## 3 Approximate Analytical Solution of the Equations of Motion of the System
\[\alpha_{1}\Omega_{2}^{2}a_{2}+\frac{3}{4}\alpha_{1}\alpha_{2}\Omega_{2}^{2}a_{ 2}^{3}+\lambda_{1}\lambda_{2}a_{2}=f\sin b \tag{28}\]1 Primary structure amplitude time course curve (ideal linear TMD and optimal design frequency NTMD). In order to study the effect of increasing nonlinear coefficients on the amplitude of the structure, frequency-amplitude curves corresponding to different nonlinear coefficients were obtained by taking \(\alpha_{2}\) values of 0.01, 0.02, 0.03, 0.04, and 0.05, as shown in \[\alpha_{1}\Omega_{2}^{2}a_{2}+\frac{3}{4}\alpha_{1}\alpha_{2}\Omega_{2}^{2}a_{ 2}^{3}+\lambda_{1}\lambda_{2}a_{2}=f\sin b \tag{28}\]2 Observing Figure 7, the horizontal coordinate represents the intrinsic frequency of the TMD, and the vertical coordinate represents the displacement amplitude of the primary structure. With the increase in the nonlinearity coefficient, the frequency-amplitude curve of the primary structure gradually shifted to the left, and the optimal frequency for structure control changed. As shown in Figure 8a, when \(\alpha_{2}\) increased to 0.04, the frequency-amplitude curve exhibited a multi-valued phenomenon in the interval of TMD frequency \(\omega_{2}\) values of 0.545-0.550. For the convenience of observation, the interval where the multi-valued phenomenon occurred was enlarged. As shown in Figure 8b, the dashed part of the blue curve corresponds to multiple amplitude values at the same TMD frequency. Similarly, Figure 9a shows the frequency-amplitude curve of the structure at \(\alpha_{2}=0.05\). From Figure 9b, the range of TMD frequencies \(\omega_{2}\) where the multi-valued phenomenon appeared was 0.505-0.515, which was expanded (compared to the range of TMD frequencies where the multi-valued phenomenon appeared at \(\alpha_{2}=0.04\)). When multiple values appear in the frequency-amplitude plot, it indicates that the structure is in an unstable state in this frequency interval, and jumps may occur, which may have adverse effects on the structure. It is possible that two or more structural amplitude values may occur at the same frequency, resulting in static bifurcation. As the nonlinear coefficient increased, the minimum amplitude value of the structure did not change; thus, increasing the nonlinear coefficient does not improve the control performance of the TMD on the structure. The nonlinear coefficient is not required to be as small as possible, and the impact of considering the nonlinearity on the control performance of the TMD could not be obviously observed during the study. Thus, the most suitable value for the nonlinear coefficient was determined to be \(\alpha_{2}=0.02\). \[\alpha_{1}\Omega_{2}^{2}a_{2}+\frac{3}{4}\alpha_{1}\alpha_{2}\Omega_{2}^{2}a_{ 2}^{3}+\lambda_{1}\lambda_{2}a_{2}=f\sin b \tag{28}\]3 Frequency–amplitude curves of the primary structure corresponding to different nonlinear coefficients \(\alpha_{2}\) (\(\alpha_{2}=0.01,0.02,\) 0.03, 0.04, 0.05). \[\alpha_{1}\Omega_{2}^{2}a_{2}+\frac{3}{4}\alpha_{1}\alpha_{2}\Omega_{2}^{2}a_{ 2}^{3}+\lambda_{1}\lambda_{2}a_{2}=f\sin b \tag{28}\]4 Frequency–amplitude curve stability of the primary structure (\(\alpha_{1}=1\), \(\alpha_{2}=0.04\)). \[\alpha_{1}\Omega_{2}^{2}a_{2}+\frac{3}{4}\alpha_{1}\alpha_{2}\Omega_{2}^{2}a_{ 2}^{3}+\lambda_{1}\lambda_{2}a_{2}=f\sin b \tag{28}\]5 Frequency–amplitude curve stability of the primary structure (\(\alpha_{1}=1\), \(\alpha_{2}=0.05\)). Figure 10 shows the frequency-amplitude curves of the NTMD compared to those of the primary structure for mass ratios \(\alpha_{1}=1,2,3,4,\) and \(5\). It can be seen from Figure 10 that the maximum amplitude of the primary structure gradually decreased with an increasing mass ratio, and the frequency-amplitude curve of the structure gradually shifted to the right. The best control effect of NTMD was achieved when the mass ratio \(\alpha_{1}=5\), i.e., the mass of NTMD was 10% of the mass of the primary structure.