Difference between critically damped systems and overdamped systems

2.1k Views Asked by At

Consider a spring system whose equation is given by $$my''+\mu y'+ky=0$$ and let $D=\mu^2-4mk$. Now there are three cases and I am considering the cases that $D=0$ and $D>0$:

  1. When $D=0$, the solution is of the form $y=(a+bt)e^{rt}$. (Critically damped)
  2. When $D>0$, the solution is of the form $y=c_1e^{r_1t}+c_2e^{r_2t}$. (Overdamped)

While I understand that these two cases are very different and the function $y=(a+bt)e^{rt}$ is very different from the function $y=c_1e^{r_1t}+c_2e^{r_2t}$, it also seems to me that the two functions have very similar graphs (in particular, very similar end behaviours).

Question

  1. Why are the graphs of the solution in these two cases so similar (while in the other case $D<0$, the graph is very different)?
  2. How to tell from the graph whether we have a critically damped system or an overdamped system?