## 4 Analysis of Nonlinear Tuned Mass Damper Systems Based on Different Conditions
### NTMD Control Performance Analysis Based on 1:1 Resonance Conditions  
The control performance of the NTMD with damping conditions was studied and analyzed in the previous subsection, and now, the control performance of the nonlinear tuned mass damper will be analyzed based on the 1:1 resonance condition. To study the effects of different mass ratios on the optimized frequency and vibration reduction effect of the NTMD, the nonlinear coefficient \(\alpha_{2}=0.3\) is used, and the mass ratios of 0.02, 0.03, 0.04 and 0.05 are considered to obtain the frequency amplitude curves of the primary structure as shown in Figure 8. Mass ratios in engineering are generally very small, so Figure 8 only shows the graph with a maximum mass ratio of 0.05, indicating that the mass of the tuned mass damper is ten percent of the mass of the controlled structure, which is already a fairly large mass ratio in engineering. As shown in the graph, the steady-state amplitude of the controlled structure decreases continuously as the mass ratio increases at the position of the 1:1 resonance. It can be observed that the mass ratio is a favorable factor affecting the amplitude of the controlled primary structure at the 1:1 resonance, and within a certain range, the larger the mass ratio, the better the vibration reduction effect. As shown in Figure 9, the horizontal axis represents the excitation frequency, and the vertical axis represents the steady-state amplitude of the main structure. The original set of differential equations is solved using the Runge-Kutta method in the numerical solution method to obtain the corresponding amplitude-frequency response curves. The control bandwidth of the tuned mass damper increases as the mass ratio increases, while other conditions remain constant. By comparing the maximum steady-state amplitude of the controlled structure at a mass ratio of 0.01 and a mass ratio of 0.02, it can be observed that increasing the mass ratio leads to a decrease in the maximum amplitude. However, the maximum steady-state amplitude of the structure remains the same for mass ratios of 0.02 and 0.03. This suggests that within a certain range, increasing the mass ratio can improve the damping effect of the NTMD without altering other parameters. When the mass ratio reaches a certain degree, the bandwidth of the amplitude-frequency curve expands, but the improvement in the damping effect becomes minimal. Therefore, selecting an appropriate mass ratio can effectively reduce the steady-state amplitude of the structure while controlling the design cost. Based on the obtained results, the mass ratio in engineering  
Figure 7: Damping parameters are \(\lambda_{2}=0.16\), \(\lambda_{2}=0\) amplitude frequency response curve. generally does not exceed 0.05. The recommended range for the mass ratio is between 0.02 and 0.05, provided that the NTMD control effect is ensured. Figure 10 shows the amplitude-frequency response curve of the structure. The blue dashed line indicates the optimized frequency obtained by the undamped frequency design Equation (40) when the mass ratio is 0.03 and \(\sigma=0\) and the amplitude response curve of the excitation frequency-controlled structure based on this optimized frequency design; the black solid line indicates the amplitude response curve of the controlled structure under the uncontrolled condition without the installation of mass dampers; the red dotted line indicates the amplitude-frequency response curve obtained by using the linear design frequency after installing the nonlinear mass damper. In comparison, it is found that both the tuned mass dampers with linear and nonlinear designs install  
Figure 8: Frequency ratio-controlled structure amplitude curves for different mass ratios. Figure 9: Amplitude–frequency response curves obtained by the Runge–Kutta method. vibration reduction effects. However, the results of the amplitude-frequency response curves obtained based on the design equations are better than those obtained by the linear design.