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content/article/claeyssen07_amplif_piezoel_actuat.md
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@ -9,7 +9,7 @@ Tags
Reference
: ([Claeyssen et al. 2007](#org3151b14))
: <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>
Author(s)
: Claeyssen, F., Letty, R. L., Barillot, F., & Sosnicki, O.
@ -17,24 +17,5 @@ Author(s)
Year
: 2007
The amplified piezo actuator APA is an external leveraged actuator based on a shell used both for the ceramic **pre stress** and for the ceramic **motion magnification**.
It is based on low voltage multilayer piezoelectric ceramics (PZT type).
In static conditions, their free strain \\(S\_p\\) is typically 0.1% when driven at 150 V.
The displacement amplification effect is related in a first approximation to the ratio of the shell long axis length to the short axis height.
The flatter is the actuator, the higher is the amplification.
Piezoceramics can bear large compressive stress but they can not bear tensile forces with a good reliability.
The usual way to solve this limitation consists in prestressing the ceramics by maintaining a compressive stress.
This introduces another force limit: if the internal dynamic forces are above the prestress, the actuator is endangered because of the ceramic goes in tensile stress and also the ceramic stack looses contact with the shell interface.
For many APA actuators, the amplitude of maximal applicable external force is close to half the actuator blocked force.
The maximum dynamic force achievable by the actuator is determined by the prestress.
The prestress design allows a peak force equal to half the blocked force.
## Bibliography {#bibliography}
<a id="org3151b14"></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>.
# Bibliography
<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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content/article/fleming10_nanop_system_with_force_feedb.md
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@ -8,7 +8,7 @@ Tags
: [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Force Sensors]({{< relref "force_sensors" >}})
Reference
: ([Fleming 2010](#org37731c2))
: <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>
Author(s)
: Fleming, A.
@ -16,8 +16,7 @@ Author(s)
Year
: 2010
## Summary {#summary}
Summary:
- The noise generated by a piezoelectric force sensor is much less than a capacitive sensor
- Dynamical model of a piezoelectric stack actuator and piezoelectric force sensor
@ -31,7 +30,7 @@ Year
## Model of a multi-layer monolithic piezoelectric stack actuator {#model-of-a-multi-layer-monolithic-piezoelectric-stack-actuator}
<a id="orgae51a2c"></a>
<a id="org3f4c96b"></a>
{{< figure src="/ox-hugo/fleming10_piezo_model.png" caption="Figure 1: Schematic of a multi-layer monolithic piezoelectric stack actuator model" >}}
@ -113,13 +112,11 @@ As piezoelectric sensors have a capacitive source impedance, the sensor noise de
The current noise density of a general purpose LM833 FET-input op-amp is \\(0.5\ pA/\sqrt{\text{Hz}}\\).
The capacitance of a piezoelectric stack is typically between \\(1 \mu F\\) and \\(100 \mu F\\).
## Bibliography {#bibliography}
<a id="org37731c2"></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>.
# 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)
## Backlinks {#backlinks}
- [Actuators]({{< relref "actuators" >}})
- [Force Sensors]({{< relref "force_sensors" >}})
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}})

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content/book/ewins00_modal.md
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@ -8,7 +8,7 @@ Tags
: [System Identification]({{< relref "system_identification" >}}), [Reference Books]({{< relref "reference_books" >}})
Reference
: ([Ewins 2000](#org84d73f8))
: <sup id="12ff508e9095d666cf081e3c5a6a4cce"><a href="#ewins00_modal" title="Ewins, Modal testing: theory, practice and application, Wiley-Blackwell (2000).">(Ewins, 2000)</a></sup>
Author(s)
: Ewins, D.
@ -141,7 +141,7 @@ The main measurement technique studied are those which will permit to make **dir
The type of test best suited to FRF measurement is shown in figure [fig:modal_analysis_schematic](#fig:modal_analysis_schematic).
<a id="orga193754"></a>
<a id="org76193b4"></a>
{{< figure src="/ox-hugo/ewins00_modal_analysis_schematic.png" caption="Figure 1: Basic components of FRF measurement system" >}}
@ -215,7 +215,7 @@ This assumption allows us to use the circular nature of a modulus/phase polar pl
This process can be **repeated** for each resonance individually until the whole curve has been analyzed.
At this stage, a theoretical regeneration of the FRF is possible using the set of coefficients extracted.
<a id="org37e66c2"></a>
<a id="org128748c"></a>
{{< figure src="/ox-hugo/ewins00_sdof_modulus_phase.png" caption="Figure 2: Curve fit to resonant FRF data" >}}
@ -253,7 +253,7 @@ Theoretical foundations of modal testing are of paramount importance to its succ
The three phases through a typical theoretical vibration analysis progresses are shown on figure [fig:vibration_analysis_procedure](#fig:vibration_analysis_procedure).
Generally, we start with a description of the structure's physical characteristics (mass, stiffness and damping properties), this is referred to as the **Spatial model**.
<a id="org00d3f58"></a>
<a id="org454ea68"></a>
{{< figure src="/ox-hugo/ewins00_vibration_analysis_procedure.png" caption="Figure 3: Theoretical route to vibration analysis" >}}
@ -298,7 +298,7 @@ Three classes of system model will be described:
The basic model for the SDOF system is shown in figure [fig:sdof_model](#fig:sdof_model) where \\(f(t)\\) and \\(x(t)\\) are general time-varying force and displacement response quantities.
The spatial model consists of a **mass** \\(m\\), a **spring** \\(k\\) and (when damped) either a **viscous dashpot** \\(c\\) or **hysteretic damper** \\(d\\).
<a id="org470c5bf"></a>
<a id="org640feed"></a>
{{< figure src="/ox-hugo/ewins00_sdof_model.png" caption="Figure 4: Single degree-of-freedom system" >}}
@ -374,7 +374,7 @@ which is a single mode of vibration with a complex natural frequency having two
The physical significance of these two parts is illustrated in the typical free response plot shown in figure [fig:sdof_response](#fig:sdof_response)
<a id="org169b90c"></a>
<a id="orga99ae3e"></a>
{{< figure src="/ox-hugo/ewins00_sdof_response.png" caption="Figure 5: Oscillatory and decay part" >}}
@ -418,7 +418,7 @@ The damping effect of such a component can conveniently be defined by the ratio
| ![](/ox-hugo/ewins00_material_histeresis.png) | ![](/ox-hugo/ewins00_dry_friction.png) | ![](/ox-hugo/ewins00_viscous_damper.png) |
|-----------------------------------------------|----------------------------------------|------------------------------------------|
| <a id="orgb3a7b8e"></a> Material hysteresis | <a id="org68fe7c2"></a> Dry friction | <a id="org03c75ad"></a> Viscous damper |
| <a id="org54caaf8"></a> Material hysteresis | <a id="org0fc2b44"></a> Dry friction | <a id="org0985c72"></a> Viscous damper |
| height=2cm | height=2cm | height=2cm |
Another common source of energy dissipation in practical structures, is the **friction** which exist in joints between components of the structure.
@ -537,7 +537,7 @@ Bode plot are usually displayed using logarithmic scales as shown on figure [fig
| ![](/ox-hugo/ewins00_bode_receptance.png) | ![](/ox-hugo/ewins00_bode_mobility.png) | ![](/ox-hugo/ewins00_bode_accelerance.png) |
|-------------------------------------------|-----------------------------------------|--------------------------------------------|
| <a id="org4673396"></a> Receptance FRF | <a id="org9f41af5"></a> Mobility FRF | <a id="org6696bcf"></a> Accelerance FRF |
| <a id="orgea747d3"></a> Receptance FRF | <a id="orgc5e3717"></a> Mobility FRF | <a id="orgcf610b2"></a> Accelerance FRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
Each plot can be divided into three regimes:
@ -560,7 +560,7 @@ This type of display is not widely used as we cannot use logarithmic axes (as we
| ![](/ox-hugo/ewins00_plot_receptance_real.png) | ![](/ox-hugo/ewins00_plot_receptance_imag.png) |
|------------------------------------------------|------------------------------------------------|
| <a id="org66926ef"></a> Real part | <a id="orgaf2afdd"></a> Imaginary part |
| <a id="org695538e"></a> Real part | <a id="org95c5960"></a> Imaginary part |
| width=\linewidth | width=\linewidth |
@ -578,7 +578,7 @@ Figure [fig:inverse_frf_mixed](#fig:inverse_frf_mixed) shows an example of a plo
| ![](/ox-hugo/ewins00_inverse_frf_mixed.png) | ![](/ox-hugo/ewins00_inverse_frf_viscous.png) |
|---------------------------------------------|-----------------------------------------------|
| <a id="org84ad953"></a> Mixed | <a id="orgc18e658"></a> Viscous |
| <a id="org9e0909f"></a> Mixed | <a id="orge2690df"></a> Viscous |
| width=\linewidth | width=\linewidth |
@ -595,7 +595,7 @@ The missing information (in this case, the frequency) must be added by identifyi
| ![](/ox-hugo/ewins00_nyquist_receptance_viscous.png) | ![](/ox-hugo/ewins00_nyquist_receptance_structural.png) |
|------------------------------------------------------|---------------------------------------------------------|
| <a id="orgfee48c0"></a> Viscous damping | <a id="org41c7d29"></a> Structural damping |
| <a id="org86b8a60"></a> Viscous damping | <a id="orgb0d3b09"></a> Structural damping |
| width=\linewidth | width=\linewidth |
The Nyquist plot has the particularity of distorting the plot so as to focus on the resonance area.
@ -1103,7 +1103,7 @@ Equally, in a real mode, all parts of the structure pass through their **zero de
While the real mode has the appearance of a **standing wave**, the complex mode is better described as exhibiting **traveling waves** (illustrated on figure [fig:real_complex_modes](#fig:real_complex_modes)).
<a id="org05c0f39"></a>
<a id="org76fb154"></a>
{{< figure src="/ox-hugo/ewins00_real_complex_modes.png" caption="Figure 6: Real and complex mode shapes displays" >}}
@ -1118,7 +1118,7 @@ Note that the almost-real mode shape does not necessarily have vector elements w
| ![](/ox-hugo/ewins00_argand_diagram_a.png) | ![](/ox-hugo/ewins00_argand_diagram_b.png) | ![](/ox-hugo/ewins00_argand_diagram_c.png) |
|--------------------------------------------|--------------------------------------------|-----------------------------------------------|
| <a id="orgc7a8526"></a> Almost-real mode | <a id="orgcd8be0a"></a> Complex Mode | <a id="orgf34a135"></a> Measure of complexity |
| <a id="orgd9e3564"></a> Almost-real mode | <a id="orgeedeefa"></a> Complex Mode | <a id="org2d21384"></a> Measure of complexity |
| width=\linewidth | width=\linewidth | width=\linewidth |
@ -1235,7 +1235,7 @@ On a logarithmic plot, this produces the antiresonance characteristic which refl
| ![](/ox-hugo/ewins00_mobility_frf_mdof_point.png) | ![](/ox-hugo/ewins00_mobility_frf_mdof_transfer.png) |
|---------------------------------------------------|------------------------------------------------------|
| <a id="org04908dc"></a> Point FRF | <a id="orgc9e36d0"></a> Transfer FRF |
| <a id="org464f787"></a> Point FRF | <a id="orgd21bcd3"></a> Transfer FRF |
| width=\linewidth | width=\linewidth |
For the plot in figure [fig:mobility_frf_mdof_transfer](#fig:mobility_frf_mdof_transfer), between the two resonances, the two components have the same sign and they add up, no antiresonance is present.
@ -1260,7 +1260,7 @@ Most mobility plots have this general form as long as the modes are relatively w
This condition is satisfied unless the separation between adjacent natural frequencies is of the same order as, or less than, the modal damping factors, in which case it becomes difficult to distinguish the individual modes.
<a id="org3342d4f"></a>
<a id="orgd6edca6"></a>
{{< figure src="/ox-hugo/ewins00_frf_damped_system.png" caption="Figure 7: Mobility plot of a damped system" >}}
@ -1281,7 +1281,7 @@ The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in figure [
| ![](/ox-hugo/ewins00_nyquist_point.png) | ![](/ox-hugo/ewins00_nyquist_transfer.png) |
|------------------------------------------|---------------------------------------------|
| <a id="org51d6859"></a> Point receptance | <a id="org49ad44a"></a> Transfer receptance |
| <a id="org5dbb609"></a> Point receptance | <a id="orgf225939"></a> Transfer receptance |
| width=\linewidth | width=\linewidth |
In the two figures [fig:nyquist_nonpropdamp_point](#fig:nyquist_nonpropdamp_point) and [fig:nyquist_nonpropdamp_transfer](#fig:nyquist_nonpropdamp_transfer), we show corresponding data for **non-proportional** damping.
@ -1296,7 +1296,7 @@ Now we find that the individual modal circles are no longer "upright" but are **
| ![](/ox-hugo/ewins00_nyquist_nonpropdamp_point.png) | ![](/ox-hugo/ewins00_nyquist_nonpropdamp_transfer.png) |
|-----------------------------------------------------|--------------------------------------------------------|
| <a id="orgbc84787"></a> Point receptance | <a id="org1fde70c"></a> Transfer receptance |
| <a id="orgae9806e"></a> Point receptance | <a id="orgb532a2f"></a> Transfer receptance |
| width=\linewidth | width=\linewidth |
@ -1450,7 +1450,7 @@ Examples of random signals, autocorrelation function and power spectral density
| ![](/ox-hugo/ewins00_random_time.png) | ![](/ox-hugo/ewins00_random_autocorrelation.png) | ![](/ox-hugo/ewins00_random_psd.png) |
|---------------------------------------|--------------------------------------------------|------------------------------------------------|
| <a id="org9b223d2"></a> Time history | <a id="orgf89ee65"></a> Autocorrelation Function | <a id="org839a4fd"></a> Power Spectral Density |
| <a id="org30bff26"></a> Time history | <a id="org7e07ced"></a> Autocorrelation Function | <a id="orgcb31329"></a> Power Spectral Density |
| width=\linewidth | width=\linewidth | width=\linewidth |
A similar concept can be applied to a pair of functions such as \\(f(t)\\) and \\(x(t)\\) to produce **cross correlation** and **cross spectral density** functions.
@ -1547,7 +1547,7 @@ Then in [fig:frf_feedback_model](#fig:frf_feedback_model) is given a more detail
| ![](/ox-hugo/ewins00_frf_siso_model.png) | ![](/ox-hugo/ewins00_frf_feedback_model.png) |
|------------------------------------------|--------------------------------------------------|
| <a id="orgf9a7bf7"></a> Basic SISO model | <a id="org258a6e2"></a> SISO model with feedback |
| <a id="orgcf49de0"></a> Basic SISO model | <a id="orgad8dce0"></a> SISO model with feedback |
| width=\linewidth | width=\linewidth |
In this configuration, it can be seen that there are two feedback mechanisms which apply.
@ -1580,7 +1580,7 @@ We obtain two alternative formulas:
In practical application of both of these formulae, care must be taken to ensure the non-singularity of the spectral density matrix which is to be inverted, and it is in this respect that the former version may be found to be more reliable.
<a id="org00c19fd"></a>
<a id="org2388f52"></a>
{{< figure src="/ox-hugo/ewins00_frf_mimo.png" caption="Figure 8: System for FRF determination via MIMO model" >}}
@ -1852,7 +1852,7 @@ The experimental setup used for mobility measurement contains three major items:
A typical layout for the measurement system is shown on figure [fig:general_frf_measurement_setup](#fig:general_frf_measurement_setup).
<a id="org76e9cb0"></a>
<a id="org1415164"></a>
{{< figure src="/ox-hugo/ewins00_general_frf_measurement_setup.png" caption="Figure 9: General layout of FRF measurement system" >}}
@ -1909,7 +1909,7 @@ This can modify the response of the system in those directions.
In order to avoid that, a drive rod which is stiff in one direction and flexible in the other five directions is attached between the shaker and the structure as shown on figure [fig:shaker_rod](#fig:shaker_rod).
Typical size for the rod are \\(5\\) to \\(\SI{10}{mm}\\) long and \\(\SI{1}{mm}\\) in diameter, if the rod is longer, it may introduce the effect of its own resonances.
<a id="orga841e57"></a>
<a id="orgbf524e6"></a>
{{< figure src="/ox-hugo/ewins00_shaker_rod.png" caption="Figure 10: Exciter attachment and drive rod assembly" >}}
@ -1930,7 +1930,7 @@ Figure [fig:shaker_mount_3](#fig:shaker_mount_3) shows an unsatisfactory setup.
| ![](/ox-hugo/ewins00_shaker_mount_1.png) | ![](/ox-hugo/ewins00_shaker_mount_2.png) | ![](/ox-hugo/ewins00_shaker_mount_3.png) |
|---------------------------------------------|-------------------------------------------------|------------------------------------------|
| <a id="org5ad1e59"></a> Ideal Configuration | <a id="orge10385d"></a> Suspended Configuration | <a id="orgf027a3a"></a> Unsatisfactory |
| <a id="orga9157bf"></a> Ideal Configuration | <a id="org4b90d28"></a> Suspended Configuration | <a id="org3061b55"></a> Unsatisfactory |
| width=\linewidth | width=\linewidth | width=\linewidth |
@ -1948,7 +1948,7 @@ The frequency range which is effectively excited is controlled by the stiffness
When the hammer tip impacts the test structure, this will experience a force pulse as shown on figure [fig:hammer_impulse](#fig:hammer_impulse).
A pulse of this type (half-sine shape) has a frequency content of the form illustrated on figure [fig:hammer_impulse](#fig:hammer_impulse).
<a id="org1e8111f"></a>
<a id="orgdb53d89"></a>
{{< figure src="/ox-hugo/ewins00_hammer_impulse.png" caption="Figure 11: Typical impact force pulse and spectrum" >}}
@ -1979,7 +1979,7 @@ By suitable design, such a material may be incorporated into a device which **in
The force transducer is the simplest type of piezoelectric transducer.
The transmitter force \\(F\\) is applied directly across the crystal, which thus generates a corresponding charge \\(q\\), proportional to \\(F\\) (figure [fig:piezo_force_transducer](#fig:piezo_force_transducer)).
<a id="orge942cb7"></a>
<a id="org93aad2e"></a>
{{< figure src="/ox-hugo/ewins00_piezo_force_transducer.png" caption="Figure 12: Force transducer" >}}
@ -1992,7 +1992,7 @@ In an accelerometer, transduction is indirect and is achieved using a seismic ma
In this configuration, the force exerted on the crystals is the inertia force of the seismic mass (\\(m\ddot{z}\\)).
Thus, so long as the body and the seismic mass move together, the output of the transducer will be proportional to the acceleration of its body \\(x\\).
<a id="orged1c285"></a>
<a id="org84766b5"></a>
{{< figure src="/ox-hugo/ewins00_piezo_accelerometer.png" caption="Figure 13: Compression-type of piezoelectric accelerometer" >}}
@ -2040,7 +2040,7 @@ Shown on figure [fig:transducer_mounting_response](#fig:transducer_mounting_resp
| ![](/ox-hugo/ewins00_transducer_mounting_types.png) | ![](/ox-hugo/ewins00_transducer_mounting_response.png) |
|-----------------------------------------------------|------------------------------------------------------------|
| <a id="org7c446c6"></a> Attachment methods | <a id="org9920b7a"></a> Frequency response characteristics |
| <a id="org796e903"></a> Attachment methods | <a id="org308a233"></a> Frequency response characteristics |
| width=\linewidth | width=\linewidth |
@ -2127,7 +2127,7 @@ Aliasing originates from the discretisation of the originally continuous time hi
With this discretisation process, the **existence of very high frequencies in the original signal may well be misinterpreted if the sampling rate is too slow**.
These high frequencies will be **indistinguishable** from genuine low frequency components as shown on figure [fig:aliasing](#fig:aliasing).
<a id="orgd434c7d"></a>
<a id="orge489af5"></a>
{{< figure src="/ox-hugo/ewins00_aliasing.png" caption="Figure 14: The phenomenon of aliasing. On top: Low-frequency signal, On the bottom: High frequency signal" >}}
@ -2144,7 +2144,7 @@ This is illustrated on figure [fig:effect_aliasing](#fig:effect_aliasing).
| ![](/ox-hugo/ewins00_aliasing_no_distortion.png) | ![](/ox-hugo/ewins00_aliasing_distortion.png) |
|--------------------------------------------------|-----------------------------------------------------|
| <a id="org6412686"></a> True spectrum of signal | <a id="orgd099bc4"></a> Indicated spectrum from DFT |
| <a id="org3c7851f"></a> True spectrum of signal | <a id="orgd31d06c"></a> Indicated spectrum from DFT |
| width=\linewidth | width=\linewidth |
The solution of the problem is to use an **anti-aliasing filter** which subjects the original time signal to a low-pass, sharp cut-off filter.
@ -2165,7 +2165,7 @@ Leakage is a problem which is a direct **consequence of the need to take only a
| ![](/ox-hugo/ewins00_leakage_ok.png) | ![](/ox-hugo/ewins00_leakage_nok.png) |
|--------------------------------------|----------------------------------------|
| <a id="org18c664c"></a> Ideal signal | <a id="org71abe57"></a> Awkward signal |
| <a id="org62f211a"></a> Ideal signal | <a id="orgd4e0fe1"></a> Awkward signal |
| width=\linewidth | width=\linewidth |
The problem is illustrated on figure [fig:leakage](#fig:leakage).
@ -2190,7 +2190,7 @@ Windowing involves the imposition of a prescribed profile on the time signal pri
The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in figure [fig:windowing_examples](#fig:windowing_examples).
<a id="org4e17829"></a>
<a id="orge28ad03"></a>
{{< figure src="/ox-hugo/ewins00_windowing_examples.png" caption="Figure 15: Different types of window. (a) Boxcar, (b) Hanning, (c) Cosine-taper, (d) Exponential" >}}
@ -2211,7 +2211,7 @@ Common filters are: low-pass, high-pass, band-limited, narrow-band, notch.
#### Improving Resolution {#improving-resolution}
<a id="orgc547d0b"></a>
<a id="org81b4f25"></a>
##### Increasing transform size {#increasing-transform-size}
@ -2247,10 +2247,10 @@ If we apply a band-pass filter to the signal, as shown on figure [fig:zoom_bandp
| ![](/ox-hugo/ewins00_zoom_range.png) | ![](/ox-hugo/ewins00_zoom_bandpass.png) |
|------------------------------------------------|------------------------------------------|
| <a id="org78b0c83"></a> Spectrum of the signal | <a id="orge62379a"></a> Band-pass filter |
| <a id="org28ed6ec"></a> Spectrum of the signal | <a id="org8a7e75c"></a> Band-pass filter |
| width=\linewidth | width=\linewidth |
<a id="org9584b09"></a>
<a id="org60b3e9b"></a>
{{< figure src="/ox-hugo/ewins00_zoom_result.png" caption="Figure 16: Effective frequency translation for zoom" >}}
@ -2322,7 +2322,7 @@ This is the traditional method of FRF measurement and involves the use of a swee
It is necessary to check that progress through the frequency range is sufficiently slow to check that steady-state response conditions are attained.
If excessive sweep rate is used, then distortions of the FRF plot are introduced as shown on figure [fig:sweep_distortions](#fig:sweep_distortions).
<a id="orgbf547e6"></a>
<a id="orgeab1f57"></a>
{{< figure src="/ox-hugo/ewins00_sweep_distortions.png" caption="Figure 17: FRF measurements by sine sweep test" >}}
@ -2440,7 +2440,7 @@ It is known that a low coherence can arise in a measurement where the frequency
This is known as a **bias** error and leakage is often the most likely source of low coherence on lightly-damped structures as shown on figure [fig:coherence_resonance](#fig:coherence_resonance).
<a id="orgb273bd2"></a>
<a id="orgb72faa8"></a>
{{< figure src="/ox-hugo/ewins00_coherence_resonance.png" caption="Figure 18: Coherence \\(\gamma^2\\) and FRF estimate \\(H\_1(\omega)\\) for a lightly damped structure" >}}
@ -2483,7 +2483,7 @@ For the chirp and impulse excitations, each individual sample is collected and p
Burst excitation signals consist of short sections of an underlying continuous signal (which may be a sine wave, a sine sweep or a random signal), followed by a period of zero output, resulting in a response which shows a transient build-up followed by a decay (see figure [fig:burst_excitation](#fig:burst_excitation)).
<a id="org4a271bc"></a>
<a id="org681a980"></a>
{{< figure src="/ox-hugo/ewins00_burst_excitation.png" caption="Figure 19: Example of burst excitation and response signals" >}}
@ -2502,7 +2502,7 @@ The chirp consist of a short duration signal which has the form shown in figure
The frequency content of the chirp can be precisely chosen by the starting and finishing frequencies of the sweep.
<a id="org9c55941"></a>
<a id="org632f8cc"></a>
{{< figure src="/ox-hugo/ewins00_chirp_excitation.png" caption="Figure 20: Example of chirp excitation and response signals" >}}
@ -2513,7 +2513,7 @@ The hammer blow produces an input and response as shown in the figure [fig:impul
This and the chirp excitation are very similar in the analysis point of view, the main difference is that the chirp offers the possibility of greater control of both amplitude and frequency content of the input and also permits the input of a greater amount of vibration energy.
<a id="org0ed8171"></a>
<a id="orgdecf769"></a>
{{< figure src="/ox-hugo/ewins00_impulsive_excitation.png" caption="Figure 21: Example of impulsive excitation and response signals" >}}
@ -2523,7 +2523,7 @@ However, it should be recorded that in the region below the first cut-off freque
On some structures, the movement of the structure in response to the hammer blow can be such that it returns and **rebounds** on the hammer tip before the user has had time to move that out of the way.
In such cases, the spectrum of the excitation is seen to have "holes" in it at certain frequencies (figure [fig:double_hits](#fig:double_hits)).
<a id="org6bd77b6"></a>
<a id="orgea279f8"></a>
{{< figure src="/ox-hugo/ewins00_double_hits.png" caption="Figure 22: Double hits time domain and frequency content" >}}
@ -2598,7 +2598,7 @@ Suppose the response parameter is acceleration, then the FRF obtained is inertan
Figure [fig:calibration_setup](#fig:calibration_setup) shows a typical calibration setup.
<a id="org5e0d830"></a>
<a id="org3a6c052"></a>
{{< figure src="/ox-hugo/ewins00_calibration_setup.png" caption="Figure 23: Mass calibration procedure, measurement setup" >}}
@ -2613,7 +2613,7 @@ This is because near resonance, the actual applied force becomes very small and
This same argument applies on a lesser scale as we examine the detail around the attachment to the structure, as shown in figure [fig:mass_cancellation](#fig:mass_cancellation).
<a id="org3d2d464"></a>
<a id="orgf6011aa"></a>
{{< figure src="/ox-hugo/ewins00_mass_cancellation.png" caption="Figure 24: Added mass to be cancelled (crossed area)" >}}
@ -2657,7 +2657,7 @@ It should be noted that the transducer's inertia is also effective not only in t
#### Significance of rotational FRF data {#significance-of-rotational-frf-data}
\\(\SI{50}{\%}\\) of all DOFs are rotations (as opposed to translations) and \\(\SI{75}{\%}\\) of all frequency response functions involve rotation DOFs.
However, it is relatively rare the find reference to methods for measurements of rotational DOFs.
However, it is relatively rate the find reference to methods for measurements of rotational DOFs.
This situation arises from a considerable difficulty which is encountered when trying to measure either rotational responses or excitations and also when trying to apply rotational excitation.
@ -2670,7 +2670,7 @@ There are two problems to be tackled:
The first of these is less difficult and techniques usually use a pair a matched conventional accelerometers placed at a short distance apart on the structure to be measured as shown on figure [fig:rotational_measurement](#fig:rotational_measurement).
<a id="org8a3adca"></a>
<a id="org6c1a993"></a>
{{< figure src="/ox-hugo/ewins00_rotational_measurement.png" caption="Figure 25: Measurement of rotational response" >}}
@ -2691,7 +2691,7 @@ First, a single applied excitation force \\(F\_1\\) corresponds to a simultaneou
Then, the same excitation force is applied at the second position that gives a force \\(F\_0 = F\_2\\) and moment \\(M\_0 = F\_2 l\_2\\).
By adding and subtracting the responses produced by these two separate excitations conditions, we can deduce the translational and rotational responses to the translational force and the rotational moment separately, thus enabling the measurement of all four types of FRF: \\(X/F\\), \\(\Theta/F\\), \\(X/M\\) and \\(\Theta/M\\).
<a id="orgd9d3238"></a>
<a id="org19d9418"></a>
{{< figure src="/ox-hugo/ewins00_rotational_excitation.png" caption="Figure 26: Application of moment excitation" >}}
@ -3031,7 +3031,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
| ![](/ox-hugo/ewins00_PRF_numerical_FRF.png) | ![](/ox-hugo/ewins00_PRF_numerical_svd.png) | ![](/ox-hugo/ewins00_PRF_numerical_PRF.png) |
|---------------------------------------------|---------------------------------------------|---------------------------------------------|
| <a id="org27a7bd2"></a> FRF | <a id="org0725348"></a> Singular Values | <a id="orgcc8943d"></a> PRF |
| <a id="org911bfc8"></a> FRF | <a id="org60f84fb"></a> Singular Values | <a id="orgdf8522b"></a> PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
<a id="table--fig:PRF-measured"></a>
@ -3042,7 +3042,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
| ![](/ox-hugo/ewins00_PRF_measured_FRF.png) | ![](/ox-hugo/ewins00_PRF_measured_svd.png) | ![](/ox-hugo/ewins00_PRF_measured_PRF.png) |
|--------------------------------------------|--------------------------------------------|--------------------------------------------|
| <a id="orgad6d59c"></a> FRF | <a id="orged00ce0"></a> Singular Values | <a id="orga025551"></a> PRF |
| <a id="org3d1c696"></a> FRF | <a id="orgeb81dac"></a> Singular Values | <a id="orgc25aeb3"></a> PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
@ -3084,7 +3084,7 @@ Associated with the CMIF values at each natural frequency \\(\omega\_r\\) are tw
- the left singular vector \\(\\{U(\omega\_r)\\}\_1\\) which approximates the **mode shape** of that mode
- the right singular vector \\(\\{V(\omega\_r)\\}\_1\\) which represents the approximate **force pattern necessary to generate a response on that mode only**
<a id="org80fd4e8"></a>
<a id="org5f7cb1f"></a>
{{< figure src="/ox-hugo/ewins00_mifs.png" caption="Figure 27: Complex Mode Indicator Function (CMIF)" >}}
@ -3179,7 +3179,7 @@ The peak-picking method is applied as follows (illustrated on figure [fig:peak_a
It must be noted that the estimates of both damping and modal constant depend heavily on the accuracy of the maximum FRF level \\(|\hat{H}|\\) which is difficult to measure with great accuracy, especially for lightly damped systems.
Only real modal constants and thus real modes can be deduced by this method.
<a id="org7d69374"></a>
<a id="org0d4b46a"></a>
{{< figure src="/ox-hugo/ewins00_peak_amplitude.png" caption="Figure 28: Peak Amplitude method of modal analysis" >}}
@ -3214,7 +3214,7 @@ A plot of the quantity \\(\alpha(\omega)\\) is given in figure [fig:modal_circle
| ![](/ox-hugo/ewins00_modal_circle.png) | ![](/ox-hugo/ewins00_modal_circle_bis.png) |
|----------------------------------------|--------------------------------------------------------------------|
| <a id="org290c571"></a> Properties | <a id="orgc059e31"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
| <a id="org187efdc"></a> Properties | <a id="org0e24b72"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
| width=\linewidth | width=\linewidth |
For any frequency \\(\omega\\), we have the following relationship:
@ -3328,7 +3328,7 @@ The sequence is:
5. **Determine modal constant modulus and argument**.
The magnitude and argument of the modal constant is determined from the diameter of the circle and from its orientation relative to the Real and Imaginary axis.
<a id="orga4f6a8d"></a>
<a id="org379e1a2"></a>
{{< figure src="/ox-hugo/ewins00_circle_fit_natural_frequency.png" caption="Figure 29: Location of natural frequency for a Circle-fit modal analysis" >}}
@ -3453,7 +3453,7 @@ However, by the inclusion of two simple extra terms (the "**residuals**"), the m
| ![](/ox-hugo/ewins00_residual_without.png) | ![](/ox-hugo/ewins00_residual_with.png) |
|--------------------------------------------|-----------------------------------------|
| <a id="orgb0a10e7"></a> without residual | <a id="org7168563"></a> with residuals |
| <a id="orge96a388"></a> without residual | <a id="org92a8b32"></a> with residuals |
| width=\linewidth | width=\linewidth |
If we regenerate an FRF curve from the modal parameters we have extracted from the measured data, we shall use a formula of the type
@ -3484,7 +3484,7 @@ The three terms corresponds to:
These three terms are illustrated on figure [fig:low_medium_high_modes](#fig:low_medium_high_modes).
<a id="org3ba03ab"></a>
<a id="org745f0a4"></a>
{{< figure src="/ox-hugo/ewins00_low_medium_high_modes.png" caption="Figure 30: Numerical simulation of contribution of low, medium and high frequency modes" >}}
@ -3785,7 +3785,7 @@ As an example, a set of mobilities measured are shown individually in figure [fi
| ![](/ox-hugo/ewins00_composite_raw.png) | ![](/ox-hugo/ewins00_composite_sum.png) |
|-------------------------------------------|-----------------------------------------|
| <a id="orgf1eae63"></a> Individual curves | <a id="org156012b"></a> Composite curve |
| <a id="org3f9a0d6"></a> Individual curves | <a id="org9ebc973"></a> Composite curve |
| width=\linewidth | width=\linewidth |
The global analysis methods have the disadvantages first, that the computation power required is high and second that there may be valid reasons why the various FRF curves exhibit slight differences in their characteristics and it may not always be appropriate to average them.
@ -4332,7 +4332,7 @@ Measured coordinates of the test structure are first linked as shown on figure [
Then, the grid of measured coordinate points is redrawn on the same plot but this time displaced by an amount proportional to the corresponding element in the mode shape vector as shown on figure [fig:static_display](#fig:static_display) (b).
The elements in the vector are scaled according the normalization process used (usually mass-normalized), and their absolute magnitudes have no particular significance.
<a id="orgaffacf3"></a>
<a id="orge0d2fb3"></a>
{{< figure src="/ox-hugo/ewins00_static_display.png" caption="Figure 31: Static display of modes shapes. (a) basic grid (b) single-frame deflection pattern (c) multiple-frame deflection pattern (d) complex mode (e) Argand diagram - quasi-real mode (f) Argand diagram - complex mode" >}}
@ -4377,7 +4377,7 @@ If we consider the first six modes of the beam, whose mode shapes are sketched i
All the higher modes will be indistinguishable from these first few.
This is a well known problem of **spatial aliasing**.
<a id="org1952587"></a>
<a id="org5c16ec7"></a>
{{< figure src="/ox-hugo/ewins00_beam_modes.png" caption="Figure 32: Misinterpretation of mode shapes by spatial aliasing" >}}
@ -4440,7 +4440,7 @@ The inclusion of these two additional terms (obtained here only after measuring
| ![](/ox-hugo/ewins00_H22_without_residual.png) | ![](/ox-hugo/ewins00_H22_with_residual.png) |
|--------------------------------------------------------|-----------------------------------------------------------|
| <a id="org7d9a13a"></a> Using measured modal data only | <a id="orgae3b985"></a> After inclusion of residual terms |
| <a id="orgee3fc43"></a> Using measured modal data only | <a id="org959e2d5"></a> After inclusion of residual terms |
| width=\linewidth | width=\linewidth |
The appropriate expression for a "correct" response model, derived via a set of modal properties is thus
@ -4495,7 +4495,7 @@ If the transmissibility is measured during a modal test which has a single excit
In general, the transmissibility **depends significantly on the excitation point** (\\({}\_iT\_{jk}(\omega) \neq {}\_qT\_{jk}(\omega)\\) where \\(q\\) is a different DOF than \\(i\\)) and it is shown on figure [fig:transmissibility_plots](#fig:transmissibility_plots).
This may explain why transmissibilities are not widely used in modal analysis.
<a id="orgd4fb092"></a>
<a id="orgb69dd65"></a>
{{< figure src="/ox-hugo/ewins00_transmissibility_plots.png" caption="Figure 33: Transmissibility plots" >}}
@ -4516,7 +4516,7 @@ The fact that the excitation force is not measured is responsible for the lack o
| ![](/ox-hugo/ewins00_conventional_modal_test_setup.png) | ![](/ox-hugo/ewins00_base_excitation_modal_setup.png) |
|---------------------------------------------------------|-------------------------------------------------------|
| <a id="org1dc5bf9"></a> Conventional modal test setup | <a id="orge8f2893"></a> Base excitation setup |
| <a id="orgfb8d62b"></a> Conventional modal test setup | <a id="orgb803ff7"></a> Base excitation setup |
| height=4cm | height=4cm |
@ -4556,7 +4556,5 @@ This is accomplished using the above equation in the form:
Because the rank of each pseudo matrix is less than its order, it cannot be inverted and so we are unable to construct stiffness or mass matrix from this approach.
## Bibliography {#bibliography}
<a id="org84d73f8"></a>Ewins, DJ. 2000. _Modal Testing: Theory, Practice and Application_. _Research Studies Pre, 2nd Ed., ISBN-13_. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.
# Bibliography
<a id="ewins00_modal"></a>Ewins, D., *Modal testing: theory, practice and application* (2000), Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell. [↩](#12ff508e9095d666cf081e3c5a6a4cce)

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content/book/fleming14_desig_model_contr_nanop_system.md
View File

@ -9,7 +9,7 @@ Tags
Reference
: ([Fleming and Leang 2014](#org611ad6b))
: <sup id="1851788e0c4aa5b06afe3362c73ea5eb"><a href="#fleming14_desig_model_contr_nanop_system" title="Andrew Fleming \&amp; Kam Leang, Design, Modeling and Control of Nanopositioning Systems, Springer International Publishing (2014).">(Andrew Fleming \& Kam Leang, 2014)</a></sup>
Author(s)
: Fleming, A. J., & Leang, K. K.
@ -17,935 +17,5 @@ Author(s)
Year
: 2014
## 1 Introduction {#1-introduction}
### 1.1 Introduction to Nanotechnology {#1-dot-1-introduction-to-nanotechnology}
### 1.2 Introduction to Nanopositioning {#1-dot-2-introduction-to-nanopositioning}
### 1.3 Scanning Probe Microscopy {#1-dot-3-scanning-probe-microscopy}
### 1.4 Challenges with Nanopositioning Systems {#1-dot-4-challenges-with-nanopositioning-systems}
#### 1.4.1 Hysteresis {#1-dot-4-dot-1-hysteresis}
#### 1.4.2 Creep {#1-dot-4-dot-2-creep}
#### 1.4.3 Thermal Drift {#1-dot-4-dot-3-thermal-drift}
#### 1.4.4 Mechanical Resonance {#1-dot-4-dot-4-mechanical-resonance}
### 1.5 Control of Nanopositioning Systems {#1-dot-5-control-of-nanopositioning-systems}
#### 1.5.1 Feedback Control {#1-dot-5-dot-1-feedback-control}
#### 1.5.2 Feedforward Control {#1-dot-5-dot-2-feedforward-control}
### 1.6 Book Summary {#1-dot-6-book-summary}
#### 1.6.1 Assumed Knowledge {#1-dot-6-dot-1-assumed-knowledge}
#### 1.6.2 Content Summary {#1-dot-6-dot-2-content-summary}
### References {#references}
## 2 Piezoelectric Transducers {#2-piezoelectric-transducers}
### 2.1 The Piezoelectric Effect {#2-dot-1-the-piezoelectric-effect}
### 2.2 Piezoelectric Compositions {#2-dot-2-piezoelectric-compositions}
### 2.3 Manufacturing Piezoelectric Ceramics {#2-dot-3-manufacturing-piezoelectric-ceramics}
### 2.4 Piezoelectric Transducers {#2-dot-4-piezoelectric-transducers}
### 2.5 Application Considerations {#2-dot-5-application-considerations}
#### 2.5.1 Mounting {#2-dot-5-dot-1-mounting}
#### 2.5.2 Stroke Versus Force {#2-dot-5-dot-2-stroke-versus-force}
#### 2.5.3 Preload and Flexures {#2-dot-5-dot-3-preload-and-flexures}
#### 2.5.4 Electrical Considerations {#2-dot-5-dot-4-electrical-considerations}
#### 2.5.5 Self-Heating Considerations {#2-dot-5-dot-5-self-heating-considerations}
### 2.6 Response of Piezoelectric Actuators {#2-dot-6-response-of-piezoelectric-actuators}
#### 2.6.1 Hysteresis {#2-dot-6-dot-1-hysteresis}
#### 2.6.2 Creep {#2-dot-6-dot-2-creep}
#### 2.6.3 Temperature Dependence {#2-dot-6-dot-3-temperature-dependence}
#### 2.6.4 Vibrational Dynamics {#2-dot-6-dot-4-vibrational-dynamics}
#### 2.6.5 Electrical Bandwidth {#2-dot-6-dot-5-electrical-bandwidth}
### 2.7 Modeling Creep and Vibration in Piezoelectric Actuators {#2-dot-7-modeling-creep-and-vibration-in-piezoelectric-actuators}
### 2.8 Chapter Summary {#2-dot-8-chapter-summary}
### References {#references}
## 3 Types of Nanopositioners {#3-types-of-nanopositioners}
### 3.1 Piezoelectric Tube Nanopositioners {#3-dot-1-piezoelectric-tube-nanopositioners}
#### 3.1.1 63mm Piezoelectric Tube {#3-dot-1-dot-1-63mm-piezoelectric-tube}
#### 3.1.2 40mm Piezoelectric Tube Nanopositioner {#3-dot-1-dot-2-40mm-piezoelectric-tube-nanopositioner}
### 3.2 Piezoelectric Stack Nanopositioners {#3-dot-2-piezoelectric-stack-nanopositioners}
#### 3.2.1 Phyisk Instrumente P-734 Nanopositioner {#3-dot-2-dot-1-phyisk-instrumente-p-734-nanopositioner}
#### 3.2.2 Phyisk Instrumente P-733.3DD Nanopositioner {#3-dot-2-dot-2-phyisk-instrumente-p-733-dot-3dd-nanopositioner}
#### 3.2.3 Vertical Nanopositioners {#3-dot-2-dot-3-vertical-nanopositioners}
#### 3.2.4 Rotational Nanopositioners {#3-dot-2-dot-4-rotational-nanopositioners}
#### 3.2.5 Low Temperature and UHV Nanopositioners {#3-dot-2-dot-5-low-temperature-and-uhv-nanopositioners}
#### 3.2.6 Tilting Nanopositioners {#3-dot-2-dot-6-tilting-nanopositioners}
#### 3.2.7 Optical Objective Nanopositioners {#3-dot-2-dot-7-optical-objective-nanopositioners}
### References {#references}
## 4 Mechanical Design: Flexure-Based Nanopositioners {#4-mechanical-design-flexure-based-nanopositioners}
### 4.1 Introduction {#4-dot-1-introduction}
### 4.2 Operating Environment {#4-dot-2-operating-environment}
### 4.3 Methods for Actuation {#4-dot-3-methods-for-actuation}
### 4.4 Flexure Hinges {#4-dot-4-flexure-hinges}
#### 4.4.1 Introduction {#4-dot-4-dot-1-introduction}
#### 4.4.2 Types of Flexures {#4-dot-4-dot-2-types-of-flexures}
#### 4.4.3 Flexure Hinge Compliance Equations {#4-dot-4-dot-3-flexure-hinge-compliance-equations}
#### 4.4.4 Stiff Out-of-Plane Flexure Designs {#4-dot-4-dot-4-stiff-out-of-plane-flexure-designs}
#### 4.4.5 Failure Considerations {#4-dot-4-dot-5-failure-considerations}
#### 4.4.6 Finite Element Approach for Flexure Design {#4-dot-4-dot-6-finite-element-approach-for-flexure-design}
### 4.5 Material Considerations {#4-dot-5-material-considerations}
#### 4.5.1 Materials for Flexure and Platform Design {#4-dot-5-dot-1-materials-for-flexure-and-platform-design}
#### 4.5.2 Thermal Stability of Materials {#4-dot-5-dot-2-thermal-stability-of-materials}
### 4.6 Manufacturing Techniques {#4-dot-6-manufacturing-techniques}
### 4.7 Design Example: A High-Speed Serial-Kinematic Nanopositioner {#4-dot-7-design-example-a-high-speed-serial-kinematic-nanopositioner}
#### 4.7.1 State-of-the-Art Designs {#4-dot-7-dot-1-state-of-the-art-designs}
#### 4.7.2 Tradeoffs and Limitations in Speed {#4-dot-7-dot-2-tradeoffs-and-limitations-in-speed}
#### 4.7.3 Serial- Versus Parallel-Kinematic Configurations {#4-dot-7-dot-3-serial-versus-parallel-kinematic-configurations}
#### 4.7.4 Piezoactuator Considerations {#4-dot-7-dot-4-piezoactuator-considerations}
#### 4.7.5 Preloading Piezo-Stack Actuators {#4-dot-7-dot-5-preloading-piezo-stack-actuators}
#### 4.7.6 Flexure Design for Lateral Positioning {#4-dot-7-dot-6-flexure-design-for-lateral-positioning}
#### 4.7.7 Design of Vertical Stage {#4-dot-7-dot-7-design-of-vertical-stage}
#### 4.7.8 Fabrication and Assembly {#4-dot-7-dot-8-fabrication-and-assembly}
#### 4.7.9 Drive Electronics {#4-dot-7-dot-9-drive-electronics}
#### 4.7.10 Experimental Results {#4-dot-7-dot-10-experimental-results}
### 4.8 Chapter Summary {#4-dot-8-chapter-summary}
### References {#references}
## 5 Position Sensors {#5-position-sensors}
### 5.1 Introduction {#5-dot-1-introduction}
### 5.2 Sensor Characteristics {#5-dot-2-sensor-characteristics}
#### 5.2.1 Calibration and Nonlinearity {#5-dot-2-dot-1-calibration-and-nonlinearity}
#### 5.2.2 Drift and Stability {#5-dot-2-dot-2-drift-and-stability}
#### 5.2.3 Bandwidth {#5-dot-2-dot-3-bandwidth}
#### 5.2.4 Noise {#5-dot-2-dot-4-noise}
#### 5.2.5 Resolution {#5-dot-2-dot-5-resolution}
#### 5.2.6 Combining Errors {#5-dot-2-dot-6-combining-errors}
#### 5.2.7 Metrological Traceability {#5-dot-2-dot-7-metrological-traceability}
### 5.3 Nanometer Position Sensors {#5-dot-3-nanometer-position-sensors}
#### 5.3.1 Resistive Strain Sensors {#5-dot-3-dot-1-resistive-strain-sensors}
#### 5.3.2 Piezoresistive Strain Sensors {#5-dot-3-dot-2-piezoresistive-strain-sensors}
#### 5.3.3 Piezoelectric Strain Sensors {#5-dot-3-dot-3-piezoelectric-strain-sensors}
#### 5.3.4 Capacitive Sensors {#5-dot-3-dot-4-capacitive-sensors}
#### 5.3.5 MEMs Capacitive and Thermal Sensors {#5-dot-3-dot-5-mems-capacitive-and-thermal-sensors}
#### 5.3.6 Eddy-Current Sensors {#5-dot-3-dot-6-eddy-current-sensors}
#### 5.3.7 Linear Variable Displacement Transformers {#5-dot-3-dot-7-linear-variable-displacement-transformers}
#### 5.3.8 Laser Interferometers {#5-dot-3-dot-8-laser-interferometers}
#### 5.3.9 Linear Encoders {#5-dot-3-dot-9-linear-encoders}
### 5.4 Comparison and Summary {#5-dot-4-comparison-and-summary}
### 5.5 Outlook and Future Requirements {#5-dot-5-outlook-and-future-requirements}
### References {#references}
## 6 Shunt Control {#6-shunt-control}
### 6.1 Introduction {#6-dot-1-introduction}
### 6.2 Shunt Circuit Modeling {#6-dot-2-shunt-circuit-modeling}
#### 6.2.1 Open-Loop {#6-dot-2-dot-1-open-loop}
#### 6.2.2 Shunt Damping {#6-dot-2-dot-2-shunt-damping}
### 6.3 Implementation {#6-dot-3-implementation}
### 6.4 Experimental Results {#6-dot-4-experimental-results}
#### 6.4.1 Tube Dynamics {#6-dot-4-dot-1-tube-dynamics}
#### 6.4.2 Amplifier Performance {#6-dot-4-dot-2-amplifier-performance}
#### 6.4.3 Shunt Damping Performance {#6-dot-4-dot-3-shunt-damping-performance}
### 6.5 Chapter Summary {#6-dot-5-chapter-summary}
### References {#references}
## 7 Feedback Control {#7-feedback-control}
### 7.1 Introduction {#7-dot-1-introduction}
### 7.2 Experimental Setup {#7-dot-2-experimental-setup}
### 7.3 PI Control {#7-dot-3-pi-control}
### 7.4 PI Control with Notch Filters {#7-dot-4-pi-control-with-notch-filters}
### 7.5 PI Control with IRC Damping {#7-dot-5-pi-control-with-irc-damping}
### 7.6 Performance Comparison {#7-dot-6-performance-comparison}
### 7.7 Noise and Resolution {#7-dot-7-noise-and-resolution}
### 7.8 Analog Implementation {#7-dot-8-analog-implementation}
### 7.9 Application to AFM Imaging {#7-dot-9-application-to-afm-imaging}
### 7.10 Repetitive Control {#7-dot-10-repetitive-control}
#### 7.10.1 Introduction {#7-dot-10-dot-1-introduction}
#### 7.10.2 Repetitive Control Concept and Stability Considerations {#7-dot-10-dot-2-repetitive-control-concept-and-stability-considerations}
#### 7.10.3 Dual-Stage Repetitive Control {#7-dot-10-dot-3-dual-stage-repetitive-control}
#### 7.10.4 Handling Hysteresis {#7-dot-10-dot-4-handling-hysteresis}
#### 7.10.5 Design and Implementation {#7-dot-10-dot-5-design-and-implementation}
#### 7.10.6 Experimental Results and Discussion {#7-dot-10-dot-6-experimental-results-and-discussion}
### 7.11 Summary {#7-dot-11-summary}
### References {#references}
## 8 Force Feedback Control {#8-force-feedback-control}
### 8.1 Introduction {#8-dot-1-introduction}
### 8.2 Modeling {#8-dot-2-modeling}
#### 8.2.1 Actuator Dynamics {#8-dot-2-dot-1-actuator-dynamics}
#### 8.2.2 Sensor Dynamics {#8-dot-2-dot-2-sensor-dynamics}
#### 8.2.3 Sensor Noise {#8-dot-2-dot-3-sensor-noise}
#### 8.2.4 Mechanical Dynamics {#8-dot-2-dot-4-mechanical-dynamics}
#### 8.2.5 System Properties {#8-dot-2-dot-5-system-properties}
#### 8.2.6 Example System {#8-dot-2-dot-6-example-system}
### 8.3 Damping Control {#8-dot-3-damping-control}
### 8.4 Tracking Control {#8-dot-4-tracking-control}
#### 8.4.1 Relationship Between Force and Displacement {#8-dot-4-dot-1-relationship-between-force-and-displacement}
#### 8.4.2 Integral Displacement Feedback {#8-dot-4-dot-2-integral-displacement-feedback}
#### 8.4.3 Direct Tracking Control {#8-dot-4-dot-3-direct-tracking-control}
#### 8.4.4 Dual Sensor Feedback {#8-dot-4-dot-4-dual-sensor-feedback}
#### 8.4.5 Low Frequency Bypass {#8-dot-4-dot-5-low-frequency-bypass}
#### 8.4.6 Feedforward Inputs {#8-dot-4-dot-6-feedforward-inputs}
#### 8.4.7 Higher-Order Modes {#8-dot-4-dot-7-higher-order-modes}
### 8.5 Experimental Results {#8-dot-5-experimental-results}
#### 8.5.1 Experimental Nanopositioner {#8-dot-5-dot-1-experimental-nanopositioner}
#### 8.5.2 Actuators and Force Sensors {#8-dot-5-dot-2-actuators-and-force-sensors}
#### 8.5.3 Control Design {#8-dot-5-dot-3-control-design}
#### 8.5.4 Noise Performance {#8-dot-5-dot-4-noise-performance}
### 8.6 Chapter Summary {#8-dot-6-chapter-summary}
### References {#references}
## 9 Feedforward Control {#9-feedforward-control}
### 9.1 Why Feedforward? {#9-dot-1-why-feedforward}
### 9.2 Modeling for Feedforward Control {#9-dot-2-modeling-for-feedforward-control}
### 9.3 Feedforward Control of Dynamics and Hysteresis {#9-dot-3-feedforward-control-of-dynamics-and-hysteresis}
#### 9.3.1 Simple DC-Gain Feedforward Control {#9-dot-3-dot-1-simple-dc-gain-feedforward-control}
#### 9.3.2 An Inversion-Based Feedforward Approach for Linear Dynamics {#9-dot-3-dot-2-an-inversion-based-feedforward-approach-for-linear-dynamics}
#### 9.3.3 Frequency-Weighted Inversion: The Optimal Inverse {#9-dot-3-dot-3-frequency-weighted-inversion-the-optimal-inverse}
#### 9.3.4 Application to AFM Imaging {#9-dot-3-dot-4-application-to-afm-imaging}
### 9.4 Feedforward and Feedback Control {#9-dot-4-feedforward-and-feedback-control}
#### 9.4.1 Application to AFM Imaging {#9-dot-4-dot-1-application-to-afm-imaging}
### 9.5 Iterative Feedforward Control {#9-dot-5-iterative-feedforward-control}
#### 9.5.1 The ILC Problem {#9-dot-5-dot-1-the-ilc-problem}
#### 9.5.2 Model-Based ILC {#9-dot-5-dot-2-model-based-ilc}
#### 9.5.3 Nonlinear ILC {#9-dot-5-dot-3-nonlinear-ilc}
#### 9.5.4 Conclusions {#9-dot-5-dot-4-conclusions}
### References {#references}
## 10 Command Shaping {#10-command-shaping}
### 10.1 Introduction {#10-dot-1-introduction}
#### 10.1.1 Background {#10-dot-1-dot-1-background}
#### 10.1.2 The Optimal Periodic Input {#10-dot-1-dot-2-the-optimal-periodic-input}
### 10.2 Signal Optimization {#10-dot-2-signal-optimization}
### 10.3 Frequency Domain Cost Functions {#10-dot-3-frequency-domain-cost-functions}
#### 10.3.1 Background: Discrete Fourier Series {#10-dot-3-dot-1-background-discrete-fourier-series}
#### 10.3.2 Minimizing Signal Power {#10-dot-3-dot-2-minimizing-signal-power}
#### 10.3.3 Minimizing Frequency Weighted Power {#10-dot-3-dot-3-minimizing-frequency-weighted-power}
#### 10.3.4 Minimizing Velocity and Acceleration {#10-dot-3-dot-4-minimizing-velocity-and-acceleration}
#### 10.3.5 Single-Sided Frequency Domain Calculations {#10-dot-3-dot-5-single-sided-frequency-domain-calculations}
### 10.4 Time Domain Cost Function {#10-dot-4-time-domain-cost-function}
#### 10.4.1 Minimum Velocity {#10-dot-4-dot-1-minimum-velocity}
#### 10.4.2 Minimum Acceleration {#10-dot-4-dot-2-minimum-acceleration}
#### 10.4.3 Frequency Weighted Objectives {#10-dot-4-dot-3-frequency-weighted-objectives}
### 10.5 Application to Scan Generation {#10-dot-5-application-to-scan-generation}
#### 10.5.1 Choosing β and K {#10-dot-5-dot-1-choosing-β-and-k}
#### 10.5.2 Improving Feedback and Feedforward Controllers {#10-dot-5-dot-2-improving-feedback-and-feedforward-controllers}
### 10.6 Comparison to Other Techniques {#10-dot-6-comparison-to-other-techniques}
### 10.7 Experimental Application {#10-dot-7-experimental-application}
### 10.8 Chapter Summary {#10-dot-8-chapter-summary}
### References {#references}
## 11 Hysteresis Modeling and Control {#11-hysteresis-modeling-and-control}
### 11.1 Introduction {#11-dot-1-introduction}
### 11.2 Modeling Hysteresis {#11-dot-2-modeling-hysteresis}
#### 11.2.1 Simple Polynomial Model {#11-dot-2-dot-1-simple-polynomial-model}
#### 11.2.2 Maxwell Slip Model {#11-dot-2-dot-2-maxwell-slip-model}
#### 11.2.3 Duhem Model {#11-dot-2-dot-3-duhem-model}
#### 11.2.4 Preisach Model {#11-dot-2-dot-4-preisach-model}
#### 11.2.5 Classical Prandlt-Ishlinksii Model {#11-dot-2-dot-5-classical-prandlt-ishlinksii-model}
### 11.3 Feedforward Hysteresis Compensation {#11-dot-3-feedforward-hysteresis-compensation}
#### 11.3.1 Feedforward Control Using the Presiach Model {#11-dot-3-dot-1-feedforward-control-using-the-presiach-model}
#### 11.3.2 Feedforward Control Using the Prandlt-Ishlinksii Model {#11-dot-3-dot-2-feedforward-control-using-the-prandlt-ishlinksii-model}
### 11.4 Chapter Summary {#11-dot-4-chapter-summary}
### References {#references}
## 12 Charge Drives {#12-charge-drives}
### 12.1 Introduction {#12-dot-1-introduction}
### 12.2 Charge Drives {#12-dot-2-charge-drives}
### 12.3 Application to Piezoelectric Stack Nanopositioners {#12-dot-3-application-to-piezoelectric-stack-nanopositioners}
### 12.4 Application to Piezoelectric Tube Nanopositioners {#12-dot-4-application-to-piezoelectric-tube-nanopositioners}
### 12.5 Alternative Electrode Configurations {#12-dot-5-alternative-electrode-configurations}
#### 12.5.1 Grounded Internal Electrode {#12-dot-5-dot-1-grounded-internal-electrode}
#### 12.5.2 Quartered Internal Electrode {#12-dot-5-dot-2-quartered-internal-electrode}
### 12.6 Charge Versus Voltage {#12-dot-6-charge-versus-voltage}
#### 12.6.1 Advantages {#12-dot-6-dot-1-advantages}
#### 12.6.2 Disadvantages {#12-dot-6-dot-2-disadvantages}
### 12.7 Impact on Closed-Loop Control {#12-dot-7-impact-on-closed-loop-control}
### 12.8 Chapter Summary {#12-dot-8-chapter-summary}
### References {#references}
## 13 Noise in Nanopositioning Systems {#13-noise-in-nanopositioning-systems}
### 13.1 Introduction {#13-dot-1-introduction}
### 13.2 Review of Random Processes {#13-dot-2-review-of-random-processes}
#### 13.2.1 Probability Distributions {#13-dot-2-dot-1-probability-distributions}
#### 13.2.2 Expected Value, Moments, Variance, and RMS {#13-dot-2-dot-2-expected-value-moments-variance-and-rms}
#### 13.2.3 Gaussian Random Variables {#13-dot-2-dot-3-gaussian-random-variables}
#### 13.2.4 Continuous Random Processes {#13-dot-2-dot-4-continuous-random-processes}
#### 13.2.5 Joint Density Functions and Stationarity {#13-dot-2-dot-5-joint-density-functions-and-stationarity}
#### 13.2.6 Correlation Functions {#13-dot-2-dot-6-correlation-functions}
#### 13.2.7 Gaussian Random Processes {#13-dot-2-dot-7-gaussian-random-processes}
#### 13.2.8 Power Spectral Density {#13-dot-2-dot-8-power-spectral-density}
#### 13.2.9 Filtered Random Processes {#13-dot-2-dot-9-filtered-random-processes}
#### 13.2.10 White Noise {#13-dot-2-dot-10-white-noise}
#### 13.2.11 Spectral Density in V/sqrtHz {#13-dot-2-dot-11-spectral-density-in-v-sqrthz}
#### 13.2.12 Single- and Double-Sided Spectra {#13-dot-2-dot-12-single-and-double-sided-spectra}
### 13.3 Resolution and Noise {#13-dot-3-resolution-and-noise}
### 13.4 Sources of Nanopositioning Noise {#13-dot-4-sources-of-nanopositioning-noise}
#### 13.4.1 Sensor Noise {#13-dot-4-dot-1-sensor-noise}
#### 13.4.2 External Noise {#13-dot-4-dot-2-external-noise}
#### 13.4.3 Amplifier Noise {#13-dot-4-dot-3-amplifier-noise}
### 13.5 Closed-Loop Position Noise {#13-dot-5-closed-loop-position-noise}
#### 13.5.1 Noise Sensitivity Functions {#13-dot-5-dot-1-noise-sensitivity-functions}
#### 13.5.2 Closed-Loop Position Noise Spectral Density {#13-dot-5-dot-2-closed-loop-position-noise-spectral-density}
#### 13.5.3 Closed-Loop Noise Approximations with Integral Control {#13-dot-5-dot-3-closed-loop-noise-approximations-with-integral-control}
#### 13.5.4 Closed-Loop Position Noise Variance {#13-dot-5-dot-4-closed-loop-position-noise-variance}
#### 13.5.5 A Note on Units {#13-dot-5-dot-5-a-note-on-units}
### 13.6 Simulation Examples {#13-dot-6-simulation-examples}
#### 13.6.1 Integral Controller Noise Simulation {#13-dot-6-dot-1-integral-controller-noise-simulation}
#### 13.6.2 Noise Simulation with Inverse Model Controller {#13-dot-6-dot-2-noise-simulation-with-inverse-model-controller}
#### 13.6.3 Feedback Versus Feedforward Control {#13-dot-6-dot-3-feedback-versus-feedforward-control}
### 13.7 Practical Frequency Domain Noise Measurements {#13-dot-7-practical-frequency-domain-noise-measurements}
#### 13.7.1 Preamplification {#13-dot-7-dot-1-preamplification}
#### 13.7.2 Spectrum Estimation {#13-dot-7-dot-2-spectrum-estimation}
#### 13.7.3 Direct Measurement of Position Noise {#13-dot-7-dot-3-direct-measurement-of-position-noise}
#### 13.7.4 Measurement of the External Disturbance {#13-dot-7-dot-4-measurement-of-the-external-disturbance}
### 13.8 Experimental Demonstration {#13-dot-8-experimental-demonstration}
### 13.9 Time-Domain Noise Measurements {#13-dot-9-time-domain-noise-measurements}
#### 13.9.1 Total Integrated Noise {#13-dot-9-dot-1-total-integrated-noise}
#### 13.9.2 Estimating the Position Noise {#13-dot-9-dot-2-estimating-the-position-noise}
#### 13.9.3 Practical Considerations {#13-dot-9-dot-3-practical-considerations}
#### 13.9.4 Experimental Demonstration {#13-dot-9-dot-4-experimental-demonstration}
### 13.10 A Simple Method for Measuring the Resolution of Nanopositioning Systems {#13-dot-10-a-simple-method-for-measuring-the-resolution-of-nanopositioning-systems}
### 13.11 Techniques for Improving Resolution {#13-dot-11-techniques-for-improving-resolution}
### 13.12 Chapter Summary {#13-dot-12-chapter-summary}
### References {#references}
## Electrical Considerations {#electrical-considerations}
### Amplifier and Piezo electrical models {#amplifier-and-piezo-electrical-models}
<a id="org393f35b"></a>
{{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="Figure 1: A voltage source \\(V\_s\\) driving a piezoelectric load. The actuator is modeled by a capacitance \\(C\_p\\) and strain-dependent voltage source \\(V\_p\\). The resistance \\(R\_s\\) is the output impedance and \\(L\\) the cable inductance." >}}
Consider the electrical circuit shown in Figure [1](#org393f35b) where a voltage source is connected to a piezoelectric actuator.
The actuator is modeled as a capacitance \\(C\_p\\) in series with a strain-dependent voltage source \\(V\_p\\).
The resistance \\(R\_s\\) and inductance \\(L\\) are the source impedance and the cable inductance respectively.
<div class="examp">
<div></div>
Typical inductance of standard RG-58 coaxial cable is \\(250 nH/m\\).
Typical value of \\(R\_s\\) is between \\(10\\) and \\(100 \Omega\\).
</div>
When considering the effects of both output impedance and cable inductance, the transfer function from source voltage \\(V\_s\\) to load voltage \\(V\_L\\) is second-order low pass filter:
\begin{equation}
\frac{V\_L(s)}{V\_s(s)} = \frac{1}{\frac{s^2}{\omega\_r^2} + 2 \xi \frac{s}{\omega\_r} + 1}
\end{equation}
with:
- \\(\omega\_r = \frac{1}{\sqrt{L C\_p}}\\)
- \\(\xi = \frac{R\_s \sqrt{L C\_p}}{2 L}\\)
### Amplifier small-signal Bandwidth {#amplifier-small-signal-bandwidth}
The most obvious bandwidth limitation is the small-signal bandwidth of the amplifier.
If the inductance \\(L\\) is neglected, the transfer function from source voltage \\(V\_s\\) to load voltage \\(V\_L\\) forms a first order filter with a cut-off frequency
\begin{equation}
\omega\_c = \frac{1}{R\_s C\_p}
\end{equation}
This is thus highly dependent of the load.
The high capacitive impedance nature of piezoelectric loads introduces phase-lag into the feedback path.
A rule of thumb is that closed-loop bandwidth cannot exceed one-tenth the cut-off frequency of the pole formed by the amplifier output impedance \\(R\_s\\) and load capacitance \\(C\_p\\) (see Table [1](#table--tab:piezo-limitation-Rs) for values).
<a id="table--tab:piezo-limitation-Rs"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:piezo-limitation-Rs">Table 1</a></span>:
Bandwidth limitation due to \(R_s\)
</div>
| | Cp = 100 nF | Cp = 1 uF | Cp = 10 uF |
|--------------|-------------|-----------|------------|
| Rs = 1 Ohm | 1.6 MHz | 160 kHz | 16 kHz |
| Rs = 10 Ohm | 160 kHz | 16 kHz | 1.6 kHz |
| Rs = 100 Ohm | 16 kHz | 1.6 kHz | 160 Hz |
The inductance \\(L\\) does also play a role in the amplifier bandwidth as it changes the resonance frequency.
Ideally, low inductance cables should be used.
It is however usually quite high compare to \\(\omega\_c\\) as shown in Table [2](#table--tab:piezo-limitation-L).
<a id="table--tab:piezo-limitation-L"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:piezo-limitation-L">Table 2</a></span>:
Bandwidth limitation due to \(R_s\)
</div>
| | Cp = 100 nF | Cp = 1 uF | Cp = 10 uF |
|-------------|-------------|-----------|------------|
| L = 25 nH | 3.2 MHz | 1 MHz | 320 kHz |
| L = 250 nH | 1 MHz | 320 kHz | 100 kHz |
| L = 2500 nH | 320 kHz | 100 kHz | 32 kHz |
### Amplifier maximum slew rate {#amplifier-maximum-slew-rate}
Further bandwidth restrictions are imposed by the maximum **slew rate** of the amplifier.
This is the maximum rate at which the output voltage can change and is usually expressed in \\(V/\mu s\\).
For sinusoidal signals, the amplifiers slew rate must exceed:
\\[ SR\_{\text{sin}} > V\_{p-p} \pi f \\]
where \\(V\_{p-p}\\) is the peak to peak voltage and \\(f\\) is the frequency.
<div class="examp">
<div></div>
If a 300kHz sine wave is to be reproduced with an amplitude of 10V, the required slew rate is \\(\approx 20 V/\mu s\\).
</div>
When dealing with capacitive loads, **the current limit is usually exceed well before the slew rate limit**.
### Current and Power Limitations {#current-and-power-limitations}
When driving the actuator off-resonance, the current delivered to a piezoelectric actuator is approximately:
\\[ I\_L(s) = V\_L(s) C\_p s \\]
For sinusoidal signals, the maximum positive and negative current is equal to:
\\[ I\_L^\text{max} = V\_{p-p} \pi f C\_p \\]
<a id="table--tab:piezo-required-current"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:piezo-required-current">Table 3</a></span>:
Minimum current requirements for a 10V sinusoid
</div>
| | Cp = 100 nF | Cp = 1 uF | Cp = 10 uF |
|-------------|-------------|-----------|------------|
| f = 30 Hz | 0.19 mA | 1.9 mA | 19 mA |
| f = 3 kHz | 19 mA | 190 mA | 1.9 A |
| f = 300 kHz | 1.9 A | 19 A | 190 A |
### Chapter Summary {#chapter-summary}
The bandwidth limitations of standard piezoelectric drives were identified as:
- High output impedance
- The presence of a ple in the voltage-feedback loop due to output impedance and load capacitance
- Insufficient current capacity due to power dissipation
- High cable and connector inductance
### References {#references}
## Bibliography {#bibliography}
<a id="org611ad6b"></a>Fleming, Andrew J., and Kam K. Leang. 2014. _Design, Modeling and Control of Nanopositioning Systems_. Advances in Industrial Control. Springer International Publishing. <https://doi.org/10.1007/978-3-319-06617-2>.
# Bibliography
<a id="fleming14_desig_model_contr_nanop_system"></a>Fleming, A. J., & Leang, K. K., *Design, modeling and control of nanopositioning systems* (2014), : Springer International Publishing. [↩](#1851788e0c4aa5b06afe3362c73ea5eb)

107
content/book/hatch00_vibrat_matlab_ansys.md
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@ -8,7 +8,7 @@ Tags
: [Finite Element Model]({{< relref "finite_element_model" >}})
Reference
: ([Hatch 2000](#org26212b4))
: <sup id="484c4fad309f6b0e866a7cacf4653d74"><a class="reference-link" href="#hatch00_vibrat_matlab_ansys" title="Hatch, Vibration simulation using MATLAB and ANSYS, CRC Press (2000).">(Hatch, 2000)</a></sup>
Author(s)
: Hatch, M. R.
@ -21,14 +21,14 @@ Matlab Code form the book is available [here](https://in.mathworks.com/matlabcen
## Introduction {#introduction}
<a id="org660282e"></a>
<a id="org4692204"></a>
The main goal of this book is to show how to take results of large dynamic finite element models and build small Matlab state space dynamic mechanical models for use in control system models.
### Modal Analysis {#modal-analysis}
The diagram in Figure [1](#org7e10f92) shows the methodology for analyzing a lightly damped structure using normal modes.
The diagram in Figure [1](#org6569db5) shows the methodology for analyzing a lightly damped structure using normal modes.
<div class="important">
<div></div>
@ -46,7 +46,7 @@ The steps are:
</div>
<a id="org7e10f92"></a>
<a id="org6569db5"></a>
{{< figure src="/ox-hugo/hatch00_modal_analysis_flowchart.png" caption="Figure 1: Modal analysis method flowchart" >}}
@ -58,7 +58,7 @@ Because finite element models usually have a very large number of states, an imp
<div class="important">
<div></div>
Figure [2](#org1c1177f) shows such process, the steps are:
Figure [2](#org2fb61c6) shows such process, the steps are:
- start with the finite element model
- compute the eigenvalues and eigenvectors (as many as dof in the model)
@ -71,14 +71,14 @@ Figure [2](#org1c1177f) shows such process, the steps are:
</div>
<a id="org1c1177f"></a>
<a id="org2fb61c6"></a>
{{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="Figure 2: Model size reduction flowchart" >}}
### Notations {#notations}
Tables [3](#org437cc66), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
Tables [3](#org3e528f9), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
<a id="table--tab:notations-modes-nodes"></a>
<div class="table-caption">
@ -127,22 +127,22 @@ Tables [3](#org437cc66), [2](#table--tab:notations-eigen-vectors-values) and [3]
## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems}
<a id="orgf7c45a9"></a>
<a id="org2522018"></a>
The origin and influence of poles are clear: they represent the resonant frequencies of the system, and for each resonance frequency, a mode shape can be defined to describe the motion at that frequency.
We here which to give an intuitive understanding for **when to expect zeros in SISO mechanical systems** and **how to predict the frequencies at which they will occur**.
Figure [3](#org437cc66) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
Figure [3](#org3e528f9) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
The degrees of freedom are numbered from left to right, \\(z\_1\\) through \\(z\_n\\).
<a id="org437cc66"></a>
<a id="org3e528f9"></a>
{{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="Figure 3: n dof system showing various SISO input/output configurations" >}}
<div class="important">
<div></div>
([Miu 1993](#org44eae27)) shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function.
<sup id="2169677da08094824a29bd3231ea1264"><a class="reference-link" href="#miu93_mechat" title="Denny Miu, Mechatronics: Electromechanics and Contromechanics, Springer-Verlag New York (1993).">(Denny Miu, 1993)</a></sup> shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function.
The resonances of the "overhanging appendages" of the constrained system create the zeros.
@ -151,12 +151,12 @@ The resonances of the "overhanging appendages" of the constrained system create
## State Space Analysis {#state-space-analysis}
<a id="orgc7d1453"></a>
<a id="orgd09cb02"></a>
## Modal Analysis {#modal-analysis}
<a id="orgaec588e"></a>
<a id="orga858b43"></a>
Lightly damped structures are typically analyzed with the "normal mode" method described in this section.
@ -196,9 +196,9 @@ Summarizing the modal analysis method of analyzing linear mechanical systems and
#### Equation of Motion {#equation-of-motion}
Let's consider the model shown in Figure [4](#org7d4f157) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
Let's consider the model shown in Figure [4](#orgc897d6a) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
<a id="org7d4f157"></a>
<a id="orgc897d6a"></a>
{{< figure src="/ox-hugo/hatch00_undamped_tdof_model.png" caption="Figure 4: Undamped tdof model" >}}
@ -297,17 +297,17 @@ One then find:
\end{bmatrix}
\end{equation}
Virtual interpretation of the eigenvectors are shown in Figures [5](#org102195c), [6](#org8c88cc4) and [7](#org033aa75).
Virtual interpretation of the eigenvectors are shown in Figures [5](#org40e1b2b), [6](#orgbe3ed46) and [7](#org766efd1).
<a id="org102195c"></a>
<a id="org40e1b2b"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="Figure 5: Rigid-Body Mode, 0rad/s" >}}
<a id="org8c88cc4"></a>
<a id="orgbe3ed46"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_2.png" caption="Figure 6: Second Model, Middle Mass Stationary, 1rad/s" >}}
<a id="org033aa75"></a>
<a id="org766efd1"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_3.png" caption="Figure 7: Third Mode, 1.7rad/s" >}}
@ -346,9 +346,9 @@ There are many options for change of basis, but we will show that **when eigenve
The n-uncoupled equations in the principal coordinate system can then be solved for the responses in the principal coordinate system using the well known solutions for the single dof systems.
The n-responses in the principal coordinate system can then be **transformed back** to the physical coordinate system to provide the actual response in physical coordinate.
This procedure is schematically shown in Figure [8](#org137f17c).
This procedure is schematically shown in Figure [8](#org9c058ac).
<a id="org137f17c"></a>
<a id="org9c058ac"></a>
{{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="Figure 8: Roadmap for Modal Solution" >}}
@ -696,7 +696,7 @@ Absolute damping is based on making \\(b = 0\\), in which case the percentage of
## Frequency Response: Modal Form {#frequency-response-modal-form}
<a id="org5db4c17"></a>
<a id="org74f8e80"></a>
The procedure to obtain the frequency response from a modal form is as follow:
@ -704,9 +704,9 @@ The procedure to obtain the frequency response from a modal form is as follow:
- use Laplace transform to obtain the transfer functions in principal coordinates
- back-transform the transfer functions to physical coordinates where the individual mode contributions will be evident
This will be applied to the model shown in Figure [9](#orgefc4430).
This will be applied to the model shown in Figure [9](#orgbbe5276).
<a id="orgefc4430"></a>
<a id="orgbbe5276"></a>
{{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="Figure 9: tdof undamped model for modal analysis" >}}
@ -888,9 +888,9 @@ Equations \eqref{eq:general_add_tf} and \eqref{eq:general_add_tf_damp} shows tha
</div>
Figure [10](#org997887a) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
Figure [10](#org4f8e313) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
<a id="org997887a"></a>
<a id="org4f8e313"></a>
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 10: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
@ -899,16 +899,16 @@ The zeros for SISO transfer functions are the roots of the numerator, however, f
## SISO State Space Matlab Model from ANSYS Model {#siso-state-space-matlab-model-from-ansys-model}
<a id="orgee24ebb"></a>
<a id="orgde113bc"></a>
### Introduction {#introduction}
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#org64b074d).
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#orgf61b7cd).
A z direction force is applied at the midpoint of the beam and z displacement at the tip is the output.
The objective is to provide the smallest Matlab state space model that accurately represents the pertinent dynamics.
<a id="org64b074d"></a>
<a id="orgf61b7cd"></a>
{{< figure src="/ox-hugo/hatch00_cantilever_beam.png" caption="Figure 11: Cantilever beam with forcing function at midpoint" >}}
@ -987,7 +987,7 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## Ground Acceleration Matlab Model From ANSYS Model {#ground-acceleration-matlab-model-from-ansys-model}
<a id="org88a2eb8"></a>
<a id="org86135e3"></a>
### Model Description {#model-description}
@ -1001,25 +1001,25 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model}
<a id="org9f31aa5"></a>
<a id="org5a61d5f"></a>
In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#org594b960)).
In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#orgc2d185d)).
### Actuator Description {#actuator-description}
<a id="org594b960"></a>
<a id="orgc2d185d"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="Figure 12: Drawing of Actuator/Suspension system" >}}
The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#orgb941e2d)).
The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#orgd4d4d64)).
<a id="orgb941e2d"></a>
<a id="orgd4d4d64"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_nodes_reduced_model.png" caption="Figure 13: Nodes used for reduced Matlab model. Shown with partial finite element mesh at coil" >}}
For reduced models, we only require eigenvector information for dof where forces are applied and where displacements are required.
Figure [13](#orgb941e2d) shows the nodes used for the reduced Matlab model.
Figure [13](#orgd4d4d64) shows the nodes used for the reduced Matlab model.
The four nodes 24061, 24066, 24082 and 24087 are located in the center of the coil in the z direction and are used for simulating the VCM force.
The arrows at the nodes indicate the direction of forces.
@ -1079,7 +1079,7 @@ From Ansys, we have the eigenvalues \\(\omega\_i\\) and eigenvectors \\(\bm{z}\\
## Balanced Reduction {#balanced-reduction}
<a id="orgc607ae9"></a>
<a id="org1ed7c19"></a>
In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods.
@ -1194,14 +1194,14 @@ The **states to be kept are the states with the largest diagonal terms**.
## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model}
<a id="org03a495e"></a>
<a id="org2ac725c"></a>
In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#orgf0d40c4)).
In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#org781c515)).
### Actuator Description {#actuator-description}
<a id="orgf0d40c4"></a>
<a id="org781c515"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="Figure 14: Drawing of actuator/suspension system" >}}
@ -1223,9 +1223,9 @@ Since the same forces are being applied to both piezo elements, they represent t
### Ansys Model Description {#ansys-model-description}
In Figure [15](#org3b2d630) are shown the principal nodes used for the model.
In Figure [15](#org6316e01) are shown the principal nodes used for the model.
<a id="org3b2d630"></a>
<a id="org6316e01"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_ansys.png" caption="Figure 15: Nodes used for reduced Matlab model, shown with partial mesh at coil and piezo element" >}}
@ -1344,11 +1344,11 @@ And we note:
G = zn * Gp;
```
<a id="orga327a57"></a>
<a id="org1a720cb"></a>
{{< figure src="/ox-hugo/hatch00_z13_tf.png" caption="Figure 16: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_3\\)" >}}
<a id="orgcdf4fe9"></a>
<a id="org9f278e9"></a>
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 17: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
@ -1446,13 +1446,13 @@ G_f = ss(A, B, C, D);
### Simple mode truncation {#simple-mode-truncation}
Let's plot the frequency of the modes (Figure [18](#org677de35)).
Let's plot the frequency of the modes (Figure [18](#orgd322a53)).
<a id="org677de35"></a>
<a id="orgd322a53"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="Figure 18: Frequency of the modes" >}}
<a id="orgad1205a"></a>
<a id="orgb88c298"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="Figure 19: Unsorted DC Gains" >}}
@ -1521,7 +1521,7 @@ dc_gain = abs(xn(i_input, :).*xn(i_output, :))./(2*pi*f0).^2;
[dc_gain_sort, index_sort] = sort(dc_gain, 'descend');
```
<a id="org78bca6c"></a>
<a id="orgf4c7beb"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="Figure 20: Sorted DC Gains" >}}
@ -1865,7 +1865,7 @@ wo = gram(G_m, 'o');
And we plot the diagonal terms
<a id="orgdf43aa3"></a>
<a id="orgb83de68"></a>
{{< figure src="/ox-hugo/hatch00_gramians.png" caption="Figure 21: Observability and Controllability Gramians" >}}
@ -1883,7 +1883,7 @@ We use `balreal` to rank oscillatory states.
[G_b, G, T, Ti] = balreal(G_m);
```
<a id="org09f35b9"></a>
<a id="org7516695"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="Figure 22: Sorted values of the Gramian of the balanced realization" >}}
@ -2126,11 +2126,12 @@ pos_frames = pos([1, i_input, i_output], :);
```
## Bibliography {#bibliography}
## Import super-element from Ansys {#import-super-element-from-ansys}
<a id="org26212b4"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
# Bibliography
<a class="bibtex-entry" id="hatch00_vibrat_matlab_ansys">Hatch, M. R., *Vibration simulation using matlab and ansys* (2000), : CRC Press.</a> [↩](#484c4fad309f6b0e866a7cacf4653d74)
<a id="org44eae27"></a>Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.
<a class="bibtex-entry" id="miu93_mechat">Miu, D. K., *Mechatronics: electromechanics and contromechanics* (1993), : Springer-Verlag New York.</a> [↩](#2169677da08094824a29bd3231ea1264)
## Backlinks {#backlinks}

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+++
title = "Active Isolation Platforms"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}})
| Manufacturers | Links | Country |
|---------------|------------------------------------------------------------------------|---------|
| TMC | [link](https://www.techmfg.com/) | USA |
| Newport | [link](https://www.newport.com/c/optical-tables-%26-isolation-systems) | USA |
<./biblio/references.bib>

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+++
title = "Amplified Piezoelectric Actuators"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}})
<./biblio/references.bib>

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:
## Piezoelectric Force Sensors {#piezoelectric-force-sensors}
### Dynamics and Noise of a piezoelectric force sensor {#dynamics-and-noise-of-a-piezoelectric-force-sensor}
## Dynamics and Noise of a piezoelectric force sensor {#dynamics-and-noise-of-a-piezoelectric-force-sensor}
An analysis the dynamics and noise of a piezoelectric force sensor is done 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" >}})).
### Manufacturers {#manufacturers}
## Manufacturers {#manufacturers}
| Manufacturers | Links |
|---------------|---------------------------------------------------------------|
| PCB | [link](https://www.pcb.com/products/productfinder.aspx?tx=17) |
### Signal Conditioner {#signal-conditioner}
The voltage generated by the piezoelectric material generally needs to be amplified.
| Manufacturers | Links |
|---------------|-----------------------------------------------|
| PCB | [link](https://www.pcb.com/products?m=482c15) |
# 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)
## Backlinks {#backlinks}
- [Collocated Control]({{< relref "collocated_control" >}})
- [Nanopositioning system with force feedback for high-performance tracking and vibration control]({{< relref "fleming10_nanop_system_with_force_feedb" >}})
- [Sensors]({{< relref "sensors" >}})
- [Collocated Control]({{< relref "collocated_control" >}})
- [Position Sensors]({{< relref "position_sensors" >}})

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+++
title = "Granite"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
:
<https://www.microplan-group.com/fr/>
<./biblio/references.bib>

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## Review of Absolute (inertial) Position Sensors {#review-of-absolute--inertial--position-sensors}
- Collette, C. et al., Review: inertial sensors for low-frequency seismic vibration measurement ([Collette, Janssens, Fernandez-Carmona, et al. 2012](#org3cd922d))
- Collette, C. et al., Comparison of new absolute displacement sensors ([Collette, Janssens, Mokrani, et al. 2012](#org8b5d5a2))
- Collette, C. et al., Review: inertial sensors for low-frequency seismic vibration measurement <sup id="dd5109075933cf543c7eba0979c0ba50"><a class="reference-link" href="#collette12_review" title="Collette, Janssens, Fernandez-Carmona, , Artoos, Guinchard, Hauviller \&amp; Preumont, Review: Inertial Sensors for Low-Frequency Seismic Vibration Measurement, {Bulletin of the Seismological Society of America}, v(4), 1289-1300 (2012).">(Collette {\it et al.}, 2012)</a></sup>
- Collette, C. et al., Comparison of new absolute displacement sensors <sup id="0b0b67de6dddc4d28031ab2d3b28cd3d"><a class="reference-link" href="#collette12_compar" title="Collette, Janssens, Mokrani, Fueyo-Roza, L, Artoos, Esposito, Fernandez-Carmona, , Guinchard \&amp; Leuxe, Comparison of new absolute displacement sensors, in in: {International Conference on Noise and Vibration Engineering
(ISMA)}, edited by (2012)">(Collette {\it et al.}, 2012)</a></sup>
<a id="org1914e49"></a>
<a id="org472a92d"></a>
{{< figure src="/ox-hugo/collette12_absolute_disp_sensors.png" caption="Figure 1: Dynamic range of several types of inertial sensors; Price versus resolution for several types of inertial sensors" >}}
## Accelerometers {#accelerometers}
| Manufacturers | Links | Country |
|--------------------|---------------------------------------------------------------------------------------------|---------|
| Micromega Dynamics | [link](https://micromega-dynamics.com/products/) | Belgium |
| MMF | [link](https://www.mmf.de/seismic%5Faccelerometers.htm) | Germany |
| PCB | [link](https://www.pcb.com/products/productfinder.aspx?tx=14) | USA |
| Guralp | [link](https://www.guralp.com/products/surface) | UK |
| Nanometric | [link](https://www.nanometrics.ca/products/accelerometers) | Canada |
| Kistler | [link](https://www.kistler.com/fr/produits/composants/accelerometres/?pfv%5Fmetrics=metric) | Swiss |
| Manufacturers | Links |
|--------------------|---------------------------------------------------------------|
| Micromega Dynamics | [link](https://micromega-dynamics.com/products/) |
| MMF | [link](https://www.mmf.de/seismic%5Faccelerometers.htm) |
| PCB | [link](https://www.pcb.com/products/productfinder.aspx?tx=14) |
Wireless Accelerometers
- <https://micromega-dynamics.com/products/recovib/miniature-vibration-recorder/>
<a id="orgf34c817"></a>
<a id="org005935d"></a>
{{< figure src="/ox-hugo/inertial_sensors_characteristics_accelerometers.png" caption="Figure 2: Characteristics of commercially available accelerometers <sup id=\"642a18d86de4e062c6afb0f5f20501c4\"><a class=\"reference-link\" href=\"#collette11_review\" title=\"Collette, Artoos, Guinchard, Janssens, , Carmona Fernandez \&amp; Hauviller, Review of sensors for low frequency seismic vibration measurement, CERN, (2011).\">(Collette {\it et al.}, 2011)</a></sup>" >}}
## Geophones and Seismometers {#geophones-and-seismometers}
## Geophones {#geophones}
| Manufacturers | Links | Country |
|-----------------------|---------------------------------------------------------------------------------------------|---------|
| Sercel | [link](http://www.sercel.com/products/Pages/seismometers.aspx) | France |
| Wilcoxon | [link](https://wilcoxon.com/) | USA |
| Geospace technologies | [link](https://www.geospace.com/sensors/#) | USA |
| Ion | [link](https://www.iongeo.com/technologies/hardware/seismic-equipment/precision-geophones/) | USA |
| Streckeisen | [link](https://streckeisen.swiss/en/products/overview/) | Swiss |
| Guralp | [link](https://www.guralp.com/products/surface) | UK |
| Nanometric | [link](https://www.nanometrics.ca/products/seismometers) | Canada |
| Manufacturers | Links |
|---------------|----------------------------------------------------------------|
| Sercel | [link](http://www.sercel.com/products/Pages/seismometers.aspx) |
| Wilcoxon | [link](https://wilcoxon.com/) |
<a id="org877de39"></a>
<a id="orgd64c709"></a>
{{< figure src="/ox-hugo/inertial_sensors_characteristics_geophone.png" caption="Figure 3: Characteristics of commercially available geophones <sup id=\"642a18d86de4e062c6afb0f5f20501c4\"><a class=\"reference-link\" href=\"#collette11_review\" title=\"Collette, Artoos, Guinchard, Janssens, , Carmona Fernandez \&amp; Hauviller, Review of sensors for low frequency seismic vibration measurement, CERN, (2011).\">(Collette {\it et al.}, 2011)</a></sup>" >}}
# Bibliography
<a class="bibtex-entry" id="collette12_review">Collette, C., Janssens, S., Fernandez-Carmona, P., Artoos, K., Guinchard, M., Hauviller, C., & Preumont, A., *Review: inertial sensors for low-frequency seismic vibration measurement*, Bulletin of the Seismological Society of America, *102(4)*, 1289–1300 (2012). http://dx.doi.org/10.1785/0120110223</a> [↩](#dd5109075933cf543c7eba0979c0ba50)
## Bibliography {#bibliography}
<a class="bibtex-entry" id="collette12_compar">Collette, C., Janssens, S., Mokrani, B., Fueyo-Roza, L., Artoos, K., Esposito, M., Fernandez-Carmona, P., …, *Comparison of new absolute displacement sensors*, In , International Conference on Noise and Vibration Engineering (ISMA) (pp. ) (2012). : .</a> [↩](#0b0b67de6dddc4d28031ab2d3b28cd3d)
<a id="org3cd922d"></a>Collette, C., S. Janssens, P. Fernandez-Carmona, K. Artoos, M. Guinchard, C. Hauviller, and A. Preumont. 2012. “Review: Inertial Sensors for Low-Frequency Seismic Vibration Measurement.” _Bulletin of the Seismological Society of America_ 102 (4):1289–1300. <https://doi.org/10.1785/0120110223>.
<a id="org8b5d5a2"></a>Collette, C, S Janssens, B Mokrani, L Fueyo-Roza, K Artoos, M Esposito, P Fernandez-Carmona, M Guinchard, and R Leuxe. 2012. “Comparison of New Absolute Displacement Sensors.” In _International Conference on Noise and Vibration Engineering (ISMA)_.
<a class="bibtex-entry" id="collette11_review">Collette, C., Artoos, K., Guinchard, M., Janssens, S., Carmona Fernandez, P., & Hauviller, C., *Review of sensors for low frequency seismic vibration measurement* (2011).</a> [↩](#642a18d86de4e062c6afb0f5f20501c4)
## Backlinks {#backlinks}
- [Sensors]({{< relref "sensors" >}})
- [Collocated Control]({{< relref "collocated_control" >}})
- [Position Sensors]({{< relref "position_sensors" >}})

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@ -12,11 +12,11 @@ Tags
Books:
- ([Higham 2017](#org311950e))
- ([Attaway 2018](#org2d4cbee))
- ([OverFlow 2018](#org84f4050))
- ([Johnson 2010](#orgd1edf93))
- ([Hahn and Valentine 2016](#org07606c6))
- <sup id="88712982e0649b89da706b6abbcbc6c2"><a href="#higham17_matlab" title="Higham, MATLAB guide, Society for Industrial and Applied Mathematics (2017).">(Higham, 2017)</a></sup>
- <sup id="15f4380b6ce8a647387d3ccea25711f1"><a href="#attaway18_matlab" title="Attaway, MATLAB : a practical introduction to programming and problem solving, Butterworth-Heinemann (2018).">(Attaway, 2018)</a></sup>
- <sup id="e770e23b0d222a65eb74f036227b13b2"><a href="#overflow18_matlab_notes_profes" title="Stack OverFlow, MATLAB Notes for Professionals, GoalKicker.com (2018).">(Stack OverFlow, 2018)</a></sup>
- <sup id="87b279fa5b4ec9b1a73abed2d00b313f"><a href="#johnson10_matlab" title="Johnson, The elements of MATLAB style, Cambridge University Press (2010).">(Johnson, 2010)</a></sup>
- <sup id="1b4159c36c5367ee0c92139fb403e7e1"><a href="#hahn16_essen_matlab" title="Hahn \&amp; Valentine, Essential MATLAB for engineers and scientists, Academic Press (2016).">(Hahn \& Valentine, 2016)</a></sup>
## Useful Commands {#useful-commands}
@ -54,33 +54,13 @@ hold off;
legend('Location', 'northeast');
```
# Bibliography
<a id="higham17_matlab"></a>Higham, D., *Matlab guide* (2017), Philadelphia: Society for Industrial and Applied Mathematics. [↩](#88712982e0649b89da706b6abbcbc6c2)
## Linux Installation {#linux-installation}
<a id="attaway18_matlab"></a>Attaway, S., *Matlab : a practical introduction to programming and problem solving* (2018), Amsterdam: Butterworth-Heinemann. [↩](#15f4380b6ce8a647387d3ccea25711f1)
If a single user is using the Matlab installation on the machine:
<a id="overflow18_matlab_notes_profes"></a>OverFlow, S., *Matlab notes for professionals* (2018), : GoalKicker.com. [↩](#e770e23b0d222a65eb74f036227b13b2)
```bash
sudo chown -R $LOGNAME: /usr/local/MATLAB/R2017b
```
<a id="johnson10_matlab"></a>Johnson, R. K., *The elements of matlab style* (2010), : Cambridge University Press. [↩](#87b279fa5b4ec9b1a73abed2d00b313f)
Then, Toolboxes can be installed by the user without any problem.
To install Toolboxes, the best is to Download the Matlab installer from mathworks and just select the wanted toolboxes.
## Used Toolboxes {#used-toolboxes}
- `vfit3` ([link](https://www.sintef.no/projectweb/vectorfitting/)): used to identify transfer functions
## Bibliography {#bibliography}
<a id="org2d4cbee"></a>Attaway, Stormy. 2018. _MATLAB : a Practical Introduction to Programming and Problem Solving_. Amsterdam: Butterworth-Heinemann.
<a id="org07606c6"></a>Hahn, Brian, and Daniel T Valentine. 2016. _Essential MATLAB for Engineers and Scientists_. Academic Press.
<a id="org311950e"></a>Higham, Desmond. 2017. _MATLAB Guide_. Philadelphia: Society for Industrial and Applied Mathematics.
<a id="orgd1edf93"></a>Johnson, Richard K. 2010. _The Elements of MATLAB Style_. Cambridge University Press.
<a id="org84f4050"></a>OverFlow, Stack. 2018. _MATLAB Notes for Professionals_. GoalKicker.com.
<a id="hahn16_essen_matlab"></a>Hahn, B., & Valentine, D. T., *Essential matlab for engineers and scientists* (2016), : Academic Press. [↩](#1b4159c36c5367ee0c92139fb403e7e1)

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### Manufacturers {#manufacturers}
| Manufacturers | Links | Country |
|---------------------|----------------------------------------------------------------------------------------------------------------|-----------|
| Cedrat | [link](http://www.cedrat-technologies.com/) | France |
| PI | [link](https://www.physikinstrumente.com/en/) | USA |
| Piezo System | [link](https://www.piezosystem.com/products/piezo%5Factuators/stacktypeactuators/) | Germany |
| Noliac | [link](http://www.noliac.com/) | Denmark |
| Thorlabs | [link](https://www.thorlabs.com/newgrouppage9.cfm?objectgroup%5Fid=8700) | USA |
| PiezoDrive | [link](https://www.piezodrive.com/actuators/) | Australia |
| Mechano Transformer | [link](http://www.mechano-transformer.com/en/products/10.html) | Japan |
| CoreMorrow | [link](http://www.coremorrow.com/en/pro-9-1.html) | China |
| PiezoData | [link](https://www.piezodata.com/piezo-stack-actuator-2/) | China |
| Queensgate | [link](https://www.nanopositioning.com/product-category/nanopositioning/nanopositioning-actuators-translators) | UK |
| Matsusada Precision | [link](https://www.matsusada.com/product/pz/) | Japan |
| Manufacturers | Links |
|---------------------|------------------------------------------------------------------------------------|
| Cedrat | [link](http://www.cedrat-technologies.com/) |
| PI | [link](https://www.physikinstrumente.com/en/) |
| Piezo System | [link](https://www.piezosystem.com/products/piezo%5Factuators/stacktypeactuators/) |
| Noliac | [link](http://www.noliac.com/) |
| Thorlabs | [link](https://www.thorlabs.com/newgrouppage9.cfm?objectgroup%5Fid=8700) |
| PiezoDrive | [link](https://www.piezodrive.com/actuators/) |
| Mechano Transformer | [link](http://www.mechano-transformer.com/en/products/10.html) |
| CoreMorrow | [link](http://www.coremorrow.com/en/pro-9-1.html) |
### Model {#model}
A model of a multi-layer monolithic piezoelectric stack actuator is described in ([Fleming 2010](#org1025f36)) ([Notes]({{< relref "fleming10_nanop_system_with_force_feedb" >}})).
Basically, it can be represented by a spring \\(k\_a\\) with the force source \\(F\_a\\) in parallel.
The relation between the applied voltage \\(V\_a\\) to the generated force \\(F\_a\\) is:
\\[ F\_a = g\_a V\_a, \quad g\_a = d\_{33} n k\_a \\]
with:
- \\(d\_{33}\\) is the piezoelectric strain constant [m/V]
- \\(n\\) is the number of layers
- \\(k\_a\\) is the actuator stiffness [N/m]
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" >}})).
## Mechanically Amplified Piezoelectric actuators {#mechanically-amplified-piezoelectric-actuators}
The Amplified Piezo Actuators principle is presented in ([Claeyssen et al. 2007](#org4de69d6)):
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>:
> The displacement amplification effect is related in a first approximation to the ratio of the shell long axis length to the short axis height.
> The flatter is the actuator, the higher is the amplification.
A model of an amplified piezoelectric actuator is described in ([Lucinskis and Mangeot 2016](#org2278a86)).
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,
author = {R. Lucinskis and C. Mangeot},
title = {Dynamic Characterization of an amplified piezoelectric
actuator},
year = 2016,
}">(Lucinskis \& Mangeot, 2016)</a></sup>.
<a id="org220f472"></a>
<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** | **Country** |
|---------------------|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-------------|
| Cedrat | [link](https://www.cedrat-technologies.com/en/products/actuators/amplified-piezo-actuators.html) | France |
| PiezoDrive | [link](https://www.piezodrive.com/actuators/ap-series-amplified-piezoelectric-actuators/) | Australia |
| Dynamic-Structures | [link](https://www.dynamic-structures.com/category/piezo-actuators-stages) | USA |
| Thorlabs | [link](https://www.thorlabs.com/newgrouppage9.cfm?objectgroup%5Fid=8700) | USA |
| Noliac | [link](http://www.noliac.com/products/actuators/amplified-actuators/) | Denmark |
| 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) | Japan |
| CoreMorrow | [link](http://www.coremorrow.com/en/pro-13-1.html) | China |
| PiezoData | [link](https://www.piezodata.com/piezoelectric-actuator-amplifier/) | China |
| Manufacturers | Links |
|---------------------|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| Cedrat | [link](https://www.cedrat-technologies.com/en/products/actuators/amplified-piezo-actuators.html) |
| PiezoDrive | [link](https://www.piezodrive.com/actuators/ap-series-amplified-piezoelectric-actuators/) |
| Dynamic-Structures | [link](https://www.dynamic-structures.com/category/piezo-actuators-stages) |
| Thorlabs | [link](https://www.thorlabs.com/newgrouppage9.cfm?objectgroup%5Fid=8700) |
| Noliac | [link](http://www.noliac.com/products/actuators/amplified-actuators/) |
| 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}
@ -130,7 +121,7 @@ with:
### Resolution {#resolution}
The resolution is limited by the noise in the [Voltage Amplifier]({{< relref "voltage_amplifier" >}}).
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
@ -144,57 +135,53 @@ For a piezoelectric stack with a displacement of \\(100\,[\mu m]\\), the resolut
### Electrical Capacitance {#electrical-capacitance}
The electrical capacitance may limit the maximum voltage that can be used to drive the piezoelectric actuator as a function of frequency (Figure [2](#org4b5f8bd)).
This is due to the fact that voltage amplifier has a limitation on the deliverable current.
The electrical capacitance gives the maximum voltage that can be used to drive the piezoelectric actuator as a function of frequency (Figure [2](#org3da123f)).
[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.
<a id="org4b5f8bd"></a>
<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](#org6e4c8b2)).
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="org6e4c8b2"></a>
<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](#orgadae726)):
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="orgadae726"></a>
<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](#org51f52cb)).
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.
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="org51f52cb"></a>
<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)
## Bibliography {#bibliography}
<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)
<a id="org4de69d6"></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>.
<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 id="org1025f36"></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>.
<a id="org2278a86"></a>Lucinskis, R., and C. Mangeot. 2016. “Dynamic Characterization of an Amplified Piezoelectric Actuator.”
<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}
- [Actuators]({{< relref "actuators" >}})
- [Voltage Amplifier]({{< relref "voltage_amplifier" >}})

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+++
Tags
: [Inertial Sensors]({{< relref "inertial_sensors" >}}), [Force Sensors]({{< relref "force_sensors" >}}), [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Signal Conditioner]({{< relref "signal_conditioner" >}}), [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio" >}})
: [Inertial Sensors]({{< relref "inertial_sensors" >}}), [Force Sensors]({{< relref "force_sensors" >}}), [Sensor Fusion]({{< relref "sensor_fusion" >}})
## Reviews of Relative Position Sensors {#reviews-of-relative-position-sensors}
- Fleming, A. J., A review of nanometer resolution position sensors: operation and performance ([Fleming 2013](#orgeaf4a0a)) ([Notes]({{< relref "fleming13_review_nanom_resol_posit_sensor" >}}))
- Fleming, A. J., A review of nanometer resolution position sensors: operation and performance <sup id="3fb5b61524290e36d639a4fac65703d0"><a class="reference-link" href="#fleming13_review_nanom_resol_posit_sensor" title="Andrew Fleming, A Review of Nanometer Resolution Position Sensors: Operation and Performance, {Sensors and Actuators A: Physical}, v(nil), 106-126 (2013).">(Andrew Fleming, 2013)</a></sup> ([Notes]({{< relref "fleming13_review_nanom_resol_posit_sensor" >}}))
<a id="table--tab:characteristics-relative-sensor"></a>
<div class="table-caption">
@ -44,7 +44,8 @@ Tags
| Interferometer | Meters | | 0.5 nm | >100kHz | 1 ppm FSR |
| Encoder | Meters | | 6 nm | >100kHz | 5 ppm FSR |
Capacitive Sensors and Eddy-Current sensors are compare [here](https://www.lionprecision.com/comparing-capacitive-and-eddy-current-sensors/).
## Strain Gauge {#strain-gauge}
## Capacitive Sensor {#capacitive-sensor}
@ -54,48 +55,42 @@ Description:
- <http://www.lionprecision.com/tech-library/technotes/cap-0020-sensor-theory.html>
- <https://www.lionprecision.com/comparing-capacitive-and-eddy-current-sensors>
| Manufacturers | Links | Country |
|----------------|--------------------------------------------------------------------------------------------------|---------|
| Micro Sense | [link](http://www.microsense.net/products-position-sensors.htm) | USA |
| Micro-Epsilon | [link](https://www.micro-epsilon.com/displacement-position-sensors/capacitive-sensor/) | Germany |
| PI | [link](https://www.physikinstrumente.com/en/technology/sensor-technologies/capacitive-sensors/) | Germany |
| Unipulse | [link](https://www.unipulse.com/product/ps-ia/) | Japan |
| Lion-Precision | [link](https://www.lionprecision.com/products/capacitive-sensors) | USA |
| Fogale | [link](http://www.fogale.fr/brochures.html) | USA |
| Queensgate | [link](https://www.nanopositioning.com/product-category/nanopositioning/nanopositioning-sensors) | UK |
| Capacitec | [link](https://www.capacitec.com/Displacement-Sensing-Systems) | USA |
| Manufacturers | Links |
|----------------|-------------------------------------------------------------------------------------------------|
| Micro Sense | [link](http://www.microsense.net/products-position-sensors.htm) |
| Micro-Epsilon | [link](https://www.micro-epsilon.com/displacement-position-sensors/capacitive-sensor/) |
| PI | [link](https://www.physikinstrumente.com/en/technology/sensor-technologies/capacitive-sensors/) |
| Unipulse | [link](https://www.unipulse.com/product/ps-ia/) |
| Lion-Precision | [link](https://www.lionprecision.com/products/capacitive-sensors) |
## Inductive Sensor (Eddy Current) {#inductive-sensor--eddy-current}
| Manufacturers | Links | |
|----------------|-------------------------------------------------------------------------------------------|---------|
| Micro-Epsilon | [link](https://www.micro-epsilon.com/displacement-position-sensors/eddy-current-sensor/) | Germany |
| Lion Precision | [link](https://www.lionprecision.com/products/eddy-current-sensors) | USA |
| Cedrat | [link](https://www.cedrat-technologies.com/en/products/sensors/eddy-current-sensors.html) | France |
| Kaman | [link](https://www.kamansensors.com/product/smt-9700/) | USA |
| Keyence | [link](https://www.keyence.com/ss/products/measure/measurement%5Flibrary/type/inductive/) | USA |
| Manufacturers | Links |
|----------------|------------------------------------------------------------------------------------------|
| Micro-Epsilon | [link](https://www.micro-epsilon.com/displacement-position-sensors/eddy-current-sensor/) |
| Lion Precision | [link](https://www.lionprecision.com/products/eddy-current-sensors) |
## Inductive Sensor (LVDT) {#inductive-sensor--lvdt}
| Manufacturers | Links | Country |
|---------------|--------------------------------------------------------------------------------------------|---------|
| Micro-Epsilon | [link](https://www.micro-epsilon.com/displacement-position-sensors/inductive-sensor-lvdt/) | Germany |
| Keyence | [link](https://www.keyence.eu/products/measure/contact-distance-lvdt/gt2/index.jsp) | USA |
| Manufacturers | Links |
|---------------|--------------------------------------------------------------------------------------------|
| Micro-Epsilon | [link](https://www.micro-epsilon.com/displacement-position-sensors/inductive-sensor-lvdt/) |
| Keyence | [link](https://www.keyence.eu/products/measure/contact-distance-lvdt/gt2/index.jsp) |
## Interferometers {#interferometers}
| Manufacturers | Links | Country |
|---------------|----------------------------------------------------------------------------------------------------------|---------|
| Attocube | [link](http://www.attocube.com/) | Germany |
| Zygo | [link](https://www.zygo.com/?/met/markets/stageposition/zmi/) | USA |
| Smaract | [link](https://www.smaract.com/interferometry) | Germany |
| Qutools | [link](https://www.qutools.com/qudis/) | Germany |
| Renishaw | [link](https://www.renishaw.com/en/fibre-optic-laser-encoder-products--6594) | UK |
| Sios | [link](https://sios-de.com/products/length-measurement/laser-interferometer/) | Germany |
| Keysight | [link](https://www.keysight.com/en/pc-1000000393%3Aepsg%3Apgr/laser-heads?nid=-536900395.0&cc=FR&lc=fre) | USA |
| Manufacturers | Links |
|---------------|----------------------------------------------------------------------------------------------------------|
| Attocube | [link](http://www.attocube.com/) |
| Zygo | [link](https://www.zygo.com/?/met/markets/stageposition/zmi/) |
| Smaract | [link](https://www.smaract.com/interferometry) |
| Qutools | [link](https://www.qutools.com/qudis/) |
| Renishaw | [link](https://www.renishaw.com/en/fibre-optic-laser-encoder-products--6594) |
| Sios | [link](https://sios-de.com/products/length-measurement/laser-interferometer/) |
| Keysight | [link](https://www.keysight.com/en/pc-1000000393%3Aepsg%3Apgr/laser-heads?nid=-536900395.0&cc=FR&lc=fre) |
<div class="table-caption">
<span class="table-number">Table 3</span>:
@ -108,33 +103,31 @@ Description:
| Renishaw | 0.2 | 1 | 6 | 1 |
| Picoscale | 0.2 | 2 | 2 | 1 |
([Jang and Kim 2017](#org5a2485c))
<sup id="7658b1219a4458a62ae8c6f51b767542"><a class="reference-link" href="#jang17_compen_refrac_index_air_laser" title="Yoon-Soo Jang \&amp; Seung-Woo Kim, Compensation of the Refractive Index of Air in Laser Interferometer for Distance Measurement: a Review, {International Journal of Precision Engineering and
Manufacturing}, v(12), 1881-1890 (2017).">(Yoon-Soo Jang \& Seung-Woo Kim, 2017)</a></sup>
<a id="orgdc4dc3c"></a>
<a id="org0399c13"></a>
{{< figure src="/ox-hugo/position_sensor_interferometer_precision.png" caption="Figure 1: Expected precision of interferometer as a function of measured distance" >}}
## Linear Encoders {#linear-encoders}
## Fiber Optic Displacement Sensor {#fiber-optic-displacement-sensor}
| Manufacturers | Links | Country |
|----------------|-----------------------------------------------------------------------|---------|
| Heidenhain | [link](https://www.heidenhain.com/en%5FUS/products/linear-encoders/) | Germany |
| MicroE Systems | [link](https://www.celeramotion.com/microe/products/linear-encoders/) | USA |
| Renishaw | [link](https://www.renishaw.com/en/browse-encoder-range--6440) | UK |
| Manufacturers | Links |
|---------------|----------------------------------------------------|
| Unipulse | [link](https://www.unipulse.com/product/atw200-2/) |
# Bibliography
<a class="bibtex-entry" id="fleming13_review_nanom_resol_posit_sensor">Fleming, A. J., *A review of nanometer resolution position sensors: operation and performance*, Sensors and Actuators A: Physical, *190(nil)*, 106–126 (2013). http://dx.doi.org/10.1016/j.sna.2012.10.016</a> [↩](#3fb5b61524290e36d639a4fac65703d0)
## Bibliography {#bibliography}
<a class="bibtex-entry" id="collette11_review">Collette, C., Artoos, K., Guinchard, M., Janssens, S., Carmona Fernandez, P., & Hauviller, C., *Review of sensors for low frequency seismic vibration measurement* (2011).</a> [↩](#642a18d86de4e062c6afb0f5f20501c4)
<a id="orgeaf4a0a"></a>Fleming, Andrew J. 2013. “A Review of Nanometer Resolution Position Sensors: Operation and Performance.” _Sensors and Actuators a: Physical_ 190 (nil):106–26. <https://doi.org/10.1016/j.sna.2012.10.016>.
<a id="org5a2485c"></a>Jang, Yoon-Soo, and Seung-Woo Kim. 2017. “Compensation of the Refractive Index of Air in Laser Interferometer for Distance Measurement: A Review.” _International Journal of Precision Engineering and Manufacturing_ 18 (12):1881–90. <https://doi.org/10.1007/s12541-017-0217-y>.
<a class="bibtex-entry" id="jang17_compen_refrac_index_air_laser">Jang, Y., & Kim, S., *Compensation of the refractive index of air in laser interferometer for distance measurement: a review*, International Journal of Precision Engineering and Manufacturing, *18(12)*, 1881–1890 (2017). http://dx.doi.org/10.1007/s12541-017-0217-y</a> [↩](#7658b1219a4458a62ae8c6f51b767542)
## Backlinks {#backlinks}
- [A review of nanometer resolution position sensors: operation and performance]({{< relref "fleming13_review_nanom_resol_posit_sensor" >}})
- [Measurement technologies for precision positioning]({{< relref "gao15_measur_techn_precis_posit" >}})
- [Inertial Sensors]({{< relref "inertial_sensors" >}})
- [Sensors]({{< relref "sensors" >}})
- [Collocated Control]({{< relref "collocated_control" >}})
- [Inertial Sensors]({{< relref "inertial_sensors" >}})

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+++
title = "Power Spectral Density"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio" >}})
Tutorial about Power Spectral Density is accessible [here](https://tdehaeze.github.io/spectral-analysis/).
A good article about how to use the `pwelch` function with Matlab ([Schmid 2012](#org8fdb443)).
## Bibliography {#bibliography}
<a id="org8fdb443"></a>Schmid, Hanspeter. 2012. “How to Use the FFT and Matlab’s Pwelch Function for Signal and Noise Simulations and Measurements.” _Institute of Microelectronics_.

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+++
title = "Shaker"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Voice Coil Actuators]({{< relref "voice_coil_actuators" >}})
## Manufacturers {#manufacturers}
| Manufacturers | Links |
|-----------------|----------------------------------------------------------------------------------|
| Labsen | [link](http://labsentec.com.au/category/products/vibrationshock/) |
| The Modal Shop | [link](http://www.modalshop.com/excitation/Electrodynamic-Exciter-Family?ID=243) |
| Deweshop | [link](https://dewesoft.com/fr/products/interfaces-and-sensors/shakers) |
| Bruel and Kjaer | [link](https://www.bksv.com/en/products/shakers-and-exciters/LDS-shaker-systems) |
| YMC | [link](http://www.chinaymc.com/product/showproduct.php?id=78&lang=en) |
<./biblio/references.bib>

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+++
title = "Signal Conditioner"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Force Sensors]({{< relref "force_sensors" >}})
Most sensors needs some signal conditioner electronics before digitize the signal.
Few examples are:
- Piezoelectric force sensors
- Geophone
- Thermocouple, ...
The signal conditioning electronics can have different functions:
- Amplification
- Isolation
- Filtering
- Excitation
- Linearization
## Charge Amplifier {#charge-amplifier}
| Manufacturers | Links |
|---------------|---------------------------------------------------------------------------------------------------------------------|
| PCB | [link](https://www.pcb.com/sensors-for-test-measurement/electronics/line-powered-multi-channel-signal-conditioners) |
## Voltage Amplifier {#voltage-amplifier}
| Manufacturers | Links |
|---------------|------------------------------------------------------------------|
| Femto | [link](https://www.femto.de/en/products/voltage-amplifiers.html) |
## Current Amplifier {#current-amplifier}
| Manufacturers | Links |
|---------------|------------------------------------------------------------------|
| Femto | [link](https://www.femto.de/en/products/current-amplifiers.html) |
<./biblio/references.bib>
## Backlinks {#backlinks}
- [Position Sensors]({{< relref "position_sensors" >}})

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Tags
: [Electronics]({{< relref "electronics" >}}), [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
## SNR to Noise PSD {#snr-to-noise-psd}
From ([Jabben 2007](#org05d266b)) (Section 3.3.2):
From <sup id="3b7899e183dba866e6a6419cf820467f"><a href="#jabben07_mechat" title="@phdthesis{jabben07_mechat,
author = {Jabben, Leon},
school = {Delft University},
title = {Mechatronic design of a magnetically suspended rotating
platform},
year = 2007,
}">@phdthesis{jabben07_mechat,
author = {Jabben, Leon},
school = {Delft University},
title = {Mechatronic design of a magnetically suspended rotating
platform},
year = 2007,
}</a></sup> (Section 3.3.2):
> Electronic equipment does most often not come with detailed electric schemes, in which case the PSD should be determined from measurements.
> In the design phase however, one has to rely on information provided by specification sheets from the manufacturer.
@ -22,90 +31,5 @@ From ([Jabben 2007](#org05d266b)) (Section 3.3.2):
> \\[ S\_{snr} = \frac{x\_{fr}^2}{8 f\_c C\_{snr}^2} \\]
> with \\(x\_{fr}\\) the full range of \\(x\\), and \\(C\_{snr}\\) the SNR.
<div class="examp">
<div></div>
Let's take an example.
- \\(x\_{fr} = 170 V\\)
- \\(C\_{snr} = 85 dB\\)
- \\(f\_c = 200 Hz\\)
The Power Spectral Density of the output voltage is:
\\[ S\_{snr} = \frac{170^2}{8 \cdot 200 \cdot {10^{\frac{2 \cdot 85}{20}}}} = 5.7 \cdot 10^{-8}\ V^2/Hz \\]
And the RMS of that noise up to \\(f\_c\\) is:
\\[ S\_{rms} = \sqrt{S\_{snr} \cdot f\_c} \approx 3.4\ mV \\]
</div>
## Convert SNR to Noise RMS value {#convert-snr-to-noise-rms-value}
The RMS value of the noise can be computed from:
\\[ N\_\text{rms} = 10^{-\frac{S\_{snr}}{20}} S\_\text{rms} \\]
where \\(S\_{snr}\\) is the SNR in dB and \\(S\_\text{rms}\\) is the RMS value of a sinus taking the full range.
If the full range is \\(\Delta V\\), then:
\\[ S\_\text{rms} = \frac{\Delta V/2}{\sqrt{2}} \\]
<div class="examp">
<div></div>
As an example, let's take a voltage amplifier with a full range of \\(\Delta V = 20V\\) and a SNR of 85dB.
The RMS value of the noise is then:
\\[ n\_\text{rms} = 10^{-\frac{S\_{nrs}}{20}} s\_\text{rms} \\]
\\[ n\_\text{rms} = 10^{-\frac{85}{20}} \frac{10}{\sqrt{2}} \approx 0.4 mV\_\text{rms} \\]
</div>
## Convert wanted Noise RMS value to required SNR {#convert-wanted-noise-rms-value-to-required-snr}
If the wanted full range and RMS value of the noise are defined, the required SNR can be computed from:
\\[ S\_{snr} = 20 \log \frac{\text{Signal, rms}}{\text{Noise, rms}} \\]
<div class="examp">
<div></div>
Let's say the wanted noise is \\(1 mV, \text{rms}\\) for a full range of \\(20 V\\), the corresponding SNR is:
\\[ S\_{snr} = 20 \log \frac{\frac{20/2}{\sqrt{2}}}{10^{-3}} \approx 77dB \\]
</div>
## Noise Density to RMS noise {#noise-density-to-rms-noise}
From ([Fleming 2010](#org8235840)):
\\[ \text{RMS noise} = \sqrt{2 \times \text{bandwidth}} \times \text{noise density} \\]
If the noise is normally distributed, the RMS value is also the standard deviation \\(\sigma\\).
The peak to peak amplitude is then approximatively \\(6 \sigma\\).
<div class="examp">
<div></div>
- noise density = \\(20 pm/\sqrt{Hz}\\)
- bandwidth = 100Hz
\\[ \sigma = \sqrt{2 \times 100} \times 20 = 0.28nm RMS \\]
The peak-to-peak noise will be approximately \\(6 \sigma = 1.7 nm\\)
</div>
## Bibliography {#bibliography}
<a id="org8235840"></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>.
<a id="org05d266b"></a>Jabben, Leon. 2007. “Mechatronic Design of a Magnetically Suspended Rotating Platform.” Delft University.
## Backlinks {#backlinks}
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}})
- [Power Spectral Density]({{< relref "power_spectral_density" >}})
- [Position Sensors]({{< relref "position_sensors" >}})
- [Voltage Amplifier]({{< relref "voltage_amplifier" >}})
- [Voltage Amplifier]({{< relref "voltage_amplifier" >}})
# Bibliography
<a id="jabben07_mechat"></a>Jabben, L., *Mechatronic design of a magnetically suspended rotating platform* (Doctoral dissertation) (2007). Delft University, . [↩](#3b7899e183dba866e6a6419cf820467f)

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title = "Slip Rings"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
:
| Manufacturers | Links |
|---------------|---------------------------------|
| Moflon | [link](https://www.moflon.com/) |
<./biblio/references.bib>

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| Beikimco | [link](http://www.beikimco.com/) |
| Electromate | [link](https://www.electromate.com/) |
| Magnetic Innovations | [link](https://www.magneticinnovations.com/) |
| Monticont | [link](http://www.moticont.com/) |
## Typical Specifications {#typical-specifications}

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title = "Voltage Amplifier"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio" >}}), [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}}), [Electronics]({{< relref "electronics" >}})
## Voltage Amplifiers to drive Capacitive Load {#voltage-amplifiers-to-drive-capacitive-load}
### Limitation in Current {#limitation-in-current}
The piezoelectric stack can be represented as a capacitance.
Let's take a capacitance driven by a voltage amplifier (Figure [1](#orgcab6e6f)).
<a id="orgcab6e6f"></a>
{{< figure src="/ox-hugo/voltage_amplifier_capacitance.png" caption="Figure 1: Piezoelectric actuator model with a voltage source" >}}
The equation linking the voltage to the current is:
\\[ I = C \frac{dU}{dt} \\]
Suppose we want to drive the piezo with a voltage:
\\[ U(t) = U\_0 \sin(\omega\_0 t) \\]
The required current is:
\\[ I = C U\_0 \omega\_0 \cos(\omega\_0 t) \\]
Thus, for a specified maximum current \\(I\_\text{max}\\), the "power bandwidth" will be:
\\[ \omega\_{0, \text{max}} = \frac{I\_\text{max}}{C U\_0}, \quad f\_{0, \text{max}} = \frac{I\_\text{max}}{C U\_0} \frac{1}{2 \pi} \\]
- Below \\(\omega\_{0, \text{max}}\\), a voltage amplitude \\(U\_0\\) can be applied to the piezoelectric load without reaching the maximum current \\(I\_\text{max}\\).
- Above \\(\omega\_{0, \text{max}}\\), the maximum current \\(I\_\text{max}\\) is reached and the maximum voltage that can be applied decreases with frequency:
\\[ U\_\text{max} = \frac{I\_\text{max}}{\omega C} \\]
The maximum voltage as a function of frequency is shown in Figure [2](#org1475933).
```matlab
Vpkp = 170; % [V]
Imax = 30e-3; % [A]
C = 1e-6; % [F]
(1/(2*pi))*Imax/(C * Vpkp/2) % Fmax [Hz]
```
```text
56.172
```
<a id="org1475933"></a>
{{< figure src="/ox-hugo/voltage_amplifier_max_V_piezo.png" caption="Figure 2: Maximum voltage as a function of the frequency for \\(C = 1 \mu F\\), \\(I\_\text{max} = 30mA\\) and \\(V\_{pkp} = 170 V\\)" >}}
Similarly, the voltage rise time is determined by the Capacitance of the piezoelectric stack and by the maximum current that the amplifier can deliver:
\\[ t\_c = \frac{\Delta U C}{I\_\text{max}} \\]
with \\(t\_c\\) in seconds, \\(\Delta U\\) in volts, \\(C\\) in Farads and \\(I\_\text{max}\\) in Amperes.
If driven at \\(\Delta U = 100V\\), \\(C = 1 \mu F\\) and \\(I\_\text{max} = 1 A\\), then:
\\[ t\_c = \frac{100 \cdot 10^{-6}}{1} = 0.1 ms \\]
### Bandwidth limitation (small signals) {#bandwidth-limitation--small-signals}
This is takken from Chapter 14 of ([Fleming and Leang 2014](#org01aad4a)).
```matlab
L = 250e-9; % Cable inductance [H]
Cp = 10e-6; % Driving capacitance [F]
Rs = 10; % Source impedance [Ohm]
G = 1/(L*Cp)/(s^2 + Rs/L*s + 1/(L*Cp));
```
### Amplifiers for Low Voltage PZT {#amplifiers-for-low-voltage-pzt}
Piezoelectric Stack Actuators are behaving like capacitor for the Amplifiers.
Specifications are usually:
- Maximum Current
- DC Gain (usually around 10)
- Output Noise or [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio" >}})
The bandwidth can be estimated from the Maximum Current and the Capacitance of the Piezoelectric Actuator.
| Manufacturers | Links | Country |
|---------------------|---------------------------------------------------------------------------------------------------------------------------------------------------------|-------------|
| Piezo Drive | [link](https://www.piezodrive.com/drivers/) | Australia |
| Thorlabs | [link](https://www.thorlabs.com/navigation.cfm?guide%5FID=2085) | USA |
| PI | [link](https://www.pi-usa.us/en/products/controllers-drivers-motion-control-software/piezo-drivers-controllers-power-supplies-high-voltage-amplifiers/) | USA |
| Micromega Dynamics | | Belgium |
| Lab Systems | [link](https://www.lab-systems.com/products/amplifier/amplifier.html) | Isreal |
| Falco System | [link](https://www.falco-systems.com/products.html) | Netherlands |
| Piezomechanics | [link](https://www.piezomechanik.com/products/) | Germany |
| Cedrat Technologies | [link](https://www.cedrat-technologies.com/en/products/piezo-controllers/electronic-amplifier-boards.html) | France |
| Trek | [link](https://www.trekinc.com/products/HV%5FAmp.asp) | USA |
| Madcitylabs | [link](http://www.madcitylabs.com/piezoactuators.html) | USA |
| Piezosystem | [link](https://www.piezosystem.com/products/controller/) | Germany |
| Matsusada Precision | [link](https://www.matsusada.com/product/pz/) | Japan |
| Mechano Transformer | [link](http://www.mechano-transformer.com/en/products/08.html) | Japan |
## Bibliography {#bibliography}
<a id="org01aad4a"></a>Fleming, Andrew J., and Kam K. Leang. 2014. _Design, Modeling and Control of Nanopositioning Systems_. Advances in Industrial Control. Springer International Publishing. <https://doi.org/10.1007/978-3-319-06617-2>.
## Backlinks {#backlinks}
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}})
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}})

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