## 2 Equation of Motion of the System and Its Dimensionless Form
The parameters \(m_{2}\), \(c_{2}\), and \(k_{2}\) are the design variables in the design of TMD systems, and the parameter \(k_{3}\) is the intrinsic coefficient required for the TMD to consider nonlinearity in practical applications. For the design variables in the TMD system, \(c_{2}\) takes a fixed value, while \(m_{2}\) and \(k_{2}\) are the parameters to be determined. By introducing dimensionless coefficients and using suitable variable substitutions, all units of parameters involving physical quantities are removed. This can clearly reflect the large parameters in the equation, making subsequent calculations easier. Furthermore, distinguishing the large and small parameters through the substitution of dimensionless coefficients can help identify the more important parameters in the design phase, making the calculation results of key parameters clear and unambiguous. First, the frequency of the controlled structure is \(k_{1}/m_{1}=\omega_{1}^{2}\), the frequency of the NTMD is \(k_{2}/m_{2}=\omega_{2}\), and the ratio of the mass of the NTMD to the mass of the controlled structure is called the mass ratio, set to \(\varepsilon\alpha_{1}\). To facilitate the study of the effect of the mass ratio parameter on the system, the product of the dimensionless minor parameter \(\varepsilon\) and the dimensionless parameter \(\alpha_{1}\) was used in this paper to denote the mass ratio, which is \(m_{2}/m_{1}=\varepsilon\alpha_{1}\). The dimensionless minor parameter \(\varepsilon\) is a specific parameter, which is set to facilitate the use of subsequent approximate analytical methods. Next, the time variable is dimensionless; a new time variable \(\tau=\omega_{1}t\) is used as the time variable of the equation, and the time variable \(\tau\) is a  
Figure 1:0 Nonlinear mass dampers and structural dynamic system. dimensionless quantity. The selection of coordinates is crucial to the use of the subsequent analysis method. In this paper, we first used \(x=u_{1}/L\) to denote the new coordinates and to carry out coordinate transformation and the dimensionless transformation of the original coordinates. Here we let \(L\) be the unit length, whose role is only dimensionless, and \(x\) can be regarded as the response of the main structure. Secondly, \(y=(u_{1}-u_{2})/L\) is used to denote another new coordinate, which is used to substitute the two coordinates of the original equation, and \(y\) can be considered as the relative displacement of the controlled structure and the mass dampers. For the damping coefficient, since the damping ratio of the controlled structure is generally less than 5%, the original damping coefficient \(c_{1}\) is replaced by the new dimensionless damping coefficient \(\varepsilon\lambda_{1}\), and the damping coefficient \(c_{2}\) is replaced by the dimensionless damping coefficient \(\lambda_{2}\). The dimensionless coefficient \(\alpha_{2}\) represents the nonlinear stiffness, and \(\omega\) is the excitation frequency. The ratio of NTMD frequency to controlled structure frequency is replaced by dimensionless quantity \(\Omega_{2}\), and the ratio of excitation frequency to controlled structure frequency is replaced by dimensionless quantity \(\Omega\). Meanwhile, the excitation force is expressed as a small quantity of the first order. The specific dimensionless coefficients are set as follows:  
Figure 1:1  
Figure 1:2  
Substituting the above dimensionless coefficients into Equations (5) and (6), the equation of motion can be written as:  
Figure 1:3  
Figure 1:4