## 2 Equation of Motion of the System and Its Dimensionless Form  
To investigate the vibration reduction performance of a nonlinear TMD, a system model was developed consisting of an ideal single-degree-of-freedom primary structure and a stiffness-geometry nonlinear TMD, with the goal of reducing the steady-state amplitude of the structure. When the primary structure is subjected to an external load, the TMD installed on it absorbs energy, and the TMD produces displacement in the horizontal direction. The spring attached to the TMD stretches and tilts, creating a geometric nonlinearity, which in turn leads to a stiffness nonlinearity in the linear spring, as shown in Figure 1.  
Figure 1: Stiffness nonlinearity leads to geometric nonlinearity. The relationship between the restoring force \(f\) and the displacement \(x\) in the horizontal direction is described by Equation (1), where \(l_{0}\) is the length of the spring at rest, \(l\) is the length of the spring after tilting, and \(k\) is the stiffness of the linear spring. The Taylor expansion is used to expand to the fifth order of \(x\), while omitting the higher order terms. The expansion of Equation (1) results in a nonlinear restoring force, with results representing the first, third, and fifth powers of stiffness, consequently yielding Equation (2):  
\[f=2 kx\left(1-\frac{l}{\sqrt{x^{2}+l_{0}^{2}}}\right) \tag{1}\]  
\[f\approx 2k\left(x-\frac{l_{0}}{l}x+\frac{l_{0}}{l^{3}}x^{3}-\frac{9l_{0}}{l^{ 5}}x^{5}\right) \tag{2}\]  
The following dimensionless coefficients are set:  
\[f_{0}=\frac{f}{kl_{0}},\ \ \ \ \ R=\frac{l}{l_{0}},\ \ \ \ \ \ W=\frac{x}{l_{0}}\]  
Substituting the dimensionless coefficients set above into Equations (1) and (2), the corresponding dimensionless forms of Equations (1) and (2) are obtained as follows:  
\[f_{0}=2W\left(1-\frac{R}{\sqrt{W^{2}+1^{2}}}\right) \tag{3}\]  
\[f_{0}\approx 2(W-RW+RW^{3}-9 RW^{5}) \tag{4}\]  
where \(W\) is the displacement produced by the motion of the tuned mass damper, \(f_{0}\) is the restoring force, and \(R\) is the relative length of the spring. From Equations (3) and (4), a graph depicting the variation of the restoring force \(f_{0}\) with the displacement \(W\) produced by the motion of the tuned mass damper can be plotted. As shown in Figure 2, the geometric nonlinearity generated by the linear spring leads to the generation of five times the stiffness nonlinearity, which can produce significant errors after moving away from the equilibrium position and has less research significance. Therefore, this paper used analyses based on the third power nonlinear stiffness. Figure 2: Comparison of actual and approximate results. From the above analysis, the force-displacement relationship for the single-degree-of-freedom system considered in this paper is represented by \(F_{d}=k_{2}x+k_{3}x^{3}\), \(k_{2}>0\), and \(k_{3}>0\), which correspond to the types of hardening considered in this paper. The mechanical model of the system is schematically shown in Figure 3, where the primary structure is subjected to harmonic excitation with an amplitude of \(F\) and an excitation frequency of \(\omega\). The kinetic equation for a single degree of freedom system with a nonlinear TMD is derived from [PERSON]'s second law as:  
\[m_{1}\ddot{u}_{1}+c_{1}\dot{u}_{1}+k_{1}u_{1}+m_{2}\ddot{u}_{2}=F\cos\omega t \tag{5}\]  
\[m_{2}\ddot{u}_{2}+c_{2}\big{(}\dot{u}_{2}-\dot{u}_{1}\big{)}+k_{2}(u_{2}-u_{1}) +k_{3}(u_{2}-u_{1})^{3}=0 \tag{6}\]  
where \(m_{1}\), \(c_{1}\), and \(k_{1}\) represent the mass, damping, and linear spring stiffness of the primary structure; \(m_{2}\), \(c_{2}\), \(k_{2}\), and \(k_{3}\) represent the mass, damping, linear stiffness, and nonlinear stiffness of the TMD; \(u_{1}\), \(u_{2}\), and \(u_{2}-u_{1}\) represent the displacement of the main structure, the displacement of the nonlinear TMD, and the relative displacement of the nonlinear TMD.