## 2 Models of Systems, Equations of Motion and Their Dimensionless Forms  
The nonlinear system composed of a controlled structure with an ideal single degree of freedom and a tuned mass damper considering stiffness nonlinearity can be simplified to the kinematic structure shown in Figure 1. As can be seen from the figure, the mass of the controlled structure is set to \(m_{1}\), the linear spring stiffness is \(k_{1}\), and the linear damping is \(c_{1}\); the equivalent mass of the tuned mass damper is set to \(m_{2}\), the linear spring stiffness between the controlled structure and the tuned mass damper is \(k_{2}\), the nonlinear spring stiffness between the controlled structure and the tuned mass damper is \(k_{3}\), and the linear damping between the controlled structure and the tuned mass damper is \(c_{2}\). The kinetic equations of the system can be obtained from Figure 1:  
\[m_{1}\ddot{u}_{1}+c_{1}\dot{u}_{1}+k_{1}u_{1}+m_{2}\ddot{u}_{2}=F\cos\omega t \tag{1}\]  
\[m_{2}\ddot{u}_{2}+c_{2}\big{(}\dot{u}_{2}-\dot{u}_{1}\big{)}+k_{2}(u_{2}-u_{1} )+k_{3}(u_{2}-u_{1})^{3}=0 \tag{2}\]  
where \(u_{1}\) is the displacement produced by the controlled structure under the action of external force \(F\), \(u_{1}\) is the first-order derivative of \(u_{1}\) with respect to time \(t\), and \(\ddot{u_{1}}\) is the second-order derivative of \(u_{1}\) with respect to time \(t\). \(u_{2}\) is the displacement produced by the nonlinear tuned mass damper under the action of external force \(F\) on the controlled structure, \(u_{2}\) is the first-order derivative of \(u_{2}\) with respect to time \(t\), and \(\ddot{u_{2}}\) is the second-order derivative of \(u_{2}\) with respect to time \(t\). The relative displacement between the controlled structure and the nonlinear tuned mass damper is \(u_{1}-u_{2}\), its first-order derivative is \(\big{(}\dot{u}_{1}-\dot{u}_{2}\big{)}\) with respect to time \(t\), and \(\omega\) is the excitation frequency of the harmonic excitation received. In order to facilitate the calculation and to reflect the more important parameters in the equation that need to be studied in this paper in the subsequent derivation process, dimensionless coefficients are introduced to remove the units of all parameters involving physical quantities in the equation by replacing the original parameters in the equation with suitable variables. Before dimensionless transformation, a unit length \(L\) is first set, which serves only for dimensionless transformation. The frequency of the controlled structure is denoted as \(k_{1}/m_{1}=\omega_{1}^{2}\), and the frequency of the tuned mass damper is denoted as \(k_{2}/m_{2}=\omega_{2}^{2}\). To study the effect of the mass ratio between the nonlinear tuned mass damper and the controlled structure, denoted as the mass ratio \(m_{2}/m_{1}=\varepsilon\alpha_{1}\), this paper employs the product of dimensionless parameters \(\varepsilon\) and \(\alpha_{1}\). Additionally, a small and specific dimensionless parameter \(\varepsilon\) is introduced for subsequent approximate analytical solutions. A new dimensionless time variable \(\tau=\omega_{1}t\) is adopted, replacing the original time variable \(t\) and transforming all time derivatives to derivatives with respect to the new time variable \(\tau\). The original coordinates of the structure are transformed into a  
Figure 1: Nonlinear tuned mass damper and controlled structural system. dimensionless form. The displacement generated by the controlled structure is denoted as \(x=u_{1}/L\), and \(y=(u_{1}-u_{2})/L\) represents the relative displacement between the controlled structure and the nonlinear tuned mass damper. The dimensionless damping coefficient \(\varepsilon\lambda_{1}\) replaces the linear damping coefficient \(c_{1}\) in the controlled structure, while \(\lambda_{2}\) replaces the linear damping coefficient \(c_{2}\) between the controlled structure and the tuned mass damper. The dimensionless coefficient \(\alpha_{2}\) indicates the nonlinear stiffness of the nonlinear tuned mass damper.