# Boundaries of Oscillatory Motion in Structures with Nonviscous Dampers
## 3 Derivation of Critical Curves: the Modal Critical Equation
Setting a mode \(j\), then, by sweeping out the parameter \(\alpha\) in the range \(0<\alpha\leq 1\), the graph of a curve arises, solving the different solutions of Equations (52) and (53) in the unknowns \(\theta_{1}\) and \(\theta_{2}\). Therefore, a family of \(n\) curves may be plotted:  
\[s^{2}\frac{\mathfrak{g}(s,\mathbf{u})-s\,\mathfrak{g}^{\prime}(s, \mathbf{u})}{\mathfrak{g}(s,\mathbf{u})+s\,\mathfrak{g}^{\prime}(s,\mathbf{u })} =s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}{g_{jj}(s) +s\,g^{\prime}_{jj}(s)+\mathcal{O}(2)}=s^{2}\frac{g_{jj}(s)-s\,g^{\prime}_{jj }(s)}{g_{jj}(s)+s\,g^{\prime}_{jj}(s)}+\mathcal{O}(2) \tag{43}\]7  
although, for large systems, the most representative modes can be taken, as shown later in Example 2. It turns out that Equations (52) and (53) constitute, themselves, an eigenvalue problem in the parameters \(\theta_{1}\) and \(\theta_{2}\). In some cases, one of these parameters may be obtained as a linear or quadratic function of the other parameter from Equation (53), and after plugging into Equation (52), an one-parameter eigenvalue problem is obtained. In other cases, this is not possible, and both parameters must be solved simultaneously as part of a two-parameter eigenvalue problem. The different scenarios arise after combining the damping parameters in pairs, leading to four possible type of curves (see Table 1). The details explaining how to transform Equations (52) and (53) into eigenvalue problems are explained in Example 1. Below, the four distinct types of curves which can be found are listed, together with a brief description of the algebraic numerical problem that arises:  
* Critical curves between two viscous coefficients \(\{c_{i}(\alpha),c_{k}(\alpha)\}\), where \(c_{i}\) and \(c_{k}\) are entries of \(\mathbf{C}_{i}\) and \(\mathbf{C}_{k}\), respectively. If \(i\
eq k\), both coefficients belong to matrices associated to different kernels. Otherwise, if \(i=k\), then both parameters are part of the same viscous matrix, a case which can be denoted as \(\{c_{i1}(\alpha),c_{i2}(\alpha)\}\). Since both Equations (52) and (53) are linear in the viscous parameters, \(\{c_{i1}(\alpha),c_{i2}(\alpha)\}\), one of these coefficients can be solved from Equation (53) and plugged into Equation (52), leading to a linear eigenvalue problem. * Critical curves that relate pairs of different nonviscous coefficients \(\{\mu_{i}(\alpha),\mu_{k}(\alpha)\}\) corresponding to different hereditary kernels. The numerical method is based on considering the auxiliary variables \(\{\mathcal{R}_{ji},\mathcal{R}_{jk}\}\) instead of \(\mu_{i}\) and \(\mu_{k}\). Due to the mathematical structure of Equations (52) and (53), both parameters \(\mathcal{R}_{ji}\) and \(\mathcal{R}_{jk}\) must be solved simultaneously by a two-parameter linear eigenvalue problem. * Critical curves between a viscous and a nonviscous coefficient, both corresponding to the same \(i\)th kernel, \(\{c_{i}(\alpha),\mu_{i}(\alpha)\}\), where \(c_{i}\) is an entree of \(\mathbf{C}_{i}\). Since both \(c_{i}\) and \(\mu_{i}\) belong to the same kernel, straight rearrangements of Equations (52) and (53) lead to a linear eigenvalue problem involving one parameter. * Critical curves of a viscous coefficient of the \(i\)th kernel, say \(c_{i}\), with the \(k\)th relaxation parameter \(\mu_{k},i\
eq k\): \(\{c_{i}(\alpha),\mu_{k}(\alpha)\}\). Again, using Equation (53) to solve for \(c_{i}\), and after straight manipulations, the matrix equation (52) can be transformed into a quadratic eigenvalue problem with one parameter. As known, any quadratic eigenvalue problem can be reduced to a double-sized linear problem.