# Boundaries of Oscillatory Motion in Structures with Nonviscous Dampers
## 3 Derivation of Critical Curves: the Modal Critical Equation
Both the eigenvalue problem, \(\mathbf{D}(s)\mathbf{u}=\mathbf{0}\), and the MCE can be expressed, in terms of this new parameter, as:  
\[\left[\left(-\frac{\omega_{j}}{\alpha}\right)^{2}\mathbf{M}-\frac{ \omega_{j}}{\alpha}\,\mathbf{G}(-\omega_{j}/\alpha)+\mathbf{K}\right]\mathbf{u} = \mathbf{0}, \tag{49}\] \[\left(-\frac{\omega_{j}}{\alpha}\right)^{2}\frac{g_{jj}(-\omega_ {j}/\alpha)+\left(\omega_{j}/\alpha\right)g^{\prime}_{jj}(-\omega_{j}/\alpha)} {g_{jj}(-\omega_{j}/\alpha)-\left(\omega_{j}/\alpha\right)g^{\prime}_{jj}(- \omega_{j}/\alpha)} = \omega_{j}^{2}\,,\quad 1\leq j\leq n. \tag{50}\]  
Considering the general form for matrix \(\mathbf{G}(s)\) consisting of \(N\) hereditary kernels (see Equation (5)), we have:  
\[s^{2}\frac{\mathfrak{g}(s,\mathbf{u})-s\,\mathfrak{g}^{\prime}(s, \mathbf{u})}{\mathfrak{g}(s,\mathbf{u})+s\,\mathfrak{g}^{\prime}(s,\mathbf{u })} =s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}{g_{jj}(s) +s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}=s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj }(s)}{g_{jj}(s)+s\,g^{\prime}_{jj}(s)}+\mathcal{O}(2) \tag{43}\]0 \[s^{2}\frac{\mathfrak{g}(s,\mathbf{u})-s\,\mathfrak{g}^{\prime}(s, \mathbf{u})}{\mathfrak{g}(s,\mathbf{u})+s\,\mathfrak{g}^{\prime}(s,\mathbf{u })} =s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}{g_{jj}(s) +s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}=s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj }(s)}{g_{jj}(s)+s\,g^{\prime}_{jj}(s)}+\mathcal{O}(2) \tag{43}\]1 \[s^{2}\frac{\mathfrak{g}(s,\mathbf{u})-s\,\mathfrak{g}^{\prime}(s, \mathbf{u})}{\mathfrak{g}(s,\mathbf{u})+s\,\mathfrak{g}^{\prime}(s,\mathbf{u })} =s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}{g_{jj}(s) +s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}=s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj }(s)}{g_{jj}(s)+s\,g^{\prime}_{jj}(s)}+\mathcal{O}(2) \tag{43}\]2 \[s^{2}\frac{\mathfrak{g}(s,\mathbf{u})-s\,\mathfrak{g}^{\prime}(s, \mathbf{u})}{\mathfrak{g}(s,\mathbf{u})+s\,\mathfrak{g}^{\prime}(s,\mathbf{u })} =s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}{g_{jj}(s) +s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}=s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj }(s)}{g_{jj}(s)+s\,g^{\prime}_{jj}(s)}+\mathcal{O}(2) \tag{43}\]3  
Plugging these expressions into Equations (49) and (50), and after further simplifications, it yields:  
\[s^{2}\frac{\mathfrak{g}(s,\mathbf{u})-s\,\mathfrak{g}^{\prime}(s, \mathbf{u})}{\mathfrak{g}(s,\mathbf{u})+s\,\mathfrak{g}^{\prime}(s,\mathbf{u })} =s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}{g_{jj}(s) +s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}=s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj }(s)}{g_{jj}(s)+s\,g^{\prime}_{jj}(s)}+\mathcal{O}(2) \tag{43}\]4 \[s^{2}\frac{\mathfrak{g}(s,\mathbf{u})-s\,\mathfrak{g}^{\prime}(s, \mathbf{u})}{\mathfrak{g}(s,\mathbf{u})+s\,\mathfrak{g}^{\prime}(s,\mathbf{u })} =s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}{g_{jj}(s) +s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}=s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj }(s)}{g_{jj}(s)+s\,g^{\prime}_{jj}(s)}+\mathcal{O}(2) \tag{43}\]5  
where:  
\[s^{2}\frac{\mathfrak{g}(s,\mathbf{u})-s\,\mathfrak{g}^{\prime}(s, \mathbf{u})}{\mathfrak{g}(s,\mathbf{u})+s\,\mathfrak{g}^{\prime}(s,\mathbf{u })} =s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}{g_{jj}(s) +s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}=s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj }(s)}{g_{jj}(s)+s\,g^{\prime}_{jj}(s)}+\mathcal{O}(2) \tag{43}\]6  
All the parameters that control the dissipative model are included within Equations (52) and (53): indeed, the relaxation coefficients \(\mu_{k}\), \(1\leq k\leq N\) are represented by the auxiliary variables \(\mathcal{R}_{kj}=\alpha/(\alpha-\omega_{j}/\mu_{k})\), and the viscous coefficients are represented as the entries of \(\mathbf{C}_{k}\). The strategy to build a critical curve starts by choosing two of these parameters of interest; let us denote them as \(\theta_{1}\) and \(\theta_{2}\). Both can be either two viscous coefficients or two nonviscous coefficients, or even one of each type (possible combinations are listed below and in Table 1).