## 4 Numerical Results and Discussion
### Parametric Analysis
#### 4.3.2 Influence of Blade off Location  
Considering that the first-stage axial compressor (1 AC) is the most susceptible to foreign objects, it is also one of the most likely locations for blade loss in the turboshaft engine. This section studies the influence of the blade off location on the dynamic characteristics. Figure 20 shows the comparison of impact factors when blade off occurs at 1 AC and the first-stage gas turbine (1 GT). It shows that the blade off location has a significant effect on the impact factor. When the blade off location is changed from 1 GT to 1 AC, the impact factor at 1# support is increased from 2.8 to 6.84. In contrast, the impact factor at 2# support and the center of gravity position does not increase but decrease, although the decrease is small. This is because the sudden unbalance excitation at different locations has a distinct  
Figure 19: Impact factors at different speeds. Figure 17: The time-domain response of 1# bearing in the horizontal direction when the rotor operates at 30,000 r/min: (**A**) 0.04–0.2 s; (**B**) 0.2–0.25 s; (**C**) 1.48–1.5 s.  
Figure 18: Frequency spectrums at different locations when the rotor operates at 30,000 r/min: (**a**) 1# bearing; (**b**) node 11 (C.G. position); (**c**) 2# bearing. sensitivity to each mode of the rotor operating at a certain speed. The details are as follows. It can be seen from the mode shapes shown in Figure 8 that the first three-order critical modes can be effectively excited by the excitation at 1 AC, while the excitation at 1 GT can only excite the first two modes. In addition, the effects of the sudden unbalance excitation at different locations on each mode are inconsistent. Namely, the first-order critical mode (Figure 8a) is easier to be excited by the excitation at 1 GT than that at 1 AC, while the second- and third-order critical modes are just the opposite, especially for the third-order mode (Figure 8c), which can hardly be excited by the excitation at 1 GT since the vibration node point is near 1 GT. The above reasons will cause the transient response excited by the blade loss of 1 AC to be greater than that caused by the loss of the 1 GT blade. From the steady-state vibration mode (Figure 21) caused by the large unbalance excitation at 1 AC, it can be seen that the steady-state amplitude of 1# support is smaller than that of the C.G. position and 2# support. As a result, when the blade loss occurs at 1 AC, the large transient response peak and the smaller steady-state vibration amplitude at 1# support cause a large impact factor. Apparently, the 1# bearing and its supporting structure are most vulnerable to damage, such that some safety design methods need to be adopted. The frequency-domain responses at the instantaneous stage (0.05-0.12 s) when blade off occurs at two different locations (1 AC and 1 GT) are shown in Figure 22. Obviously, there are big differences in the frequency components between these two cases. It can be seen from Figure 22 that the sudden unbalance excitation at 1 AC can excite the first three-order natural modes simultaneously (\(f_{\text{n1}}\),\(f_{\text{n2-}}\),\(f_{\text{n2+}}\),\(f_{\text{n3-}}\),\(f_{\text{n3+}}\)), while the excitation at 1 GT can only excite the first two modes (\(f_{\text{n1}}\),\(f_{\text{n2-}}\),\(f_{\text{n2+}}\)). Furthermore, the amplitudes of the second-order modes (\(f_{\text{n2-}}\),\(f_{\text{n2+}}\)) excited by the sudden unbalance excitation at 1 AC are larger than those excited by 1 GT. For example, when the blade off event changes from 1 GT to 1 AC, the amplitude of the second backward precession frequency component increases from 5.27 um to 12.3 um. Another significant vibration difference between these two cases of blade off can also be observed in Figure 22a,c, i.e., in all the natural modes excited by sudden unbalance excitation, when the blade off occurs at 1 GT, the first natural frequency component has the largest amplitude; however, the amplitude of the second forward natural frequency component is the largest when the blade off occurs at 1 AC.