## 2 Models of Systems, Equations of Motion and Their Dimensionless Forms
\begin{array}{c}\Big{(}\dot{A}_{2}+\frac{1}{2}i\Omega A_{2}+\frac{1}{2i\Omega }A_{2}\Big{)}\\ \qquad\qquad+\varepsilon\delta\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{1}{2i} A_{1}-\frac{1}{2i}A_{2}\bigg{)}=0\end{array}  \tag{20}\]  
Influenced by the nonlinearity, the response mechanism of the excitation frequency near the frequency of the controlled structure also becomes very complex. Considering the engineering reality that the mass of the tuned mass damper is much smaller than the mass of the controlled structure, vibration reduction is significant in such cases. Therefore, the vibration reduction effect of the nonlinear mass damper is taken into account when the maximum mass ratio is set to 0.2. Additionally, the case of 1:1 resonance between the excitation frequency and the controlled structure frequency is considered, and the difference between the external excitation frequency and the controlled structure frequency is controlled within the same order of \(\varepsilon^{1}\), ensuring that \(\Omega\) satisfies the relationship described by \(\Omega=1+\varepsilon\sigma\). In the field of structural vibration control, the introduction of nonlinearity through the implementation of tuned mass dampers using nonlinear stiffness devices can directly exert forces on the structure, leading to stability problems. As a result, the dynamics exhibit a more complex response. In comparison to linear tuned mass dampers, the performance of nonlinear tuned mass dampers is primarily determined by the design parameters \(\Omega_{2}\) and \(\alpha_{2}\). This study focuses on investigating the steady-state response of the structure, where the first-order derivatives in Equations (19) and (20) are set to zero. Through transformation, the equations can be simplified to the following form:  
\[\mathrm{i}\sigma A_{1}+\alpha_{1}\lambda_{2}A_{2}-\mathrm{i}\alpha_{1}\Omega_ {2}^{2}A_{2}-\mathrm{i}\frac{3}{4}\alpha_{1}\alpha_{2}\Omega_{2}^{2}|A_{2}|^{ 2}A_{2}+\lambda_{1}A_{1}-f=0 \tag{21}\]  
\[-i\Omega_{2}^{2}A_{2}-i\frac{3}{4}\alpha_{2}\Omega_{2}^{2}|A_{2}|^{2}A_{2}+ \lambda_{2}A_{2}-iA_{1}+iA_{2}=0 \tag{22}\]  
It can be observed that \(A_{1}\) can be expressed by the equation provided in (22). By combining Equations (21) and (22), the expressions of \(A_{1}\) and \(A_{2}\) can be decoupled, resulting in the following:  
\[A_{1}=-\Omega_{2}^{2}A_{2}-\frac{3}{4}\alpha_{2}\Omega_{2}^{2}|A_{2}|^{2}A_{2} -i\lambda_{2}A_{2}+A_{2} \tag{23}\]  
\[ \begin{array}{c}-i\sigma\Omega_{2}^{2}A_{2}-i\sigma_{3}^{3}\alpha_{2}\Omega _{2}^{2}|A_{2}|^{2}A_{2}+\sigma\lambda_{2}A_{2}\\ \qquad+i\sigma A_{2}-\lambda_{1}\Omega_{2}^{2}A_{2}-\lambda_{1}\frac{3}{4} \alpha_{2}\Omega_{2}^{2}|A_{2}|^{2}A_{2}\\ \qquad-i\lambda_{1}\lambda_{2}A_{2}+\lambda_{1}A_{2}+\alpha_{1}\lambda_{2}A_{ 2}-i\alpha_{1}\Omega_{2}^{2}A_{2}-\mathrm{i}\frac{3}{4}\alpha_{1}\alpha_{2} \Omega_{2}^{2}|A_{2}|^{2}A_{2}-f=0\end{array}  \tag{24}\]  
Expand the expressions of \(A_{1}\) and \(A_{2}\) into the plural form:  
\[A_{1}=a_{1}(\cos b_{1}+i\sin b_{1});\;A_{2}=a_{2}(\cos b_{2}+i\sin b_{2}) \tag{25}\]  
where \(a_{1}\) and \(a_{2}\) denote the steady-state amplitudes obtained by the complex variable averaging method, and \(b_{1}\) and \(b_{2}\) are the steady-state phases. Substituting the expression of Equation (25) into Equations (23) and (24), and separating the real and imaginary parts, we obtain:  
\[a_{1}\cos(b_{1}-b_{2})=-\Omega_{2}^{2}a_{2}-\frac{3}{4}a_{2}\Omega_{2}^{2}a_{2 }^{3}+a_{2} \tag{26}\]  
\[a_{1}\sin(b_{1}-b_{2})=-\lambda_{2}a_{2} \tag{27}\]\[\sigma\lambda_{2}a_{2}-\lambda_{1}\Omega_{2}^{2}a_{2}-\lambda_{1}\frac{3}{4} \alpha_{2}\Omega_{2}^{2}a_{2}^{3}+\lambda_{1}a_{2}+\alpha_{1}\lambda_{2}a_{2}=f \cos b \tag{28}\]  
\[\sigma\Omega_{2}^{2}a_{2}+\sigma\frac{3}{4}\alpha_{2}\Omega_{2}^{2}a_{2}^{3}- \sigma a_{2}+\lambda_{1}\lambda_{2}a_{2}+\alpha_{1}\Omega_{2}^{2}a_{2}+\frac{3 }{4}\alpha_{1}\alpha_{2}\Omega_{2}^{2}a_{2}^{3}=f\sin b \tag{29}\]