# Boundaries of Oscillatory Motion in Structures with Nonviscous Dampers
## 4 Numerical Examples  
### Example 1: Discrete System  
In the first example, a discrete four-degrees-of-freedom mass-spring structure will be considered. Figure 1 represents the distribution of the four masses \(m=10\) kg linked by linear springs with coefficient \(k=10^{3}\) N/m. The mass and stiffness matrices yield:  
\[\mathbf{M}=m\mathbf{I}_{4}\;,\quad\mathbf{K}=k\left[\begin{array}{rrrr}2&-1& 0&0\\ -1&2&-1&0\\ 0&-1&2&-1\\ 0&0&-1&2\end{array}\right] \tag{56}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Curve** & **Parameters \(\{\theta_{1},\theta_{2}\}\)** & **Equivalent Eigenvalue Problem** \\ \hline Type I & \(\{c_{i},c_{k}\}\) & one-parameter linear \\ Type II & \(\{\mu_{i},\mu_{k}\}\) & two-parameter linear \\ Type III & \(\{c_{i},\mu_{i}\}\) & one-parameter linear \\ Type IV & \(\{c_{i},\mu_{k}\}\) & one-parameter quadratic \\ \hline \hline \end{tabular}
\end{table}
Table 1: Types of critical curves and the corresponding eigenvalue problem required for their determination (see Example 1 for details about the procedure to obtain such eigenvalue problems).  
Damping is introduced through two types of nonviscous dampers, distributed as shown in Figure 1 and modeled, respectively, with two different hereditary exponential functions, leading to the damping matrix:  
\[\mathbf{\mathcal{G}}(t)=\mu_{1}e^{-p_{1}t}\mathbf{C}_{1}+\mu_{2}e^{-p_{2}t}\mathbf{C }_{2}, \tag{57}\]  
where:  
\[\mathbf{C}_{1}=c_{1}\left[\begin{array}{cccc}1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array}\right]\equiv c_{1}\,\mathbf{Q}_{1}\,\quad\mathbf{C}_{2}=c_{2} \left[\begin{array}{cccc}1&-1&0&0\\ -1&2&-1&0\\ 0&-1&2&-1\\ 0&0&-1&2\end{array}\right]\equiv c_{2}\,\mathbf{Q}_{2}. \tag{58}\]  
Above, the set \(\{c_{1},c_{2},\mu_{1},\mu_{2}\}\) collects the four damping parameters of the viscoelastic model. The main goal is to find critical relationships between different pairs, fixing the rest of them. Since the undamped modes play an important role in the construction of such curves, the linear eigenvalue problem \(\left(-\omega^{2}\mathbf{M}+\mathbf{K}\right)\mathbf{u}=\mathbf{0}\) needs to be solved, resulting in the natural frequencies of Table 2, expressed in terms of a reference frequency, \(\omega_{r}=\sqrt{k/m}=10\) rad/s.  
In the context of this example, Equations (52) and (53) can be particularized, yielding:  
\[\mathcal{R}_{j1}\mathbf{C}_{1}\,\mathbf{u}+\mathcal{R}_{j2}\mathbf{C }_{2}\,\mathbf{u} = \mathfrak{W}_{j}\,\mathbf{u}, \tag{59}\] \[\mathcal{R}_{j1}^{2}\,\mathbf{\phi}_{j}^{T}\mathbf{C}_{1}\mathbf{\phi}_{ j}+\mathcal{R}_{j2}^{2}\,\mathbf{\phi}_{j}^{T}\mathbf{C}_{2}\mathbf{\phi}_{j} = \frac{2}{\alpha^{2}+1}\Big{(}\mathcal{R}_{j1}\,\mathbf{\phi}_{j}^{T} \mathbf{C}_{1}\mathbf{\phi}_{j}+\mathcal{R}_{j2}\,\mathbf{\phi}_{j}^{T}\mathbf{C}_{2} \mathbf{\phi}_{j}\Big{)}\,\quad 1\leq j\leq n, \tag{60}\]  
where:  
\[\mathcal{R}_{j1}=\frac{\alpha}{\alpha-\frac{\omega_{j}}{\mu_{1}}}\,\qquad \mathfrak{W}_{j}=\frac{\omega_{j}}{\alpha}\mathbf{M}+\frac{\alpha}{\omega_{j} }\mathbf{K}\,\quad 1\leq j\leq n. \tag{61}\]  
The procedures for addressing the curves differ depending on the type under consideration (see classification above in Table 1). Although an eigenvalue problem arises in every case, it changes the way in which it is derived, and this example will help clarify how to carry out the solution in each case.