# Vibration Properties of Dual-Rotor Systems under Base Excitation, Mass Unbalance and Gravity
## 3 Results and Discussions
### Forced Vibration Due to Mass Unbalance with Base Rotations  
The influences of the constant base rotations on the mass unbalance responses of the dual-rotor system are analyzed in this section. As mentioned above, the rotation speed relationship between the rotors is set at \(\Omega_{HP}=-1.2\Omega_{LP}\).  
For the mass unbalance on a low-pressure rotor, the mass unbalances on the three disks are set as \(m_{u1}e_{1}=m_{u3}e_{3}=0\) and \(m_{u2}e_{2}=1\times 10^{-4}\) kg \(\cdot\) m. The frequency responses are shown in Figure 6. Only the forward whirling modes are excited when the base is fixed (see the blue lines). For the pitching base motion shown in Figure 6a,b, the backward whirling modes were also excited when the pitching base motion is concerned. The resonant frequencies decreased a little with increasing angular velocity of the pitching base motion. For the rolling base motion shown in Figure 6c,d, only the forward whirling modes were excited. The resonant frequencies decreased remarkably, when the angular velocity of the rolling base motion increased.  
For the mass unbalance on a high-pressure rotor, the mass unbalances are set at \(m_{u1}e_{1}=m_{u2}e_{2}=0\) and \(m_{u3}e_{3}=1\times 10^{-4}\)kg \(\cdot\) m. The frequency responses are shown in Figure 7. Only the backward whirling modes were excited when the base was fixed (see the blue lines). For the pitching base motion shown in Figure 7a,b, the forward whirling modes were also excited when the pitching base motion was concerned. The resonant frequencies  
Figure 6: Responses of the dual rotor system excited by mass unbalance on disk 2 with base motions: (**a**) Pitching base motion; (**b**) Pitching base motion; (**c**) Rolling base motion; (**d**) Rolling base motion.  
decreased a little with increasing angular velocity of the pitching base motion. For the rolling base motion shown in Figure 7c,d, only the backward whirling modes were excited. The resonant frequencies decreased remarkably, when the angular velocity of the rolling base motion increased.  
Here are some explanations for the above phenomenon. When the base was fixed, the system possessed perfect symmetry along the two linear displacement directions (\(O_{b}X_{b}\) and \(O_{b}Z_{b}\)). Even the amplitudes of the two components of the unbalance force along these directions were equal. Thus, the unbalanced force was only in accordance with whirling modes along one direction: backward for the mass unbalance on the low-pressure rotor and forward for the mass unbalance on the high-pressure rotor. The rolling base motion did not break this symmetry, and the amplitudes of the two components of the unbalance force remained equal (see Equations (32), (36), (39), (64), (65) and (70)). However, when the pitching base motion was involved, the stiffness symmetry along \(O_{b}X_{b}\) and \(O_{b}Z_{b}\) was not maintained and the amplitudes of the two components of the unbalance force were not equal (see Equations (35), (58) and (63)).  
Figure 7: Responses of the dual-rotor system excited by mass unbalance on disk 3 with base motions: (**a**) Pitching base motion; (**b**) Pitching base motion; (**c**) Rolling base motion; (**d**) Rolling base motion.