# Three-Dimensional Identification for Unbalanced Mass of Rotor Systems in Operation
## 3 Experiment  
The experimental setup is shown in Figure 4. A high-speed dual camera image acquisition system includes two high-speed cameras (Flare-CL offered by IO Industries in Canada) equipped with 25 mm lenses, two Xcelera-CL image acquisition cards, a ring light, a signal generator and a computer. The rotor machine, eddy current sensors and Xcelera-CL image acquisition cards are all offered by Donghua Testing Technology Compony in Jiangsu of China. The laser displacement sensors are offered by Micro-Epsilon (Beijing) Measurement Company in Beijing of China. Two high-speed cameras are mounted on a crossbar atop a tripod and they are able to capture images with spatial resolution of 1280 \(\times\) 1024 pixel at a rate of 300 fps (frames per second). A signal generator is used to transmit the trigger signal for the purpose to ensure sufficiently high synchronization accuracy. Figure 4c is a brief schematic of the whole testing system. The high-speed cameras are placed in front of the end face of the rotor bearing systems as shown in Figure 4c to capture images. To define the \(XOY\) plane of the world coordinate system parallel to the end face of the rotor bearing systems, the lens of left camera are set perpendicular to the shaft because the origin of the world coordinate system is the optical center of left camera. The rotation of the shaft is in \(XOY\) plane of the world coordinate system and the out-of-plane movement is considered as turbulence caused by imbalance and defects in shaft [33; 34]. During the experiments, the rotating speed of the shaft is adjusted by the control cabinet and the cameras work at 300 fps. The experimental setup can be modelled as a rotor bearing systems, which is illustrated in Figure 5. The major excitation of the system is a radial unbalance at the disk. The rotor is connected to the stator through sliding bearings and the flexible coupling between the rotor and the motor is regarded as a radial linear spring. The length of the shaft is about 800 mm, and the diameter is approximately 10 mm. The thickness of disk 1 is 25 mm and the weight is 800 g, while the thickness of disk 2 is 19 mm and the weight is 500 g. The speed of this rotor varies from 200 to 8000 rpm (rotation per minute). A feature point is attached to the end face of the shaft. The vibration is three-dimensional and the dynamic model is as follows. Figure 3: (**a**) 3D Reconstruction results; and (**b**) the projection in plane. The governing equations of the 3D model of rotor bearing system can be expressed as Equation (4):  
\[M\ddot{u}+(C_{b}+\Omega C_{cor})\dot{u}+(K_{s}+K_{b}-\Omega^{2}K_{d})u=f_{ext} \tag{4}\]  
where \(M\) and \(K_{s}\) are the mass and elastic stiffness matrices, \(C_{b}\) and \(K_{b}\) are the periodical damping and stiffness matrices due to bearings, \(\Omega C_{cor}\) denotes the Coriolis matrix, \(\Omega^{2}K_{d}\) is the spin softening matrix due to the rotation of structure, \(u\) is the vector of nodal displacement in the rotating frame, and \(f_{ext}\) denotes the external force vector. Figure 4: Experiments setup: (**a**) laser displacement sensors; (**b**) photograph; (**c**) and schematic. Figure 5: The model of rotor bearing system.