf.

+
+ +

+The goal of this test bench is to extract all the important parameters of the Amplified Piezoelectric Actuator APA300ML. +

+ +

+This include: +

+
    +
  • Stroke
  • +
  • Stiffness
  • +
  • Hysteresis
  • +
  • Gain from the applied voltage \(V_a\) to the generated Force \(F_a\)
  • +
  • Gain from the sensor stack strain \(\delta L\) to the generated voltage \(V_s\)
  • +
  • Dynamical behavior
  • +
+ + +
+

apa300ML.png +

+

Figure 1: Picture of the APA300ML

+
+ +
+

1 Model of an Amplified Piezoelectric Actuator and Sensor

+
+

+Consider a schematic of the Amplified Piezoelectric Actuator in Figure 2. +

+ + +
+

apa_model_schematic.png +

+

Figure 2: Amplified Piezoelectric Actuator Schematic

+
+ +

+A voltage \(V_a\) applied to the actuator stacks will induce an actuator force \(F_a\): +

+\begin{equation} + F_a = g_a \cdot V_a +\end{equation} + +

+A change of length \(dl\) of the sensor stack will induce a voltage \(V_s\): +

+\begin{equation} + V_s = g_s \cdot dl +\end{equation} + +

+We wish here to experimental measure \(g_a\) and \(g_s\). +

+ +

+The block-diagram model of the piezoelectric actuator is then as shown in Figure 3. +

+ + +
+

apa-model-simscape-schematic.png +

+

Figure 3: Model of the APA with Simscape/Simulink

+
+
+
+ +
+

2 Test-Bench Description

+
+
+

+Here are the documentation of the equipment used for this test bench: +

+ + +
+ + +
+

test_bench_apa_alone.png +

+

Figure 4: Schematic of the Test Bench

+
+
+
+ +
+

3 Measurement Procedure

+
+
+
+

3.1 Stroke Measurement

+
+

+Using the PD200 amplifier, output a voltage: +\[ V_a = 65 + 85 \sin(2\pi \cdot t) \] +To have a quasi-static excitation between -20 and 150V. +

+ +

+As the gain of the PD200 amplifier is 20, the DAC output voltage should be: +\[ V_{dac}(t) = 3.25 + 4.25\sin(2\pi \cdot t) \] +

+ +

+Verify that the voltage offset of the PD200 is zero! +

+ +

+Measure the output vertical displacement \(d\) using the interferometer. +

+ +

+Then, plot \(d\) as a function of \(V_a\), and perform a linear regression. +Conclude on the obtained stroke. +

+
+
+ +
+

3.2 Stiffness Measurement

+
+

+Add some (known) weight \(\delta m g\) on the suspended mass and measure the deflection \(\delta d\). +This can be tested when the piezoelectric stacks are open-circuit. +

+ +

+As the stiffness will be around \(k \approx 10^6 N/m\), an added mass of \(m \approx 100g\) will induce a static deflection of \(\approx 1\mu m\) which should be large enough for a precise measurement using the interferometer. +

+ +

+Then the obtained stiffness is: +

+\begin{equation} + k = \frac{\delta m g}{\delta d} +\end{equation} +
+
+ +
+

3.3 Hysteresis measurement

+
+

+Supply a quasi static sinusoidal excitation \(V_a\) at different voltages. +

+ +

+The offset should be 65V, and the sin amplitude can range from 1V up to 85V. +

+ +

+For each excitation amplitude, the vertical displacement \(d\) of the mass is measured. +

+ +

+Then, \(d\) is plotted as a function of \(V_a\) for all the amplitudes. +

+ + +
+

expected_hysteresis.png +

+

Figure 5: Expected Hysteresis (poel10_explor_activ_hard_mount_vibrat)

+
+
+
+ +
+

3.4 Piezoelectric Actuator Constant

+
+

+Using the measurement test-bench, it is rather easy the determine the static gain between the applied voltage \(V_a\) to the induced displacement \(d\). +Use a quasi static (1Hz) excitation signal \(V_a\) on the piezoelectric stack and measure the vertical displacement \(d\). +Perform a linear regression to obtain: +

+\begin{equation} + d = g_{d/V_a} \cdot V_a +\end{equation} + +

+Using the Simscape model of the APA, it is possible to determine the static gain between the actuator force \(F_a\) to the induced displacement \(d\): +

+\begin{equation} + d = g_{d/F_a} \cdot F_a +\end{equation} + +

+From the two gains, it is then easy to determine \(g_a\): +

+\begin{equation} + g_a = \frac{F_a}{V_a} = \frac{F_a}{d} \cdot \frac{d}{V_a} = \frac{g_{d/V_a}}{g_{d/F_a}} +\end{equation} +
+
+ +
+

3.5 Piezoelectric Sensor Constant

+
+

+From a quasi static excitation of the piezoelectric stack, measure the gain from \(V_a\) to \(V_s\): +

+\begin{equation} + V_s = g_{V_s/V_a} V_a +\end{equation} + +

+Note here that there is an high pass filter formed by the piezo capacitor and parallel resistor. +The excitation frequency should then be in between the cut-off frequency of this high pass filter and the first resonance. +

+ +

+Alternatively, the gain can be computed from the dynamical identification and taking the gain at the wanted frequency. +

+ +

+Using the simscape model, compute the static gain from the actuator force \(F_a\) to the strain of the sensor stack \(dl\): +

+\begin{equation} + dl = g_{dl/F_a} F_a +\end{equation} + +

+Then, the static gain from the sensor stack strain \(dl\) to the general voltage \(V_s\) is: +

+\begin{equation} + g_s = \frac{V_s}{dl} = \frac{V_s}{V_a} \cdot \frac{V_a}{F_a} \cdot \frac{F_a}{dl} = \frac{g_{V_s/V_a}}{g_a \cdot g_{dl/F_a}} +\end{equation} + +

+Alternatively, we could impose an external force to add strain in the APA that should be equally present in all the 3 stacks and equal to 1/5 of the vertical strain. +This external force can be some weight added, or a piezo in parallel. +

+
+
+ +
+

3.6 Capacitance Measurement

+
+

+Measure the capacitance of the 3 stacks individually using a precise multi-meter. +

+
+
+ +
+

3.7 Dynamical Behavior

+
+

+Perform a system identification from \(V_a\) to the measured displacement \(d\) by the interferometer and by the encoder, and to the generated voltage \(V_s\). +

+ +

+This can be performed using different excitation signals. +

+ +

+This can also be performed with and without the encoder fixed to the APA. +

+
+
+ +
+

3.8 Compare the results obtained for all 7 APA300ML

+
+

+Compare all the obtained parameters for all the test APA. +

+
+
+
+ +
+

4 Measurement Results

+
+ +

Bibliography

+
+
Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” CEAS Space Journal 10 (2). Springer:157–65.
+
+
+
+

Author: Dehaeze Thomas

+

Created: 2021-02-02 mar. 19:29

+
+ + diff --git a/test-bench-apa300ml.org b/test-bench-apa300ml.org new file mode 100644 index 0000000..c556437 --- /dev/null +++ b/test-bench-apa300ml.org @@ -0,0 +1,1157 @@ +#+TITLE: Amplifier Piezoelectric Actuator APA300ML - Test Bench +:DRAWER: +#+LANGUAGE: en +#+EMAIL: dehaeze.thomas@gmail.com +#+AUTHOR: Dehaeze Thomas + +#+HTML_LINK_HOME: ../index.html +#+HTML_LINK_UP: ../index.html + +#+HTML_HEAD: +#+HTML_HEAD: + +#+BIND: org-latex-image-default-option "scale=1" +#+BIND: org-latex-image-default-width "" +#+BIND: org-latex-bib-compiler "biber" + +#+LaTeX_CLASS: scrreprt +#+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full] +#+LaTeX_HEADER_EXTRA: \input{preamble.tex} +#+LATEX_HEADER_EXTRA: \addbibresource{ref.bib} + +#+PROPERTY: header-args:matlab :session *MATLAB* +#+PROPERTY: header-args:matlab+ :comments org +#+PROPERTY: header-args:matlab+ :exports both +#+PROPERTY: header-args:matlab+ :results none +#+PROPERTY: header-args:matlab+ :eval no-export +#+PROPERTY: header-args:matlab+ :noweb yes +#+PROPERTY: header-args:matlab+ :mkdirp yes +#+PROPERTY: header-args:matlab+ :output-dir figs + +#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}") +#+PROPERTY: header-args:latex+ :imagemagick t :fit yes +#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150 +#+PROPERTY: header-args:latex+ :imoutoptions -quality 100 +#+PROPERTY: header-args:latex+ :results file raw replace +#+PROPERTY: header-args:latex+ :buffer no +#+PROPERTY: header-args:latex+ :tangle no +#+PROPERTY: header-args:latex+ :eval no-export +#+PROPERTY: header-args:latex+ :exports results +#+PROPERTY: header-args:latex+ :mkdirp yes +#+PROPERTY: header-args:latex+ :output-dir figs +#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png") +:END: + +#+begin_export html +
+

This report is also available as a pdf.

+
+#+end_export + +* Introduction :ignore: + +The goal of this test bench is to extract all the important parameters of the Amplified Piezoelectric Actuator APA300ML. + +This include: +- Stroke +- Stiffness +- Hysteresis +- Gain from the applied voltage $V_a$ to the generated Force $F_a$ +- Gain from the sensor stack strain $\delta L$ to the generated voltage $V_s$ +- Dynamical behavior + +#+name: fig:apa300ML +#+caption: Picture of the APA300ML +#+attr_latex: :width 0.8\linewidth +[[file:figs/apa300ML.png]] + +* Model of an Amplified Piezoelectric Actuator and Sensor + +Consider a schematic of the Amplified Piezoelectric Actuator in Figure [[fig:apa_model_schematic]]. + +#+name: fig:apa_model_schematic +#+caption: Amplified Piezoelectric Actuator Schematic +[[file:figs/apa_model_schematic.png]] + +A voltage $V_a$ applied to the actuator stacks will induce an actuator force $F_a$: +\begin{equation} + F_a = g_a \cdot V_a +\end{equation} + +A change of length $dl$ of the sensor stack will induce a voltage $V_s$: +\begin{equation} + V_s = g_s \cdot dl +\end{equation} + +We wish here to experimental measure $g_a$ and $g_s$. + +The block-diagram model of the piezoelectric actuator is then as shown in Figure [[fig:apa-model-simscape-schematic]]. + +#+begin_src latex :file apa-model-simscape-schematic.pdf + \begin{tikzpicture} + \node[block={2.0cm}{2.0cm}, align=center] (model) at (0,0){Simscape\\Model}; + \node[block, left=1.0 of model] (ga){$g_a(s)$}; + \node[block, right=1.0 of model] (gs){$g_s(s)$}; + + \draw[<-] (ga.west) -- node[midway, above]{$V_a$} node[midway, below]{$[V]$} ++(-1.0, 0); + \draw[->] (ga.east) --node[midway, above]{$F_a$} node[midway, below]{$[N]$} (model.west); + \draw[->] (model.east) --node[midway, above]{$dl$} node[midway, below]{$[m]$} (gs.west); + \draw[->] (gs.east) -- node[midway, above]{$V_s$} node[midway, below]{$[V]$} ++(1.0, 0); + \end{tikzpicture} +#+end_src + +#+name: fig:apa-model-simscape-schematic +#+caption: Model of the APA with Simscape/Simulink +#+RESULTS: +[[file:figs/apa-model-simscape-schematic.png]] + +* Test-Bench Description + +#+begin_note +Here are the documentation of the equipment used for this test bench: +- Voltage Amplifier: [[file:doc/PD200-V7-R1.pdf][PD200]] +- Amplified Piezoelectric Actuator: [[file:doc/APA300ML.pdf][APA300ML]] +- DAC/ADC: Speedgoat [[file:doc/IO131-OEM-Datasheet.pdf][IO313]] +- Encoder: [[file:doc/L-9517-9678-05-A_Data_sheet_VIONiC_series_en.pdf][Renishaw Vionic]] and used [[file:doc/L-9517-9862-01-C_Data_sheet_RKLC_EN.pdf][Ruler]] +- Interferometer: [[https://www.attocube.com/en/products/laser-displacement-sensor/displacement-measuring-interferometer][Attocube IDS3010]] +#+end_note + +#+name: fig:test_bench_apa_alone +#+caption: Schematic of the Test Bench +[[file:figs/test_bench_apa_alone.png]] + +* Measurement Procedure +** Introduction :ignore: + +** Stroke Measurement + +Using the PD200 amplifier, output a voltage: +\[ V_a = 65 + 85 \sin(2\pi \cdot t) \] +To have a quasi-static excitation between -20 and 150V. + +As the gain of the PD200 amplifier is 20, the DAC output voltage should be: +\[ V_{dac}(t) = 3.25 + 4.25\sin(2\pi \cdot t) \] + +Verify that the voltage offset of the PD200 is zero! + +Measure the output vertical displacement $d$ using the interferometer. + +Then, plot $d$ as a function of $V_a$, and perform a linear regression. +Conclude on the obtained stroke. + +** Stiffness Measurement + +Add some (known) weight $\delta m g$ on the suspended mass and measure the deflection $\delta d$. +This can be tested when the piezoelectric stacks are open-circuit. + +As the stiffness will be around $k \approx 10^6 N/m$, an added mass of $m \approx 100g$ will induce a static deflection of $\approx 1\mu m$ which should be large enough for a precise measurement using the interferometer. + +Then the obtained stiffness is: +\begin{equation} + k = \frac{\delta m g}{\delta d} +\end{equation} + +** Hysteresis measurement + +Supply a quasi static sinusoidal excitation $V_a$ at different voltages. + +The offset should be 65V, and the sin amplitude can range from 1V up to 85V. + +For each excitation amplitude, the vertical displacement $d$ of the mass is measured. + +Then, $d$ is plotted as a function of $V_a$ for all the amplitudes. + +#+name: fig:expected_hysteresis +#+caption: Expected Hysteresis (cite:poel10_explor_activ_hard_mount_vibrat) +#+attr_latex: :width 0.8\linewidth +[[file:figs/expected_hysteresis.png]] + +** Piezoelectric Actuator Constant + +Using the measurement test-bench, it is rather easy the determine the static gain between the applied voltage $V_a$ to the induced displacement $d$. +Use a quasi static (1Hz) excitation signal $V_a$ on the piezoelectric stack and measure the vertical displacement $d$. +Perform a linear regression to obtain: +\begin{equation} + d = g_{d/V_a} \cdot V_a +\end{equation} + +Using the Simscape model of the APA, it is possible to determine the static gain between the actuator force $F_a$ to the induced displacement $d$: +\begin{equation} + d = g_{d/F_a} \cdot F_a +\end{equation} + +From the two gains, it is then easy to determine $g_a$: +\begin{equation} + g_a = \frac{F_a}{V_a} = \frac{F_a}{d} \cdot \frac{d}{V_a} = \frac{g_{d/V_a}}{g_{d/F_a}} +\end{equation} + +** Piezoelectric Sensor Constant + +From a quasi static excitation of the piezoelectric stack, measure the gain from $V_a$ to $V_s$: +\begin{equation} + V_s = g_{V_s/V_a} V_a +\end{equation} + +Note here that there is an high pass filter formed by the piezo capacitor and parallel resistor. +The excitation frequency should then be in between the cut-off frequency of this high pass filter and the first resonance. + +Alternatively, the gain can be computed from the dynamical identification and taking the gain at the wanted frequency. + +Using the simscape model, compute the static gain from the actuator force $F_a$ to the strain of the sensor stack $dl$: +\begin{equation} + dl = g_{dl/F_a} F_a +\end{equation} + +Then, the static gain from the sensor stack strain $dl$ to the general voltage $V_s$ is: +\begin{equation} + g_s = \frac{V_s}{dl} = \frac{V_s}{V_a} \cdot \frac{V_a}{F_a} \cdot \frac{F_a}{dl} = \frac{g_{V_s/V_a}}{g_a \cdot g_{dl/F_a}} +\end{equation} + +Alternatively, we could impose an external force to add strain in the APA that should be equally present in all the 3 stacks and equal to 1/5 of the vertical strain. +This external force can be some weight added, or a piezo in parallel. + +** Capacitance Measurement + +Measure the capacitance of the 3 stacks individually using a precise multi-meter. + +** Dynamical Behavior + +Perform a system identification from $V_a$ to the measured displacement $d$ by the interferometer and by the encoder, and to the generated voltage $V_s$. + +This can be performed using different excitation signals. + +This can also be performed with and without the encoder fixed to the APA. + +** Compare the results obtained for all 7 APA300ML + +Compare all the obtained parameters for all the test APA. + +* Measurement Results + +* TODO Compare with the FEM/Simscape Model :noexport: +:PROPERTIES: +:header-args:matlab+: :tangle matlab/APA300ML.m +:END: + +** Introduction :ignore: +In this section, the Amplified Piezoelectric Actuator APA300ML ([[file:doc/APA300ML.pdf][doc]]) is modeled using a Finite Element Software. +Then a /super element/ is exported and imported in Simscape where its dynamic is studied. + +A 3D view of the Amplified Piezoelectric Actuator (APA300ML) is shown in Figure [[fig:apa300ml_ansys]]. +The remote point used are also shown in this figure. + +#+name: fig:apa300ml_ansys +#+caption: Ansys FEM of the APA300ML +[[file:figs/apa300ml_ansys.jpg]] + +** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab :tangle no + addpath('matlab/'); + addpath('matlab/APA300ML/'); +#+end_src + +#+begin_src matlab :eval no + addpath('APA300ML/'); +#+end_src + +#+begin_src matlab + open('APA300ML.slx'); +#+end_src + +** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates +We first extract the stiffness and mass matrices. +#+begin_src matlab + K = readmatrix('APA300ML_mat_K.CSV'); + M = readmatrix('APA300ML_mat_M.CSV'); +#+end_src + +#+begin_src matlab :exports results :results value table replace :tangle no + data2orgtable(K(1:10, 1:10), {}, {}, ' %.1g '); +#+end_src + +#+caption: First 10x10 elements of the Stiffness matrix +#+RESULTS: +| 200000000.0 | 30000.0 | -20000.0 | -70.0 | 300000.0 | 40.0 | 10000000.0 | 10000.0 | -6000.0 | 30.0 | +| 30000.0 | 30000000.0 | 2000.0 | -200000.0 | 60.0 | -10.0 | 4000.0 | 2000000.0 | -500.0 | 9000.0 | +| -20000.0 | 2000.0 | 7000000.0 | -10.0 | -30.0 | 10.0 | 6000.0 | 900.0 | -500000.0 | 3 | +| -70.0 | -200000.0 | -10.0 | 1000.0 | -0.1 | 0.08 | -20.0 | -9000.0 | 3 | -30.0 | +| 300000.0 | 60.0 | -30.0 | -0.1 | 900.0 | 0.1 | 30000.0 | 20.0 | -10.0 | 0.06 | +| 40.0 | -10.0 | 10.0 | 0.08 | 0.1 | 10000.0 | 20.0 | 9 | -5 | 0.03 | +| 10000000.0 | 4000.0 | 6000.0 | -20.0 | 30000.0 | 20.0 | 200000000.0 | 10000.0 | 9000.0 | 50.0 | +| 10000.0 | 2000000.0 | 900.0 | -9000.0 | 20.0 | 9 | 10000.0 | 30000000.0 | -500.0 | 200000.0 | +| -6000.0 | -500.0 | -500000.0 | 3 | -10.0 | -5 | 9000.0 | -500.0 | 7000000.0 | -2 | +| 30.0 | 9000.0 | 3 | -30.0 | 0.06 | 0.03 | 50.0 | 200000.0 | -2 | 1000.0 | + + +#+begin_src matlab :exports results :results value table replace :tangle no + data2orgtable(M(1:10, 1:10), {}, {}, ' %.1g '); +#+end_src + +#+caption: First 10x10 elements of the Mass matrix +#+RESULTS: +| 0.01 | -2e-06 | 1e-06 | 6e-09 | 5e-05 | -5e-09 | -0.0005 | -7e-07 | 6e-07 | -3e-09 | +| -2e-06 | 0.01 | 8e-07 | -2e-05 | -8e-09 | 2e-09 | -9e-07 | -0.0002 | 1e-08 | -9e-07 | +| 1e-06 | 8e-07 | 0.009 | 5e-10 | 1e-09 | -1e-09 | -5e-07 | 3e-08 | 6e-05 | 1e-10 | +| 6e-09 | -2e-05 | 5e-10 | 3e-07 | 2e-11 | -3e-12 | 3e-09 | 9e-07 | -4e-10 | 3e-09 | +| 5e-05 | -8e-09 | 1e-09 | 2e-11 | 6e-07 | -4e-11 | -1e-06 | -2e-09 | 1e-09 | -8e-12 | +| -5e-09 | 2e-09 | -1e-09 | -3e-12 | -4e-11 | 1e-07 | -2e-09 | -1e-09 | -4e-10 | -5e-12 | +| -0.0005 | -9e-07 | -5e-07 | 3e-09 | -1e-06 | -2e-09 | 0.01 | 1e-07 | -3e-07 | -2e-08 | +| -7e-07 | -0.0002 | 3e-08 | 9e-07 | -2e-09 | -1e-09 | 1e-07 | 0.01 | -4e-07 | 2e-05 | +| 6e-07 | 1e-08 | 6e-05 | -4e-10 | 1e-09 | -4e-10 | -3e-07 | -4e-07 | 0.009 | -2e-10 | +| -3e-09 | -9e-07 | 1e-10 | 3e-09 | -8e-12 | -5e-12 | -2e-08 | 2e-05 | -2e-10 | 3e-07 | + + +Then, we extract the coordinates of the interface nodes. +#+begin_src matlab + [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA300ML_out_nodes_3D.txt'); +#+end_src + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) + data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f '); +#+end_src + +#+caption: Coordinates of the interface nodes +#+RESULTS: +| Node i | Node Number | x [m] | y [m] | z [m] | +|--------+-------------+---------+-------+--------| +| 1.0 | 697783.0 | 0.0 | 0.0 | -0.015 | +| 2.0 | 697784.0 | 0.0 | 0.0 | 0.015 | +| 3.0 | 697785.0 | -0.0325 | 0.0 | 0.0 | +| 4.0 | 697786.0 | -0.0125 | 0.0 | 0.0 | +| 5.0 | 697787.0 | -0.0075 | 0.0 | 0.0 | +| 6.0 | 697788.0 | 0.0125 | 0.0 | 0.0 | +| 7.0 | 697789.0 | 0.0325 | 0.0 | 0.0 | + +#+begin_src matlab :exports results :results value table replace :tangle no + data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f '); +#+end_src + +#+caption: Some extracted parameters of the FEM +#+RESULTS: +| Total number of Nodes | 7 | +| Number of interface Nodes | 7 | +| Number of Modes | 120 | +| Size of M and K matrices | 162 | + +Using =K=, =M= and =int_xyz=, we can now use the =Reduced Order Flexible Solid= simscape block. + +** Piezoelectric parameters +#+begin_src matlab + Ga = 1; % [N/V] + Gs = 1; % [V/m] +#+end_src + +#+begin_src matlab + m = 0.1; % [kg] +#+end_src + +** Simscape Model +The flexible element is imported using the =Reduced Order Flexible Solid= simscape block. + +Let's say we use two stacks as a force sensor and one stack as an actuator: +- A =Relative Motion Sensor= block is added between the nodes A and C +- An =Internal Force= block is added between the remote points E and B + +The interface nodes are shown in Figure [[fig:apa300ml_ansys]]. + +One mass is fixed at one end of the piezo-electric stack actuator (remove point F), the other end is fixed to the world frame (remote point G). + +** Identification of the APA Characteristics +*** Stiffness +#+begin_src matlab :exports none + m = 0.0001; +#+end_src + +The transfer function from vertical external force to the relative vertical displacement is identified. + +#+begin_src matlab :exports none + %% Name of the Simulink File + mdl = 'APA300ML'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; + + G = linearize(mdl, io); +#+end_src + +The inverse of its DC gain is the axial stiffness of the APA: +#+begin_src matlab :results replace value + 1e-6/dcgain(G) % [N/um] +#+end_src + +#+RESULTS: +: 1.753 + +The specified stiffness in the datasheet is $k = 1.8\, [N/\mu m]$. + +*** Resonance Frequency +The resonance frequency is specified to be between 650Hz and 840Hz. +This is also the case for the FEM model (Figure [[fig:apa300ml_resonance]]). + +#+begin_src matlab :exports none + freqs = logspace(2, 4, 5000); + + figure; + hold on; + plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Amplitude'); + hold off; +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/apa300ml_resonance.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:apa300ml_resonance +#+caption: First resonance is around 800Hz +#+RESULTS: +[[file:figs/apa300ml_resonance.png]] + +*** Amplification factor +The amplification factor is the ratio of the vertical displacement to the stack displacement. + +#+begin_src matlab :exports none + %% Name of the Simulink File + mdl = 'APA300ML'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1; + + G = linearize(mdl, io); +#+end_src + +The ratio of the two displacement is computed from the FEM model. +#+begin_src matlab :results replace value + abs(dcgain(G(1,1))./dcgain(G(2,1))) +#+end_src + +#+RESULTS: +: 5.0749 + +This is actually correct and approximately corresponds to the ratio of the piezo height and length: +#+begin_src matlab :results replace value + 75/15 +#+end_src + +#+RESULTS: +: 5 + +*** Stroke + +Estimation of the actuator stroke: +\[ \Delta H = A n \Delta L \] +with: +- $\Delta H$ Axial Stroke of the APA +- $A$ Amplification factor (5 for the APA300ML) +- $n$ Number of stack used +- $\Delta L$ Stroke of the stack (0.1% of its length) + +#+begin_src matlab :results replace value + 1e6 * 5 * 3 * 20e-3 * 0.1e-2 +#+end_src + +#+RESULTS: +: 300 + +This is exactly the specified stroke in the data-sheet. + +*** TODO Stroke BIS +- [ ] Identified the stroke form the transfer function from V to z + +#+begin_src matlab :exports none + %% Name of the Simulink File + mdl = 'APA300ML'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/V'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1; + + G = linearize(mdl, io); + + 1e6*170*abs(dcgain(G)) +#+end_src + +** Identification of the Dynamics from actuator to replace displacement +We first set the mass to be approximately zero. +#+begin_src matlab :exports none + m = 0.01; +#+end_src + +The dynamics is identified from the applied force to the measured relative displacement. +#+begin_src matlab :exports none + %% Name of the Simulink File + mdl = 'APA300ML'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; + + Gh = -linearize(mdl, io); +#+end_src + +The same dynamics is identified for a payload mass of 10Kg. +#+begin_src matlab + m = 10; +#+end_src + +#+begin_src matlab :exports none + %% Name of the Simulink File + mdl = 'APA300ML'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; + + Ghm = -linearize(mdl, io); +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(0, 4, 5000); + + figure; + tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + ax1 = nexttile([2,1]); + hold on; + plot(freqs, abs(squeeze(freqresp(Gh, freqs, 'Hz'))), '-'); + plot(freqs, abs(squeeze(freqresp(Ghm, freqs, 'Hz'))), '-'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude'); set(gca, 'XTickLabel',[]); + hold off; + + ax2 = nexttile; + hold on; + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gh, freqs, 'Hz')))), '-', ... + 'DisplayName', '$m = 0kg$'); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Ghm, freqs, 'Hz')))), '-', ... + 'DisplayName', '$m = 10kg$'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + yticks(-360:90:360); + ylim([-360 0]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); + legend('location', 'southwest'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/apa300ml_plant_dynamics.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:apa300ml_plant_dynamics +#+caption: Transfer function from forces applied by the stack to the axial displacement of the APA +#+RESULTS: +[[file:figs/apa300ml_plant_dynamics.png]] + +The root locus corresponding to Direct Velocity Feedback with a mass of 10kg is shown in Figure [[fig:apa300ml_dvf_root_locus]]. +#+begin_src matlab :exports none + figure; + + gains = logspace(0, 5, 500); + + hold on; + plot(real(pole(Ghm)), imag(pole(G)), 'kx'); + plot(real(tzero(Ghm)), imag(tzero(G)), 'ko'); + for k = 1:length(gains) + cl_poles = pole(feedback(Ghm, gains(k)*s)); + plot(real(cl_poles), imag(cl_poles), 'k.'); + end + hold off; + axis square; + xlim([-500, 10]); ylim([0, 510]); + + xlabel('Real Part'); ylabel('Imaginary Part'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/apa300ml_dvf_root_locus.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:apa300ml_dvf_root_locus +#+caption: Root Locus for Direct Velocity Feedback +#+RESULTS: +[[file:figs/apa300ml_dvf_root_locus.png]] + +** Identification of the Dynamics from actuator to force sensor +Let's use 2 stacks as a force sensor and 1 stack as force actuator. + +The transfer function from actuator voltage to sensor voltage is identified and shown in Figure [[fig:apa300ml_iff_plant]]. +#+begin_src matlab :exports none + m = 10; +#+end_src + +#+begin_src matlab :exports none + %% Name of the Simulink File + mdl = 'APA300ML'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; + + Giff = -linearize(mdl, io); +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(0, 4, 5000); + + figure; + tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + ax1 = nexttile([2,1]); + hold on; + plot(freqs, abs(squeeze(freqresp(Giff, freqs, 'Hz'))), '-'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude'); set(gca, 'XTickLabel',[]); + hold off; + + ax2 = nexttile; + hold on; + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Giff, freqs, 'Hz')))), '-'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + yticks(-360:90:360); + ylim([-180 180]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/apa300ml_iff_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:apa300ml_iff_plant +#+caption: Transfer function from actuator to force sensor +#+RESULTS: +[[file:figs/apa300ml_iff_plant.png]] + +For root locus corresponding to IFF is shown in Figure [[fig:apa300ml_iff_root_locus]]. +#+begin_src matlab :exports none + figure; + + gains = logspace(0, 5, 500); + + hold on; + plot(real(pole(Giff)), imag(pole(Giff)), 'kx'); + plot(real(tzero(Giff)), imag(tzero(Giff)), 'ko'); + for k = 1:length(gains) + cl_poles = pole(feedback(Giff, gains(k)/s)); + plot(real(cl_poles), imag(cl_poles), 'k.'); + end + hold off; + axis square; + xlim([-500, 10]); ylim([0, 510]); + + xlabel('Real Part'); ylabel('Imaginary Part'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/apa300ml_iff_root_locus.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:apa300ml_iff_root_locus +#+caption: Root Locus for IFF +#+RESULTS: +[[file:figs/apa300ml_iff_root_locus.png]] + +** Identification for a simpler model +The goal in this section is to identify the parameters of a simple APA model from the FEM. +This can be useful is a lower order model is to be used for simulations. + +The presented model is based on cite:souleille18_concep_activ_mount_space_applic. + +The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure [[fig:souleille18_model_piezo]]). +The parameters are shown in the table below. + +#+name: fig:souleille18_model_piezo +#+caption: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator +[[file:./figs/souleille18_model_piezo.png]] + +#+caption:Parameters used for the model of the APA 100M +| | Meaning | +|-------+----------------------------------------------------------------| +| $k_e$ | Stiffness used to adjust the pole of the isolator | +| $k_1$ | Stiffness of the metallic suspension when the stack is removed | +| $k_a$ | Stiffness of the actuator | +| $c_1$ | Added viscous damping | + +The goal is to determine $k_e$, $k_a$ and $k_1$ so that the simplified model fits the FEM model. + +\[ \alpha = \frac{x_1}{f}(\omega=0) = \frac{\frac{k_e}{k_e + k_a}}{k_1 + \frac{k_e k_a}{k_e + k_a}} \] +\[ \beta = \frac{x_1}{F}(\omega=0) = \frac{1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \] + +If we can fix $k_a$, we can determine $k_e$ and $k_1$ with: +\[ k_e = \frac{k_a}{\frac{\beta}{\alpha} - 1} \] +\[ k_1 = \frac{1}{\beta} - \frac{k_e k_a}{k_e + k_a} \] + +#+begin_src matlab :exports none + m = 10; +#+end_src + +#+begin_src matlab :exports none + %% Name of the Simulink File + mdl = 'APA300ML'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; % External Vertical Force [N] + io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m] + io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force [N] + io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m] + io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor [V] + io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1; % Stack Displacement [m] + + G = linearize(mdl, io); + + G.InputName = {'Fd', 'w', 'Fa'}; + G.OutputName = {'y', 'Fs', 'd'}; +#+end_src + +From the identified dynamics, compute $\alpha$ and $\beta$ +#+begin_src matlab + alpha = abs(dcgain(G('y', 'Fa'))); + beta = abs(dcgain(G('y', 'Fd'))); +#+end_src + +$k_a$ is estimated using the following formula: +#+begin_src matlab + ka = 0.8/abs(dcgain(G('y', 'Fa'))); +#+end_src +The factor can be adjusted to better match the curves. + +Then $k_e$ and $k_1$ are computed. +#+begin_src matlab + ke = ka/(beta/alpha - 1); + k1 = 1/beta - ke*ka/(ke + ka); +#+end_src + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) + data2orgtable(1e-6*[ka; ke; k1], {'ka', 'ke', 'k1'}, {'Value [N/um]'}, ' %.1f '); +#+end_src + +#+RESULTS: +| | Value [N/um] | +|----+--------------| +| ka | 40.5 | +| ke | 1.5 | +| k1 | 0.4 | + +The damping in the system is adjusted to match the FEM model if necessary. +#+begin_src matlab + c1 = 1e2; +#+end_src + +The analytical model of the simpler system is defined below: +#+begin_src matlab + Ga = 1/(m*s^2 + k1 + c1*s + ke*ka/(ke + ka)) * ... + [ 1 , k1 + c1*s + ke*ka/(ke + ka) , ke/(ke + ka) ; + -ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 , -ke/(ke + ka)*(m*s^2 + c1*s + k1)]; + + Ga.InputName = {'Fd', 'w', 'Fa'}; + Ga.OutputName = {'y', 'Fs'}; +#+end_src + +And the DC gain is adjusted for the force sensor: +#+begin_src matlab + F_gain = dcgain(G('Fs', 'Fd'))/dcgain(Ga('Fs', 'Fd')); +#+end_src + +The dynamics of the FEM model and the simpler model are compared in Figure [[fig:apa300ml_comp_simpler_model]]. + +#+begin_src matlab :exports none + freqs = logspace(0, 5, 1000); + + figure; + tiledlayout(2, 3, 'TileSpacing', 'None', 'Padding', 'None'); + + ax1 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G( 'y', 'w'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Ga('y', 'w'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); + ylabel('$x_1/w$ [m/m]'); + ylim([1e-6, 1e2]); + + ax2 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fa'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Ga('y', 'Fa'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); + ylabel('$x_1/f$ [m/N]'); + ylim([1e-14, 1e-6]); + + ax3 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fd'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Ga('y', 'Fd'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); + ylabel('$x_1/F$ [m/N]'); + ylim([1e-14, 1e-4]); + + ax4 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'w'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'w'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); + ylabel('$F_s/w$ [m/m]'); + ylim([1e2, 1e8]); + + ax5 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fa'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'Fa'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); + ylabel('$F_s/f$ [m/N]'); + ylim([1e-4, 1e1]); + + ax6 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fd'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'Fd'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); + ylabel('$F_s/F$ [m/N]'); + ylim([1e-7, 1e2]); + + linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/apa300ml_comp_simpler_model.pdf', 'width', 'full', 'height', 'full'); +#+end_src + +#+name: fig:apa300ml_comp_simpler_model +#+caption: Comparison of the Dynamics between the FEM model and the simplified one +#+RESULTS: +[[file:figs/apa300ml_comp_simpler_model.png]] + +The simplified model has also been implemented in Simscape. + +The dynamics of the Simscape simplified model is identified and compared with the FEM one in Figure [[fig:apa300ml_comp_simpler_simscape]]. +#+begin_src matlab :exports none + %% Name of the Simulink File + mdl = 'APA300ML_simplified'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; % External Vertical Force [N] + io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m] + io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force [N] + io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m] + io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor [V] + + Gs = linearize(mdl, io); + + Gs.InputName = {'Fd', 'w', 'Fa'}; + Gs.OutputName = {'y', 'Fs'}; +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(0, 5, 1000); + + figure; + tiledlayout(2, 3, 'TileSpacing', 'None', 'Padding', 'None'); + + ax1 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G( 'y', 'w'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Gs('y', 'w'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); + ylabel('$x_1/w$ [m/m]'); + ylim([1e-6, 1e2]); + + ax2 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fa'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Gs('y', 'Fa'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); + ylabel('$x_1/f$ [m/N]'); + ylim([1e-14, 1e-6]); + + ax3 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fd'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Gs('y', 'Fd'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); + ylabel('$x_1/F$ [m/N]'); + ylim([1e-14, 1e-4]); + + ax4 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'w'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'w'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); + ylabel('$F_s/w$ [m/m]'); + ylim([1e2, 1e8]); + + ax5 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fa'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'Fa'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); + ylabel('$F_s/f$ [m/N]'); + ylim([1e-4, 1e1]); + + ax6 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fd'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'Fd'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); + ylabel('$F_s/F$ [m/N]'); + ylim([1e-7, 1e2]); + + linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/apa300ml_comp_simpler_simscape.pdf', 'width', 'full', 'height', 'full'); +#+end_src + +#+name: fig:apa300ml_comp_simpler_simscape +#+caption: Comparison of the Dynamics between the FEM model and the simplified simscape model +#+RESULTS: +[[file:figs/apa300ml_comp_simpler_simscape.png]] + +** Integral Force Feedback +In this section, Integral Force Feedback control architecture is applied on the APA300ML. + +First, the plant (dynamics from voltage actuator to voltage sensor is identified). +#+begin_src matlab :exports none + Kiff = tf(0); +#+end_src + +The payload mass is set to 10kg. +#+begin_src matlab + m = 10; +#+end_src + +#+begin_src matlab :exports none + %% Name of the Simulink File + mdl = 'APA300ML_IFF'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/APA300ML'], 1, 'openoutput'); io_i = io_i + 1; + + G_ol = linearize(mdl, io); + G_ol.InputName = {'w', 'f', 'F'}; + G_ol.OutputName = {'x1', 'Fs'}; + + G = G_ol({'Fs'}, {'f'}); +#+end_src + +The obtained dynamics is shown in Figure [[fig:piezo_amplified_iff_plant]]. + +#+begin_src matlab :exports none + freqs = logspace(1, 5, 1000); + + figure; + tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + ax1 = nexttile([2,1]); + hold on; + plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude'); set(gca, 'XTickLabel',[]); + hold off; + + ax2 = nexttile; + hold on; + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G, freqs, 'Hz'))))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + yticks(-360:90:360); + ylim([-390 30]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/piezo_amplified_iff_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:piezo_amplified_iff_plant +#+caption: IFF Plant +#+RESULTS: +[[file:figs/piezo_amplified_iff_plant.png]] + +The controller is defined below and the loop gain is shown in Figure [[fig:piezo_amplified_iff_loop_gain]]. +#+begin_src matlab + Kiff = -1e3/s; +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(1, 5, 1000); + + figure; + tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + ax1 = nexttile([2,1]); + hold on; + plot(freqs, abs(squeeze(freqresp(G*Kiff, freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude'); set(gca, 'XTickLabel',[]); + hold off; + + ax2 = nexttile; + hold on; + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G*Kiff, freqs, 'Hz'))))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + yticks(-360:90:360); + ylim([-180 180]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/piezo_amplified_iff_loop_gain.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:piezo_amplified_iff_loop_gain +#+caption: IFF Loop Gain +#+RESULTS: +[[file:figs/piezo_amplified_iff_loop_gain.png]] + +Now the closed-loop system is identified again and compare with the open loop system in Figure [[fig:piezo_amplified_iff_comp]]. + +It is the expected behavior as shown in the Figure [[fig:souleille18_results]] (from cite:souleille18_concep_activ_mount_space_applic). + +#+begin_src matlab :exports none + clear io; io_i = 1; + io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/APA300ML'], 1, 'output'); io_i = io_i + 1; + + Giff = linearize(mdl, io); + Giff.InputName = {'w', 'f', 'F'}; + Giff.OutputName = {'x1', 'Fs'}; +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + tiledlayout(2, 3, 'TileSpacing', 'None', 'Padding', 'None'); + + ax1 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G_ol('x1', 'w'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Giff('x1', 'w'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); ylabel('$x_1/w$ [m/m]') + + ax2 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G_ol('x1', 'f'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Giff('x1', 'f'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); ylabel('$x_1/f$ [m/N]'); + + ax3 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G_ol('x1', 'F'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Giff('x1', 'F'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); ylabel('$x_1/F$ [m/N]'); + + ax4 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G_ol('Fs', 'w'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'w'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('$F_s/w$ [N/m]'); + + ax5 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G_ol('Fs', 'f'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'f'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('$F_s/f$ [N/N]'); + + ax6 = nexttile; + hold on; + plot(freqs, abs(squeeze(freqresp(G_ol('Fs', 'F'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'F'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('$F_s/F$ [N/N]'); + + linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/piezo_amplified_iff_comp.pdf', 'width', 'full', 'height', 'full'); +#+end_src + +#+name: fig:piezo_amplified_iff_comp +#+caption: OL and CL transfer functions +#+RESULTS: +[[file:figs/piezo_amplified_iff_comp.png]] + +#+name: fig:souleille18_results +#+caption: Results obtained in cite:souleille18_concep_activ_mount_space_applic +[[file:figs/souleille18_results.png]] + +* Bibliography :ignore: +#+latex: \printbibliography diff --git a/test-bench-apa300ml.pdf b/test-bench-apa300ml.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1962e35d79951d998da9b6bfabe45a7c59193a93 GIT binary patch literal 1713805 zcmeFZXH-+`x;AWC;!;FxfYKEKg+S=Ns4NsjKsrd18d`|bL$!i{bg2O%HPTDy5I}kl z5PGi(y@i$#;G6D!*52!k^W#0=kF|fD$p|AOnPkrAZqIdH_Z>K2{_&JgkYDsNXYWAs 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