For ζ < 1 {\displaystyle \zeta <1\,} , the corresponding equation should be:
y ( t ) = k ( 1 − e − ζ ω 0 t cos ω d t − ζ 1 − ζ 2 e − ζ ω 0 t sin ω d t ) {\displaystyle y(t)=k{\Big (}1-e^{-\zeta \omega _{0}t}\cos {\omega _{d}t}-{\frac {\zeta }{\sqrt {1-\zeta ^{2}}}}e^{-\zeta \omega _{0}t}\sin {\omega _{d}t}{\Big )}} .
--Shuo