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### Manufacturers {#manufacturers}
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| Manufacturers | Links |
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|---------------------|------------------------------------------------------------------------------------|
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| Cedrat | [link](http://www.cedrat-technologies.com/) |
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| PI | [link](https://www.physikinstrumente.com/en/) |
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| Piezo System | [link](https://www.piezosystem.com/products/piezo%5Factuators/stacktypeactuators/) |
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| Noliac | [link](http://www.noliac.com/) |
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| Thorlabs | [link](https://www.thorlabs.com/newgrouppage9.cfm?objectgroup%5Fid=8700) |
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| PiezoDrive | [link](https://www.piezodrive.com/actuators/) |
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| Mechano Transformer | [link](http://www.mechano-transformer.com/en/products/10.html) |
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| CoreMorrow | [link](http://www.coremorrow.com/en/pro-9-1.html) |
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| Manufacturers | Links |
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|---------------------|----------------------------------------------------------------------------------------------------------------|
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| Cedrat | [link](http://www.cedrat-technologies.com/) |
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| PI | [link](https://www.physikinstrumente.com/en/) |
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| Piezo System | [link](https://www.piezosystem.com/products/piezo%5Factuators/stacktypeactuators/) |
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| Noliac | [link](http://www.noliac.com/) |
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| Thorlabs | [link](https://www.thorlabs.com/newgrouppage9.cfm?objectgroup%5Fid=8700) |
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| PiezoDrive | [link](https://www.piezodrive.com/actuators/) |
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| Mechano Transformer | [link](http://www.mechano-transformer.com/en/products/10.html) |
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| CoreMorrow | [link](http://www.coremorrow.com/en/pro-9-1.html) |
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| PiezoData | [link](https://www.piezodata.com/piezo-stack-actuator-2/) |
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| Queensgate | [link](https://www.nanopositioning.com/product-category/nanopositioning/nanopositioning-actuators-translators) |
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### Model {#model}
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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" >}})).
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A model of a multi-layer monolithic piezoelectric stack actuator is described in ([Fleming 2010](#org7ef2e50)) ([Notes]({{< relref "fleming10_nanop_system_with_force_feedb" >}})).
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Basically, it can be represented by a spring \\(k\_a\\) with the force source \\(F\_a\\) in parallel.
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The relation between the applied voltage \\(V\_a\\) to the generated force \\(F\_a\\) is:
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\\[ F\_a = g\_a V\_a, \quad g\_a = d\_{33} n k\_a \\]
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with:
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- \\(d\_{33}\\) is the piezoelectric strain constant [m/V]
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- \\(n\\) is the number of layers
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- \\(k\_a\\) is the actuator stiffness [N/m]
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## Mechanically Amplified Piezoelectric actuators {#mechanically-amplified-piezoelectric-actuators}
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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, \& Sosnicki, Amplified Piezoelectric Actuators: Static \& Dynamic Applications, {Ferroelectrics}, v(1), 3-14 (2007).">(Frank Claeyssen {\it et al.}, 2007)</a></sup>:
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The Amplified Piezo Actuators principle is presented in ([Claeyssen et al. 2007](#orgc110fa4)):
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> The displacement amplification effect is related in a first approximation to the ratio of the shell long axis length to the short axis height.
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> The flatter is the actuator, the higher is the amplification.
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A model of an amplified piezoelectric actuator is described in <sup id="849750850d9986ed326e74bd3c448d03"><a class="reference-link" href="#lucinskis16_dynam_charac" title="@misc{lucinskis16_dynam_charac,
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author = {R. Lucinskis and C. Mangeot},
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title = {Dynamic Characterization of an amplified piezoelectric
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actuator},
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year = 2016,
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}">(Lucinskis \& Mangeot, 2016)</a></sup>.
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A model of an amplified piezoelectric actuator is described in ([Lucinskis and Mangeot 2016](#orge1d2714)).
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<a id="orgd9b1a8d"></a>
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<a id="org5a5d286"></a>
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{{< 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, \& 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>" >}}
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@@ -57,6 +64,7 @@ A model of an amplified piezoelectric actuator is described in <sup id="84975085
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| Noliac | [link](http://www.noliac.com/products/actuators/amplified-actuators/) |
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| 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) |
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| CoreMorrow | [link](http://www.coremorrow.com/en/pro-13-1.html) |
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| PiezoData | [link](https://www.piezodata.com/piezoelectric-actuator-amplifier/) |
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## Specifications {#specifications}
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@@ -121,7 +129,7 @@ with:
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### Resolution {#resolution}
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The resolution is limited by the noise in the voltage amplified.
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The resolution is limited by the noise in the [Voltage Amplifier]({{< relref "voltage_amplifier" >}}).
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Typical [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio" >}}) of voltage amplifiers is \\(100dB = 10^{5}\\).
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Thus, for a piezoelectric stack with a displacement \\(L\\), the resolution will be
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@@ -135,53 +143,57 @@ For a piezoelectric stack with a displacement of \\(100\,[\mu m]\\), the resolut
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### Electrical Capacitance {#electrical-capacitance}
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The electrical capacitance gives the maximum voltage that can be used to drive the piezoelectric actuator as a function of frequency (Figure [2](#org3da123f)).
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The electrical capacitance may limit the maximum voltage that can be used to drive the piezoelectric actuator as a function of frequency (Figure [2](#orgebd19c2)).
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This is due to the fact that voltage amplifier has a limitation on the deliverable current.
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<a id="org3da123f"></a>
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[Voltage Amplifier]({{< relref "voltage_amplifier" >}}) with high maximum output current should be used if either high bandwidth is wanted or piezoelectric stacks with high capacitance are to be used.
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<a id="orgebd19c2"></a>
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{{< 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" >}}
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## Piezoelectric actuator experiencing a mass load {#piezoelectric-actuator-experiencing-a-mass-load}
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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)).
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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](#orgb64bc37)).
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<a id="orgab6e282"></a>
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<a id="orgb64bc37"></a>
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{{< figure src="/ox-hugo/piezoelectric_mass_load.png" caption="Figure 3: Motion of a piezoelectric stack actuator under external constant force" >}}
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## Piezoelectric actuator in contact with a spring load {#piezoelectric-actuator-in-contact-with-a-spring-load}
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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)):
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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](#org944d760)):
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\begin{equation}
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\Delta L = \Delta L\_f \frac{k\_p}{k\_p + k\_e}
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\end{equation}
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<a id="orgcf60838"></a>
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<a id="org944d760"></a>
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{{< figure src="/ox-hugo/piezoelectric_spring_load.png" caption="Figure 4: Motion of a piezoelectric stack actuator in contact with a stiff environment" >}}
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For piezo actuators, force and displacement are inversely related (Figure [5](#orga8ee6e8)).
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For piezo actuators, force and displacement are inversely related (Figure [5](#org0a60bcb)).
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Maximum, or blocked, force (\\(F\_b\\)) occurs when there is no displacement.
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Likewise, at maximum displacement, or free stroke, (\\(\Delta L\_f\\)) no force is generated.
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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.
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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.
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<a id="orga8ee6e8"></a>
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<a id="org0a60bcb"></a>
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{{< figure src="/ox-hugo/piezoelectric_force_displ_relation.png" caption="Figure 5: Relation between the maximum force and displacement" >}}
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# Bibliography
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<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)
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<a class="bibtex-entry" id="claeyssen07_amplif_piezoel_actuat">Claeyssen, F., Letty, R. L., Barillot, F., & Sosnicki, O., *Amplified piezoelectric actuators: static \& dynamic applications*, Ferroelectrics, *351(1)*, 3–14 (2007). http://dx.doi.org/10.1080/00150190701351865</a> [↩](#5decd2b31c4a9842b80c58b56f96590a)
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## Bibliography {#bibliography}
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<a class="bibtex-entry" id="lucinskis16_dynam_charac">Lucinskis, R., & Mangeot, C. (2016). *Dynamic characterization of an amplified piezoelectric actuator*. Retrieved from [](). .</a> [↩](#849750850d9986ed326e74bd3c448d03)
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<a id="orgc110fa4"></a>Claeyssen, Frank, R. Le Letty, F. Barillot, and O. Sosnicki. 2007. “Amplified Piezoelectric Actuators: Static & Dynamic Applications.” _Ferroelectrics_ 351 (1):3–14. <https://doi.org/10.1080/00150190701351865>.
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<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)
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<a id="org7ef2e50"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” _IEEE/ASME Transactions on Mechatronics_ 15 (3):433–47. <https://doi.org/10.1109/tmech.2009.2028422>.
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<a id="orge1d2714"></a>Lucinskis, R., and C. Mangeot. 2016. “Dynamic Characterization of an Amplified Piezoelectric Actuator.”
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## Backlinks {#backlinks}
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- [Actuators]({{< relref "actuators" >}})
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- [Voltage Amplifier]({{< relref "voltage_amplifier" >}})
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