The existence of strong solutions to the abstract \textsc{Cauchy} problem for the nonlinearly damped inertial system \begin{align*} \begin{cases} u''(t)+\partialΨ_{u(t)}(u'(t))+ \mathrm{D} E_t(u(t))+B(t,u(t),u'(t)) \ni f(t), &\quad t\in (0,T),\\ u(0)=u_0, \quad u'(0)=v_0, \end{cases} \end{align*} on a Gelfand triplet type of framework is proven. Under mild assumptions on the dissipation potential $Ψ_{u}$ and differentiablility assumption on the energy functional, the existence of strong solutions is shown by establishing the convergence of a semi-implicite time discretization scheme.