# Three-Dimensional Identification for Unbalanced Mass of Rotor Systems in Operation
## 4 Results and Discussion
### Different Unbalanced Mass Experiments  
Different unbalanced mass is added to the rotor disk at a fixed angle for the detection experiments. Figure 8 indicates the in-plane displacement components with different additional weight varied with time and the corresponding rotation speeds are 600, 1200 and 1800 rpm, respectively. During the experiments, the same feature point is accurately positioned repeatedly in dynamic sequential images due to stable performance of the algorithm. The dynamic responses of the unbalanced system are reflected by monitoring the changes of trajectories. The hollow points in Figure 8 correspond to the condition without additional weight, while the solid points are the results with 1 g additional weight and the triangle points are the results with 2 g unbalanced weight. Since the rotor shaft rotates at constant speeds, these plots are approximately sinusoidal curves which are consistent with the motion equations (Equations (11) and (12)). Moreover, the results illustrated that the coordinate values in x-direction tend to decrease with the increasing load, while the coordinate values in direction y have a tread to increase. In other words, the in-plane trajectories move to the upper left according to the definition of the word coordinate system. Meanwhile, the amplitude in horizontal direction appears larger with the increase of the unbalanced mass, similar to the right y-displacement of Figure 8c. The corresponding out-plane displacement components of the above experiments are compared in Figure 9. Unlike in-plane motions, the out-plane vibrations are no longer sinusoidal curves. However, from these curves, the vibration in the axial direction is periodic. The vibration amplitude increases significantly with the rising of the additional weight and speed. These vibrations in z-direction cannot be detected using traditional eddy current sensors. As shown in Figure 9, the out-of-plane motion is distinct from the other responses when the additional weight is 2 g and the rotating speed arrives 1800 rpm. This result agrees with the phenomena that the rotor can be treated as rigid at low rotating speed, and the flexibility such as the bending of the shaft cannot be ignored at high rotating speed. The measurement results also indicate that the proposed method has  
Figure 6: The raw images captured by binocular stereo video system: (**a**) left camera; and (**b**) right camera. Figure 7: The 3D motion trajectories of feature point at different rotation speed: (**a**) 600 rpm; (**b**) 1200 rpm; and (**c**) 1500 rpm. potential applications not only in the unbalanced mass identification, but also in the rotor bow and rotor misalignment estimation. The comparison results for the measurements by laser displacement sensors, eddy current sensors and the proposed method are plotted in Figure 10, where the hollow points are the data of no additional weight, the asterisks represent the data at 1 g additional weight, and the triangle points correspond to 2 g additional weight. The trajectory of the feature point is approximately a circle when the disk  
Figure 8: The in-plane motion with different additional weight (left: x direction; and right: y direction): (a) 600 rpm; (b) 1200 rpm; and (c) 1800 rpm. Figure 9: The out-plane motion at different rotation speed with different additional weight at different speed: (**a**) 600 rpm; (**b**) 1200 rpm; and (c) 1800 rpm. is in ballast. Then, the trajectory gradually becomes an ellipse with the increase of the additional weights. By calculating Equations (11)-(13), the deflection angles and offsets caused by various loading mass in the plane are plotted in Figure 11. The arrow points to the phase of unbalance and the length of the lines indicate the weight of the unbalanced mass. These results are consistent with theoretical model in Equation (13) that the displacement response is an ellipse and the ellipticity increases with the adding unbalanced mass.