## 2 Models of Systems, Equations of Motion and Their Dimensionless Forms
The dimensionless quantity \(\Omega_{2}\) represents the ratio of the nonlinear tuned mass damper frequency to the controlled structure frequency, and \(\Omega\) represents the ratio of the excitation frequency to the controlled structure frequency. The specific dimensionless coefficients are defined as follows:  
\[ \begin{array}{l}\frac{m_{2}}{m_{1}}=\varepsilon\alpha_{1},x=\frac{u_{1}}{L}, y=\frac{u_{1}-u_{2}}{L},\frac{k_{1}}{m_{1}}=\omega_{1}^{2},\frac{k_{2}}{m_{2}}= \omega_{2}^{2},\frac{c_{1}}{m_{1}\omega_{1}}=\varepsilon\lambda_{1}\\ \frac{\varepsilon}{Lk_{1}}=\varepsilon f,\frac{c_{2}}{m_{2}\omega_{1}}=\lambda _{2},\frac{k_{2}}{k_{2}}=\alpha_{2},\frac{\omega}{\omega_{1}}=\Omega,\frac{ \omega_{2}}{\omega_{1}}=\Omega_{2},\omega_{1}t=\tau\end{array}  \tag{3}\]  
After the dimensionless transformation, Equations (1) and (2) become of the following form:  
\[(1+\varepsilon\alpha_{1})\ddot{x}+\varepsilon\lambda_{1}\dot{x}+x-\varepsilon \alpha_{1}\ddot{y}=\varepsilon f\cos\Omega t \tag{4}\]  
\[\ddot{y}-\ddot{x}+\lambda_{2}\dot{y}+\Omega_{2}^{2}y+\alpha_{2}\Omega_{2}^{2}y ^{3}=0 \tag{5}\]  
Equations (4) and (5) both contain both \(\ddot{x}\) and \(\ddot{y}\) terms, which is not easy to solve. Combining the two equations yields Equation (6) with only \(\ddot{x}\) and Equation (7) with only \(\ddot{y}\):  
\[\frac{\ddot{x}}{\alpha_{1}}+\varepsilon\lambda_{2}\dot{y}+\varepsilon\Omega_{ 2}^{2}y+\frac{\varepsilon\lambda_{1}}{\alpha_{1}}\dot{x}+\frac{x}{\alpha_{1}} +\frac{\varepsilon\alpha_{2}\Omega_{2}^{2}}{\alpha_{1}}x^{3}=\frac{\varepsilon f }{\alpha_{1}}\cos\Omega t \tag{6}\]  
\[\frac{1}{1+\varepsilon\alpha_{1}}\ddot{y}+\lambda_{2}\dot{y}+\Omega_{2}^{2}y +\alpha_{2}\Omega_{2}^{2}y^{3}+\frac{\varepsilon\lambda_{1}\dot{x}}{1+ \varepsilon\alpha_{1}}+\frac{x}{1+\varepsilon\alpha_{1}}=\frac{\varepsilon f \cos\Omega t}{1+\varepsilon\alpha_{1}} \tag{7}\]  
For Equations (6) and (7), a direct analytical solution is more difficult, and the original equations are treated by Taylor expansion. Equations (6) and (7) are treated as functions of small parameters, expanded at \(r(0+\varepsilon)\) and retained up to the primary term. Since \(\varepsilon\) itself is small, the first-order expansion is sufficient for the subsequent analysis:  
\[r(\varepsilon)=r(0)+\varepsilon r^{\prime}(0) \tag{8}\]  
Expanding Equation (6) in the form of Equation (8) yields \(r(0)\) and \(r^{\prime}(0)\) of Equation (6):  
\[\left\{ \begin{array}{l}r(0)=\frac{\ddot{x}+x}{\alpha_{1}}\\ r^{\prime}(0)=\lambda_{2}\dot{y}+\Omega_{2}^{2}y+\alpha_{2}\Omega_{2}^{2}y^{3}+ \frac{\lambda_{1}}{\alpha_{1}}\dot{x}-\frac{f}{\alpha_{1}}\cos\omega t\end{array} \right. \tag{9}\]  
In the same way, Equation (7) is expanded in the same way to obtain \(r(0)\) and \(r^{\prime}(0)\) of Equation (7):  
\[\left\{ \begin{array}{l}r(0)=\ddot{y}+\lambda_{2}\dot{y}+\Omega_{2}^{2}y+ \alpha_{2}\Omega_{2}^{2}y^{3}+x\\ r^{\prime}(0)=-\dot{\alpha}_{1}\ddot{y}+\lambda_{2}\dot{y}+\lambda_{1}\dot{x}-x- f\cos\Omega t\end{array} \right. \tag{10}\]  
With Equations (9) and (10), Equations (6) and (7) can be transformed into a simpler way with the following:  
\[\frac{\ddot{x}+x}{\alpha_{1}}+\varepsilon\bigg{(}\lambda_{2}\dot{y}+\Omega_{2 }^{2}y+\alpha_{3}\Omega_{2}^{2}y^{3}+\frac{\lambda_{1}}{\alpha_{1}}\dot{x}- \frac{f}{\alpha_{1}}\cos\Omega t\bigg{)}=0 \tag{11}\]  
\[\ddot{y}+y+\varepsilon\delta\bigg{(}\lambda_{2}\dot{y}+\Omega_{2}^{2}y+\alpha_{ 2}\Omega_{2}^{2}y^{3}+x-y\bigg{)}=0 \tag{12}\]  
Equation (12) is in the form after omitting the small quantities, where \(\varepsilon\delta=1\). The system of nonlinear equations, which was dimensionlessized and Taylor-expanded, is already in a form suitable for subsequent solutions using the complex variable averaging method. Such a transformation is performed mainly to transform the system of nonlinear differential equations of higher order into lower order, which is beneficial for subsequent analysis. The differential equations governing the system under the control of nonlinear tuned mass dampers are coupled nonlinear equations.