Second Order Underdamped

This page demonstrates the steps in finding the step response of an underdamped second order system. As is shown, on my underdamped page, it is possible to find the response by adapting the overdamped response. This page however, shows the conventional approach. Again it is assumed that initial conditions are zero.

Here we assume the roots of the auxiliary equation are $r_{1} = a + ib$ and $r_{2} = a - ib$, so the response includes $e^{(a+ib)t}$ which using Euler's identity is $e^{at}(cos(bt) + i sin(bt))$.

As, the step response must be real, it is reasonable to assume it has the form:

$O = 1 + e^{at} (k_{1} cos(bt) + k_{2}sin(bt))$

The constants are determined by finding $\frac{dO}{dt}$, evaluating that and $O$ when $t = 0$, forming two simultaneous equations in $k_{1}$ and $k_{2}$ and solving them, as is shown here.

Finding general response

Here we assume the roots are $r_{1} = a + ib$ and $r_{2} = a - ib$, so : \begin{align} O = 1 + e^{at} (k_{1} cos(bt) + k_{2}sin(bt)) \\ \cssId{Step1}{{}\frac{dO}{dt} = a e^{at} (k_{1} cos(bt) + k_{2}sin(bt)) + e^{at} (-b k_{1} sin(bt) + bk_{2}cos(bt))} \\[3px] \cssId{Step2}{{}At \ t = 0, O = 0, so \ 0 = 1 + k_{1} } \\[3px] \cssId{Step3}{{}Thus \ k_{1} = -1 } \\[3px] \cssId{Step4}{{}At \ t = 0, \frac{dO}{dt} = 0, so \ 0 = a k_{1} + b k_{2} = - a + b k_{2}} \\[3px] \cssId{Step5}{{}Thus \ k_{2} = \frac{a}{b} } \\[3px] \cssId{Step6}{{}O = 1 - e^{at} (cos(bt) - \frac{a}{b} sin(bt)) } \end{align}