## 2 Modeling of Mounting System
### 9-DOF Modeling
\begin{array}{ll}I_{s}\ddot{\theta}_{s,y}+(k_{m2}^{*}l_{s2}-k_{m1}^{*}l_{s1 })\varepsilon_{s,g1,z}+k_{m1}^{*}l_{s1}\varepsilon_{st1,z}-k_{m2}^{*}l_{s2} \varepsilon_{st2,z}+(k_{m2}^{*}l_{s2}^{2}+k_{m1}^{*}l_{s1}^{2})\theta_{s,y}=w_{ z}(t)d\end{array}  \tag{6}\]  
\[ \begin{array}{ll}I_{r}\ddot{\theta}_{r,y}+(k_{m5}^{*}l_{s2}-k_{m4}^{*}l_{s2 }+k_{br2}^{*}l_{r2}-k_{br1}^{*}l_{r1})\varepsilon_{r,g1,z}\\ +k_{m4}^{*}l_{s1}\varepsilon_{st1,z}-k_{m5}^{*}l_{s2}\varepsilon_{st2,z}+(k_{ m4}^{*}l_{s1}^{2}+k_{m5}^{*}l_{s2}^{2}+k_{br1}^{*}l_{r1}^{2}+k_{br2}^{*}l_{r2}^{2}+k_{ br3}^{*}l_{r3}^{2}+k_{br4}^{*}l_{r4}^{2})\theta_{r,y}=0\end{array}  \tag{7}\]  
\[ \begin{array}{ll}I_{s}\ddot{\theta}_{sf,x}+(k_{m1}^{*}l_{f1}+k_{m2}^{*}l_{f1 }-k_{m3}^{*}l_{f2})\varepsilon_{s,g1,z}\\ +k_{m3}^{*}l_{f2}\varepsilon_{st3,z}-k_{m1}^{*}l_{f1}\varepsilon_{st1,z}-k_{m2}^ {*}l_{f1}\varepsilon_{st2,z}+(k_{m1}^{*}l_{f1}^{2}+k_{m2}^{*}l_{f1}^{2}+k_{m3} ^{*}l_{f2}^{2})\theta_{sf,x}=w_{z}(t)d_{2}\end{array}  \tag{8}\]  
\[ \begin{array}{ll}I_{r}\ddot{\theta}_{rf,x}&+(k_{m4}^{*}l_{f1}+k_{m5}^{*}l_{f1 }-k_{m6}^{*}l_{f2}-k_{br1}^{*}l_{rf1}-k_{br2}^{*}l_{rf3}+k_{br3}^{*}l_{rf2}+k_{ br4}^{*}l_{rf4})\varepsilon_{r,g1,z}\\ &+k_{m6}^{*}l_{f2}\varepsilon_{st3,z}-k_{m4}^{*}l_{f1}\varepsilon_{st1,z}-k_{ m5}^{*}l_{f1}\varepsilon_{st2,z}\\ &+(k_{m4}^{*}l_{f1}^{2}+k_{m5}^{*}l_{f1}^{2}+k_{m6}^{*}l_{f2}^{2}+k_{br1}^{*}l_{ r1}^{2}+k_{br2}^{*}l_{rf3}^{2}+k_{br3}^{*}l_{rf2}^{2}+k_{br4}^{*}l_{rf4}^{2}) \theta_{rf,x}=0\end{array}  \tag{9}\]  
To express the equation of motion in a simple matrix form, the mass, stiffness, forces, and displacement vector are defined in Equations (10)-(14). \[\mathbf{M}=diag(\left\{m_{s}\;m_{r}\;m_{st1}\;m_{st2}\;m_{st3}\;l_{s,y}\;l_{r,y} \;l_{s,x}\;l_{r,x}\right\}) \tag{10}\]  
\[\mathbf{K}^{*}=\left[\begin{array}{cccccc}k_{m2}^{*}+k_{m3}^{*}+k_{m4}^{*}&0& -k_{m1}^{*}&-k_{m2}^{*}&-k_{m3}^{*}\\ 0&k_{m4}^{*}+k_{m5}^{*}+k_{m6}^{*}+k_{br1}^{*}+k_{br2}^{*}&-k_{m6}^{*}&-k_{m5}^{*} &-k_{m6}^{*}\\ -k_{m1}^{*}&-k_{m4}^{*}&k_{m1}^{*}+k_{m4}^{*}&0&0&0\\ -k_{m2}^{*}&-k_{m5}^{*}&0&k_{m2}^{*}+k_{m5}^{*}&0&0\\ -k_{m3}^{*}l_{s2}-k_{m6}^{*}&0&0&0&k_{m3}^{*}+k_{m6}^{*}\\ k_{m3}^{*}l_{s2}-k_{m1}^{*}l_{s1}  
\[\mathbf{q}(t)=\left\{ \begin{array}{l}\varepsilon_{s,g1,z}(t)\\ \varepsilon_{r,g1,z}(t)\\ \varepsilon_{sti,z}(t)\\ \theta_{s,y}(t)\\ \theta_{r,y}(t)\\ \theta_{s,f,x}(t)\\ \theta_{rf,x}(t)\end{array} \right\} \tag{12}\]  
\[\mathbf{w}(t)=\left\{ \begin{array}{l}w_{z}(t)\\ 0\\ 0\\ 0\\ w_{z}(t)d\\ 0\\ w_{z}(t)d_{2}\\ 0\end{array} \right\} \tag{13}\]  
\[\mathbf{F}(t)=\left\{ \begin{array}{l}0\\ 0\\ f_{st1,z}(t)\\ f_{st2,z}(t)\\ f_{st3,z}(t)\\ 0\\ 0\\ 0\\ 0\end{array} \right\} \tag{14}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variable** & **Values** & **Units** & **Variable** & **Values** & **Units** \\ \hline \(m_{s}=m_{r}\) & 1.081 & kg & \(k_{m5}^{*}\) & 0.53(1 + i0.256) & kN mm\({}^{-1}\) \\ \(m_{st1}=m_{st3}\) & 0.067 & kg & \(k_{m6}^{*}\) & 0.61(1 + i0.300) & kN mm\({}^{-1}\) \\ \(m_{st2}\) & 0.075 & kg & \(k_{brn\,(n=1,2)}^{*}\) & 0.42(1 + i0.300) & kN mm\({}^{-1}\) \\ \(I_{s,y}=I_{r,y}\) & 16.172 & g m\({}^{2}\) & \(l_{sn\,(n=1,2)}\) & 100 & mm \\ \(I_{sf,x}=I_{rf,x}\) & 5.940 & g m\({}^{2}\) & \(l_{rn\,(n=1,2)}\) & 140 & mm \\ \(k_{m1}^{*}\) & 5.46(1 + i0.034) & kN mm\({}^{-1}\) & \(l_{fn\,(n=1,2)}\) & 80 & mm \\ \(k_{m2}^{*}\) & 5.46(1 + i0.034) & kN mm\({}^{-1}\) & \(l_{rfn\,(n=1\sim 4)}\) & 100 & mm \\ \(k_{m3}^{*}\) & 2.48(1 + i0.036) & kN mm\({}^{-1}\) & \(d\) & 50 & mm \\ \(k_{m4}^{*}\) & 0.61(1 + i0.300) & kN mm\({}^{-1}\) & \(d_{2}\) & 10 & mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variable values and units for 9 degrees of freedom (DOF). The focus of this study was on the vibration reduction effect by the active path. To investigate the vibration reduction performance of the part where each active path is located, a coordinate transformation was performed using the transfer matrix \(\prod\) defined in Equation (15). \[\prod=\left[