digital-brain/content/zettels/mass_spring_damper_systems.md

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title = "Mass Spring Damper Systems"
author = ["Thomas Dehaeze"]
draft = false
+++
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## Actuated Mass Spring Damper System {#actuated-mass-spring-damper-system}
Let's consider Figure [1](#orgeec8f0f) where:
- \\(m\\) is the mass in [kg]
- \\(ḱ\\) 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="orgeec8f0f"></a>
{{< figure src="/ox-hugo/mass_spring_damper_system.png" caption="Figure 1: 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
## 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}
<./biblio/references.bib>