Update Content - 2023-01-24

content/phdthesis/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md

@@ -1,13 +1,12 @@
 +++
 title = "Element and system design for active and passive vibration isolation"
-author = ["Dehaeze Thomas"]
 draft = false
 ref_author = "Zuo, L."
 ref_year = 2004
 +++
 
 Tags
-: [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
+: [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Eddy Current Damping]({{< relref "eddy_current_damping.md" >}})
 
 Reference
 : (<a href="#citeproc_bib_item_1">Zuo 2004</a>)
@@ -28,21 +27,47 @@ Year
 > They found that coupling from flexible modes is much smaller than in soft active mounts in the load (force) feedback.
 > Note that reaction force actuators can also work with soft mounts or hard mounts.
 
+
+## Passive Vibration Isolation {#passive-vibration-isolation}
+
+
+### The Role of damping and its practical constructions {#the-role-of-damping-and-its-practical-constructions}
+
+
+#### Viscous damping {#viscous-damping}
+
+
+#### Eddy-current damper {#eddy-current-damper}
+
+<a id="figure--fig:zuo04-eddy-current-magnets"></a>
+
+{{< figure src="/ox-hugo/zuo04_eddy_current_magnets.png" caption="<span class=\"figure-number\">Figure 1: </span>(left) Magnetic field and conductor plates assemblies, (right) magnet arrays" >}}
+
+<a id="figure--fig:zuo04-eddy-current-setup"></a>
+
+{{< figure src="/ox-hugo/zuo04_eddy_current_setup.png" caption="<span class=\"figure-number\">Figure 2: </span>Single DoF system damped by eddy current damper" >}}
+
+
+## Elements and configurations for active vibration systems {#elements-and-configurations-for-active-vibration-systems}
+
+
+### System architectures {#system-architectures}
+
 <a id="figure--fig:zuo04-piezo-spring-series"></a>
 
 {{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="<span class=\"figure-number\">Figure 1: </span>PZT actuator and spring in series" >}}
 
 <a id="figure--fig:zuo04-voice-coil-spring-parallel"></a>
 
-{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="<span class=\"figure-number\">Figure 2: </span>Voice coil actuator and spring in parallel" >}}
+{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="<span class=\"figure-number\">Figure 1: </span>Voice coil actuator and spring in parallel" >}}
 
 <a id="figure--fig:zuo04-piezo-plant"></a>
 
-{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="<span class=\"figure-number\">Figure 3: </span>Transmission from PZT voltage to geophone output" >}}
+{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="<span class=\"figure-number\">Figure 1: </span>Transmission from PZT voltage to geophone output" >}}
 
 <a id="figure--fig:zuo04-voice-coil-plant"></a>
 
-{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="<span class=\"figure-number\">Figure 4: </span>Transmission from voice coil voltage to geophone output" >}}
+{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="<span class=\"figure-number\">Figure 1: </span>Transmission from voice coil voltage to geophone output" >}}
 
 
 ## Bibliography {#bibliography}

content/zettels/eddy_current_damping.md

@@ -1,6 +1,5 @@
 +++
 title = "Eddy Current Damping"
-author = ["Dehaeze Thomas"]
 draft = false
 +++
 
@@ -15,7 +14,96 @@ Tags
 <https://www.mceproducts.com/articles/magnets-in-vacuum-applications>
 
 
+## Estimate the damping {#estimate-the-damping}
+
+
+### Formulas {#formulas}
+
+From (<a href="#citeproc_bib_item_1">Zuo 2004</a>):
+The empirical formula for damping coefficient (Ns/m) of an eddy current damper is:
+
+\begin{equation} \label{eq:damping\_formula}
+C = C\_0 B^2 t A \sigma
+\end{equation}
+
+with:
+
+-   \\(B\\) is the magnetic flux density in [T] or in [Vs/m2]
+-   \\(t\\) is the thickness of the conductor plate in [m]
+-   \\(A\\) is the area of the conductor intersected by the magnetic field in [m2]
+-   \\(\sigma\\) is the electrical conductivity of the conductor material [S/m]
+-   \\(C\_0\\) is a dimensionless coefficient to account for the shapes and sizes of the conductor and magnetic field
+
+\\(C\_0 = 1\\) corresponds to a conductor with conductivity \\(\sigma\\) inside a uniform magnetic field and conductivity infinite outside this field.
+A typical value of \\(C\_0\\) is about 0.25-0.4 for a conductor plate with area 2 to 5 times that of the magnetic field.
+
+From <eq:damping_formula>, we see that the damping coefficient is proportional to:
+
+-   the square of the magnetic flux density \\(B\\). Therefore it is very important to have large magnetic field strengh
+-   the thickness \\(t\\) of the conductor. However due to **skin depth effect**,  the benefit of increasing the thickness is limited.
+    The apparent conductivity \\(\sigma\_e\\) is:
+
+    \begin{equation}
+      \sigma\_e = \frac{2\delta\_s}{t}(1 - e^{-\frac{t}{2\delta\_s}})\sigma
+    \end{equation}
+
+    where \\(\delta\_s\\) is the skin depth in [m] of the conductor with permeability \\(\mu\\) in [H/m] at frequency \\(f\\) in [Hz]:
+
+    \begin{equation}
+      \delta\_s = \sqrt{\frac{2}{2 \pi f \cdot \mu \cdot \sigma}}
+    \end{equation}
+
+An eddy current damper is developed in (<a href="#citeproc_bib_item_1">Zuo 2004</a>).
+The magnets have alternating poles to optimize the eddy current damping (stronger varying magnetic field).
+See Figures [1](#figure--fig:zuo04-eddy-current-magnets) and [2](#figure--fig:zuo04-eddy-current-setup).
+
+<a id="figure--fig:zuo04-eddy-current-magnets"></a>
+
+{{< figure src="/ox-hugo/zuo04_eddy_current_magnets.png" caption="<span class=\"figure-number\">Figure 1: </span>(left) Magnetic field and conductor plates assemblies, (right) magnet arrays" >}}
+
+<a id="figure--fig:zuo04-eddy-current-setup"></a>
+
+{{< figure src="/ox-hugo/zuo04_eddy_current_setup.png" caption="<span class=\"figure-number\">Figure 1: </span>Single DoF system damped by eddy current damper" >}}
+
+
+### Numerical Simulation {#numerical-simulation}
+
+It is possible to estimate that with FEM simulation: <https://www.youtube.com/watch?v=_1pgyj4lD7Q>
+
+An approximation is done bellow.
+
+```matlab
+B = 1.0; % Magnetic Flux Density [T]
+t = 5e-3; % Thickness [m]
+A = 50e-3*50e-3; % Area [m2]
+sigma = 6e7; % Copper conductivity [S/m]
+C0 = 0.5; % [-]
+```
+
+```matlab
+C = C0*B^2*t*A*sigma; % Damping in [N/(m/s)]
+```
+
+```text
+C = 375 [N/(m/s)]
+```
+
+```matlab
+m = 10; % [kg]
+k = m*(2*pi*10)^2; % [N/m]
+```
+
+```matlab
+xi = 1/2*C/sqrt(k*m);
+```
+
+```text
+xi = 0.298
+```
+
+
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology.</div>
 </div>

content/zettels/encoders.md

@@ -22,6 +22,7 @@ There are two main types of encoders: optical encoders, and magnetic encoders.
 | [RLS](https://www.rls.si/eng/products/rotary-magnetic-encoders)          | Slovenia    |
 | [AMO](https://www.amo-gmbh.com/en/)                                      | Australia   |
 | [NumerikJena](https://www.numerikjena.de/en/)                            | Germany     |
+| [RSF Elektronik](https://www.rsf.at/en/)                                 | Austria     |
 
 
 ## Bibliography {#bibliography}

content/zettels/mass_spring_damper_systems.md

@@ -7,7 +7,10 @@ Tags
 :
 
 
-## Actuated Mass Spring Damper System {#actuated-mass-spring-damper-system}
+## One Degree of Freedom {#one-degree-of-freedom}
+
+
+### Model and equation of motion {#model-and-equation-of-motion}
 
 Let's consider Figure [1](#figure--fig:mass-spring-damper-system) where:
 
@@ -23,57 +26,150 @@ Let's consider Figure [1](#figure--fig:mass-spring-damper-system) where:
 
 {{< figure src="/ox-hugo/mass_spring_damper_system.png" caption="<span class=\"figure-number\">Figure 1: </span>Mass Spring Damper System" >}}
 
-Let's write the transfer function from \\(F\\) to \\(x\\):
+Transmissibility:
 
 \begin{equation}
-  \frac{x}{F}(s) = \frac{1}{m s^2 + c s + k}
+  \frac{x}{w}(s) = \frac{c s + k}{m s^2 + c s + k} = \frac{2 \xi \frac{s}{\omega\_0} + 1}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1}
 \end{equation}
 
-This can be re-written as:
+Compliance:
 
 \begin{equation}
-  \frac{x}{F}(s) = \frac{1/k}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1}
+  \frac{x}{F}(s) = \frac{x}{F\_d}(s) = \frac{1}{m s^2 + c s + k} = \frac{1/k}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1}
 \end{equation}
 
 with:
 
--   \\(\omega\_0\\) the natural frequency in [rad/s]
--   \\(\xi\\) the damping ratio
+-   \\(\omega\_0 = \sqrt{k/m}\\) the natural frequency in [rad/s]
+-   \\(\xi = \frac{1}{2} \frac{c}{\sqrt{km}}\\) the damping ratio [unit-less]
 
 
-## Transfer function {#transfer-function}
-
-
-### Voice Coil Actuator with flexible guiding {#voice-coil-actuator-with-flexible-guiding}
+### Matlab model {#matlab-model}
 
 ```matlab
 %% Mechanical properties
 m = 1; % Mobile mass [kg]
 k = 1e6; % stiffness [N/m]
-xi = 0.01; % Modal Damping
+xi = 0.1; % Modal Damping
 
 c = 2*xi*sqrt(k*m);
 ```
 
 ```matlab
-%% Transfer function from F [N] to x [m]
-G = 1/(m*s^2 + c*s + k);
+%% Compliance: Transfer function from F [N] to x [m]
+Gf = 1/(m*s^2 + c*s + k);
+
+%% Transmissibility: Transfer function from w [m] to x [m]
+Gw = (c*s + k)/(m*s^2 + c*s + k);
 ```
 
+<a id="figure--fig:mass-spring-damper-1dof-compliance"></a>
 
-### Transmissibility {#transmissibility}
+{{< figure src="/ox-hugo/mass_spring_damper_1dof_compliance.png" caption="<span class=\"figure-number\">Figure 2: </span>1dof Mass spring damper system - Compliance" >}}
+
+<a id="figure--fig:mass-spring-damper-1dof-transmissibility"></a>
+
+{{< figure src="/ox-hugo/mass_spring_damper_1dof_transmissibility.png" caption="<span class=\"figure-number\">Figure 1: </span>1dof Mass spring damper system - Transmissibility" >}}
+
+
+## Two Degrees of Freedom {#two-degrees-of-freedom}
+
+
+### Model and equation of motion {#model-and-equation-of-motion}
+
+Consider the two degrees of freedom mass spring damper system of Figure [1](#figure--fig:mass-spring-damper-2dof).
+
+<a id="figure--fig:mass-spring-damper-2dof"></a>
+
+{{< figure src="/ox-hugo/mass_spring_damper_2dof.png" caption="<span class=\"figure-number\">Figure 1: </span>2 DoF Mass Spring Damper system" >}}
+
+We can write the Newton's second law of motion to the two masses:
+
+\begin{align}
+m\_2 s^2 x\_2 &= F\_2 + (k\_2 + c\_2 s) (x\_1 - x\_2) \\\\
+m\_1 s^2 x\_1 &= F\_1 + (k\_1 + c\_1 s) (x\_0 - x\_1)  + (k\_2 + c\_2 s) (x\_2 - x\_1)
+\end{align}
+
+The goal is to have \\(x\_1\\) and \\(x\_2\\) as a function of \\(F\_1\\), \\(F\_2\\) and \\(x\_0\\).
+
+When, we have:
 
 \begin{equation}
-  \frac{x}{w}(s) = \frac{1}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1}
+\boxed{x\_1 = \frac{(m\_2 s^2 + c\_2 s + k\_2) F\_1 + (k\_1 + c\_1 s) (m\_2 s^2 + c\_2 s + k\_2) x\_0  + (k\_2 + c\_2 s) F\_2}{(m\_1 s^2 + c\_1 s + k\_1)(m\_2 s^2 + c\_2 s + k\_2) + m\_2 s^2 (c\_2 s + k\_2)}}
 \end{equation}
 
-
-### Compliance {#compliance}
-
 \begin{equation}
-  \frac{x}{F\_d}(s) = \frac{1/k}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1}
+\boxed{x\_2 = \frac{(c\_2s + k\_2)F\_1 + (c\_2s + k\_2)(k\_1 + c\_1 s) x\_0  + (m\_1 s^2 + c\_1 s + k\_1 + c\_2 s + k\_2) F\_2}{(m\_1 s^2 + c\_1 s + k\_1)(m\_2 s^2 + c\_2 s + k\_2) + m\_2 s^2 (c\_2 s + k\_2)}}
 \end{equation}
 
+We can see that the effects of \\(x\_0\\) and \\(F\_1\\) are related with a factor \\((c\_1 s + k\_1)\\).
+
+If we are interested by \\(x\_2-x\_1\\):
+
+\begin{equation}
+(x\_2 - x1) = \frac{- m\_2 s^2 F\_1 - (m\_2 s^2)(k\_1 + c\_1 s) x\_0  + (m\_1 s^2 + c\_1 s + k\_1) F\_2}{(m\_1 s^2 + c\_1 s + k\_1)(m\_2 s^2 + c\_2 s + k\_2) + m\_2 s^2 (c\_2 s + k\_2)}
+\end{equation}
+
+|    | x1                          | x2                         | x2-x1              |
+|----|-----------------------------|----------------------------|--------------------|
+| x0 | (c1s + k1)(m2s2 + c2s + k2) | (c1s + k1)(c2s + k2)       | - m2s2\*(k1 + c1s) |
+| F1 | m2s2 + c2s + k2             | c2s + k2                   | - m2s2             |
+| F2 | c2s + k2                    | m1s2 + c1s + k1 + c2s + k2 | m1s2 + c1s + k1    |
+
+
+### Matlab model {#matlab-model}
+
+```matlab
+%% Values for the 2dof Mass-Spring-Damper system
+m1 = 5e2; % [kg]
+k1 = 2e6; % [N/m]
+c1 = 2*0.01*sqrt(m1*k1); % [N/(m/s)]
+
+m2 = 10; % [kg]
+k2 = 1e6; % [N/m]
+c2 = 2*0.01*sqrt(m2*k2); % [N/(m/s)]
+```
+
+```matlab
+%% Transfer functions
+G_x0_to_x1 = (c1*s + k1)*(m2*s^2 + c2*s + k2)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2));
+G_F1_to_x1 = (m2*s^2 + c2*s + k2)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2));
+G_F2_to_x1 = (c2*s + k2)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2));
+
+G_x0_to_x2 = (c1*s + k1)*(c2*s + k2)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2));
+G_F1_to_x2 = (c2*s + k2)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2));
+G_F2_to_x2 = (m1*s^2 + c1*s + k1 + c2*s + k2)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2));
+
+G_x0_to_d2 = -m2*s^2*(c1*s + k1)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2));
+G_F1_to_d2 = -m2*s^2/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2));
+G_F2_to_d2 = (m1*s^2 + c1*s + k1)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2));
+```
+
+From Figure [1](#figure--fig:mass-spring-damper-2dof-x0-bode-plots), we can see that:
+
+-   the low frequency transmissibility is equal to one
+-   the high frequency transmissibility to the second mass is smaller than to the first mass
+
+<a id="figure--fig:mass-spring-damper-2dof-x0-bode-plots"></a>
+
+{{< figure src="/ox-hugo/mass_spring_damper_2dof_x0_bode_plots.png" caption="<span class=\"figure-number\">Figure 1: </span>Transfer functions from x0 to x1 and x2 (Transmissibility)" >}}
+
+The transfer function from \\(F\_1\\) to the mass displacements (Figure [1](#figure--fig:mass-spring-damper-2dof-F1-bode-plots)) has the same shape than the transmissibility (Figure [1](#figure--fig:mass-spring-damper-2dof-x0-bode-plots)).
+
+However, the low frequency gain is now equal to \\(1/k\_1\\).
+
+<a id="figure--fig:mass-spring-damper-2dof-F1-bode-plots"></a>
+
+{{< figure src="/ox-hugo/mass_spring_damper_2dof_F1_bode_plots.png" caption="<span class=\"figure-number\">Figure 1: </span>Transfer functions from F1 to x1 and x2" >}}
+
+The transfer functions from \\(F\_2\\) to the mass displacements are shown in Figure [1](#figure--fig:mass-spring-damper-2dof-F2-bode-plots):
+
+-   the motion \\(x\_1\\) is smaller than \\(x\_2\\)
+
+<a id="figure--fig:mass-spring-damper-2dof-F2-bode-plots"></a>
+
+{{< figure src="/ox-hugo/mass_spring_damper_2dof_F2_bode_plots.png" caption="<span class=\"figure-number\">Figure 1: </span>Transfer functions from F2 to x1 and x2" >}}
+
 
 ## Bibliography {#bibliography}
 

content/zettels/tuned_mass_damper.md

@@ -1,11 +1,10 @@
 +++
 title = "Tuned Mass Damper"
-author = ["Dehaeze Thomas"]
 draft = false
 +++
 
 Tags
-: [Passive Damping]({{< relref "passive_damping.md" >}})
+: [Passive Damping]({{< relref "passive_damping.md" >}}), [Mass Spring Damper Systems]({{< relref "mass_spring_damper_systems.md" >}})
 
 Review: (<a href="#citeproc_bib_item_1">Elias and Matsagar 2017</a>), (<a href="#citeproc_bib_item_2">Verbaan 2015</a>)
 
@@ -48,7 +47,10 @@ The optimal parameters of the tuned mass damper can be roughly estimated as foll
 
 ## Simple TMD model {#simple-tmd-model}
 
-Let's consider a primary system that is represented by a mass-spring-damper system with the following parameters: \\(m\_1\\), \\(k\_1\\), \\(c\_1\\).
+
+### Model {#model}
+
+Let's consider a primary system that is represented by a [Mass Spring Damper Systems]({{< relref "mass_spring_damper_systems.md" >}}) with the following parameters: \\(m\_1\\), \\(k\_1\\), \\(c\_1\\).
 The TMD is also represented by a mass-spring-damper system with parameters \\(m\_2\\), \\(k\_2\\), \\(c\_2\\).
 The system is schematically represented in Figure [1](#figure--fig:tuned-mass-damper-schematic).
 

content/zettels/voice_coil_actuators.md

@@ -1,6 +1,5 @@
 +++
 title = "Voice Coil Actuators"
-author = ["Dehaeze Thomas"]
 draft = false
 category = "equipment"
 +++
@@ -37,7 +36,6 @@ As the force is proportional to the current, a [Transconductance Amplifiers]({{<
 | [H2tech](https://www.h2wtech.com/)                                                                                                  | USA                                                                                                                                                      |
 | [Beikimco](http://www.beikimco.com/)                                                                                                | USA                                                                                                                                                      |
 | [Monticont](http://www.moticont.com/)                                                                                               | USA                                                                                                                                                      |
-| [Thorlabs](https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=14116)                                                         | USA                                                                                                                                                      |
 | [Celera](https://www.celeramotion.com/applimotion/products/direct-drive-frameless-linear-motors/voice-coil/juke-series-round-body/) |                                                                                                                                                          |