## 2 Modeling of Mounting System  
### 9-DOF Modeling  
The overall schematic of the 9-DOF mounting system is based on Figure 1 and modeled as shown in Figure 2.  
The 3D model was created on the basis of the lumped parameter model and expressed using orthographic projection. The floor plan shows the location of the active structural paths; \(m_{s}\), \(m_{r}\), and \(m_{st1}\) represent the source, receiver, and actuator mass, respectively, that were actually measured and used; \(I_{s,y}\) and \(I_{r,y}\) represent the inertia of the source and receiver, respectively, in the y-direction. \(I_{sf,x}\) and \(I_{rf,x}\) represent the inertia of the source and receiver, respectively, in the \(x\)-direction and were derived from calculations; \(l_{si}\) and \(l_{ri}\) represent the distance between each path on the front of the source and receiver, respectively; \(l_{fi}\) and \(l_{rfi}\) indicate the distance between each path on the sides of the source and receiver, respectively; \(d\) and \(d_{2}\) are the distances of the shaker relative to each direction; \(\theta_{s,y}\), \(\theta_{r,y}\), \(\theta_{sf,x}\), and \(\theta_{rf,x}\) represent the rotational displacement of the source and receiver relative to the front and side, respectively; \(k^{*}_{mi}\) and \(k^{*}_{bi}\) represent the complex stiffness for each path; \(\epsilon_{s,g1,z}\) and \(\epsilon_{r,g2,z}\) indicate the translational motion of the source and receiver, respectively; \(\epsilon_{sti,z}\) and \(f_{sti}\) represent the motion and force, respectively, of the ith actuator; \(w_{z}\) represents the force of the shaker. Figure 1: Powertrain with three active structural paths. Figure 2: Modeling of source-active structural path–receiver. In this study, the translational motion assumed only the \(z\)-direction. The rotational displacement in the \(y\)-direction was assumed to be on the front. Only the \(x\)-direction was assumed to be on the sides. Moreover, the rotational displacement was assumed to be very small. The values of each parameter are summarized in Table 1. Equations (1)-(9) express the equations of motion of the front and sides:  
\[ \begin{array}{ll}m_{s}\ddot{\varepsilon}_{s,g1,z}&+(k_{m1}+k_{m3}+k_{m2}) \varepsilon_{s,g1,z}-k_{m1}\varepsilon_{st1,z}-k_{m2}\varepsilon_{st2,z}-k_{m3 }\varepsilon_{st3,z}\\ &+(k_{m2}l_{s2}-k_{m1}\varepsilon_{s1})\theta_{s,y}+\left(k_{m1}l_{f1}+k_{m2}l_ {f1}-k_{m3}l_{f2}\right)\theta_{sf,x}=w_{z}(t)\end{array}  \tag{1}\]  
\[ \begin{array}{ll}m_{r}\ddot{\varepsilon}_{r,g1,z}&+(k_{m4}^{*}+k_{m5}^{*}+k_{ m6}^{*}+k_{r1}^{*}+k_{br2}^{*}+k_{br3}^{*}+k_{br4}^{*})\varepsilon_{r,g1,z}-k_{m4}^{*} \varepsilon_{st1,z}-k_{m5}^{*}\varepsilon_{st2,z}-k_{m6}^{*}\varepsilon_{st3, z}\\ &+(k_{m5}^{*}l_{s1}-k_{m4}^{*}l_{s1}-k_{br1}^{*}l_{r1}+k_{br2}^{*}l_{r2}-k_{ br3}^{*}l_{r3}+k_{br4}^{*}l_{r4})\theta_{r,y}\\ &+(k_{m4}^{*}l_{f1}+k_{m5}^{*}l_{f1}-k_{m6}^{*}l_{f2}-k_{br1}^{*}l_{rf1}-k_{br2 }^{*}l_{rf2}+k_{br3}^{*}l_{rf3}+k_{br4}^{*}l_{rf4})\theta_{rf,x}=0\end{array}  \tag{2}\]  
\[ \begin{array}{ll}m_{st1}\ddot{\varepsilon}_{st1,g1,z}+(k_{m4}^{*}+k_{m4}^{*} )\varepsilon_{st1,z}\\ -k_{m1}^{*}\varepsilon_{s,g1,z}-k_{m4}^{*}\varepsilon_{r,g1,z}+k_{m1}^{*}l_{s1} \theta_{s,y}+k_{m4}^{*}l_{s1}\theta_{r,y}-k_{m1}^{*}l_{f1}\theta_{sf,x}-k_{m4} ^{*}l_{f1}\theta_{rf,x}=f_{st1,z}(t)\end{array}  \tag{3}\]  
\[ \begin{array}{ll}m_{st2}\ddot{\varepsilon}_{st2,z}+(k_{m2}^{*}+k_{m5}^{*}) \varepsilon_{st2,z}\\ -k_{m2}^{*}\varepsilon_{s,g1,z}-k_{m5}^{*}\varepsilon_{r,g1,z}-k_{m2}^{*}l_{s2 }\theta_{s,y}-k_{m5}^{*}l_{s2}\theta_{r,y}-k_{m2}^{*}l_{f1}\theta_{sf,x}-k_{m5 }^{*}l_{f1}\theta_{rf,x}=f_{st2,z}(t)\end{array}  \tag{4}\]  
\[ \begin{array}{ll}m_{st3}\ddot{\varepsilon}_{st3,z}+(k_{m3}^{*}+k_{m6}^{*}) \varepsilon_{st3,z}-k_{m3}^{*}\varepsilon_{s,g1,z}-k_{m6}^{*}\varepsilon_{r,g1, z}-k_{m3}^{*}l_{f2}\theta_{sf,x}-k_{m6}^{*}l_{f2}\theta_{rf,x}=f_{st3,z}(t), \end{array}  \tag{5}\]  
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