Update Content - 2020-10-26

content/zettels/mass_spring_damper_systems.md

@@ -7,4 +7,52 @@ 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>