digital-brain/content/zettels/mass_spring_damper_systems.md

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title = "Mass Spring Damper Systems"
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## Actuated Mass Spring Damper System {#actuated-mass-spring-damper-system}

Let's consider Figure [1](#figure--fig:mass-spring-damper-system) where:

-   \\(m\\) is the mass in [kg]
-   \\(k\\) is the spring stiffness in [N/m]
-   \\(c\\) is the damping coefficient in [N/(m/s)]
-   \\(F\\) is the actuator force in [N]
-   \\(F\_d\\) is external force applied to the mass in [N]
-   \\(w\\) is ground motion
-   \\(x\\) is the absolute mass motion

<a id="figure--fig:mass-spring-damper-system"></a>

{{< 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\\):

\begin{equation}
  \frac{x}{F}(s) = \frac{1}{m s^2 + c s + k}
\end{equation}

This can be re-written as:

\begin{equation}
  \frac{x}{F}(s) = \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


## Transfer function {#transfer-function}


### Voice Coil Actuator with flexible guiding {#voice-coil-actuator-with-flexible-guiding}

```matlab
%% Mechanical properties
m = 1; % Mobile mass [kg]
k = 1e6; % stiffness [N/m]
xi = 0.01; % 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);
```


### Transmissibility {#transmissibility}

\begin{equation}
  \frac{x}{w}(s) = \frac{1}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1}
\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}
\end{equation}


## Bibliography {#bibliography}

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