## 3 Approximate Analytical Solution of the Equations of Motion of the System
For various combinations of \(\alpha_{1}\) and \(\alpha_{2}\), the approximate analytical solution of the optimal design frequency was obtained using Equation (30) to compare with the solution obtained by the conventional numerical method. From the data in Table 1, it was concluded that the average error between the approximate  
Figure 4: Frequency–amplitude curves obtained using the complex variable averaging method and numerical method. analytical solution and the numerical method was 0.0017, and the average error rate was 0.188%. Although there were some errors between the results of the complex variable averaging method and the traditional numerical method, the errors were very small and even negligible. The accuracy of Equation (30) was proven to be very high; thus, the correctness of the modulation demodulation equation obtained by the complex variable averaging method is known, and the equation can be used for subsequent analysis. Inherent frequency is a key parameter in TMD design. Choosing a suitable frequency enables the TMD to achieve better control performance, and the corresponding response amplitude of the primary structure will be relatively small. In the frequency-amplitude curve, the frequency \(\omega_{2}\) corresponding to the maximum amplitude point generated by the TMD and the minimum amplitude point generated by the primary structure represent the theoretical optimum design value of TMD frequency. Figure 4 shows the frequency-amplitude curves of the primary structure and the TMD obtained by numerical and complex variable averaging methods of the system equations of motion, from which it was concluded that when \(\omega_{2}\) = 0.821, the corresponding TMD amplitude is maximum, and the amplitude of the primary structure is minimum. That is, the theoretical optimum design value of the TMD frequency of the system can be considered 0.821. Figure 5 shows the comparison of the steady-state displacement time diagrams obtained for the primary structure without TMD and the primary structure under nonlinear TMD control when \(\omega_{2}\) = 0.821 and other parameters are constant. As can be seen in Figure 5, the steady-state displacement of the primary structure with the non-linear TMD installed at this frequency was significantly smaller than the steady-state displacement of the primary structure without the TMD. The steady-state displacement was reduced by about 35.90%, which fully proves the effectiveness of the nonlinear TMD control effect at this frequency. Figure 6 shows the comparison of the steady-state displacement time plots of the first-stage structure under the control of the ideal linear TMD and the optimized nonlinear TMD. From Figure 6, it can be seen that the steady-state displacement of the first-stage structure with the nonlinear TMD installed at this frequency was smaller than the steady-state displacement of the first-stage structure with the optimal linear TMD. The transient response of the primary structure was partially the same, and the steady-state displacement was reduced by about 9.05%. The ideal linear TMD is a hypothetical linear TMD without any nonlinear characteristics generated during the vibration, which cannot exist in reality. It can be seen that when the design frequency \(\omega_{2}\) = 0.821, the nonlinear TMD achieved good control effectiveness, and subsequent steady-state analysis was performed based on this design frequency. \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{\(\mathbf{\kappa_{1}}\)} & \multirow{2}{*}{\(\mathbf{\kappa_{2}}\)} & **Numerical Solution Method** & **Complex Variable Averaging Method** \\  & & **(rad/s)** & **(rad/s)** \\ \hline
1 & 0.01 & 0.817 & 0.821 \\
1 & 0.02 & 0.708 & 0.711 \\
2 & 0.01 & 0.916 & 0.918 \\
2 & 0.02 & 0.846 & 0.848 \\
3 & 0.01 & 0.958 & 0.957 \\
3 & 0.02 & 0.914 & 0.912 \\
4 & 0.01 & 0.978 &\[\alpha_{1}\Omega_{2}^{2}a_{2}+\frac{3}{4}\alpha_{1}\alpha_{2}\Omega_{2}^{2}a_{ 2}^{3}+\lambda_{1}\lambda_{2}a_{2}=f\sin b \tag{28}\]0 Primary structure amplitude time course curve (\(\omega_{2}=0.821\)).