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Cracked rotors: Industrial machine case histories and nonlinear effects shown by simple Jeffcott rotor
ARTICLE IN PRESS Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 22 (2008) 805–817 www.elsevier.com/locate/jnlabr/ymssp
Cracked rotors: Industrial machine case histories and nonlinear effects shown by simple Jeffcott rotor Yukio Ishida Department of Electronic-Mechanical Engineering, Nagoya University Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan Available online 17 November 2007
Abstract Horizontal rotors are always imposed to periodic stresses and, therefore, a crack due to a fatigue is unavoidable. The proper diagnosis of machinery is necessary to prevent tragic accidents and the vibration monitoring is the most important tool for such a diagnosis system. In order to develop a monitoring system that can detect a crack in an early stage of propagation, it is important to know the vibration characteristics of a cracked rotor. A crack opens or closes due to the direction of the lateral deflection. Therefore, a cracked rotor has nonlinear spring characteristics of a piecewise linear type. In order to diagnose the vibration characteristics properly, it is essential to understand the behavior caused by the nonlinearity. These piecewise linear characteristics make a directional difference in stiffness and this difference rotates with the rotor. As the result, the coefficients of linear and nonlinear terms in restoring forces become time dependent. According to the physical characteristics, cracked rotors can be classified into a class of nonlinear parametrically excited system. At first, this article introduces case histories of cracks found in industrial machines. Secondly, it explains the vibration characteristics of various kinds of resonances due to cracks using simple Jeffcott rotor with nonlinear spring characteristics. The utilization of the nonstationary vibrations for monitoring system is also explained. r 2007 Elsevier Ltd. All rights reserved.
1. Introduction In a horizontal rotor where gravitational force works, the tension and compression work in the upper and lower sides of the rotor, respectively. Since these stresses change periodically as a rotor rotates, the rotor has high probability to be destroyed by a fatigue crack. In order to develop a vibration monitoring system that enables detecting a crack during the operation, vibration characteristics of cracked rotors have been investigated by many researchers. The accidents due to cracks in turbine generators have been reported since 1950s but theoretical and numerical investigations on the vibration characteristics of cracked rotors started from 1970s. The overview of the past studies on this field until 1989 is reviewed by Wauer [1]. When a horizontal rotor has a transverse crack, the crack area opens or closes due to the self-weight bending as it rotates. This is called breathing and this makes the characteristics of a cracked rotor nonlinear. This mechanism can be considered when a physical rotor model is constructed. In 1976, Gasch [2,3], Henry and Oka-Avae [4] considered this breathing nonlinear mechanism by using different flexibilities for open and closed condition and solved the equations of motion by E-mail address: [email protected] 0888-3270/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2007.11.005
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an analog computer. Many researchers investigated vibration characteristics of cracked rotors in detail qualitatively and quantitatively using various kinds of physical models. Since it is considered that the essential characteristics can be understood more easily by a simple physical model, the author clarified various kinds of resonance which may occur in cracked rotors using models composed of a massless elastic shaft and a disc mounted at the center of the shaft. This paper introduces case histories of cracked rotors in industrial machines and presents nonlinear effects on various kinds of resonance in cracked rotors using a simple rotor model similar to Jeffcott rotor. Even though the characteristics of this model cannot be considered rigorously as Jeffcott rotor model, for simplicity, this model is called as ‘‘Jeffcott rotor model’’ in this model. Stationary and nonstationary phenomena are both investigated. 2. Case history in industrial machines Turbine generators developed very rapidly during 1950s–1970s and their sizes increased from about 60 MW to about 500 MW in this period. Since many turbines are constructed without consideration of the experiences of the foregoing products sufficiently, turbine rotors suffered from cracks and many tragic accidents occurred in those days. Since 1950s, cracks have been found in turbine shafts or in turbine blades and some kind of tragic accidents which probably happened due to cracks have been reported. Generally, it is very difficult to specify the cause of the accidents after the rotor system was destroyed completely. In 1953, the 1800 rpm steam turbine at the Tanners Creek power station in USA suddenly went into vibration, and it was found that a segment of approximately 1601 had broken out in a wheel due to a crack [5]. In March 1954, a 3600 rpm generator rotor in Arizona, USA, burst while being balanced in the factory [6]. Several cracks were found before bursting in this rotor. In September 1954, a generator rotor in Cromby, USA, burst while running at 3780 rpm [6]. In December 1954, the LP steam turbine in Ridgeland, USA, burst during a routine over-speed trip test [7]. In 1954, a generator rotor in Pittsburg, USA, burst during an overrun test at 3920 rpm [8]. In 1970, a LP turbine burst during a running test in Nagasaki, Japan [9]. More than 60 people died and injured in this tragic accident [9]. The investigation team concluded that the cause of this collapse is cracks due to a stress concentration at grinded holes. There are many cases where cracks were found luckily before a disaster. Coyle and Watson [10] reported that, in 1956–1957, the shafts of three large turbine rotors developed fatigue cracks under very low nominal gravitational bending stresses in Castle Donnington, UK. The fatigue cracks propagated over 75% of the shaft section in two cases, and the cracking started all around the circumference and propagated no deeper than 1/16 inch in the other case. Yoshida [9] reported four cases that cracks are found in four steam turbine rotors during 1970–1971 in Japan. He concluded that the cyclic thermal stress which appeared during the frequent start up and shut down made cracks. Jack and Paterson [11] reported that three cracks were discovered in three LP shafts in 500 MW turbines at Ferrybridge power station during 1972–1974. Muszynska [12] reported that at least 28 cracked failures happened within the period 1970s. Laws [13] in Bently Nevada Co. reported four cracked rotor incidents happened in UK during 1971–1981. These incidents are a 40% transverse crack of 500 MW turbine generator, a circumferential 30 in long crack of MP rotor of 60 MW turbine generator, a 45% transverse crack of the main generator rotor of 660 MW turbine generator and a crack extending around complete circumference of the MP rotor of 350 MW turbine at main HP coupling end. Cracks of other kinds of machines are also reported. For example, a rotor crack which expanded of 1201 of the cross section of the shaft of a wind tunnel fan for a car testing was reported in 1990 in Japan [14]. In many cases, the occurrence of a crack is noted by an abnormal increase of vibrations. However, the development of vibration diagnosis system, which was installed to monitor the turbines started to reveal the detailed symptoms of cracks. Dimarogonas and Paradopoulos reported a crack failure found in 300 MW turbine generator in 1983 in Lavrion power plant in Greek [15]. He reported vibration spectrum where characteristic peaks are admitted. From 1970s, cracks were found in turbines of several atomic power plants. This led to a much more serious situation for the people. Ziebarth and Baumgartner [16] reported the vibration histories after cracks occurred in the 1300 MW turbine in Cumberland power plant in USA and in Wuergassen nuclear power plant in Germany. In the former, an indication appeared and started to increase before 3 days prior to the shutdown. A double-frequency excitation was noted upon the coastdown. In the latter, the horizontal and vertical
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vibrations of the LP turbine started to increase from 6 days prior to the shutdown. A pump shaft in Crystal River nuclear power plant broke due to a fatigue crack in a groove in 1986 and also due to crack in another part in 1989 [17]. In 1990, a crack distributed 25% of the cross section was found in a generator rotor of the 935 MW turbine generator of Darlington nuclear power plant. Sanderson [18] reported the characteristic of the data obtained by the monitoring system in detail. It shows that the overall vibration started to increase about 5 days before the shutdown and the amplitudes of the harmonic resonance and the double-frequency resonance increased due to the occurrence of a crack. It can be imagined that there are many other cases, which are not reported to the public. Besides a turbine shaft or a generator shaft, there are many reports on the crack found in other parts such as turbine blades. Except the following important case, other all incidents are omitted in this paper. In 1889, a ring of the recurrent pump in Fukushima atomic power station in Japan broke due to cracks made by a resonance. 3. Modeling and equations of motion 3.1. Physical models of a cracked rotor When a disk is mounted on a massless elastic shaft, the deflection motion and the inclination motion of the rotor are generally coupled. However, if the disk is mounted at the center of the shaft, the system is separated into two independent 2DOF rotor systems that execute only a deflection motion or an inclination motion as shown in Fig. 1 [19]. The former is the so-called ‘‘Jeffcott rotor’’ and its natural frequencies are constant and independent of the rotational speed. Since a forward resonance and a backward resonance of the same kind occur in the same rotational speed, this model is not suitable for the analysis of nonlinear vibrations where a mutual interaction among components often occur through the nonlinearity. Cracked rotors are typical nonlinear systems as mentioned below, the latter model is mainly used in this analysis. Fig. 2 shows an inclination model with a transverse crack. The z-axis of the static rectangular coordinate system O-xyz coincides with the bearing centerline. The inclination angle of the elastic shaft at the disk position is expressed by y and its projections to the xz and yz planes are expressed by yx and yy, respectively. It is assumed that the crack developed in the area of half of the shaft cross section. We consider the rotating coordinate system O-x0 y0 z0 in which x0 -axis coincides with the boundary of the crack. When y0y 40, the crack opens and the shaft stiffness becomes small. On the contrary, when y0y o0, the crack closes and the shaft stiffness becomes large. As a result, the restoring moment has piecewise spring characteristics and its
Fig. 1. Natural frequency diagrams of a deflection model and an inclination model: (a) deflection model and (b) inclination model.
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Fig. 2. Inclination model with a transverse crack.
Fig. 3. Spring characteristics of a cracked rotor.
components M 0yx and M 0yy in the x0 z and y0 z planes, respectively, are shown in Fig. 3. These components are represented by M 0yx ¼ d1 y0x , M 0yy ¼ ðd2 Dd2 Þy0y ðy0y 40Þ, M 0yy ¼ ðd2 þ Dd2 Þy0y ðy0y o0Þ.
ð3:1Þ
3.2. Equations of motion Let the ratio of the polar moment of inertia to the diametric moment of inertia of the rotor be ip, the rotating speed o, the damping coefficient c, and the magnitude and the phase angle of the dynamic unbalance t and a, respectively. Corresponding to the gravitational force working in the experimental setup, we consider the constant moment M0 in the yy-axis. Transferring Eq. (3.1) into the expression in the stationary coordinate system and connecting them with the equations of motion for the nonstationary rotor with no crack [20], we obtain the following equations of motion: € y þ ip c _ y_ y þ cy_ x þ ð1 D2 Þyx þ ðD1 D2 Þðyx C 2 þ yy S 2 Þ y€ x þ ip cy 2 € sinðc þ aÞg, ¼ ð1 ip Þtfc_ cosðc þ aÞ þ c € x ip c _ y_ x þ cy_ y þ ð1 D2 Þyy þ ðD1 D2 Þðyx S 2 yy C 2 Þ y€ y ip cy 2 ¼ ð1 ip Þtfc_ sinðc þ aÞ c€ cosðc þ aÞg þ M 0 ,
ð3:2Þ
where M ¼ (1ip)to2, d ¼ (d1+d2)/2, D1 ¼ (d1d2)/2d, D2 ¼ (Dd2/2d). Concerning the symbol ‘‘7’’ in these equations, we use all the upper signs for y0y 40 and all the lower signs for y0y o0.
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These equations of motion possess the following dynamical characteristics: (a) Time dependent coefficients similar to an asymmetrical rotor (parametrically excited system). (b) Rotating nonlinearity of a piecewise linear type (nonlinear system). (c) Excitation by the unbalance (forced oscillation system). The corresponding equations of motion of the 2DOF deflection system are obtained by putting the terms corresponding gyroscopic moment in Eq. (3.2). 4. Numerical integration (PWL model) This section overviews various kinds of resonances, which appear in the steady state condition. When the _ ¼ o is constant, the angular position of the rotor is given by rotational speed C C ¼ ot þ C0 .
(4.1)
Figs. 4(a) and (b) show resonances which are obtained by integrating piecewise linear model (PWL model) given by Eq. (3.2) numerically with this condition. These are the cases in which the unbalance is located in the direction at the center on the crack side. It is known that an unstable range appears at the major critical speed when an unbalance is located in the same side as the crack and disappears when it is located in the opposite side of the crack [3,20]. Fig. 4(a) shows a case of a vertical rotor. Only a harmonic resonance [+o] occurs at the major critical speed. Fig. 4(b) shows a case of a horizontal rotor. In addition to a harmonic resonance, a backward harmonic resonance [o], superharmonic resonances [+2o], [+3o], a subharmonic resonance [+(1/2)o], a supersubharmonic resonance [+(3/2)o], and a combination resonance [pfpb] occur. This is because the system becomes to have more complex nonlinear spring characteristics and more complex parametric excitation characteristics due to the shift of the equilibrium position under the action of gravity. Among these
Fig. 4. Resonance curves of a cracked rotor (numerical results): (a) vertical rotor (M0 ¼ 0) and (b) horizontal rotor.
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resonances, the harmonic resonance [+o] changes its characteristics remarkably depending on the direction of the unbalance. 5. Steady-state oscillations (PS-model) The piecewise linear expression of Eq. (3.1) is inconvenient for the theoretical treatment because we have to use different expressions for the restoring moment depending on the direction of the deflection. Therefore, the spring characteristics are approximated by a power series for convenience (PS-model). In this section, theoretical results for various kinds of resonances are introduced. 5.1. Power series approximation of spring characteristics Let assume that piecewise linear spring characteristics are approximated to the following power series: M 0yx ¼ d0x y0x , 2
3
4
M 0yy ¼ d0t y0y þ 2 y0 y þ b3 y0 y þ 4 y0 y þ .
ð5:1Þ
By the transformation of coordinates, we obtain the following expressions for the restoring moment in the static coordinates: 2 M yx ¼ yx þ Dðyx C 2 þ yy S2 Þ þ fð3S 1 þ S3 Þy2x þ 2ðC 1 C 3 Þyx yy ðS 1 þ S 3 Þy2y g þ , 4 2 M yy ¼ yy þ Dðyx S 2 yy C 2 Þ þ fðC 1 C 3 Þy2y 2ðS1 þ S3 Þyx yy þ ð3C 1 þ C 3 Þy2y g þ , ð5:2Þ 4 where only the terms up to the second order are expressed explicitly. The notations C n ¼ cos not and Sn ¼ sin not are used for simplicity. 5.2. Harmonic resonance (vertical rotor) [20] If the rotor is supported vertically, the static deflection is not caused by gravity. The response at the major critical speed depends on whether the unbalance is located on the same side or the side opposite to the crack. The theoretical response curves (left) are compared with the experimental results (right) in Fig. 5. In this theoretical analysis, the deflection model (ip ¼ 0) is used and the nonlinearity is approximated by the power series up to the fourth order. In these figures, both amplitudes and phase angles are illustrated. As already known well, the response curves change remarkably depending on the direction of the unbalance. When the unbalance is located in the same side as the crack, the resonance curves spread in the large amplitude region and, as a result, an unstable range appears. On the other hand, when the unbalance is located in the opposite side to the crack, the unstable range disappears. 5.3. Harmonic resonance (horizontal rotor) [21] If the rotor is supported horizontally, the harmonic resonance becomes more complicated due to the influence of the static gravitational deflection. In this analysis, the nonlinearity is approximated by the power series up to the second order. The equations governing the amplitude and phase angle of the harmonic solution yx ¼ P cosðot þ bÞ, yy ¼ P sinðot þ bÞ are given approximately by Af Pb_ ¼ GP þ DP cos 2b þ 142 ð3P2 þ 2M 20 Þ sin b 142 P2 sin 3b M cosða bÞ, Af P_ ¼ coP þ DP sin 2b þ 12 ðP2 þ 2M 2 Þ cos b þ 12 P2 cos 3b þ M sinða bÞ, 4
0
4
ð5:3Þ
where Af ¼ (2ip)o40 and G(o) ¼ 1+ipo2o2. Putting the left-hand sides of this expression equal to zero, we can obtain the solutions for the steady-state response. The analytical results are shown in Figs. 6 and 7. Numerical results for a piecewise linear model are also illustrated in duplicate. Fig. 6 shows a response when the unbalance is comparatively large. Similar to the case of the vertical rotor mentioned above, the shape of
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Fig. 5. Harmonic resonance of a vertical rotor: (a) unbalance is located in the same side as the crack and (b) unbalance is located in the opposite side to the crack.
Fig. 6. Harmonic resonance of a horizontal rotor (large unbalance).
resonance curve depends remarkably on the direction of the unbalance. Fig. 7 shows a case that the unbalance is comparatively small. The unstable range always appears for any direction of the unbalance. This is because that the effect of the parametric excitation becomes predominant to the effect of the force excitation. 5.4. Superharmonic resonance of order 2 (horizontal rotor) [22] In Fig. 1, when the natural frequency pf and the rotational speed o have the relationship pf ¼ 2o, a superharmonic resonance with frequency 2o occurs. Resonance curves are shown in Fig. 8 where a theoretical result and a numerical result by a power series model (PS model) with the second order terms are illustrated. Since this kind of resonance occurs in a horizontal asymmetrical shaft with linear spring characteristics, it is
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Fig. 7. Harmonic resonance of a horizontal rotor (small unbalance).
Fig. 8. Superharmonic resonance [+2o].
easily understood that it occurs also in a horizontal cracked rotor. However, the characteristics of this resonance in a cracked rotor are different from that of an asymmetrical rotor. Unlike the case of a linear asymmetrical rotor, the amplitude changes remarkably depending on the angular position of the unbalance in a cracked rotor as shown in Fig. 8. The amplitude becomes maximum when the unbalance is located at the center of the crack and does minimum when it is located at the center of the uncracked area. 5.5. Superharmonic resonance of order 3 (horizontal rotor) A superharmonic resonance with frequency 3o occurs when the relationship pf ¼ 3o holds. This resonance occurs only in a horizontal rotor. A resonance curve is shown in Fig. 9 where a theoretical result, a numerical result by a piecewise linear model (PWL model) and that by a power series model (PS model) with the second power terms are illustrated. The shape of resonance curve is similar to that of a superharmonic resonance of order 2. However, different from the case of that of order 2, its shape is not influenced by the direction of the unbalance appreciably.
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Fig. 9. Superharmonic resonance [+3o].
Fig. 10. Subharmonic resonance [+(1/2)o].
5.6. Subharmonic resonance of order 1/2 (horizontal rotor) [23] In Fig. 1, when the relationship pf ¼ +(1/2)o, holds, a subharmonic resonance of a forward whirling mode with frequency +(1/2)o occurs. The resonance curve obtained by a theoretical analysis is compared with the numerical results using a piecewise liner (PWL) model and a power series (PS) model shown in Fig. 9. In this approximate analysis, the second and fourth power terms are considered. This resonance occurs only in a horizontal shaft. Since this resonance occurs due to the parametric excitation, it occurs even if the unbalance does not exists. Unlike the harmonic and superharmonic resonances, the resonance curve bifurcates from the trivial solution with zero amplitude. Although they are related to the parametric resonance, its amplitude does not increase infinitely but converges to finite amplitude due to nonlinearity. It is noted that the amplitude of this steady-state oscillation decreases as the unbalance increases. 5.7. Supersubharmonic resonance of order 3/2 (horizontal rotor) [23] A forward supersubharmonic resonance of order 3/2 occurs when the relationship pf ¼ +(3/2)o holds. In this approximate analysis, the second and fourth power terms are considered. The shape of this resonance curve shown in Fig. 11 is similar to that of subharmonic resonance shown in Fig. 10. It is noted that, among various kinds of supersubharmonic resonance of types [7(m/n)o], only this kind of resonance occurs.
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Fig. 11. Subharmonic resonance [+(3/2)o].
Fig. 12. Combination resonance [pf–pb].
5.8. Combination resonance (horizontal rotor) In Fig. 1(b), when the relationship pfpb ¼ o holds among the natural frequencies pf and pb and the rotational speed o, a combination resonance of the type [pfpb] occurs. This type of resonance does not occur the deflection model given in Fig. 1(a) since its resonance speed coincides with that of a subharmonic resonance [+(1/2)o]. The responses of each components whose frequencies are near pf and pb are shown in Fig. 12. The characteristic of this resonance is almost the same as that of the subharmonic resonance of order 1/2 (Fig. 10).
6. Nonstationary oscillations [24] The increase of vibrations during start up or shut down of a rotor system often becomes a clue to find a crack. In this section, nonstationary characteristics of a cracked rotor are explained. If a rotor is accelerated or decelerated with a constant acceleration, the angular position of the unbalance is given by C ¼ ð1=2Þlt2 þ os t þ C0 .
(6.1)
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6.1. Harmonic resonance (vertical rotor) Suppose that the rotor passes the harmonic resonance at the major critical speed shown in Fig. 4(a) in a vertical rotor. The equations of motion is obtained by putting the constant moment M0 ¼ 0 in Eq. (3.2). Fig. 13 shows curves of the amplitude variation obtained by numerical integrations for several different accelerations. Fig. 13(a) is a case that the unbalance is located in the same side as the crack. Small circles in the figure are the amplitudes of the steady-state solutions obtained by numerical integration. Since the rotor passes through the unstable range, the amplitude becomes very large when the angular acceleration is small. Fig. 13(b) is a case that the unbalance is located in the opposite side to the crack. Since an unstable range does not exist, the maximum amplitude is small for any value of angular acceleration. The relationship between the maximum amplitude rmax and the angular acceleration is shown in Fig. 14. This result shows that we can detect the occurrence of a crack from the increase of the maximum amplitude if the unbalance is located in the same side as the crack. However, it is difficult to detect it if the unbalance is located in the opposite side to the crack.
Fig. 13. Amplitude variation during passage through the major critical speed: (a) unbalance is located in the crack side and (b) unbalance is located in the opposite side.
Fig. 14. Maximum radius for angular acceleration (vertical rotor).
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Fig. 15. Comparison of relationships between the maximum amplitude and the angular acceleration: (a) major critical speed (small unbalance) and (b) various kinds of sub-critical speeds.
6.2. Harmonic resonance (horizontal rotor) In the case of a horizontal rotor shown in Fig. 4(b), various kinds of resonance occur due to a crack. The nonstationary vibrations are more complex than that of a vertical rotor. 6.2.1. Case of a large unbalance This is the case that the rotor passes the resonance range shown in Fig. 6. The phenomenon is qualitatively the same as the case of a vertical rotor mentioned above. Therefore, the occurrence of a crack can be detected only in the case in which the unbalance is located on the same side of the crack. 6.2.2. Case of a small unbalance This is the case that the rotor passes the resonance range shown in Fig. 7. As explained in Section 5.3, the characteristics of a parametrically excited system due to the breathing of a crack appear predominantly when the rotor is well balanced. As the result, an unstable range always exists regardless to the angular position of the unbalance. Correspondingly, as shown in Fig. 15 (left), the maximum amplitude becomes large for any angular position of the unbalance if the acceleration is small. This result suggests that, if the rotor is well balanced, the occurrence of the crack can be detected by observing the increase of the amplitude during passage through the major critical speed for any direction of the unbalance. Fig. 15 (right) shows the maximum amplitude rmax for the angular acceleration in various kinds of resonance caused by cracks. Among these characteristic curves, those of the subharmonic resonance [+(1/2)o] and the combination resonance change remarkably depending on the initial angular position of the unbalance [25,26]. Since the initial angular position is given at random, the maximum amplitude varies at every operation even the same direction of the unbalance and the same angular acceleration is given. For these two cases, the curves in Fig. 15 are written for certain initial conditions. Fig. 15 (right) shows that, for large acceleration, it becomes difficult to detect the subharmonic, supersubharmonic, and combination resonances and utilize them as the proof of the occurrence of a crack. 7. Conclusions In this article, vibration characteristics of the various kinds of resonances caused by cracks are explained with special reference to the nonlinear effect. Although the rotor models used in this study are very simple, they enable to understand the fundamental and most important natures of the cracked rotor. For the development of more sophisticated monitoring systems, more complicated models are required. However, in order to develop such a sophisticated monitoring system, this article provides the essential knowledge on the characteristic of phenomena.
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Cracked rotors: Industrial machine case histories and nonlinear effects shown by simple Jeffcott rotor Yukio Ishida Department of Electronic-Mechanical Engineering, Nagoya University Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan Available online 17 November 2007
Abstract Horizontal rotors are always imposed to periodic stresses and, therefore, a crack due to a fatigue is unavoidable. The proper diagnosis of machinery is necessary to prevent tragic accidents and the vibration monitoring is the most important tool for such a diagnosis system. In order to develop a monitoring system that can detect a crack in an early stage of propagation, it is important to know the vibration characteristics of a cracked rotor. A crack opens or closes due to the direction of the lateral deflection. Therefore, a cracked rotor has nonlinear spring characteristics of a piecewise linear type. In order to diagnose the vibration characteristics properly, it is essential to understand the behavior caused by the nonlinearity. These piecewise linear characteristics make a directional difference in stiffness and this difference rotates with the rotor. As the result, the coefficients of linear and nonlinear terms in restoring forces become time dependent. According to the physical characteristics, cracked rotors can be classified into a class of nonlinear parametrically excited system. At first, this article introduces case histories of cracks found in industrial machines. Secondly, it explains the vibration characteristics of various kinds of resonances due to cracks using simple Jeffcott rotor with nonlinear spring characteristics. The utilization of the nonstationary vibrations for monitoring system is also explained. r 2007 Elsevier Ltd. All rights reserved.
1. Introduction In a horizontal rotor where gravitational force works, the tension and compression work in the upper and lower sides of the rotor, respectively. Since these stresses change periodically as a rotor rotates, the rotor has high probability to be destroyed by a fatigue crack. In order to develop a vibration monitoring system that enables detecting a crack during the operation, vibration characteristics of cracked rotors have been investigated by many researchers. The accidents due to cracks in turbine generators have been reported since 1950s but theoretical and numerical investigations on the vibration characteristics of cracked rotors started from 1970s. The overview of the past studies on this field until 1989 is reviewed by Wauer [1]. When a horizontal rotor has a transverse crack, the crack area opens or closes due to the self-weight bending as it rotates. This is called breathing and this makes the characteristics of a cracked rotor nonlinear. This mechanism can be considered when a physical rotor model is constructed. In 1976, Gasch [2,3], Henry and Oka-Avae [4] considered this breathing nonlinear mechanism by using different flexibilities for open and closed condition and solved the equations of motion by E-mail address: [email protected] 0888-3270/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2007.11.005
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an analog computer. Many researchers investigated vibration characteristics of cracked rotors in detail qualitatively and quantitatively using various kinds of physical models. Since it is considered that the essential characteristics can be understood more easily by a simple physical model, the author clarified various kinds of resonance which may occur in cracked rotors using models composed of a massless elastic shaft and a disc mounted at the center of the shaft. This paper introduces case histories of cracked rotors in industrial machines and presents nonlinear effects on various kinds of resonance in cracked rotors using a simple rotor model similar to Jeffcott rotor. Even though the characteristics of this model cannot be considered rigorously as Jeffcott rotor model, for simplicity, this model is called as ‘‘Jeffcott rotor model’’ in this model. Stationary and nonstationary phenomena are both investigated. 2. Case history in industrial machines Turbine generators developed very rapidly during 1950s–1970s and their sizes increased from about 60 MW to about 500 MW in this period. Since many turbines are constructed without consideration of the experiences of the foregoing products sufficiently, turbine rotors suffered from cracks and many tragic accidents occurred in those days. Since 1950s, cracks have been found in turbine shafts or in turbine blades and some kind of tragic accidents which probably happened due to cracks have been reported. Generally, it is very difficult to specify the cause of the accidents after the rotor system was destroyed completely. In 1953, the 1800 rpm steam turbine at the Tanners Creek power station in USA suddenly went into vibration, and it was found that a segment of approximately 1601 had broken out in a wheel due to a crack [5]. In March 1954, a 3600 rpm generator rotor in Arizona, USA, burst while being balanced in the factory [6]. Several cracks were found before bursting in this rotor. In September 1954, a generator rotor in Cromby, USA, burst while running at 3780 rpm [6]. In December 1954, the LP steam turbine in Ridgeland, USA, burst during a routine over-speed trip test [7]. In 1954, a generator rotor in Pittsburg, USA, burst during an overrun test at 3920 rpm [8]. In 1970, a LP turbine burst during a running test in Nagasaki, Japan [9]. More than 60 people died and injured in this tragic accident [9]. The investigation team concluded that the cause of this collapse is cracks due to a stress concentration at grinded holes. There are many cases where cracks were found luckily before a disaster. Coyle and Watson [10] reported that, in 1956–1957, the shafts of three large turbine rotors developed fatigue cracks under very low nominal gravitational bending stresses in Castle Donnington, UK. The fatigue cracks propagated over 75% of the shaft section in two cases, and the cracking started all around the circumference and propagated no deeper than 1/16 inch in the other case. Yoshida [9] reported four cases that cracks are found in four steam turbine rotors during 1970–1971 in Japan. He concluded that the cyclic thermal stress which appeared during the frequent start up and shut down made cracks. Jack and Paterson [11] reported that three cracks were discovered in three LP shafts in 500 MW turbines at Ferrybridge power station during 1972–1974. Muszynska [12] reported that at least 28 cracked failures happened within the period 1970s. Laws [13] in Bently Nevada Co. reported four cracked rotor incidents happened in UK during 1971–1981. These incidents are a 40% transverse crack of 500 MW turbine generator, a circumferential 30 in long crack of MP rotor of 60 MW turbine generator, a 45% transverse crack of the main generator rotor of 660 MW turbine generator and a crack extending around complete circumference of the MP rotor of 350 MW turbine at main HP coupling end. Cracks of other kinds of machines are also reported. For example, a rotor crack which expanded of 1201 of the cross section of the shaft of a wind tunnel fan for a car testing was reported in 1990 in Japan [14]. In many cases, the occurrence of a crack is noted by an abnormal increase of vibrations. However, the development of vibration diagnosis system, which was installed to monitor the turbines started to reveal the detailed symptoms of cracks. Dimarogonas and Paradopoulos reported a crack failure found in 300 MW turbine generator in 1983 in Lavrion power plant in Greek [15]. He reported vibration spectrum where characteristic peaks are admitted. From 1970s, cracks were found in turbines of several atomic power plants. This led to a much more serious situation for the people. Ziebarth and Baumgartner [16] reported the vibration histories after cracks occurred in the 1300 MW turbine in Cumberland power plant in USA and in Wuergassen nuclear power plant in Germany. In the former, an indication appeared and started to increase before 3 days prior to the shutdown. A double-frequency excitation was noted upon the coastdown. In the latter, the horizontal and vertical
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vibrations of the LP turbine started to increase from 6 days prior to the shutdown. A pump shaft in Crystal River nuclear power plant broke due to a fatigue crack in a groove in 1986 and also due to crack in another part in 1989 [17]. In 1990, a crack distributed 25% of the cross section was found in a generator rotor of the 935 MW turbine generator of Darlington nuclear power plant. Sanderson [18] reported the characteristic of the data obtained by the monitoring system in detail. It shows that the overall vibration started to increase about 5 days before the shutdown and the amplitudes of the harmonic resonance and the double-frequency resonance increased due to the occurrence of a crack. It can be imagined that there are many other cases, which are not reported to the public. Besides a turbine shaft or a generator shaft, there are many reports on the crack found in other parts such as turbine blades. Except the following important case, other all incidents are omitted in this paper. In 1889, a ring of the recurrent pump in Fukushima atomic power station in Japan broke due to cracks made by a resonance. 3. Modeling and equations of motion 3.1. Physical models of a cracked rotor When a disk is mounted on a massless elastic shaft, the deflection motion and the inclination motion of the rotor are generally coupled. However, if the disk is mounted at the center of the shaft, the system is separated into two independent 2DOF rotor systems that execute only a deflection motion or an inclination motion as shown in Fig. 1 [19]. The former is the so-called ‘‘Jeffcott rotor’’ and its natural frequencies are constant and independent of the rotational speed. Since a forward resonance and a backward resonance of the same kind occur in the same rotational speed, this model is not suitable for the analysis of nonlinear vibrations where a mutual interaction among components often occur through the nonlinearity. Cracked rotors are typical nonlinear systems as mentioned below, the latter model is mainly used in this analysis. Fig. 2 shows an inclination model with a transverse crack. The z-axis of the static rectangular coordinate system O-xyz coincides with the bearing centerline. The inclination angle of the elastic shaft at the disk position is expressed by y and its projections to the xz and yz planes are expressed by yx and yy, respectively. It is assumed that the crack developed in the area of half of the shaft cross section. We consider the rotating coordinate system O-x0 y0 z0 in which x0 -axis coincides with the boundary of the crack. When y0y 40, the crack opens and the shaft stiffness becomes small. On the contrary, when y0y o0, the crack closes and the shaft stiffness becomes large. As a result, the restoring moment has piecewise spring characteristics and its
Fig. 1. Natural frequency diagrams of a deflection model and an inclination model: (a) deflection model and (b) inclination model.
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Fig. 2. Inclination model with a transverse crack.
Fig. 3. Spring characteristics of a cracked rotor.
components M 0yx and M 0yy in the x0 z and y0 z planes, respectively, are shown in Fig. 3. These components are represented by M 0yx ¼ d1 y0x , M 0yy ¼ ðd2 Dd2 Þy0y ðy0y 40Þ, M 0yy ¼ ðd2 þ Dd2 Þy0y ðy0y o0Þ.
ð3:1Þ
3.2. Equations of motion Let the ratio of the polar moment of inertia to the diametric moment of inertia of the rotor be ip, the rotating speed o, the damping coefficient c, and the magnitude and the phase angle of the dynamic unbalance t and a, respectively. Corresponding to the gravitational force working in the experimental setup, we consider the constant moment M0 in the yy-axis. Transferring Eq. (3.1) into the expression in the stationary coordinate system and connecting them with the equations of motion for the nonstationary rotor with no crack [20], we obtain the following equations of motion: € y þ ip c _ y_ y þ cy_ x þ ð1 D2 Þyx þ ðD1 D2 Þðyx C 2 þ yy S 2 Þ y€ x þ ip cy 2 € sinðc þ aÞg, ¼ ð1 ip Þtfc_ cosðc þ aÞ þ c € x ip c _ y_ x þ cy_ y þ ð1 D2 Þyy þ ðD1 D2 Þðyx S 2 yy C 2 Þ y€ y ip cy 2 ¼ ð1 ip Þtfc_ sinðc þ aÞ c€ cosðc þ aÞg þ M 0 ,
ð3:2Þ
where M ¼ (1ip)to2, d ¼ (d1+d2)/2, D1 ¼ (d1d2)/2d, D2 ¼ (Dd2/2d). Concerning the symbol ‘‘7’’ in these equations, we use all the upper signs for y0y 40 and all the lower signs for y0y o0.
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These equations of motion possess the following dynamical characteristics: (a) Time dependent coefficients similar to an asymmetrical rotor (parametrically excited system). (b) Rotating nonlinearity of a piecewise linear type (nonlinear system). (c) Excitation by the unbalance (forced oscillation system). The corresponding equations of motion of the 2DOF deflection system are obtained by putting the terms corresponding gyroscopic moment in Eq. (3.2). 4. Numerical integration (PWL model) This section overviews various kinds of resonances, which appear in the steady state condition. When the _ ¼ o is constant, the angular position of the rotor is given by rotational speed C C ¼ ot þ C0 .
(4.1)
Figs. 4(a) and (b) show resonances which are obtained by integrating piecewise linear model (PWL model) given by Eq. (3.2) numerically with this condition. These are the cases in which the unbalance is located in the direction at the center on the crack side. It is known that an unstable range appears at the major critical speed when an unbalance is located in the same side as the crack and disappears when it is located in the opposite side of the crack [3,20]. Fig. 4(a) shows a case of a vertical rotor. Only a harmonic resonance [+o] occurs at the major critical speed. Fig. 4(b) shows a case of a horizontal rotor. In addition to a harmonic resonance, a backward harmonic resonance [o], superharmonic resonances [+2o], [+3o], a subharmonic resonance [+(1/2)o], a supersubharmonic resonance [+(3/2)o], and a combination resonance [pfpb] occur. This is because the system becomes to have more complex nonlinear spring characteristics and more complex parametric excitation characteristics due to the shift of the equilibrium position under the action of gravity. Among these
Fig. 4. Resonance curves of a cracked rotor (numerical results): (a) vertical rotor (M0 ¼ 0) and (b) horizontal rotor.
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resonances, the harmonic resonance [+o] changes its characteristics remarkably depending on the direction of the unbalance. 5. Steady-state oscillations (PS-model) The piecewise linear expression of Eq. (3.1) is inconvenient for the theoretical treatment because we have to use different expressions for the restoring moment depending on the direction of the deflection. Therefore, the spring characteristics are approximated by a power series for convenience (PS-model). In this section, theoretical results for various kinds of resonances are introduced. 5.1. Power series approximation of spring characteristics Let assume that piecewise linear spring characteristics are approximated to the following power series: M 0yx ¼ d0x y0x , 2
3
4
M 0yy ¼ d0t y0y þ 2 y0 y þ b3 y0 y þ 4 y0 y þ .
ð5:1Þ
By the transformation of coordinates, we obtain the following expressions for the restoring moment in the static coordinates: 2 M yx ¼ yx þ Dðyx C 2 þ yy S2 Þ þ fð3S 1 þ S3 Þy2x þ 2ðC 1 C 3 Þyx yy ðS 1 þ S 3 Þy2y g þ , 4 2 M yy ¼ yy þ Dðyx S 2 yy C 2 Þ þ fðC 1 C 3 Þy2y 2ðS1 þ S3 Þyx yy þ ð3C 1 þ C 3 Þy2y g þ , ð5:2Þ 4 where only the terms up to the second order are expressed explicitly. The notations C n ¼ cos not and Sn ¼ sin not are used for simplicity. 5.2. Harmonic resonance (vertical rotor) [20] If the rotor is supported vertically, the static deflection is not caused by gravity. The response at the major critical speed depends on whether the unbalance is located on the same side or the side opposite to the crack. The theoretical response curves (left) are compared with the experimental results (right) in Fig. 5. In this theoretical analysis, the deflection model (ip ¼ 0) is used and the nonlinearity is approximated by the power series up to the fourth order. In these figures, both amplitudes and phase angles are illustrated. As already known well, the response curves change remarkably depending on the direction of the unbalance. When the unbalance is located in the same side as the crack, the resonance curves spread in the large amplitude region and, as a result, an unstable range appears. On the other hand, when the unbalance is located in the opposite side to the crack, the unstable range disappears. 5.3. Harmonic resonance (horizontal rotor) [21] If the rotor is supported horizontally, the harmonic resonance becomes more complicated due to the influence of the static gravitational deflection. In this analysis, the nonlinearity is approximated by the power series up to the second order. The equations governing the amplitude and phase angle of the harmonic solution yx ¼ P cosðot þ bÞ, yy ¼ P sinðot þ bÞ are given approximately by Af Pb_ ¼ GP þ DP cos 2b þ 142 ð3P2 þ 2M 20 Þ sin b 142 P2 sin 3b M cosða bÞ, Af P_ ¼ coP þ DP sin 2b þ 12 ðP2 þ 2M 2 Þ cos b þ 12 P2 cos 3b þ M sinða bÞ, 4
0
4
ð5:3Þ
where Af ¼ (2ip)o40 and G(o) ¼ 1+ipo2o2. Putting the left-hand sides of this expression equal to zero, we can obtain the solutions for the steady-state response. The analytical results are shown in Figs. 6 and 7. Numerical results for a piecewise linear model are also illustrated in duplicate. Fig. 6 shows a response when the unbalance is comparatively large. Similar to the case of the vertical rotor mentioned above, the shape of
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Fig. 5. Harmonic resonance of a vertical rotor: (a) unbalance is located in the same side as the crack and (b) unbalance is located in the opposite side to the crack.
Fig. 6. Harmonic resonance of a horizontal rotor (large unbalance).
resonance curve depends remarkably on the direction of the unbalance. Fig. 7 shows a case that the unbalance is comparatively small. The unstable range always appears for any direction of the unbalance. This is because that the effect of the parametric excitation becomes predominant to the effect of the force excitation. 5.4. Superharmonic resonance of order 2 (horizontal rotor) [22] In Fig. 1, when the natural frequency pf and the rotational speed o have the relationship pf ¼ 2o, a superharmonic resonance with frequency 2o occurs. Resonance curves are shown in Fig. 8 where a theoretical result and a numerical result by a power series model (PS model) with the second order terms are illustrated. Since this kind of resonance occurs in a horizontal asymmetrical shaft with linear spring characteristics, it is
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Fig. 7. Harmonic resonance of a horizontal rotor (small unbalance).
Fig. 8. Superharmonic resonance [+2o].
easily understood that it occurs also in a horizontal cracked rotor. However, the characteristics of this resonance in a cracked rotor are different from that of an asymmetrical rotor. Unlike the case of a linear asymmetrical rotor, the amplitude changes remarkably depending on the angular position of the unbalance in a cracked rotor as shown in Fig. 8. The amplitude becomes maximum when the unbalance is located at the center of the crack and does minimum when it is located at the center of the uncracked area. 5.5. Superharmonic resonance of order 3 (horizontal rotor) A superharmonic resonance with frequency 3o occurs when the relationship pf ¼ 3o holds. This resonance occurs only in a horizontal rotor. A resonance curve is shown in Fig. 9 where a theoretical result, a numerical result by a piecewise linear model (PWL model) and that by a power series model (PS model) with the second power terms are illustrated. The shape of resonance curve is similar to that of a superharmonic resonance of order 2. However, different from the case of that of order 2, its shape is not influenced by the direction of the unbalance appreciably.
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Fig. 9. Superharmonic resonance [+3o].
Fig. 10. Subharmonic resonance [+(1/2)o].
5.6. Subharmonic resonance of order 1/2 (horizontal rotor) [23] In Fig. 1, when the relationship pf ¼ +(1/2)o, holds, a subharmonic resonance of a forward whirling mode with frequency +(1/2)o occurs. The resonance curve obtained by a theoretical analysis is compared with the numerical results using a piecewise liner (PWL) model and a power series (PS) model shown in Fig. 9. In this approximate analysis, the second and fourth power terms are considered. This resonance occurs only in a horizontal shaft. Since this resonance occurs due to the parametric excitation, it occurs even if the unbalance does not exists. Unlike the harmonic and superharmonic resonances, the resonance curve bifurcates from the trivial solution with zero amplitude. Although they are related to the parametric resonance, its amplitude does not increase infinitely but converges to finite amplitude due to nonlinearity. It is noted that the amplitude of this steady-state oscillation decreases as the unbalance increases. 5.7. Supersubharmonic resonance of order 3/2 (horizontal rotor) [23] A forward supersubharmonic resonance of order 3/2 occurs when the relationship pf ¼ +(3/2)o holds. In this approximate analysis, the second and fourth power terms are considered. The shape of this resonance curve shown in Fig. 11 is similar to that of subharmonic resonance shown in Fig. 10. It is noted that, among various kinds of supersubharmonic resonance of types [7(m/n)o], only this kind of resonance occurs.
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Fig. 11. Subharmonic resonance [+(3/2)o].
Fig. 12. Combination resonance [pf–pb].
5.8. Combination resonance (horizontal rotor) In Fig. 1(b), when the relationship pfpb ¼ o holds among the natural frequencies pf and pb and the rotational speed o, a combination resonance of the type [pfpb] occurs. This type of resonance does not occur the deflection model given in Fig. 1(a) since its resonance speed coincides with that of a subharmonic resonance [+(1/2)o]. The responses of each components whose frequencies are near pf and pb are shown in Fig. 12. The characteristic of this resonance is almost the same as that of the subharmonic resonance of order 1/2 (Fig. 10).
6. Nonstationary oscillations [24] The increase of vibrations during start up or shut down of a rotor system often becomes a clue to find a crack. In this section, nonstationary characteristics of a cracked rotor are explained. If a rotor is accelerated or decelerated with a constant acceleration, the angular position of the unbalance is given by C ¼ ð1=2Þlt2 þ os t þ C0 .
(6.1)
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6.1. Harmonic resonance (vertical rotor) Suppose that the rotor passes the harmonic resonance at the major critical speed shown in Fig. 4(a) in a vertical rotor. The equations of motion is obtained by putting the constant moment M0 ¼ 0 in Eq. (3.2). Fig. 13 shows curves of the amplitude variation obtained by numerical integrations for several different accelerations. Fig. 13(a) is a case that the unbalance is located in the same side as the crack. Small circles in the figure are the amplitudes of the steady-state solutions obtained by numerical integration. Since the rotor passes through the unstable range, the amplitude becomes very large when the angular acceleration is small. Fig. 13(b) is a case that the unbalance is located in the opposite side to the crack. Since an unstable range does not exist, the maximum amplitude is small for any value of angular acceleration. The relationship between the maximum amplitude rmax and the angular acceleration is shown in Fig. 14. This result shows that we can detect the occurrence of a crack from the increase of the maximum amplitude if the unbalance is located in the same side as the crack. However, it is difficult to detect it if the unbalance is located in the opposite side to the crack.
Fig. 13. Amplitude variation during passage through the major critical speed: (a) unbalance is located in the crack side and (b) unbalance is located in the opposite side.
Fig. 14. Maximum radius for angular acceleration (vertical rotor).
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Fig. 15. Comparison of relationships between the maximum amplitude and the angular acceleration: (a) major critical speed (small unbalance) and (b) various kinds of sub-critical speeds.
6.2. Harmonic resonance (horizontal rotor) In the case of a horizontal rotor shown in Fig. 4(b), various kinds of resonance occur due to a crack. The nonstationary vibrations are more complex than that of a vertical rotor. 6.2.1. Case of a large unbalance This is the case that the rotor passes the resonance range shown in Fig. 6. The phenomenon is qualitatively the same as the case of a vertical rotor mentioned above. Therefore, the occurrence of a crack can be detected only in the case in which the unbalance is located on the same side of the crack. 6.2.2. Case of a small unbalance This is the case that the rotor passes the resonance range shown in Fig. 7. As explained in Section 5.3, the characteristics of a parametrically excited system due to the breathing of a crack appear predominantly when the rotor is well balanced. As the result, an unstable range always exists regardless to the angular position of the unbalance. Correspondingly, as shown in Fig. 15 (left), the maximum amplitude becomes large for any angular position of the unbalance if the acceleration is small. This result suggests that, if the rotor is well balanced, the occurrence of the crack can be detected by observing the increase of the amplitude during passage through the major critical speed for any direction of the unbalance. Fig. 15 (right) shows the maximum amplitude rmax for the angular acceleration in various kinds of resonance caused by cracks. Among these characteristic curves, those of the subharmonic resonance [+(1/2)o] and the combination resonance change remarkably depending on the initial angular position of the unbalance [25,26]. Since the initial angular position is given at random, the maximum amplitude varies at every operation even the same direction of the unbalance and the same angular acceleration is given. For these two cases, the curves in Fig. 15 are written for certain initial conditions. Fig. 15 (right) shows that, for large acceleration, it becomes difficult to detect the subharmonic, supersubharmonic, and combination resonances and utilize them as the proof of the occurrence of a crack. 7. Conclusions In this article, vibration characteristics of the various kinds of resonances caused by cracks are explained with special reference to the nonlinear effect. Although the rotor models used in this study are very simple, they enable to understand the fundamental and most important natures of the cracked rotor. For the development of more sophisticated monitoring systems, more complicated models are required. However, in order to develop such a sophisticated monitoring system, this article provides the essential knowledge on the characteristic of phenomena.
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