## 3 Introduction to the Generalized Inverse Cascade Method
### Identify Design Objectives and Design Variables  
According to the mounting system vibration isolation hierarchical decomposition system shown in Figure 1, the weighted force transmission rate that measures the vibration isolation performance of the mount, and the weighted decoupling rate that characterizes the degree of energy decoupling in each direction, are used as design objectives, and the stiffness values of mounts in the X-, Y-, and Z-directions are used as design variables. The force transmission rate of mount is defined as the ratio of the support reaction force at each mount to the acceleration at the center of mass of the powertrain. This is calculated by Equation (2):  
\[T_{i,j}=\frac{F_{i,j}}{RMS_{f}(a_{j})} \tag{2}\]  
where; \(T_{i,j}\) is the transmission rate of the \(i\)th (\(i\) =1,2,3) mount in direction \(j\) (vehicle coordinate system X, Y, Z), and the unit is N-s\({}^{2}\)/m; \(F_{j}\) is the support reaction force in direction \(j\) (X-, Y-, and Z-directions) at the \(i\)th mount, and the unit is N; \(a_{j}\) is the vibration acceleration in direction \(j\) at the center of mass of the powertrain system, and the unit is m/s\({}^{2}\); and \(RMS_{f}\) represents the effective value of the vibration acceleration in the frequency range \(f\), and the unit is m/s\({}^{2}\).  
The mount-weighted force transmission rate is a combination of the force transmission rates in each of the three directions, according to a certain weight. Considering the higher vibration in the Z-direction, the weight is higher. Based on engineering experience, the weight coefficients of the three mounts in the vehicle coordinate system's X-, Y-, and Z-directions are 0.065, 0.065, and 0.2, respectively. Therefore, the mount weighted force transmission rate and the decoupling rate can be calculated by Equations (3) and (4):  
\[T=\sum_{i=1}^{3}0.065*T_{i,X}+0.065*T_{i,Y}+0.2*T_{i,Z} \tag{3}\]  
\[DR=\sum_{i=1}^{6}k_{i}*DR_{i} \tag{4}\]  
where; \(DR\) represents the weighted decoupling rate; \(DR_{i}\) represents the primary decoupling rate of the \(i\)th order rigid body mode; and \(k_{i}\) is the corresponding weight coefficient. Since the unbalanced force in the Z-direction of the vehicle coordinate system is the largest source of excitation for the engine, the torque fluctuation around the X axis is also relatively large when the engine is placed longitudinally. In order to consider that the decoupling rates of the two main directions can be increased more, the weight coefficients of these two directions can be set higher, the weight coefficient of the Z-direction is set to 0.35, the weight coefficient around the X axis is set to 0.25, and the weight coefficients of the other four are each taken as 0.1.  
For different vehicles, the fluctuations in the stiffness of each mount have their variation pattern. Although the initial interval of the mount stiffness is different for each vehicle,  
Figure 5: The schematic diagram of the mounting system connections.  
the idea of applying the algorithm is the same. The stiffness of the three powertrain mounts in three directions is chosen as the design variable. Combined with engineering experience, the initial value interval is obtained by increasing and decreasing the original stiffness value by 30%. The ranges for each variable are shown in Table 1.