110 lines
3.6 KiB
Markdown
110 lines
3.6 KiB
Markdown
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title = "Eddy Current Damping"
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draft = false
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Tags
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: [Passive Damping]({{< relref "passive_damping.md" >}})
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<https://courses.lumenlearning.com/suny-physics/chapter/23-4-eddy-currents-and-magnetic-damping/>
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## Vacuum compatible magnets {#vacuum-compatible-magnets}
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<https://www.mceproducts.com/articles/magnets-in-vacuum-applications>
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## Estimate the damping {#estimate-the-damping}
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### Formulas {#formulas}
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From (<a href="#citeproc_bib_item_1">Zuo 2004</a>):
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The empirical formula for damping coefficient (Ns/m) of an eddy current damper is:
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\begin{equation} \label{eq:damping\_formula}
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C = C\_0 B^2 t A \sigma
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\end{equation}
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with:
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- \\(B\\) is the magnetic flux density in [T] or in [Vs/m2]
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- \\(t\\) is the thickness of the conductor plate in [m]
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- \\(A\\) is the area of the conductor intersected by the magnetic field in [m2]
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- \\(\sigma\\) is the electrical conductivity of the conductor material [S/m]
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- \\(C\_0\\) is a dimensionless coefficient to account for the shapes and sizes of the conductor and magnetic field
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\\(C\_0 = 1\\) corresponds to a conductor with conductivity \\(\sigma\\) inside a uniform magnetic field and conductivity infinite outside this field.
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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.
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From <eq:damping_formula>, we see that the damping coefficient is proportional to:
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- the square of the magnetic flux density \\(B\\). Therefore it is very important to have large magnetic field strengh
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- the thickness \\(t\\) of the conductor. However due to **skin depth effect**, the benefit of increasing the thickness is limited.
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The apparent conductivity \\(\sigma\_e\\) is:
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\begin{equation}
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\sigma\_e = \frac{2\delta\_s}{t}(1 - e^{-\frac{t}{2\delta\_s}})\sigma
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\end{equation}
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where \\(\delta\_s\\) is the skin depth in [m] of the conductor with permeability \\(\mu\\) in [H/m] at frequency \\(f\\) in [Hz]:
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\begin{equation}
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\delta\_s = \sqrt{\frac{2}{2 \pi f \cdot \mu \cdot \sigma}}
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\end{equation}
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An eddy current damper is developed in (<a href="#citeproc_bib_item_1">Zuo 2004</a>).
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The magnets have alternating poles to optimize the eddy current damping (stronger varying magnetic field).
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See Figures [1](#figure--fig:zuo04-eddy-current-magnets) and [2](#figure--fig:zuo04-eddy-current-setup).
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<a id="figure--fig:zuo04-eddy-current-magnets"></a>
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{{< 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" >}}
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<a id="figure--fig:zuo04-eddy-current-setup"></a>
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{{< 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" >}}
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### Numerical Simulation {#numerical-simulation}
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It is possible to estimate that with FEM simulation: <https://www.youtube.com/watch?v=_1pgyj4lD7Q>
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An approximation is done bellow.
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```matlab
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B = 1.0; % Magnetic Flux Density [T]
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t = 5e-3; % Thickness [m]
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A = 50e-3*50e-3; % Area [m2]
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sigma = 6e7; % Copper conductivity [S/m]
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C0 = 0.5; % [-]
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```
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```matlab
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C = C0*B^2*t*A*sigma; % Damping in [N/(m/s)]
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```
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```text
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C = 375 [N/(m/s)]
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```
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```matlab
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m = 10; % [kg]
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k = m*(2*pi*10)^2; % [N/m]
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```
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```matlab
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xi = 1/2*C/sqrt(k*m);
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```
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```text
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xi = 0.298
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```
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## Bibliography {#bibliography}
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<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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<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>
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</div>
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