# Boundaries of Oscillatory Motion in Structures with Nonviscous Dampers
## 4 Numerical Examples
### Example 2: Continuous Systems  
This example is intended to illustrate the application of the method of continuous systems with a higher number of degrees of freedom, in particular, a beam modeled using finite elements. Consider the beam of length \(l=5\) m, shown in Figure 6, with simple and clamped supports as boundary conditions, and with three local viscoelastic dampers, placed at sections \(l/3,\,\,2l/3\), and \(l\), constraining the vertical displacements. A total number of \(n_{e}=12\) two-node Euler-Bernoulli finite elements are used. Each node has three degrees of freedom (two displacements and rotation) resulting in a 36-dof system. Figure 5: Critical curves of the four-dof discrete system of Example 1 in the domain of parameters \((\zeta_{1},\
u_{2})\) (viscous ratio of the first kernel and nonviscous parameters of the second kernel). Other parameters: \(\zeta_{2}=0.80\) and \(\
u_{1}=0.02\). The assumed material has a Young modulus \(E=210\) GPa and density \(\rho=7.85\) t/m\({}^{3}\). The cross section is constant with a flexural stiffness of \(EI=224\) kNm\({}^{2}\) and a mass per unit of length of \(\rho A=62.8\) kg/m. Each viscoelastic link is formed by a linear spring together with a nonviscous exponential damping function. The stiffness of the springs are \(k_{1}=k_{2}=3.584\) kN/m and \(k_{3}=7.168\) kN/m. According to the hypothesis assumed in this paper, the three associated exponential kernels (\(N=3\)) are controlled under three relaxation parameters \(\mu_{k}\). Hence, \(\mathcal{G}_{k}(t)=\mu_{k}\,c_{k}\,e^{-\mu_{k}t}\), \(1\leq k\leq 3\). The damping model is then a function of three viscous matrices  
\[\mathbf{\mathcal{G}}(t)=\mathbf{C}_{1}\,\mu_{1}\,e^{-\mu_{1}t}+\mathbf{C}_{2}\,\mu _{2}\,e^{-\mu_{2}t}+\mathbf{C}_{3}\,\mu_{3}\,e^{-\mu_{3}t}, \tag{83}\]  
where the matrix \(\mathbf{C}_{k}\) has a non-zero element equal to \(c_{k}\) in the entries corresponding to those degrees of freedom that are affected by the \(k\)th damper. The parameters will be written, preferably in dimensionless form, as \(\zeta_{k}=c_{k}/2m_{r}\omega_{r}\) and \(\
u_{k}=\omega_{r}/\mu_{k}\), where \(m_{r}=314\) kg is the total mass of the beam and \(\omega_{r}=\sqrt{EI/ml^{3}}=2.3889\) rad/s is the reference frequency. We propose drawing the critical curves of the parameters \((\zeta_{1},\
u_{1})\) for the fixed values of the rest of parameters, say \(\{\zeta_{2},\zeta_{3},\
u_{2},\
u_{3}\}\). According to the classification given above, this is a type-III curve. Parameters \(c_{1}\) and \(\mu_{1}\) are part of the matrix \(\mathbf{C}_{1}=c_{1}\mathbf{Q}_{1}\) and the auxiliary variable \(\mathcal{R}_{j1}=\alpha/(\alpha-\omega_{j}/\mu_{1})\), respectively. A derivation of the associated eigenvalue problem is carried out by repeating the process as in the previous example (type-III curves), but considering a greater number of hereditary nuclei. Thus, Equations (52) and (53), after some simplifications, yield:  
\[\mathbf{Q}_{1}\,\mathbf{u}=\lambda\,\mathbf{\mathcal{B}}_{j}\,\mathbf{u}, \tag{84}\]  
where:  
\[\mathbf{\mathcal{B}}_{j}=\mathfrak{M}_{j}-\sum_{k=2}^{N}\mathcal{R}_{jk}\mathbf{C }_{k}, \tag{85}\]  
and the new parameter \(\lambda\) is just a change of the variable of \(\mu_{1}\) under the expression:  
\[\lambda=-\frac{\left[(\alpha^{2}+1)\mathcal{R}_{j1}-2\right]\mathbf{\phi}_{j}^{T} \mathbf{Q}_{1}\mathbf{\phi}_{j}}{\sum_{k=2}^{N}\mathcal{R}_{jk}\left[(\alpha^{2}+ 1)\mathcal{R}_{jk}-2\right]\mathbf{\phi}_{j}^{T}\mathbf{C}_{k}\mathbf{\phi}_{j}}. \tag{86}\]  
As in previous cases, the curves are obtained by sweeping out the interval \(0<\alpha\leq 1\) for each mode \(1<j<n\). The list of so-determined \(\lambda\) values are first transformed into \(\mu_{1}\) by Equation (86), and then to \(c_{1}\), knowing that \(c_{1}=1/(\mathcal{R}_{j1}\,\lambda)\). The graphical results are shown in Figure 7, where it can be noticed that taking all modes in structures with a large number of degrees of freedom into consideration is not necessary.