## 3 Approximate Analytical Solution of the Equations of Motion of the System
Figure 11 shows a comparison of frequency-amplitude curves for NTMD and primary structure mass ratios of \(\alpha_{1}=6,7,\) 8, 9, and 10. After the mass ratio of NTMD exceeded 10%, the improvement of the control effect for each 2% increase in the mass ratio of NTMD was not obvious, and too large of a mass ratio has influence on the control performance of the structure. In summary, the NTMD to primary structure mass ratio of \(\alpha_{1}=5\) is more suitable. The response of the system was solved using the Runge-Kutta numerical integration method. The response characteristics of the solution were analyzed using a time course diagram for the parameter conditions of \(\omega_{2}=0.821\), \(\alpha_{1}=5\), and \(\alpha_{2}=0.02\). As shown in Figure 12, the transient amplitude of the structure was reduced by about 44.75%, and the steady-state amplitude of the structure was reduced by about 36.62% compared to the initial design. The vibration reduction performance of the structure was effectively improved. In summary, the optimal theoretical design parameters for a series of structures were obtained using the complex variable averaging method. Under these design parameters, the structural control performance was greatly improved, which provides a valuable reference for subsequent studies on the nonlinear characteristics of the TMD. Figure 13 shows the frequency response curves of the primary structure with NTMD installed, designed using different methods. The frequency response curves allowed us to observe whether the steady-state amplitude of the structure decreased at different excitation frequencies after optimizing the NTMD. The black, blue, and red curves indicate the structures without TMD installed, NTMD with traditional linear design installed, and NTMD with nonlinear design installed. Figure 13 shows that the control performance obtained by using the traditional linear design method to optimize the NTMD was not  
\[\alpha_{1}\Omega_{2}^{2}a_{2}+\frac{3}{4}\alpha_{1}\alpha_{2}\Omega_{2}^{2}a_{ 2}^{3}+\lambda_{1}\lambda_{2}a_{2}=f\sin b \tag{28}\]6 Frequency–amplitude curves of the primary structure corresponding to different mass ratios \(\alpha_{1}\) (\(\alpha_{1}=1,2,3,4,5\)). satisfactory. The peak difference between the amplitude bimodal peaks was large, and the optimization effect was not good. In contrast, NTMD designed with nonlinear optimal parameters showed an equal amplitude at both left and right peaks. The control performance of the primary structure significantly improved compared to the linearly designed NTMD. \[\alpha_{1}\Omega_{2}^{2}a_{2}+\frac{3}{4}\alpha_{1}\alpha_{2}\Omega_{2}^{2}a_{ 2}^{3}+\lambda_{1}\lambda_{2}a_{2}=f\sin b \tag{28}\]7 Frequency-amplitude curves of the primary structure corresponding to different mass ratios \(\alpha_{1}\) (\(\alpha_{1}=6,7,8,9,10\)). \[\alpha_{1}\Omega_{2}^{2}a_{2}+\frac{3}{4}\alpha_{1}\alpha_{2}\Omega_{2}^{2}a_{ 2}^{3}+\lambda_{1}\lambda_{2}a_{2}=f\sin b \tag{28}\]8 Comparison of the amplitude time course curves of the primary structure with various parameter designs. Figure 14 compares the control performance of the TMD and the NTMD. As shown in the figure, the red, blue, and green curves indicate the frequency response curves of the TMD with conventional optimal linear design, the TMD with nonlinear design, and the NTMD with nonlinear design, respectively. Compared to the conventional optimal linear design, the control performance of the TMD with nonlinear design was slightly weaker, but it still achieved nearly 58.33% of the optimized effect. Therefore, the nonlinear design conducted in the previous section is still valid for the linear TMD. As seen from Figure 14, the control performance obtained from the NTMD designed according to the nonlinear design method was comparable to that of the TMD designed using traditional linear design. Moreover, the TMD designed using traditional linear design and the NTMD designed using the nonlinear design method possessed 70.87% and 69.67%, respectively, of the optimization effect.