# Vibration Reduction of a Timoshenko Beam with Multiple Parallel Nonlinear Energy Sinks
## 3 Analysis Results
### Parameter Analysis  
To further improve the performance of NESs, the effects of NES parameters are investigated by varying the value of \(m\), \(c\), and \(k_{3}\), respectively. The NESs are mounted randomly on a beam in this section. The parameter influence of the mass ratio \(\varepsilon\) is demonstrated in Figure 5. The system parameters are fixed to \(\zeta=0.1\), \(k_{3}=10^{7}\) N/m\({}^{3}\), \(d_{1}=0.5\) m, \(d_{2}=0.75\) m, \(d_{3}=0.8\) m. As \(\varepsilon\) increases from 0.05 to 0.2, better performances at the first two resonance frequencies are obtained. However, there is usually a limit to the total attached mass.  
The parameter influence of the damping ratio \(\zeta\) is demonstrated in Figure 6. The system parameters are fixed to \(k_{3}=10^{7}\) N/m\({}^{3}\), \(\varepsilon=0.1\), \(d_{1}=0.5\) m, \(d_{2}=0.75\) m, and \(d_{3}=0.8\) m. As the damping ratio \(\zeta\) increases from 0.05 to 0.15, the amplitude of the beam decreases in all frequency ranges. However, due to material limits, there is usually a limit to damping ratios as well.  
Figure 4: (**a**) Numerical verification; (**b**) local magnified view.  
Figure 5: (**a**) Response at the first resonance frequency under varying mass ratio; (**b**) response at the second resonance frequency under varying mass ratio.  
The parameter influence of cubic stiffness \(k_{3}\) is shown in Figure 7. The system parameters are fixed to \(\varepsilon=0.1\), \(\zeta=0.1\), \(d_{1}=0.5\) m, \(d_{2}=0.75\) m, and \(d_{3}=0.8\) m. With cubic stiffness ranges from 100 to \(10^{7}\) N/m\({}^{3}\), the resonant amplitudes are almost the same. This indicates that it is not possible to improve vibration reduction performance by adjusting cubic stiffness within this range. With the cubic stiffness of \(10^{8}\) N/m\({}^{3}\), a larger reduction is achieved for the first resonant amplitude, together with a slight change of resonant frequency. With the cubic stiffness of \(10^{9}\) N/m\({}^{3}\), an unstable response is exhibited due to strong nonlinearity.  
To obtain reliable vibration reduction performance, in the following section, all the optimization results are based on the stable response of the NESs. However, trying to find the optimal value of \(k_{3}\) with the method of exhaustion is very time-consuming. To obtain the optimal value in a short time, the genetic algorithm is used to solve this problem. The optimization function is the energy dissipated portion \(\eta_{\rm NES}\). The searching range of \(k_{3}\) is set at \(k_{3}=[10^{5},10^{9}]\) N/m\({}^{3}\), where the steady-state results can be guaranteed. The constant parameters of the genetic algorithm are fixed at P(\(N\)) = 500, Gen\({}_{\rm max}\) = 20. After iterating for 20 generations, the optimal \(k_{3}\) value for three NESs is founded at \(k_{3}=9\times 10^{7}\) N/m\({}^{3}\).  
Figure 6: (**a**) Response at the first resonance frequency under varying damping ratio; (**b**) response at the second resonance frequency under varying damping ratio.  
Figure 7: (**a**) Response at the first resonance frequency under varying cubic stiffness; (**b**) response at the second resonance frequency under varying cubic stiffness.