## 3 Approximate Analytical Solution of the Equations of Motion of the System  
To facilitate the use and calculation of subsequent analytical methods, Equations (7) and (8) are decoupled through [PERSON]'s formula. Equations (7) and (8) are considered to be functions of the mass ratio \(\varepsilon\) and are expanded by [PERSON]'s formula near \(\varepsilon=0\). Only the primary term is retained because \(\varepsilon\) itself is small, and the first order expansion is sufficient for subsequent analysis. The above expressions can be written as follows:  
\[r(\varepsilon)=r(0)+\varepsilon r^{\prime}(0) \tag{9}\]  
Substituting Equation (9) into Equations (7) and (8), the following new equation can be obtained:  
\[\ddot{x}+x+\varepsilon\Big{(}\alpha_{1}\lambda_{2}\dot{y}+\alpha_{1}\Omega_{2 }^{2}y+\alpha_{1}\alpha_{2}\Omega_{2}^{2}y^{3}+\lambda_{1}\dot{x}-f\cos\Omega \tau\Big{)}=0 \tag{10}\]  
\[\ddot{y}+y+\varepsilon\delta\Big{(}\Omega_{2}^{2}y+\alpha_{2}\Omega_{2}^{2}y^ {3}+\lambda_{2}\dot{y}+x-y\Big{)}=0 \tag{11}\]  
Among them: \(\varepsilon\delta=1\). Higher-order terms are eliminated from Equations (10) and (11) because they are less significant in the current study, and only the necessary parameter terms are retained. The complex variable averaging method can simplify the process of operation by assuming the steady-state response of the nonlinear TMD as a first-order harmonic approximation with the same equivalent excitation frequency. By setting the solution of the equation in the following form and substituting the set solution into Equations (10) and (11), the second-order system of equations can be reduced to a first-order system for subsequent analysis. The complex variable averaging method is mainly in the form of complexification, i.e., replacing the displacement and velocity variables of the system with complex variables. Therefore, it can reduce the original system equations from second order to first order. This approach simplifies the representation based on expansion methods, such as Fourier series expansion or Taylor expansion. The problem studied in this paper is related to the phenomena arising from the 1:1 resonance of the system. The calculation based on the averaging method of complex variables can focus on the 1:1 resonance frequency of the system and facilitate the relationship between the nonlinear coefficients of the system and the vibration reduction effect of the tuned mass dampers. The averaging method takes the amplitude and phase of the solution as time-dependent parameters and averages the derivatives corresponding to the amplitude and phase of the system. The complex variable-averaging method also follows this approach, but it sets the solution in the form of a complex number, which is more conducive to the solution process. The complex variable-averaging method assumes the vibration as a harmonic wave in complex form with the same frequency as the external excitation. This assumption is similar to the derivation of the averaging method, and the first-order approximate solutions of Equations (10) and (11) can be set as follows:  
\[x=\frac{1}{2i\Omega}A_{1}e^{i\Omega t}+cc \tag{12}\]  
\[y=\frac{1}{2i\Omega}A_{2}e^{i\Omega t}+cc \tag{13}\]  
where \(A_{1}\) is the complex variable, consisting of the steady-state amplitude and phase of the main structure; \(A_{2}\) is the complex variable, consisting of the steady-state amplitude and phase of the nonlinear TMD; \(\Omega\) is the ratio of the excitation frequency to the frequency of the controlled structure; and \(cc\) substitutes the conjugate complex in the first half of the equation. Correspondingly, the two sides of Equations (12) and (13) give expressions for \(\dot{x}\) and \(\dot{y}\) by means of derivatives for time \(t\). The same is true for \(\ddot{x}\) and \(\ddot{y}\).