3.6 KiB
+++ title = "Eddy Current Damping" draft = false +++
- Tags
- [Passive Damping]({{< relref "passive_damping.md" >}})
https://courses.lumenlearning.com/suny-physics/chapter/23-4-eddy-currents-and-magnetic-damping/
Vacuum compatible magnets
https://www.mceproducts.com/articles/magnets-in-vacuum-applications
Estimate the damping
Formulas
From (Zuo 2004): 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:
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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. 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 (Zuo 2004). The magnets have alternating poles to optimize the eddy current damping (stronger varying magnetic field). See Figures 1 and 2.
{{< figure src="/ox-hugo/zuo04_eddy_current_magnets.png" caption="<span class="figure-number">Figure 1: (left) Magnetic field and conductor plates assemblies, (right) magnet arrays" >}}
{{< figure src="/ox-hugo/zuo04_eddy_current_setup.png" caption="<span class="figure-number">Figure 1: Single DoF system damped by eddy current damper" >}}
Numerical Simulation
It is possible to estimate that with FEM simulation: https://www.youtube.com/watch?v=_1pgyj4lD7Q
An approximation is done bellow.
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; % [-]
C = C0*B^2*t*A*sigma; % Damping in [N/(m/s)]
C = 375 [N/(m/s)]
m = 10; % [kg]
k = m*(2*pi*10)^2; % [N/m]
xi = 1/2*C/sqrt(k*m);
xi = 0.298