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 $$ \begin{equation} [mD_tD_t \uex + \beta D_{2t} \uex + s(\uex)=F + R]^n \tp \end{equation} $$