# A New Concept of Instantaneous Whirling Speed for Cracked Rotor's Axis Orbit
## 3 The Definition of Instantaneous Whirling Speed  
The dynamic equation, Equation (1), is solved in the parameter \(e=0.05\), \(\xi=0.006\), \(\beta=90^{\circ}\), \(\lambda=0.7\), \(\mu=0\) and \(\mu=0.2\). The axis orbits of normal and cracked rotors are compared in Figure 2.  
It can be seen that the normal rotor's axis orbit is a standard circular trajectory, and the cracked rotor's axis orbit is approximately a circular trajectory. It is shown that the axis orbit of the cracked rotor may be very similar to that of the normal rotor under the condition of certain system parameters. Figure 1: Schematic diagram of rotor system: (**a**) The Jeffcott rotor with a transverse crack, (**b**) schematic diagram of the cracked cross section. At this special time, if only the morphological characteristics of the axis orbit are used, it is easy to miss the diagnosis of cracks. In order to increase the effect of axis orbit for distinguishing normal and cracked rotor, the new attribute (instantaneous whirling speed) are given to axis orbit, as shown in Figures 3 and 4. Figure 4: Steady state vibration response of cracked system at \(\mu=0.2\): (**a**) discrete axis orbit, (**b**) relative whirling speed. Figure 3: Steady state vibration response of normal system at \(\mu=0\): (**a**) discrete axis orbit, (**b**) relative whirling speed. Figure 2: The axis orbits of normal and cracked rotors: (**a**) \(\mu=0\), (**b**) \(\mu=0.2\). The plot diagram is shown in Figure 5. First, the vibration signal should be obtained at equal intervals. Then, the sampling points during a whirling cycle in the X-direction and Y-direction can construct a discrete axis orbit, as shown in Figures 3a and 4a. After that, the whirling center can be obtained, that is the square \"\(\Box\)\" in Figures 3a and 4a. Next, the whirling center is set as the origin of the polar coordinates and the angle differences between the two adjacent sampling points are calculated. The angle difference is defined as the instantaneous whirling speed. Finally, the relative instantaneous whirling speed (RWS) is defined as the ratio of the instantaneous whirling speed to its maximum value and the RWS within a whirling cycle can be obtained, as shown in Figures 3b and 4b. Figure 3a is a discrete axis orbit which is composed of equal time interval sampling points in a whirling cycle. As it can be seen from Figure 3a that the sampling points are uniformly distributed, indicating that the whirling angles are equal in the same time interval when \(\mu\) = 0. Figure 3b is the RWS within a whirling cycle (\(0\leq\varphi\leq 2\pi\)). In Figure 3b, it can be seen that the RWS of a normal rotor system is constant. The discrete axis orbit and RWS when \(\mu\) = 0.2 are shown in Figure 4. In Figure 4a, it can be seen that the distribution of the sampling points in the discrete axis orbit of the cracked system is extremely uneven, and the sampling point is very dense near the position 4. In Figure 4b, it has four fluctuations in the RWS in a whirling cycle. At position 1, RWS is at the local maximum value. Then RWS decreases from position 1 to position 2, reached the local minimum value at position 2. After that, RWS increases from position 2 to position 3, reached the local maximum value at position 3. From position 3 to position 4, RWS decreases again, and the minimum value occurs at position 4. The RWS varies transiently in a whirling cycle. The above analysis shows that the addition of this new attribute significantly increases the effect of axis orbit for distinguishing normal and cracked states of rotor systems. In order to study the application effects of RWS in experimental data, the vibration response test of a rotor system is carried out on the Bently KR4 Rotor Test Rig. The parameters of the experiment are the same as that of [30]. As showed in Figure 6, the rotor test rig mainly includes speed controller, motor, shaft, and base. The data acquisition equipment mainly includes eddy current displacement probe, signal regulator, and DAS (data acquisition system).