# Vibration Reduction of a Timoshenko Beam with Multiple Parallel Nonlinear Energy Sinks
## 4 Optimization Results  
In this section, the total attached mass of NESs is a constant, and the optimal number and distribution of NESs are investigated. The effect of NES distribution on vibration reduction is considered. For three NESs, the system parameters are fixed to \(\varepsilon=0.1\), \(\zeta=0.1\) and \(k_{3}=9\times 10^{7}\) N/m\({}^{3}\). The constant parameters of the genetic algorithm are fixed to P(N) = 500, \(\text{Gen}_{\text{max}}=20\). The optimal locations of three NESs, obtained with the genetic algorithm, are mounted at \(d_{1}=0.5\) m, \(d_{2}=0.6\) m, \(d_{3}=0.75\) m. To demonstrate the effect of the optimal distribution, a uniform distribution of three NESs is set as a reference group, i.e., \(d_{1}=0.25\) m, \(d_{2}=0.5\) m, \(d_{3}=0.75\) m. The optimization process of the genetic algorithm for three NESs is listed in Table 2. The optimal results are achieved with three iterations. The vibration reduction performances are listed in Table 3. To verify the convergence of the optimal results, an optimization with initial population P(N) = 1000 was carried out, and the same optimal results were obtained. The performances of optimal distribution and uniform distribution are shown in Figure 8. Compared with the uniform distribution, the optimal distribution decreased by 5.19% more at the first resonance. This not only demonstrates the effect of distribution on performance but also proves the efficiency of the genetic algorithm. The effect of the NES number on vibration reduction is considered. Here, \(N\) varies from one to five, and the corresponding \(k_{3}\) is optimized. The optimal locations for different \(N\) NESs are obtained, and the optimal performances of \(N\) NESs are shown in Figure 9. From Figure 9, it is found that under the premise of not changing the total attached mass, \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Iteration** & \(\mathbf{d_{i}}\)** **(m)** & **The 1 st Peak** & **Reduction** & **The 2 nd Peak** & **Reduction** \\ \hline
0 & / & 0.077 & / & 0.0062 & / \\
1 & [0.35, 0.75, 0.8] & 0.044 & 42.86\% & 0.0058 & 7.05\% \\
2 & [0.45, 0.5, 0.8] & 0.042 & 45.45\% & 0.0058 & 7.05\% \\
3 & [0.5, 0.6, 0.75] & 0.040 & 48.05\% & 0.0057 & 8.06\% \\
4 & [0.5, 0.6, 0.75] & 0.040 & 48.05\% & 0.0057 & 8.06\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The distribution optimization process of the genetic algorithm. Figure 8: Comparison of the performance between the optimal distribution and the uniform distribution. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Distribution** & \(\mathbf{d_{i}}\)** **(m)** & **The 1 st Peak** & **Reduction** & **The 2 nd Peak** & **Reduction** \\ \hline No NES & / & 0.077 & / & 0.0062 & / \\ Optimal & [0.5, 0.6, 0.75] & 0.040 & 48.05\% & 0.0057 & 8.06\% \\ Uniform & [0.25, 0.5, 0.75] & 0.044 & 42.86\% & 0.0056 & 9.67\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the performance between different distributions of parallel NESs. maximum vibration reduction occurs at \(N=2\). However, when the number of parallel nonlinear energy sinks continues to increase, the vibration reductions decrease. It is not the case that more is better. The mounting locations of nonlinear energy sinks and their performances are listed in Table 4. Both the vibration reduction performance at the first resonance and the second resonance are analyzed. According to the results of the genetic algorithm, the optimal locations and cubic stiffness are different for different numbers of nonlinear energy sinks. When there is only one nonlinear energy sink attached to the beam, the optimal mounting location is at \(d_{1}=0.5\) m, where the peak of the first modal shape occurs. When \(N\) increases to two, the optimal mounting locations are \(d_{1}=0.5\) m, \(d_{2}=0.75\) m, where the peaks of the first two modes occur, respectively. When \(N\) increases to three, the optimal mounting locations are \(d_{1}=0.5\) m, \(d_{2}=0.75\) m, \(d_{3}=0.6\) m, and \(d_{3}\) is the location where the performance computed.