## 3 Approximate Analytical Solution of the Equations of Motion of the System
\begin{split}&\alpha_{1}\lambda_{2}A_{2}-i\alpha_{1}\Omega_{2}^{2 }A_{2}-\\ &\frac{3}{4}i\alpha_{1}\alpha_{2}\Omega_{2}^{2}|A_{2}|^{2}A_{2}- \lambda_{1}\Omega_{2}^{2}A_{2}-\frac{3}{4}\alpha_{2}\lambda_{1}\Omega_{2}^{2} |A_{2}|^{2}A_{2}-i\lambda_{1}\lambda_{2}A_{2}+\lambda_{1}A_{2}-f=0\end{split}  \tag{26}\]  
By expanding \(A_{2}\) in Equation (26) into the form of complex numbers, substituting the expansion into Equation (26), and then separating the real and imaginary parts, we could obtain nonlinear Equations (27) and (28) with respect to amplitude and phase as follows:  
\[\alpha_{1}\lambda_{2}a_{2}-\lambda_{1}\Omega_{2}^{2}a_{2}-\frac{3}{4}\lambda_ {1}\alpha_{2}\Omega_{2}^{2}a_{2}^{3}+\lambda_{1}a_{2}=f\cos b \tag{27}\]  
\[\alpha_{1}\Omega_{2}^{2}a_{2}+\frac{3}{4}\alpha_{1}\alpha_{2}\Omega_{2}^{2}a_{ 2}^{3}+\lambda_{1}\lambda_{2}a_{2}=f\sin b \tag{28}\]By adding Equations (27) and (28) squared, the redundant trigonometric functions can be eliminated to obtain a sixfold equation related only to the tuned mass damper amplitude \(a_{2}\). After organizing the equations, we can obtain:  
\[\left\{\left[\frac{\left(3\alpha_{1}\alpha_{2}a_{2}^{3}\right)}{4}+ \alpha_{1}a_{2}\right]^{2}+\left[\frac{\left(3\lambda_{1}\alpha_{2}a_{2}^{3} \right)}{4}+\lambda_{1}a_{2}\right]^{2}\right\}\Omega_{2}^{4}+ \tag{29}\] \[\left\{\left(2\lambda_{1}\lambda_{2}a_{2}\right)\left[\frac{ \left(3\alpha_{1}\alpha_{2}a_{2}^{3}\right)}{4}+\alpha_{1}a_{2}\right]-2( \lambda_{1}a_{2}+\lambda_{2}\alpha_{1}a_{2})\left[\frac{\left(3\lambda_{1} \alpha_{2}a_{2}^{3}\right)}{4}+\lambda_{1}a_{2}\right]\right\}\Omega_{2}^{2}\] \[+(\lambda_{1}a+\lambda_{2}\alpha_{1}a_{2})^{2}+\lambda_{1}^{2} \lambda_{2}^{2}a_{2}^{2}-f^{2}=0\]  
Equation (29) has a more complex and confusing form, which does not facilitate subsequent calculations and analysis. Equation (29) can be rectified and transformed into an equation about \(z\), where \(z=a_{2}^{2}\):  
\[\left[\left(\frac{9\alpha_{1}^{2}\alpha_{2}^{2}\Omega_{2}^{4}}{16} \right)+\left(\frac{9\lambda_{1}^{2}\alpha_{2}^{2}\Omega_{2}^{4}}{16}\right) \right]z^{3}+\] (30) \[\left\{\frac{\left[3\alpha_{1}\alpha_{2}\Omega_{2}^{2}\left( \alpha_{1}\Omega_{2}^{2}+\lambda_{1}\lambda_{2}\right)\right]}{2}-\frac{\left[ 3\lambda_{1}\alpha_{2}\Omega_{2}^{2}\left(-\lambda_{1}\Omega_{2}^{2}+\lambda_ {1}+\lambda_{2}\alpha_{1}\right)\right]}{2}\right\}z^{2}\] \[+\left[\left(\alpha_{1}\Omega_{2}^{2}+\lambda_{1}\lambda_{2} \right)^{2}+\left(-\lambda_{1}\Omega_{2}^{2}+\lambda_{1}+\lambda_{2}\alpha_{1 }\right)^{2}\right]z-The mass ratio of the TMD to the primary structure was tentatively set at 0.02, and the remaining design parameters were determined based on the optimized equation for linear TMD design with damping:  
\[m_{2}=0.02\ \text{kg},\ c_{2}=0.0031\ \text{N}\cdot\text{s}/\text{m},\ k_{2}=0.0161\ \text{N}/\text{m}\]  
The corresponding dimensionless parameters could also be obtained as:  
\[\lambda_{1}=5,\ \lambda_{2}=0.155,\ f=15\]  
This resulted in a frequency-amplitude curve related to the TMD frequency. Figure 4 shows the frequency-amplitude curves of the structure, where the horizontal coordinate is the intrinsic frequency of the TMD and the vertical coordinate is the amplitude. The solid black line is the result obtained by the complex variable averaging method, and the red dashed line is the result obtained by the numerical method. The graph shows that the curves obtained by the complex variable averaging method and the numerical method are close to each other with almost no error. It can be seen that the modulation demodulation equation obtained by the complex variable averaging method is highly accurate and meets the accuracy requirement. Table 1 shows ten combinations (columns 1 and 2 of the table) where the structural mass ratios \(\alpha_{1}\) are 1, 2, 3, 4, and 5, and the nonlinear stiffnesses \(\alpha_{2}\) are 0.01 and 0.02. Other parameters were held constant.