# Boundaries of Oscillatory Motion in Structures with Nonviscous Dampers
## 2 New Results on Critical Damping of Structures with Viscoelastic Dampers
If the actual state of the damping model, represented by \(\mathbf{\theta}\), induces light damping, then there exist \(n\) conjugate-complex eigensolutions of the form \((s_{j},\mathbf{u}_{j}),\,(s_{j}^{*},\mathbf{u}_{j}^{*}),\,1\leq j\leq n\), and \(r\) real nonviscous eigenmodes \((s_{j},\mathbf{u}_{j}),\,2n+1\leq j\leq 2n+r\), where \(r=\sum_{k=1}^{N}\text{rank}(\mathbf{C}_{k})\)[7]. As the damping level (through the variation of parameters \(\mathbf{\theta}\)), some of the conjugate-complex eigensolutions may come close to the real axis. If one of these eigenvalues drops into the real axis, then the two pairs of conjugate-complex eigenvalues are transformed into a double real eigenvalue, and then \(\mathbf{\theta}\) will lie exactly on a critical manifold. At this point, the resulting negative eigenvalue is critical and double; therefore, it will be the root, simultaneously, of both expressions in Equation (4). If the level of damping continues to increase, the array \(\mathbf{\theta}\) will be completely inside an overdamped region and the double root will be split into two overcritical (real and negative) eigenvalues with non-oscillatory natures. At this point, it may be useful to distinguish between _overdamped modes_ and _nonviscous modes_. Both are properly non-oscillatory modes because they correspond to real and negative eigenvalues. However, while the presence of the former strongly depends on the level of damping, the amount of the latter, \(r=\sum_{k=1}^{N}\text{rank}(\mathbf{C}_{k})\), depends on the spatial distribution of the viscous matrices \(\mathbf{C}_{k}\), rather than the value of their coefficients. Therefore, the reader must be aware that nonviscous modes without an oscillatory nature will be present always in nonviscously damped structures based on kernels with exponential decay, no matter the damping level. Some works specifically devoted to the study of nonviscous modes can be found in the references [8; 20; 21; 22], where, in particular, [PERSON] and [PERSON] propose a mathematical characterization [20] and study their distribution [21]. However, in the context of the current investigation, overdamped modes are those whose non-oscillatory nature (as negative real numbers) can be affected by the damping level, so that they can be transformed into complex underdamped modes with oscillatory natures for low damping conditions. Let us assume certain combination of damping parameters \(\mathbf{\theta}=\mathbf{\theta}_{0}\), lying within the overdamped region and leading, consequently, to (at least) two overdamped modes. Let us denote, respectively, by \(s_{0}\in\mathbb{R}^{-}\) and by \(\mathbf{u}_{0}\in\mathbb{R}^{n}\), the eigenvalue and eigenvector of one such modes. The \((\mathbf{n}+1)\) - tuple \((s_{0},\mathbf{u}_{0})\in\mathbb{R}^{n+1}\) is, then, a particular solution of Equation (9). Small variations of \(\mathbf{\theta}\) around \(\mathbf{\theta}_{0}\) lead to a functional dependence of \(s\) and \(\mathbf{u}\) with to \(\mathbf{\theta}\). Mathematically, it is said, then, that there exist two functions:  
\[s=s(\mathbf{\theta}):\mathbb{R}^{p}\to\mathbb{R}\,\quad\mathbf{u}=\mathbf{u}(\mathbf{ \theta}):\mathbb{R}^{p}\to\mathbb{R}^{n}, \tag{10}\]  
which are implicitly defined by the expression in Equation (9) around the point \((s_{0},\mathbf{u}_{0})\), holding that \(s_{0}=s(\mathbf{\theta}_{0})\) and \(\mathbf{u}_{0}=\mathbf{u}(\mathbf{\theta}_{0})\).