Add stuff about piezoelectric actuators

content/zettels/piezoelectric_actuators.md

@@ -30,59 +30,6 @@ Tags
 A model of a multi-layer monolithic piezoelectric stack actuator is described in <sup id="c823f68dd2a72b9667a61b3c046b4731"><a class="reference-link" href="#fleming10_nanop_system_with_force_feedb" title="Fleming, Nanopositioning System With Force Feedback for  High-Performance Tracking and Vibration Control, {IEEE/ASME Transactions on Mechatronics}, v(3), 433-447 (2010).">(Fleming, 2010)</a></sup> ([Notes]({{< relref "fleming10_nanop_system_with_force_feedb" >}})).
 
 
-### Specifications {#specifications}
-
-Typical specifications of piezoelectric stack actuators are usually in terms of:
-
--   Displacement/ Travel range \\([\mu m]\\)
--   Blocked force \\([N]\\)
--   Stiffness \\([N/\mu m]\\)
--   Resolution \\([nm]\\)
--   Length \\([mm]\\)
-
-
-#### Displacement and Length {#displacement-and-length}
-
-The maximum displacement specified is the displacement of the actuator when the maximum voltage is applied and when no load is added.
-
-Typical strain of Piezoelectric Stack Actuators is \\(0.1\%\\), the free displacement \\(d\\) is then related to the length of piezoelectric stack:
-\\[ d \approx \frac{L}{1000} \\]
-
-
-#### Blocked Force {#blocked-force}
-
-The blocked force is measured by first applying the maximum voltage to the piezoelectric stack without any load.
-Thus, the piezoelectric stack experiences its maximum displacement.
-
-A force is then applied to return the actuator to its original length.
-This force is measured and recorded as the blocking force.
-
-The blocking force is also the maximum force that can produce the piezoelectric stack in contact with an infinitely stiff environment.
-
-
-#### Stiffness {#stiffness}
-
-
-#### Resolution {#resolution}
-
-The resolution is limited by the noise in the voltage amplified.
-
-Typical [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio" >}}) of voltage amplified is \\(100dB = 10^{5}\\).
-Thus, for a piezoelectric stack with a displacement \\(L\\), the resolution will be
-
-\begin{equation}
-  r = \frac{L}{10^5}
-\end{equation}
-
-For a piezoelectric stack with a displacement of \\(100\,[\mu m]\\), the resolution will be \\(\approx 1\,[nm]\\).
-
-
-### Piezoelectric Stack experiencing a mass load {#piezoelectric-stack-experiencing-a-mass-load}
-
-
-### Piezoelectric Stack in contact with a spring load {#piezoelectric-stack-in-contact-with-a-spring-load}
-
-
 ## Mechanically Amplified Piezoelectric actuators {#mechanically-amplified-piezoelectric-actuators}
 
 The Amplified Piezo Actuators principle is presented in <sup id="5decd2b31c4a9842b80c58b56f96590a"><a class="reference-link" href="#claeyssen07_amplif_piezoel_actuat" title="Frank Claeyssen, Le Letty, Barillot, \&amp; Sosnicki, Amplified Piezoelectric Actuators: Static \&amp; Dynamic  Applications, {Ferroelectrics}, v(1), 3-14 (2007).">(Frank Claeyssen {\it et al.}, 2007)</a></sup>:
@@ -97,6 +44,10 @@ A model of an amplified piezoelectric actuator is described in <sup id="84975085
   year            = 2016,
 }">(Lucinskis \& Mangeot, 2016)</a></sup>.
 
+<a id="orgd9b1a8d"></a>
+
+{{< figure src="/ox-hugo/ling16_topology_piezo_mechanism_types.png" caption="Figure 1: Topology of several types of compliant mechanisms <sup id=\"d9e8b33774f1e65d16bd79114db8ac64\"><a class=\"reference-link\" href=\"#ling16_enhan_mathem_model_displ_amplif\" title=\"Mingxiang Ling, Junyi Cao, Minghua Zeng, Jing Lin, \&amp; Daniel J Inman, Enhanced Mathematical Modeling of the Displacement  Amplification Ratio for Piezoelectric Compliant Mechanisms, {Smart Materials and Structures}, v(7), 075022 (2016).\">(Mingxiang Ling {\it et al.}, 2016)</a></sup>" >}}
+
 | Manufacturers       | Links                                                                                                                                                                                                                                             |
 |---------------------|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
 | Cedrat              | [link](https://www.cedrat-technologies.com/en/products/actuators/amplified-piezo-actuators.html)                                                                                                                                                  |
@@ -107,6 +58,120 @@ A model of an amplified piezoelectric actuator is described in <sup id="84975085
 | Mechano Transformer | [link](http://www.mechano-transformer.com/en/products/01a%5Factuator%5F5.html), [link](http://www.mechano-transformer.com/en/products/01a%5Factuator%5F3.html), [link](http://www.mechano-transformer.com/en/products/01a%5Factuator%5Fmtkk.html) |
 | CoreMorrow          | [link](http://www.coremorrow.com/en/pro-13-1.html)                                                                                                                                                                                                |
 
+
+## Specifications {#specifications}
+
+
+### Typical Specifications {#typical-specifications}
+
+Typical specifications of piezoelectric stack actuators are usually in terms of:
+
+-   Displacement/ Travel range \\([\mu m]\\)
+-   Blocked force \\([N]\\)
+-   Stiffness \\([N/\mu m]\\)
+-   Resolution \\([nm]\\)
+-   Length \\([mm]\\)
+-   Electrical Capacitance \\([nF]\\)
+
+
+### Displacement and Length {#displacement-and-length}
+
+The maximum displacement specified is the displacement of the actuator when the maximum voltage is applied without any load.
+
+Typical maximum strain of Piezoelectric Stack Actuators is \\(0.1\%\\).
+The free displacement \\(\Delta L\_{f}\\) is then related to the length \\(L\\) of piezoelectric stack by:
+
+\begin{equation}
+  \Delta L\_f \approx \frac{L}{1000}
+\end{equation}
+
+> A “free” actuator — one that experiences no resistance to movement — will produce its maximum displacement, often referred to as “free stroke,” and generate zero force.
+
+Note that this maximum displacement is only attainable at DC.
+For dynamical applications, the electrical capacitance of the piezoelectric actuator is an important factor (see bellow).
+
+
+### Blocked Force {#blocked-force}
+
+The blocked force \\(F\_b\\) is measured by first applying the maximum voltage to the piezoelectric stack without any load.
+Thus, the piezoelectric stack experiences its maximum displacement.
+
+A force is then applied to return the actuator to its original length.
+This force is measured and recorded as the blocking force.
+
+The blocking force is also the maximum force that can produce the piezoelectric stack in contact with an infinitely stiff environment.
+
+> When an actuator is blocked from moving, it will produce its maximum force, which is referred to as the blocked, or blocking, force.
+
+
+### Stiffness {#stiffness}
+
+The stiffness of the actuator is the ratio of the blocking force to the free stroke:
+
+\begin{equation}
+  k\_p = \frac{F\_b}{\Delta L\_f}
+\end{equation}
+
+with:
+
+-   \\(k\_p\\): stiffness of the piezo actuator
+-   \\(F\_b\\): blocking force
+-   \\(\Delta L\_f\\): free stroke
+
+
+### Resolution {#resolution}
+
+The resolution is limited by the noise in the voltage amplified.
+
+Typical [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio" >}}) of voltage amplifiers is \\(100dB = 10^{5}\\).
+Thus, for a piezoelectric stack with a displacement \\(L\\), the resolution will be
+
+\begin{equation}
+  r \approx \frac{L}{10^5}
+\end{equation}
+
+For a piezoelectric stack with a displacement of \\(100\,[\mu m]\\), the resolution will be \\(\approx 1\,[nm]\\).
+
+
+### Electrical Capacitance {#electrical-capacitance}
+
+The electrical capacitance gives the maximum voltage that can be used to drive the piezoelectric actuator as a function of frequency (Figure [2](#org3da123f)).
+
+<a id="org3da123f"></a>
+
+{{< figure src="/ox-hugo/piezoelectric_capacitance_voltage_max.png" caption="Figure 2: Maximum sin-wave amplitude as a function of frequency for several piezoelectric capacitance" >}}
+
+
+## Piezoelectric actuator experiencing a mass load {#piezoelectric-actuator-experiencing-a-mass-load}
+
+When the piezoelectric actuator is supporting a payload, it will experience a static deflection due to its finite stiffness \\(\Delta l\_n = \frac{mg}{k\_p}\\), but its stroke will remain unchanged (Figure [3](#orgab6e282)).
+
+<a id="orgab6e282"></a>
+
+{{< figure src="/ox-hugo/piezoelectric_mass_load.png" caption="Figure 3: Motion of a piezoelectric stack actuator under external constant force" >}}
+
+
+## Piezoelectric actuator in contact with a spring load {#piezoelectric-actuator-in-contact-with-a-spring-load}
+
+Then the piezoelectric actuator is in contact with a spring load \\(k\_e\\), its maximum stroke \\(\Delta L\\) is less than its free stroke \\(\Delta L\_f\\) (Figure [4](#orgcf60838)):
+
+\begin{equation}
+  \Delta L = \Delta L\_f \frac{k\_p}{k\_p + k\_e}
+\end{equation}
+
+<a id="orgcf60838"></a>
+
+{{< figure src="/ox-hugo/piezoelectric_spring_load.png" caption="Figure 4: Motion of a piezoelectric stack actuator in contact with a stiff environment" >}}
+
+For piezo actuators, force and displacement are inversely related (Figure [5](#orga8ee6e8)).
+Maximum, or blocked, force (\\(F\_b\\)) occurs when there is no displacement.
+Likewise, at maximum displacement, or free stroke, (\\(\Delta L\_f\\)) no force is generated.
+When an external load is applied, the stiffness of the load (\\(k\_e\\)) determines the displacement (\\(Delta L\_A\\)) and force (\\(\Delta F\_A\\)) that can be produced.
+
+<a id="orga8ee6e8"></a>
+
+{{< figure src="/ox-hugo/piezoelectric_force_displ_relation.png" caption="Figure 5: Relation between the maximum force and displacement" >}}
+
 # Bibliography
 <a class="bibtex-entry" id="fleming10_nanop_system_with_force_feedb">Fleming, A., *Nanopositioning system with force feedback for high-performance tracking and vibration control*, IEEE/ASME Transactions on Mechatronics, *15(3)*, 433–447 (2010).  http://dx.doi.org/10.1109/tmech.2009.2028422</a> [↩](#c823f68dd2a72b9667a61b3c046b4731)
 
@@ -114,6 +179,8 @@ A model of an amplified piezoelectric actuator is described in <sup id="84975085
 
 <a class="bibtex-entry" id="lucinskis16_dynam_charac">Lucinskis, R., & Mangeot, C. (2016). *Dynamic characterization of an amplified piezoelectric actuator*. Retrieved from [](). .</a> [↩](#849750850d9986ed326e74bd3c448d03)
 
+<a class="bibtex-entry" id="ling16_enhan_mathem_model_displ_amplif">Ling, M., Cao, J., Zeng, M., Lin, J., & Inman, D. J., *Enhanced mathematical modeling of the displacement amplification ratio for piezoelectric compliant mechanisms*, Smart Materials and Structures, *25(7)*, 075022 (2016).  http://dx.doi.org/10.1088/0964-1726/25/7/075022</a> [↩](#d9e8b33774f1e65d16bd79114db8ac64)
+
 
 ## Backlinks {#backlinks}