The influence of crack breathing and imbalance orientation angle on the characteristics of the critical speed of a cracked rotor

Journal of Sound and Vibration 330 (2011) 2031–2048

Contents lists available at ScienceDirect

Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

The influence of crack breathing and imbalance orientation angle on the characteristics of the critical speed of a cracked rotor Li Cheng a,b,n, Ning Li b, Xue-Feng Chen a, Zheng-Jia He a a b

State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China Air Force Engineering University, Xi’an 710038, China

a r t i c l e in f o

abstract

Article history: Received 3 March 2010 Received in revised form 12 November 2010 Accepted 12 November 2010 Handling Editor: L.N. Virgin Available online 8 December 2010

By analyzing the limitations of weight dominance and by taking the complicated whirl of the rotor into account, general equations of motion have been developed in case of a Jeffcott rotor with a transverse crack. The angle between the crack direction and the shaft deformation direction is used to determine the closing and opening of the crack, allowing one to study the dynamic response without assuming weight dominance. Using the new equations, the dynamic response of a cracked rotor near its critical speed has been computed via a numerical method to investigate the influence of nonlinear breathing of the crack and that of the imbalance orientation angle b on the stability, critical speed and peak response of the rotor. The results show that nonlinear breathing can improve the stability of a rotor in contrast to a rotor with an open crack, and, with a reversed imbalance (701 o b o 2701), that it can reduce the vibration response in contrast to an uncracked rotor. The basic characteristics of a cracked rotor near its critical speed are similar to those of an uncracked rotor. The critical speed can be determined by measuring the rotation of the center of gravity. The critical speed of a cracked rotor is located between the natural frequencies of the fully open crack and those of the fully closed crack and depends on the imbalance orientation angle. Its value is lowest at b E901 and highest at b E2701. The peak in the response at the critical speed is mainly determined by the imbalance orientation angle. At b E01 and 1801, the peak corresponds to the maximum and minimum response, respectively. & 2010 Elsevier Ltd. All rights reserved.

1. Introduction Rotating machines, such as high-speed compressors, steam and gas turbines, generators, and pumps, are widely applied in many fields. Fatigue cracking of rotor shafts is an important phenomenon that can lead to catastrophic failure and great economic loss if not detected in time. Cracked rotors have been the focus of investigations since the 1970s. Important progress has been made in the last 30 years in detecting cracks and in stopping cracked shafts before catastrophic failure. Nearly 1000 papers have been published on various topics related to cracks in rotating shafts, with comprehensive reviews by Gasch [1] and Dimarogonas [2]. More recent studies have been reviewed by Papadopoulos [3], Ishida [4] and Bachschmid et al. [5]. Two issues related to the critical speed of a cracked rotor have been emphasized. One of these is the stability of a cracked rotor near its critical speed and that at subcritical speeds [1–15]. The other is the transient response when the rotor passes through its critical speed or a subcritical speed [16–26].

n

Corresponding author at: State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China. E-mail address: [email protected] (L. Cheng).

0022-460X/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2010.11.012

2032

L. Cheng et al. / Journal of Sound and Vibration 330 (2011) 2031–2048

Nomenclature a c C e E= e/dst fbreathing fky fkz fkx fkZ Fbreathing Feei(y + b) Fk Fs g k kxx(c) kxZ(c) KHC KOP KUC m M o o0 o  xyz

depth of the transverse crack external damping ratio damping matrix (including the gyroscopic effect) eccentricity of the disk dimensionless eccentricity breathing function of the crack projection of the elastic force in y direction projection of the elastic force in z direction projection of the elastic force in x direction projection of the elastic force in Z direction breathing function of the crack centrifugal force vector of the unbalanced mass elastic force vector due to shaft deformation weight force vector acceleration due to gravity stiffness of the non-cracked shaft stiffness of the cracked shaft in deformation direction cross-coupled stiffness in deformation perpendicular direction stiffness matrix of the half-open half-closed crack condition stiffness matrix of the fully open crack condition stiffness matrix of the uncracked condition mass of the disk mass matrix center of the bearing center of the disk stationary coordinates

ox ouxu o  xZ

shaft deformation direction crack direction rotational coordinates based on the shaft deformation direction rotates with whirling angle speed oxuZu the rotational coordinates based on the crack direction rotates with rotating speed R radius of the shaft t time T period of rotor rotating u displacement vector u_ velocity vector u€ acceleration vector u= oo0 displacement of the rotor y ¼ y=dst dimensionless deflection z ¼ z=dst dimensionless deflection b imbalance orientation angle dst = mg/k static weight deflection Dk relative reduction in shaft stiffness in the crack direction Dkox(c) stiffness change of the cracked shaft in x direction DkoZ(c) stiffness change of the cracked shaft in Z direction f whirling angle of the rotor center m Floquet multiplier y rotating angle of the rotor t = oct dimensionless time o pffiffiffiffiffiffiffiffiffi rotating speed ffi oc ¼ k=m critical speed of non-cracked rotor X ¼ o=oc dimensionless rotating speed c = y  f crack reference angle z = c/2moc dimensionless damping factor

Rotor instability can be induced by changes in various parameters. Just as the location, size, effective bending and shear stiffness of damaged regions affect the stability of a rotor, the location and breathing of a crack are also significant parameters that impact the dynamic instability of a system. Papadopoulos and Dimarogonas [6] showed that a surface crack in a rotating shaft can yield a variety of unstable areas of operation owing to the coupling of the lateral and longitudinal vibrations. Huang et al. [7] found, via the Floquet theory, that instability occurs in an undamped shaft when the rotation frequency is close to an integer fraction or an integer multiple of the bending frequency of the shaft if the crack depth values reaches half of the radius. Meng and Gasch [8] investigated the stability and the stability degree of a flexible cracked rotor supported on journal bearings, and found that the larger the speed ratio and the stiffness change ratio are, the wider the crack ridge zone is. Taking into account the various parameters of the crack, the internal damping of the shaft and the geometric parameters, Sekhar and Dey [9] studied the stability threshold of a rotor-bearing system having a transverse crack. These authors reported that the speed instability decreases considerably with increasing crack depth and is influenced more by hysteretic damping than by viscous damping. Zhu et al. [11] analyzed theoretically the dynamic characteristics of a cracked rotor with an active magnetic bearing. They showed that if the effect of a crack is not taken into account in the design of the controller, the rotor-bearing system will lose its stability in some cases. Chen et al. [13] investigated numerically the effects of a crack on the dynamic stability of a rotor system with asymmetric viscoelastic supports. Sinou [14] reported that the areas of instability increase considerably as the crack deepens, and that the crack’s position and depth are the main factors affecting not only the nonlinear behavior of the rotor system but also the various zones of dynamic instability in the periodic solution of the cracked rotor. As reported by Jun and Gadala [15], there exists some speed range near the critical speed, where temporary whirl direction reversal and phase shift exist. When an unbalance is applied, the peculiar features, such as the whirl direction reversal and phase shift, disappear. A crack reduces the critical speed of the rotor system and leads to changes in the amplitude of the first-, second- and thirdorder harmonic vibrations. Plaut et al. [16] used Galerkin’s method and numerical integration of the resulting bilinear equations to obtain approximate time histories during the passage of a cracked shaft through its critical speed. By comparing the results of a breathing crack with those of a crack that is always open and those of an uncracked shaft, they observed, surprisingly, that the peak response may sometimes be smaller with a cracked shaft than it is with an uncracked shaft. According to Sekhar and Prabhu [17], the transient vibration response of a cracked rotor develops oscillations near the critical

L. Cheng et al. / Journal of Sound and Vibration 330 (2011) 2031–2048

2033

speed, and, for deeper cracks, the vibration is violent and there is no definite critical speed but there is a zone with severe vibration. Prabhakar et al. [18] analyzed the vibration of a rotor with a slanted crack as the rotor passed through its flexural critical speed and found that the subharmonic frequency and superharmonic frequency components in an interval of frequency corresponding to the torsional frequency were centered on the critical speed of the cracked rotor system. Sawicki et al. [20,21] used the angle between the crack centerline and the shaft vibration (whirl) vector to determine the breathing of the crack, and investigated analytically and numerically the dynamic response of a cracked Jeffcott rotor as it passed through the critical speed at a constant acceleration, with or without the assumption of dominance by the rotor weight. Darpa et al. [22] analyzed a cracked Jeffcott rotor past through its critical speed and the subharmonic resonances. They observed that the breathing behavior and the peak response were strongly influenced by the imbalance orientation angle relative to the crack direction, and that during passage through the subharmonic resonances the orientation of the rotor orbit changed rather noticeably. They suggested that the variations in the peak response typical of slotted and cracked rotors can be helpful in distinguishing asymmetric rotors from cracked rotors. Go´mez-Mancilla et al. [23] emphasized local resonances and orbital evolution around one-half, one-third and one-quarter of the critical speed of various crack–imbalance orientations and concluded that the orbital evolution around that around one-half and one-third of the first resonance can be used to detect cracks in a rotor, even if the crack–imbalance orientation is unknown. Sinou and Lees [24,25] investigated the evolution of the orbit of a cracked rotor near half and that near one-third of the first critical speed. These authors studied the influence of crack–imbalance interactions and, more particularly, that of the relative orientation between the crack front and the imbalance, with consideration of various crack depths and imbalance magnitudes. Many investigators have highlighted the fact that it is easier to detect cracks during passage through the critical speed than it is at a steady speed. Yang et al. [10,12] applied a wavelet-based method and the consideration of instantaneous frequencies to the interpretation and characterization of bifurcations and to the evolution of instabilities induced by a transverse crack in a rotary system. They constructed a dynamic failure curve, differentiating zones of stability and bifurcation instability from zones of dynamic failure. Sekhar [19] found subharmonic resonant peaks using a continuous wavelet transform when a cracked rotor passed through its critical speed. These peaks were not apparent in either the frequency spectrum or in the time response. The continuous wavelet transform is more powerful than the time response for detecting cracks at high accelerations and at low crack depths. Ramesh Babu et al. [26] applied a Hilbert–Huang transform to the transient response of a cracked rotor and observed a few interesting results. In all cases it was found that the Hilbert–Huang transform appeared to be a better tool than either the fast Fourier transform or the continuous wavelet transform for detection of cracks in a rotor using the transient response. After the pioneering work of Gasch [1], Dimarogonas [2] and so on, almost all studies [1–3,6–19,23–26,29] on cracked rotors with the assumption of weight dominance have identified little change in the form of the equations of motion of cracked rotors. With weight dominance, the crack-breathing behavior in the equations of motion has been expressed mostly in three forms [1–3]. In the first form [1–3,8,10–12,16,17,19,23], regardless of the number of degrees of freedom, the crack breathing is represented by a periodic time-varying system, and the governing equations of the cracked rotor are bilinear equations, as expressed in Eqs. (12) and (14) below. In the second form [14,24,25], a breathing function is multiplied by the stiffness matrix of the shaft directly, as seen in Eq. (15) below. In the third form, [6,9,13,18,26], the variation in the stiffness of the cracked shaft is expressed by a truncated cosine series, as seen in Eq. (17) below. Although the dynamic response of a cracked rotor has been studied without the assumption of weight dominance in some papers [20,21], the equations of motion remained bilinear equations. When a cracked rotor rotates near its critical speed, and also when it passes through the critical speed, the vibration (whirl) of the shaft is extremely complicated and is not synchronized with the rotation of the shaft; the dynamic response of the rotor increases rapidly and stability may even be lost. Although the critical speed of a cracked rotor has been addressed in many studies, the equations of motion for a cracked rotor that are used at present, derived with the assumption of weight dominance, are not suitable for studying the vibrations near the critical speed, and hence the results obtained from these equations are not reliable and the basic characteristics of a cracked rotor near its critical speed are still not clear. Therefore, the critical speed, one of the most important dynamic characteristics of a cracked rotor, deserves further research, especially regarding the influence of nonlinear crack breathing on it. The paper is organized as follows. The general equations of motion for a Jeffcott rotor with a transverse crack are developed in Section 2. The angle between the crack direction and the shaft deformation direction is used to determine the closing and opening of the crack, allowing us to study the dynamic response without assuming weight dominance. Using the new equations derived here, the influence of nonlinear breathing of the crack and that of the imbalance orientation angle on the stability, critical speed and peak response of the rotor are investigated via a numerical method in Section 3. Finally, Section 4 presents our conclusions from this work. 2. Equations of motion of a cracked rotor 2.1. Limitations of weight dominance Fig. 1 shows a Jeffcott rotor with a transverse crack. The stationary coordinates are o–xyz, where the origin o is the intersection of the plane of the disk and the line between the bearing centers; o0 is the center of the rotor; R is the radius of the shaft; a is the transverse depth of the crack; y is the rotation angle; and f is the whirling angle. c = y  f is the angle between the direction of the shaft deformation ox and the crack direction o0 x0 , called the crack reference angle. b is the angle between

2034

L. Cheng et al. / Journal of Sound and Vibration 330 (2011) 2031–2048

η′

η

o o

z

ξ′

ω a

x

z

o′

oc

R

′

mg ξ ′

y

y Fig. 1. Geometry of a crack in a Jeffcott rotor using stationary and rotational coordinates.

the imbalance eccentricity o0 oc and the crack direction o0 x0 , called the imbalance orientation angle. The rotational coordinates o–xZ are based on the shaft deformation direction, which rotates at the speed of the whirling angle, and the rotational coordinates o–x0 Z0 are based on the crack direction, which rotates at the speed o of the rotation angle. The bilinear equations of motion for a rotor with a harmonically breathing crack can be obtained from Eqs. (1), (19) and (44) in Gasch’s paper [1], as follows: 

m

( )  c y€ þ m z€

  (  y_ k þ c z_

 k

" fbreathing ðotÞDk

cos2 ðotÞ

sinðotÞcosðotÞ

sinðotÞcosðotÞ

sin2 ðotÞ

fbreathing ðotÞ ¼

  1 þ cosðotÞ 2

#)  y z

 ¼

mg 0



( þ emo2

cosðot þ bÞ sinðot þ bÞ

)

(1)

The steady response of the cracked rotor around its critical speed O = o/oc = 1.00 (where oc =O(k/m)) was calculated from Eq. (1) with Dk/k= 0.05, b =01, ek/mg= 0.10 and c/(2O(mk)) =0.01, as shown in Fig. 2. The solid curves denote the orbits of the movement of the rotor center o0 , and the position of the origin o is shown by a square in each part of the figure. The arrows indicate the instantaneous crack direction, and their lengths show the instantaneous breathing state. A longer arrow means that the crack is open to a greater extent at that position. The crack has its maximum opening at the point A and its minimum opening at the point B. The assumption of weight dominance is found to work when the speed is far from the critical speed (as seen in Fig. 2(a), where O = 0.92, and in Fig. 2(d), where O = 1.06). But when the speed is near the critical speed (as seen in Fig. 2(b), where O = 0.973, and in Fig. 2(c), where O = 1.01), the breathing state of the crack contradicts this assumption of weight dominance. (At point B of the orbit in Fig. 2(b), the crack is shown as closed, but should be open when the crack direction is along ox in Fig. 1(b). At point C in Fig. 2(c), the crack is shown as partly open, but should be closed when the crack direction is along xo in Fig. 1(b)). So, weight dominance does not work. In fact, the assumption of weight dominance is only suitable for heavy rotors with low critical speeds and low imbalance. This is very commonly the case with large turbine-generator rotors. Fig. 3(a) shows the ‘‘breathing’’ of a crack under the influence of weight when the shaft is turned slowly [1]. When the rotor center o0 is at o1u, the crack is fully open; when the rotor center is at o2u and at o4u , the crack is partially open; and when the rotor center is at o3u, the crack is fully closed. But in light or vertical rotors, the vibration response governs the breathing mechanism, and the opening and closing of a crack is a function of this vibration. Almost all rotors are outside the range of applicability of weight dominance near their critical speed, and the lower damping expands the range within which dynamic behavior dominates [1]; an example is shown in Fig. 3(b). Therefore equations of motion like Eq. (1) are not suitable for analysis near the critical speed.

2.2. The model of a breathing crack In a Jeffcott rotor with a transverse crack, the forces applied to the disk consist of inertial forces (caused by relative acceleration, transport acceleration and Coriolis acceleration), damping forces, weight forces and elastic forces, in which only the elastic force exerted by the shaft is related to the crack. The elastic force of a cracked shaft can be calculated from various breathing-crack models, as reported in literatures. It has been proven that the breathing behavior of a crack in a rotating shaft is quite complicated. As the shaft whirls, if the crack lies completely in the compression zone of the shaft it is fully closed (Fig. 4(a)), and if the crack lies completely in the tension zone, it is fully open (Fig. 4(b)). However, if the crack lies in another zone then it may be partially open (Fig. 4(c)). Chasalevris and Papadopoulos [28,29] developed the strain energy release rate method to estimate, for the first time, the local compliance due to the crack as a function of both the crack depth and the angle of rotation, integrating over the angle of rotation (see Fig. 5).

L. Cheng et al. / Journal of Sound and Vibration 330 (2011) 2031–2048

1 0.5

1

Ω=0.92

0.5

B

B

-0.5 -1 -Y

-1 -Y

Ω=0.973

0

0 -0.5

-1.5

-1.5

A

-2

-2

A

-2.5

-2.5 -3

-3

-3.5

-3.5 -4

-4 -2

-1

0

1

2

-2

-1

Z

1

Ω=1.01

0

1

2

1

2

Z

1

A

0.5

C

Ω=1.06

0.5

0

0

-0.5

-0.5

-1

-1 -Y

-Y

2035

-1.5

A

-1.5

B

-2

-2

-2.5

-2.5

B

-3

-3 -3.5

-3.5 -4

-4 -2

-1

0

1

2

-2

-1

Z

0 Z

Fig. 2. The steady response of the cracked rotor about its critical speed with the bilinear equations of motion and the harmonic crack model.

o

o′3

z o

ψ

o′4

ω

o′3 o′4

o′2

or Δu

ω

or

z Δu o 1′

o′2

whirling

y

whirling

o′1

y Fig. 3. (a) With weight dominance and (b) without weight dominance.

By comparing a 3D nonlinear finite element model, the strain energy release rate model and a simplified linear model, Bachschmid et al. [5,27] showed that the breathing mechanism in rotating shafts can be accurately reproduced by 3D nonlinear finite element models and that a simple approximate model can also accurately simulate the breathing behavior. Therefore, the elastic force of a cracked shaft is usually expressed by a breathing function to increase the efficiency of the dynamic analysis.

2036

L. Cheng et al. / Journal of Sound and Vibration 330 (2011) 2031–2048

f kξ

Fk

η′

f kξ ′ O

η

f kη ′

R

a

O′

Fk

ξ

ψ

ξ′

Fig. 4. The displacement of disk and the elastic force of crack shaft.

0°

Dimensionless compliance C for crack depth 30%

30° 0.40 0.35 0.30 0.25 60° 0.20 0.15 90° 0.10

120° 180°

137°

0.05 0.00 -0.05 0

20

40

60

80

100

120

140

160

180

Rotational angle ϕ, deg Fig. 5. The B-spline fitting between the defined areas (crack depth a/R= 40%) [28].

It was shown in Table 1 and Figs. 6 and 11 in Ref. [28] that the stiffness kxx(c) of a cracked shaft in the deformation direction at an arbitrary angle is hundreds of times larger than the cross-coupled stiffness kxZ(c) in the direction perpendicular to the deformation. So the direction of the elastic force of a cracked shaft is opposite to the deformation direction. If the disk moves from o to o0 , in Fig. 1(b) and 4(c), the elastic force Fk of the cracked shaft in the deformation direction oo0 can be expressed generally by   (f ) kð1fbreathing ðcÞDkÞu kx ¼ Fk ¼ Fk ðu,a=R, cÞ ¼ kð1fbreathing ðcÞDkða=RÞÞu ¼ (2) fkZ 0 where u = oo0 is the displacement of the disk; k is the stiffness of the uncracked shaft; Dk, corresponding to the relative crack depth a/R, is the relative reduction in the stiffness of the shaft in the crack direction; and fbreathing(c) is a breathing function that describes the nonlinear breathing of the crack. fbreathing(c)=1 means that the crack is fully open, fbreathing(c) =0 means that the crack is fully closed. Unlike in the case of weight dominance, fbreathing(c) is a function of the reference angle c of the crack here. Because fbreathing(c)= fbreathing(c + 2p), cos(c) can be calculated directly using Eq. (3). So, the breathing function can be expressed directly as Fbreathing(cos(c)): 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ycosðyðtÞÞ þz sinðyðtÞÞ > < pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 y2 þ z2 a0 2 þz2 y cosðcÞ ¼ cosðyfÞ ¼ cosðyÞcosðfÞ þ sinðyÞsinðfÞ ¼ (3) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : 1 8 y2 þ z2 ¼ 0

L. Cheng et al. / Journal of Sound and Vibration 330 (2011) 2031–2048

2037

Two main types of method are used to establish the breathing function. One type is model methods, such as the assumption method. The simplest representation of a crack is the switching model, or hinge model [10,16,17,19]. Although the switching model might be an appropriate representation for very small cracks, Mayes and Davies [30,31] proposed a harmonic model with a smooth transition between the opening and closing of the crack, which is more appropriate for larger cracks. So, this harmonic model has been used in the following study. In this case the harmonic model [14,20,21,24,25], or the Mayes modified model, takes the following form: Fbreathing ðcosðcÞÞ ¼

1 þcosðcÞ 2

(4)

The other type of method is numerical fitting, such as in the cases of the strain energy release rate method [28,29], the 3D nonlinear finite element method [5,27] and the use of experimental results [5,27]. The stiffness changes Dkox(c), DkoZ(c) and DkxZ(c) of the cracked shaft in an arbitrary direction can be calculated by the strain energy release rate method, and the elastic forces are expressed as Dfox = Dkox(c)u = fbreathing(c) Dkox(c = 0)u. Using the results [28] of the strain energy release rate method, the breathing function can be fitted as a function based on (1+ cos(c))/2:  4  6  10 1 þ cosðcÞ 1 þcosðcÞ 1 þ cosðcÞ Fbreathing ðcosðcÞÞ ¼ 3:0992 10:392 þ36:6247 2 2 2  14  20  21 1 þ cosðcÞ 1þ cosðcÞ 1 þcosðcÞ 55:192 þ106:86 80 2 2 2

(5)

2.3. The general equations of motion of a cracked rotor The equations of motion of a cracked rotor with a constant rotation speed (i.e. y = ot) can be obtained by applying Newton’s Second Law to the disk and using stationary coordinates as seen in Fig. 1(b): )  ( )  ( )     ( fky ðu, Dk, cÞ cosðy þ bÞ y_ m mg c y€ 2 ¼ þ emo þ þ fkz ðu, Dk, cÞ sinðy þ bÞ z_ m 0 c z€ Mu€ þ Cu_ þ Fk ðu, Dk, cÞ ¼ Fs þ Fe eiðy þ bÞ

(6)

€ u_ and u are the acceleration, velocity and displacement vectors, respectively; M and C are the mass and where u, damping matrices (including the gyroscopic effect), respectively; Fs is the weight force vector; Feei(y + b) is the centrifugal force vector of the unbalanced mass; Fk(u, Dk, c) is the vector of the elastic force due to deformation of the shaft; and fky(u, Dk, c) and fkz(u, Dk, c) are the projections onto the stationary coordinates y and z, respectively, which are functions of the deformation of the shaft. The transformation between the stationary and rotational coordinates can be obtained directly from Fig. 1: ( ) ( ) ( ) ( ) cosðfÞ cosðfÞ u cosðfÞ fky ðu, Dk, cÞ ¼ Fk ¼ kð1Fbreathing ðcosðcÞÞDkÞu ¼ kð1Fbreathing ðcosðcÞÞDkÞ sinðfÞ sinðfÞ u sinðfÞ fkz ðu, Dk, cÞ   y ¼ kð1Fbreathing ðcosðcÞÞDkÞ : (7) z If the vectors of the elastic force are replaced using Eq. (7), Eq. (6) can be simplified to ( )      ( )    cosðot þ bÞ y y_ mg m c y€ 2 ¼ þ emo þ þ kð1Fbreathing ðcosðcÞÞDkÞ sinðot þ bÞ z z_ 0 m c z€

(8)

The steady response of the cracked rotor around its critical speed O = o/oc = 1.00 obtained with Eq. (8) and the same parameters as in Section 2.1 is shown in Fig. 6. Fig. 6(a) and (d) are almost the same as Fig. 2(a) and (d). But, from Fig. 6(b) and (c), it can be seen that the breathing state of the crack is quite different from the case of weight dominance. In Fig. 6(b), the rotation speed is lower than the critical speed, and the crack breathes weakly twice during one revolution. The crack has its maximum opening at the points A1 and A2, and its minimum opening at the points B1 and B2. In Fig. 6(c), the rotation speed is higher than the critical speed, and the crack breathes strongly. From B1 to B3, the crack is almost closed. Introducing the following dimensionless parameters: the natural frequency, oc =O(k/m); the damping factor, z = c/2moc; the static weight deflection, dst =mg/k; the dimensionless time, t = oct; the rotational speed factor, O = o/oc; y ¼ y=dst ; z ¼ z=dst ; and the eccentricity, E =e/dst. Eq. (4) can then be written in a dimensionless form: ( ) ( ) ( )     cosðt þ bÞ y 1 y00 yu 2 ¼ þ 2z (9) þ ð1Fbreathing ðcosðcÞÞDkÞ þE sinðt þ bÞ z 0 z00 zu

2038

L. Cheng et al. / Journal of Sound and Vibration 330 (2011) 2031–2048

1

1

Ω=0.92

0.5

0.5

0

0

B

-0.5

A1 B2

B1

-0.5 -1 -Y

-1 -Y

Ω=0.973

-1.5

A

-2

-1.5 -2

-2.5

-2.5

-3

-3

-3.5

-3.5

A2

-4

-4 -2

-1

0

1

2

-2

-1

0

Z

1

1

Ω=1.01

B3

0.5 0

2

1

2

Ω=1.06

0.5 0

A

B2

-0.5

A

-0.5 -1

B1

-1.5

-Y

-1 -Y

1

Z

-1.5

-2

-2

-2.5

-2.5

-3

-3

-3.5

-3.5

B

-4

-4 -2

-1

0 Z

1

2

-2

-1

0 Z

Fig. 6. The steady response of the cracked rotor about its critical speed with Eq. (8) and the harmonic crack model.

Eq. (9) is the general equation of motion for a Jeffcott rotor with a transverse crack, without any restrictions or assumptions. 2.4. Discussion There are two points of difference between Eqs. (1) and (8). The first is that the breathing behavior in Eq. (1) is expressed as function of the rotation angle y, whereas the same in Eq. (8) is expressed as a function of the crack reference angle c. The second is that Eq. (8) is not a bilinear equation and that it is simpler than Eq. (1). 2.4.1. Breathing behavior of crack The expression of the breathing behavior of a crack as a function of the rotation angle y with the assumption of weight dominance, as has been done in most of the previous work, neglects the fact that a cracked rotor whirls in a complicated way and that the rotation angle y is not usually equal to the whirling angle f (see Fig. 1), especially near the critical speed. The opening and closing, or breathing, of the crack is determined by its position relative to the shaft deformation (see Fig. 3) and is independent of whether the shaft rotates or not. In order not to be restricted to weight dominance, the crack reference angle c (see Fig. 1(b)) has been used to judge the opening and closing of the crack in the present work. This angle describes the breathing of the crack more practically by taking into consideration the influence of the whirling of the rotor on the breathing of the crack [20,21].

L. Cheng et al. / Journal of Sound and Vibration 330 (2011) 2031–2048

2039

2.4.2. Calculation of the elastic force 2.4.2.1. Bilinear form of the elastic force. According to Eqs. (13), (14) and (19) in Gasch’s paper [1], the following bilinear form of the elastic force exerted by the shaft can be obtained [1,10,12,17,19]: ( )   ( )  foxu xu Dk k ¼ fbreathing ðtÞ foZu Zu 0 k Fuk ¼ fKfbreathing ðtÞDKguu

(10)

From Fig. 1(b), it is possible to obtain the following transformation matrix between the stationary and the rotational coordinates directly: ( ) " #  xu cosðyÞ sinðyÞ y ¼ Zu sinðyÞ cosðyÞ z (11)

uu ¼ TðyÞu

If the displacements in Eq. (10) are replaced using the transformation given in Eq. (11) and then multiplied from the left by TT(y), the elastic force of the shaft in the stationary coordinates is provided by (

fky fkz

)

( ¼

k

Fk ¼ TT ðyÞFuk ¼ TT ðyÞðKfbreathing ðtÞDKÞTðyÞu " #)  (  cos ðyÞ sinðyÞcosðyÞ 1 þ cosð2yÞ y k 1 ¼  fbreathing ðtÞDk fbreathing ðtÞDk 2 sinð2yÞ 2 z k k sinðyÞcosðyÞ sin ðyÞ "



2

sinð2yÞ 1cosð2yÞ

#)  y z

(12) In the above derivation, a crack in the shaft introduces dissimilar flexibilities in the direction of the crack tip and in the direction perpendicular to it. As the shaft rotates, the stiffness in any fixed direction will change with time; more precisely, it will be a periodic function of time. This situation is similar mathematically to that of a two-pole rotor, which has dissimilar moments of inertia, and thus the response should be similar [2]. This approach has been used by many researchers in this field, and the conclusion drawn is that there is a harmonic, periodic change in the compliance [3]. 2.4.2.2. Common bilinear form of the elastic force. According to Eqs. (3) and (12) in Meng and Gasch’s paper [8], the bilinear form of the elastic force of the shaft can be obtained regarding a more common situation [8,11,20,21,23], as shown below ( ) ( " #)( )  foxu Dkxu xu k ¼ fbreathing ðtÞ foZu DkZu Zu k Fuk ¼ fKfbreathing ðtÞDKguu

(13)

As in the derivation of Eq. (12) (

fky fkz

)

Fk ¼ TT ðyÞFuk ¼ TT ðyÞðKfbreathing ðtÞDKÞTðyÞu 2 39  Dkxu cos2 ðyÞ þ DkZu sin2 ðyÞ ðDkxu DkZu ÞsinðyÞcosðyÞ = y  4 5

¼ fbreathing ðtÞDk : k Dkxu DkZu sinðyÞcosðyÞ Dkxu sin2 ðyÞ þ DkZu cos2 ðyÞ ; z ( " #)   Dk1 þ Dk2 cosð2yÞ Dk2 sinð2yÞ y k 1 ¼  fbreathing ðtÞDk Dk2 sinð2yÞ Dk1 Dk2 cosð2yÞ 2 z k 8 < k

Dk1 ¼ Dkxu þ DkZu , Dk2 ¼ Dkxu DkZu

(14)

In Eq. (13) and Fig. 3, Dfoxu ¼ Dkxu ðtÞu cosðcÞ ¼ fbreathing ðtÞDkxu u cosðcÞ is the nonlinear component of the elastic force in the direction of the crack tip, DfoZu ¼ DkZu ðtÞu sinðcÞ ¼ fbreathing ðtÞDkZu u sinðcÞ is the component perpendicular to the direction of the crack tip and Dfox ¼ Dkox ðcÞu ¼ Fbreathing ðcosðcÞÞDku is the component in an arbitrary direction. Regarding the stiffness of an asymmetric shaft, Dfox can be obtained directly from the resultant of Dfox0 and Dfox. But the crack-breathing behavior is a nonlinear process; the crack-opening state for calculation of Dkxu and DkZu is usually different from that for Dkox , and the rule of calculating the change of Dkxu and DkZu with c is different as well [2,28,29]. For example, in the calculation of Dkxu ¼ Dkxu ðc ¼ 0Þ, the crack is fully open, and in the calculation of DkZu ¼ DkZu ðc ¼ 903 Þ the crack is half-open and half-closed, but in the calculation of Dkox ¼ Dkox ðcÞ the crack is partially open and partially closed. Therefore the linear resultant and the resolution of forces and displacements are used wrongly in the above derivation. 2.4.2.3. Another form of the elastic force. Some investigators [6,9,13,14,18,24,25] have assumed directly that the elastic force is a function of the rotation speed o: Fk ¼ ðKfbreathing ðotÞDKÞu

(15)

Sinou and Lees [14,24,25] replaced fbreathing(ot) by the form given in Eq. (4) and expressed u by a finite Fourier series with respect to o. The nonlinear elastic force of the shaft was also approximated by a finite Fourier series with

2040

L. Cheng et al. / Journal of Sound and Vibration 330 (2011) 2031–2048

respect to o: Fk ¼ Fk0 þ

m X

fFck cosðotÞ þ Fsk sinðotÞg

(16)

k¼1

Papadopoulos and other authors [6,9,13,18,26] expressed the stiffness as a truncated cosine series: Fk ¼ KðotÞu ¼ fK0 þK1 cosðotÞ þ K2 cosð2otÞ þ K3 cosð3otÞ þ K4 cosð4otÞgu

(17)

where 8 K0 ¼ ð5KOP þ 5KUC þ 6KHC Þ=16 > > > > > K ¼ 9ðKUC KOP Þ=16 > < 1 K2 ¼ ðKOP þ KUC 2KHC Þ=4 > > > > K3 ¼ ðKOP KUC Þ=16 > > : K ¼ ð2K K K Þ=16 4

UC

OP

HC

and KUC, KOP and KHC are the stiffness matrices of the uncracked condition, for a fully open crack and for a half-open and halfclosed crack, respectively. Eqs. (16) and (17) are similar in form and are suitable only in the case of weight dominance. 2.4.3. Assumption of weight dominance In fact, the assumption of weight dominance is accurate and useful in some cases, such as large turbine-generator rotors with a low critical speed and low imbalance. In Figs. 1 and 3, it can be seen that weight dominance implies cos(c) E1, sin(c)E0 and Dfox E Dfox0 ; therefore Eqs. (12) and (14) are conditionally correct. 3. Analysis of the characteristics of the critical speed of a cracked rotor The dynamic response of a cracked rotor was computed from Eqs. (4) and (9) using a fourth-order Runge–Kutta method to investigate the influence of nonlinear breathing of the crack and that of the imbalance orientation angle on the critical speed. The parameter values Dk= 0.05 and E= 0.10 were used in the simulations. An uncracked rotor and a rotor with an open crack with Dk= 0.05 were also analyzed for comparison. 3.1. Influence of breathing behavior on the stability of a cracked rotor 3.1.1. Inherent instability of a rotor with an open crack A rotor with an open crack, which had equations of motion corresponding to a rotor with an asymmetric stiffness, was found to have two natural frequencies (fcmin ¼ 0:9747 and fcmax ¼ 1:0) and to have a divergent instability in the region between the rotation speeds corresponding to these frequencies (0.9747o O o1.0) [32]. As described previously, an inherent instability exists in the periodically time-varying system that is used to describe cracked rotor systems in most of the literature [7]. The process of stability loss of t = 0–400 and the movement orbit at t = 250 are shown regarding a rotor with an open crack at a speed O = 0.99 in Fig. 7. In Fig. 7(c), the solid curves denote the movement orbits and the arrows indicate the instantaneous imbalance direction. It is evident that the amplitude of vibration increases continually to breakdown. Therefore, studies have often been restricted to subcritical speeds or have passed through the critical speed quickly; few publications [33] have reported on the influence of nonlinear crack breathing on stability near the critical speed. 3.1.2. Instability and stability of a rotor with a breathing crack 3.1.2.1. Floquet theory. The equations of motion of a cracked rotor (Eq. (9)) can be expressed as a nonautonomous system dv ¼ Fðv,t, lÞ ¼ Fðv,t þ T, lÞ dt

ðv,t, lÞ 2 Rn  R  Rm

(18)

where v is the state vector of the system, l is a system parameter, F is a periodic function of the time t, and T is the rotation period of the rotor. If U(t) is a solution of Eq. (18), then so is U(t +T). A discrete state transition matrix C is considered as follows:

Uðt þ TÞ ¼ CðtÞUUðtÞ

(19)

From the Floquet theory [7,11], the stability criterion of the system can be determined from the eigenvalues of Eq. (19): 9mICðtÞ9 ¼ 0

(20)

where m is called the Floquet multiplier. If the moduli of every eigenvalue 9m9 are less than unity, the system is stable. Otherwise, the system is unstable.

L. Cheng et al. / Journal of Sound and Vibration 330 (2011) 2031–2048

Vibration Amp

300

2041

60

200

40

100

20

Steering Function

0

50

100

150

200

250

300

350

400

-Y

0

0

1

-20

0.5

-40

0

-60 0

50

100

150

200

– Time

250

300

350

400

-80

-60

-40

-20

0

20

40

60

80

Z

Fig. 7. A rotor with an open crack b =01 and z = 0.001, at speed O = 0.990: (a) the process of stability losing and (b) breathing behavior at t = 0–400. (c) The movement orbit at t = 250.

There are two types of rotor instability, namely orbit structural instability at subcritical speeds and divergence instability near the critical speed, where the transverse vibrational energy transferred from the rotational energy is greater than the energy consumed by damping, and the transverse vibration of the rotor increases continually to breakdown. Using Eq. (9), the Floquet theory was used to analyze the stability of a cracked rotor near the critical speed. The results showed that with a lower damping factor (z =0.001), a cracked rotor with b =01 was unstable at 0.98o O o1.0 and a cracked rotor with b = 1801 was always stable, whereas a rotor with a breathing crack and a suitable damping factor (z =0.01) was stable at all imbalance orientation angles. The energy transfer was well represented by a numerical simulation of the lower damping factor (z = 0.001) as follows. 3.1.2.2. Instability of a rotor with a breathing crack. The process of stability loss at t = 0–1300 and the movement orbit at t =850 are shown in Fig. 8 for a rotor with a breathing crack with b =01 at a speed O =0.99. In Fig. 8(c) and (d), the solid curves denote the movement orbits and the arrows indicate the instantaneous imbalance direction. The breathing behavior of the points A1, B1, A2 and B2 on the orbit in Fig. 8(c) is marked by the same letters in the breathing function shown in Fig. 8(d). It should be noted that the crack breathes twice during one revolution. Comparing Figs. 7 and 8, it is evident that the processes of stability loss are similar and that the breathing behavior slows down the divergence by a factor of about three. 3.1.2.3. Stability of a rotor with a breathing crack. Two long time histories (up to t =4000) of the response amplitude and breathing behavior are shown in Fig. 9 regarding a rotor with a breathing crack with b = 1801. The amplitude of vibration decreases step by step in the sequence of peaks A1, A2, A3, A4, A5 and A6 in Fig. 9(a), and the behavior tends to a steady wave. At the peaks A1, A2, A3, A4, A5 and A6, the breathing almost stops. The same is true in Fig. 9(b). The steady response of the cracked rotor was defined as the peak of the steady wave as follows. The peak value of the steady response was found at about O = 0.9876. It is interesting that the difference DO in the speed between Fig. 9(a) and (b) is only 0.0015; the breathing of the crack changes direction between the two speeds. The behavior of the crack changes drastically from a situation where the crack is almost fully closed at O =0.9875 to a situation where it is almost fully open fully at O =0.98765; the breathing does not stop, and the rotor remains stable. It is evident that the direction of the crack reverses from O = 0.9875 to 0.98765, and the nonlinear breathing improves the stability of the rotor by destroying or weakening the mechanism by which rotational energy is transformed into transverse vibrational energy. These features of the vibrational behavior are different from those of a rotor with an open crack, and have not been previously mentioned in the literature because the equations of motion of a cracked rotor with weight dominance are not suitable for studying the present situation. 3.2. Influence of breathing and imbalance orientation angle on the speed at the peak of steady response [34] In an uncracked rotor, there are three basic characteristic features of the critical speed [32]. First, the peak response is found at the critical speed. Second, the critical speed is equal to the natural frequency of the rotor, i.e. oc =fc =1.0. Third, the center ;of gravity rotates by 901 when the critical speed is passed through. The steady response and breathing behavior of a cracked rotor are shown in Figs. 10 and 11, respectively, at eight imbalance orientation angles and at speeds in the region 0.95o O o1.05.

L. Cheng et al. / Journal of Sound and Vibration 330 (2011) 2031–2048

Steering Function

2042

Vibration Amp

300 200 100

1

0.5

0

0 0

200

400

800 600 – Time

1000

0

1200

200

400

600

800

1000

1200

T–Time

1 40 0.9

A1

30

0.8 Breathing Function

B1

20

-Y

10 0 -10 -20 -30

A2

A1

0.6 0.5 0.4 0.3

F

B2

A2

A1

0.7

0.2

A2

B1

0.1

-40

B1

B2

0

-50 -60

-40

-20

0

20

40

60

846

848

850

Z

852

854

856

858

T–Time

Fig. 8. A rotor with breathing crack b = 01 and z = 0.001, at speed O = 0.990: (a) the process of stability losing and (b) breathing behavior at t =0–1300. (c) The movement orbit and (d) breathing behavior at t =850.

B1

25 A1

20

A2

15

A3 A4

10

A5

A6

5

Vibration Amp

Vibration Amp

25

0

20

B2

15

B3

10

B5

B6

5 0

0

500

1000

1500

2000

2500

3000

3500

0

4000

500

1000

1500

T–Time

1

0.5

0 0

500

1000

1500

2000 T–Time

2000

2500

3000

3500

4000

2000 2500 T–Time

3000

3500

4000

T–Time

Breathing Function

Breathing Function

B4

2500

3000

3500

4000

1

0.5

0

0

500

1000

1500

Fig. 9. Influence of crack breathing on rotor stability at b =1801 and z = 0.001: (a) O = 0.9875 (fully closed) and (b) O = 0.98765 (fully open).

Fig. 10 shows that the characteristics of a cracked rotor are similar to those of an uncracked rotor; there is a peak in the steady response in the region of the critical speed, but the speed corresponding to the peak of the steady response varies with b and is lowest at b = 901 (Oc = 0.975) and is highest at b = 2701 (Oc = 1.000). Although the speed is almost the same at b = 01

L. Cheng et al. / Journal of Sound and Vibration 330 (2011) 2031–2048

10

10

β=135

5

1

10

1

1.05 Ω=0.996

Max=5.8068

1

10

5 0 0.95

1

5 0 0.95

1.05 Ω=1.000

Max=5.9559

Ω=0.979

Max=5.7495

10

5 0 0.95

1.05

5 0 0.95

1.05 Ω=0.988

Max=5.6474

1

10

Ω=0.975

1

Ω=0.979

Max=6.0882 5 0 0.95

1.05

β=225

β=90

Max=5.8827

0 0.95

β=180

1

β=315

β=0

5 0 0.95

β=270

β=45

Ω=0.988

Max=6.2276

10

2043

1.05

1.05 Ω=0.996

Max=6.1203 5 0 0.95

1

1.05

Fig. 10. Steady response of a cracked rotor. The x-axis denotes the rotating speed O and the y-axis denotes the response amplitude. D indicates the peak of steady response.

and 1801 (Oc = 0.998), there is a difference in the response amplitude, which has its maximum (6.2276) and minimum (5.6474) values, respectively, at these values of b, as a function of the imbalance orientation angle. In Fig. 11, solid and dashed curves show the upper and lower limits of breathing function, respectively. It is evident that the breathing function varies between 0 and 1 far from the peak of the steady response in Fig. 10, indicating that the breathing behavior is similar to that in the case of weight dominance. The breathing function changes in a complex manner near the peak of the steady response; see, for example, the states at points C (b =1351), D (b = 1801) and E (b = 2251). The breathing amplitude tends to decrease on the whole, indicating that the breathing behavior weakens near the peak response. The value of the breathing function is approximately 1 at the peak of the steady response at b = 901 (at point A), corresponding to a fully open crack (minimum stiffness and lowest speed), and is approximately 0 at the peak at b = 2701 (at point B), corresponding to a fully closed crack (maximum stiffness and highest speed). 3.3. Movement of the center of gravity on passing through the peak of the steady response The movement orbit and breathing behavior around the speed of the peak of the steady response were analyzed regarding a cracked rotor with a variable imbalance orientation angle. The dynamic response at imbalance orientation angles b = 01 and 901 is shown in Figs. 12 and 13, respectively. The solid curves denote the movement orbits and the arrows indicate the instantaneous imbalance direction. The breathing behaviors of points A and B on the orbit are marked by the same letters on plots of the breathing function. The crack opens to its maximum extent at A and to its minimum extent at B. The variation in breathing behavior is different at different imbalance orientation angles on passing through the peak of the steady response. But the behavior is similar to that of an uncracked rotor in that the center of gravity rotates on passing through the peak of the steady response. 3.4. Influence of breathing and imbalance orientation angle on the peak of the steady response An analysis of the peak of the steady response, combining the effects of a crack and an imbalance, with different damping factors (Fig. 14, t = 500) showed no obvious variation in the peak of the steady response with the imbalance orientation angle for higher damping factors (z = 0.01–0.04). But with lower damping factors (z =0.001–0.005), the variation is obvious. The peaks of the steady response at b E701 and 2701 are approximately equal to that of an uncracked rotor. Moreover, the peak of the steady response for a reversed imbalance (701o b o2701) is lower than that of an uncracked rotor, with a minimum at b E1801.

2044

L. Cheng et al. / Journal of Sound and Vibration 330 (2011) 2031–2048

1 β=45

β=0

1 0.5 0 0.95

0 0.95

1.05

A

0.5 0 0.95

1

D

1

1.05

1

1.05

1

1.05

1

1.05

C

E

1 β=315

β=270

0.5 0 0.95

1 0.5 B 0 0.95

1.05

1 β=225

β=180

0 0.95

0.5 0 0.95

1.05

1 0.5

1

1 β=135

β=90

1

1

0.5

1

1.05

0.5 0 0.95

Fig. 11. Breathing behavior of a cracked rotor. The x-axis denotes the rotating speed O and the y-axis denotes the breathing function. Solid and dashed curves show the upper and lower limits of breathing function, respectively.

3.5. Discussion Plaut et al. [16] investigated the influence of the imbalance orientation angle and, surprisingly, observed that the peak response during passage through the critical speed may sometimes be smaller in a cracked shaft than in an uncracked shaft. Go´mez-Mancilla et al. [23] emphasized that when a crack and a residual imbalance (both being described by unknown vectors) are present, the imbalance can mask the crack’s presence and make traditional detection techniques difficult. But they concluded that the orbital evolution around one-half and that around one-third of the first resonance frequency can be used to detect cracks in a rotor, even if the crack–imbalance orientation is unknown. Sinou and Lees [24,25] investigated the influence of crack–imbalance interactions and, more particularly, that of the relative orientation between the crack front and the imbalance, with consideration of various crack depths and imbalance magnitudes. But no general rule was derived. Sekhar and Prabhu [17] showed that when a crack occurs in the direction opposite to that of the imbalance eccentricity, the vibrations are severe compared with the case when b =901. Darpa et al. [22] observed that breathing-crack models gave a single maximum at b =22.51 and a single minimum at b =157.51, and that the response due to only a crack (with negligible imbalance) appeared to have a maximum at b = 01 and a minimum at b = 1801. The addition of an imbalance force modified the variation in the response of each model. These authors observed experimentally that a cracked rotor had a minimum peak response between b = 157.51 and 1801 and a maximum peak response at b = 451. The results presented in Sections 3.2 and 3.3 show that the critical speed is located between the natural frequencies of a fully open crack and those of a fully closed crack, and depends on the imbalance orientation angle. The results presented in Section 3.4 show that the peak of the steady response is determined mainly by the imbalance orientation angle. Sometimes, at a reversed imbalance (701o b o2701), the peak of the steady response of a cracked rotor may be smaller than that of an uncracked rotor with the same parameters. Some of the above mentioned rules have not been reported before.

4. Conclusions 4.1. General equations of motion for a Jeffcott rotor with a transverse crack Numerical solutions and mathematical analysis have shown that the equations of motion of a cracked rotor derived with the assumption of weight dominance are not suitable for studying the vibration of a cracked rotor near its critical speed. Hence the results obtained using these equations are not reliable, and the basic characteristics of a cracked rotor near its critical speed are not clear.

L. Cheng et al. / Journal of Sound and Vibration 330 (2011) 2031–2048

1 2

A

A

0.9 0.8

B

0

F–Breathing Function

1

-Y

2045

A

-1 -2

0.7 0.6

B

0.5

B

0.4 0.3 0.2

-3

0.1 -4

0 -4

-3

-2

-1

0 Z

1

2

3

4

488

490

492

494

496

498

500

T–Time 4

1

3

A2

0.9

2

A1

B1

A2

B2

A1

B1

A2

F–Breathing Function

0.8

1 0 -Y

B2

B1

B2

-1 -2 -3

0.7 0.6 0.5 0.4 0.3 0.2

-4

0.1

A1

-5

0

-6 -6

-4

-2

0

2

4

488

6

490

492

494

496

498

500

T–Time

Z

1 2

A

-Y

0

F–Breathing Function

1

B

-1 -2 -3

0.8

A

A

0.9

B

B

0.7 0.6 0.5 0.4 0.3 0.2

-4

0.1

-5

0 -5

0 Z

5

488

490

492

494

496

498

500

T–Time

Fig. 12. Dynamic response of a rotor with b = 01 crack. (a) Before the critical speed: O = 0.980 (direction of the center of gravity is outwards). (b) Near the critical speed: O = 0.990 (direction of the center of gravity is tangential). (c) After the critical speed: O = 1.000 (direction of the center of gravity is inwards).

2046

L. Cheng et al. / Journal of Sound and Vibration 330 (2011) 2031–2048

1 2

B

0

F–Breathing Function

1

-Y

A

A

0.9

A

-1 -2 -3

0.8 0.7 0.6

B

0.5

B

0.4 0.3 0.2 0.1

-4

0 -4

-3

-2

-1

0

1

2

3

488

4

490

492

Z 4

A2

0.9 F–Breathing Function

2 1 0 -Y

496

498

500

1

3

B1

B2

-1 -2 -3 -4 -5

A1

B2

A1

A2

B1

B2

A1

B1

496

498

A2

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

-6

0 -6

-4

-2

0

2

4

6

488

490

492

Z

0.9

2

0.8

0

F–Breathing Function

1

3

A

494

500

T–Time

4

1 -Y

494

T–Time

B

-1 -2 -3 -4 -5

0.7

A

A B

B

0.6 0.5 0.4 0.3 0.2 0.1

-6

0 -5

0 Z

5

488

490

492

494

496

498

500

T–Time

Fig. 13. Dynamic response of a rotor with b =901 crack. (a) Before the critical speed: O = 0.965 (direction of the center of gravity is outwards). (b) Near the critical speed: O = 0.975 (direction of the center of gravity is tangential). (c) After the critical speed: O = 0.985 (direction of the center of gravity is inwards).

L. Cheng et al. / Journal of Sound and Vibration 330 (2011) 2031–2048

2047

25 β =70°

β =270°

Peak of steady response

20 ζ=0.001 15 ζ=0.005 10 ζ=0.010 5

ζ=0.040

0 0

50

100

150

200

250

300

350

Imbalance orientation angle Fig. 14. Peak of steady response of a cracked rotor for different damping factors. The dashed line is the response of a non-cracked rotor.

Taking the complicated whirl of the rotor into account, general equations of motion have been developed for a Jeffcott rotor with a transverse crack. The angle between the crack direction and the direction of the shaft deformation has been used to determine the closing and opening of the crack, allowing the dynamic response to be studied with and without weight dominance. Using the new equations, the steady response of a cracked rotor near its critical speed has been analyzed via a numerical method. 4.2. Influence of nonlinear breathing on the stability of a cracked rotor near its critical speed By destroying or weakening the mechanism by which rotational energy is transformed into transverse vibrational energy, the breathing of a crack can improve the stability of a rotor in contrast to that of a rotor with an open crack and reduce the vibration response for a reversed imbalance (701o b o2701) in contrast to an uncracked rotor. 4.3. Influence of a crack and imbalance on the behavior at the critical speed The basic characteristics of a cracked rotor near its critical speed are similar to those of an uncracked rotor. When the rotation speed coincides with the critical speed, the vibration response is maximized and the center of gravity rotates by 901. The critical speed can be determined by measuring the rotation of the center of gravity. The critical speed of a cracked rotor is located between the natural frequencies in the cases of a fully open crack and those of a fully closed crack and depends on the imbalance orientation angle. Its value is lowest at b E901 and highest at b E2701. The peak of the response at the critical speed is also determined mainly by the imbalance orientation angle. When b E01 and 1801, the peak corresponds to the maximum and minimum, respectively. Sometimes, the peak of the response of a cracked rotor may be smaller than that of an uncracked rotor with the same parameters. Therefore, the characteristics of a rotor with a breathing crack are quite different from those of a rotor with an asymmetric stiffness (which corresponds to an open crack), and the diagnosis of a cracked rotor based only on variations in the critical speed may lead to incorrect results.

Acknowledgements This work is supported by National Natural Science Foundation of China (No. 51035007), FOK YING TUNG Education Foundation (No.121052) and the Fundamental Research Funds for the Central Universities. References [1] R. Gasch, A survey of the dynamic behaviour of a simple rotating shaft with a transverse crack, Journal of Sound and Vibration 160 (1993) 313–332. [2] A.D. Dimarogonas, Vibration of cracked structures: a state of the art review, Engineering Fracture Mechanics 55 (1996) 831–857. [3] C.A. Papadopoulos, The strain energy release approach for modeling cracks in rotors: a state of the art review, Mechanical Systems and Signal Processing 22 (2008) 763–789. [4] Yukio Ishida, Cracked rotors: Industrial machine case histories and nonlinear effects shown by simple Jeffcott rotor, Mechanical Systems and Signal Processing 22 (2008) 805–817.

2048

L. Cheng et al. / Journal of Sound and Vibration 330 (2011) 2031–2048

[5] Nicolo Bachschmid, Paolo Pennacchi, Ezio Tanzi, Some remarks on breathing mechanism, on non-linear effects and on slant and helicoidal cracks, Mechanical Systems and Signal Processing 22 (2008) 879–904. [6] C.A. Papadopoulos, A.D. Dimarogonas, Stability of cracked rotors in the coupled vibration mode, Journal of Vibration, Acoustics, Stress, and Reliability in Design 110 (1988) 356–359. [7] S.C. Huang, Y.M. Huang, S.M. Shieh, Vibration and stability of a rotating shaft containing a transverse crack, Journal of Sound and Vibration 162 (1993) 387–401. [8] G. Meng, R. Gasch, Stability and stability degree of a cracked flexible rotor supported on journal bearings, Journal of Vibration and Acoustics 122 (2000) 116–125. [9] A.S. Sekhar, J.K. Dey, Effects of cracks on rotor system instability, Mechanism and Machine Theory 35 (2000) 1657–1674. [10] B. Yang, C.S. Suh, A.K. Chan, Characterization and detection of crack-induced rotary instability, Journal of Vibration and Acoustics 124 (2002) 40–48. [11] Changsheng Zhu, D.A. Robb, D.J. Ewins, The dynamics of a cracked rotor with an active magnetic bearing, Journal of Sound and Vibration 265 (2003) 469–487. [12] B. Yang, C.S. Suh, On fault induced nonlinear rotary response and instability, International Journal of Mechanical Sciences 48 (2006) 1103–1125. [13] Changping Chen, Liming Dai, Yiming Fu, Nonlinear response and dynamic stability of a cracked rotor, Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1023–1037. [14] J.J. Sinou, Effects of a crack on the stability of a non-linear rotor system, International Journal of Non-Linear Mechanics 42 (2007) 959–972. [15] Oh Sung Jun, Mohamed S. Gadal, Dynamic behavior analysis of cracked rotor, Journal of Sound and Vibration 309 (2008) 210–245. [16] R.H. Plaut, R.H. Andruet, S. Suherman, Behavior of a cracked rotating shaft during passage through a critical speed, Journal of Sound and Vibration 173 (1994) 577–589. [17] A.S. Sekhar, B.S. Prabhu, Condition monitoring of cracked rotors through transient response, Mechanism and Machine Theory 33 (1998) 1167–1175. [18] S. Prabhakar, A.S. Sekhar, A.R. Mohanty, Transient lateral analysis of a slant-cracked rotor passing through its flexural critical speed, Mechanism and Machine Theory 37 (2002) 1007–1020. [19] A.S. Sekhar, Crack detection through wavelet transform for a run-up rotor, Journal of Sound and Vibration 259 (2003) 461–472. [20] J.T. Sawicki, Xi Wu, G.Y. Baaklini, A.L. Gyekenyesi, Vibration-based crack diagnosis in rotating shafts during acceleration through resonance, Proceedings of SPIE 5046 (2003) 1–10. [21] J.T. Sawicki, A.L. Gyekenyesi, G.Y. Baaklini, Analysis of transient response of cracked flexible rotor, Proceedings of SPIE 5393 (2004) 142–150. [22] A.K. Darpe, K. Gupta, A. Chawla, Transient response and breathing behaviour of a cracked Jeffcott rotor, Journal of Sound and Vibration 272 (2004) 207–243. [23] J. Go´mez-Mancilla, J.J. Sinou, V.R. Nosov, F. Thouverez, A. Zambrano, The influence of crack-imbalance orientation and orbital evolution for an extended cracked Jeffcott rotor, Comptes Rendus Mecanique 332 (2004) 955–962. [24] J.J. Sinou, A.W. Lees, A non-linear study of a cracked rotor, European Journal of Mechanics A/Solids 26 (2007) 152–170. [25] J.J. Sinou, Detection of cracks in rotor based on the 2  and 3  super-harmonic frequency components and the crack-unbalance interactions, Communications in Nonlinear Science and Numerical Simulation 13 (2008) 2024–2040. [26] T. Ramesh Babu, S. Srikanth, A.S. Sekhar, Hilbert–Huang transform for detection and monitoring of crack in a transient rotor, Mechanical Systems and Signal Processing 22 (2008) 905–914. [27] N. Bachschmid, E. Tanzi, Deflections and Strains in cracked shafts due to rotating loads: a numerical and experimental analysis, International Journal of Rotating Machinery 10 (2004) 283–291. [28] A.C. Chasalevris, C.A. Papadopoulos, A continuous model approach for cross-coupled bending vibrations of a rotor-bearing system with a transverse breathing crack, Mechanism and Machine Theory 44 (2009) 1176–1191. [29] A.C. Chasalevris, C.A. Papadopoulos, Identification of multiple cracks in beams under bending, Mechanical Systems and Signal Processing 20 (2006) 1631–1673. [30] I.W. Mayes, W.G.R. Davies, A method of calculating the vibrational behavior of coupled rotating shafts containing a transverse crack, Proceedings of the International Conference on Vibrations in Rotating Machinery, Paper C254/80, IMechE, 1980, pp. 12–27. [31] I.W. Mayes, W.G.R. Davies, Analysis of the response of a multi-rotor-bearing system containing a transverse crack in a rotor, Journal of Vibration, Acoustics, Stress, and Reliability in Design, Transactions of ASME 106 (1984) 139–145. [32] A. Tondl, Some Problems in Rotor Dynamics, Chapman & Hall, London, 1965. [33] L. Cheng, Z.W. Qian, B.B. Zhao, et al., Influence of nonlinear breathing on the stability of a Jeffcot rotor with transverse crack, Chinese Journal of Applied Mechanics 27 (2010) 219–225. [34] L. Cheng, Z.J He, N. Li, et al., Influence of crack nonlinear breathing behavior on critical speed of roto, Journal of Vibration and Shock 29 (2010) 44–47.