## 4 Analysis of Nonlinear Tuned Mass Damper Systems Based on Different Conditions
### Analysis of NTMD Control Performance Based on Damping Conditions  
For the undamped tuned mass damper system, the analysis is performed by considering the case where the excitation frequency is equal to the frequency of the controlled structure. Setting the damping of the tuned mass damper to zero and the excitation frequency equal to the controlled structure frequency, we have \(\sigma=0\) and \(\lambda_{2}=0\), and then Equation (39) can be reduced to  
\[\Omega_{2}=\sqrt{\frac{4\alpha_{1}^{2}}{3\alpha_{2}f^{2}+4\alpha_{1}^{2}}} \tag{40}\]  
Equation (40) represents the undamped frequency design equation of the nonlinear tuned mass damper derived in this paper. Upon observing Equation (40), we can see that when \(\alpha_{2}=0\), there is no nonlinearity in the structure, resulting in \(\omega_{2}=\omega_{1}\), and the TMD exhibits an optimal damping effect. This finding aligns with the conclusions drawn by [PERSON] in his previous work. The aforementioned results are verified through the numerical simulations of Equations (1) and (2). The parameters are set as follows:  
\[\alpha_{1}=1,\ \alpha_{2}=0.001,\ f=15\]  
The optimal parameter \(\Omega_{2}=0.925\) is obtained from Equation (40), and the initial conditions are set as follows: \(\left.x\right|_{t=0}=0,\ \dot{x}\right|_{t=0}=0,\left.y\right|_{t=0}=0\), and \(\left.\dot{y}\right|_{t=0}=0\). The time-history diagram obtained is shown in Figure 3. It is observed that when the damping coefficient is zero but nonlinearity exists, the steady-state amplitude of the controlled structure obtained from the design equation is approximately 0.6% of the transient amplitude, indicating a non-zero value. This discrepancy is attributed to inaccuracies in the calculation of the approximate solution expansion. Figure 3 illustrates that at the resonant frequency, the NTMD significantly absorbs the energy of the structure over time and dissipates it through damping. Figure 4 demonstrates that using the design Equation (40) for optimization can substantially reduce the steady-state amplitude of the structure. Initially, the amplitude increases and then gradually decreases until eventually stabilizing, with the NTMD absorbing most of the energy generated during vibration in the steady-state phase. Figure 5 presents the time curve of the steady-state amplitude of the controlled structure, which is magnified. It can be observed that despite the small steady-state amplitude (approximately 0.6% of the transient amplitude), the optimization effect is non-zero, confirming the mean \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{\alpha_{1}}\) & \(\mathbf{\alpha_{2}}\) & **Numerical Method** & **Approximate Analysis Method** \\ \hline
1 & 0.01 & 0.817 & 0.816 \\
1 & 0.02 & 0.707 & 0.706 \\
2 & 0.01 & 0.916 & 0.910 \\
2 & 0.02 & 0.846 & 0.840 \\
3 & 0.01 & 0.958 & 0.948 \\
3 & 0.02 & 0.914 & 0.903 \\
4 & 0.01 & 0.978 & 0.966 \\
4 & 0.02 & 0.947 & 0.936 \\
5 & 0.01 & 0.987 & 0.977 \\
5 & 0.02 & 0.965 & 0.955 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimal frequency of tuned mass damper, comparison between numerical method and analytical method. ingfulness of these results. Figure 6 shows that both steady-state and transient amplitudes of the controlled structure, controlled with undamped nonlinear tuned mass dampers, are reduced. This reflects the excellent vibration reduction effect achieved by the nonlinear TMD with a damped condition design at the 1:1 resonance. Figure 4: Displacement time curve of the controlled structure with nonlinear TMD installed. Figure 3: Displacement time course curves of the controlled structure and TMD mass block with nonlinear TMD installed. Figure 7 illustrates the amplitude-frequency response curves of the primary structure. The solid black line in the figure represents the curve without damping, i.e., when the damping coefficient \(\lambda_{2}=0\), and the red dashed line corresponds to the curve with the damping coefficient \(\lambda_{2}=0.16\).