# Boundaries of Oscillatory Motion in Structures with Nonviscous Dampers
## 2 New Results on Critical Damping of Structures with Viscoelastic Dampers  
In this paper, nonviscous damping, based on [PERSON]'s model [19], will be considered. In general, the damping matrix can be written as the superposition of \(N\) hereditary exponential functions, expressed both in the time and frequency domains as:  
\[\mathbf{\mathcal{G}}(t)=\sum_{k=1}^{N}\mathbf{C}_{k}\,\mu_{k}\,e^{-\mu_{k}t}\quad, \quad\mathbf{G}(s)=\mathcal{L}\{\mathbf{\mathcal{G}}(t)\}=\sum_{k=1}^{N}\frac{\mu _{k}}{s+\mu_{k}}\mathbf{C}_{k}, \tag{5}\]  
where \(\mu_{k}>0\), \(1\leq k\leq N\) stand for the relaxation (or nonviscous) parameters, and \(\mathbf{C}_{k}\in\mathbb{R}^{n\times n}\) are the symmetric damping matrices of the limited viscous model, obtained as the relaxation parameters tend to infinite; that is,  
\[\sum_{k=1}^{N}\mathbf{C}_{k}=\lim_{\mu_{1}\ldots\mu_{N}\rightarrow\infty} \mathbf{G}(s). \tag{6}\]  
The coefficients \(\mu_{k}\) control the time- (and frequency-) dependence of the damping model, while the spatial location and the level of damping are modeled via the matrices \(\mathbf{C}_{k}\). From now on, the matrix \(\mathbf{G}(s)\) will be assumed to depend on a set of damping parameters which monitor the dissipative behavior. Thus, \(\mathbf{\mathcal{G}}(t)\) and \(\mathbf{G}(s)\), from Equation (5), depend at most on \(p_{\text{max}}=N+Nn(n+1)/2\)-independent parameters: \(N\) distinct nonviscous parameters, say \(\mu_{k}\), and \(1\leq k\leq N\) plus \(n(n+1)/2\) matrix entries within each \(\mathbf{C}_{k}\), \(1\leq k\leq N\):  
\[\mu_{1},\ldots,\mu_{N},\qquad\mathbf{C}_{1}=\left[ \begin{array}{ccc}C_{11}^ {(1)}&\cdots&C_{1n}^{(1)}\\ \vdots&\ddots&\vdots\\ C_{1n}^{(1)}&\cdots&C_{nn}^{(1)}\end{array} \right],\ldots,\mathbf{C}_{N}=\left[ \begin{array}{ccc}C_{11}^{(N)}&\cdots&C_{1n}^{(N)}\\ \vdots&\ddots&\vdots\\ C_{1n}^{(N)}&\cdots&C_{nn}^{(N)}\end{array} \right] \tag{7}\]  
where \(C_{ij}^{(k)}=C_{ji}^{(k)}\) is the \(ij\)-entree of \(\mathbf{C}_{k}\), assumed to be symmetric. Real applications depend, in general, on less parameters, say \(p<<p_{\text{max}}\). Just for the shake of simplicity in our exposition, the array \(\mathbf{\theta}=\left\{\theta_{1},\ldots,\theta_{p}\right\}\) is introduced to denote the set of independent damping parameters. Hence, both the viscoelastic and dynamical stiffness matrices can be written as functional arrays of \(s\) and \(\mathbf{\theta}\), denoted as \(\mathbf{G}(s,\mathbf{\theta})\) and \(\mathbf{D}(s,\mathbf{\theta})\), respectively, such that:  
\[\mathbf{D}(s,\mathbf{\theta})=s^{2}\mathbf{M}+s\,\mathbf{G}(s,\mathbf{\theta})+ \mathbf{K}. \tag{8}\]Let us consider the following system of \(n+1\) equations expressed in terms of the unknowns given by the \((n+1)\) - tuple \((s,u_{1},\ldots,u_{n})\):  
\[F_{j}(s,u_{1},\ldots,u_{n},\mathbf{\theta}) \equiv \sum_{k=1}^{n}D_{jk}(s,\mathbf{\theta})\,u_{k}=0,\quad 1\leq j\leq n,\] \[F_{n+1}(s,u_{1},\ldots,u_{n},\mathbf{\theta}) \equiv f(u_{1},\ldots,u_{n})=0, \tag{9}\]  
where \(D_{jk}(s,\mathbf{\theta})\) is the entry \((j,k)\) of the dynamical stiffness matrix \(\mathbf{D}(s,\mathbf{\theta})\), and \(f(u_{1},\ldots,u_{n})\) is a real-valued function of the \(n\) components \(u_{1},\ldots,u_{n}\), which enables fixing the eigenvector \(\mathbf{u}=\{u_{1},\ldots,u_{n}\}^{T}\). Thus, some normalization forms can be used by the function \(f(\mathbf{u})\), for instance, \(f(\mathbf{u})=\sum_{k=1}^{n}u_{k}^{*}\,u_{k}-1\) (unit vector), \(f(\mathbf{u})=\sum_{k=1}^{n}\sum_{j=1}^{n}u_{j}^{*}\,M_{jk}\,u_{k}-1\) (mass-normalized vector), or \(f(\mathbf{u})=u_{j}-1\) (\(j\)th unit component), where the above \(\mathbf{u}^{*}\) denotes the conjugate-complex vector of \(\mathbf{u}\). Equation (9) can be read as a system of \(n+1\) equations with \(n+1\) unknowns, say \((s,u_{1},\ldots,u_{n})\), which, in turn, are functions of the \(p\) parameters via \(\mathbf{\theta}\in\mathbb{R}^{p}\).