\(\displaystyle M\frac{d^2x}{dt^2} + f_v\frac{dx}{dt} + kx = f(t)\)
I have that equation and when I apply laplace transform assuming all initial conditions are zero, I get this.
transfer function \(\displaystyle = \frac{X(s)}{F(s)} = \frac{1}{Ms^2 + f_vs + k}\)
Now I have to find the state-space representation of that.
According to this video, I get this.
\(\displaystyle \dot{X} = \begin{bmatrix}\dot{x_1} \\\dot{x_2} \end{bmatrix} = \begin{bmatrix}0 & 1 \\-k & -f_v \end{bmatrix} \begin{bmatrix}x_1 \\x_2 \end{bmatrix} + \begin{bmatrix}0 \\1 \end{bmatrix}f(t)\)
According to my notes, I get this.
\(\displaystyle \dot{X} = \begin{bmatrix}\dot{x_1} \\\dot{x_2} \end{bmatrix} = \frac{1}{M}\begin{bmatrix}0 & 1 \\k & f_v \end{bmatrix} \begin{bmatrix}x_1 \\x_2 \end{bmatrix} + \begin{bmatrix}0 \\1 \end{bmatrix}f(t)\)
Which method is the correct one?
Any help would be appreciated!
I have that equation and when I apply laplace transform assuming all initial conditions are zero, I get this.
transfer function \(\displaystyle = \frac{X(s)}{F(s)} = \frac{1}{Ms^2 + f_vs + k}\)
Now I have to find the state-space representation of that.
According to this video, I get this.
\(\displaystyle \dot{X} = \begin{bmatrix}\dot{x_1} \\\dot{x_2} \end{bmatrix} = \begin{bmatrix}0 & 1 \\-k & -f_v \end{bmatrix} \begin{bmatrix}x_1 \\x_2 \end{bmatrix} + \begin{bmatrix}0 \\1 \end{bmatrix}f(t)\)
According to my notes, I get this.
\(\displaystyle \dot{X} = \begin{bmatrix}\dot{x_1} \\\dot{x_2} \end{bmatrix} = \frac{1}{M}\begin{bmatrix}0 & 1 \\k & f_v \end{bmatrix} \begin{bmatrix}x_1 \\x_2 \end{bmatrix} + \begin{bmatrix}0 \\1 \end{bmatrix}f(t)\)
Which method is the correct one?
Any help would be appreciated!