## 3 HBM with Modified Input Conditions  
### Basic Formulation of the HBM  
From Equation (1), the [PERSON] scheme of the governing equations is expressed as follows [2;12]:  
\[-\omega^{2}m\underline{\underline{\mathbf{H}}\mathbf{P}}^{\prime\prime}\underline {\underline{\varrho}}_{\underline{c}}+\omega c\underline{\underline{\mathbf{H }}\mathbf{P}}^{\prime}\underline{\underline{\varrho}}_{\underline{c}}+\underline {f_{n}}\Big{(}\underline{\theta_{f}},\dot{\underline{\theta}_{f}}\Big{)}- \underline{\underline{F}_{\underline{E}}}(t)=\underline{0}. \tag{7}\]  
Here, its corresponding formulas are defined as follows:  
\[\underline{\theta_{f}}(t)=\underline{\underline{\mathbf{H}}\underline{\varrho }_{\underline{c}}}, \tag{8a}\] \[\underline{\theta_{f}}(t)=\big{[} \begin{array}{ccccccccc}\theta_{f}(t_{0})& \theta_{f}(t_{1})&\cdots&\theta_{f}(t_{m-2})&\theta_{f}(t_{m-1})\end{array}  \big{]}^{T},\] (8b) \[\underline{\theta_{\underline{c}}}=\Big{[} \begin{array}{ccccccccc}\theta_{m}& \theta_{a(1)}&\theta_{b(1)}&\cdots&\theta_{a(k)}&\theta_{b(k)}&\cdots&\theta_{a (yN_{\max})}&\theta_{b(yN_{\max})}\end{array} \big{]}^{T}. \tag{8c}\]\[\underline{\underline{\mathbf{H}}}=\left[ \begin{array}{ccccc}1&\cdots&cos(k \Psi_{0})&sin(k\Psi_{0})&\cdots\\ 1&\cdots&cos(k\Psi_{1})&sin(k\Psi_{1})&\cdots\\ &\ddots&&&\ddots\\ 1&\cdots&cos(k\Psi_{N-2})&sin(k\Psi_{N-2})&\cdots\\ 1&\cdots&cos(k\Psi_{N-1})&sin(k\Psi_{N-1})&\cdots\end{array} \right],\ \underline{\underline{ \mathbf{H}}}^{\prime}=\omega\underline{\underline{\mathbf{H}}}\underline{ \mathbf{p}}^{\prime},\ \underline{\underline{\mathbf{H}}}^{\prime}=-\omega^{2}\underline{ \underline{\mathbf{H}}}\underline{\mathbf{p}}^{\prime\prime}, \tag{8d}\]  
\[\underline{\underline{\mathbf{p}}}^{\prime}=\left[ \begin{array}{ccccc}0&&&\\ &\ddots&&&\\ &&&\left[\begin{array}{cc}0&k\\ -k&0\end{array}\right]&\\ &&&\ddots\end{array} \right], \tag{8e}\]  
\[\underline{\underline{\mathbf{P}}}^{\prime\prime}=\left[ \begin{array}{ccccc}0 &&&&\\ &\ddots&&&\\ &&&\left[\begin{array}{cc}k^{2}&0\\ 0&k^{2}\end{array}\right]&\\ &&&\ddots\end{array} \right]. \tag{8f}\]  
Likewise, its nonlinear and input functions are expressed using the same schemes as follows:  
\[\underline{f_{n}}\Big{(}\underline{\theta_{f}},\underline{\theta_{f}}\Big{)}= \underline{\underline{\mathbf{H}}}\underline{f_{nc}}, \tag{9a}\] \[\underline{\underline{F}_{E}}(t)=\underline{\underline{\mathbf{H}}}\underline{F_ {Ec}},\] (9b) \[\underline{f_{nc}}=\Big{[} \begin{array}{ccccccccc}f_{m}&f_{a(1)}&f_{b(1)}& \cdots&f_{a(k)}&f_{b(k)}&\cdots&f_{a(\eta N_{\max})}&f_{b(\eta N_{\max})}\\ \end{array} \Big{]}^{T},\] (9c) \[\underline{F_{Ec}}=\Big{[} \begin{array}{ccccccccc}F_{m}&F_{a(1)}&F_{b(1)}& \cdots&F_{a(k)}&F_{b(k)}&\cdots&F_{a(\eta N_{\max})}&F_{b(\eta N_{\max})}\\ \end{array} \Big{]}^{T}. \tag{9d}\]  
Here, let \(\varpi t=\Psi\) and  
non-dimensionalize the time scale and normalize the frequency values. Based on this relationship and by considering the effective time period with the sub-harmonic effect, the relevant time ranges are defined as \(0\leq t<T\rightarrow\), where \(0\leq\Psi<\frac{2\pi}{\omega_{n}}T=\eta\tau\), where \(\eta\) is a sub-harmonic index and \(\tau\) is the fundamental frequency. Thus, the index \(k\) in Equation (8) is incremented by \(k=\omega_{n},\ 2\omega_{n},\ 3\omega_{n}\cdots\) which includes the non-dimensionlized factor \(\omega_{n}\). From this relationship, \(\dot{\theta}(t)=\frac{d\theta}{dt}=\varpi\frac{d\theta}{d\Psi}=\varpi\theta^{\prime}\) because \(\varpi dt=d\Psi\). Likewise, \(\ddot{\theta}(t)=\varpi^{2}\theta^{\prime\prime}\). The matrices and coefficient vectors are constructed along with the incremental indices \(k\) and \(\eta\). In addition, the range of the normalized time period should be adapted based on \(\eta\). By substituting Equations (8) and (9) into Equation (7), the overall Galerkin scheme of the basic equation is defined as follows.