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<h1 class="title">Amplifier Piezoelectric Actuator APA300ML - Test Bench</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org7de9329">1. Model of an Amplified Piezoelectric Actuator and Sensor</a></li>
<li><a href="#orgee5ee06">2. Test-Bench Description</a></li>
<li><a href="#org532e46b">3. Measurement Procedure</a>
<ul>
<li><a href="#org1fa2bb1">3.1. Stroke Measurement</a></li>
<li><a href="#orge53dfac">3.2. Stiffness Measurement</a></li>
<li><a href="#orgd7e3e7b">3.3. Hysteresis measurement</a></li>
<li><a href="#org444da20">3.4. Piezoelectric Actuator Constant</a></li>
<li><a href="#org027bf4a">3.5. Piezoelectric Sensor Constant</a></li>
<li><a href="#org0de7709">3.6. Capacitance Measurement</a></li>
<li><a href="#org66f2e6f">3.7. Dynamical Behavior</a></li>
<li><a href="#orgc275f3f">3.8. Compare the results obtained for all 7 APA300ML</a></li>
</ul>
</li>
<li><a href="#org4ce78ab">4. Measurement Results</a></li>
</ul>
</div>
</div>
<p>
The goal of this test bench is to extract all the important parameters of the Amplified Piezoelectric Actuator APA300ML.
</p>
<p>
This include:
</p>
<ul class="org-ul">
<li>Stroke</li>
<li>Stiffness</li>
<li>Hysteresis</li>
<li>Gain from the applied voltage \(V_a\) to the generated Force \(F_a\)</li>
<li>Gain from the sensor stack strain \(\delta L\) to the generated voltage \(V_s\)</li>
<li>Dynamical behavior</li>
</ul>
<div id="org084a571" class="figure">
<p><img src="figs/apa300ML.png" alt="apa300ML.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Picture of the APA300ML</p>
</div>
<div id="outline-container-org7de9329" class="outline-2">
<h2 id="org7de9329"><span class="section-number-2">1</span> Model of an Amplified Piezoelectric Actuator and Sensor</h2>
<div class="outline-text-2" id="text-1">
<p>
Consider a schematic of the Amplified Piezoelectric Actuator in Figure <a href="#org2231a2d">2</a>.
</p>
<div id="org2231a2d" class="figure">
<p><img src="figs/apa_model_schematic.png" alt="apa_model_schematic.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Amplified Piezoelectric Actuator Schematic</p>
</div>
<p>
A voltage \(V_a\) applied to the actuator stacks will induce an actuator force \(F_a\):
</p>
\begin{equation}
F_a = g_a \cdot V_a
\end{equation}
<p>
A change of length \(dl\) of the sensor stack will induce a voltage \(V_s\):
</p>
\begin{equation}
V_s = g_s \cdot dl
\end{equation}
<p>
We wish here to experimental measure \(g_a\) and \(g_s\).
</p>
<p>
The block-diagram model of the piezoelectric actuator is then as shown in Figure <a href="#orge718081">3</a>.
</p>
<div id="orge718081" class="figure">
<p><img src="figs/apa-model-simscape-schematic.png" alt="apa-model-simscape-schematic.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Model of the APA with Simscape/Simulink</p>
</div>
</div>
</div>
<div id="outline-container-orgee5ee06" class="outline-2">
<h2 id="orgee5ee06"><span class="section-number-2">2</span> Test-Bench Description</h2>
<div class="outline-text-2" id="text-2">
<div class="note" id="org5799347">
<p>
Here are the documentation of the equipment used for this test bench:
</p>
<ul class="org-ul">
<li>Voltage Amplifier: <a href="doc/PD200-V7-R1.pdf">PD200</a></li>
<li>Amplified Piezoelectric Actuator: <a href="doc/APA300ML.pdf">APA300ML</a></li>
<li>DAC/ADC: Speedgoat <a href="doc/IO131-OEM-Datasheet.pdf">IO313</a></li>
<li>Encoder: <a href="doc/L-9517-9678-05-A_Data_sheet_VIONiC_series_en.pdf">Renishaw Vionic</a> and used <a href="doc/L-9517-9862-01-C_Data_sheet_RKLC_EN.pdf">Ruler</a></li>
<li>Interferometer: <a href="https://www.attocube.com/en/products/laser-displacement-sensor/displacement-measuring-interferometer">Attocube IDS3010</a></li>
</ul>
</div>
<div id="org6705e31" class="figure">
<p><img src="figs/test_bench_apa_alone.png" alt="test_bench_apa_alone.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Schematic of the Test Bench</p>
</div>
</div>
</div>
<div id="outline-container-org532e46b" class="outline-2">
<h2 id="org532e46b"><span class="section-number-2">3</span> Measurement Procedure</h2>
<div class="outline-text-2" id="text-3">
</div>
<div id="outline-container-org1fa2bb1" class="outline-3">
<h3 id="org1fa2bb1"><span class="section-number-3">3.1</span> Stroke Measurement</h3>
<div class="outline-text-3" id="text-3-1">
<p>
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.
</p>
<p>
As the gain of the PD200 amplifier is 20, the DAC output voltage should be:
\[ V_{dac}(t) = 3.25 + 4.25\sin(2\pi \cdot t) \]
</p>
<p>
Verify that the voltage offset is zero!
</p>
<p>
Measure the output vertical displacement \(d\) using the interferometer.
</p>
<p>
Then, plot \(d\) as a function of \(V_a\), and perform a linear regression.
Conclude on the obtained stroke.
</p>
</div>
</div>
<div id="outline-container-orge53dfac" class="outline-3">
<h3 id="orge53dfac"><span class="section-number-3">3.2</span> Stiffness Measurement</h3>
<div class="outline-text-3" id="text-3-2">
<p>
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.
</p>
<p>
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.
</p>
<p>
Then the obtained stiffness is:
</p>
\begin{equation}
k = \frac{\delta m g}{\delta d}
\end{equation}
</div>
</div>
<div id="outline-container-orgd7e3e7b" class="outline-3">
<h3 id="orgd7e3e7b"><span class="section-number-3">3.3</span> Hysteresis measurement</h3>
<div class="outline-text-3" id="text-3-3">
<p>
Supply a quasi static sinusoidal excitation \(V_a\) at different voltages.
</p>
<p>
The offset should be 65V, and the sin amplitude can range from 1V up to 85V.
</p>
<p>
For each excitation amplitude, the vertical displacement \(d\) of the mass is measured.
</p>
<p>
Then, \(d\) is plotted as a function of \(V_a\) for all the amplitudes.
</p>
</div>
</div>
<div id="outline-container-org444da20" class="outline-3">
<h3 id="org444da20"><span class="section-number-3">3.4</span> Piezoelectric Actuator Constant</h3>
<div class="outline-text-3" id="text-3-4">
<p>
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:
</p>
\begin{equation}
d = g_{d/V_a} \cdot V_a
\end{equation}
<p>
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 \(d\):
</p>
\begin{equation}
d = g_{d/F_a} \cdot F_a
\end{equation}
<p>
From the two gains, it is then easy to determine \(g_a\):
</p>
\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}
</div>
</div>
<div id="outline-container-org027bf4a" class="outline-3">
<h3 id="org027bf4a"><span class="section-number-3">3.5</span> Piezoelectric Sensor Constant</h3>
<div class="outline-text-3" id="text-3-5">
<p>
From a quasi static (1Hz) excitation of the piezoelectric stack, measure the gain from \(V_a\) to \(V_s\):
</p>
\begin{equation}
V_s = g_{V_s/V_a} V_a
\end{equation}
<p>
Using the simscape model, compute the static gain from the actuator force \(F_a\) to the strain of the sensor stack \(dl\):
</p>
\begin{equation}
dl = g_{dl/F_a} F_a
\end{equation}
<p>
Then, the static gain from the sensor stack strain \(dl\) to the general voltage \(V_s\) is:
</p>
\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}
<p>
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.
</p>
</div>
</div>
<div id="outline-container-org0de7709" class="outline-3">
<h3 id="org0de7709"><span class="section-number-3">3.6</span> Capacitance Measurement</h3>
<div class="outline-text-3" id="text-3-6">
<p>
Measure the capacitance of the 3 stacks individually using a precise multi-meter.
</p>
</div>
</div>
<div id="outline-container-org66f2e6f" class="outline-3">
<h3 id="org66f2e6f"><span class="section-number-3">3.7</span> Dynamical Behavior</h3>
<div class="outline-text-3" id="text-3-7">
<p>
Perform a system identification from \(V_a\) to the measured displacement \(d\) by the interferometer and by the encoder, and to the general voltage \(V_s\).
</p>
<p>
This can be performed using different excitation signals.
</p>
<p>
This can also be performed with and without the encoder fixed to the APA.
</p>
</div>
</div>
<div id="outline-container-orgc275f3f" class="outline-3">
<h3 id="orgc275f3f"><span class="section-number-3">3.8</span> Compare the results obtained for all 7 APA300ML</h3>
<div class="outline-text-3" id="text-3-8">
<p>
Compare all the obtained parameters for all the test APA.
</p>
</div>
</div>
</div>
<div id="outline-container-org4ce78ab" class="outline-2">
<h2 id="org4ce78ab"><span class="section-number-2">4</span> Measurement Results</h2>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2021-01-04 lun. 14:44</p>
</div>
</body>
</html>

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#+TITLE: Amplifier Piezoelectric Actuator APA300ML - Test Bench
:DRAWER:
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas
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: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

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@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,
}

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<h1 class="title">Amplifier Piezoelectric Actuator APA300ML - Test Bench</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org3671a24">1. Model of an Amplified Piezoelectric Actuator and Sensor</a></li>
<li><a href="#org83e51bc">2. Test-Bench Description</a></li>
<li><a href="#orgb8f415f">3. Measurement Procedure</a>
<ul>
<li><a href="#org6a1e417">3.1. Stroke Measurement</a></li>
<li><a href="#org96ea50c">3.2. Stiffness Measurement</a></li>
<li><a href="#orgd47f2df">3.3. Hysteresis measurement</a></li>
<li><a href="#orgf91088f">3.4. Piezoelectric Actuator Constant</a></li>
<li><a href="#org8d23151">3.5. Piezoelectric Sensor Constant</a></li>
<li><a href="#org3560597">3.6. Capacitance Measurement</a></li>
<li><a href="#org8113f87">3.7. Dynamical Behavior</a></li>
<li><a href="#org25802eb">3.8. Compare the results obtained for all 7 APA300ML</a></li>
</ul>
</li>
<li><a href="#org1f490aa">4. Measurement Results</a></li>
</ul>
</div>
</div>
<hr>
<p>This report is also available as a <a href="./test-bench-apa300ml.pdf">pdf</a>.</p>
<hr>
<p>
The goal of this test bench is to extract all the important parameters of the Amplified Piezoelectric Actuator APA300ML.
</p>
<p>
This include:
</p>
<ul class="org-ul">
<li>Stroke</li>
<li>Stiffness</li>
<li>Hysteresis</li>
<li>Gain from the applied voltage \(V_a\) to the generated Force \(F_a\)</li>
<li>Gain from the sensor stack strain \(\delta L\) to the generated voltage \(V_s\)</li>
<li>Dynamical behavior</li>
</ul>
<div id="orgb0d5679" class="figure">
<p><img src="figs/apa300ML.png" alt="apa300ML.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Picture of the APA300ML</p>
</div>
<div id="outline-container-org3671a24" class="outline-2">
<h2 id="org3671a24"><span class="section-number-2">1</span> Model of an Amplified Piezoelectric Actuator and Sensor</h2>
<div class="outline-text-2" id="text-1">
<p>
Consider a schematic of the Amplified Piezoelectric Actuator in Figure <a href="#org3bbcf6a">2</a>.
</p>
<div id="org3bbcf6a" class="figure">
<p><img src="figs/apa_model_schematic.png" alt="apa_model_schematic.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Amplified Piezoelectric Actuator Schematic</p>
</div>
<p>
A voltage \(V_a\) applied to the actuator stacks will induce an actuator force \(F_a\):
</p>
\begin{equation}
F_a = g_a \cdot V_a
\end{equation}
<p>
A change of length \(dl\) of the sensor stack will induce a voltage \(V_s\):
</p>
\begin{equation}
V_s = g_s \cdot dl
\end{equation}
<p>
We wish here to experimental measure \(g_a\) and \(g_s\).
</p>
<p>
The block-diagram model of the piezoelectric actuator is then as shown in Figure <a href="#org12e874e">3</a>.
</p>
<div id="org12e874e" class="figure">
<p><img src="figs/apa-model-simscape-schematic.png" alt="apa-model-simscape-schematic.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Model of the APA with Simscape/Simulink</p>
</div>
</div>
</div>
<div id="outline-container-org83e51bc" class="outline-2">
<h2 id="org83e51bc"><span class="section-number-2">2</span> Test-Bench Description</h2>
<div class="outline-text-2" id="text-2">
<div class="note" id="orgdde2181">
<p>
Here are the documentation of the equipment used for this test bench:
</p>
<ul class="org-ul">
<li>Voltage Amplifier: <a href="doc/PD200-V7-R1.pdf">PD200</a></li>
<li>Amplified Piezoelectric Actuator: <a href="doc/APA300ML.pdf">APA300ML</a></li>
<li>DAC/ADC: Speedgoat <a href="doc/IO131-OEM-Datasheet.pdf">IO313</a></li>
<li>Encoder: <a href="doc/L-9517-9678-05-A_Data_sheet_VIONiC_series_en.pdf">Renishaw Vionic</a> and used <a href="doc/L-9517-9862-01-C_Data_sheet_RKLC_EN.pdf">Ruler</a></li>
<li>Interferometer: <a href="https://www.attocube.com/en/products/laser-displacement-sensor/displacement-measuring-interferometer">Attocube IDS3010</a></li>
</ul>
</div>
<div id="org2ca856d" class="figure">
<p><img src="figs/test_bench_apa_alone.png" alt="test_bench_apa_alone.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Schematic of the Test Bench</p>
</div>
</div>
</div>
<div id="outline-container-orgb8f415f" class="outline-2">
<h2 id="orgb8f415f"><span class="section-number-2">3</span> Measurement Procedure</h2>
<div class="outline-text-2" id="text-3">
</div>
<div id="outline-container-org6a1e417" class="outline-3">
<h3 id="org6a1e417"><span class="section-number-3">3.1</span> Stroke Measurement</h3>
<div class="outline-text-3" id="text-3-1">
<p>
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.
</p>
<p>
As the gain of the PD200 amplifier is 20, the DAC output voltage should be:
\[ V_{dac}(t) = 3.25 + 4.25\sin(2\pi \cdot t) \]
</p>
<p>
Verify that the voltage offset of the PD200 is zero!
</p>
<p>
Measure the output vertical displacement \(d\) using the interferometer.
</p>
<p>
Then, plot \(d\) as a function of \(V_a\), and perform a linear regression.
Conclude on the obtained stroke.
</p>
</div>
</div>
<div id="outline-container-org96ea50c" class="outline-3">
<h3 id="org96ea50c"><span class="section-number-3">3.2</span> Stiffness Measurement</h3>
<div class="outline-text-3" id="text-3-2">
<p>
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.
</p>
<p>
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.
</p>
<p>
Then the obtained stiffness is:
</p>
\begin{equation}
k = \frac{\delta m g}{\delta d}
\end{equation}
</div>
</div>
<div id="outline-container-orgd47f2df" class="outline-3">
<h3 id="orgd47f2df"><span class="section-number-3">3.3</span> Hysteresis measurement</h3>
<div class="outline-text-3" id="text-3-3">
<p>
Supply a quasi static sinusoidal excitation \(V_a\) at different voltages.
</p>
<p>
The offset should be 65V, and the sin amplitude can range from 1V up to 85V.
</p>
<p>
For each excitation amplitude, the vertical displacement \(d\) of the mass is measured.
</p>
<p>
Then, \(d\) is plotted as a function of \(V_a\) for all the amplitudes.
</p>
<div id="orgab4aadf" class="figure">
<p><img src="figs/expected_hysteresis.png" alt="expected_hysteresis.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Expected Hysteresis (<a class='org-ref-reference' href="#poel10_explor_activ_hard_mount_vibrat">poel10_explor_activ_hard_mount_vibrat</a>)</p>
</div>
</div>
</div>
<div id="outline-container-orgf91088f" class="outline-3">
<h3 id="orgf91088f"><span class="section-number-3">3.4</span> Piezoelectric Actuator Constant</h3>
<div class="outline-text-3" id="text-3-4">
<p>
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:
</p>
\begin{equation}
d = g_{d/V_a} \cdot V_a
\end{equation}
<p>
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 \(d\):
</p>
\begin{equation}
d = g_{d/F_a} \cdot F_a
\end{equation}
<p>
From the two gains, it is then easy to determine \(g_a\):
</p>
\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}
</div>
</div>
<div id="outline-container-org8d23151" class="outline-3">
<h3 id="org8d23151"><span class="section-number-3">3.5</span> Piezoelectric Sensor Constant</h3>
<div class="outline-text-3" id="text-3-5">
<p>
From a quasi static excitation of the piezoelectric stack, measure the gain from \(V_a\) to \(V_s\):
</p>
\begin{equation}
V_s = g_{V_s/V_a} V_a
\end{equation}
<p>
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.
</p>
<p>
Alternatively, the gain can be computed from the dynamical identification and taking the gain at the wanted frequency.
</p>
<p>
Using the simscape model, compute the static gain from the actuator force \(F_a\) to the strain of the sensor stack \(dl\):
</p>
\begin{equation}
dl = g_{dl/F_a} F_a
\end{equation}
<p>
Then, the static gain from the sensor stack strain \(dl\) to the general voltage \(V_s\) is:
</p>
\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}
<p>
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.
</p>
</div>
</div>
<div id="outline-container-org3560597" class="outline-3">
<h3 id="org3560597"><span class="section-number-3">3.6</span> Capacitance Measurement</h3>
<div class="outline-text-3" id="text-3-6">
<p>
Measure the capacitance of the 3 stacks individually using a precise multi-meter.
</p>
</div>
</div>
<div id="outline-container-org8113f87" class="outline-3">
<h3 id="org8113f87"><span class="section-number-3">3.7</span> Dynamical Behavior</h3>
<div class="outline-text-3" id="text-3-7">
<p>
Perform a system identification from \(V_a\) to the measured displacement \(d\) by the interferometer and by the encoder, and to the generated voltage \(V_s\).
</p>
<p>
This can be performed using different excitation signals.
</p>
<p>
This can also be performed with and without the encoder fixed to the APA.
</p>
</div>
</div>
<div id="outline-container-org25802eb" class="outline-3">
<h3 id="org25802eb"><span class="section-number-3">3.8</span> Compare the results obtained for all 7 APA300ML</h3>
<div class="outline-text-3" id="text-3-8">
<p>
Compare all the obtained parameters for all the test APA.
</p>
</div>
</div>
</div>
<div id="outline-container-org1f490aa" class="outline-2">
<h2 id="org1f490aa"><span class="section-number-2">4</span> Measurement Results</h2>
</div>
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><h2 class='citeproc-org-bib-h2'>Bibliography</h2>
<div class="csl-bib-body">
<div class="csl-entry"><a name="citeproc_bib_item_1"></a>Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” <i>CEAS Space Journal</i> 10 (2). Springer:157–65.</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2021-02-02 mar. 19:29</p>
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