# Boundaries of Oscillatory Motion in Structures with Nonviscous Dampers
## 2 New Results on Critical Damping of Structures with Viscoelastic Dampers
In addition, we will assume that:  
\[\
abla f(\mathbf{u}_{0})=\left\{\frac{\partial f}{\partial u_{1}},\ldots,\frac {\partial f}{\partial u_{n}}\right\}_{\mathbf{u}=\mathbf{u}_{0}}^{\top}\
eq \mathbf{0}. \tag{11}\]The expressions in Equation (9) can be written in a more compact form as:  
\[\mathbf{F}(\mathbf{y},\mathbf{\theta})=\mathbf{0}, \tag{12}\]  
where \(\mathbf{F}(\mathbf{y},\mathbf{\theta}):\mathbb{R}^{n+1+p}\to\mathbb{R}^{n+1}\) is a vector field defined as:  
\[\mathbf{F}(\mathbf{y},\mathbf{\theta})=\{F_{1}(\mathbf{y},\mathbf{\theta}),\dots,F_{n} (\mathbf{y},\mathbf{\theta}),F_{n+1}(\mathbf{y},\mathbf{\theta})\}^{T}, \tag{13}\]  
with \(\mathbf{y}=(s,\mathbf{u})\in\mathbb{R}^{n+1}\) and \(\mathbf{\theta}=\{\theta_{1},\dots,\theta_{p}\}\in\mathbb{R}^{p}\). Sufficient conditions to guarantee the existence of the functions of Equation (10) are provided by the implicit function theorem. Assuming that \(\mathbf{F}(\mathbf{y},\mathbf{\theta})\) is a continuously differentiable function in a neighborhood of \((\mathbf{y}_{0},\theta_{0})\), where \(\mathbf{y}_{0}=\{s_{0},\mathbf{u}_{0}\}^{T}\) and \(\mathbf{F}(\mathbf{y}_{0},\mathbf{\theta}_{0})=\mathbf{0}\), then if the Jacobian matrix \(\mathbf{\mathcal{J}}_{\mathbf{F},\mathbf{y}}(\mathbf{y}_{0},\mathbf{\theta}_{0})\) is invertible, it can be ensured that there exists an open set \(\mathcal{U}\subset\mathbb{R}^{p}\) around \(\mathbf{\theta}_{0}\) and a unique continuously differentiable function \(\mathbf{y}=\mathbf{g}(\mathbf{\theta}):\mathcal{U}\to\mathbb{R}^{n+1}\), such that \(\mathbf{g}(\mathbf{\theta}_{0})=\mathbf{y}_{0}\) and \(\mathbf{F}(\mathbf{g}(\mathbf{\theta}),\mathbf{\theta})=\mathbf{0}\) for all \(\mathbf{\theta}\in U\). The existence of the function \(\mathbf{g}(\mathbf{\theta})=\{s(\mathbf{\theta}),\mathbf{u}(\mathbf{\theta})\}\) is directly related to the location of \(\mathbf{\theta}\) within the domain of the damping parameters. Thus, if \(\mathbf{\theta}_{0}\) lies on a critical surface, the above functions are not well defined since small variations of the damping parameters lead to an indefinite state with two possible conjugate-complex solutions. An assessment of the conditions under which the Jacobian matrix becomes non-invertible will provide valuable information about critical damping. According to the definition of the Jacobian matrix, and after some straight operations, it yields:  
\[\mathbf{\mathcal{J}}_{\mathbf{F},\mathbf{y}}=\left[\frac{\partial\mathbf{F}}{ \partial\mathbf{y}}\right]=\left[ \begin{array}{c|ccc}\frac{\partial \mathbf{F}_{1}}{\partial s}&\frac{\partial\mathbf{F}_{1}}{\partial u_{1}}& \dots&\frac{\partial\mathbf{F}_{1}}{\partial u_{n}}\\ &\\ \frac{\partial\mathbf{F}_{2}}{\partial s}&\frac{\partial\mathbf{F}_{2}}{ \partial u_{1}}&\dots&\frac{\partial\mathbf{F}_{2}}{\partial u_{n}}\\ &\vdots&\vdots&\ddots&\vdots\\ \frac{\partial\mathbf{F}_{n}}{\partial s}&\frac{\partial\mathbf{F}_{n}}{ \partial u_{1}}&\dots&\frac{\partial\mathbf{F}_{n}}{\partial u_{n}}\\ &\\ \frac{\partial\mathbf{F}_{n+1}}{\partial s}&\frac{\partial\mathbf{F}_{n+1}}{ \partial u_{1}}&\dots&\frac{\partial\mathbf{F}_{n+1}}{\partial u_{n}}\\ \end{array} \right]=\left[ \begin{array}{c|ccc}\frac{\partial\mathbf{D}}{ \partial s}\mathbf{u}&\mathbf{D}\\ &\\ &\\ 0&\mathbf{\
abla}^{\mathsf{T}}f\\ \end{array} \right]\in\mathbb{R}^{(n+1)\times(n+1)} \tag{14}\]  
This result shows that the Jacobian matrix is formed by four matrix blocks. In ref. [18], it was demonstrated that if \(\mathbf{D}^{\prime}(s)\mathbf{u}=\mathbf{0}\) (denoting \((\bullet)^{\prime}=\partial(\bullet)/\partial s\)), then \(\operatorname{rank}\bigl{[}\mathbf{\mathcal{J}}_{\mathbf{F},\mathbf{y}}\bigr{]} <n+1\), and therefore, \(\mathbf{\mathcal{J}}_{\mathbf{F},\mathbf{y}}\) is non-invertible and the functions \(s(\mathbf{\theta})\) and \(\mathbf{u}(\mathbf{\theta})\) are not well defined at that point.