# A New Concept of Instantaneous Whirling Speed for Cracked Rotor's Axis Orbit
## 1 Introduction  
Rotor system is an important part of the rotating machinery which is widely used in the area of aerospace, rail transportation, and energy fields. However, because of poor working condition and the cyclic loading, fatigue cracks often occur in the position which has initial damages or stress concentration. Accidents caused by crack growth are catastrophic, thus the monitoring of cracks is widely considered. The research of the vibration characteristic and monitoring index can provide significant references for crack diagnosis. Scholars have carried out extensive research on rotor crack diagnosis. The study of the breathing mechanism promotes the further understanding of crack mechanical behavior [1]. The vibration analysis provides optimal indicators for crack detection [2] and harmonic component extraction obtains more fault information [3; 4]. In addition, model updating method [5], modal state observer method [6], and modal force identification method [7] for identifying cracks have also been deeply studied. Meanwhile, the feature extraction method for acoustic signals [8], the classification methods for motor faults [9], and the sensor fusion approach for acoustic, electric, and vibration signals [10]can provide important reference for rotor crack diagnosis. However, different methods have their own advantages and applicable objects. Axis orbits and instantaneous angular speeds are the two important diagnostic criteria for crack diagnosis of shaft. Axis orbits can be obtained by two vertical displacement sensors and have important values for the condition monitoring of rotor systems. [PERSON] [11] used axis orbits to investigate the stability of a twin-crew refrigeration compressor. Axis orbits are also used as information source to identify the torque load of rotor system and to balance the rotor system [12; 13]. For the diagnosis of rotor cracks, [PERSON] [14] found that the axis orbit for each harmonic component is an ellipse, the shape and orientation orbit of which depend on the crack size. [PERSON] [15] indicated that both of the orbit evolution at half of the first critical speed and the magnitude of the response at the first critical speed and at half of the first critical speed may be considered as the unique characteristics of the system with a breathing crack. [PERSON] [16] demonstrated that the distortion of the orbit at the 1/2 first critical speed could be considered as one of the most practical indicators of the presence of a transversal crack. [PERSON] [17] presented an experimental study on the axis orbit response for different rotor-related faults. [PERSON] [18] studied the whirl orbits evolution around the 1/3 subcritical speed zone and showed that two inner loops appear in the whirl orbits of cracked rotor. [PERSON] [19] presented the number and orientation changes of inner loops during the subcritical speed region of rotor system with crack. [PERSON] [20] discovered new backward whirl zones which are affected by the crack damage in the shaft. The above reviews show that axis orbit is a key characteristic in rotor system monitoring and provide important references for crack identification. However, axis orbit only reflects the trajectory information of a whirling cycle and ignored the instantaneous characteristics of the rotation during one whirling cycle. The instantaneous characteristic during a whirling cycle and a spinning cycle contains a lot of fault characteristic information. [PERSON] [21] measured the shaft instantaneous angular speed (IAS) and they showed a distinct difference between the healthy and the faulty blade conditions. [PERSON] [22] indicated that instantaneous angular speed is strictly synchronized to shaft rotation and much less dependent on the transfer path between the defect and the sensor. They also showed that IAS measurement is a promising tool for health monitoring of very low speed machines.