1.9 KiB
1.9 KiB
+++ title = "Mass Spring Damper Systems" draft = false +++
Tags :
Actuated Mass Spring Damper System
Let's consider Figure 1 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
{{< figure src="/ox-hugo/mass_spring_damper_system.png" caption="<span class="figure-number">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
Transfer function
Voice Coil Actuator with flexible guiding
%% Mechanical properties
m = 1; % Mobile mass [kg]
k = 1e6; % stiffness [N/m]
xi = 0.01; % Modal Damping
c = 2*xi*sqrt(k*m);
%% Transfer function from F [N] to x [m]
G = 1/(m*s^2 + c*s + k);
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
\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}