# Boundaries of Oscillatory Motion in Structures with Nonviscous Dampers
## 3 Derivation of Critical Curves: the Modal Critical Equation
Indeed, \(\mathcal{O}(\bullet)\)-notation will be considered to distinguish this order of magnitude. Thus, in general, a certain vector or scalar will be of order \(h\), say \(\mathcal{O}(h)\), if its components depends on the \(h\)th order of the entries \(g_{ik}(s)\) and their \(s\)-derivatives, \(g^{\prime}_{ik}(s)\). For instance, products of the type \(g_{ik}(s)g^{\prime}_{lm}(s)\) or \(g^{\prime}_{ik}(s)g^{\prime}_{lm}(s)\) are both considered to be of order \(\mathcal{O}(2)\). If the system exhibits lightly nonproportional damping, it is expected that terms of a higher order involved in the equations could be neglected with respect to those ones of order zero. Based on the above considerations, we are interested in addressing the nonproportionality order defined above for each of the terms involved in Equation (38) and, ultimately, the order of the entire equation. Thus, for \(\mathfrak{g}(s,\mathbf{u})\), it yields:  
\[\mathfrak{g}(s,\mathbf{u})=\mathbf{u}^{T}\mathbf{G}(s)\mathbf{u}=\left( \boldsymbol{\phi}_{j}^{T}+\sum_{ \begin{subarray}{c}k=1\\ k\
eq j\end{subarray} }^{n}a_{jk}\boldsymbol{\phi}_{k}^{T}\right)\mathbf{G}(s )\left(\boldsymbol{\phi}_{j}+\sum_{ \begin{subarray}{c}k=1\\ k\
eq j\end{subarray} }^{n}a_{jk}\boldsymbol{\phi}_{k}\right)=g_{jj}(s)+ \mathcal{O}(2). \tag{41}\]  
Similarly, and using the orthogonality relations, it is straightforward that:  
\[\mathfrak{g}^{\prime}(s,\mathbf{u})=g^{\prime}_{jj}(s)+\mathcal{O}(2)\,\quad \mathfrak{K}(\mathbf{u})=\omega_{j}^{2}+\mathcal{O}(2)\,\quad\mathfrak{M}(\mathbf{u})=1+\mathcal{O}(2). \tag{42}\]  
Plugging the above relations into both sides of Equation (38), it yields:  
\[s^{2}\frac{\mathfrak{g}(s,\mathbf{u})-s\,\mathfrak{g}^{\prime}(s, \mathbf{u})}{\mathfrak{g}(s,\mathbf{u})+s\,\mathfrak{g}^{\prime}(s,\mathbf{u })} =s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}{g_{jj}(s) +s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}=s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj }(s)}{g_{jj}(s)+s\,g^{\prime}_{jj}(s)}+\mathcal{O}(2) \tag{43}\] \[\frac{\mathfrak{K}(\mathbf{u})}{\mathfrak{M}(\mathbf{u})} =\frac{\omega_{j}^{2}+\mathcal{O}(2)}{1+\mathcal{O}(2)}=\omega_ {j}^{2}+\mathcal{O}(2).\]  
Finally, we find that:  
\[s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj}(s)}{g_{jj}(s)+s\,g^{\prime}_{jj}(s)}= \omega_{j}^{2}+\mathcal{O}(2). \tag{44}\]Encouraged by this result, we postulate that, in those cases where the eigenvector \(\mathbf{u}\) is close to \(\mathbf{\phi}_{j}\), it can be stated that:  
\[g^{2}\,\frac{g_{jj}(s)-s\,g^{\prime}_{jj}(s)}{g_{ji}(s)+s\,g^{\prime}_{jj}(s)} \approx\omega_{j}^{2}. \tag{45}\]  
In the context of this article, this equation will be called the _Modal Critical Equation (MCE)_ associated to the \(j\)th mode. This scalar equation, together with the eigenvalue problem \(\mathbf{D}(s)\mathbf{u}=\mathbf{0}\), makes up the system of equations for solving the critical curve. We find, in this equation, the two particular cases that have been already solved in the literature: (a) purely viscous damping, studied by [PERSON] and [PERSON] (2010), and (b) proportional nonviscous damping, investigated by [PERSON] and [PERSON] (2016). Both cases deserve some comments before addressing the proposed strategy to solve the general case of nonproportional nonviscously damped systems. Mathematically, the purely viscous case is characterized by a frequency-independent damping matrix, \(\mathbf{G}(s)\equiv\mathbf{C}\). Equation (45) degenerates into the \(n\) equations \(s^{2}\approx\omega_{j}^{2}\) for all modes \(1\leq j\leq n\). The results obtained fit the actual overdamped regions, generally, quite well, as is reflected in ref. (2010). One of the numerical examples of the current paper shows the outcomes for viscous systems, reinforcing those results. The second particular case which can be derived from Equation (45) is that of proportional damping.