# Boundaries of Oscillatory Motion in Structures with Nonviscous Dampers
## 4 Numerical Examples
### Example 1: Discrete System
#### 4.1.3 Critical Curves between \(c_{2}\) and \(\mu_{2}\) (Type III)  
The procedure to solve the critical curves between the viscous and nonviscous coefficients of the same kernel will be described now. In particular, and without loss of generality, the parameters \(\{c_{2},\mu_{2}\}\) will be considered now as variables to be solved (or unknowns); meanwhile, parameters \(\mu_{1}\) and \(c_{1}\) are assumed to be fixed with known values. Denote, by \(\mathbf{C}_{2}=c_{2}\,\mathbf{Q}_{2}\); then, Equations (59) and (60) yield:  
\[\mathcal{R}_{j2}\,c_{2}\,\mathbf{Q}_{2}\,\mathbf{u} = \mathfrak{M}_{j}\,\mathbf{u}-\mathcal{R}_{j1}\mathbf{C}_{1}\, \mathbf{u}, \tag{73}\] \[+\mathcal{R}_{j2}\,c_{2}\,\Big{[}(\alpha^{2}+1)\mathcal{R}_{j2}- 2\Big{]}\,\boldsymbol{\phi}_{j}^{T}\mathbf{Q}_{2}\boldsymbol{\phi}_{j} = -\mathcal{R}_{j1}\Big{[}(\alpha^{2}+1)\mathcal{R}_{j1}-2\Big{]} \,\boldsymbol{\phi}_{j}^{T}\mathbf{C}_{1}\boldsymbol{\phi}_{j}. \tag{74}\]  
Solving, now, for \(\mathcal{R}_{j2}c_{2}\) in Equation (74), and plugging it into Equation (73), this latter leads, after some manipulations, to the linear eigenvalue problem:  
\[\mathbf{Q}_{2}\,\mathbf{u}=\lambda\,\boldsymbol{\mathcal{B}}_{j}\,\mathbf{u}, \tag{75}\]  
where:  
\[\boldsymbol{\mathcal{B}}_{j}=\mathfrak{M}_{j}-\mathcal{R}_{j1}\mathbf{C}_{1}. \tag{76}\]  
The new \(\lambda\) parameter maps \(\mu_{2}\) under the following change of variable, reminding that \(\mathcal{R}_{j2}=\alpha/(\alpha-\frac{\omega_{j}}{\mu_{2}})\), and:  
\[\lambda=-\frac{\big{[}(\alpha^{2}+1)\mathcal{R}_{j2}-2\big{]}\,\boldsymbol{ \phi}_{j}^{T}\mathbf{Q}_{2}\boldsymbol{\phi}_{j}}{\mathcal{R}_{j1}\big{[}( \alpha^{2}+1)\mathcal{R}_{j1}-2\big{]}\,\boldsymbol{\phi}_{j}^{T}\mathbf{C}_{1 }\boldsymbol{\phi}_{j}}. \tag{77}\]  
If \(\alpha\), \(\omega_{j}\), \(\boldsymbol{\phi}_{j}\), and the parameters \(c_{1}\) and \(\mu_{1}\) are fixed, Equation (77) establishes a straight relationship between \(\lambda\) and \(\mu_{2}\). For a shake of clarity in the notation, the \(\alpha\)-dependency is omitted in all matrices. Equation (75) produces as many solutions in \(\lambda\) as \(\text{rank}(\mathbf{Q}_{2})\) (in the current example \(\text{rank}(\mathbf{Q}_{2})=4\)). Hence, \(\mu_{2}\) can be found from Equation (77) and \(c_{2}\) from Equation (74). Curves between \(c_{1}\) and \(v_{1}\) could be drawn similarly, just changing  
Figure 3: Critical curves \((v_{1},v_{2})\) of the four-degrees-of-freedom discrete system of Example 1 with \(\zeta_{1}=2.50\), \(\zeta_{2}=3.00\). Shaded regions represent the exact critical regions. The different levels of gray color stands for the number of overdamped modes.  
the roles of the parameters. Examples of these curves are shown in Figure 4. Since the overdamped regions are somehow enveloped by critical curves, it is good practice to plot these curves for all modes, improving the visualization of the result. In addition, the presence of loop-type shapes is observed in the curves, something that is associated with the presence of non-viscous eigenvalues, as in the previous case.