## 3 HBM with Modified Input Conditions
### Basic Formulation of the HBM
\[-\varpi^{2}m\underline{\underline{\mathbf{H}}}\underline{\mathbf{P}}^{\prime \prime}\underline{\underline{\underline{\mathbf{\theta}}}}+\varpi\underline{ \underline{\mathbf{H}}}\underline{\mathbf{P}}^{\prime}\underline{\underline{ \underline{\mathbf{\theta}}}}+\underline{\underline{\mathbf{H}}}\underline{f _{nc}}-\underline{\underline{\mathbf{H}}}\underline{F_{Ec}}=\underline{0}. \tag{10}\]  
Thus,  
\[\underline{\underline{\mathbf{H}}}\underline{\mathbf{Y}}=\underline{0}, \tag{11a}\] \[\underline{\underline{\mathbf{Y}}}=-\varpi^{2}m\underline{\underline{\mathbf{P}}} ^{\prime\prime}\underline{\underline{\underline{\mathbf{\theta}}}}_{c}+\varpi \underline{\underline{\mathbf{P}}}^{\prime\prime}\underline{\underline{ \underline{\mathbf{\theta}}}}_{c}+\underline{f_{nc}}-\underline{F_{Ec}}= \underline{0}. \tag{11b}\]  
To determine the solutions for \(\underline{\underline{\mathbf{\theta}}}_{c}\) in Equations (10) and (11) with their relevant values \(\varpi\), the Newton-Raphson method is used by considering the condition \(\underline{\underline{\mathbf{Y}}}\rightarrow\underline{0}\) where \(\underline{\underline{\mathbf{Y}}}\) is considered as a function of \(\underline{\underline{\mathbf{\theta}}}_{c}\) and \(\varpi\) such as \(\underline{\underline{\mathbf{Y}}}(\underline{\underline{\mathbf{\theta}}}_{c},\varpi)\). With respect to the implementation of the Newton-Raphson method, the Jacobian matrices for \(\underline{\underline{\mathbf{\theta}}}_{c}\) and \(\varpi\) must be calculated for  
the NS results well. This indicates that the sub-harmonic index \(\eta\) does not work properly. As for the detailed employment of the NS, it concerned research that can be referred [1; 21]. For example, the super-harmonic responses from both the HBM and NS are well-correlated, as shown in Figure 2b. However, the sub-harmonic regime in Figure 2c shows that the two results of HBM and NS are slightly different because the HBM does not capture the sub-harmonic responses satisfactorily, and its solutions generally reflects the unstable conditions based on [PERSON]'s method [2; 12; 17]. However, the NS still projects sub-harmonic responses satisfactorily since the resonance from the calculated results are clearly obtained in Figure 2. Thus, to overcome these discrepancies in the HBM, techniques to trigger the sub-harmonic effects should be implemented [10]. From Equation (9) and its employed values in Table 2, the valid components of the input torque vector with \(\eta=2\) and \(N_{\text{max}}=\)12 are given as, \(F_{m}=168.9\), \(F_{a(2)}=-87.97\), and \(F_{b(2)}=235.65\). To trigger the sub-harmonic responses, the components of the input torque pertaining to the sub-harmonic locations, such as \(F_{a(1)}\) and, \(F_{b(1)}\) can be assigned with small values. In this study, \(F_{a(1)}=\varepsilon F_{a(2)}\) and \(F_{b(1)}=\varepsilon F_{b(2)}\) were used with \(\varepsilon=1\times 10^{-5}\). With a small range of values multiplied by the fundamental input torque components, the sub-harmonic responses are effectively triggered, as shown in Figure 3. The results in Figure 3a,b successfully capture the sub-harmonic effects with the modified input conditions. Here, regardless of the number of \(N_{\text{max}}\) employed, such as \(N_{\text{max}}=1\) and \(N_{\text{max}}=12\), the sub-harmonic effects are clearly detectable. Figure 2: Nonlinear frequency responses with RMS values: (**a**) comparison of HBM with NS; (**b**) super-harmonic regions; (**c**) sub-harmonic region. Key:, HBM result with \(\eta=2\) and \(N_{\text{max}}=12\);, NS result with frequency up-sweeping: +, NS result with frequency down-sweeping.