Model with damping and nonlinearity

Linear damping $\beta u'$, nonlinear spring force $s(u)$, and excitation $F$: $$\begin{equation} mu'' + \beta u' + s(u) =F(t)\tp \tag{38} \end{equation}$$ Central difference discretization: $$\begin{equation} [mD_tD_t u + \beta D_{2t} u + s(u)=F]^n \tp \end{equation}$$ Truncation error is defined by $$\begin{equation} [mD_tD_t \uex + \beta D_{2t} \uex + s(\uex)=F + R]^n \tp \end{equation}$$