A student asked me how to find the time delay, $t_d$  for a second-order system by applying mathematical equation.

Let $c_{final}$ denote the final value of the waveform. Then, the rise time $t_r$ will be the time duration between $0.1c_{final}$ to $0.9c_{final}$.

Since the time delay $t_d$ equals to the time for the waveform to reach $0.5c_{final}$. My rough idea for the mathematical equation is:
$t_d \approx 1.25 * t_r / 2$ .

Just an approximation as the waveform between $0.1c_{final}$ to $0.9c_{final}$ is not a linear line.