# Boundaries of Oscillatory Motion in Structures with Nonviscous Dampers
## 1 Introduction  
Currently, there are many applications, from micro-scale systems to large structures, that require the control of vibration, sound, and wave propagation for proper operation. New techniques to optimize energy dissipation have been proposed in recent years, from the consolidation of time-dependent viscoelastic materials [1; 2; 3; 4], to the emergent phenomenon of metadamping [5; 6]. In part, the advances in new dissipation mechanisms have been carried out thanks to the constant increase in computing capacity, since the new discoveries are generally accompanied by more complex mathematical models. In the context of vibration control with nonviscously damped materials, it is especially important to know how to choose the characteristics of the dissipative devices or materials to ensure that the system becomes overcritically damped in one or more modes. In this article, we investigate criticality in nonviscously damped multiple-degrees-of-freedom (dof) systems, considering any number of exponential hereditary kernels. Nonviscous damping is characterized by dissipative forces which depend on the past history of the velocity response via convolution integrals over hereditary kernel functions. Denoting, with \(\mathbf{u}(t)\in\mathbb{R}^{n}\), the array with degrees of freedom, the equations of motion are expressed in integro-differential form, as [1]:  
\[\mathbf{M}\ddot{\mathbf{u}}+\int_{-\infty}^{t}\boldsymbol{\mathcal{G}}(t- \tau)\,\dot{\mathbf{u}}\,\mathrm{d}\tau+\mathbf{Ku}=\boldsymbol{\mathfrak{f}}( t), \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{K}\in\mathbb{R}^{n\times n}\) are the mass and stiffness matrices assumed to be positive definite and positive semidefinite, respectively; \(\boldsymbol{\mathcal{G}}(t)\in\mathbb{R}^{n\times n}\) represents the time-domain damping matrix, assumed to be symmetric. Viscous damping arises just as a particular form ofEquation (1), with \(\mathbf{\mathcal{G}}(t)\equiv\mathbf{C}\,\delta(t)\), where \(\mathbf{C}\) is the viscous damping matrix and \(\delta(t)\) the Dirac's delta function. Checking for solutions of the form \(\mathbf{u}(t)=\mathbf{u}\,e^{st}\) in Equation (1) transforms the time-domain equations into a nonlinear eigenvalue problem with the form:  
\[\Big{[}s^{2}\mathbf{M}+s\mathbf{G}(s)+\mathbf{K}\Big{]}\mathbf{u}\equiv\mathbf{D}(s )\,\mathbf{u}=\mathbf{0}, \tag{2}\]  
where \(\mathbf{G}(s)=\mathcal{L}\{\mathbf{\mathcal{G}}(t)\}\in\mathbb{C}^{n\times n}\) is the frequency-domain damping matrix, \(\mathbf{D}(s)\) is the dynamical stiffness matrix, and \(s=i\omega\) is the Laplace parameter. The roots of the characteristic equation:  
\[\det[\mathbf{D}(s)]=0, \tag{3}\]  
are the eigenvalues of Equation (2). Viscoelastically damped structures are characterized as a frequency-dependent damping matrix. The time-domain response will be affected by the nature of the eigenvalues of Equation (2). In turn, complex eigenvalues are associated to oscillatory motion; meanwhile, real negative eigenvalues lead to nonoscillatory modes. When the nonviscous damping model is based on hereditary exponential kernels, something that will be assumed in the current investigation, the number of eigenvalues will be invariably greater than \(2n\) because there exist \(p\) real nonviscous modes associated to hereditary exponential kernels [7; 8]. Thus, undamped or lightly damped systems present \(2n\) oscillatory modes with a relatively small real part in magnitude, together with \(p\) real nonviscous eigenvalues. However, as the damping level increases, the complex eigenvalues move away from the undamped ones in the complex plane. For high-damping forces, some conjugate-complex pair of eigenvalues may drop into the real axis, vanishing the oscillatory nature. The root locus in the complex plane depends on the damping parameters presented in the matrix \(\mathbf{G}(s)\). In general, viscoelastic models are mathematically defined by several parameters so that the set of all of them defines a multidimensional parametric domain.