The transfer function for the charge amplifier is
$$T(s)=\frac{Gs}{\tau s+1}$$
and the Transfer function for the mechanical system is
$$T(s)=\frac{1}{ms^2+bs+k}$$
When we take the inverse Laplace transform of these equations multiplied together we get
$$
\begin{matrix}
\ddot{x}(t)=-\frac{k}{m}x(t)-\frac{b}{m}\dot{x}(t)+F \\
\dot{v}(t)=\frac{1}{\tau}v(t)+G\dot{x}(t)
\end{matrix}$$
Without states being \(x(t), \dot{x}(t),v(t)\) the state-space model of the accelerometer becomes
$$\begin{bmatrix}
\dot{x}(t)\\ \ddot{x}(t)\\ \dot{v}(t)
\end{bmatrix}=
\begin{bmatrix}
0&1&0\\
-\frac{k}{m}&-\frac{b}{m}&0\\
0&G&-\frac{1}{\tau}
\end{bmatrix}
\begin{bmatrix}
x(t)\\ \dot{x}(t)\\ v(t)
\end{bmatrix}+
\begin{bmatrix}
0\\ 0\\ 1
\end{bmatrix}F$$
$$y(t)=
\begin{bmatrix}
-\frac{k}{m}C&-\frac{b}{m}C&0
\end{bmatrix}
\begin{bmatrix}
x(t)\\ \dot{x}(t)\\ v(t)
\end{bmatrix}+CF$$
- candyAccelerometerMode
General General Accelerometer State-Space Model
[[Piezoelectric Accelerometers]]
[[State-Space Model]]