## 4 Comparison of the Sub-Harmonic Responses from HBM and NS  
Figure 4 compares the results of the HBM under modified input conditions with the NS solutions. When the sub-harmonic responses are examined carefully, as illustrated in Figure 4b, the HBM still reflects the discrepancies. As marked with red lines and characters (A) and (B) in Figure 4b, the calculated results of the HBM at regimes less than (A) or greater than (B) correlate well with those from NS. However, the HBM solutions in the regimes between (A) and (B) are under unstable conditions, and their results do not show good agreement with the NS solutions. To examine the dynamic differences between both methods, the time histories and FFT results can be compared, as shown in Figures 5-7. Figure 5 compares the time histories of the HBM and NS at two different frequencies. Figure 5 shows the time histories based on both HBM and NS at \(\omega=111.2\left(\frac{rad}{s}\right)\) (or 17.7 Hz) that are observed in the regimes below (A) marked in Figure 4b. Figure 4: \(Cont.\)  
Figure 3: Nonlinear frequency responses of HBM reflecting sub-harmonic effects with maximum, mean and minimum values of \(\theta_{f}\) (t): (**a**) frequency responses of max, mean and min values with \(\eta=2\) and \(N_{\max}=1\); (**b**) frequency responses of max, mean and min values with \(\eta=2\) and \(N_{\max}=12\). As shown in Figure 4, the two calculated time histories coincide with each other. However, the time histories of the HBM at \(\omega=118.1\)\(\left(\frac{nd}{s}\right)\)\(\left(\text{or 18.8 Hz}\right)\) do not show good agreement with the NS results, as shown in Figure 5b, because the HBM generally obtains solutions based on integer-based incremental harmonics. These differences between the HBM and NS are clearly observed in Figures 4b and 5b. In addition, the FFT results reflect the discrepancies between the two methods in unstable regimes. Figure 6a shows that the two FFT results at \(\omega=111.2\)\(\left(\frac{nd}{s}\right)\)\(\left(\text{or 17.7 Hz}\right)\) are correlated with each other. However, the FFT results in the unstable regime with \(\omega=118.1\)\(\left(\frac{rad}{s}\right)\)\(\left(\text{or 18.8 Hz}\right)\) show discrepancies between the HBM and NS results. The two results still include the sub-harmonic terms clearly identifiable at \(\omega=59.1\)\(\left(\frac{nd}{s}\right)\)\(\left(\text{or 9.4 Hz}\right)\), as shown in Figure 6b. Thus, the unstable conditions estimated by the HBM should have a practical meaning that the dynamic behaviors are affected by more complex factors, even  
Figure 4: Comparison of nonlinear frequency responses with RMS values: (**a**) HBM vs. NS in a whole range of frequency value; (**b**) HBM vs. NS focused on a sub-harmonic range. Key:, stable solution from HBM with \(N_{\text{max}}=12\) and \(\rho=2\);, unstable solution from HBM with \(N_{\text{max}}=12\) and \(\rho=2\);, NS result with frequency up-sweeping; +, NS result with frequency down-sweeping. Figure 5: Comparison of time histories between HBM (\(N_{\text{max}}=12\) and \(\rho=2\)) and NS at different excitation conditions: (**a**) time histories at 17.7 Hz; (**b**) time histories at 18.8 Hz. Key:, HBM;, NS. though the HBM has limitations owing to the integer-based harmonic term simulation. These phenomena result in quasi-harmonic or chaotic responses [22]. Figure 6: Comparison of FFT results between HBM (\(N_{\rm max}=12\) and \(\rho=2\)) and NS at different excitation conditions: (**a**) FFT results at 17.7 Hz; (**b**) FFT results at 18.8 Hz. Key:, HBM;, NS. Figure 7: Comparisons of phase diagrams calculated by HBM (\(N_{\rm max}=12\) and \(\rho=2\)) and NS at different excitation conditions: (**a**) phase diagrams at 17.7 Hz; (**b**) phase diagrams at 18.8 Hz. Key:, HBM;, NS. Figure 7 compares the phase-plane diagrams for two different frequency regimes with \(\theta_{f}\) and \(\theta_{f}\). As seen previously, the phase planes at \(\omega=111.2\left(\frac{nd}{s}\right)\) (or 17.7 Hz) from the two methods are correlated with each other.