Single Degree of Freedom Vibrational Motion Equation
Hello, I’m Hiroki🐶, a data scientist.
A month has almost passed since the start of 2024.
Time flies, doesn’t it? Writing this blog allows me to forget about my busy days and express my own worldview, which is an important time for me.
Now, in my previous blog, I introduced the Matlab code for solving the single degree of freedom vibrational motion equation shown in the diagram below:
The input f is given as a force input like this:
$$\begin{align*}
f=c\dot{y}+ky& \
\end{align*}$$
However, I believe you might be interested in the state space representation when displacement input is provided, as shown in the diagram.
Therefore, this time,
- I will introduce how to express the behavior of a single degree of freedom vibration model when displacement input is provided, using state space representation
First, this single degree of freedom vibration model can be written using the motion equation as follows:
$$\begin{align*}
m\ddot{x}+c\dot{x}+kx&=c\dot{y}+ky& \
\end{align*}$$
If we apply the Laplace transform and find the transfer function G (output/input), we get:
$$\begin{align*}
(ms^2+cs+k)x=(cs+k)y \
\end{align*}$$
Therefore,
$$\begin{align*}
G=\frac{x}{y}=\frac{cs+k}{ms^2+cs+k} \
\end{align*}$$
Once the transfer function is known, all that’s left is to convert it into a state space representation. Actually, there is a formula for this conversion.
In the future posts, I plan to explain this in detail using equations.
That’s all for today. Thank you very much for reading until the end.
Love&Respect♡
Hiroki🐶