## 2 Models of Systems, Equations of Motion and Their Dimensionless Forms
In this paper, the complex variable averaging method is chosen to solve these equations. The complex variable averaging method primarily replaces the displacement and velocity variables of the system with complex variables in order to reduce the order of the original system equations. This simplification allows for a more straightforward representation using techniques such as Fourier series expansion or Taylor expansion. By employing the complex variable averaging method, the analysis can narrow its focus to the 1:1 resonance of the system studied in this paper and facilitate the understanding of the relationship between the nonlinear coefficients of the system and the damping effect of the tuned mass dampers. The averaging method treats the amplitude and phase of the solution as time-dependent parameters and averages the derivatives corresponding to these parameters. Similarly, the complex variable averaging method performs the same procedure, except that it represents the solution in the form of complex numbers, which offers greater convenience in solving. The complex variable averaging method assumes an oscillation in the form of a complex harmonic with the same frequency as the external excitation, similar to the derivation of the averaging method. Therefore, in this analysis, the steady-state response of the nonlinear tuned mass damper is approximated as a first-order harmonic with the same frequency as the equivalent excitation. The first-order approximate solutions for Equations (11) and (12) are set as follows:  
\[x=\frac{1}{2i\Omega}A_{1}e^{i\Omega t}+cc \tag{13}\]  
\[y=\frac{1}{2i\Omega}A_{2}e^{i\Omega t}+cc \tag{14}\]  
where \(A_{1}\) is the complex variable that consists of the steady-state amplitude and phase of the controlled structure, \(A_{2}\) represents the complex variable consisting of the steady-state amplitude and phase of the nonlinear tuned mass damper, and \(cc\) is the conjugate complex representing the first half of the solution. By taking the first- and second-order derivatives of the time variable \(t\) on both sides of Equations (13) and (14), we obtain the expressions for \(\dot{x}\), \(\dot{y}\), \(\ddot{x}\) and \(\ddot{y}\) as follows:  
\[\dot{x}=\frac{1}{2}A_{1}e^{i\Omega t}+cc \tag{15}\]  
\[\dot{y}=\frac{1}{2}A_{2}e^{i\Omega t}+cc \tag{16}\]  
\[\ddot{x}=\dot{A}_{1}e^{i\Omega t}+\frac{1}{2}i\Omega A_{1}e^{i\Omega t}+cc \tag{17}\]  
\[\ddot{y}=\dot{A}_{2}e^{i\Omega t}+\frac{1}{2}i\Omega A_{2}e^{i\Omega t}+cc \tag{18}\]  
The results obtained through the complex variable averaging method are simpler compared to the traditional averaging method, and its complex form is shorter than the trigonometric form, which facilitates analysis. By substituting the set solutions, i.e., Equations (13)-(18), into Equations (11) and (12) and considering only the \(e^{0}\) and \(\epsilon\) terms while omitting higher-order terms and all other terms except \(e^{i\Omega t}\), we obtain a system of two coupled first-order differential equations. It is important to note here that the term \(e^{i\Omega t}\) in the equation, referred to as the long-term term, keeps increasing over time and renders the approximation method invalid. Thus, the long-term term needs to be eliminated in the process of using the complex averaging method. Only the coefficients before this term are considered to sum to zero, and therefore, the coefficients before the long-term term are retained to obtain the following equation:  
\[ \begin{array}{c}\frac{1}{a_{1}}\Big{(}A_{1}+\frac{1}{2}i\Omega A_{1}+\frac{1}{2 i\Omega}A_{1}\Big{)}\\ \qquad\qquad+\varepsilon\bigg{(}\frac{\lambda_{2}}{2}A_{2}+\frac{\Omega_{2}^{ 2}}{2i}A_{2}+\frac{3a_{2}\Omega_{2}^{2}}{8i}|A_{2}|^{2}A_{2}+\frac{\lambda_{1 }}{2a_{1}}A_{1}-\frac{f}{2a_{1}}\bigg{)}=0\end{array}  \tag{19}\]  
\[