Abstract
Damages in rotor systems have severe impact on their functionality, safety, running durability and their industrial productivity, which usually leads to unavoidable economical and human losses. Rotor systems are employed in extensive industrial applications such as jet engines, gas and steam turbines, heavyduty pumps and compressors, drilling tools, and in other machineries. One of the major damages in such systems is the propagation of fatigue cracks. The heavyduty and recurrent cyclic fatigue loading in rotor systems is one of the main factors leading to fatigue crack propagation. For the past few decades, numerous research have been conducted to study crack related damages and various methodologies were proposed or employed for damage detection in rotor systems. Therefore, the purpose of the present review article is to provide a thorough analysis and evaluation regarding the associated research related to the modeling aspects of rotor systems that are associated with various kinds of (rotor related) damages. Based on this review, it is observed that the crack modeling, especially with the breathing crack type in rotor systems, is still based on few primary models. Several researchers, based on different assumptions, have extended and modified such models to be more reliable for analysis. Moreover, the arising demand for early crack detection has led to utilization of various tools such as Fast Fourier transform, Hilbert Huang transform, wavelet transform, whirling analysis, energy methods, and the correlation between backward whirling and rotor faults etc. In addition, the significant impact of nonsynchronous whirl within resonance zones of rotor systems on postresonance backward whirl, under various rotor related faults, is also highlighted in the present review. Therefore, the review provides an evaluation and comparison between several crack models and detection methodologies in rotor systems. Moreover, this review could help in identifying the gaps in modeling, simulation, and dynamical analysis of cracked rotor systems to establish robust research platform on cracked rotor systems.
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1 Introduction
It is imperative to understand and study various faults related to rotor systems to avoid catastrophic failure and prevent economic and human losses. The present section provides a brief summary of initial studies related to the subject of rotor dynamics and crack damages in rotor systems. Various studies were employed to rotor systems since the end of the nineteenth century where some of major contributions are summarized in Table 1. One of the earlier works related to cracks in rotor systems was conducted by Gasch [1], and Henry and OkahAvae [2]. They studied the vibrations with consideration of the nonlinear stiffness due to closing and opening of the crack. They reported about the presence of an unstable region at major critical speed based on the relative orientation of unbalance forces. Later in 1976, a breathing crack model, which took into account the periodic opening and closing of a crack, was developed by Mayes and Davies [3]. In addition, more realistic crack models were developed later which will be further assessed in later sections. Subsequently, research related to crack detection has been active since the 1980’s with various studies available in the literature [4,5,6,7,8]. The studies in [4,5,6,7,8] focused on the existence of first and second harmonic components in lateral vibration for crack detection since the crack induces notable changes in these components. The effect of cracks was also studied in [9, 10] of rotor systems exhibiting torsional vibration. Finite element methods (FEMs) were applied to a cracked rotor system by Chen [11]. The Floquet’s stability analysis was employed by Gasch [12] for identifying the effect of a crack on the stability of rotor systems. A brief overview of the advancements in research related to rotor dynamics is summarized in Fig. 1. It is evident from this figure and Table 1 that there has been abundant and progressive research in the field of rotor dynamics since the beginning of the industrial revolution. Further studies on diverse topics about the rotor systems during the twentieth century can be also found in [13,14,15,16,17].
Although there are some review articles published in the literature in [55,56,57,58,59] accounting for the various relevant analysis related to cracked rotor systems, the present article focuses on being more comprehensive on the subject of crack modeling and detection techniques. In the subsequent sections, various aspects related to the cracked rotor system will be established. In Sect. 2, the faults that occur in rotor systems are described while in Sect. 3 the types of cracks are discussed. Section 4 covers various crack models utilized across different research articles. Section 5 focuses on major crack detection methodologies. Section 6 addresses the nonlinear dynamics of a cracked rotor system while the conclusion are provided in Sect. 7.
2 Faults in rotor system
The area of fault diagnosis in rotor systems is an everrelevant area of research. With continuous heavyduty operations of rotating machineries, there will be the development of various kind of faults and fatigue damages. Therefore, the overarching goal of structural health monitoring in rotor systems is the development of early fault diagnosis and prognosis. The diagnosis can help in identifying and localizing the fault and its extent while prognosis can help in optimizing equipment reliability, durability, and maintainability. The major faults in rotor systems, that were actively investigated in the past decades, are summarized as the following.

1.
Cracks propagation in machinery shafts or rotor blades.

2.
Misalignment of machinery shafts.

3.
Rotor–stator rubbing.

4.
Bearing anisotropy and other related faults.

5.
Bowing of the rotor shaft.

6.
Unbalance forces in the rotating parts.

7.
Fluidinduced instability.
The cause and attribution of each fault can be different, but there are cases when the faults occur simultaneously. In the following subsections, various faults will be discussed along with a review of the associated literature for identification of such faults.
2.1 Rotor–stator rubbing
Rotor–stator rubbing occurs as a secondary effect due to various primary factors that minimize clearance distance between the rotor and stator [14]. The simply supported rotor model in Fig. 2, in which a rigid disk is attached to an elastic massless shaft, is provided to demonstrate the rotor–stator rubbing. The rubimpact can either be in the form of partial or full annular rub. Partial rubbing occurs when the rotor–stator contact occurs for a short time duration in the vibration cycle while annular rubbing has longer and continuous duration. The presence of rub has been analyzed using signal processingbased techniques which included empirical mode decomposition (EMD) [60], Wavelet transform (WT) [61], Hilbert Huang transform (HHT) [62]. Most of these techniques were employed to identify rubbing in rotor systems based on frequency response analysis. The effect of rub impact in rotor systems has also been studied extensively as can be seen from the review articles in [63,64,65].
2.2 Unbalance
Unbalance forces can arise in rotor systems when the shaft’s mass center does not coincide with the centerline of its rotation. As such, the rotor exhibits periodic forcing in its whirling plane, which elevates the whirl amplitudes during the passage through resonance rotational speeds. Depending on unbalance force magnitude, that rotor system could develop destructive levels of vibration amplitudes that might cause a catastrophic failure to the whole system. A nonlinear mathematical model was developed in [66] for identifying the plane of the rotor in which the unbalance occurs that has a direct implication in turbomachinery operations. There are various modelbased techniques used to analyze the unbalance effect in rotor systems. In [67] the unbalance in a rotor system was detected based on its location and severity by using equivalent loads minimization method accompanied with a theoretical fault model and vibration minimization method. A Kalman filter based method has been applied in [68], to detect the presence of unbalance in the rotor system. Methods based on modal expansion, optimization, and inverse problem have been used to analyze and predict the unbalance related parameters as in [69]. Moreover, unbalance parameters identification based on unbalance force reconstruction was employed in [70].
2.3 Rotor bow
The rotor bow occurs when the shaft centerline is in a bent condition as demonstrated in the schematic plot in Fig. 3b when compared to that of the intact rotor in Fig. 3a. The rotor bow can be categorized as elastic, temporary and permanent (plastic) based on the initiating parameters [71]. Elastic bow is generated by static radial loads that has magnitudes below the shaft’s yield strength. Temporary bow is caused by uneven heating of the rotor surface or by the anisotropic material thermal properties whereas the permanent plastic bow is caused when the stress in certain parts of the rotor exceeds the yield strength limits. Although both rotor bow and unbalance forces generate 1X harmonic components, the rotor bow produces significant forward whirling 1X component at slow roll speed [71]. In [72], polynomial functions have been proposed to identify and predict the shape of the thermal bow in an aeroengine based on temperature and displacement results. A novel identification method has been used to identify the shaft bow, unbalance effect, and the flexible supports effect in [73] during transient operations. A thermal bow in a generator shaft with various fault initiation conditions has been studied in [74]. Machine learning tools were also used for identification of rotor bowing in [75,76,77]. A more detailed review of various research related to rotor bow can be found in [78].
2.4 Rotors misalignment
Misalignment can be categorized as an internal or external condition in rotor systems. Internal misalignment occurs when the average shaft centerline of a single machine is not at the design position. However, external misalignment occurs when two or more shafts are not perfectly aligned at coupling as demonstrated in Fig. 4. External misalignment can be further categorized as either parallel or angular misalignment. Parallel misalignment occurs when two or more shafts are displaced in the transverse directions and angular misalignment occurs when one of the shafts is angularly displaced with respect to the other shaft [71]. It was observed in [79] that the presence of misalignment in rotors can be diagnosed by using the second and fourth harmonic components. In [80, 81], an estimation method was utilized for identifying both unbalance and misalignment faults during rotor transient operations. Variational mode decomposition and probabilistic principal component analysis in conjunction with convolutional neural network was used to characterize shaft misalignment and crack propagation in [82]. In [83], the authors have studied the presence of both angular and parallel misalignment between the rotor and the active magnetic bearings. The major theme of the study in [83] was to understand the severity of misalignment faults by using modelbased identification algorithm that employs Fast Fourier Transform (FFT) generated response harmonics to estimate the fault parameters.
2.5 Fluidinduced instability
Fluidinduced instability occurs when the rotor and surrounding fluid interact with each other. This interaction may result in producing large selfexcited amplitudes of vibration during rotor operation. The main phenomena associated with fluidinduced instability are the oil whip and the oil whirl. Oil whip occurs when the angular speed of the rotor is close to that of the first critical speed while oil whirl occurs when the whirling frequency is near to onehalf of the rotational angular speed. In [84], the effect of oil whip and oil whirl was studied with the help of vertical and horizontal test rig by varying the journal bearing parameters (diameter ratio, bearing length, oil film viscosity, and radial clearance) and unbalance moments. In [85], a full Hilbert spectrum has been used to study the coexistence of oil whip and dry whip that caused by rotor–stator rubbing. Recently, a polynomial chaos expansion model was used in [86] to study the stochastic response of rotorbearing model with oil whip and oil whirl. Moreover, in [87] a new technique in the form of multisensor fusion approach was experimentally employed for early detection of oil whip and oil whirl phenomenon.
2.6 Bearing related faults
Bearings are integral components in rotor systems where various faults could exist across several types of bearing components. A few widely used bearings include journal bearing, magnetic bearing, roller bearing, etc. One of early research that addressed bearing faults diagnosis was in [88] wherein cyclostationary statistics approach was applied for identification of rotor element and inner race faults in roller bearings. A comparison between this cyclostationary statistics’ approach and minimum entropy deconvolutionspectral kurtosis has been conducted in [89] for roller bearing fault identification. Faults in rotormagnetic bearings were investigated by using WT in [90]. A failure detection mechanism based on correlation functions and artificial neural networks (ANN) was proposed in [91] to identify both mechanical and electrical failures of the active magnetic bearings. Accordingly, the authors detected the corresponding electrical faults generated from the magnetic actuators, position sensors, and unbalancerelated faults. Moreover, EMD along with HHT was used for fault identification of journal bearings in [92].
2.7 Cracks faults
Rotors usually undergo periodical change in stress distribution during their rotation where such change can be synchronous or nonsynchronous with the shaft rotational speed. Rotors usually experience low fatigue cyclic loading during their synchronous rotation. On the other hand, nonsynchronous rotation of the shaft can induce high fatigue cyclic loading which causes a stress reversal [93]. This cyclic stress fluctuations can result in fatigue cracks’ initiation and propagation in rotating equipment. Once a crack is initiated, it can propagate and results in failure of the whole rotor system. As such, an early detection of cracks propagation in rotor systems is of high priority to save human and equipment. Blade tip timing method and convolutional neural network (CNN) was utilized in [94] to detect cracks in turbomachinery blades. Accordingly, the authors reported that they were able to identify the depth of the crack and its relative location on the shaft with an accuracy exceeding 85% in multiple trials. Additionally, in [95] kernel independent component analysis together with wavelet neural network were used for identifying the crack location in turbine blades. An acoustic emission approach using an airborne sound has been used in [96] for damage detection in wind turbine blades. The authors reported good damage identification rates. Compared to cracks in rotor blades, cracks in shafts have been an active area of research and the topic of the present review.
3 Types of cracks in rotor systems
There are several types of cracks that exist in rotating shafts based on the crack orientation with respect to the shaft’s centerline. Accordingly, cracks are usually classified as either slant, longitudinal or transverse as shown in Fig. 5.
3.1 Slant crack
When a slant crack appears in the shaft, it propagates at a specific angle with respect to shaft’s centerline (Fig. 5a). The slant crack usually propagates due to fatigue under high torsional moment as addressed in [97]. The rotor’s stiffness matrix inclusive of the slant crack was developed to be superimposed in the FEM analysis. A comparison approach between eigenfrequencies of rotor systems with slant against transverse cracks was established in [98]. It was observed that the eigenfrequencies of a shaft with slant crack decreased slightly in comparison with a shaft with a transverse one. In addition, a comparison of the mechanical impedance between systems with slant and transverse cracks was considered in [99]. The modeling, detection techniques, and dynamical response of the considered rotor systems with both types of cracks have been discussed. The mechanical impedance was found to be more sensitive to the transverse crack rather than the slant crack. Furthermore, the rotor model with slant crack was used to study the vibration response of the rotating system in [100], where a comparison of stiffness coefficients between slant and transverse cracks was addressed. It was found that the slant crack had higher crosscoupled stiffness magnitude compared to the transverse crack.
3.2 Longitudinal crack
Longitudinal cracks are oriented in a direction parallel to the rotational axis of the shaft (Fig. 5b) and they are less common in occurrence when compared with transverse and slant cracks. A study in [101] was performed to investigate the vibration whirl response of a rotating shaft with longitudinal crack. It was observed that vibration whirl amplitudes of the rotating shaft closely correlate with the longitudinal crack location and its depth. Many researchers have employed the longitudinal crack to study the vibration whirl response for the purpose of developing crack detection methods as discussed in [102, 103]. A wind turbine rotor with a longitudinal crack was tested for investigating the dynamical behavior of the cracked rotor system in [104]. It was observed that the strain, displacement, and stress are directly proportional to the crack size. Additionally, an increment in the initial six natural frequencies was observed when the corresponding crack size decreased.
3.3 Transverse crack
Transverse crack is the most common type of crack that has been extensively analyzed in literature. The transverse crack is oriented in a direction that is orthogonal to the rotor’s rotational axis in the transverse direction (Fig. 5c). The transverse crack is usually caused by shaft’s fatigue due to heavy cyclic loading, which generates high bending moment. This crack is considered to be one of the most damaging scenarios that can occur in a rotor system, altering the shaft’s stiffness integrity which can further lead to failure of the whole rotor system. Transverse cracks can be classified into two types open cracks and fatigue cracks. Fatigue cracks are further subcategorized into breathing and switching cracks. When a crack continuously opens and closes during the shaft’s rotation, due to fluctuations between tensile and compression stress fields, it is considered as a breathing crack. Therefore, this crack leads to nonlinear dynamical behavior of the rotor system. The switching crack is described by a scenario when the shaft undergoes an abrupt fullopen and fullclosed within complete revolution. Open crack is described by the scenario when the crack remains continuously open during the shaft’s rotation. Furthermore, the transverse crack front could have straight front [105] or an elliptical front [106] (shown in Fig. 6). It was stipulated that in [106] the straight edge crack was a special case when the crack front shape factor became zero.
4 Models of rotor cracks
There are many factors that contribute to cracks initiation and propagation in rotor systems. Some of these can be attributed to the presence of a sudden change in geometry leading to stress concentrations (such as holes and threads), thermal stress (because of altered operational regime), cyclic fatigue loading, etc. This can be substantiated with some of the industrial examples reported in the literature. It was found in [107] that fatigue cracks developed in three turbine rotor shafts as a result of the operation during lower magnitude of gravitational bending stresses. In two cases, the depth of the fatigue cracks extended higher than 75% of the shaft’s cross section and for 1/16 inch deeper of the diameter in the other case. It was also reported that the cyclic thermal stresses, caused by frequent startup and shutdown of the steam turbine rotors, have generated cracks in the systems [108]. The development of cracks was also reported to be associated with high engine vibrations that lead to fatigue in rotor systems [109,110,111]. Similar examples of such industrial instances can be found in [112, 113]. The research on the crack modeling in rotor shafts has been active since the 1970’s [58, 114]. Although there are many researchers who studied the propagation of cracks in machineries, the modeling aspect of utilizing variation of stiffness in the equations of motion was pioneered by Gasch [115], Henry and OkahAvae [2], Mayes and Davies [3], and Dimarogonas [116]. The models in [2, 115] introduced varying flexibility for open and closed cracks and applied to a Jeffcott (JC) rotor model. The work in [3, 116] has utilized the fracture mechanics concept to model cracks by varying stiffness of the system. In the following sections, various crack models used in literature for studying their effect on rotor whirling dynamics are presented.
4.1 Dimarogonas’ crack model
The presence of a crack on the rotor shaft produces varying flexibilities along the crack tip in the orthogonal direction to the shaft’s centerline. As such, when the shaft rotates, the stiffness in fixed coordinates will vary resulting in a system with timevarying moments of inertia. One of the early studies to develop a crack model for rotor systems was reported by Dimarogonas in which he utilized the theory of fracture mechanics to model the crack [116]. The shaft’s crosssection was assumed to consist of multiple rectangular strips of different heights which are orthogonal to the tip of the crack and parallel to the axis of symmetry of the circular rotor shaft. It was assumed that there was no traction between these strips. The compliance was found by integrating the strain energy release rate (SERR) function parallel to the transverse xaxis that was coincident with the crack tip. An open crack model in JC rotor shaft was also studied by using the local flexibility from the ParisErdogan equation in [117]. The presence of a crack introduced variation in the local flexibility which is calculated by considering the plane strain assumption. Accordingly, the resultant formulation of the local flexibility was deduced as follows
where P indicates the applied load, \(\nu\) is the Poisson ratio, \(K_{I}\) represents the modeI stress intensity factor (SIF), E is the modulus of elasticity, b is the halfwidth of the crack, \(\xi\) and \(\eta\) are the rotational coordinates. The compliance function was then converted into a nondimensional form and compared with the results produced by Grabowski in [118]. For the cases of which the crack is near vertical, there is abrupt change of stiffness from crackfree state to cracked one. During these situations, the stiffness was determined experimentally at a crack angle with respect to the vertical axis and given as Fourier series of this angle as follows [117]
Additionally, an analytical approach was also introduced for a closing crack based on the higher magnitude of the static deflections which rationally replicates turbomachinery applications. SERR rates for a circular shaft with edge crack under bending were studied in [119]. It was found that the SERR for a circular shaft was lower than that of a rectangular bar with similar crack size undergoes bending moment. In [120], the coupled bending and longitudinal vibrations for a JC rotor containing a transverse crack (in an open state) was studied for damped and undamped models. The 6 × 6 local flexibility matrix for a transverse crack was developed based on the SIFs and superimposed with the global flexibility matrix for modeling the corresponding equations of motion. The local flexibility matrix due to the crack was given in [120] as
where \(\overline{c}_{ij} \left( {i,j = 1,2,3, \ldots 6} \right)\) represents the corresponding nondimensional compliance coefficients, E is the Young’s modulus, R is the radius of the shaft and \(\nu\) is the Poisson ratio. The authors assumed that the crack remains always open whenever the vibrational amplitude of shaft’s selfdeflection was dominant in comparison with other components. A similar crack model was applied in [121] to analyze the effect of a crack in Timoshenkotype beam. The breathing effect of the crack was simulated by the varying the SIFs. In [122], a modified version of Eq. (3) was used for modeling the compliance matrix due to the appearance of open crack. This was given as follows
In a later study in [123], the compliance matrix for a halfopen or halfclosed crack was proposed as
Equations (4) and (5) were applied in [124,125,126] to model two open transverse cracks. For cracked rotors with two different crack depths, the larger crack shows more influence on the eigenfrequency, especially for low slenderness ratio shafts. In [127], a variation of the Dimarogonas 6 × 6 stiffness matrix model was proposed and applied to a rotor with the breathing crack. Timoshenko beam was used for modeling the rotor system while also accounting for shear forces. The proposed model was further applied in [128] for studying transverse cracks in rotating shafts subjected to axial impulses. The research in [129] utilized the model in [120] where energy variation in a cracked shaft during a single full revolution was evaluated.
The open crack model was proposed in [130] using FEM and fracture mechanics taking into account torsional and bending interaction. In addition, the presence of shear forces along with the bending moment was considered in the model where the resultant stiffness matrix had a size of 10 × 10 considering 5 degreeoffreedom (DOF) at each node. TMM in conjunction with Timoshenko beam theory was employed in [131] to model for the open transverse crack of a stepped shaft in multidisk rotor. The crack was modeled as a local spring with constant stiffness coefficient value derived from the strain energy and SIF like [117]. This approach was also extended to a multistep multicrack rotor system in [132]. In [133], the modeling of a straight edge and an elliptical arc crack was proposed by using SIF and FEM. This crack model also incorporated the fatigue crack growth in [134]. It was reported that the SIF of the crack is indirectly proportional to the crack depth ratio, and directly proportional to the crack aspect ratio. Accordingly, the fatigue crack growth was given in [134] as
where C, n, p and q are material constants, f is a function of crack opening, R is the stress ratio, \(K_{c}\) is the critical SIF, and \(\Delta K_{th}\) is the threshold of SIF.
In [135], a modification to the original breathing crack model by Dimarogonas was proposed. However, Papadopoulos [136] indicated a certain discrepancy in the updated model in [135]. There was an appearance of singularity when the cracked depth exceeds the radius of the shaft and a value of \(0.90 \le \zeta \le 0.95\) where, \(\zeta\) is the ratio of integration length of the compliance to the crack length. In [137] the Dimarogonas’ model was applied to transverse as well as slant cracks in rotor systems subjected to torsional excitations. The slant crack resulted in a static coupling of the bending, torsion, and tension stiffnesses. However, the transverse crack resulted in a static coupling of only the bending and tension stiffnesses. Experimental and numerical studies were conducted in [137] where the coupled vibration was found to be stronger for the rotor with a slant crack than the transverse crack. Furthermore, the presence of combination of frequencies \(\left {\Omega \pm \omega_{T} } \right\), where \(\omega_{T}\) is the torsional excitation frequency and \(\Omega\) is the angular speed, was used to differentiate between the transverse and slant crack in rotor systems. The Dimarogonas’ model was extended in [138] by accounting for the contribution of the transverse general forces to the crack SIF which further resulted in additional ten offdiagonal terms in the flexibility matrix. The authors observed that these nondiagonal terms had a peak value for a partially open crack and disappeared for a fully open crack.
In [139], threedimensional (3D) timedependent FEM was adopted to study the local flexibility terms in the offdiagonal direction like the ones in Eq. (3) by considering the frictional crack closure. The resultant method allowed for computations of the respective offdiagonal terms in the local flexibility matrix for a periodic breathing crack. The breathing mechanism of the crack was simulated using the size of the contact and the friction between the crack faces. The authors described three states for the crack: open, stick and slip cracks. This was an extension to the study in [140] where a nonlinear contactFEM procedure was proposed for the timevarying flexibility of the cracked shaft. An analytical method that calculated the corresponding torsional, longitudinal, and bending local stiffness values of an asymmetric circumferential crack inclusive of a contact condition was provided in [141]. The model in [124, 126] was used to generate the breathing crack model in [142] by using an updated integration limit like that in [135]. The multifaults case that consists of rotor–stator rub, rotor cracks and misalignment faults was studied in [143] using the proposed model presented in [123].
4.2 Breathing and switching crack models
A crack is said to be “breathing” when it periodically opens and closes as triggered by unbalance excitation. For a fully closed crack state, the resultant stiffness matrix is like that of a healthy shaft. Different models have been utilized to analyze the breathing behavior of the crack. In [118], the breathing effect of the crack was simulated based on the angular location of the crack on a JC rotor where the closing and opening of the crack was based on the static deflection of the shaft. The time dependent stiffness matrix due to crack appearance was calculated at each timebased iteration based on the angular orientation of the crack. In [144], a transverse crack in a multirotorbearing system was analyzed by using the FEM. The main objective was to understand the minimum crack depth at which the vibrational changes occur in the system. The variation of stiffness (due to the crack) was formulated by using the simple breathing function \(f(t) = (1  \sin \phi )/2\) where \(\phi\) is the angle of shaft rotation that measured with respect to the transverse horizontal axis. Moreover, a method for modeling the stiffness reduction, due to a crack, was also presented in [144] and applied to two types of turbogenerators.
A hinge crack model was used in [12] for modeling low depth crack (nondimensional crack depth less than 0.5). In the hinge crack model, the crosscoupled stiffness terms are not included because of its smaller magnitudes at low crack depths for both hollow and solid shafts. Because the crack model switches between open and close states, the model has also been interchangeably termed as a switching crack model. The breathing effect was also observed to be dependent on the static deflection of the rotor. Accordingly, the resultant stiffness matrix was given as
where \(S_{0}\) and \(s_{0}\) are the stiffness matrix and the stiffness coefficient of the uncracked rotor shaft, respectively. \(\Delta S(t)\) is the stiffness matrix contribution due to the crack, \(\Delta s_{\xi }\) is the crack stiffness component, \(\Omega\) is the angular speed, and \(f(t)\) is the steering function given as
where \(w_{\xi }\) being the position in the vertical direction based on the rotational coordinate \(\xi\). Similar model was also employed in [145] which is shown in Fig. 7. The stepwise variation in stiffness shown in the figure was modeled by using Fourier series in which \(\theta\) represented the angular position during the shaft’s rotation.
Equation (8) was used in [146] for studying the cracked rotor systems with flexible bearings, in [147] for a multidisk rod fastening rotor system with multiple faults and in [148, 149] for cracked JC rotor systems with active magnetic bearings (AMB). The hinge crack model was also applied in [150, 151] along with the method of disturbance rejection to develop a modelbased method for crack detection in turbomachinery rotors. The closing and opening conditions of the crack were dependent on the curvature at the crack position. The hinge crack function in conjunction with the flexibility matrix was applied in [152] for a JC rotor with AMB. The FEM implementation of similar model can be found in [153]. Moreover, the model in [152] was extended to include the shaft initial bow, internal damping, and the gyroscopic effect. The authors categorized the faults into either additive or multiplicative faults. Later in [154], the hinge crack model was used for studying the coupled vibrations of torque and lateral modes using the JC rotor model. Moreover, the hinge crack model was used for an extended JC rotor model in [155].
The hinge crack model was applied in [156, 157] to study the effect of multiple faults in rotor system including statorrotor rubimpact, and oilwhirl. The steering function of the hinge crack was applied in [158, 159] for a JC rotor with multiple faults including unbalance, misalignment, and statorrotor rub. The cosine steering function (\({{f(t) = (1  \cos \theta )} \mathord{\left/ {\vphantom {{f(t) = (1  \cos \theta )} 2}} \right. \kern0pt} 2}\)) was used in [160, 161] to analyze the nonlinear dynamic characteristics of a rotor system containing a transverse crack using FEM. The cosine steering function was also employed in [162] to study the super harmonic frequency excitations due to the breathing crack and unbalance. It was reported that when the magnitude of the unbalance increased, the \({1 \mathord{\left/ {\vphantom {1 {n{\text{th}}}}} \right. \kern0pt} {n{\text{th}}}}\) \((n = 1,2,3, \ldots )\) of critical resonant peaks also increased mainly because of the nonlinear interaction between the unbalance, gravity forces and crack of the rotor. A modelbased identification algorithm based on hinge crack model, leastsquares method, and regression techniques was used to approximate some parameters such as crack stiffness, eccentricity, and damping in [163]. The hinge model was also used for uncertainty modeling in [164].
Multiple research has modified the original cosine steering function to account for various effects of the crack. A modified version of the hinge model was used in [165] with extended Kalman filters to develop a modelbased method for crack localization. This modified version of the hinge model involved modeling of the crack as an external disturbance that accounts for the loss of stiffness. Furthermore, the hinge crack model was utilized in [166] for studying the transient operation of a JC rotor with crack. The vibrational characteristics of a cracked rotor in a hover flight was studied in [167] using equations of motion from [168]. For the nondimensional crack depth less than 0.5 where Eq. (8) was used. Otherwise, the cosine steering function was alternatively used. The synchronous breathing function \(\left( {{{f(\omega t) = (1 + \sin \omega t)} \mathord{\left/ {\vphantom {{f(\omega t) = (1 + \sin \omega t)} 2}} \right. \kern0pt} 2}} \right)\) with shaft’s rotation was used in [169] to analyze the stability characteristics of a cracked JC rotor whirling motion based on the variation on the crack size.
In [170], a modified steering function in the form of \(f(t) = 0.8 \times \frac{1 + \cos \phi }{2} + 0.2\) was used for modeling the breathing effect of the crack such that the crack does not completely close. An alternative steering function was also proposed in [171] to account for the saturation of closing and opening behavior of the cracked rotor system. The resultant model was applied to two breathing crack rotor systems and was given as
where \(\chi_{r} = \left {\phi_{r}  \phi_{1} } \right\) with \(\phi_{r} (r = 1,2, \ldots )\) represents the angle between the crack edge normal line and the rotational \(\xi\) axis. The original cosine steering function was either fully open or closed at \(\theta = \pm {{n\pi } \mathord{\left/ {\vphantom {{n\pi } 2}} \right. \kern0pt} 2}(n = 0,1,2, \ldots )\) whereas the modified function in Eq. (9) makes the crack to reach fully open or closed state at various shaft rotational angles. The \(\cos (3(\Omega t + \chi_{r} ))\) term in Eq. (9) resulted in the distribution of the energy dissipation originated from the crack saturation to additional lateral response harmonic components.
A modified steering function was used in [172] to account for the crack depth as \({{f(t) = [(1 + \cos \phi )} \mathord{\left/ {\vphantom {{f(t) = [(1 + \cos \phi )} 2}} \right. \kern0pt} 2}]^{A}\) with \(A\) representing the nondimensional crack depth. This modified model was also employed to study multifault characteristics of a dualdisk bearing rotorsystem using JC rotor in [173, 174], respectively. Moreover, the new model was also used in [175] for studying the dynamic behavior of a cracked rotor system with oilfilm force where an “eye of chaos” phenomenon was observed because of the limitation–accumulation of forward and reverse perioddoubling bifurcation sequences. A steering function that combines both switching and cosine breathing function was utilized for a turboexpander rotor system in [176] and a cracked rotorball bearing system in [177]. The proposed breathing function of this model was given as
where \(\theta = \omega t + \beta  \varphi\), \(\omega\) is the rotational speed of the rotor, \(\beta\) is the angle between crack and unbalance, \(\varphi\) is the arctan of the transverse coordinates, and \(\alpha\) is onehalf of the crack angle. In a recent work, the cosine breathing function was further updated in [178] to form as softly clipped cosine function which is given as
The authors stated that the breathing function in Eq. (11) could describe crack’s saturation (while fully closed or opened state) more efficiently within a complete revolution cycle. The variation of the cosine steering function, the modified breathing functions in Eq. (9), and the function in Eq. (11) are compared in Fig. 8 for a full shaft revolution.
The switching crack model by O.S. Jun was proposed in [179] wherein the stiffness variation (because of the crack) was calculated using the fracture mechanics. In this model, the crack undergoes either a completely open or completely closed state. Such scenario results in statically uncoupled stiffness matrices. As the crack either closes or opens, the direct stiffness terms increase or decrease, respectively. The resultant equations of motion of O.S. Jun model were given as follows
where \(m\) is the mass of the disk located at the midspan, \(c\) is the external damping, \(\Omega\) is the rotational speed, \(\xi\) and \(\eta\) are the rotational coordinates, \(\beta\) is the orientation of the eccentricity in the direction of shaft rotation, \(e\) is the eccentricity of the disk, \(g\) is the acceleration due to gravity, \(k_{0}\) is the stiffness of the shaft without the crack, \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{\xi }\) and \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{\eta }\) are the stiffnesses in the \(\xi\) and \(\eta\) direction when the crack is fully open. The total SIFs was derived from the summation of SIFs due to forces acting in the \(\xi\) and \(\eta\) directions. Moreover, the breathing behavior was based on the sign convention of the total SIFs at each point along the crack front. The deflection in \(\xi\) and \(\eta\) directions, because of the open crack, were determined using Castigliano’s theorem by utilizing the numerical integration over the open crack area. This model was used in [180] where the breathing mechanism was considered based on the initial open or initial closed states as shown in Fig. 9.
Experimental analysis for the switching crack model was conducted in [181] where comparable results were observed with the numerical simulation results. Furthermore, the breathing crack model in [179] was extended to include the shaft’s axial coordinate in [182]. This was due to axial impulses resulting in high frequency axial vibrations. Subsequently, the stress present on the crack edge because of the transverse forces (\(Q_{\eta } ,Q_{\xi }\)) and axial force (\(Q_{u}\)) dictates the breathing effect of the crack. The resultant equations of motion were given as
In [183], the sign convention of the SIF was used for the breathing effect of the crack in a rotor shaft with bowing failure mode. The negative SIF was associated with a compressive stress at which the crack remains close while positive SIF indicates a tensile stress field at which the crack stays open. The corresponding SIF was calculated assuming pure bending without including the shear deformation. In [184], a comparison study between the breathing crack in Eq. (14), switching crack in Eq. (15), and the open crack model in Eq. (16) was presented. The major difference between these crack models was that the breathing crack model allowed for the smooth opening and closure of the crack while the switching crack model is restricted to be fully closed or fully open. In the open crack model, the crack remains open during the shaft rotation. The equations of the rotor containing a breathing crack were given as follows [184]
Further, the equations of motion with switching crack model were updated as
In addition, the equations with the open crack model were given as
It was observed that the crack remained fully opened when the rotational speed was closer to the critical speed zone and undergoes a breathing effect before and after passing the critical rotational speed zone. Further, it was also reported that the existence of the crack (in a shaft) is like the addition of an “out of balance” force at a certain angle with respect to the corresponding initial unbalance angle. Therefore, when the respective unbalance forces are in phase with the orientation of the crack, there was a superposition of combined nonlinear effect. Similarly, when both the crack and unbalance are out of phase to each other, the nonlinear response produced by one force is opposite to the other.
Fracture mechanics was used for developing the breathing crack model for twocrack rotor system in [185]. The authors studied the influence of unbalance force orientation angles on the cracks as well as the combination of open and breathing cracks during the shaft’s rotation. The opening and closing of cracks were dictated based on their respective modeI SIF and the crack closure line. The sign convention of the SIF was used to calculate the crack closure line position (CCLP) during each iteration of the numerical problem. The CCLP value varied from 1 to 100 where the CCLP of 1 and 100 indicated a fully closed crack state; CCLP of 50 indicated a fully open crack; CCLP of 25 indicated a half openhalf closed crack; CCLP of 75 indicated a half closedhalf open crack. This breathing crack mechanism, as described in [185], is shown in Fig. 10.
The angular acceleration and the stress forces produced by the unbalance force excitation resulting in generation of external torsional moments in the cracked cross section, and the corresponding effect of unbalance force orientation on the stiffness of the transverse crack were investigated in [186]. The authors utilized the SIF and the SERR for an accelerated rotor with a breathing crack model. The adopted breathing model incorporated partial closing and opening in which the effect of stress generated forces and moments on the cracked cross section is similar to that in [187]. Fracture mechanics and an additional slope at the crack region were used in [188] for modeling the breathing crack in a simple rotor. The direction and magnitude of the bending moment were utilized to understand the closing and opening of the crack. The additional slope was calculated by integrating the bending moment (due to the gravity forces) over the open crack region. It was observed that when the crack depth is increased, the peak value of SIF was shifted towards the ends of the crack centerline which was comparable with that in [189]. Thus, it was concluded that as the crack propagates further, the maximum stress is shifted from the crack center to its ends.
4.3 Synchronous breathing crack model
A breathing crack model, based on rotor shaft selfweight static deflection, was assumed schncrounous with the shaft angular rotation in [105]. The resultant model was based on expansion of the Fourier series to simulate timedependent moments of area for the rotor shaft’s cracked crosssection. This was further used to calculate the corresponding timedependent stiffness matrix of the cracked crosssectional element. The breathing functions were formulated to approximate the area moments of inertia about the fixed transverse \(\overline{X}\) and \(\overline{Y}\) centroidal axes as shown in Fig. 11.
Based on static load conditions, crack’s behavior is dictated by the stress distribution on the periphery of the shaft neutral axis (N.A). It can be said that when portion of the crosssection is above the N.A, it exhibits a compression stress, while the bottom portion exhibits a tensile stress field. At any instant of time when the crack (or portion of it) falls above the N.A, it is considered as closed. Conversely, if the crack (or portion of it) falls underneath the N.A, it will be considered as open as reflected in Fig. 11a–f. Accordingly, the cracked crosssectional moments of area \(I_{{\overline{X}}}\) and \(I_{{\overline{Y}}}\) with respect to the fixed centriodal coordinates are calculated as
\(\overline{I}_{1}\) and \(\overline{I}_{2}\) are the crosssectional area moments of inertia about the the centroidal rotating \(\overline{x}\) and \(\overline{y}\) axes at \(t \ge 0\), n is a positive even number, \(\theta_{1}\) defines the angle at which the crack begins closing, and \(\theta_{2}\) defines the angle at which the crack is fully closed. These angles were obtained by the following expressions
where \(R\) is the radius of the shaft and \(\mu = h/R\) is the normalized crack depth. The crack is assumed to be fully at \( \theta_{1} \le \theta \le \theta_{1}\), partially open at \(\theta_{1} \le \theta < \left( {\pi + \phi /2} \right)/2\) and \(\pi  \phi /2 \le \theta < 2\pi  \theta_{1}\), and fully closed at \(\left( {\pi + \phi /2} \right)/2 \le \theta \le \left( {3\pi  \phi /2} \right)/2\) where \(\phi = 2\cos^{  1} (1  \mu )\), and \(\theta = \Omega t\). In addition, the crosscoupling stiffness was developed in [190] as follows
where \(\overline{\theta }_{2} = 0.8 \times \theta_{2}\), \(p\) is a positive integer, and \(I_{{\overline{X}\overline{Y}}}^{{A_{1} }}\) is calculated as
where \(A_{1}\) is the crosssectional area of the cracked element crosssection and \(e\) is the location of centroid about the fixed X and Y coordinates. These functions were presumed to be more accurate in resembling the actual breathing mechanisim when compared to the cosine and the other considered steering functions in the previous sections. The variaion of the area moments of inertia \(I_{{\overline{X}}}\), \(I_{{\overline{Y}}}\) and \(I_{{\overline{X}\overline{Y}}}\) in Eqs. (17) and (19) are shown in Fig. 12 for a full shaft revelotion. More details regarding these breathing functions and comparison to traditional cosine breathing functions were discussed in depth in [105, 190, 191].
The synchronous breathing crack model was applied for studying cracked rotors with both open and breathing cracks along with timedependent base motions in [192]. In [193, 194], experimental investigations were conducted to verify breathing crack results by using a cracked JC rotor model. The study in [195] analyzed the interaction between varying unbalance force orientations and the crack depth on the effective stiffness content and potential energy for the steadystate and transient operations of a JC rotor system with crack. Moreover, the effect of unbalance force orientation angles with respect to crack opening direction was studied in [196] using FEM and experimental techniques. The authors observed a complete elimination of the resonance whirl amplitude at certain unbalance orientation angles. Stability analysis of a rotor shaft with breathing crack, journal bearing anisotropy, and the speeddependent characteristics was performed in [197]. Instability regions of the cracked rotorbearing system widened because of bearings anisotropy and the instability ranges became narrow because of the speeddependent characteristic of bearings. A comparison between breathing and open crack models was addressed in [198] based on whirl orbits analysis. An uncertainty modeling of cracked hollow shaft was studied in [199, 200] using the aforementioned synchronous model. The stiffness matrix for the rotor system with crack was modeled using the Timoshenko beam axis model in both [199, 200]. In the former, Yongfeng et al. [199], the influence of uncertain parameters was found to intensify proportionally with crack depth for a breathing crack model. While in the latter, Fu et al. [200], it was reported that the presence of uncertainties can lead to resonance peak amplitude shift, resulting in inaccuracy of crack’s diagnostics’ methods. A dual diskrotorsystem with transverse breathing crack was studied in [201] using breathing crack functions given in Eq. (15). A similar approach was also presented in [202] for a cracked rotor containing unbalance and misalignment faults, and in [203] for studying the nonlinear vibrational response of a cracked dualdisk rotor in a hollow shaft.
Modifications to the synchronous breathing crack model was established in a few literatures. The synchronous breathing crack model application was extended to include an elliptical crack front in [106] as shown in Fig. 13.
According to the model in [106], the orientation of the N.A was observed in [204] to be nonhorizontal based on the 3D FEM simulations performed. The crack closing angle, \(\theta_{1}\), given in Eq. (18), was updated in [204] as
where \(\overline{I}_{1}\) and \(\overline{I}_{2}\) were obtained from the original synchronous breathing crack model in [105] and were formulated as follows
A comparison between the synchronous breathing crack models in [105, 204] is provided in Fig. 14. The difference between the proposed model in [204] and the original model was found to be only apparent at normalized crack depths above 0.5.
The effect of unbalance force on breathing crack behavior was explored in [205]. The authors defined an effectual bending angle, which describes the angle between the bending direction and the orientation of the crack, that caused the breathing behavior of the crack. Similarly, in [206], the effect of unbalance excitation force on the breathing crack model was studied without considering the weightdominance. The effectual bending angle replaced the “\(\Omega \,\,t\)” terms for breathing functions \(f_{1} (t)\) and \(f_{2} (t)\) in Eq. (17) and for the product of moment of inertia in Eq. (19). A breathing crack model for a laminated viscoelastic composite shaft consisting of journal bearings was presented in [207]. A comparison between the proposed model and the original synchronous breathing crack model can be seen in Fig. 15a, b.
The stability characteristics of two transverse breathing crack in a rotor system was modeled using the synchronous crack model and the Generalized Bolotin Method in [208]. Moreover, the stability characteristics between the synchronous breathing crack model and the cosine breathing crack model was performed. The dynamic stability of the synchronous model for a rotor system containing an open crack was studied in [209]. It was reported that the change in sign convention of the determinant of the scaled submatrix of the semiinfinite coefficient matrix have characterized most of unstable regions and their respective boundaries. The results obtained from the matrix method were comparable to the results obtained via Floquet’s method. Moreover, the presence of an open crack in the shaft was observed to excite backward whirl phenomena. In [196], the dependency of unbalance force vector on the corresponding whirl orbital response of open cracked rotors was analyzed. The authors reported certain instances of unbalance orientation angles at which the peak whirl amplitudes were nearly eliminated. The latter study was conducted for doubledisk and singledisk configuration of the rotor system. The FEM of the synchronous crack model was also used for studying backward whirling phenomena in [210, 211].
4.4 Nonsynchronous breathing crack model
The breathing crack model in [105, 190] was derived by assuming that the breathing crack mechanism is synchronous with the shaft’s rotation when the shaft’s whirl is dominant by the static deflection. However, in [212, 213], it was observed that the synchronous breathing mechanism is not necessarily effective in modeling breathing mechanism of a crack, especially in the neighborhood of resonance rotational speeds. Therefore, the nonsynchronous whirl response generates a relative whirling angle between the shaft’s rotational speed and its whirling angular speed in the transverse plane to the shaft centerline. Accordingly, the breathing mechanism in Sect. 4.4 was updated in [214] to be dependent on the relative whirl angle rather than the synchronous whirl angle of the shaft rotation. Consequently, this updated breathing mechanism is controlled by the nonsynchronous whirl of the shaft in the neighborhood of its critical rotational speed rather than the assumption of synchronous whirl. The nonsynchronous whirl that occurs between the transverse component of whirling velocity and the rotational speed of shaft in the whirl orbit is affected by the angular acceleration rate of the shaft. For \(\theta_{s} (t)\) denotes to the angular motion of the rotor shaft with respect to its deflected centerline and \(\theta_{w} (t)\) represent the corresponding whirl angle of the shaft’s deflected centerline as shown in Fig. 16a, b. Additionally, the angular velocity, \(\Omega_{s} (t)\) of the shaft about its centerline deflection represents the rate of change of the angle \(\theta_{s} (t)\). Similarly, let \(\Omega_{w} (t)\) be the whirling velocity representing the rate of change in \(\theta_{w} (t)\). When \(\Omega_{s} \left( t \right) = \Omega_{w} \left( t \right)\), the whirl is synchronous with the shaft’s rotational speed. For the case when the angular acceleration rate of the shaft is nonzero, the whirl occurring in the resonance region becomes nonsynchronous, i.e., \(\Omega_{s} \left( t \right) \ne \Omega_{w} \left( t \right)\) which results in activation of the breathing mechanism of the crack.
During breathing crack propagation in the shaft, the mechanism of breathing shown in Fig. 16 becomes reliant on the relative whirl angle \(\theta (t) = \theta_{s} (t)  \theta_{w} (t)\). Accordingly, the cracked shaft timevarying crosssectional moments of area in [105, 190] were updated in [214] to be a function of the relative nonsynchronous whirl angle \(\theta (t)\) instead of the shaft’s rotational angle. The updated formulation is given as follows
The variations of the cracked crosssection stiffness values of \(K_{X}\), \(K_{Y}\) and \(K_{XY}\) with respect to \(\theta (t)\), based on the nonsynchronous breathing crack model in an accelerated shaft in Eq. (23), are shown in Fig. 17.
4.5 Other synchronous crack models
In addition to the crack models discussed in the previous sections, there are various other models that were also utilized to study the nonlinear vibrational aspects of cracked rotor systems. In [112], an inclination model of a JC rotor containing a transverse crack was proposed. The resultant model was used to study various resonance characteristics for horizontal and vertical rotor systems. ANSYS was used in [215] for modeling various faults such as unbalance, transverse cracks, bent shafts, misalignment, and combination of faults. A 3D lumped mass crack model was introduced in [216] using energy formulations. The model accounted for a local prestressing around the tip of the fatigue crack (observed in [217]) and included the breathing aspects of the crack. An extension to the model in [216] was provided in [218] to account for the torsional behavior. The authors also provided experimental results using elliptic and helixshaped cracks. The model was also extended in [213] to include a nondimensional flexibility term developed as a result of the crack dependent only on the properties of the crack (geometry, relative depth) and without the requirement of additional 3D computations. The combined Dimarogonas’ flexibility matrix and cosine breathing crack function was employed in [219] to analyze the transient operation of rotor system containing a crack. In [220], the breathing stiffness terms (\(k_{{x^{*} }}\), \(k_{{y^{*} }}\), \(k_{{x^{*} y^{*} }}\)) were developed as continuous functions in the following forms
where \(\psi\) is the rotationwhirling difference angle, and a, b, and c represent fitting coefficients at crack depth. The crack model in Eq. (24) has nearly similar effect on the direct and coupled stiffness values in the cracked section to the breathing crack model in [221]. In another study, the fracture mechanics and strain energy concept were employed to develop the stiffness matrix of the cracked element. Accordingly, the breathing crack mechanism was derived based on the flexibility coefficients of the cracked element where the total flexibility matrix represents the stiffness matrix of the element incorporating the crack [222]. In addition, this crack model has utilized the crack closure line method in [100]. The nonlinear contactbased breathing crack model along with 3D FEM of the rotor was employed in [223] to study the nonlinear aspects of unbalance forces on the relative dynamics of cracked rotor system.
The comparisons between direct and coupled stiffness plots for full cycle of shaft rotation based on the crack models in [213, 220,221,222,223] are provided in Figs. 18 and 19. It is apparent that these crack models are nearly in agreement with those of the synchronous breathing crack model in [105, 190, 191]. All aforementioned synchronous crack models were capable to approximate the breathing crack behavior which helped in revealing the underlying nonlinear dynamical behavior of the cracked rotors whirl. These advanced breathing cracked models replaced the simple cosine breathing crack model and provided higher accuracy analysis of cracked rotor systems.
Furthermore, A rigid FEM was used in [224] for modeling a breathing crack. Several spring–damping elements connected between rigid finite elements (representing crackfree elements) were utilized for modeling the crack. In order to incorporate the breathing effect, the stiffness of an individual spring–damping element was either given a value of zero or \(k\) based on the sign convention of the deformation of the element in a direction orthogonal to the crack surface. As an alternative to the fracture mechanicsbased breathing crack model, Cohesive zone modeling (CZM) was proposed in [225]. This model was developed under the assumption of modeI plane strain conditions and considered the triaxiality of the stress state by utilizing the cohesive zone relation. In CZM, the fracture emanated as a discontinuity surface that extended once the maximum value of the principal stress overcomes a critical magnitude known as the cohesive strength of the material. The normalized stiffness of the CZM model was compared with that of [179]. The authors observed a harmonic change in stiffness in [225] in contrast to [179] where the partial opening and closing was linked to the sign convention of the SIF (Fig. 20). Further, the authors compared the natural frequency of the fully opencrack system (without the disk) with that estimated in [144]. The natural frequencies were smaller, but the mode shapes of the initial three were observed to be the same.
4.6 Comments on crack models
The fracture mechanics utilization in crack modeling has certain limitations. For instance, it is only valid when there is an initial crack development in the rotor. The compliance matrix attains singularity when the crack depth magnitude is extended over the radius of the shaft thereby making it only applicable to small crack depth [225]. Furthermore, for unilateral contact problems, the modeI SIF has to vanish at the limit of the contact zone because of the singularities of the stress [216]. The aforementioned limitations apply to Dimarogonas’ model and the switching crack model by O.S. Jun et al. The Hinge crack model does not correlate the shaft’s stiffness to the crack depth although it is the simplest model to implement. It is widely known that whenever the crack depth increases, the nonlinearities associated with the cracked rotor system also increases. Therefore, a generalized crack model should have a dependency on the crack’s depth. The switching crack model by O.S. Jun et al. was found to produce chaotic, quasiperiodic, and subharmonic vibrations which was not the case in the corresponding breathing crack model (with similar parameters) reported in [226]. The authors further stated that the switching crack model yielded rich nonlinear dynamic response features because of the particular mathematical model utilized in the simulations (i.e., unbalancefree) and was not found in reallife situations or experimental studies. The CZM described in [225] is a discrete model which is inferior to continuum based FEMs. Moreover, crack propagation path is normally unpredictable.
One of drawbacks in crack models discussed so far is the assumption that the whirl response is dominated by the static deflection of the rotor. Note that the assumption of selfdeflection being dominant is valid in certain aspects of rotor operation. As stated in previous section for the transient operation, the nonsynchronous whirl component dominates the selfdeflection of the shaft during the resonance speed. Therefore, when the analysis is conducted for transient operations, it is recommended that the updated breathing crack model in [214] is used. Otherwise, steadystate operation can be employed away from resonance zone. The JC rotor model and the FEM were the prominent methods used for analysis except for a few literatures which employed TMM or other tools. Since modern machineries operate in a range of highfrequencies resulting in elevated levels of vibration, there is a requirement of a generalized, universal model that does not only consider the assumption of weight dominance, rather it can be adapted for various operational scenarios of rotor including steadystate and transient operation as well [213].
4.7 Uncertainty in cracked rotor modeling
Uncertainties are inherent in the operation of any machinery. Materials can expand or contract because of varying thermal gradients. Material’s geometry and physical properties can also alter due to constant wear and tear. Finished products can also possess inherent manufacturing defects that affect overall system operational integrity. Therefore, it is imperative to consider uncertainties in mathematical or experimental analysis. For a rotor system, literature has indicated that various uncertainties could develop in form of physical parameters (shaft length and diameter [227], disc diameter and mass [228], unbalance [229] etc.), boundary conditions [86, 230, 231], external loads [232, 233], and measurementbased observational errors and noise [234]. The methods for quantifying uncertainties can be categorized into probabilistic or stochastic methods. Some of the probabilistic methods utilized in rotodynamic applications include Monte Carlo simulation [235], polynomial chaos expansion [236], and random matrix theory [237]. Fuzzy theory [238], interval methods [239], and imprecise theory [240] are examples of stochastic approach methods. Although the topic of uncertainty modeling is a vast area of research, the present focus will be more of modeling related to cracked rotor systems. More details about the state of art in uncertainty modeling for rotor systems can be found in [241].
In [242], whitenoise, introduced to the rotor system containing the crack and the corresponding effects on the change in stiffness and rotational speed ratios, was analyzed using MonteCarlo method. The Polynomial Chaos Expansion (PCE) method was used in [199] to model for the stiffness of the shaft and the unbalanced excitation forces related to cracked hollow shafts. The resultant model was also compared using Monte Carlo simulation method. Furthermore in [200], an uncertain response surrogate function was utilized to calculate the dynamic behavior for varying cases of parametric uncertainties in an open cracked hollow shaft rotor system. Moreover, uncertain parameters such as modulus of elasticity, crack depth, and mass unbalance were represented using the nonprobabilistic interval variables.
The research in [201] utilized derivativefree interval approach to analyze the uncertain dynamic response of a doubledisk rotor containing a breathing crack. Chebyshev inclusion function was further used to build the surrogate model associated with the interval problem. Crack depth, shaft stiffness, and unbalance were considered as uncertain parameters. It was concluded that the cracked rotor system response is sensitive to the crack size, material properties such as density of the shaft, and Young’s modulus. On the contrary, it was reported that shaft’s response is not sensitive to the support stiffness boundary conditions. PCE along with kriging was also utilized in [243] to analyze the effect of uncertainty parameters on the harmonic components in terms of breathing crack depth and position. The parameters that were kept uncertain include the Young’s modulus, density of the shaft, thicknesses of the two discs, right bearing supports’ horizontal and vertical stiffnesses. It was reported that the system uncertainties led to the variation in the variance and mean of the vibration amplitude and critical speed. Moreover, the detection of small cracks and their localizations were challenging when uncertainties were present in the system.
Further, in [164] uncertain parameters were utilized to define the nonprobabilistic uncertainties that occur in a breathing cracked rotor system. The uncertain parameters were quantified using the polynomial surrogate model. The crack depth, mass unbalance, unbalance initial angle, bearing support’s damping and stiffnesses, shaft’s density and Young’s modulus were kept as uncertain parameters with varying levels of uncertainty. Chebyshev collocation surrogate method, along with Harmonic Balance Method (HBM), was used in [244] to study opencrack rotor system with uncertainties. It was reported that Chebyshev order was a key factor to reduce spurious peaks and overestimation near critical speed range of the rotor including uncertainties. In [245], a method consisting of PCE, stochastic FEM, and HBM was proposed for studying uncertainty analysis with respect to breathing cracked rotor. The stiffness and unbalance forces were assumed to be random using the gaussian law.
5 Crack detection methodologies
Literature is replete with many techniques utilized for the detection of cracks in rotor systems. In this section, a few of those most common techniques (Fig. 21) shall be addressed.
5.1 Application of nondestructive testing (NDT)
It is challenging to detect the crack growth in a rotor shaft before it reaches severe magnitudes which might lead to destructive damage in the system. Therefore, authors in [246] used an acoustic emission transducer (ultrasonic technique) for detecting cracks in the early stage. Further, envelope analysis and discrete WT were also used in [246] to identify characteristics of the crack. In [13], NDT’s in the form of colored dye penetrative testing, florescent dye testing, ultrasonic testing, magnetic particle testing, and eddy current testing were addressed. Additionally, in [113], static and dynamics tests were also identified for crack’s detection, as well as localization. A method was proposed for detecting defects in a rotor system using infrared thermography in [247]. The aforementioned methods are not rendered as realtime methods for detecting cracks in a rotor system wherein a complete unit disassembly and a thorough inspection is required.
5.2 Application of signal processing based methods
It is imperative to constantly monitor the system to detect its functional state. There are various methods related in the literature to extract such information. One of the most common methods is to employ the usage of signal processing techniques. Typically, a vibration signal contains high and low frequency components. While the low frequency components describe the fundamental information of the signal, the high frequency components indicate the nuances in a system [248]. For a faulty rotor system, the faults generate nuances in the high frequency components which can be further utilized to substantiate its presence. Literature is abundant with methods that have been utilized to study high frequency components. Among which the most common of all are the FFT and Full Spectrum Analysis (FSA) [149, 153, 193, 194, 211, 220, 249,250,251,252,253,254,255,256,257,258,259,260,261], Short time Fourier transform (STFT) [142, 143, 260], WT [170, 248, 262,263,264,265,266,267,268,269,270,271], and HHT [143, 190, 219, 272,273,274,275,276]. Other methods have also been employed to identify the faults in a system, which will be presented in subsequent sections.
5.2.1 Fast Fourier transform
FFT is an algorithm that calculates the Discrete Fourier Transforms in \(O(N\log_{2} N)\). The FFT helps in converting an input signal into spectral components thereby providing information in the frequency domain. While FFT is a linear process, any shifting of the timedomain signal results in changes in the phase only with no influence on the amplitude [277]. FFT has been employed in multiple research [149, 153, 193, 194, 222, 250, 251, 258,259,260] for crack detection purposes, some of the results can be found in Table 2.
It is factual that multiple faults can occur simultaneously in a rotor system during its nominal operation albeit these faults may be within the operational limit. The research in [261] uses FFT and HHT to study faults including rotor–stator rub, crack, and unbalance. For the combined case of rubimpact and crack in a rotor system, the 1X harmonic component was excited in the FFT plot of the lateral vibration response with higher harmonic component excited whenever there is the rub phenomenon. Further, the IF characteristics of the combined rubimpact and crack faults was studied in [249] and it was concluded that the presence of the 1X, 2X, and 3X modulation frequencies are the indicators for the presence of cracks present in the rotor system. The authors in [249, 261] also studied intermittent rubbing effects between rotor–stator contact region and observed the triggering of higher harmonic components when simultaneous crack and rub faults occur in a system. The study in [220] involved crack identification in a “disturbed state” wherein the steadystate operation of a rotor system was perturbed to reach a motion state with free damped vibration and forced vibration. The concept of modulation frequencies is similar to the Intrawave Frequency Modulation approach mentioned in [249]. Moreover, the idea of using a disturbed state is not practical since it involves changing a normal motion of a rotor system to a transient operation abruptly, although for a short period.
Higher harmonics can be used to signify crack presence in vibration monitoring during transient and steadystate operations [278]. The rotor–stator rub contact can also be diagnosed using low higherorder harmonic components in the frequency spectrum [252]. Various subharmonic and superharmonic oscillations as a result of rubimpact and its occurrence has been addressed in [253] using nonlinear contact models. It was reported that rotor–stator rub impact induces excitation frequencies of 2X harmonic and higher [254]. The main drawback of using FFT is that the directionality of the signal's orbital whirling information is never addressed. The FFT always handles the vibration signals as real quantities. This implies that the negative (\( nX\)) and positive (\(+ nX\)) parts of the FFT are similar to each other and only the positive half (\(+ nX\)) is plotted [251]. FSA can be used to address such an issue as indicated in [255]. FSA preserves the relative information of the phase associated with the measured vibration signals and thus making it a useful tool for fault identification. FSA was used to study the dynamics of rotor–stator rub impact in a cracked rotor system during steadystate operation in [255]. The absence of backward spectral peak is evident in Fig. 22a for the FFT plot while it is captured accurately in the FSA plot (Fig. 22b). With increasing crack depth, the amplitude of the 2X response was found to increase significantly when compared with the 3X response. Further, when rub was initiated due to the presence of crack in the rotor system, the backward 2X component dominated the FSA response whereas it was predominantly forward 2X component when rub was absent. FSA was also employed in [211, 256, 257] for studying the backward whirling zones for a cracked rotor system using numerical and experimental data. The authors were able to capture backward whirling zones accurately in varying rotordisk configurations during transient operations.
5.2.2 Shorttime Fourier transform
Signals that are observed in industrial machinery could be categorized as stationary or nonstationary signals [279]. Stationary signals can be attributed to timeinvariant statistical properties, whereas the statistical characteristics of nonstationary signals vary with respect to time. FFT employs a timeaveraging approach transforming the concealed time information and displays only spectrum which is averaged over the entire time of analysis [280]. Time–frequency techniques such as STFT and WT derives the twodimensional (2D) properties of the onedimensional input signals (in time domain) by utilizing the changes of the characteristic frequencies inherent to the signal, thus providing a valid and robust tool for analyzing nonstationary signals. STFT was used for breathing crack detection during transient operations in [142]. The author was able to observe multiple inclined lines in the STFT plots for the case of breathing crack whereas only one inclined line was reported for the case of uncracked rotor (shown in Fig. 23a, b). In addition, it was reported that the magnitude of these lines is directly proportional with the depth of the crack. Further, the research in [281] focused on using the STFT to detect small cracks in a rotorball bearing system.
5.2.3 Wavelet transform
The main difference between WT and STFT is that the latter uses fixed window size for its analysis, whereas the former utilizes short windows for high frequency components and longer window functions for low frequency components [282]. The parameters which describe the WT are the timeshift variable and the scale variable. WT can be either Continuous Wavelet Transform (CWT) or Discrete Wavelet Transform (DWT) subject to application. The WT employs a scaled version of a single wavelet called the mother wavelet. The main advantage of using WT is that it can detect any discontinuities and transitory effects, if compared with FFT, and locate it in time with respective frequency components. Some of the results from the studies related to WT [170, 248, 264,265,266,267,268, 270, 271] can be found in Table 2.
A modelbased approach based on the FEM of Bspline wavelet was proposed in [262]. The method has been characterized as Wavelet Finite Element Method (WFEM) and the objective was to specify crack’s size and location in shafts. The main advantage of using the WFEM was due to multiresolution properties and the availability of various structural analysis tools. Further, a model parameter identification based on the EMD and Laplace Wavelet correlation filtering was proposed in [263] to be applied in conjunction with the WFEM. The signal from the multiDOF system was decomposed into three intrinsic mode functions (IMFs) and the modal parameters were identified using Laplace wavelet correlation filtering.
5.2.4 Hilbert–Huang transform
Various faults such as a breathing crack and rotor–stator rub exhibit nonlinear and nonstationary vibration responses during steadystate operation of the rotor system. For these nonstationary signals, the spectral properties of the signal change with time [283] wherein the usage of Instantaneous Frequency (IF) is adopted. The IF defines the position of the input signal’s spectral peak when it varies with time. Conventional signal processing methods produce inaccurate fault identification features during transient operation since the frequency changes dynamically. Additionally, WT suffers from energy leakage in the neighboring modes because of varying frequencytime resolution as defined by the Heisenberg–Gabor inequality [261]. As such, HHT can be employed to achieve the frequencytimeenergy distribution from vibration signals due to its adaptive and equal resolution properties at all time and frequency domains. One main benefit of using HHT is that its most computation consuming step, the EMD operation does not incorporate convolution or any other laborious timedependent operations. Therefore, the HHT method can deal with signals of large size. Additionally, the Hilbert–Huang spectrum does not include the concept of time and frequency resolutions and is only limited to IF calculation. A few results of the research [143, 190, 273,274,275,276] that studied HHT for crack detection in rotor systems is reported in Table 2. The IMFs and the corresponding WT of a cracked rotor system at 1/2 of the critical speed are shown in Fig. 24a–c. It can be seen that the dominant component at 1/2 of the critical speed is the 2X harmonic component when compared with the 3X harmonic component.
HHT was used for crack detection in [272] for transient operation of a cracked rotor system. For a small nondimensional crack depth of 0.1, it was observed that small disturbances occur in the HHT plot, which were negligible in FFT and CWT plots. This is because HHT utilizes the concept of IF thereby making it more sensitive than CWT and FFT methods. Further, the effect of propagating crack on HHT was analyzed in [219]. The researchers were able to deduce that the propagating crack resulted in the increase of resonance time and reduction in the resonance peak amplitude during the transient response. Moreover, they also observed nonharmonic 2X and 3X components which can be attributed to the IMF reported in [249]. Eventually, a few disadvantages for employing EMD method were reported in [284]. These disadvantages can be summarized as follows. Firstly, EMD generates unwanted IMFs of low amplitude in the lowfrequency region resulting in additional nonuseful frequency components. Secondly, the initial IMF can contain a broad range of frequency in the highfrequency region thereby failing the definition of a monocomponent. Third, the EMD operation sometimes cannot distinguish the lowenergy components (from the analyzed signal) thereby making them absent in the time–frequency plane. In order to overcome shortcomings of EMD, an improved Variational Mode Decomposition (VMD) along with Generalized Demodulation Technology and ZeroPhase shift filter is employed in [285] for studying cracks in the rotor system. The study’s purpose was to separate harmonic components that were close by and the crosscoupled terms near transient operations. This was achieved by first converting the broadband input signal to a narrowband frequency range using Generalized Demodulation Technology, and then ZeroPhase shift filter was used to filter certain components and improve the signaltonoise ratio. Finally, VMD was used to attenuate the coupling between the damped synchronous vibration and free vibration.
5.3 Whirling behavior and whirl orbits analysis
When the rotor is operated at supercritical speed range, the amplitude of whirl orbit becomes independent of crack regardless of its depth [286]. The variation in the shape of whirl orbits can be used to signify the breathing crack diagnostic [193]. If there is any propagation of the transverse crack in an asymmetric rotor system, it will be evident in the whirl orbit plot of the rotor system [183]. Any increment in the resultant ellipticity of the whirl orbit could be used as indicators for crack prognostics [287]. An elliptic orbit was observed at 2X, 3X, and 4X harmonic components with an overall increase in ellipticity due to increased crack depth. The distortion of the whirl orbit and the development of inner and outer loops in the rotor whirl orbit at the nth critical speed (\(n = 2,3,4\)) can be utilized as a diagnostic tool for existence of transverse cracks [288]. The inner loops in the whirl orbit diagrams along with the phase development are reported to be valid for crack’s diagnostics [145]. It was reported that the existence of 2X harmonic resonance component excited at 1/2 of the critical speed becomes more visible in proportion to crack’s propagation.
As the rotor speed approached 1/2, 1/3 and 1/4 of resonance forward speed, a single, double, and triple inner loops appeared in the whirl orbit. Conversely, these inner loops vanish at speed range that is beyond critical speed [105]. Additionally, for an open transverse crack, the presence of inner loops was not evident [198]. Open crack was found to excite the backward critical and subcritical speeds only with an outer loop pattern in the whirl orbital plots. Therefore, it can be concluded that the inner loop pattern at the subcritical forward speeds is unique to the breathing crack phenomena. The presence of inner and outer loop patterns at 2X and 3X harmonics were also established in [161]. Further, it was reported that the “outsideinside loop” phenomenon would not be observed if the crack located closer to the bearings, away from shaft’s midspan. Experimental analysis was conducted in [159] for studying faults such as rotor unbalance, bow, misalignment, crack, bearing looseness, and rotor–stator rub using whirl orbital plots. The study reported the absence of “inner loop” patterns. Further, a double “outer loop” was observed at the 2X harmonic component. Moreover, the relative magnitude of the amplitude at 1X harmonic due to unbalance and 2X harmonic due to the crack dictated the presence of inner loop at the 1X harmonic whirl orbit [289].
Experimental verification of inner loops as the rotor traverses 1/2, 1/3, 1/4, and 1/5 of critical speed was also conducted in [194]. It was observed that closer the rotational speed was to the center of the subcritical speed, the harmonic component was found to be significant. Whirl orbits were also employed for detecting coupled twocrack rotor system in [222]. The single and double inner loop pattern at 1/2 and 1/3 of the critical speed was still prevalent for the coupled twocrack rotor system (Fig. 25a, b). Moreover, the experimental results in Fig. 25c, d confirmed the numerical results. In a recent numerical and experimental study in [290], it was found that the excitation of the single inner loop within the fundamental whirl orbit is not restricted to appear at 1/2 of the critical rotational speed in a cracked rotor. This pattern is found strongly dependent on the magnitude of the unbalance force rather than the 1/2X subharmonic component. Accordingly, the whirl orbits with single inner loop pattern were excited at rotational speeds away from the 1/2X in preresonance and postresonance regions when the unbalance force magnitude is low. This was also experimentally verified as shown in Fig. 26a–h for a cracked rotor system with 58 Hz critical resonance speed.
For a rotor system composed of soft bearings, the influence of cracks on whirl orbits and the corresponding diagnostic forces technique was utilized in identifying the presence of cracks in [291]. For a rotor system with stiff bearing, the whirl orbit had an inner loop pattern due to the excitation of the 2X harmonic component. The effect of rub, crack, and oil film forces were analyzed in [173] where the phase change for the whirl orbit inner loop presents with rubimpact fault. The study in [292] employed instantaneous whirling speed of the whirl orbit in order to distinguish between healthy and cracked rotor systems. It was reported that the minimum value of the relative whirling speed decreased linearly with increased crack depth, thereby indicating crack severity. Whirl orbits were also employed recently in [178] for transverse and [106] for elliptical breathing crack detection purposes.
5.4 Application of energy methods
Various energy related methods were also employed for the purposes of crack identification. Negative potential energy content was used for damage detection in [195] for a JC rotor system with crack during transient and steady state operations. Both breathing crack and open crack models were employed for analysis, and the direct integration of the equations of motion resulted in the calculation of the effective stiffness. When the rotational speed changed from preresonance to postresonance zones, the amount of negative effective stiffness content was substantially influenced for both the steady state and transient operations.
In [293], energy analysis for experimental vibration response spectrum of a cracked rotor system was conducted. It was found out that the rotational speed parameter had a significant effect on the relative energy of the wavelet packets. It was reported that as the rotational speed increases, the relative energy also increases. A similar study was also conducted in [170]. From the analytical model, the authors observed that 1X harmonic had higher absolute energy increments followed by 2X harmonic and finally 3X harmonic. However, in terms of relative energy increments, the highest value was observed at 3X, then 2X, and finally 1X harmonic. Wavelet packet energy was also employed for detecting cracks in [294].
A 3D energy method based on transverse vibration was used for identification of shallow crack (nondimensional depth of 0.05) in [295]. Transverse vibration was used over torsional vibration mainly because of the lower energy magnitude for torsional vibration (shown in Fig. 27a, b). Shallow cracks have been substantiated to affect only the peak of the energy response rather than the energy response period. Moreover, the presence of cracks led to a deformed energy track curve projection (in 2D) in the shape of numeral “8” utilizing the 3D energy model (Fig. 27d). Recently, harmonic balanceenergy power flow analysis was used in [296] to identify the presence of cracks and the corresponding crack position.
5.5 Application of operational deflection shapes
Operational Deflection Shapes (ODS) can be described as the deflection of a structure at a certain speed or frequency. The difference between ODS and the mode shape is that the former can be applied for a structural motion that is nonlinear and nonstationary, whereas the latter is defined for the linear and stationary cases only [297]. Moreover, ODS depends on forces or loads that are applied to the structure. Crack detection using ODS, and the kurtosis of the vibration response was addressed numerically and experimentally in [298]. In a system that is rotating, the presence of crack and its breathing nature affects the vibration behavior over the entire periods of shaft’s rotation. Depending on the configuration of the rotor and the relative position of the crack, the vibration response is different when measured at distinct locations of the rotor. Therefore, it can be concluded that the kurtosis value of the vibration signal also changes accordingly. The authors in [298] plotted the Slope Deviation Curve (SDC) from the ODS in order to evaluate the effects due to the crack’s presence. A deviation in slope was reported due to the presence of crack in the SDC. Moreover, the kurtosis values showed a deviation from the nominal value for the cracked shaft and higher deviation near the element where crack was present. The study in [298] was an extension of [299] where SDC was used to identify cracks in a twocrack rotor system. For the smallest nondimensional crack depth of 0.1, the authors in [299] were able to observe significant difference in the SDC.
ODS, along with DWT and ANN, was used in [300] for evaluating crack’s depth and localization. Once the ODS shapes were extracted in both rotational and translational directions, the DWT was directly applied to it and used as feature vector to train the ANN. DWT in conjunction with Adaptive NeuroFuzzy Inference Systems (ANFIS) was utilized to denoise signals and identify open cracks in an overhung rotor system [301]. The trained ANFIS’s were optimally used for identifying the proximity of the crack’s location and the corresponding depth of the crack. Additionally, optimal control based on linear quadratic gaussian algorithm was used for multicrack identification in [302]. Modal parameters such as frequency, damping, and amplitude were identified initially along with the modal shape for the study. A weighted Approximate Waveform Capacity Dimension (AWCD) was used in combination with ODS for shallow crack detection and crack’s localization in [303]. AWCD is a fractal dimensionbased diagnostic algorithm applied to the ODS curve as a postprocessing tool. ODS was also used for identifying imbalance and misalignment in [304].
5.6 Application of machine learning
Machine Learning (ML) algorithms are effective in identifying signal patterns with less time consumption than traditional methods. In rotor systems, the vibration response data is obtained from experimental or numerical analysis of various parameters such as acceleration, displacement, etc. This can be employed in finding suitable feature space vectors. As such, it can be said that the rotor system fault identification is a timeseries classification problem. There are various algorithms that are available in the classification problem some of which include Naïve Bayes, Support Vector Machines (SVM), KNearest Neighbor, and Decision Tree [305,306,307]. All the aforementioned algorithms utilize mathematical operations to find a suitable boundary between various classes in different engineering problems. It is evident from Fig. 28 that a rotor system can be classified either as healthy or faulty. This can be done through mathematical operations on the timevarying input signal which can be univariate or multivariate as shown in Fig. 29.
One of the earliest studies using neural networks for crack identification was in [308] wherein two and three layer feedforward neural network with backpropagation was used for crack identification in rotors. The network was trained using experimental vibration signals of cracked and intact rotors. Later, ANN was used for identification of crack position, depth, and orientation in [309]. Optimization algorithms such as genetic algorithm and fuzzy logic were also used to obtain the crack characteristics from the objective functions. In [310], identification of multiple transverse cracks in a rotor shaft was performed based on decreasing the respective natural frequency (because of crack propagation) using ANN and FEM. A modelbased crack localization procedure using neural networks and the 3D FEM was also presented in [311]. In addition, an ANN based method for crack detection was proposed in [312] for a composite and steel shaft. For the purposes of binary classification of healthy and faulty rotor systems as well as multiclass classification of cracks at varying depths, the SVM was used in [313]. In [314], CNN and deep metric learning (CNNC) were used for finding the location of a breathing crack in a twodisk hollow shaft rotor system. It was found out that the CNNC had the highest accuracy with the testing datasets in crack detection. CNN was also used in [315, 316] for shallow crack detection purposes in a multifault system and crack location identification in a multispan rotor system respectively. In [317, 318], a multiinput CNN and an ensemble deep learning method was used for multifault classification.
Various techniques have been employed for obtaining patterns in the vibration signals generated from experiments or numerical simulations. In [319], ANN along with Power Spectral Density (PSD) and Peak Position Component Method (PPCM) was used for predicting crack propagation at varying depths. PSD was employed to detect the cracks based on the variation of the spectral content in the vibrational signals. PPCM was used to reduce data transfer by statistically characterizing the peak positions in the PSD. Multiresolution analysis was used in [270] for generating patterns from the DWT of the vibration signals from the breathing crack. The resultant output from the multiresolution analysis was fed into the ANNradial basis function network. Later in [320], vibrational signals from an experimental setup during steadystate operation were processed using WPT and utilized as an input feature into an ANNradial basis function. In [321], a feature space was extracted from the experimental vibrational signals using the relative wavelet energy and wavelet entropy of the DWT as an input to the ANN. ML was also coupled with acoustic emission for finding crack intensity in [322]. An interval least squares one class SVM was used in conjunction with glowworm swarm optimization for crack identification in [323]. The main concept of glowworm swarm optimization is based on the bioluminescent behavior of glowworms. In the initial stages, many glowworms are distributed randomly like particle swarm optimization. Eventually, the glowworms gather at a certain area that has brighter illumination representing the optimal solution.
ML algorithms were also utilized for studying bearing related faults related to rotor systems in multiple studies. A fault diagnosis method based on random forests and improved multiscale attention entropy was proposed in [324] and applied to a hydroelectric unit. Bearing fault diagnosis method based on a wild horse optimizer optimized VMD and correlation coefficient weight threshold denoising entropy feature fusion was proposed in [325]. The main highlight of the study was the requirement of minimal training samples for higher accuracy of the ML classifier. Multiple localized faults in gearbox and bearings related to a rotor system were identified using an adaptive feature mode decomposition in [326]. The resultant model utilized an autoregressive method and an adaptive finite impulse response filter bank for preprocessing and signal decomposition respectively. Other studies related to bearing faults diagnosis using ML algorithms can also be found in [327,328,329,330].
5.7 Analysis of pre and post resonance backward whirling
Whirling is the rotation of the bent shaft’s geometric centerline in the transverse plane. The shaft undergoes a deflection from its initial rotational axis due to unbalance force. If the shaft whirling speed is equal to the applied angular speed, the shaft is said to be in synchronous whirling. On the contrary, if the whirling speed is not equal to the applied angular speed, the shaft is said to be in nonsynchronous whirling. The whirling direction can be either forward whirling or backward whirling as seen in Fig. 30. In the case of forward whirling (FW), the direction of the shafts angular rotation coincides with its angular whirling in the whirl orbit. Otherwise, the whirl is referred to as a backward whirling (BW) motion.
The BW is classified as preresonance backward whirl (PrBW) which appears before the passage through resonance while the postresonance backward whirl (PoBW) immediately appears after passing the resonance speed in faulty rotor systems [210, 211, 214, 256].
5.7.1 Preresonance backward whirling
The physical mechanisms that induce split resonance and PrBW in a JC rotor system without gyroscopic effect was investigated in [331]. The dominating parameter that resulted in PrBW was reported to be the stiffness asymmetry in the two transverse directions followed by rotor’s unbalance and damping. Further, the study also reported that gyroscopic effect does not contribute to the PrBW phenomena. Unbalance excited PrBW modes for a JC rotor was also explored in [332]. It was concluded that the main contributor to PrBW modes’ presence was the stiffness asymmetry. The effect of unbalance, rotation direction, intershaft bearing, and speed ratio on the PrBW of a counterrotating dualrotor system was studied in [333]. It was concluded that the mandatory condition for exciting the PrBW in a dual rotor system is a negative speed ratio. In another study, the existence of PrBW between two critical speeds of a flexible rotor undergoing synchronous whirl was reported when both crosscoupled spring coefficients of hydrodynamic bearings were positive [334]. For a flexible rotor system containing viscous damping bearings and linear stiffness, the effect of internal damping and shear deformation on both the PrBW and FW speeds was studied in [335]. The PrBW modes were reported to be unstable with the presence of hysteretic and viscous internal damping at supercritical speed range. This was further confirmed in [336] when hysteretic internal damping was incorporated into the dynamic model. The effect of adding noise to JC rotor to stimulate motion shift from PrBW to FW was explored in [337]. In [338], the stability criterion of the whirling mode for a flywheel rotor system that was magnetically suspended was studied. In [339], an empirical study was used to gauge the presence of PrBW in a JC rotor. The authors observed the existence of PrBW zones during coastdown operation. The nonlinear lateral vibrations of a horizontal JC rotor, along with nonlinear spring characteristic and asymmetric stiffness behavior, was investigated in [340]. It was reported that when the (nondimensional) eccentricity of a symmetric rotordisk increased beyond 0.015, the corresponding spinning speed interval was increased with either FW or PrBW motion exhibited during the interval subject to initial conditions.
The appearance of the PrBW phenomena is associated with systems having anisotropic supports, rotor–stator rub, certain effects of fluid films, and cracks leading to stiffness anisotropy [341]. Support flexibility influences both PrBW and FW critical speeds, which was numerically and experimentally studied in [342]. Moreover, the excitation and intensity of PrBW is directly proportional to crack depth. This was verified in [343] with a numerical model simulating the cracked rotor using local flexibility matrix. It was reported in [344] that harmonics of rotor response undergoes a whirl reversal procession in the neighborhood of subcritical resonances. Additionally, the nonsynchronous whirl response provided evidence for crack detection in subcritical range but was too small to be detected in the supercritical range [287]. In a separate study, it was reported that PrBW was first observed at the disk, and then propagates to the shaft, which finally returns to the disk and then dies out [345]. Moreover, it was reported that at certain frequencies the outboard disk undergoes PrBW precession, while the midspan and inboard part of the rotor are dominated by FW procession. Higher system damping has resulted in elimination of the PrBW precession as indicated in [93].
The vibrational response of a cracked rotor is inherently nonsynchronous when rotational speed is less than 2/3 of critical speed according to [287]. The nonsynchronous component is directly proportional to the crack depth, but the synchronous component may vary depending on the crack depth subject to the crack’s angle [287]. Moreover, the critical speeds and natural whirl frequencies of a rotorbearing system are indirectly proportional to crack depth for certain crack locations and slenderness ratios [346]. Further, the effect of crack on FW and PrBW critical speeds of a tapered shaft was studied in [347]. In a rotor system with internal resonance, FW and PrBW are reported to be coupled with each other [348]. The studies in [287, 346, 347] have reported that the PoBW zone preceded the FW zone where the PoBW zones assumed to have higher whirl amplitude. This was not the case in [210] for which the appearance of PoBW zone was associated with abrupt reduction in whirl amplitude following the passage through the resonance speed. Additionally, the combination of crack and stiffness anisotropy in bearings excites the PoBW orbits immediately as the rotational speed traverses the critical FW speed during both coastdown and startup operations.
5.7.2 Postresonance backward whirling
It was found in [210, 211, 214, 256] that open and breathing cracks along with rotors with snubbing effect [349] excites PoBW zones. In [211], the PrBW and PoBW was analyzed for three different rotor configurations. It was found that the PoBW was easily observed and had a higher extent in the doubledisk cracked rotor system. The excitation of the PrBW and PoBW is evident in the whirl amplitude response in Fig. 31a which was confirmed using FSA in Fig. 31b for an overhung rotor with snubbing contact. Moreover, the excitation of the PoBW was confirmed using experimental analysis for a cracked singledisk rotor based on whirl amplitude response and FSA (Fig. 32). In addition, the difference between PrBW and PoBW was addressed in [211]. A few conclusions were reported as follows.

1.
The PrBW was generally in accordance with the predictions from the Campbell diagram during steadystate operations while PoBW was associated with transient operations during the passage through the forward critical resonance frequency.

2.
For an intact rotor system with isotropic bearings, the PoBW is not excited during the passage through resonance rotational speeds.

3.
The recurrence and extent of PoBW excitation zones increases at higher angular acceleration rate in faulty rotor systems.

4.
The intensity of PoBW zones is significantly influenced by orientation of the unbalance force.

5.
The PoBW excitation zones were more prominent and had higher intensities in doubledisk than singledisk cracked rotor configurations.
Recently, the PoBW excitation zones were accurately captured with a breathing crack model for an overhung rotor system in [256] and for a vertical rotor systems in [214]. In the latter, it was observed that the orientation angle of the unbalance forces had minimal impact on the excitation of PoBW zones. Moreover, the author verified the existence of PoBW zones at a relatively small nondimensional crack depth. This proves high fidelity for the usage of PoBW as an early indicator for crack propagation in rotor systems.
5.8 Application of other rotor fault detection methods
In addition to methods described till now, other miscellaneous methods were also employed for early rotor crack diagnostics and prognostics. A frequency domain leastsquare identification method was utilized to find both the location of the crack and its corresponding depth in [350]. Higherorder spectra method was used in [351] for detecting crack and misalignment faults in a rotor system. The method consisted of utilizing the trispectrum and bispectrum signal characteristics to detect higher harmonic components and phase relationships. In [352], the presence of crack, its depth, and location were identified using an evolutionary optimization method in conjunction with mathematical models of the rotor system containing a crack. The 2X harmonic component of the angular vibration response was used for crack’s diagnostics in [353, 354]. The resultant 2X harmonic component was reported to have indirect proportional relation with crack depth. Uncertain response surrogate function was utilized in [200] for studying open crack on a hollow shaft containing uncertainties. A crack localization method based on the Bayesian fusion of multiscale super harmonic characteristic deflection shapes (SCDS) for stepped rotors was proposed in [355]. In the proposed method, local shape distortion in the SCDS’s (due to the crack) were analyzed by frequency domain decomposition at super harmonic frequencies. The aforementioned research work was a followup study to [356] where singular value decomposition was utilized for minimizing noise effects and optimizing the accuracy of the SCDS. An identification algorithm utilizing multiharmonic influence coefficient method was proposed in [357] to identify multifault parameters in a rotor system with AMB and crack. The algorithm utilized vibration response and multiharmonic magnetic excitation of the AMB. Parameters such as internal damping, residual unbalances, and crack stiffness were estimated.
6 Nonlinear dynamical analysis in faulty rotor systems
Floquet’s stability analysis was extensively used in the literatures [186, 191, 212, 358,359,360,361,362] for cracked rotor systems. Based on Floquet’s method, the information about stability was obtained by using the monodromy matrix that projected the solution vector at the end of the fundamental period. The calculation of the monodromy matrix for fullmodal order system involves tedious calculation time and exhaustive memory capacity. As such, alternate methods were used in [186, 191, 212, 360] wherein Bolotin’s method for parametric instability is employed. The Bolotinbased instability method was 500 times faster than the Floquet’s theory. The HBM was utilized in [209] to find the solution of the FEM of a periodic rotor system containing a crack. The change in sign convention of the determinant of the scaled submatrix of the semiinfinite coefficient matrix helps in identifying substantial number of unstable regions and their respective boundaries. The method generated results similar to that from Floquet’s theory. The author in [209] further concluded that the proposed method was efficient and computationally faster than Floquet’s theory, even in the case of largescale parametrically excited dynamical systems.
The stability of a rotor system containing a crack is dependent on the rotational speed and crack’s depth [12, 145]. Moreover, the influence of nonlinear breathing of crack and orientation angle of the unbalance force on system’s stability was studied in [212]. Poincaré map and bifurcation diagrams were utilized in [363] to study the effect of crack’s depth on the whirling motion during the subharmonic, primary, and super harmonic resonance cases of a rotor system containing a crack. The crack’s depth was utilized as a bifurcation control parameter and the corresponding bifurcation behaviors of the rotor system was studied. The Periodm (\(m = 1,2,3\)) motions for the case of a buckled nonlinear JC rotor system was studied in [364, 365], respectively.
The effect of perturbation on the periodic solution of a cracked rotor was studied in [366]. A small zone of dynamic instability was reported when the crack was simulated at midspan and onethird of the shaft. The same results were further substantiated in [360]. Moreover, it was concluded in [148] that the optimal control could not always result in the stability of a cracked rotor–AMB system. Stability behavior of two transverse cracks in a functionally graded system was studied in [367]. In [368], a tribocrackdynamic modelbased faults such as oil film forces, rotor breathing crack, and rotor–stator rub was proposed. The main objective was to study the FW and BW aspects with full spectrum analysis. The presence of negative frequency components such as − 1/2X, − 1X, − 2X and − 3X was reported to be more prominent in the combined rubcrack faults than rubimpact only. Therefore, it was concluded that multifault cases such as combined rub and crack induces BW phenomena.
The nonlinear aspects of rotor system containing a crack and fractional order damping were analyzed in [288]. The presence of breathing crack resulted in the development of 1/2X, 3/2X harmonic components along with 2X and 3X harmonics. Moreover, the introduction of fractional damping resulted in the development of quasiperiodic motion, periodic motion, and chaos in the bifurcation diagrams. The dynamic vibrational response of a cracked rotor system with hydrodynamic bearings was studied using bifurcation and Poincare maps in [369]. A similar study was conducted in [370] for a cracked rotor system with ball bearings. During transient operations with constant angular acceleration rate, the statespace matrix (which represents the equations of motion of a cracked rotor system) becomes lineartime varying [214]. Such timevarying systems produce dynamic eigenvalues, as opposed to static eigenvalues [371]. Moreover, these dynamic eigenvalues contain more information than the Floquet’s numbers and the Lyapunov exponents.
7 Conclusion
The present review article has addressed the techniques utilized in modeling and identification of cracks in rotor systems. Most crack modeling techniques have been observed through this review to be an extension to some primary models. These primary models can be categorized as Dimarogonas’ crack model, breathing and switching crack model, and synchronous crack model. The Dimarogonas’ and the switching crack model by O.S Jun utilized the concept of fracture mechanics for the respective modeling of the crack. The major drawback was that the depth of the crack cannot be modeled beyond a certain range below the radius of the shaft. The hinge crack model (some literature termed it also switching crack model) utilized a harmonic steering function to model the opening and closing of the crack. However, the model does not have any correlation between the crack depth and the shaft’s stiffness although it is the simplest model to implement. Subsequent studies extended these primary models to account for various effects such as partial closing (or opening) of the crack, axial forces, etc. The synchronous crack model is the relatively newer model that accurately captures the breathing effect of the crack accounting for various states of crack closure or opening. The synchronous crack model utilized static deflection for the initiation of crack breathing which isn’t valid when the operational speed of the rotor system is at the neighborhood of resonance. As such, a nonsynchronous crack model was recently developed and applied to cracked rotor systems. It is highly likely that various components related to a rotor system contain uncertainties in the form of manufacturing defects, expansion due to thermal gradient, and damage due to wear and tear. Therefore, multiple studies accounted for such uncertainties in rotor system components using mathematical models such as PCE, Monte Carlo simulation, Chebyshev collocation surrogate method, stochastic FEM, and HBM. The variables that were kept uncertain include Young’s modulus and density (of the rotor shaft), unbalance mass and orientation, and bearing support stiffness. The resultant models were later used to detect cracks in the presence of uncertainty in rotor system.
Moreover, various crack detection methodologies were employed in the literature and evaluated in this review. Some common methodologies include the FFT, WT, HHT, whirl orbits analysis, ODS, energy methods, machine learning, and pre and post resonance backward whirling. Based on the FFT analysis conducted on cracked rotor systems, the 2X and 3X harmonic components were found suitable for crack detection with the 1X component excited for shallow crack cases. Based on the FSA, forward as well backward 2X and 3X harmonic components were observed for the cracked case. From the WT and HHT analysis, the 2X and 3X harmonic components remained the key indicator for the presence of crack with the magnitude (of harmonic components) proportional to crack depth. From the whirl orbits, the inner/outer loop(s) critical rotational speed were found to be the indicators of the crack. Generating the inner/outer loop whirl orbit plots is the easiest way to capture the presence of cracks in a rotor system. The whirl orbits and the energy methods also reiterate the utilization of 1X, 2X, and 3X harmonics for crack detection. The main drawback of the aforementioned methods is the robustness when there are multiple faults present in the rotor system. It was found that rub and misalignment related faults also excite the 2X and 3X harmonic components. Consequently, more reliable methods are needed for accurate multifault differentiation. The PoBW is a potential candidate that could accurately identify the whirl signature of multiple faults. Machine learning algorithms require relevant feature training datasets for its fault detection methodology which also is its main drawback. It was specified in the FSA that faults generate varying levels of BW and FW components. PoBW captures the unique whirl patterns in a faulty rotor system which can be used for fault detection. The method was applied to a cracked rotor and a rotor system undergoing rubbing or snubbing contact. The major theme for future research might focus on developing more advanced and reliable methodologies for single and multifault identification compared with the existing literature.
Data availability
No datasets were generated or analysed during the current study.
Abbreviations
 FEM:

Finite element methods
 TMM:

Transfer matrix method
 EMD:

Empirical mode decomposition
 WT:

Wavelet transform
 HHT:

Hilbert Huang transform
 FFT:

Fast Fourier transform
 ANN:

Artificial neural network
 CNN:

Convolutional neural network
 JC:

Jeffcott
 SERR:

Strain energy release rate
 SIF:

Stress intensity factor
 DOF:

Degree of freedom
 3D:

Threedimensional
 2D:

Twodimensional
 AMB:

Active magnetic bearings
 CCLP:

Crack closure line position
 CZM:

Cohesive zone modeling
 PCE:

Polynomial chaos expansion
 HBM:

Harmonic balance method
 NDT:

Nondestructive testing
 FSA:

Full spectrum analysis
 STFT:

Short time Fourier transform
 CWT:

Continuous wavelet transform
 DWT:

Discrete wavelet transform
 WFEM:

Wavelet finite element method
 IMF:

Intrinsic mode functions
 IF:

Instantaneous frequency
 VMD:

Variational mode decomposition
 WVD:

Wigner–Ville distribution
 WPT:

Wavelet packet transforms
 ODS:

Operational deflection shapes
 SDC:

Slope deviation curve
 ANFIS:

Adaptive neuroFuzzy inference systems
 AWCD:

Approximate waveform capacity dimension
 ML:

Machine learning
 SVM:

Support vector machines
 PSD:

Power spectral density
 PPCM:

Peak position component method
 FW:

Forward whirling
 BW:

Backward whirling
 PrBW:

Preresonance backward whirl
 PoBW:

Postresonance backward Whirl
 SCDS:

Super harmonic characteristic deflection shapes
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