# Boundaries of Oscillatory Motion in Structures with Nonviscous Dampers
## 2 New Results on Critical Damping of Structures with Viscoelastic Dampers
Discarding positive solutions, \(s\leq-\sqrt{\frac{\mathfrak{K}(\mathbf{u})}{\mathfrak{M}(\mathbf{u})}}\). It is known [25] that:  
\[\mathbf{D}^{\prime}(s,\boldsymbol{\theta})\mathbf{u}=\mathbf{D}(s,\boldsymbol{ \theta})\,\boldsymbol{\alpha}, \tag{16}\]5  
whence the inequality \(s\leq-\omega_{1}\) holds. (ii) Assume that the eigenvector \(\mathbf{u}\) is close to the \(j\)th normal mode \(\boldsymbol{\phi}_{j}\). Without loss of generality, \(\mathbf{u}\) can be written as:  
\[\mathbf{u}=\boldsymbol{\phi}_{j}+\sum_{ \begin{subarray}{c}k=1\\ k\
eq j\end{subarray} }^{n}a_{jk}\boldsymbol{\phi}_{k}, \tag{29}\]  
where \(\sum_{ \begin{subarray}{c}k=1\\ k\
eq j\end{subarray} }^{n}a_{jk}^{2}\ll 1\). Hence:  
\[ \begin{split}\mathbf{u}^{T}\mathbf{M}\mathbf{u}&= \left(\boldsymbol{\phi}_{j}^{T}+\sum_{k\
eq j}a_{jk}\boldsymbol{\phi}_{k}^{T} \right)\mathbf{M}\left(\boldsymbol{\phi}_{j}+\sum_{k\
eq j}a_{jl} \boldsymbol{\phi}_{l}^{T}\right)\\ &=\boldsymbol{\phi}_{j}^{T}\mathbf{M}\boldsymbol{\phi}_{j}+2 \sum_{k\
eq j}a_{jk}\boldsymbol{\phi}_{j}^{T}\mathbf{M}\boldsymbol{\phi}_{k}+ \sum_{k\
eq j}\sum_{l\
eq j}a_{jk}a_{jl}\boldsymbol{\phi}_{k}^{T}\mathbf{M} \boldsymbol{\phi}_{l}\\ \mathbf{u}^{T}\mathbf{K}\mathbf{u}&=\left(\boldsymbol{ \phi}_{j}^{T}+\sum_{k\
eq j}a_{jk}\boldsymbol{\phi}_{j}^{T}\right)\mathbf{K} \left(\boldsymbol{\phi}_{j}+\sum_{k\
eq j}a_{jk}\boldsymbol{\phi}_{j}\right)\\ &=\boldsymbol{\phi}_{j}^{T}\mathbf{K}\boldsymbol{\phi}_{j}+2\sum_{ k\
eq j}a_{jk}\boldsymbol{\phi}_{j}^{T}\mathbf{K}\boldsymbol{\phi}_{k}+\sum_{k\
eq j} \sum_{l\
eq j}a_{jk}a_{jl}\boldsymbol{\phi}_{k}^{T}\mathbf{K}\boldsymbol{\phi} _{l}\end{split}  \tag{30}\]  
Using, to the modal orthogonality, relations \(\boldsymbol{\phi}_{j}^{T}\mathbf{M}\boldsymbol{\phi}_{k}=\delta_{jk}\) and \(\boldsymbol{\phi}_{j}^{T}\mathbf{K}\boldsymbol{\phi}_{k}=\omega_{k}^{2} \delta_{jk}\), the above expressions can be simplified, yielding:  
\[\mathbf{u}^{T}\mathbf{M}\mathbf{u}=1+\sum_{k\
eq j}a_{jk}^{2}\,,\quad \mathbf{u}^{T}\mathbf{K}\mathbf{u}=\omega_{j}^{2}+\sum_{k\
eq j}a_{jk}^{2} \omega_{k}^{2}. \tag{31}\]  
Plugging these expressions into the Rayleigh quotient and rearranging the resulting expression:  
\[\frac{\mathfrak{K}(\mathbf{u})}{\mathfrak{M}(\mathbf{u})}=\frac{\omega_{j}^{2} +\sum_{k\
eq j}a_{jk}^{2}\omega_{k}^{2}}{1+\sum_{k\
eq j}a_{jk}^{2}}=\omega_{j} ^{2}+\frac{\sum_{k\
eq j}a_{jk}^{2}\omega_{k}^{2}-\omega_{j}^{2}\sum_{k\
eq j}a_ {jk}^{2}}{1+\sum_{k\
eq j}a_{jk}^{2}}. \tag{32}\]According to the hypothesis of Theorem 2, it is \(\sum_{k\
eq j}a_{jk}^{2}\omega_{k}^{2}\geq\omega_{j}^{2}\sum_{k\
eq j}a_{jk}^{2}\); therefore:  
\[s^{2}\geq\frac{\delta(\mathbf{u})}{\mathfrak{M}(\mathbf{u})}\geq\omega_{j}^{2}. \tag{33}\]  
Finally, since \(s<0\), it leads to the searched inequality \(s\leq-\omega_{j}\):  
(iii) If the system is purely viscous, then \(\mathbf{G}^{\prime}(s)\equiv\mathbf{0}\); therefore, it is \(\mathfrak{g}^{\prime}(s,\mathbf{u})=0\). Thus, from Equation (25), it yields, straightforwardly:  
\[s=-\sqrt{\frac{\mathbf{u}^{T}\mathbf{K}\mathbf{u}}{\mathbf{u}^{T}\mathbf{M} \mathbf{u}}}. \tag{34}\]  
The point (iii) of this theorem, proved here as a particular case of nonviscous systems, was already deduced by [23] in the context of the decoupling of defective linear dynamical systems. According to the first part of Theorem 2, the presence of critical eigenvalues within the range \(-\omega_{1}<s<0\) can be discarded. Perhaps more important is the point (ii) of the theorem, which allows for bounding critical eigenvalues associated to each undamped mode, provided that the system can be considered as slightly nonproportional. In the next section, this result will be used to derive the proposed methodology.