# Boundaries of Oscillatory Motion in Structures with Nonviscous Dampers
## 4 Numerical Examples
### Example 1: Discrete System
#### 4.1.2 Critical Curves between \(\mu_{1}\) and \(\mu_{2}\) (Type II)  
The nonviscous coefficients \(\mu_{1}\) and \(\mu_{2}\) are represented in Equations (59) and (60) by the auxiliary variables \(\mathcal{R}_{j1}\) and \(\mathcal{R}_{j2}\). In addition, let us consider \(v_{1}\in\mathbb{R}\), \(v_{1}\
eq 0\), and denote them by \(v_{2}\) and \(v_{3}\), such that:  
\[v_{2}=\left[(\alpha^{2}+1)\mathcal{R}_{j1}-2\right]v_{1},\qquad v_{3}=\left[( \alpha^{2}+1)\mathcal{R}_{j2}-2\right]v_{1}. \tag{68}\]  
Then, Equation (60) can be written as:  
\[\mathcal{R}_{j1}\,\boldsymbol{\phi}_{j}^{T}\mathbf{C}_{1}\boldsymbol{\phi}_{ j}\,v_{2}+\mathcal{R}_{j2}\,\boldsymbol{\phi}_{j}^{T}\mathbf{C}_{2} \boldsymbol{\phi}_{j}\,v_{3}=0. \tag{69}\]  
Equations (68) and (69) can be written in matrix form as:  
\[\mathcal{R}_{j1}\left[\begin{array}{cc}0&\boldsymbol{\phi}_{j}^{T}\mathbf{C }_{1}\boldsymbol{\phi}_{j}&0\\ \alpha^{2}+1&0&0\\ 0&0&0\end{array}\right]\left\{\begin{array}{c}v_{1}\\ v_{2}\\ \alpha^{2}+1&0&0\end{array}\right]\left\{\begin{array}{c}v_{1}\\ v_{2}\\ v_{3}\end{array}\right\}=\left[\begin{array}{cc}0&0&0\\ 2&1&0\\ 2&0&1\end{array}\right]\left\{\begin{array}{c}v_{1}\\ v_{2}\\ v_{3}\end{array}\right\}. \tag{70}\]  
Therefore, Equations (59) and (70) can be written together as:  
\[\mathcal{R}_{j1}\,\mathbf{C}_{1}\mathbf{u}+\mathcal{R}_{j2}\, \mathbf{C}_{2}\mathbf{u} = \boldsymbol{\mathfrak{M}}_{j}\,\mathbf{u},\] \[\mathcal{R}_{j1}\,\mathbf{E}_{j1}\mathbf{v}+\mathcal{R}_{j2}\, \mathbf{E}_{j2}\mathbf{v} = \mathbf{H}\,\mathbf{v}, \tag{71}\]  
where \(\mathbf{v}=\{v_{1},v_{2},v_{3}\}^{T}\), and:  
\[\mathbf{E}_{j1}=\left[\begin{array}{ccc}0&\boldsymbol{\phi}_{j}^{T}\mathbf{ C}_{1}\boldsymbol{\phi}_{j}&0\\ \alpha^{2}+1&0&0\\ 0&0&0\end{array}\right],\quad\mathbf{E}_{j2}=\left[\begin{array}{ccc}0&0& \boldsymbol{\phi}_{j}^{T}\mathbf{C}_{2}\boldsymbol{\phi}_{j}\\ 0&0&0\\ \alpha^{2}+1&0&0\end{array}\right],\quad\mathbf{H}=\left[\begin{array}{ccc}0& 0&0\\ 2&1&0\\ 2&0&1\end{array}\right]. \tag{72}\]  
With a fixed value of \(\alpha\in(0,1]\) and a mode \(j\), Equation (71) represents a two-parameter eigenvalue problem with eigenvalues \(\{\mathcal{R}_{j1}\), \(\mathcal{R}_{j2}\}\) and eigenvectors \(\mathbf{u}\), \(\mathbf{v}\), which can be solved using Jacobi-Davidson-based algorithms proposed by [PERSON] et al. [27] and by [PERSON] et al. or [28] for nonsingular and singular problems, respectively. Undoing the change of variable, parametric curves of the form \(\{v_{1}^{(j)}(\alpha),v_{2}^{(j)}(\alpha),\,0<\alpha\leq 1,\,1\leq j\leq 4\}\) can be drawn, where \(v_{k}=\omega_{r}/\mu_{k}\). The graphs of these curves have been plotted in Figure 3, where, as above, the domain has been meshed in a \(500\times 500\) grid, where the total spectrum of eigenvalues has been determined in order to shade exact critical regions. It can be seen that the curves calculated with the proposed method enclose nearly almost the entire overdamped region--even those boundaries separating subregions with different numbers of overdamped modes (inside boundaries), characterized in the graph with different grayscales.