59 lines
1.3 KiB
Markdown
59 lines
1.3 KiB
Markdown
+++
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title = "Mass Spring Damper Systems"
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author = ["Thomas Dehaeze"]
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draft = false
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## Actuated Mass Spring Damper System {#actuated-mass-spring-damper-system}
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Let's consider Figure [1](#orgeec8f0f) where:
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- \\(m\\) is the mass in [kg]
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- \\(ḱ\\) is the spring stiffness in [N/m]
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- \\(c\\) is the damping coefficient in [N/(m/s)]
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- \\(F\\) is the actuator force in [N]
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- \\(F\_d\\) is external force applied to the mass in [N]
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- \\(w\\) is ground motion
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- \\(x\\) is the absolute mass motion
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<a id="orgeec8f0f"></a>
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{{< figure src="/ox-hugo/mass_spring_damper_system.png" caption="Figure 1: Mass Spring Damper System" >}}
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Let's write the transfer function from \\(F\\) to \\(x\\):
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\begin{equation}
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\frac{x}{F}(s) = \frac{1}{m s^2 + c s + k}
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\end{equation}
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This can be re-written as:
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\begin{equation}
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\frac{x}{F}(s) = \frac{1/k}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1}
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\end{equation}
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with:
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- \\(\omega\_0\\) the natural frequency in [rad/s]
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- \\(\xi\\) the damping ratio
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## Transmissibility {#transmissibility}
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\begin{equation}
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\frac{x}{w}(s) = \frac{1}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1}
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\end{equation}
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## Compliance {#compliance}
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\begin{equation}
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\frac{x}{F\_d}(s) = \frac{1/k}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1}
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\end{equation}
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<./biblio/references.bib>
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