# Boundaries of Oscillatory Motion in Structures with Nonviscous Dampers
## 4 Numerical Examples
### Example 1: Discrete System
#### 4.1.1 Critical Curves between \(c_{1}\) and \(c_{2}\) (Type I)  
Consider \(\mu_{1}\) and \(\mu_{2}\) as fixed. We are interested in points of type \((c_{1},c_{2})\), for which both Equations (59) and (60) hold simultaneously. This will be achieved by eliminating one \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(k\) (N/m) & \(m\) (kg) & \(\mathbf{\omega_{1}}\) & \(\mathbf{\omega_{2}}\) & \(\mathbf{\omega_{3}}\) & \(\mathbf{\omega_{4}}\) \\ \hline \(10^{3}\) & 10.00 & 6.1803 & 11.7557 & 16.1803 & 19.0211 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Undamped natural frequencies of Example 1 in rad/s. Figure 1: The four-degrees-of-freedom discrete system of Example 1, with nonproportional damping, \(\mathcal{G}_{1}(t)\) and \(\mathcal{G}_{2}(t)\) represent the hereditary function of nonviscous dampers. Masses \(m=10\) kg; rigidities \(k=10^{3}\) N/m. of the parameters (\(c_{1}\) or \(c_{2}\)) from Equation (60) and plugging the result into Equation (59). Indeed, introducing some auxiliary variables, Equations (59) and (60) yield:  
\[c_{1}\,\mathbf{R}_{j1}\mathbf{u}+c_{2}\,\mathbf{R}_{j2}\mathbf{u} = \mathfrak{M}_{j}\,\mathbf{u}, \tag{62}\] \[c_{1}\,S_{j1}+c_{2}\,S_{j2} = 0, \tag{63}\]  
where:  
\[ \begin{array}{rcl}\mathbf{R}_{j1}=&\mathcal{R}_{j1}\,\mathbf{Q}_{1}&S_{j1}=& \mathcal{R}_{j1}\left[(\alpha^{2}+1)\mathcal{R}_{j1}-2\right]\,\boldsymbol{ \phi}_{I}^{T}\mathbf{Q}_{1}\boldsymbol{\phi}_{j},\\ \mathbf{R}_{j2}=&\mathcal{R}_{j2}\,\mathbf{Q}_{2}&S_{j2}=&\mathcal{R}_{j2} \left[(\alpha^{2}+1)\mathcal{R}_{j2}-2\right]\,\boldsymbol{\phi}_{I}^{T} \mathbf{Q}_{2}\boldsymbol{\phi}_{j}.\end{array} \]  
After straight manipulations, Equation (62) can be written exclusively as function of \(c_{1}\), giving rise to a classical linear eigenvalue problem:  
\[\left[\boldsymbol{\mathcal{A}}_{j}(\alpha)-c_{1}\,\boldsymbol{\mathcal{B}}_{ j}(\alpha)\right]\mathbf{u}=\,\mathbf{0}, \tag{64}\]  
with:  
\[\boldsymbol{\mathcal{A}}_{j}(\alpha)=S_{j2}\,\mathfrak{M}_{j}\,\qquad \boldsymbol{\mathcal{B}}_{j}(\alpha)=S_{j2}\,\mathbf{R}_{j1}-S_{j1}\,\mathbf{ R}_{j2}. \tag{65}\]  
Above, the dependency on the considered mode \(j\) and on the parameter \(\alpha\) has been highlighted. Precisely, by sweeping this latter in the range \(0<\alpha\leq 1\), pairs of the form \(\{c_{1}^{(j)}(\alpha),c_{2}^{(j)}(\alpha)\}\) can be listed and plotted. For this example, \(n=4\) and, additionally, the number of non-zero and noninfinite solutions for \(c_{1}\) in Equation (64) depends on the rank of matrices \(\boldsymbol{\mathcal{A}}_{j}(\alpha),\boldsymbol{\mathcal{B}}_{j}(\alpha)\), leading to several branches for each mode. Results are shown in Figure 2 in nondimensional form, say:  
\[\zeta_{1}=\frac{c_{1}}{2m\omega_{r}}\,\quad\zeta_{2}=\frac{c_{2}}{2m\omega_{r}} \,\quad\upsilon_{1}=\frac{\omega_{r}}{\mu_{1}}\,\quad\upsilon_{2}=\frac{ \omega_{r}}{\mu_{2}}, \tag{66}\]  
where \(\omega_{r}=\sqrt{k/m}=10\) rad/s is a reference frequency. Plots of Figure 2a-d have been built using four different pairs of non-viscous parameters, shown on top of each figure, including the purely viscous case in (a). Furthermore, in order to contrast the proposed solution with the exact critical region, the visible domain has been meshed by a 500 \(\times\) 500-point grid, and the eigenvalue problem \(\mathbf{D}(s)\mathbf{u}=\mathbf{0}\) has been solved. The grayscale shows the total number of overdamped modes. Both the system matrices' assembly code and the numerical solution of the eigenvalue problems have been implemented in Matlab(r) software. This environment has also been used for the rest of the numerical cases within this manuscript. The critical curves for the purely viscous model are depicted in Figure 2a, where \(\upsilon_{1}=\upsilon_{2}=0\) and \(\mathbf{G}(s)\equiv\mathbf{C}=c_{1}\mathbf{Q}_{1}+c_{2}\mathbf{Q}_{2}\). In this case, the modal critical equation is not a function of \(\alpha\), and degenerates into \(s=-\omega_{j}\) (\(\alpha=1\)).