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

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A continuous model approach for cross-coupled bending vibrations of a rotor-bearing system with a transverse breathing crack A.C. Chasalevris, C.A. Papadopoulos * Department of Mechanical Engineering and Aeronautics, University of Patras, GR-265.04 Patras, Greece

a r t i c l e

i n f o

Article history: Received 27 July 2007 Received in revised form 1 August 2008 Accepted 1 September 2008 Available online 15 October 2008

Keywords: Continuum rotor model Transverse crack Rotating crack Coupling Breathing Rotor-bearing system Resilient bearing

a b s t r a c t In this paper, the cross-coupled bending vibrations of a rotating shaft, with a breathing crack, mounted in resilient bearings are investigated. The equations of motion of the continuum and isotropic rotating model of the shaft follow the theory of Rayleigh. The governing equations are coupled in the two main directions, and the partial solution is obtained by solving a linear system of equations, for each time step, taking into account the non-linearity due to the breathing crack. The coupling is introduced in three different ways: the equations of motion, the resilient bearings and the crack. A main focus is made in the coupling introduction due to crack compliance variance while rotation with the cross-coupling terms of the local compliance matrix due to the crack to be calculated analytically as functions of the rotational angle. The three causes of coupling between the vertical and horizontal vibrations should be distinguished with regard to the effects that each one of them has on the dynamic response of the rotor. Inversely, the existence of each type of coupling in the frequency response could be used to identify the respective cause. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The vibration of cracked rotors is an issue that has been continuously investigated since 1970. Many turbine rotors failures started appearing in the 1970s in the USA and elsewhere, because many of them were approaching an operating life of 30 years. Such a case was that observed by Dimarogonas in [1,2] and Pafelias [3] at the turbine department of the General Electric Company in Schenectady. That failure was due to a fatigue propagating crack. Since that time many researchers have investigated the dynamic behavior of rotors with open cracks that as phenomenon are close enough to the case of rotors with dissimilar moments of inertia, as Dimentberg [4] and Tondl [5] have extensively treated them. Dimarogonas [6] suggested that the existence of higher harmonics and sub-harmonics, as well as the presence of longitudinal and torsional harmonics in the start-up lateral vibration spectrum due to the coupling, as potential methods for crack detection. It is known that when a cracked shaft rotates, the stiffness in a fixed direction changes with time due to the local compliance that the crack introduces. The precise computational of local flexibility was confronted by Dimarogonas [1,2] and he gave functions for the change of local compliance during rotation. Chondros and Dimarogonas [7] modeled a transverse crack as a local elasticity and related the crack depth to the decrease in the natural frequency, and Gounaris and Dimarogonas [8] developed a finite cracked Euler–Bernoulli element. The precise model of the crack compliance and the shaft vibration are two objects that have been researched in strong relationship with each other, in order for more precise theories of cracked shaft vibration to be found and a better approximation to the problem of crack identification to be feasible. Jun and Kim [9]

* Corresponding author. Tel.: +30 261 096 9426; fax: +30 261 099 6258. E-mail address: [email protected] (C.A. Papadopoulos). 0094-114X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2008.09.001

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Nomenclature R: shaft radius Y(x, t): vertical response a, ax: crack depth Z(x, t): horizontal response a : dimensionless crack depth U(x, t): complex response bi: boundaries of crack in cracked section, i = 1, 2 H(x, t): slope cij : dimensionless crack compliance V(x, t): shearing force Cij: crack compliance M(x, t): bending moment C: Local compliance matrix Ug(x): gravity response Ctot: total flexibility P: characteristic matrix E: Young modulus Kb: bearing stiffness coefficient matrix G: shear modulus Cb: bearing damping coefficient matrix g: gravity acceleration t: time I: shaft polar moment of inertia Dt: time interval i: complex quantity T: power transmission torque k: form factor DX: rotational speed interval Pi: bending load m: Poisson ratio L: shaft length q: material density md: disk mass u: rotational angle Ld: disk width ucl: rotational angle of crack closure Rd: disk radius x: whirling frequency r0: radius of gyration, X; X: rot, frequency/dimensionless

made a precise study of the free bending vibration of a multi-step rotor using a Timoshenko beam model. Collins et al. [10] used axial impulses in order to detect a breathing crack in a rotating Timoshenko shaft. Collins et al. [11] investigated longitudinal vibrations of a cantilever bar with transverse breathing crack and compared results with those of case of open crack and those without crack. Jun et al. [12] derived the equations of motion of a simple shaft with a breathing crack concluding that the vibration characteristics of a cracked rotor are best identified from the second horizontal harmonic components measured near to the second harmonic resonant speed. Imam et al. [13], Wauer [14], Gasch [15], Dimarogonas [6], Edwards et al. [16], and Sabnavis et al. [17] presented excellent reviews in the field of dynamics of cracked rotors and suggested different procedures for diagnosing fracture damage. Mayes et al. [18] analyzed the response of a multi-rotor-bearing system containing a transverse crack. When passing through critical speed, the transient vibration of a cracked rotor was analyzed by Sekhar and Prabhu [19], using the finite element method. Ishida and Hirokawa [20] represented the internal resonance of a linear cracked rotor and non-stationary oscillation of a non-linear rotor when passing through the major critical speed. Tsai and Wang [21] presented a free vibration mode analysis of a multi-cracked rotor. Results in many papers were obtained only by computer simulation and experimental studies were relatively few. Lee et al. [22] correlated experimental results, using propagating transverse cracks, with their theoretical analysis. Zhou and Xu [23] demonstrated many non-linear dynamic characteristics of a cracked rotor in their experiment. A recent approach to the problem of cracked rotor dynamics studies the coupling between lateral, axial, and torsional vibrations that the crack provokes. Papadopoulos [24] and Papadopoulos and Dimarogonas [25] studied the coupling of bending and torsional vibrations in a cracked Timoshenko shaft under the assumption that the crack remains open. The presence of bending vibrations in the torsional spectra had been cited as a crack indicator. Papadopoulos and Dimarogonas [26] studied coupling of lateral and longitudinal vibrations and proposed the coexistence of lateral and longitudinal vibration frequencies in the same spectrum as an unambiguous crack indicator. The same phenomenon was confirmed also by Darpe et al. [27] when they analyzed the response of a cracked Jeffcott rotor with a transverse crack in the mid-span. They also calculated [28] the cross-coupling terms of the local compliance matrix as a function of rotational angle of the crack and studied the coupled longitudinal bending and torsional vibrations of a cracked rotor. The case of coupled vibrations of all degrees of freedom has been investigated by Papadopoulos and Dimarogonas [29], and Gounaris and Papadopoulos [30] used coupled response measurements of a rotating Timoshenko shaft to identify a crack. The aim of the present study is to present a new calculation for the change in the local compliance matrix during rotation. Computation of crack local compliance is done for every shaft rotational step, in order to take into account the exact form of the crack while breathing. Also, the Timoshenko theory for continuous beams is used to model the shaft vibration under the effect of gravity and unbalance. The new model of compliance change approximates the phenomenon of coupling between bending vibrations which exist only in a specific rotational range. The analysis includes the torque that is applied at the ends of the shaft due to power transmission, and the gyroscopic effect due to shaft rotation. Two perpendicular displacements are expressed as one complex displacement and this demands the formulation of complex boundary conditions. The solution of the equation of motion is achieved for every rotational step of the shaft and thus makes the model dependent on time. For simplicity reasons, the boundary conditions assume rigid bearings at both ends of the shaft and the suitable continuity conditions at the crack position. The coupled linear system of equations with periodically varying boundary conditions becomes solvable for every rotational angle. The characteristic equation can be calculated for every value of rotational speed

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and the natural frequencies are obtained. The natural frequency is dependent on the variation of the stiffness coefficients due to the crack. The frequency response and the orbit of the shaft in the cross-section of the disk are also calculated for variable crack depths. The aim is to show the differences in the dynamic properties of the model, due to the existence of the coupling phenomenon the crack provokes, and also the effect of a deeper crack on the amplitude of vibration under the action of coupling, and to distinguish this coupling from these of the coupled differential equations and of the coupling due to the anisotropy of the bearings. The resulting diagrams of response and natural frequency as functions of crack depth present the dynamic behavior of this continuous cracked rotating shaft model. 2. The model of the breathing crack It is known from the literature that the existence of a transverse crack in a shaft induces a local compliance that differs in each direction. A cracked shaft under two bending loads is shown in Fig. 1; one load is applied in the vertical direction and the other in the horizontal direction. The planes of bending are defined as plane ZOX (load index 5) and plane YOX (load index 4) correspondently. Compliance cij is the additional transposition, due to the crack, in the direction i caused by a bending load in the j direction. When the crack is located in a non-symmetric location to the plane of loading (here, the vertical plane), it induces an additional coupling between the vertical and horizontal vibrations. These vibrations are already coupled, as will be described by the complex differential equations presented below. The coupling due to the crack is introduced in the model by the non-diagonal (cross-coupled) terms of the compliance matrix C (bar indicates dimensionless quantities). The coupling effects are proportional to the magnitude of the above coupling terms. For the problem of bending in two directions, the following matrix is defined as a local bending compliance matrix:

C¼



c44 c54

c45 c55



As the shaft rotates, the crack takes on different angular positions at the cracked section (Fig. 2) and the breathing can be simply expressed as the crack opening when it is located under the centroidal axis (Fig. 1), and closing when the crack is located over the centroidal axis. The strain energy release rate (SERR) is used here to calculate the local compliance matrix that is used to model the crack. The material of the shaft is considered to be homogeneous and isotropic. The radius of the  ¼ a=R in cross-section is R and the shaft is subjected to vertical P5 and horizontal P4 bending loads. The crack depth is a  ; uÞ and b2 ða  ; uÞ, and in the y direction by ð a dimensionless form. The crack can be bounded in the Z direction by b 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi (k + h2) and 1  z2 (Fig. 3), where, from the geometry of Fig. 3, one can obtain

b1 ¼ b cos u  s;

b2 ¼ b cos u þ s;

h1 ¼ b sin u þ ðR  aÞ cos u;

s ¼ ðR  aÞ sin u;

k ¼ ðb2  zÞ sin u

h2 ¼ b sin u þ ðR  aÞ cos u

 and u. The values of local compliance It is shown in the same figure that the boundaries of integration depend on values of a  and u using the strain energy density factor expressed by Eqs. (1)–(6). The calculation is feaare calculated as functions of a sible only for some regions of the rotational angle u. This is due to the restrictions introduced by the assumption made about the compliance calculation: the circular cracked cross-section is divided into vertical orthogonal cracked cross-sections (strips) that are not assumed to interact with each other. The compliance due to the crack is calculated for each crack depth and for each angle of rotation (bold line in Fig. 3b), following the way described below. In some transient locations (normal

Fig. 1. The model of the cracked shaft.

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Fig. 2. The breathing condition of crack section while rotation under bending moment of gravity.

Fig. 3. The geometry of cracked section.

line of Fig. 3b) the functions used (F1 and F2), that describe the strain energy density, are not accurate enough, due the fact that the crack passes from stress state caused by the vertical moment to that of horizontal moment. For these transient locations, the compliance functions are interpolated using B-splines interpolation [31], in an attempt to achieve more exact calculation of local compliances. A different approach is given by Jun et al. [12], where the sign of the total stress intensity factor, K I ¼ K In þ K Ig at each point along the crack front determines whether the crack is open or closed at the point. Jun’s results are compared with current approach in Figs. 5a and b, giving thus the comparison of both approaches. The diagram in Fig. 4 shows the change in four compliances as a function of rotational angle for the crack  ¼ 0:4. depth a

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Fig. 4. Compliance variation as a function of rotational angle and comparison with two different approximation for: (a) c55, (b) c44 and (c) c45 in the case of 40% crack depth.

z , then the following double integrals for the compliance are obtained , and ky ¼ y 1 ¼ y 1þa 1 =h If one defines y

ER3 32  ; uÞ ¼ c55 ða  ; uÞ c55 ða ¼ 1  m2 p

Z

ER3 16 ¼ 1  m2 p

Z

ER3 32 ¼ 1  m2 p

Z

 ; uÞ ¼ c44 ða  ; uÞ c44 ða  ; uÞ ¼ c45 ða  ; uÞ c45 ða

 ða fb 1  ;uÞ

 ða fb 2  ;uÞ  ða fb 2  ;uÞ

 ða fb 1  ;uÞ  ;uÞ fb2 ða

 ;uÞ fb1 ða

Z

pffiffiffiffiffiffiffiffi 1z2

 h  kþ 2

Z

pffiffiffiffiffiffiffiffi 1z2

 h  kþ 2

Z

pffiffiffiffiffiffiffiffi 1z2

 h  kþ 2



 2  1 F 2 ky dy  dz 1  z2 y

ð1Þ

  z2 y  dz 1 F 21 ky dy

ð2Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi     zy  dz 1 1  z2 F 1 ky F 2 ky dy

ð3Þ

where f = 1 for a/R 6 1 and f = 0.95 for a/R > 1 [32], and the bar on each distance variable indicates normalization in respect to R.

  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    ffi   F 0 ky ¼ tan pky =2 = pky =2 = cos pky =2    h     3 i F 1 ky ¼ F 0 ky 0:752 þ 2:02 ky þ 0:37 1  sin pky =2    h   4 i F 2 ky ¼ F 0 ky 0:923 þ 0:199 1  sin pky =2

ð4Þ ð5Þ ð6Þ

Using Fig. 4, matrix C can be determined for every value of u = Xt. See Table 1 and [31] for specific angles. The local compliance values of current approach are compared in Figs. 4 and 5, to the respective approaches of the literature, giving good agreement. The value of the total flexibility Ctot of the cracked shaft can be calculated for every value of time t so as to define the breathing of the crack. The flexibility of the uncracked shaft is defined for every direction (vertical and horizontal) as Cuncr55 = Cuncr44 = =L3/48EI, and the additional flexibility due to the crack is defined for the vertical and horizontal directions as C55 = c55L2/16 and C44 = c44L2/16 respectively. Also, the additional flexibility due to the coupling the crack provokes

A.C. Chasalevris, C.A. Papadopoulos / Mechanism and Machine Theory 44 (2009) 1176–1191

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Fig. 5. Total stiffness of the shaft, as a function of the rotational angle of the crack.

is related to the compliances, defined as C54 = c54L2/16 for the vertical plane and as C45 = c45L2/16 for the horizontal plane. The total direct flexibility is defined as C totii ¼ C uncrii þ C ii for both main directions, the vertical (5) and the horizontal (4), considering that the shaft and the crack are springs in a sequence. The cross-coupled total stiffness is expressed as C totij ¼ C ij since the initial cross-coupled stiffness of the intact system does not exist. The inverse of total flexibility (total stiffness) is plotted in Fig. 5 in dimensionless form and presents the breathing of a crack of depth a/R = 0.4. A comparison is made in Fig. 5 with corresponding results from [12]. The dimensionless stiffness value is expressed as ðC totii =C uncrii Þ1 . A small deviation is noticed between current approximation and reference [12] during the breathing due to the different assumption of crack geometry while rotation. In the cases that the crack is assumed as fully open the agreement is good since both approximations of crack compliance are based in the same assumptions. 3. A continuous model approach for the rotating cracked shaft 3.1. Equation of motion In this approach to the motion of the rotating shaft, the rotary inertia, the shear deformation, the torque of power transmission and the gyroscopic effect are taken into consideration. Assume a uniform homogenous and cracked rotating Timoshenko shaft (Fig. 1) of Young’s modulus of elasticity E = 205.8 GPa, shear modulus G = 79.76 GPa, moment of inertia of the shear factor k = 10/9 for circular cross-section, cross-section about X axis I = 3.06  107 m4, mass density q = 7860 kg m3 p, ffiffiffiffiffiffiffi length L = 2 m, radius of cross-section R = 0.025 m, radius of gyration r0 ¼ I=A ¼ 0:0125 m, and Poisson ratio m = 0.29. The shaft is rotating with a rotational speed X, whirling with a frequency x, and transmitting a power with an axial torque T. Also consider a transversely located disk in the mid-span (x = L/2) of the shaft of the same material, with radius Rd = =0.15 m, mass  ¼ a=R, exists in the mid-span, just next to the disk. If md = 27.78 kg, and thickness Ld = 0.05 m. The breathing crack, of depth a Y(x, t) and Z(x, t) are the vertical and horizontal response of the axial coordinate x and time t, respectively, then by supposing the complex notation Ui(x, t) = Yj(x, t) + i Zj (x, t), the coupled governing equation of motion is written as [9,33]:

EI

  4 o4 U j o3 U j EIq o Uj o3 U j T q o3 U j q2 Ar20 o4 U j q2 Ar20 X o3 U j o2 U j 2  iT  q Ar þ 2iqAr 20 X 2 þ i þ  2i þ qA 2 ¼ 0 þ 0 2 2 4 3 2 ox4 ox3 kG ox ot kG kG kG ox ot oxot ot ot ot ð7Þ

where j = 1 for the first part of the shaft from the left end up to the crack and j = 2 for the part from the crack up to the right end (Fig. 1). Eq. (7) is a fourth order complex partial differential equation that has a solution of the form:

U j ðx; tÞ ¼ uj ðxÞeixt ¼ pj ekj x eixt

ð8Þ

Substitution of Eq. (8) into Eq. (7) yields the complex characteristic Eq. (9), a fourth degree polynomial equation of kj,





x2 iRTkj þ q EIk2j þ Ar20 qxðx  2XÞ





þ kG Aqx2 þ k2j iTkj þ EIk2j þ Ar 20 qxðx  2XÞ ¼ 0

ð9Þ

Two of the roots of (9) are complex numbers; meanwhile, the other two are imaginary. Actually the two imaginary roots are complex with near zero real parts. Let us set the roots as kj,1, kj,2, kj,3, kj,4. Then the partial solution becomes [9],

uj ðxÞ ¼ qj;1 ekj;1 x þ qj;2 ekj;2 x þ qj;3 ekj;3 x þ qj;4 ekj;4 x ;

j ¼ 1; 2

ð10Þ

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Coefficients qj,1, qj,2, qj,3, and qj,4, are also complex numbers that are defined using the boundary conditions for displacement, slope, bending moment, and shearing force, which in a Timoshenko shaft with gyroscopic effect are defined as in [9,33] and [34], Displacement: Uj(x, t) oU ðx;tÞ

Slope: Hj ðx; tÞ ¼ jox 3

o3 U ðx;tÞ o2 U ðx;tÞ o U j ðx;tÞ o2 U j ðx;tÞ Shearing force: V j ðx; tÞ ¼ EI oxj 3  iT oxj 2  qAr 20 oxot  2iX oxot 2 Bending moment: Mj ðx; tÞ ¼ EI

o2 U j ðx;tÞ ox2

 iT

oU j ðx;tÞ ox

3.2. Boundary conditions The cracked rotor is assumed to be mounted in hinged supports at both ends. The disk and the crack are assumed to be in the mid-span (L1 = L/2 = 1 m) with the crack in the right side of the disk. Also Ip ¼ 0:5md R2d ¼ 0:31252 kg m2 and Is ¼ 0:25md R2d þ 1=12md L2d ¼ 0:162 kg m2 are the mass polar moments of inertia about axis Ox and diametric axis of the disk _

_

respectively (see Fig. 1); Hj ðx; tÞ and Mj ðx; tÞ are the conjugate complex numbers of Hj(x, t) and Mj(x, t), respectively. For this specific problem, mu = 0.01 kg, ru = 0.1 m, and uu are the unbalance mass, the distance from the centre of the rotation, and the angle between the vector of unbalance and the horizontal axis, respectively. For simplicity, ru is a constant distance and uu = 0. Due to the fact that the excitation of the system is the unbalance force, the whirling of the shaft is synchronous with the rotation, which means x = X. Therefore the boundary condition (BC) of shearing force, for the case of a disk with and without unbalance, becomes as in Eqs. (11) and (12), respectively.

  V 2 ðL1 ; tÞ ¼ V 1 ðL1 ; tÞ  md x2 U 1 ðL1 ; tÞ þ mu r u x2 eixt   V 2 ðL1 ; tÞ ¼ V 1 ðL1 ; tÞ  md x2 U 1 ðL1 ; tÞ

ð11Þ ð12Þ

At both ends, the displacement and the bending moment are equal to zero:

U 1 ð0; tÞ ¼ U 2 ðL; tÞ ¼ 0

ð13Þ

M1 ð0; tÞ ¼ M 2 ðL; tÞ ¼ 0

ð14Þ

In the mid-span: (a) The bending moment at the left side of the crack (the same as the right side of the disk) results in a slope discontinuity in the two parts of the shaft which is described by Eq. (15): _

ðe þ idÞM1 ðL1 ; tÞ þ ðb þ icÞ M1 ðL1 ; tÞ ¼ ðH2 ðL1 ; tÞ  H1 ðL1 ; tÞÞ

ð15Þ

where e = 0.5(c55 + c44), b = 0.5(c55  c44), c = 0.5(c45 + c54), and d = 0.5(c45  c54) (b) The bending moment on both sides of the disk follows the equation: _

M2 ðL1 ; tÞ ¼ M 1 ðL1 ; tÞ þ i H1 ðL1 ; tÞðIp Xx  Is x2 Þ

ð16Þ

(c) The displacements at both sides of the disk are equal:

U 1 ðL1 ; tÞ ¼ U 2 ðL1 ; tÞ

ð17Þ

When the crack is closed (see Figs. 2 and 4), the local compliances become zero and the boundary condition of Eq. (15) is transformed into Eq. (18), describing the continuity in slopes at both sides of the disk:

H1 ðL1 ; tÞ ¼ H2 ðL1 ; tÞ

ð18Þ

3.3. Calculation of gravity response As mentioned above, the crack breathes due to a gravity effect in the elastodynamic behavior of the shaft. The gravity response is assumed to surpass the unbalance response so as to set the crack condition (breathing). In theory, this happens when the rotational speed is not near the critical speed, but in practice the crack breathing due to gravity can be observed at every speed under special geometric characteristics of large rotating machines. In this paper, the gravity is assumed to be static (independent of time) and the gravity response Ugi(x) is obtained by the ordinary differential equation (19) as in [34]: 4

EI

3

d Ug j ðxÞ d Ug j ðxÞ  iT ¼ qAg; dx4 dx3

i ¼ 1; 2

ð19Þ

By neglecting the torque for simplicity, when the response due to the gravity is evaluated, and for hinged-hinged boundary conditions with the disk at the mid-span, the solution for the two parts of the shaft becomes

Ug 1 ðxÞ ¼ 

qAg 24EI

x4 þ

gmd þ 2AgLq 3 3gL2 md þ 4AgL3 q x  x 12EI 12EI

ð20Þ

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for 0 6 x 6 L=2 Ug 2 ðxÞ ¼ 

qAg

24EI for L=2 6 x 6 L:

x4 

gmd  2AgLq 3 gLmd 2 9gL2 md þ 4AgL3 q gL3 md x þ x  xþ 12EI 2EI 12EI 6EI

ð21Þ

4. Dynamic analysis 4.1. Calculation of critical speeds In order to present the effect of coupling at critical speeds, the cracked and the uncracked models are analyzed. When no crack exists, the boundary condition of Eq. (15) is substituted by Eq. (18). By substituting the general solution

U j ðx; tÞ ¼ ðqj;1 ekj;1 x þ qj;2 ekj;2 x þ qj;3 ekj;3 x þ qj;4 ekj;4 x Þeixt

ð22Þ

into the eight boundary conditions, Eqs. (11)–(18) the homogeneous system (8  8) for the variables qj,1 qj,2 qj,3 qj,4 for j = 1,2 of Eq. (23) is obtained

 ½P q1;1

q1;2

q1;3

q1;4

q2;1

q2;2

q2;3

q2;4

T

T

¼ f0 0 ... 0g |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð23Þ

8

The real and imaginary roots of jPj = 0 for x = X are the critical speeds. The roots of real part of characteristic equation are the vertical critical speeds, and the roots of the imaginary part of the characteristic equation are the critical speeds corresponding to the horizontal plane. For the cracked model, the boundary condition of Eq. (15) introduces the parameter of time into the model. Time is a factor that specifies the crack local compliance in accordance with the breathing condition. This fact makes the characteristic determinant also a function of time. For example, if t = 1/4 (2p/X), the crack is semi-closed (Fig. 2, u = 90°) and the local compliances have different values from those of any other value of time in the same period. As shown in Fig. 4, there is no value of time for which all compliances are equal to each other, except if t 2 [ucl/X,ucl/X] where  ¼ 0:4 starts to open. The characteristic determinant is calculated for ucl = 36° is the angle of rotation in which a crack of a a specific value of time (i.e in the 1/4 of period), and plotted as real parts in Fig. 6a and as imaginary parts in Fig. 6b as a function of X/x0. The value x0 is used in this paper to express frequency in dimensionless form and is defined as x0 = (EI/qAL4)1/2. For the geometry of this section x0  16. The zero crossing values express the eigenfrequencies for the instant form of the crack (here the crack is instantaneous semi-open in the 1/4 of period) and the logarithmic magnitude is used here so as to achieve the visibility of function progress since in areas near roots it tends rapidly from / to +/. 4.2. Calculation of the frequency response The response in the frequency domain is calculated by an iterative procedure. In each iteration a rotating/whirling frequency value is defined and the time response for one period of time, in steps of Dt = 0.01(2p/X)s, is calculated. In each time response, the amplitude is measured and diagrams of amplitude as a function of frequency are obtained. The cases for the uncracked and cracked shaft are investigated. The procedure is repeated for a frequency range higher than the third critical speed with a frequency step change of DX = 1 rad s1, so as to define the frequency areas of critical speeds (see Fig. 7), and then a step change of DX = 0.001 rad/s is used in order to examine the effect of coupling from both sides of first critical speed (see Fig. 8). Note that the resonance in the 2nd critical speed does not appear in this response because the measurement is

Fig. 6. (a) Real and (b) imaginary part of the characteristic determinant as a function of X.

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Fig. 7. (a) Vertical and (b) horizontal amplitude as a function of rotating/whirling speed.

Fig. 8. (a) Vertical and (b) horizontal amplitude as a function of rotating/whirling speed near first critical speed.

taken in the mid-span, which is a nodal point of the second eigenmode. For simplicity, only forward whirling will be examined. An obvious difference in amplitude change is observed in Fig. 8 when comparing the cases of uncracked and cracked shafts. The amplitude of the uncracked shaft has one peak in the first critical speed and this also applies for horizontal and vertical amplitude in the same rotating frequency, as long as the system is isotropic. When the crack is introduced, each resonance has two peaks at a lower rotating speed than that of the uncracked one. The resonance frequencies that correspond to the vertical amplitude are a bit smaller compared to those of the horizontal amplitude. This is due to the local stiffness of the crack, which always remains larger in the horizontal plane than in the vertical. These additional peaks are observed in both planes at the same value of the rotating frequency because of the coupling. Note that the magnitude is larger for smaller DX and Dt. It is a fact that the critical speed shift (cracked/uncracked) calculated from frequency response of Fig. 8 is miniscule and may not be noticed in practice. However the case with compliance coupling terms neglected provide a greater frequency shift since the coupling is not always an additional magnitude in the compliance since their signs vary positively and negatively as many researchers have presented [27,35]. Additionally a recent experimental work in cracked beam [36] provides the miniscule eigenfrequency shift when the compliance coupling terms are in their large values (crack orientation in u = 90°; see Fig. 3). On the other hand an appropriate time–frequency decomposition using wavelet transform encourage this miniscule eigenfrequency shift notification since this decomposition can localize it is components in such a ‘‘narrow” time domain between the double peak in the frequency response. 4.3. Numerical example 1 Under an unbalance excitation of 0.001 kg m, the cracked rotor vibrates with the effect of coupling. When an unbalance exists, the general solution (22) is substituted in the boundary condition of Eq. (11) instead of that of Eq. (12), and the following non-homogeneous (8  8) system of Eq. (24) is obtained

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Fig. 9. (a) Vertical and (b) horizontal time response in mid-span for the two cases, in rotational speed X = x = 90.0 rad/s.

 ½P q1;1

q1;2

q1;3

q1;4

q2;1

q2;2

q2;3

q2;4

T

 ¼

0 0 . . . 0 m u r u x2 |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

T

ð24Þ

7

As in Section 4.2, the cracked and uncracked cases are compared. For each case, the forward synchronous (X = x) whirling time response is calculated in the mid-span for both planes of vibration and for rotational speeds different from the first critical, in order to avoid singularities. The rotational speed is set to X ¼ 5:625 (Fig. 9), X ¼ 5:831 (a speed very close to critical (Fig. 10)), and X ¼ 5:937 (a speed after the first critical (Fig. 11)). It is clear that the response due to coupling is observed for crack rotational angles near 90, 180 and 270° (Fig. 2) or 1/4, 1/2 and 3/4 of the time period. In these angles of rotation, the coupling terms c54 and c45 get the higher absolute values. This difference due to coupling has been also observed by Darpe et al. [27] in 2002. The opening of the crack permits an additional response in the vertical direction near 180°, but also an additional response near 90 and 270° as shown in Figs. 9–11. The time response, treated as a signal, can be analyzed in its components with the fast Fourier transform (FFT) in order to more clearly observe the effect of the coupling in additional harmonic development. Each of the signals of Fig. 9, are trans-

Fig. 10. (a) Vertical and (b) horizontal time response in mid-span for the two cases, in rotational speed X = x = 93.3 rad/s.

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Fig. 11. (a) Vertical and (b) horizontal time response in mid-span for the two cases, in rotational speed X = x = 95.0 rad/s.

formed using FFT and the respective results are shown in Fig. 12. The component of X ¼ 5:625, is synchronous with the rotational/whirling frequency and, as a result, has the highest amplitude. Also, the components with frequencies of 2/rev, 3/rev, 4/rev and more are seen to have lower amplitudes as the frequency increases [28]. It is known that a crack introduces such harmonics as in [28], but in the case where the crack introduces a coupling, these harmonics have higher amplitudes. The coupling effect and the additional harmonic development become more intense as the vibration amplitude increases and this happens as the system gets near to resonance. At these speeds the coupling phenomenon provokes additional components with their amplitude is greater with respect to this of synchronous component than this of cases far from resonance. The domination of synchronous response is of course observed but the relative amplitude becomes greater. The same results are also verified in the signals of Fig. 10, (see Fig. 13) where the highest amplitude is observed at the synchronous frequency X ¼ 5:831, while the other components appear in higher harmonics with lower amplitude. The FFT of signals for X ¼ 5:937, in Fig. 11, are judged as not necessary for plotting such as those of Fig. 9. Note that at the speed of X ¼ 5:937 there is a phase inversion in the time domain response. A general conclusion derived from spectra given in Figs. 12 and 13 is that since the coupling is introduced mainly two times in the period duration the 2Xrev harmonic is mainly amplified. The development/ amplification of the 2Xrev harmonic is of course the main result of the crack breathing no matter what modeling of breathing is used but the co-existence of coupling increases the amplification/development of 2Xrev. 5. The resilient bearing model In the previous numerical example, the bearing was assumed to be rigid in order to focus on the crack effect. The coupling introduced by the crack is presented in the same way as the coupling due to the cross-coupled bearing stiffness and damping

Fig. 12. FFT of (a) vertical and (b) horizontal response for X = x = 90.0 rad/s.

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Fig. 13. FFT of (a) horizontal and (b) vertical response for X = x = 93.3 rad/s.

Fig. 14. The resilient bearing model.

coefficients. This is also shown in the frequency response domain by double peaks. In this numerical example, a cracked rotor model in resilient bearings is analyzed, and results for frequency response are obtained. In Fig. 14, the rotor-bearing system characteristics are shown, as given by Hong and Park [37] for comparison purposes. A shaft of diameter D = 25 mm, length L = 1200 mm, made of steel with E = 200 GPa and density q = 8000 kg m3, is supported by two bearings at its two ends, with the following dynamic properties:

 Kb ¼ Cb ¼



kyy kzy cyy czy



"

0 1:2  106 ¼ kzz 0 1:2  106    cyz 6 0 ¼ Nsm1 czz 0 6 kyz

#

Nm1

In this model, the boundary conditions at both ends of the shaft are calculated in complex form by combining the boundary conditions for the spring and damper given as in (25) and (26). The forces applied by the bearing are proportional to the displacement Ui(x, t) = Yj(x, t) + i Zj (x, t), as well as to its first time derivative. Finally, the two boundary conditions take the form as in (25) and (26).

o o _ U 1 ð0; tÞ þ ðp2 þ ip3 Þ U 1 ð0; tÞ ot ot _ o o _ V 2 ðL; tÞ ¼ ðs1  is4 ÞU 2 ðL; tÞ þ ðs2 þ is3 Þ U 2 ðL; tÞ þ ðp1  ip4 Þ U 2 ðL; tÞ þ ðp2 þ ip3 Þ U 2 ðL; tÞ ot ot _

V 1 ð0; tÞ ¼ ðs1  is4 ÞU 1 ð0; tÞ þ ðs2 þ is3 Þ U 1 ð0; tÞ þ ðp1  ip4 Þ

ð25Þ ð26Þ

where, s1 = 0.5(kyy + kzz), s2 = 0.5(k  kzz), s3 = 0.5(kyz + kzy), s4 = 0.5(kyz  kzy), p1 = 0.5(cyy + czz), p2 = 0.5(cyy  czz), p3 = 0.5 _ yy (cyz + czy), p4 = 0.5(cyz  czy) and U i ðx; tÞ ¼ Y j ðx; tÞ  iZ j ðx; tÞ is the complex conjugate value of the displacement Ui(x, t). In the present case, where an unbalance excitation is taken at the right end, the respective boundary condition of Eq. (26) becomes _

V 2 ðL; tÞ ¼ ðs1  is4 ÞU 2 ðL; tÞ þ ðs2 þ is3 Þ U 2 ðL; tÞ þ ðp1  ip4 Þ

o o _ U 2 ðL; tÞ þ ðp2 þ ip3 Þ U 2 ðL; tÞ  mu r u x2 eixt ot ot

ð27Þ

If a crack is introduced in the isotropic system of Fig. 14, then the frequency response obtained (Fig. 15) has double peaks in the 1st, the 3rd and the 5th resonance, as presented in Fig. 16 (these figures zoom in on the respective resonances). Due to the location of crack at the mid-span, the crack has no effect in the 2nd and the 4th modes since the mid-span in these modes is a nodal point.

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Fig. 15. Left journal amplitude (continuous line: uncracked, dotted line: cracked 40% of the radius).

Fig. 16. Left journal amplitude for the case of isotropic system (continuous line: uncracked, dotted line: cracked 40% of the radius) for (a) the 1st, (b) the 3rd and (c) the 5th resonance.

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Fig. 17. Left journal amplitude for the case of anisotropic system without crack (continuous line: vertical, dashed line: horizontal).

Fig. 18. Left journal amplitude for the case of anisotropic system with a crack (continuous line: vertical, dashed line: horizontal).

Fig. 19. Left journal amplitude for the case of anisotropic system (continuous line: uncracked, dotted line: cracked 40%).

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5.1. Numerical example 2 In the case where the bearing coefficients are not equal to each other in the two main directions, the frequency response displays double peaks, as in the case of the crack, but at much different eigenfrequencies since the system stiffness shift due to bearing anisotropy is much more greater than that due to crack rotation. If the horizontal stiffness is changed to kzz = 8  105 Nm1, the frequency response is changed to that of Fig. 17. When the crack is introduced, then the frequency response is formed as in Fig. 18. A detailed look at Fig. 18 shows that when the crack is introduced, the existing coupling (due to anisotropy) is presented in a different way combined with that due to crack coupling. In Fig. 19, zooming in on the 3rd resonance shows that when no crack exists, the coupling due to anisotropy is shown by two peaks. Note that this coupling does not need cross-coupling bearing properties in order to be provoked, but can be provoked even with zero cross-coupling bearing properties due to the coupled equations of the shaft motion. When the crack is present, the additional coupling due to the crack arises by introducing two more peaks in the frequency response (see Fig. 19). Then, there are a total of four peaks in each resonance; two of them are present due to crack breathing coupling and two due to coupled bearing properties and coupled equations of motion. 6. Conclusions In this paper, the problem of coupled bending vibrations in a cracked rotor mounted in resilient bearings was investigated. The phenomenon of coupling introduced by a transverse crack forces a significant change to the dynamic properties that are already affected by the crack existence. As was shown, the total stiffness in the two main directions experiences a change dependent on the crack local compliance change during rotation, making thus the system non-linear. The proposed method in this study offers a continuous model for a cracked rotor-bearing system, including three parameters of coupling: the coupled equations of motion, the coupling due to the breathing crack and the coupling due to cross-coupled bearing coefficients for stiffness and damping. The observations and the conclusions made from current analysis can be regarded as in following: 1. The coupling affects the response in the time and frequency domains and amplifies the higher harmonics that the crack introduces as components in the vibration. 2. The resilient bearing analysis, including cross-coupling stiffness and damping coefficients, provides evidence that the coupling due to a crack is negligible in respect to the coupling that the bearings introduce. 3. The higher harmonics that are introduced in the vibration spectrum due to the stiffness change caused by the breathing of the crack clearly indicate the existence of a crack since the amplification of 2Xrev harmonic amplitude is increased from the crack cross-coupling compliances. Future work combining the current continuous shaft model with finite bearings could investigate the effect of the coupling due to a breathing crack when the stiffness and damping properties of the bearing are not set for a specific equilibrium position, but instead for a real journal trajectory inside the bearing, since this motion is much more closer to the reality when the system passes through a critical speed. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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