# 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)
Hence, the proposed critical curves \((c_{1},c_{2})\) can be determined solving the eigenvalue problem \((\omega_{j}^{2}\mathbf{M}-\omega_{j}\mathbf{C}+\mathbf{K})\mathbf{u}=\mathbf{0}\) in terms of \(c_{1}\) and \(c_{2}\), leading to:  
\[\left[\omega_{j}^{2}\mathbf{M}+\mathbf{K}\right]\mathbf{u}=\omega_{j}\,c_{1} \mathbf{Q}_{1}\mathbf{u}+\omega_{j}\,c_{2}\mathbf{Q}_{2}\mathbf{u}. \tag{67}\]  
By sweeping the values of \(c_{1}\) over a preset interval, the values of \(c_{2}\) can be determined by solving the above eigenvalue problem. All the solutions \((c_{1},c_{2})\) of the previous problem have \(s=-\omega_{j}\) as a common (overdamped) eigenvalue. Consequently, they will always lie within an overdamped region. Furthermore, those solutions whose eigenvectors \(\mathbf{u}\) fullfill \(\mathbf{u}^{T}\mathbf{K}\mathbf{u}/\mathbf{u}^{T}\mathbf{M}\mathbf{u}\approx \omega_{j}^{2}\) will be those closer to the critical boundaries, holding the condition of critical damping. As shown in Figure 2a, the so-determined curves fit the critical regions with good agreement, something that was already observed in [10]. As shown in the theoretical results, it is expected that inaccuracies of the proposed approach arise as the viscoelasticity increases. It is interesting to study how this loss accuracy evolves in the current example. To this end, the proposed curves have been evaluated for different values of the nonviscous parameters \(\
u_{1},\
u_{2}\), in increasing order, leading to Figure 2b-d. In Figure 2b, the proposed critical curves fit the critical region accurately. Some curves lie inside the gray region; meanwhile, others fit satisfactorily with the critical boundaries. The former are associated to overcritically damped eigenvalues; in other words, the eigenvalue \(s=-\omega_{j}/\alpha\) is overcritical. On the other hand, the latter define the proper critical points between underdamped and one-mode overdamped regions, or even along the internal thresholds separating one-mode and two-mode overdamped regions. Figure 2b shows a case with relatively low viscoelasticity. The layout of the proposed critical curves is smooth and fits the overdamped regions quite accurately. In contrast, Figure 2c,d presents poorer results as parameters \(\
u_{1}\) and \(\
u_{2}\) become larger. Two different aspects can be highlighted with respect to Figure 2c,d: (i) on one side, the curves do not seem to regularly follow the exact contours of the regions, resulting in loop-like shapes. This seems to be caused after increasing the parameters \(v_{1}\) and \(v_{2}\), something that leads the system far away from the purely viscous case. (ii) On the other side, some curves lie outside the overdamped region. Remember that each point of the curves is associated to a real eigenvalue of the form \(s=-\omega_{j}/\alpha\), and this does not exclude the so-called nonviscous modes, also presented in the problem \(\mathbf{D}(s)\mathbf{u}=\mathbf{0}\). These nonviscous eigenvalues become more relevant as the viscoelasticity increases [8; 21], although they are not properly classified as overcritical; consequently they may lay outside the overdamped region as, indeed, it occurs (see Figure 2d). Therefore, the presence of non-viscous eigenvalues somehow makes the identification of critical regions more difficult.