## 4 Numerical Analysis of Vibration Localization Characteristics
### Numerical Results  
#### 4.2.1 Validation of Mode Localization  
The node masses of the oscillators and the center are set to 17 g and 165 g, respectively. The stiffnesses of the main and coupling springs are set to \(3.6\times 10^{3}\) N/m and 36 N/m, respectively; the coupling relationship between the oscillators can be calculated as \(R=0.1\). In order to be consistent with the parameters of the theoretical model, ignore the damping coefficient of the springs. The numerical results of the mode shapes of the CTS are presented in Figure 10.  
Figure 8: Numerical model of coupled spring oscillators.  
Figure 9: Numerical analysis flowchart.  
Comparing the numerical and theoretical results of the mode shapes, it can be found that the 1 st to 5 th and 10 th order modes of the CTS are basically the same under the two calculation methods. The physical properties of the 6 th to 9 th modes are consistent, with the difference being that the amplitude distribution of the modes manifests as different combinations of oscillators. This phenomenon may be attributed to the fact that the serial numbers and positions of the oscillators in the numerical model are constant, and the software sorts the eigenvalues based on an internal algorithm to identify the corresponding mode shapes, whereas the serial numbers of the oscillators in the theoretical model are arbitrary, and their rotations will not affect the physical properties of the structure.  
The numerical results of the mode shapes under different mistuning strengths are shown in Figure 11. It can be found that when the mistuning strengths are 7.2%, 10.9%, and 14.8%, the mode shapes of the cyclic structure of each order are basically consistent under both calculation methods. However, when the mistuning strength is 3.6%, a discernible deviation exists between the numerical and theoretical results, which is due to the fact that the vibration localization of the cyclic structure is weakened. When the mistuning strength is small, the influence of the coupling strength on the mode shapes becomes more pronounced, thereby amplifying the error introduced by the simplified coupling relationship in the theoretical model.  
The main spring stiffness is set to \(3.6\times 10^{3}\) N/m, and the coupling spring stiffness is set to be 0 N/m, 36 N/m, and 144 N/m, which corresponds to \(R=0\), \(R=0.1\), and \(R=0.2\), respectively. The mode shapes of the CMS are calculated for different coupling strengths, as shown in Figure 12. It can be seen that as the coupling strength increases, the mode localization decreases. When \(R=0\) and \(R=0.1\), the coupling strength is significantly smaller than the mistuning strength. Consequently, the error stemming from the simplified coupling relationship in the theoretical model becomes negligible, and the numerical results closely resemble the theoretical results. When \(R=0.2\), the influence of the coupling strength  
Figure 10: Numerical results of mode shapes of the CTS.  
becomes more pronounced, leading to a discernible deviation between the numerical and theoretical results.  
Figure 11: Numerical results of mode shapes for different mistuning strengths.  
Figure 12: Numerical results of mode shapes for different coupling strengths.