International Journal of Engineering Science

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1 International Journal of Engineering Science 48 () Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: wwwelseviercom/locate/ijengsci General harmonic balance solution of a cracked rotor-bearing-disk system for harmonic and sub-harmonic analysis: Analytical and experimental approach Mohammad A AL-Shudeifat a, Eric A Butcher a, *, Carl R Stern b a Department of Mechanical and Aerospace Engineering, New Mexico State University, PO Box 3, MSC 345, Las Cruces, NM 883, USA b Management Sciences Inc, 6 Constitution Ave NE, Albuquerque, NM 87, USA article info abstract Communicated by M Kachanov Keywords: Damage detection Structural health monitoring Breathing crack model Cracked rotor Harmonic balance method The effect of crack depth of a rotor-bearing-disk system on vibration amplitudes and whirl orbit shapes is investigated through a general harmonic balance technique and experimental verification Two models of the crack, which are the breathing and the open crack models, are considered Finite element models and general harmonic balance solutions are derived for breathing and open cracks which are valid for damped and undamped rotor systems It is found via waterfall plots of the system with a breathing crack that there are large vibration amplitudes at critical values of crack depth and rotor speed for a slight unbalance in the system The high vibration amplitudes at the backward whirl appear at earlier crack depths than those of the forward whirl for both crack models Resonance peaks at the second, third and fourth subcritical speeds emerge as the crack depth increases It is shown that the unique signature of orbits for the breathing crack model which have been verified experimentally can be used as an indication of a breathing crack in the shaft In addition, the veering in the critical frequencies has been noticed in the open crack case Ó Elsevier Ltd All rights reserved Introduction Damage detection in rotor dynamic systems has had a great deal of attention in the past few decades Destructive vibration amplitudes may appear in rotating shafts that are used in different industrial applications due to propagating cracks These high amplitudes of vibration, which appear due to cracks, yield an unpredicted failure and a possible damage of machine components A breathing crack in the transverse direction of the shaft is one of these dangerous damage scenarios in rotor-dynamic systems The breathing mechanism of this type of crack in heavy-duty rotating machineries is mainly due to the gravity force that caused by the shaft weight and leads to crack propagation with time This crack model has been used extensively in modeling damage in beams and shafts Some studies have focused on the open crack model while several studies have focused on the switching and breathing crack models in rotating shafts The coupling of longitudinal and bending vibration in a cracked shaft with an open transverse crack model has been studied by deriving the local flexibility matrix of the cracked shaft [] As a result, the frequency equation was derived and solved for the natural frequencies of the system It has been noticed that an instability region exists and that there are variations in the critical frequencies as the crack depth increases The same issue has been studied again in [] but with a stationary shaft that has two breathing cracks The cracks were modeled with a compliance matrix where the breathing mechanism was found to depend on the excitation load direction The analytical and experimental results have verified the effect of the * Corresponding author addresses: shdefat@nmsuedu (MA AL-Shudeifat), eab@nmsuedu (EA Butcher), Carl_Stern@mgtsciencescom (CR Stern) -75/$ - see front matter Ó Elsevier Ltd All rights reserved doi:6/jijengsci5

2 9 MA AL-Shudeifat et al / International Journal of Engineering Science 48 () coupling on both vertical and horizontal vibrations The nonlinear dynamics of the flexible cracked Jeffcott rotor on a simple rigid support was studied in [3] with both switching crack and breathing crack models Chaos and bifurcation were observed only in the case of a switching crack However, the breathing crack model can represent the physical crack mechanism in the rotor better than the switching crack and often describes crack propagation in real life applications better than other models In [4 6] the characteristics of the sub- and super-harmonics of the cracked rotor with breathing cracks were used for crack detection in rotor systems The harmonic balance method was employed in solving the cracked rotor system with a breathing crack model It was found that with an increase in the crack size, new resonance peaks emerged at the second, third and fourth super-harmonic frequency components which can be used as an indication of crack propagation The nonlinear behavior of the cracked rotor was also studied in [7] on a well known simple rotor The results showed that the peaks appear at half and one third of the critical frequencies which are caused by the second and third super-harmonics The flexibility matrices of the cracked element of the shaft were utilized for modeling the breathing crack [8 ] In addition, the finite element method (FEM) was employed in solving the cracked rotor system In [] it was shown that the transverse breathing crack can be detected through the characteristics of the second and third harmonic components while the slant rotor crack can be detected by observing the sub- and super-harmonic components It is also observed that the transverse breathing crack is highly sensitive to the mechanical impedance when compared with a slant crack Some researchers employed the transfer matrix method in the cracked rotor analysis The global and local asymmetry transverse crack models have been employed to predict the rotor system response characteristics via the transfer matrix method where the second harmonic characteristics are used in monitoring the crack in the system [] In addition, the transfer matrix method was utilized to find the cracked rotor response of a simple rotor model where the temporary whirl reversal and phase shift were observed to occur near the critical and subcritical speeds since there is an unstable range at some neighborhood of these critical speeds [] An experimental analysis of a cracked rotor was performed in [3] where the effect of the crack depth and the additional eccentricity was verified experimentally via the orbits, time histories and waterfall plots of the shaft with an open crack Most of the above techniques have considered discrete depths of the crack at different locations along the shaft In [4] the sign change of the stress intensity functions (SIFs) was used in a breathing crack modeling for a FEM of a rotor-disk system with a fatigue transverse crack The flexibility matrix was calculated and the FEM equations of motion were solved using the Newmark method of direct numerical integration The effect of coupled torsional and lateral vibration has been investigated In addition, wavelet transforms (WTs) was also employed for investigating the transient features of bending vibration at resonance A theoretical cracked beam model is used for detecting cracks in power plant rotating machines in [5] The study included theoretical, numerical and experimental analysis of the model The bending vibration amplitudes in the neighborhood of the first subcritical speed have allowed the detection of a crack in which a good match was found between the numerical and experimental results A review of strain energy release rate approach (SERR) for different modeling techniques of open, switching and breathing cracks and their corresponding methods of solution was introduced in [6] In this study the harmonic balance technique is employed for finding the critical and subcritical vibration speeds of a rotor-disk-bearing system for harmonic and sub-harmonic analysis The solution is employed in studying the behavior of the shaft with either open or breathing cracks The theoretical results are experimentally verified using Spectra-Quest rotor-dynamic simulator system with the same parameters used in the theoretical model The development of methods to track more severe crack depths and their corresponding orbit shapes is addressed in this research in addition to the veering phenomenon in the critical frequencies of the cracked rotor where an exchange of modes takes place [7 9] It is found that there are slight breathing crack depths at which high vibration amplitudes with a unique whirl orbit shapes appear These amplitudes of vibration, which appear at these low crack depths, are larger at the backward whirling frequency than those at the forward whirling frequency The unique whirl orbit shapes that appear at low breathing crack depths and are verified experimentally can be used as an early indication of the breathing crack propagation Thus, the appearance of a breathing crack in a rotor system may explain a sudden and destructive damage in rotor-dynamic systems Rotor-disk-bearing system modeling Rotor modeling The finite element method (FEM) is employed in modeling the rotor system as follows The rotor of mass M and length L is divided to N-elements with N + nodes along the z-axis as shown in Fig The finite element equation of motion of the N- element rotor with N + nodes is given by [,] Fig Finite element model of the rotor

3 MA AL-Shudeifat et al / International Journal of Engineering Science 48 () M qðtþþðcþgþ _qðtþþkqðtþ ¼F u ðtþþf g ; ðþ where qðtþ ¼ q T qt qt i q T T Nþ is the 4(N +) dimensional nodal displacement vector, q T i ðtþ ¼ u i m i / x i / y i is the single node displacement vector consisting of the translational and rotational displacements about the stationary X and Y axes for i =,,, N +,F u ðtþ is the 4(N +) unbalance force vector, and F g is the 4(N +) gravity force vector The M, K, C, G are the global mass, stiffness, damping, and gyroscopic matrices, respectively, where each is of dimension 4(N +)4(N + ) The jth element has two nodes at i = j and i = j + The single element equation of motion is therefore obtained as m j ^q j ðtþþðc j þ g j Þ _^q j ðtþþk j^q j ðtþ ¼^f uj ðtþþ^f gj ; ðþ where ^q j ðtþ ¼ u i m i / x i / y i u iþ m iþ / x T iþ /y iþ ; ^f uj ðtþ and ^f gj are the 8 unbalance force and gravity force vectors at the jth element nodes, ^f T T T uj ðtþ ¼ F i u ðtþ F iþ u ðtþ where F i uðtþ and Fiþ u ðtþ are the 4 unbalance force vectors at nodes i and i +, respectively, m j ¼ m j þ m j where m j is the jth element classical mass matrix and m j is the jth element matrix of the secondary effect of the rotary inertia, k j is the jth element stiffness matrix, c j is the jth element damping matrix where c j ¼ ck j and c is the proportional damping coefficient, g j is the jth element gyroscopic matrix The expressions for m j, k j and g j are found in references [,] Disk modeling The disk is placed at a given node that has four degrees of freedom Therefore its nodal displacement vector is q T i ðtþ ¼ u i m i / x i / y i which is the same as that of the disk center If the rotor has rigid disks, the mass center of each rigid disk is assumed to lie at the node that is shared between two elements as shown in Fig Ifthere is a disk at node i of mass m i, thickness d hi, outer radius R i, inner radius Ri in, dimetral moment of inertia Ii d ¼ mi = d 3 R i in þ 3 R i i o þ h and polar moment of inertia I i p ¼ mi = d R i in þ R i o Hence, the equations of motion of the disk at node i are given by [,] m i d u i 3 _u i m i d m i 6 4 I i d / x 7 i 5 þ X _m I i i p 5 _/ x 7 4 i 5 ¼ ; ð3þ I i / y d i I i p _/ y i where the first matrix is the classical mass matrix and the second one is the gyroscopic matrix of the disk and X is the shaft rotational speed Since the disks are assumed to be rigid, the stiffness matrix vanishes 3 Bearing modeling The bearings can be either journal or ball bearings The model of a bearing is shown in Fig 3 The equations of the forces at the bearings are obtained by neglecting the influence of slopes and the bending moment [] Hence, if there is a bearing at node i, the equations of forces at that node are given by F u k xx u i c xx _u i F v k yy m i 6 7 ¼ F / x / x 7 5 þ c yy _m i i 56 _/ x 7 4 i 5 : ð4þ F / y i / y i _/ y i The first matrix is the stiffness matrix and the second is the viscous damping matrix These matrices are diagonal and symmetric Fig The rotor-disk finite element model

4 94 MA AL-Shudeifat et al / International Journal of Engineering Science 48 () Fig 3 Bearing model of the rotor-disk-bearing system 4 Cracked element modeling The transverse crack in the rotor system can be modeled as a transverse breathing crack, which opens and closes in a synchronous manner as the shaft rotates The breathing mechanism of the crack comes from the fact that the shaft static deflection is much greater than the deflection due to the dynamic response of the cracked rotor in heavy-duty rotating machinery [4 6,8,4] Fig 4(a) shows the open and closed states of the crack at different time values based on the crack geometry in Fig 4(b) It can be seen that there is a reduction in the cross-sectional area moment of inertia of the element that includes a crack We derived the moment of inertia formulas about the x and y axes of the crack cross-section with crack depth h as shown in Fig 4(b) as I c x ¼ pr4 8 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R b br b 3 R 4 sin b ; ð5aþ 4 R I c y ¼ 3R4 sin s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ R s R ð3sr 6s 3 Þþ8bs 3 ; ð5bþ where l ¼ h is the non-dimensional crack depth, h is the crack depth of the shaft in the radial direction, a = R cos ( l) is the angle of the crack, s = R sin (a/) and b = R cos (a/) In addition, the area and the centroid of the of the cracked element cross-section at the crack location are respectively given by A ce ¼ R p a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ b R b ; ð6aþ y ce ¼ 3A R b 3 : ð6bþ Uncracked segment I, I uc x uc y t = π t = t = π 3π t = t = π Crack segment c I, c x I y (a) (b) Fig 4 Breathing crack model: (a) crack state variations for different time values, (b) geometry of the cracked element

5 MA AL-Shudeifat et al / International Journal of Engineering Science 48 () The area moments of inertia of the cracked element cross-section about its centroidal axes are written as I ce x ¼ I Ic x þ A ce y ce ; ð7aþ I ce y ¼ I Ic y ; ð7bþ where I = pr 4 /4 For I ¼ðI c x þ A ce y ce Þ and I ¼ I c y the stiffness matrix that include the effect of the crack can be given as [4,6] 3 I 6lI li 6lI I 6lI I 6lI 6lI 4l I 6lI l I k j c ¼ E 6lI 4l I 6lI l I l 3 : ð8þ li 6lI I 6lI I 6lI I 6lI 6 4 6lI l I 6lI 4l I 7 5 6lI l I 6lI 4l I Hence, if there is a breathing crack in element j, the new cracked element stiffness matrix k j ce can be expressed based on Eq (7) and the use of the breathing function in [4 6] by k j ce ¼ kj f ðtþk j c ¼ kj cos Xt ð Þkj c ; ð9þ where k j is the 8 8 element stiffness matrix when the crack is fully closed and f(t) expresses the breathing function of the crack Therefore, the equations of motion of the cracked rotor-disk-bearing system in physical coordinates can be rewritten as M qðtþþðc þ GÞ _qðtþþðk f ðtþk c ÞqðtÞ ¼F u ðtþþf g ; ðþ where K c is a 4(N +) 4(N + ) stiffness matrix of the cracked shaft of zero entries except at the cracked element where the corresponding entries are equal to k j c 3 Final model of the cracked rotor-bearing-disk system and solution A mass unbalance m i e can be situated at time t = at one or more of the disks of the rotor If this mass situated in a disk located at node i, then the 4 unbalance force vector at that node becomes: 3 m i e dx cosðxt þ hþ F i u ðtþ ¼ m i e dx sinðxt þ hþ ; ðþ where d is the distance between the mass and the rotor center and h is the angle with the x coordinate This can be rewritten as F i u ðtþ ¼Fi cosðxtþþfi sinðxtþ; ðþ where the 4 unbalance amplitude force vectors at node i are 3 3 cosðhþ sinðhþ F i ¼ sinðhþ mi e dx ; Fi ¼ cosðhþ mi e dx : ð3þ Hence, the total force vectors of a rotor-disk-system with unbalance masses at different nodes are T T T T T F ¼ F F F i F Nþ ; ð4aþ T T T T T F ¼ F F F i F Nþ : ð4bþ Both F and F have dimension 4(N +) For the nodes that do not have unbalance masses, the corresponding force vectors in F and F are zeros The equation of motion in () can thus be rewritten as

6 96 MA AL-Shudeifat et al / International Journal of Engineering Science 48 () M qðtþþc b _qðtþþ K e K c cos Xt qðtþ ¼F cos Xt þ F sin Xt þ F g ; ð5þ where e K ¼ K K c and b C ¼ C þ G The solution is expressed as a finite Fourier series as qðtþ ¼A þ Xn ða k cos ðkxtþþb k sin ðkxtþþ: ð6þ k¼ Inserting this solution in Eq (5) yields: Fig 5 Finite element model of rotor-disk-bearing system There are 8 elements and the crack location is at element 6 Table Physical parameters of the MFS RDS rotor-dynamic simulator Description Value Description Value Length of the rotor, L 74 m Disk outer radius, R o 76 m Radius of the rotor, R 588 mm Disk inner radius, R i 588 mm Density of rotor, q 78 kg/m 3 Density of disk, q 7 kg/m 3 Modulus of elasticity, E N/m Mass of the disk, m d 57 kg Bearing stiffness, (k xx, k yy ) 7 7 N/m Mass unbalance, m e d 6 kg m Bearing damping, (c xx, c yy ) 5 Ns/m Mass unbalance angle, h rad Fig 6 Frequency veering phenomena in the first eight pairs of the natural frequencies of the critical forward whirl (solid) and the critical backward whirl (dashed) for the cracked rotor-disk-system versus crack depth for open crack in element 6

7 MA AL-Shudeifat et al / International Journal of Engineering Science 48 () bk C ðþ C ðþ C ðþ C ðþ C ðþ C ðþ C ðþ C ðþ C ðþ C ðþ C ðþ C ðþ C ðþ C ð3þ C ð3þ C ð3þ C ð3þ C ð3þ C ð3þ C ð3þ C ð4þ C ðn Þ C ðn Þ C ðn Þ C ðnþ C ðnþ C ðnþ 7 5 C ðnþ C ðnþ C ðnþ A B A B A 3 A n B n A n B n 3 3 ef F ¼ ; ð7þ where b K ¼ e K X M 8 K c e K K c ; C ðjþ ¼ e K ðjxþ M; C ðjþ ¼ Xb C; C ðjþ ¼ 4 K c and e F ¼ F K c e K F g ; j ¼ ; ; ; n, and n is the number of harmonics used Eq (7) is solved for the coefficient matrices A k and B k (k =,,, n) Four harmonics were found sufficient for solving the system, while A is found as A ¼ K e F g 4 K ca : ð8þ 4 Fully open crack model solution The equations of motion of the cracked shaft with open crack model in which the stiffness matrix is assumed to be constant (K K c ) are given as M qðtþþ b C _qðtþþðk K c ÞqðtÞ ¼F cos Xt þ F sin Xt þ F g : ð9þ Inserting the Fourier series solution of Eq (6) into Eq (9) yields: Fig 7 Waterfall plot of the vibration amplitudes versus the non-dimensional crack depth and the rotor speeds for the breathing crack that located in element 6 The vibration amplitudes were calculated at node

8 98 MA AL-Shudeifat et al / International Journal of Engineering Science 48 () " ðk K c Þ X M XG # A XG ðk K c Þ X M B ¼ F ; ðþ F where all A k and B k are zeros for k =,, n and A =(K K c ) F g The eigenvalues of the fully open crack problem are found from the eigen solution of M qðtþþ b C _qðtþþðk K c ÞqðtÞ ¼: ðþ 5 Theoretical results The undamped rotor-bearing-disk system is divided into 8 elements where the unbalance mass is attached to the right disk as shown in Fig 5 The values of the physical parameters are given in Table Slight unbalance of m e d ¼ 6 kg m is assumed while the crack is fully closed at t = The results in Fig 6 of the frequency versus crack depth for an open crack located at element 6 show the veering phenomena when the non-dimensional crack depth is near l Fig 7 shows the waterfall diagram for the system with breathing crack, which was obtained via the Fig 8 Waterfall plot of the vibration amplitudes versus the non-dimensional crack depth and the rotor speeds for the open crack that located in element 6 The vibration amplitudes were calculated at node Fig 9 Waterfall plot of the vibration amplitudes versus the non-dimensional breathing crack depth at the first critical backward whirling frequency based on the harmonic balance solution The vibration amplitudes were calculated at node

9 MA AL-Shudeifat et al / International Journal of Engineering Science 48 () Fig Critical rotor speeds of high vibration amplitudes versus the non-dimensional crack depth l of the rotor-bearing-disk system for different crack models: (a) first critical forward whirling speeds, (b) first critical backward whirling speeds general harmonic balance solution The plot shows the vibration amplitudes versus crack depths and shaft speeds using four harmonics which was found to be sufficient to solve the equations of motion up to l = 5 The waterfall plot of the system with an open crack model is plotted in Fig 8 The following observations are noted: () The first pair of the natural frequencies is more sensitive to changes in crack depth for the open crack than for the breathing crack () The amplitude increase appears in a wider range of rotor speed in the breathing crack model than in the open crack model (3) The sub-harmonic amplitudes in the neighborhood of the forward or backward whirl become clear as the breathing crack depth increases (4) The waterfall plot for the breathing crack model gives the exact critical speeds of the highest vibration amplitudes in the cracked system Fig 9 shows the behavior of the cracked system at the backward whirl as the crack starts to appear at very low crack depths while it is clear that the change in the amplitude with crack depth is fluctuating between high and low values The shift in the critical forward and backward whirling speeds of both crack models versus crack depth is plotted in Fig based on the general harmonic balance solution and waterfall plots It is shown that the critical forward and backward whirling rotor speeds of the open crack model are more sensitive to changes in crack depth than that in the breathing crack model especially at high crack depths In addition, the harmonic balance solution is efficient in showing the shift in the first, second and third pairs of the critical sub-harmonic rotor speeds as shown in the waterfall plots and the plots of the critical sub-harmonic speeds versus crack depth in Fig The vibration amplitudes of subcritical speeds increase as the crack depth increases as shown in Fig Fig 3 show the orbit changes at node for different rotor speeds in the neighborhood of the first critical forward whirling speed corresponding to breathing crack depth of l = 5 An interesting phenomenon has been noticed in Figs 4 and 5 as the shaft speed passes through the subcritical speed of X = 53 rpm corresponding to the forward whirl and X = 467 rpm corresponding to backward whirl for a very small breathing crack depth l = 5 These shapes of whirl orbits that appear in a neighborhood of the subcritical speeds are nearly similar to those in [5 7,] for breathing crack model In addition, at the subcritical speed X = 53 rpm that corresponds to crack depth l = 5, it is found that as the crack starts to propagate the orbit starts to have interesting shapes as shown in Fig 6 Hence these orbits are unique signatures of the breathing crack and their changes at very low crack depths can be used as an early indication of a propagating crack 6 Experimental results The rotor-dynamics simulator (MFS RDS) that was supplied by Spectra-Quest, Inc was used in performing the experiments on the rotor-bearing-disk system A series of experiments have been done on a cracked shaft of the same parameters in Table except for the unbalance amplitude and location where m e d ¼ 6:3 5 kg m was installed on node 6 Two pairs of proximity probes were installed to the system: the first pair was installed at node and other was installed at node 8 to measure the horizontal and vertical amplitudes of vibration Two cracked shafts have been used in the experiments which are named as shaft A and shaft B For shaft A the transverse crack was made in element 6 using an EDM wire machine The

10 93 MA AL-Shudeifat et al / International Journal of Engineering Science 48 () Fig waterfall plots and the corresponding plots of rotor speed versus non-dimensional crack depth for the first three subcritical rotor speeds, (a) and (b) first subcritical rotor speeds, (c) and (d) second subcritical rotor speeds, (e) and (f) third subcritical rotor speeds of forward and backward whirling speeds crack depth was increased after each experiment to obtain data at different crack depths (ie, l = 5, l = 3, l = 45, l = 6, l = 76 and l = ) and the transverse thickness of each crack was inch Shaft B, which was supplied by Spectra-Quest, has a built-in breathing crack at element 6 of depth l = 45 Fig 7 shows the shift in theoretical rotor speeds of the highest vibration amplitudes versus l of shaft A for both breathing and open crack models The experimental values of rotor speeds for shaft A of the highest vibration amplitudes are plotted on the same figure where the shift in these values versus l was found to be close to the theoretical values of the breathing crack model especially at relatively high crack depths Fig 8 shows a good match between the theoretical and the experimental orbits in the neighborhood of the subcritical rotor speeds for the breathing crack model in shaft B This match appears even though the vibration amplitudes in the neighborhood of the subcritical speeds exceeded the static deflection range It is interesting to note that these orbits appear theoretically and experimentally for a wide range of rotor speeds in the neighborhood of the subcritical speeds such that they can be easily obtained experimentally We conclude that the breathing crack corresponds to unique orbit shapes for a wide range in the neighborhood of subcritical speed which can be used in detecting and identifying the crack type in the rotor-bearing-disk system In addition improved breathing crack model and the proposed general harmonic balance solution were found to be efficient in predicting the behavior of the breathing crack in the rotor-bearing-disk system for critical and subcritical analysis which is verified experimentally

11 MA AL-Shudeifat et al / International Journal of Engineering Science 48 () Fig The vibration amplitudes of node of the critical harmonic and subcritical rotor speeds for different non-dimensional breathing crack depths: (a) l = 5, (b) l =, and (c) l = 5 Fig 3 Shaft whirl orbits of node for different rotor speeds in the neighborhood of the first critical forward speed x f = 343 rpm corresponding to breathing crack depth l = 5

12 93 MA AL-Shudeifat et al / International Journal of Engineering Science 48 () Fig 4 Shaft whirl orbits of node for different rotor speeds in the neighborhood of the first critical forward subcritical rotor speeds speed X ffi (/ )x f = 53 rpm corresponds to breathing crack depth l = 5 Fig 5 Shaft whirl orbits of node for different rotor speeds in the neighborhood of the first critical backward subcritical rotor speeds speed X ffi (/ )x b = 467 rpm corresponds to breathing crack depth l = 5

13 MA AL-Shudeifat et al / International Journal of Engineering Science 48 () Fig 6 Shaft whirl orbits of node for different non-dimensional crack depths at the first critical forward subcritical rotor speed X = 53 rpm that correspond to l = 5 Fig 7 First theoretical critical rotor speeds of high vibration amplitudes versus non-dimensional crack depth l of the system for different crack models and the corresponding experimental values for breathing crack in shaft A

14 934 MA AL-Shudeifat et al / International Journal of Engineering Science 48 () Fig 8 Theoretical and experimental whirl orbits of node for l = 453 of shaft B of the system in the neighborhood of the subcritical whirl with m e d = 63 5 kg m at node 5 and the breathing crack is in element 6; (a c) theoretical whirl orbits and (d f) experimental whirl orbits of the shaft 7 Conclusions This study introduces an efficient technique for solving and studying the behavior of the cracked rotor system The general harmonic balance solution of the cracked rotor-bearing-disk system with breathing crack has been derived for studying the behavior of the system The results of this method show important observations of the behavior of the whirl orbits, vibration amplitudes and frequencies of a damaged rotor-disk-bearing system This behavior may help in detecting the crack at the beginning of its growth The behavior of the vibration amplitudes at different rotor speeds and crack depths is found via generating waterfall plots In addition, this method of solution predicts the changes in the orbit shapes at critical and subcritical rotor speeds at very low crack depths Hence, tracking the change in orbit shapes in the neighborhoods of subcritical rotor speeds at low crack depths can be used as an earlier indication of a propagating breathing crack Moreover, the exact critical and subcritical speeds of the system with a breathing crack have been obtained from the waterfall plots Consequently, the plots of the shift in the critical and subcritical rotor speeds of high vibration amplitudes at different crack depths have been determined using this method of solution which may help in damage detection The considerable differences in the results between the breathing crack model and the open crack model in the cracked system have been verified via waterfall plots The vibration amplitudes and the orbits of the system can be detected experimentally by using proximity probes which help in earlier detection of the crack in the rotor-disk system The accuracy in the experimental results can be increased if the crack thickness in the transverse direction of the shaft is considerably decreased and the shaft length and diameter are increased such that the breathing of the crack becomes more accurate Therefore, the experimental orbits are expected to be more close to the theoretical orbits In addition, the theoretical model has been assumed to be linear which explains the small deviation between the experimental and theoretical results References [] CA Papadopoulos, AD Dimarogonas, Coupled longitudinal and vertical vibrations of a rotating shaft with an open crack, Journal of Sound and Vibration 7 () (987) 8 93 [] AC Chasalevris, CA Papadopoulos, Coupled horizontal and vertical bending vibrations of stationary shaft with two cracks, Journal of Sound and Vibration 39 (7) [3] TH Petal, AK Darpe, Influence of crack breathing model on nonlinear dynamics of a cracked rotor, Journal of Sound and Vibration 3 (8) [4] J Sinou, Detection of cracks in rotor based on the and 3 super-harmonics frequency components and the crack unbalance interactions, Communications in Nonlinear Science and Numerical Simulation 3 (8) 4 4 [5] J Sinou, AW Lees, A non-linear study of a cracked rotor, European Journal of Mechanics 6 (7) 5 7 [6] J Sinou, AW Lees, The influence of cracks in rotating shafts, Journal of Sound and Vibration 85 (5) 5 37 [7] L Xiao-feng, X Ping-yong, S Tie-lin, Y Shu-zi, Nonlinear analysis of a cracked rotor with whirling, Applied Mathematics and Mechanics 3 () 7 73 [8] AS Sekhar, BS Prabhu, Transient analysis of a cracked rotor passing through critical speeds, Journal of Sound and Vibration 73 (3) (994) 45 4 [9] AS Sekhar, Crack identification in a rotor system: a model-based approach, Journal of Sound and Vibration 7 (4) [] AS Sekhar, AR Monhanty, S Prabhakar, Vibration of cracked rotor system: transverse crack versus slant crack, Journal of Sound and Vibration 79 (5) 3 7 [] I Green, C Casy, Crack detection in a rotor dynamic system by vibration monitoring Part I: analysis, Journal of Engineering for Gas Turbine and Power 7 (5) [] OS Jun, MS Gadala, Dynamic behavior analysis of cracked rotor, Journal of Sound and Vibration 39 (8) 45

15 MA AL-Shudeifat et al / International Journal of Engineering Science 48 () [3] T Zhou, Z Sun, J Xu, W Han, Experimental analysis of a cracked rotor, Journal of Dynamic Systems, Measurements, and Control 7 (5) 33 3 [4] AK Darpe, A novel way to detect transverse surface crack in a rotating shaft, Journal of Sound and Vibration 35 (7) 5 7 [5] CA Papadopoulos, The strain energy release approach for modeling cracks in rotors: a state of the art review, Mechanical Systems and Signal Processing (8) [6] CM Stoisser, S Audebert, A comprehensive theoretical, numerical and experimental approach for crack detection in power plant rotating machinery, Mechanical Systems and Signal Processing (8) [7] NC Perkins, CD Mote, Comments on curve veering in eigenvalue problems, Journal of Sound and Vibration 6 (3) (986) [8] C Pierre, Mode localization and eigenvalue loci veering phenomena in disordered structure, Journal of Sound and Vibration 6 (3) (988) [9] PT Chen, JH Ginsberg, On the relationship between veering of eigenvalue loci and parameter sensitivity of eigenfunctions, Journal of Vibration and Acoustics 4 (99) 4 48 [] M Lalanne, G Ferraris, Rotor Dynamics Prediction in Engineering, second ed, John Willy and Sons, New York, 998 [] T Yamamoto, Y Ishida, Linear and Nonlinear Rotordynamics, second ed, John Willy and Sons, New York, [] AK Darpe, K Gupta, A Chawla, Transient response and breathing behavior of a cracked Jeffcott rotor, Journal of Sound and Vibration 7 (4) 7 43