# Three-Dimensional Identification for Unbalanced Mass of Rotor Systems in Operation
## 4 Results and Discussion
### Different Unbalanced Mass Experiments
The results by laser and eddy current methods are similar to the conventional unbalance estimation, which only detect 2D vibrations. Figures 10c and 11c are the results of 3D trajectories by the proposed method. The changes in operating conditions of rotor shaft due to imbalance are contrasted more obviously. Figure 11a,b shows that there is almost no difference between deflection angles caused by different loading mass. The rotor bow is apparently not detectable by 2D measurements, which can be clearly identified in Figures 10c and 11c. To explain Figure 11 in more details, the eccentricity and balance accuracy of both 2D and 3D methods are calculated in Table 2. As the unbalanced mass increases, the eccentricity and balance accuracy increase. In comparison with 2D current eddy method and laser displacement method, the eccentricity of videometric method is closer to the ground truth, the same as the balance accuracy. Normally, according to ISO1940, the balance levels of small-motor rotor systems are G0.4, G1, and G2.5. The balance level of the rotor system is about G1.3, between G1 and G2.5, which is in an acceptable range. The comparison in Table 2 indicates that the average relative error of the videometric method is 2.2%, much lower than 30.6% of current eddy method and 41.3% of laser displacement method. The error of the measurement by laser displacement method is the largest of the three methods, which is because the laser measurement is very sensitive to the measurement environment. When the rotor systems are in operation, the measurement environment is not suitable for the laser displacement sensors. Figure 11: The comparison of three methods with unbalance schematic diagrams: (**a**) laser displacement method; (**b**) eddy current method; and (**c**) the proposed method. Figure 10: The trajectories of three methods with different additional weight at 2400 rpm rotating speed: (**a**) laser displacement method; (**b**) eddy current method; and (**c**) the proposed method. Frequency analysis is essential in vibration analysis, and has the potential to deal with critical speed problems of rotor systems. Thus, to investigate the 3D dynamic performances of the established system, the out-plane displacement components are subject to Fourier transform. The frequency spectrum of out-plane displacement under 600 rpm rotating speed with different additional weights is in Figure 12. The dominant frequency components are marked and the results show that the amplitude of the marked peak decreases with the adding unbalanced mass. There is an increasing additional high-frequency harmonic component at the same time. Figures 12 and 13 indicate that the amplitude of out-plane displacement components becomes smaller with the increasing speed. This illustrates that the rotor bearing systems are in a relatively stable running status before reaching natural frequency. Figure 14 gives the frequency spectrum in three directions with the rotating speed at 3000 rpm, which demonstrates that the proposed method is able to identify high frequency vibration. Only one major peak appears in the spectrum for the in-plane vibrations, which is due to the constant rotating motion of the shaft. The frequency components of out-plane displacement are more complex and the main frequency is the same as the in-plane vibrations. Monitoring the work condition of rotary structure is critical because rotating parts are the most vulnerable components subjected to the harsh working environment such as fluctuating loading, high pressure and high temperature. Therefore, the method proposed in this paper can be applied effectively in fault diagnosis. Notably, using sensors, such as eddy current sensors, can monitor the vibration of the rotating machinery in real time. The procedures of data collecting and data processing are nearly completed simultaneously. The proposed method can measure the vibration when the rotating machinery is in operation, but the data processing is off-line in the computer.