An object M is attached to a spring and damper, the spring and damper are attached to a fixed reference point and there is a force applied to the mass. We can model the system as follows.
The total force acting on the mass is the combined force of the Spring, Damper, and external force
$$F=F_{spring}+F_{damper}+F_{applied}$$
where the spring force is
$$F_{spring}=-k_sx$$
with \(k_s\) being the spring constant. The damper force can be described with a different constant:
$$F_{damper}=-k_d \dot{x}$$
where \(k_d\) is the damping constant. When combined with Newton’s second law \(F=ma\) then the system dynamic can be described by
$$m \ddot{x}=F_{applied} – k_d\dot{x}-k_sx$$
which can be rearranged to
$$F_{applied}=m\ddot{x}+k_d\dot{x}+k_sx$$
This gives us a second order linear ordinary differential equation, which has the general solution of
$$x=x(t,x(0),\dot{x}(0),F(t))$$
where \(t\) is the time, \(x(0)\) is the initial position, \(\dot{x}(0)\) is the initial velocity, and \(F(t)\) is the external applied force. We can then use this to make a state-space model where our states are position and velocity, with the input being force. The state variables are:
$$x_1=x$$
$$x_2=\dot{x}$$
We then differentiate these to states by using the equations above.
$$\dot{x}_1=\dot{x}$$
$$\dot{x_2}=\ddot{x}=-\frac{k_d}{m}\dot{x}-\frac{k_s}{m}x+\frac{F}{m}$$
We can then put these equations into the state space form :
$$\begin{bmatrix}
\dot{x}_1 \\
\dot{x}_2
\end{bmatrix} =
\begin{bmatrix}
0 & 1 \\
-\frac{k_s}{m} & -\frac{k_d}{m}
\end{bmatrix}
\begin{bmatrix}
x_1 \\ x_2
\end{bmatrix}+
\begin{bmatrix}
0 \\ \frac{1}{m}
\end{bmatrix}F$$
$$y=
\begin{bmatrix}
1 & 0
\end{bmatrix}x$$
The transfer function of a spring-mass-damper is
$$\frac{X(s)}{F(s)}=\frac{1}{ms^2+bs+k}$$
[[Order of a Differential Equation]]
[[Forward Euler]]
[[Python State-Space Spring Mass Damper System]]
[[Double Spring-mass-damper System]]
[[Dynamic Inversion Control for a Spring Mass damper System]]
[[Spring Mass System]] – same as the spring-mass damper system but without the damper.
[[Accelerometer Mechanical Equations]] – modeled as a spring-mass-damper system
[[Bond Graph of a Spring Mass Damper System]]
[[Equivalent Reduced Wing Structure]] – lumped-parameter model for a discretized spanwise wing section using individual spring-mass-damper models
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Backlinks:
[[Dynamic Systems]]
[[Second-Order Systems]]
State-Space Model
[[Transfer Functions]]