Second-order step response

I know wn and zeta From s^2 + a s + b

Natural frequency wn (rad/s)Damping ratio zeta

Steady-state gain K

G(s) = K wn^2 / (s^2 + 2 zeta wn s + wn^2), unit step input.

Type	underdamped	wn = 4, zeta = 0.4
Percentage overshoot	25.4%	exp(-zeta pi / sqrt(1 - zeta^2))
Peak time t_p	0.8569 s	pi / wd
Settling time (2%)	2.5 s	4 / (zeta wn)
Settling time (5%)	1.875 s	3 / (zeta wn)
Rise time (10-90%)	0.3675 s	from the curve
Damped frequency wd	3.666 rad/s	wn sqrt(1 - zeta^2)
Poles	-1.6 + 3.666j, -1.6 - 3.666j	left half plane: stable

Settling times use the usual envelope approximations, which are only good for zeta below about 0.7.