# A New Concept of Instantaneous Whirling Speed for Cracked Rotor's Axis Orbit
## 4 The Qualitative Mechanism Explanation  
To further understand the cause of this transient characteristic, this section studies the relationship between the breathing effect of the crack and the transient variation characteristics of the RWS with regard to the work of the additional stiffness excitation. First, the line speed is introduced as the intermediate state variable. The whirling speed can be decomposed into the linear speed \(LS_{x}\) and \(LS_{y}\) along the X-direction and Y-direction respectively. The line speeds of the normal rotor system and the cracked rotor system are shown in Figure 8. In Figure 8a, it can be seen that the line speeds \(LS_{x}\) and \(LS_{y}\) of the normal rotor are standard sine curves, and there is a 90\({}^{\circ}\) phase difference between these two curves. Therefore, the whirling speed of the synthetic motion is a constant, as shown in Figure 3b. However, in Figure 8b, the line speeds of the cracked system are not the standard sine curve anymore. At position 4, the line speeds \(LS_{x}\) and \(LS_{y}\) are close to 0. So, at this time, the whirling speed of the synthetic motion reaches the minimum value, as shown in Figure 4b. Figure 6: Bently KR4 Rotor Test Rig and data acquisition system. Figure 7: Comparison of RWS: (**a**) no crack, (**b**) with crack. In the steady state response of forced vibrations, the line speeds are mainly controlled by the mutual transformations of the kinetic energy and the potential energy. However, in the cracked rotor system, the non-uniform work of the additional stiffness excitation leads to the distortions of the line speed curves. In Figure 4b, it can be seen that the RWS has two maxima and two minima. The change mechanism of these four special positions will be analyzed in the following part. Equation (1) can be transformed into Equation (3):  
\[ \begin{array}{l}\left[\begin{array}{cc}m&0\\ 0&m\end{array}\right]\left[\begin{array}{c}\ddot{x}\\ \ddot{y}\end{array}\right]+\left[\begin{array}{cc}c&0\\ 0&c\end{array}\right]\left[\begin{array}{c}\dot{x}\\ \dot{y}\end{array}\right]+\left[\begin{array}{cc}k&0\\ 0&k\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]\\ =me\omega^{2}\left[\begin{array}{cc}\cos(\theta+\beta)\\ \sin(\theta+\beta)\end{array}\right]-\left[\begin{array}{cc}0\\ mg\end{array}\right]+\left[\begin{array}{cc}k-k_{x}(x,y,t)&-k_{xy}(x,y,t)\\ -k_{xy}(x,y,t)&k-k_{y}(x,y,t)\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]\end{array}  \tag{3}\]  
Compared to the normal rotor system, additional stiffness excitation forces of cracked rotor system are shown in Equations (4) and (5):  
\[F_{X}=F_{x}+F_{xy}=(k-k_{x}(x,y,t))x-k_{xy}(x,y,t)y \tag{4}\]  
\[F_{Y}=F_{y}+F_{yx}=\left(k-k_{y}(x,y,t)\right)y-k_{xy}(x,y,t)x \tag{5}\]  
Figure 9a shows the curves of the additional stiffness excitation forces \(F_{X}\) and \(F_{Y}\) within a whirling cycle. In Figure 9b, \(P_{x}\) and \(P_{y}\) are the instantaneous power of the additional stiffness excitation forces. At position 1, from Figure 8b we can see that the line speed \(LS_{x}\) close to 0, and the line speed \(LS_{y}\) reaches the maximum value at this time. Therefore, in Figure 4b, the relative whirling speed of the synthetic motion closes to 1. In this paper, position 1 is used as the reference position of relative whirling speed. From position 1 to position 2 (process I), as shown in Figure 9a, the additional stiffness excitation forces are close to 0. Therefore, in Figure 9b, the instantaneous powers of additional stiffness excitation forces are close to 0. Therefore, the linear speeds are determined by the conversions of the kinetic energy and potential energy. It can be known from Figure 4a that the kinetic energy is turned into the potential energy in Y-direction during this process, and the line speed \(LS_{y}\) is close to 0 in the end. On the other hand, in X-direction, the potential energy is turned into the kinetic energy, and the line speed \(LS_{x}\) reaches the negative maximum value in the end which has been shown in Figure 8b.