# Boundaries of Oscillatory Motion in Structures with Nonviscous Dampers
## 3 Derivation of Critical Curves: the Modal Critical Equation  
In the previous section, a new theoretical characterization of critical eigenmodes has been derived, considering critical modes as those whose functional dependency on the damping parameters is not well defined. According to one such result, a combination of damping parameters is said to be critical if it leads to (at least) one mode \((s,\mathbf{u})\) verifying the following relations:  
\[\mathbf{D}(s)\,\mathbf{u} = \mathbf{0}, \tag{35}\] \[\mathbf{u}^{T}\frac{\partial\mathbf{D}}{\partial s}\,\mathbf{u} = 0. \tag{36}\]  
The aim of this section is to develop a numerical method based on the above equations to determine points of the critical curves. These curves graphically represent thresholds in the parametric domain where the oscillatory nature of a complex mode is lost, giving rise to two overdamped distinct real modes. Consider the two scalar relationships, Equations (23) and (24). Now, dividing both equations by \(s\,\mathfrak{g}(s,\mathbf{u})\) and \(\mathfrak{g}(s,\mathbf{u})+s\,\mathfrak{g}^{\prime}(s,\mathbf{u})\), respectively, and subtracting, yields:  
\[\left(\frac{s^{2}}{s\mathfrak{g}(s,\mathbf{u})}-\frac{2s}{\mathfrak{g}(s, \mathbf{u})+s\,\mathfrak{g}^{\prime}(s,\mathbf{u})}\right)\mathfrak{M}( \mathbf{u})+\frac{1}{s\,\mathfrak{g}(s,\mathbf{u})}\,\delta(\mathbf{u})=0. \tag{37}\]  
After some straight operations, this equation can be rearranged in a more compact form:  
\[s^{2}\,\frac{\mathfrak{g}(s,\mathbf{u})-s\,\mathfrak{g}^{\prime}(s,\mathbf{u })}{\mathfrak{g}(s,\mathbf{u})+s\,\mathfrak{g}^{\prime}(s,\mathbf{u})}=\frac{ \delta(\mathbf{u})}{\mathfrak{M}(\mathbf{u})}. \tag{38}\]  
The starting point of the developments is focused on this equation. For purely viscous systems, with \(\mathfrak{g}^{\prime}(s,\mathbf{u})\equiv 0\), Equation (38) is reduced to \(s^{2}=\mathbf{u}^{T}\,\mathbf{K}\,\mathbf{u}/\mathbf{u}^{T}\,\mathbf{M}\, \mathbf{u}\), as proved above. [PERSON] and [PERSON] pointed out [10] that, for nonproportional systems, the approach:  
\[s^{2}\approx\frac{\boldsymbol{\phi}_{j}^{T}\mathbf{K}\boldsymbol{\phi}_{j}}{ \boldsymbol{\phi}_{j}^{T}\mathbf{M}\boldsymbol{\phi}_{j}}=\omega_{j}^{2}\]  
can be assumed with accurate results. This affirmation can be justified directly from the approximation of Equation (32), which states \(s\approx-\omega_{j}\) with an error of the second order of magnitude in terms of the coefficients \(a_{jk}\) of Equation (29). These coefficients, in turn, are directly related to the nonproportionality of the system. Indeed, [PERSON] [1] proved that \(a_{jk}\) is directly related to the off-diagonal terms of the modal damping matrix, i.e. :  
\[a_{jk}=-s\frac{g_{jk}(s)}{\mathcal{D}_{kk}(s)}+s^{2}\ \ \sum_{ \begin{subarray}{c}l=1\\ j\
eq l\
eq k\end{subarray} }^{n}\frac{g_{kl}(s)g_{jl}(s)}{\mathcal{D}_{kk}(s) \,\mathcal{D}_{ll}(s)}+\cdots\,\qquad j\
eq k, \tag{39}\]  
provided that the matrix \(\mathbf{S}^{(j)}=\left[S^{(j)}_{ik}\right]\in\mathbb{R}^{n\times n}\) fulfills the condition \(\left\|\mathbf{S}^{(j)}\right\|<1\), where:  
\[S^{(j)}_{ik} = -s\frac{g_{ik(s)}}{\mathcal{D}_{ii}}(1-\delta_{ik})(1-\delta_{kj} )(1-\delta_{ij})\,\qquad 1\leq i,k\leq n,\] \[g_{ik}(s) = \boldsymbol{\phi}_{l}^{T}\mathbf{G}(s)\boldsymbol{\phi}_{k},\] \[\mathcal{D}_{ii}(s) = s^{2}+sg_{ii}(s)+\omega_{i}^{2}, \tag{40}\]  
and \(\delta_{ij}\) denotes the Kronecker delta function. Equation (38) depends on \(\mathbf{u}\) on both sides. It is clear that if the system is proportional, then \(g_{ik}(s)\equiv 0,\ 1\leq i,k\leq n\), and such an equation will depend on \(s\) and on the given mode \(\boldsymbol{\phi}_{j}\). Proportional systems admit closed-form analytical solutions for critical curves, as proved in reference [26]. However, the presence of \(\mathbf{u}\) in Equation (38) requires some assumptions to reach approximate solutions. A more detailed inspection of the nature of equation Equation (38) enables a determination of the dependence of the off-diagonal terms of the damping matrix.