+++
title = "Eddy Current Damping"
draft = false
+++
Tags
: [Passive Damping]({{< relref "passive_damping.md" >}})
## Vacuum compatible magnets {#vacuum-compatible-magnets}
## Estimate the damping {#estimate-the-damping}
### Formulas {#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 , 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 (Zuo 2004).
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).
{{< figure src="/ox-hugo/zuo04_eddy_current_magnets.png" caption="Figure 1: (left) Magnetic field and conductor plates assemblies, (right) magnet arrays" >}}
{{< figure src="/ox-hugo/zuo04_eddy_current_setup.png" caption="Figure 1: Single DoF system damped by eddy current damper" >}}
### Numerical Simulation {#numerical-simulation}
It is possible to estimate that with FEM simulation:
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}
Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology.