Strojniški vestnik - Journal of Mechanical Engineering 63(2017)6, 394-404 © 2017 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2017.4332 Original Scientific Paper Received for review: 2017-01-19 Received revised form: 2017-04-05 Accepted for publication: 2017-04-24 A Practical Method to Detect a Transverse Cracked Rotor Using Transient Response Xiaofeng Wang - Jun Liu* - Weimin Ge Tianjin University of Technology, Tianjin Key Laboratory of the Design and Intelligent Control of the Advanced Mechanical System, China To detect a transverse crack caused by fatigue or creep, most of the research has thus far paid attention only to resonances of steady-state oscillations created by the crack and proposed diagnosis systems utilizing these vibration phenomena. However, from a practical view point, these diagnosis systems have the following flaws: (1) the probability that a resonance occurs due to a crack in the rated rotational speed range is a lower position; (2) It is very dangerous to observe vibration characteristics in resonance ranges. In order to solve these problems, this paper uses a practical detection method utilizing the characteristic changes in a transient oscillation during the start-up, the shutdown, or the variable running speeds of rotating machinery. This method has great advantages, because it can check the occurrence signals of a crack in a wide speed range using a single sweep and avoid the operation in dangerous resonance ranges. Non-stationary characteristics during passages through the main resonance and various kinds of resonances are studied numerically and experimentally. Keywords: transient response, nonlinear rotor, cracked rotor, crack identification, experiments Highlights • A practical method for the diagnosis of the cracked rotor has been investigated theoretically and verified systematically in many experiments. • The characteristics of the transient response have been shown on the cracked rotor, and there is not only the harmonic resonance, but also the sub-harmonic resonance and the super harmonic resonance are analysed under the transient state. • A sensitive and accurate experimental setup has been developed with a high-quality data acquisition system to obtain precise measurement data. • The experimental results have been verified well above the analyses from theoretical simulations. 0 INTRODUCTION A transverse crack occurs due to fatigue, creep or both in the rotating machinery during operation, To prevent a serious accident caused by a cracked rotor, it is very important to discover the crack at the early stage of the crack propagation. Therefore, various kinds of diagnosis system have been developed to detect a crack in the rotating machinery. Verney and Green presented an on-line crack diagnosis regimen hinging on the accuracy of the crack model which should account for the crack's depth and location [1]. Silani et al. introduced a new finite element (FE) approach to detect small cracks and calculated the flexibility matrix of crack elements with modified integration limits [2]. Li and Chu developed an HHT signal processing technique on the AE feature extraction of natural fatigue cracks in rotating shafts [3]. Guang and Chen introduced a FE model for the crack identification of a static rotor with an open crack [4]. Ma et al. also used a FE model to calculate time-varying mesh stiffness for the effects of profile shift and tooth crack in a gear rotor system [5]. Darpe presented a novel method of the transient torsion excitation to detect fatigue transverse cracks in rotating shafts [6]. Xie et al. studied the motion stability of the flexible rotor-bearing system under the unsteady oil-film force and other faults by calculating the maximum Lyapunov exponent of the system [7]. Zhu et al. [8] and Ishida and Inoue [9] theoretically and experimentally analysed the dynamic characteristics of a cracked rotor with an active magnetic bearing. Ferjaoui et al. investigated the effect of the presence of a transverse crack in a rotor supported by two hydrodynamic journal bearings [10]. In general, the detection methods of a crack are classified into two groups. One, such as [4], is the static examination where a rotating machine is dissolved, and the parts are examined independently. The other is the dynamic examination in which the changes of the vibration characteristics could be observed during the operation. Aiming at a practical generator rotor, Chu and Wang reported that the magnitude of the harmonic resonance at the main critical speed increased and a super-harmonic resonance at the secondary critical speed occurred due to a crack [11]. Concerning other kinds of resonances, there is no report on practical rotors. However, Ishida et al. [12] and Ishida and Hirokawa [13] observed a sub-harmonic oscillation of order 1/2, a super sub-harmonic oscillation of order 3/2 and a summing-and-differential harmonic oscillation owning to a crack in the experimental 394 *Corr. Author's Address: Tianjin University of Technology, No. 391 Binshuixi Road, Tianjin 300384, China, liujunjp@tjut.edu.cn Strojniski vestnik - Journal of Mechanical Engineering 63(2017)6, 394-404 setup. Guo et al. examined the identification of the early crack propagation with the empirical mode deposition (EMD) method [14]. Gomez et al. analysed the vibration signals based on energy utilizing the wavelet theory. The results demonstrated the good reliability of crack diagnosis with the 3* energy [15]. Ishida et al. investigated transient responses at the 1/2 order sub-harmonic oscillation [16]. Li and Zhang used the Hilbert-Huang Transform (HHT) to identify a crack in a rotor-bearing system under transient oscillations [17] and [18]. Wang etc. proposed the application of order tracking to investigate a crack when the rotor system has a varying speed [19]. Therefore, there is a possibility that those oscillations also occur in practical machines under stationary responses or transient responses. Although the static method is more reliable, it requires much time and money. Therefore, the dynamic examination is preferable for the early detection of a crack. Most of the dynamic monitoring systems focus on the changes of vibration characteristics in the steady-state oscillation. They have the following defects from the perspective of practice: (1) Many symptoms due to a crack do not occur in the rotational speed range; therefore, it is impossible to detect them during the normal operation; (2) When some changes occur due to a crack, it is dangerous to investigate the characteristics of the resonance range because there is a possibility that the crack develops rapidly during the investigation. This study focuses on a practical detection method using non-stationary vibrations to overcome those defects. When a rotating machine starts up or shuts down, the rotor sweeps all over the rotational speed range below the rated rotational speed. If these non-stationary data are used to detect a crack, defects (1) and (2) could be avoided. In addition, there are some studies to illustrate systematically the transient response with simulations. In this paper, a typical open-close model is used to investigate the characteristics of non-stationary oscillations of a cracked rotor during the passages through the main critical speed and various kinds of subcritical speeds. In particular, the study focused attention on the influences of an angular acceleration and the magnitude and phase of an unbalance on the maximum amplitude. The effectiveness of this method is verified systematically through simulations and many experiments. To solve the above problems, the next section proposes the theoretical modelling and the motion equations. The resonances of steady-state oscillations with a crack are investigated in Section 2. The method to detect a crack using the change of the characteristics of non-stationary oscillations passing through the harmonic resonance is explained in Sections 3 and 4. Non-stationary oscillation during passages through a forward super-harmonic and a forward sub-harmonic resonance are interpreted in Section 5. The characteristics of non-stationary oscillations during passages through a variety of resonances are summarized and shown in Section 6. The experimental setup is explained in Section 7, and the experiment results are presented in Section 8. Finally, the concluding discussion is given in Section 9. 1 THEORETICAL MODELLING AND MOTION EQUATIONS 1.1 Theoretical Modelling and Spring Characteristics The theoretical model and the coordinate systems are shown in Fig. 1. In the experimental setup mentioned in Section 7, the rotor system where the deflection and the inclination couple each other is a four-degree-of-freedom (4DOF). The disk is not located at the shaft centre. On the contrary, if the disk is located at the shaft centre, it can be divided into two separate 2DOF systems, that is, a deflection model (the Jeffcott rotor) and an inclination model. In the latter system, the natural frequencies change due to the gyroscopic moment similar to the 4DOF model [20]. Therefore, we use the inclination model in the theoretical analysis. The origin of the static Cartesian coordinate system O-xyz is at the midpoint of the bearing centreline (the connecting line of the right and left bearings). The z-axis coincides with the breath ring centreline. The inclination angle of the elastic shaft at the disk mounting position can be expressed by 6 and its projection angles of 6 to the xz- and yz-planes can be expressed by 6x and 6y , respectively. It is supposed that a crack appeared on the half of the shaft's length. The rotating coordinate system O-x'y'z' is also considered where the x'-axis coincides with the crack boundary. The projection angles of 6 to x'z- and y'z-planes can be represented by 0'x and Q'y, respectively. When the crack is opened and Q'y > 0, the stiffness of the shaft becomes small. When the crack is closed and 6'y < 0, the shaft stiffness returns to the same value as the shaft with no crack. Therefore, the restoring moment has the spring characteristics with a piecewise linearity and its components M' and M'y in the x'z'-and y'z'-planes, respectively, are shown in Fig. 2. They are represented as follows. A Practical Method to Detect a Transverse Cracked Rotor Using Transient Response 395 Strojniski vestnik - Journal of Mechanical Engineering 63(2017)6, 394-404 -Mx = 8[0'x -M'y = (82 -A8'2)0'y -M' = (82 + A82)0V (oy > o) (oy < o) (i) where, 8/ and S'2 are the spring constants and A8'2 is the directional difference. Crack y y Fig. 1. Model of a cracked rotor and coordinate systems Fig. 2. Spring characteristics of a cracked rotor 1.2 Motion Equations The ratio of the polar inertia moment to the diametric inertia moment can be represented by ip , the rotational speed and the damping coefficient can be represented by w and c, respectively. The dynamic unbalance's magnitude and its phase angle can be expressed by t and a, respectively. The rotational angle of the x'-axis is represented by y. Corresponding to the gravitational force, the constant moment M0 that works in the ^-direction is considered. The motion equations governing non-stationary oscillations in a symmetrical 2DOF inclination model with no crack is given by Ishida et al. [12] and Ishida and Yamamoto [20]. First, we transfer Eq. (1) into the expression of the stationary coordinate system. By replacing the part representing the restoring force in symmetrical system by this expression, we can obtain the non-dimensional motion equations for a cracked rotor as follows. (2) 0x + ipy/èy + ipnj/0y + cèx + (1T A 2)0X + (Aj ± A2)(èx coslrnt + èy sin2rnt) = = (1 - ip )t {/2 cos (y/ +a) + yz sin(^ +a)} èy - ipy/èx - ipy/èx + cèy + (1 t A2)èy + +(A1 ± A2)(èx sin2mt -èy coslrnt) = = (1 - ip )t {2 ^^^^ + a)-y/ cos(i/ + a) + Mc where 8' = (8'-+8'2) /2, A1 = (8{-8'2) / 28', A2 =A8'2/28'. As for the symbol "±" in the motion equations, all upper signs are shown for 0' > 0 and all lower signs are also shown for 0 < 0. The above motion equations have the dynamic characteristics as follows: (a) time-varying coefficients similar to the asymmetrical rotor, (b) rotating piecewise nonlinearity and (c) unbalance excitation. 2 STEADY-STATE OSCILLATIONS The resonances of steady-state oscillations are investigated before studying non-stationary oscillations. When the rotational speed y} =œ is a constant, the angular position of the x'-axis can be expressed by: w =rnt + w0), (3) where y0 is the initial angle. With this condition, Eq. (2) is integrated numerically by the Adams method. Let pf and pb are the natural frequencies of a forward and a backward whirling motions, respectively. In view of a vertical rotor, only a harmonic resonance [pf= a>] appears in the vicinity of the main critical speed. In the following, the notation [20] is used to show the relationship between the natural frequency and the rotational speed when the resonance occurs. When an unbalance and the crack are on the same side, there exists an unstable range at the main critical speed. Otherwise, the unstable range will disappear [20]. Fig. 3 shows the case of a horizontal rotor. The symbol 0 represents the amplitude obtained numerically. These values are connected smoothly by a full line. Various kinds of resonances occur in a wide rotational speed range due to a crack in view of a horizontal rotor. In addition to the harmonic resonance [pf= a>], the backward harmonic resonance [pb=-w], the super-harmonic resonances [pf= 2a>] and [pf= 3w], the sub-harmonic resonance [pf= (1/2)w], the super-sub-harmonic resonance [p= (3/2)w] and 396 Wang, X. - Liu, J. - Ge, W. Strojniski vestnik - Journal of Mechanical Engineering 63(2017)6, 394-404 the combination resonance [m = pf - pb] occur. This is because the equilibrium position of the rotor shifts due to gravity and, as a result, the rotor system has both characteristics of more complex nonlinearity and more complex parametric excitation. Within all the above resonances, only harmonic resonances of these [pf = m ] can vary these characteristics significantly, depending on the angular position of the unbalance. 3 NON-STATIONARY OSCILLATIONS (A VERTICAL ROTOR) This section explains the non-stationary characteristics of a cracked rotor. The governing motion equations of it are given by putting M0 = 0 in Eq. (2). The acceleration of a rotor is a constant X, regardless of acceleration or deceleration. The angular position of the x'-axis can be obtained by: (1/2)Xt2 +wt +y0. (4) The response curves are shown by full lines in Fig. 4, and the curves are obtained numerically via the Adams method. The results for three kinds of angular acceleration X are also shown in Fig. 4. For comparison, the amplitudes of steady-state oscillations (X = 0) are shown by the symbol When the unbalance and the crack are on the same side, the result is shown in Fig. 4a. In this case, an unstable range exists. If the rotational speed of a rotor sweeps this unstable range, the large amplitude appears for the small angular acceleration X. If they are on the opposite side, another result is shown in Fig. 4b. Since there no exists an unstable range, the amplitude is comparatively small regardless of any value of the angular acceleration X. Fig. 5 shows that the maximum amplitude rmax changes with the X. If the unbalance and the crack are on the same side, the cracked rotor cannot pass the main critical speed range due to having the very large amplitude when the angular acceleration X is less than a certain critical value. Different from the case of steady-state oscillations, the whirling speed and the rotational speed are different from each other, and the crack opens and closes repeatedly during the passage through the main critical speed. Therefore, it is imagined that the difference in the angular position of the unbalance does not influence the maximum amplitude in the case 1.5 2.0 2.5 Rotational speed Fig. 3. Amplitude variation curve of a horizontal cracked rotor a) b) Fig. 4. Amplitude variation curves In the main critical speed with a vertical rotor; a) unbalance within the crack side, and b) unbalance without the crack side A Practical Method to Detect a Transverse Cracked Rotor Using Transient Response 397 Strojniski vestnik - Journal of Mechanical Engineering 63(2017)6, 394-404 of non-stationary oscillations. But Fig. 5 shows that the maximum amplitude rmax changes remarkably due to the direction of the unbalance. This means that the repetition of the opening and closing states do not occur frequently in the resonant range. 2 3 4 Angular acceleration A(/\0~4) Fig. 5. Maximum amplitudes with acceleration X These results can be interpreted from the viewpoint of vibration diagnosis as follows: Since the maximum amplitude increases remarkably as shown in Fig. 4a, the appearance of a crack can be detected from the incremental amplitude. In contrast, if they are on the opposite side, it is very difficult to find it. 4 NON-STATIONARY OSCILLATIONS (A HORIZONTAL ROTOR) In this section, non-stationary oscillations during the passage through the main critical speed are investigated when the rotor system is supported horizontally. Time histories obtained by numerical integration are more complicated than that of a vertical rotor. A time history is shown in Fig. 6a for the case that the unbalance and the crack are on the same side. In addition, spectrums obtained by the complex-FFT method [20] are shown in Fig. 6b, in which the positive abscissa represents the forward whirling motion and the negative abscissa represents the backward whirling motion. Because of the rotor passing through the unstable range, the amplitude changes remarkably. Different from a harmonic component case of a vertical rotor, many frequency components exist in the spectrum diagram. In addition to the harmonic component between 1 and 1.5, a constant component, a backward harmonic component and a forward super-harmonic component with a frequency of two times the rotational speed coexist. Therefore, it is impossible to obtain the amplitude of the harmonic component by calculating