Nonlinear Dyn (2019) 98:671–690https://doi.org/10.1007/s11071-019-05221-0

ORIGINAL PAPER

Method of shaft crack detection based on squared gain ofvibration amplitude

Rafal Gradzki · Zbigniew Kulesza ·Blazej Bartoszewicz

Received: 26 June 2018 / Accepted: 26 August 2019 / Published online: 4 September 2019© The Author(s) 2019

Abstract Rotating machines are exposed to differ-ent faults such as shaft cracks, bearing failures, rotormisalignment, stator to rotor rub, etc. Therefore, turbo-generators, aircraft engines, compressors, pumps, andmany other rotating machines should be constantlydiagnosed to warn about the probable appearance of apossible rotor failure. Unfortunately, despite the ongo-ingwork on various rotor fault detectionmethods, thereare still very few techniques that can be considered asreliable and applicable in practical problems. The dif-ficulty lies in the fact that usually the fault introducesvery subtle local changes in the overall structure of therotor. The symptoms of these changes must be isolatedand extracted from a wide spectrum of vibration dataobtained from sensors measuring the vibrations of themachine. The measured data are usually disturbed withsome noise or other disturbances, and that is why thedetection of a possible rotor fault is even more difficult.The paper presents a new rotor fault detection method.Themethod is based on a new diagnosticmodel of rotorsignals and external disturbances. The model utilizesauto-correlation functions of measured rotor’s vibra-tions. By proper processing of the measured vibrationdata, the influence of environmental disturbances iscompletely compensated and reliable indications of thepossible rotor fault are obtained. The method has beentested numerically using the finite element model of

R. Gradzki (B) · Z. Kulesza · B. BartoszewiczFaculty of Mechanical Engineering, Bialystok Universityof Technology, ul. Wiejska 45C, 15-351 Bialystok, Polande-mail: r.gradzki@pb.edu.pl

the rotor and then verified experimentally at the shaftcrack detection test rig. The results are presented in areadable graphical form and confirm high sensitivityand reliability of the method.

Keywords Auto-correlation function · Power spectraldensity function · Signal processing · Fault detection ·Damage map

1 Introduction

Different faults such as shaft cracks, bearing failures,rotor misalignment, stator to rotor rub, etc., can affectthe normal operation of rotating machinery. If notdetected early, such failures can lead to dangerous dam-ages or even catastrophic accidents. Therefore, it isimportant to constantly monitor the technical condi-tion of a given rotating machine and quickly react topossible changes resulting from developing failures.

Traditional rotor fault detection methods are basedon vibration signal analysis. Vibrations of the rotor aremeasured at the bearings by proximity probes (eddycurrent, reluctance, fiberoptic or laser sensors) and thenamplified and analyzed at dynamic signal analyzers orspecialized data acquisition devices equippedwith ded-icated software. Usually, fast Fourier transform (FFT)[1–8] is applied. This way, additional componentsat Fourier spectra increase [3–5,9–12] in the overallamplitude of vibrations or phase variations [1,7,8,11]can be observed and used as indications for rotor failure

123

672 R. Gradzki et al.

detection and warning. For these purposes steady-state(obtained at the constant rotating speed of themachine)[2,3,5,6,11] or transient (e.g., obtained during run-upor run-down of the machine) [7,13–18] vibration datamay be applied.

Soffker et al. [19] undertook comparison of the twoapproaches used for crack detection in rotatingmachin-ery: signal-based and model-based. As a modernmodel-based technique, the PI-observer-based methodwas used. Also a novel signal-based approach, basedon support vector machine (SVM) and wavelets as anexample for amodernmachine learning technique, wasintroduced. When compared to complicated model-based [20–25],modal testing [26,27] or non-traditional[13,14,28–32] techniques, themethods based on vibra-tion signal analysis are quick, lowcost and simple.Theyutilize popular, commonly used measurement equip-ment and well-established computational proceduresand software. Theymay be applied continuously onlineduringnormal (steady-state)machine operation.There-fore, theyhave already found and are expected to extendtheir practical applications in different real-life rotordamage detection problems.

First research reports on signal-based methodsappeared at early eighties. Bently and Muszynska [11]observed changes in absolute shaft position, vibrationamplitude and steadily increasing 1× component inthe FFT spectra. The appearance of 1× componenthas been also confirmed by Werner [8]. Saavedra andCuitino [6] conducted a theoretical and experimentalanalysis of horizontally rotating shafts demonstratingthat the 2× component at the FFT spectrum appears athalf the first critical speed value if the transverse shaftcrack develops. The 2× component has been also stud-ied by Allen and Bohanick [9], Bently and Muszyn-ska [1], Lazzeri et al. [12], Kulesza and Sawicki [2].Additional components in the vibration spectra dueto the crack have been reported by Bachschmid et al.[10], Ishida et al. [33], and Sinou and Lees [27]. Pateland Darpe [5] reported a stronger 1× component inaxial and torsional frequency responses in case of par-allel misalignment and very strong 3× component incase of angular misalignment. Szolc et al. [34] usedMonte Carlo numerical simulations of coupled nonlin-ear lateral–torsional–axial vibrations of the rotor withtransverse crack to generate dynamic responses for var-ious possible crack locations and depths. To detect andlocalize the crack, Szolc et al. applied and comparedfour identification methods: the nearest-point method

(NP), the Nedler–Mead method (NM), the orthogonalprojection method (OP) and the local sample method(LS).

Another signal-based method for shaft crack detec-tion has been proposed by Gasch and Liao [35]. Thismethod analyzes changes in orbit shapes of the crackedrotating shaft. Vibration signal of the shaft is decom-posed into forward and backward orbits of 1×, 2× and3× frequencies, and a continuous monitoring of thebackward harmonics can indicate the crack. Changesin orbit shapes indicating different rotor faults havebeen also considered by Patel and Darpe [5], Goldmanet al. [36], Sinou and Lees [27], Wang et al. [37], Al-Shudeifat et al. [38], Han and Chu [39].

Investigatingdifferent variants of themethodof orbitshape changes in a two-disk accelerating/deceleratingshaft, Al-Shudeifat [40] recently observed an interest-ing phenomenon of additional backwardwhirl frequen-cies. These frequencies appeared not below any criticalforward whirl frequency (as usually accepted), but justabove the forward whirl frequencies, and only in thepresence of shaft crack. These observations have beenverified experimentally and explained theoretically bysign changes in gyroscopic matrices of the disks.

Nevertheless, the fast Fourier transform remains acommon approach, when signal-based damage detec-tion methods are applied. More sophisticated signalprocessing techniques, like auto-/cross-correlation orpower spectral density functions, i.e., the techniquesbased on statistical data analysis, are less popular.

Ha et al. [41] utilized auto-correlation functionswithproperly selected windowing and time synchronousaveraging for condition monitoring of planetary gearsin window turbines. They were able to identify a faultsignature in the planet’s tooth even in a case whenthe amount of the available stationary vibration datawas limited. Ni et al. [42] considered auto-/cross-correlation functions of multiple excitations appliedto a mechanical structure. The functions were dividedinto two parts: the time-variant one associatedwith unitresponse functions depending on structural parameters,and the time-invariant one depending on the energyof the excitation force. By using these auto-/cross-correlation functions and the model updating tech-nique, Ni et al. [42] were able to identify small cracksin a steel structure. A simple technique based on auto-correlation functions of the vibration data obtainedfrom the cracked bar was proposed by Ramsagar andPardue [43]. The effectiveness of this method was

123

Method of shaft crack detection based on squared gain 673

demonstrated experimentally for a set of aluminumbars with different depths of a crack. Similar techniquewas used by Zubaydi et al. [44] to detect small cracks instiffened plates being themodels of ship hulls. Compar-ing auto-correlation functions of healthy and crackedplates, they confirmed the effectiveness of this simplemethod, especially in the presence of external distur-bances and measurement noise. Zhang and Schmidt[45] introduced a correlation function matrix, consist-ing of cross-correlation functions between combina-tions of the vibration responses measured at differentpoints of the 12-bar mechanical structure. Defining aspecial damage index for this matrix-based damageindicator, they confirmed numerically the ability of themethod not only to detect but also to locate the localstiffness changes in the structure.

The application of correlation function-based signalprocessing techniques for rotor fault detection prob-lems is not reported frequently. A possible use ofauto-correlation functions to average vibration signalsobtained from the rotating shaft was suggested byGosiewski and Sawicki in [46]. In [47] the authors usedthe decorrelation technique, based on power spectraldensity functions, to separate vibration signals fromthe superposed signal. Seibold and Weinert [25] uti-lized auto-correlation functions to compare the out-puts of extended Kalman filters in a filter bank andthus to locate the crack along the length of a shaft.Sekhar [31] defined the coherence measure based onauto-correlation functions to estimate the probabilityof shaft cracks in a rotor.

The present paper introduces a new signal-basedapproach for rotor fault detection [48–51]. The methodis based on auto-correlation and power spectral den-sity functions of the vibration signals measured at thebearings of the rotating shaft. Environmental signals(e.g., external disturbances, sensor noise etc.) affect-ing the operation of the machine are also included inthe diagnostic model. By proper signal transformationsincluding the calculation of squared amplitude gains ofmeasured signals, the method is able to eliminate theinfluence of adverse disturbances on a diagnosticmodelof the machine. This way, the effectiveness of damageindications can be considerably improved. The abilityto reject the influence of external disturbance signalson the diagnostic model is a unique feature of the pro-posed method which differs it from other traditional,signal-based diagnostic methods.

The method was developed by Lindstedt [48,49]and applied successfully to various diagnostic prob-lems of turbine blades [48–53] and jet engines [54].The present paper unifies themathematical foundationsof the method and explains its possible application forrotor fault detection problems. Provided numerical andexperimental results confirm its high effectiveness andreliability when applied to shaft crack detection.

2 Mathematical foundations

2.1 Squared amplitude gain of signals

In diagnostics of machines, the technical conditionA(ϑ) of a given mechanical structure can be evaluatedbased on vector y(t) of operational signals yk(t) mea-sured by sensors, k = 1, 2, . . . , nm. The informationabout the ambient environment affecting the operationof this structure (external disturbances, sensor noise,etc.) can be presented as vector x(t) of environmentalsignals xl(t), l = 1, 2, . . . , nu. Here, nm and nu are thenumbers of operational and environmental signals.

Environmental signals x(t) disturb the normal oper-ation of the machine and affect also operational sig-nals y(t). Therefore, the technical condition A(ϑ) ofthe machine can be defined by the mutual relationshipbetween signals y(t) and x(t) at the current moment ϑ1

of the diagnostic time interval and at the moment ϑ0 ofthe beginning of this time interval [49]

A(ϑ) = f(y(t)ϑ0 , x(t)ϑ0 , y(t)ϑ1 , x(t)ϑ1) (1)

where A(ϑ) is the matrix of parameters that define thetechnical condition of the machine, t is time in terms ofNewton’s definition (for diagnostic examinations) andϑ is time in terms of Bergson’s definition (for diagnos-tic inference).

This relation between signals y(t) and x(t) can bepresented in the form of the following state equation

y(t)ϑ = A(ϑ)y(t)ϑ + B(ϑ)x(t)ϑ (2)

whereB(ϑ) is thematrix of coefficients that define howintensely the environment affects the given machine.

For a given pair of signals yk(t) and xl(t), the rela-tions given by Eq. (2) can be transformed to the fol-lowing form

yk(s) = Gkl(s)xl(s) (3)

where yk(s) and xl(s) are Laplace’s transforms of timesignals yk(t) and xl(t) (having initial conditions equal

123

674 R. Gradzki et al.

to zero) and Gkl(s) is a transfer function defining therelation between yk(s) and xl(s).

From the basic properties of power spectral den-sity analysis, the squared amplitude gainW 2(ω) of theoperational signal yk(t) related to the environmentalsignal xl(t) can be obtained as [55]

W 2(ω) = |Gkl( jω)|2 = Syy(ω)

Sxx (ω)(4)

where Gkl( jω) is the frequency response version oftransfer function Gkl(s), jω is the imaginary part ofcomplex variable s and Syy(ω), Sxx (ω) are power spec-tral density functions of signals yk(t) and xl(t).

Using Wiener–Khinchin theorem, power spectraldensity functions Syy(ω) and Sxx (ω) can be calcu-lated as Fourier transforms of the corresponding auto-correlation functions Ryy(τ ) and Rxx (τ ) [55]

Syy(ω) = F[Ryy(τ )

], Sxx (ω) = F [Rxx (τ )] (5)

where

Ryy(τ ) =∞∫

−∞yk(t)yk(t − τ)dτ ,

Rxx (τ ) =∞∫

−∞xl(t)xl(t − τ)dτ (6)

and F [ f (τ )] denotes Fourier transform of a time func-tion f (τ ) and τ is time shift variable.

Equations (2)–(4) may be seen as different diagnos-tic models of a given machine as they express relationsbetween operational yk(t) and environmental xl(t) sig-nals on the one hand and technical parameters of themachine A(ϑ) on the other.

The relation defined with Eq. (4) is particularlyimportant as it can be used to eliminate environmentalsignals xl(t) from the diagnostic model of the machine.This is explained in detail in the next section.

2.2 Elimination of environmental signals from thediagnostic model

Consider two time intervals �t1 and �t2 (�t1 = �t2)at which operational yk(t) and environmental xk(t) sig-nals are analyzed (Fig. 1). According to Eq. (4) thesquares of amplitude gains W 2

1 (ω), W 22 (ω) of signal

yk(t) related to signal xl(t) in time intervals �t1, �t2can be calculated as

W 21 (ω) = Syy1(ω)

Sxx1(ω), W 2

2 (ω) = Syy2(ω)

Sxx2(ω)(7)

where Syy1(ω), Sxx1(ω) and Syy2(ω), Sxx2(ω) arepower spectral density functions of yk(t) and xl(t) sig-nals in subsequent time intervals �t1 and �t2. Notethat Sxx1(ω) and Sxx2(ω) functions cannot be calcu-lated since environmental signal xl(t) is not measureddirectly. However, if a very short time distance �t0between time intervals �t1 and �t2 is assumed, thenpower spectral density functions Sxx1(ω) and Sxx2(ω)

can be considered as equal, i.e.,

Sxx1(ω) = Sxx2(ω) (8)

This assumption can be explained by the observationthat if the time distance�t0 between time intervals�t1and �t2 is short, then environmental signal xl(t) inthose time intervals may be considered as unchanged.

Fig. 1 Operational and environmental signals in two time intervals

123

Method of shaft crack detection based on squared gain 675

Including Eq. (7), a new diagnostic model W 221(ω)

describing the technical condition of the testedmachinecan be introduced in the following form

W 221(ω) = W 2

2 (ω)

W 21 (ω)

(9)

Applying Eqs. (8) and (9) yields

W 221(ω) = Syy2(ω)

Syy1(ω)(10)

Note that by taking the quotient of W 22 (ω) and W 2

1 (ω)

the power spectral density functions Sxx1(ω), Sxx2(ω)

of environmental signals xl(t) have been removed fromthe new diagnostic model W 2

21(ω). This way, the influ-ence of environmental signals xl(t) on the new modelW 2

21(ω) has been eliminated. Of course, the newmodelstill describes the technical condition of the testedmachine, as it relates operational yk(t), environmen-tal xk(t) signals and parameters A(ϑ) of the machine(retrace the sequel of Eqs. (3), (4), (7) and (10). How-ever, this model is evaluated using only measured oper-ational signals yk(t), with no need to include directmeasurements of environmental signals xl(t). What ismore important, environmental signals do not disturbdiagnostic indications of the new model, although theystill disturb operational signals yk(t).

2.3 Condition monitoring based on the newdiagnostic model

In order to use the proposed diagnostic model for con-dition monitoring, the operational signals yk(t) of agiven machine are measured and sampled with timeperiod h > 0. Then, prescriptive numbers N of sig-nal samples in two adjacent time intervals �t1 and �t2are collected into two separable sets yk1(n) and yk2(n),n = 0, 1, 2, . . . , N . The number N of signal samplesin each set must be chosen carefully to ensure properstatistical assessment of measured operational signalsyk1(n) and yk2(n). Next, to reduce frequency leakage,signals yk1(n) and yk2(n) are scaled by Hanning win-dow Hw(n)

Hw(n)= 1

2

(1−cos

(2π

n

N

)), n = 0, 1, 2, . . . , N

(11)

to form new signals yHk1(n) and yHk2(n)

yHk1(n) = Hw(n)yk1(n), yHk2(n) = Hw(n)yk2(n)

(12)

Then, discrete auto-correlation functions Ryy1(m),Ryy2(m) of the scaled signals yHk1(n) and yHk2(n) arecalculated, as follows

Ryy1(m) =N−1∑

n=0

yHk1(n)yHk1(n − m),

Ryy2(m) =N−1∑

n=0

yHk2(n)yHk2(n − m) (13)

where m = − (N − 1), . . . ,−1, 0, 1, . . . ,+(N − 1).Discrete auto-correlation functions Ryy1(m),

Ryy2(m) are then approximated with smooth approxi-mating functions, to obtain analytical representationsRyy1(τ ), Ryy2(τ ). As approximating functions thepolynomials may be used [50,51], resulting in the fol-lowing forms of the auto-correlation functions:

Ryy1(τ ) = arτr + ar−1τ

(r−1) + · · · + a1τ + a0,

Ryy2(τ ) = brτr + br−1τ

(r−1) + · · · + b1τ + b0

(14)

where a j , b j are coefficients of the polynomials, j =0, 1, 2, . . . , r .

The order r of the polynomials should be selectedcarefully—too low order will result in inaccurateapproximations, too high order—in an excessive num-ber of polynomial coefficients and longer calculationtimes.

Based on analytical representations, the auto-correlation functions Ryy1(τ ), Ryy2(τ ) Laplace trans-forms F

[Ryy1(τ )

], F

[Ryy2(τ )

], and power spectral

density functions Syy1(ω), Syy2(ω) are calculatedaccording to Eq. (5). Next, the power spectral den-sity functions Syy1(ω), Syy2(ω) are introduced intoEq. (15). For polynomial approximation, the diagnosticmodel W 2

12(ω) is obtained in the following form

W 221(ω) = Syy2(ω)

Syy1(ω)

= Br sr + Br−1sr−1 + · · · + B1s + B0

Ar sr + Ar−1sr−1 + · · · + A1s + A0

(15)

where

Bj = br− j (r − j)!, A j = ar− j (r − j)!,j = 0, 1, 2, . . . , r (16)

123

676 R. Gradzki et al.

The technical condition of the machine is then eval-uated by calculating relative changes �A j , �Bj inmodel parameters, defined as follows

�A j = A j − A j

A j, �Bj = Bj − B j

B j(17)

where A j , Bj are values of the W 221(ω) model param-

eters evaluated at the moment ϑ1 and A j , B j are meanvalues of these parameters evaluated during subsequentmeasurementswithin the period betweenϑ0 (beginningof the monitoring process) and ϑ1 (the current momentat the monitoring process).

2.4 Damage maps

To simplify the evaluation of technical condition of agiven machine, relative changes �A j , �Bj ofW 2

21(ω)

diagnostic model are classified into three damagethreshold ranges and presented in a convenient graphi-cal way, termed as a damage map [52,53]. The damagemap of the machine is created as described below.

The three damage threshold ranges are defined withmean �A j , �B j and standard deviation σA j , σBj val-ues of relative changes �A j , �Bj , where

�A j = 1

K

K∑

k

�A j (k), �B j = 1

K

K∑

k

�Bj (k)

(18)

and

σA j =√√√√ 1

K

K∑

k

(�A j (k) − �A j

)2,

σBj =√√√√ 1

K

K∑

k

(�Bj (k) − �B j

)2(19)

Here, �A j (k) and �Bj (k) are relative changes inmodel parameters evaluated at time moment ϑk , ϑ0 ≤ϑk ≤ ϑ1, and K is the number of those evaluations.

The three damage threshold ranges are assumed as

�A j ± σA j , �A j ± 2σA j , �A j ± 3σA j (20)

for �A j , and

�B j ± σBj , �B j ± 2σBj , �B j ± 3σBj (21)

for �Bj .The damage map is then created in a form of a

color table of K rows and 2r columns, where each

row contains color indications of whether at a giventime moment ϑk the value of a given relative change�A j and �Bj falls within one of the three damageranges. The colors in subsequent cells in a given roware assigned as follows:

1. If at a given timemoment ϑk , relative changes�A j

or�Bj are within the first damage threshold range,i.e., if(�A j − σA j

) ≤ �A j ≤ (�A j + σA j

)or

(�B j − σBj

) ≤ �Bj ≤ (�B j + σBj

)(22)

then the indication of the respective relative change�A j or �Bj in a given kth row, j th columnbecomes green.

2. If at a given timemoment ϑk , relative changes�A j

or �Bj are within the second damage thresholdrange, then the indication of the respective relativechange�A j or�Bj in a given kth row, j th columnbecomes dark blue.

3. If at a given time moment ϑk , relative changes�A j or�Bj are within the third damage thresholdrange, then the indication of the respective relativechange�A j or�Bj in a given kth row, j th columnbecomes red.

4. If at a given time moment ϑk , relative changes�A j or �Bj are above the third damage thresholdrange, then the indication of the respective relativechange�A j or�Bj in a given kth row, j th columnbecomes black.

It is clear that damage maps defined in this way shouldbe predominantly green for a healthy machine. For themachinewith any damage in structure, the predominantcolors should be red and black.

3 Experimental test rig

The proposed damage detection method has been veri-fied experimentally at a shaft crack detection test rig uti-lized at Bialystok University of Technology (Fig. 2a).The main component of the rig is a rotor supported bytwo ball bearings 2 and driven by an adjustable-speedelectric motor 9 (Fig. 2b). The rotor consists of threeparts connected by cones 5, flanges 6 and screws toensure axial symmetry of the rotor on the one hand,and quick and simple reconfiguration of tested shafts 7on the other. The outermost parts of the rotor are twoshort shafts with balancing disks 4 attached to them.

123

Method of shaft crack detection based on squared gain 677

Fig. 2 Test rig: a photograph, b schematic of the rotor, c schematic of tested shafts: 2-ball bearings; 3-eddy current position sensors;7-tested shaft

The shafts are the unchanging components of the rotor,and they are supported by the bearings. Themiddle partof the rotor is the tested shaft 7, which can be changed.Shafts of different diameters and lengths with or with-out transverse cracks can be attached to the unchangingparts of the rotor (Fig. 2c). This way, different config-urations of the rotor can be tested. Radial positions ofthe shaft near the left bearing aremeasured by two eddycurrent sensors 3, rotated by 45◦ from horizontal andvertical axes (Fig. 2a).

During the experiments, three configurations of therotor have been tested (2c): Cnf1 with uncracked shaftandCnf2, Cnf3with cracked shafts. InCnf2 the crack islocated at the 1/4 shaft length to the left, and in Cnf3—at the 1/4 shaft length to the right. The length of thetested shaft is 250 mm, and its diameter is 16 mm. Athin notch simulating the crack is cut perpendicularly to

the shaft axis using the electrodischarging machining.The relative depth of the crack is μ = 25%, whereμ = a/d, and a is absolute crack depth, d is diameterof shaft cross section.

4 Model of the rotor

Simulation tests of afinite elementmodel of the crackedrotor have been performed to numerically evaluate theeffectiveness of the proposedmethod. Themodel of therotor (Fig. 3) has been formulated using the methodol-ogy presented in [2,56–58].

The shaft is discretized into n = 53 elements ofequal lengths having 6 degrees of freedom at each node(Fig. 3a). These are beam finite elements with circularcross sections. The two disks are modeled as rigid bod-ies of given masses and moments of inertia. The rotor

123

678 R. Gradzki et al.

Fig. 3 Finite element model of the rotor: a discretization of the shaft, b shaft cross section at the crack, c models of tested shafts

is supported by two ball bearings located between 10thand 11th, and 44th and 45th nodes. Vertical and hori-zontal displacements of the shaft are measured by twoproximity sensors located at the 15th and 43rd nodes.

Breathing cracks can be introduced into the 24thor 32nd finite elements (Cnf2 or Cnf3 configuration,see Figs. 2c, 3c), by using the stress energy releaserate approach proposed in [2,56–58]. The model of thecrack is schematically presented in Fig. 3b.

After assembling shaft elements and including bear-ing anddisk dynamics, the rotor-bearing system in localcoordinates can be presented with the following equa-tion of motion:

Mq + (Dd + ΩDg)q + K[q(t)]q = G + Pu (23)

where M, Dd, Dg,K[q(t)] are mass, damping, gyro-scopic and stiffness matrices; G and Pu are vectorsof gravity and unbalance; q is a vector of rotor dis-placements; and Ω is the rotational speed. Here M,Dd,Dg,K[q(t)] are 6n× 6n dimensional matrices, andG, Pu, q are 6n × 1 dimensional vectors. Due to the

breathing mechanism of the crack, the stiffness matrixK[q(t)] is not constant but depends on rotor displace-ments q(t) which are in turn time dependent. To eval-uate the forces on the crack edge, the new responsevector q is used to find the response in the rotor-fixedcoordinates qr . The nodal forces P are then presentedby:

[P] = [K ]r {q}r (24)

Details about the breathing mechanism of the crackare well explained in [2,56]. The components of thematrices are given in “Appendix 1” section.

5 Simulation results

Using the FE model of the rotor presented in Fig. 3,vibration responses of the rotor have been calculated atthe constant rotational speed of Ω = 800 rpm.

The parameters of the rotor have been assumed asfollows: Young’s modulus E = 208 GPa, density ρ =

123

Method of shaft crack detection based on squared gain 679

Fig. 4 Simulation frequency response of the uncracked rotor, μ = 0% (Cnf1): a no noise, b σ = 10−7 m noise

Fig. 5 Simulation frequency response of the cracked rotor μ = 25% (Cnf2): a no noise, b σ = 10−7 m noise

Fig. 6 Simulation frequency response of the cracked rotor μ = 25% (Cnf3): a no noise, b σ = 10−7 m noise

123

680 R. Gradzki et al.

Fig. 7 Simulated shaft positions at Ω = 800 rpm, a no noise, b σ = 10−7 m noise

7850 kg/m3, Poisson’s ratio ν = 0.3, radial stiffnessand damping coefficients of the bearings kb = 3.4×106

N/m, db = 10 Ns/m, and rotor eccentricity ε = 7 ×10−5 m. The simulations have been conducted for therotor without the shaft crack (Cnf1) and with the crackof depth valuesμ = 25% (Cnf2 and Cnf3). To simulatemeasurement disturbances, additional white noise ofstandard deviation σ = 10−7 m has been added to thecalculated vibration responses.

5.1 Simulation frequency response

To initially validate the correctness of the model, sim-ulation frequency responses of the rotor have been cal-culated and are presented in Figs. 4a, b, 5a, b and 6a,b. Note that the results shown in Figs. 4b, 5b and 6bhave been disturbed with an additional white noise ofσ = 10−7 m.

As can be seen in Fig. 4a, b simulation vibra-tion response of the uncracked rotor contains onlyone strong frequency component located at 1× syn-chronous frequency of the rotor, i.e., at f = 13.3 Hz(Ω = 800 rpm). This is a typical response of a linearmodel of the rotor with no crack.

In the simulated response of the cracked rotor, addi-tional 2× and 4× frequency peaks appear, as can beseen in Figs. 5a, b and 6a, b. These 2× and 4× compo-nents result from the nonlinear breathing mechanismof the crack.

5.2 Simulation damage maps

Simulated shaft positions in y axis with and withoutnoise are presented in Fig. 7a, b. These positions havebeen calculated for the rotor rotating with a constantspeed of Ω = 800 rpm. It can be seen (the zoomedportions of Fig. 7) that the vibration amplitudes of theuncracked as well as the cracked rotor are almost thesame, and the 25% deep crack cannot be detected bysearching for possible changes in vibration amplitudeor in the orbits. Therefore, the proposed method basedon auto-correlation functions is applied and tested bysimulations.

According to the method, the vibration response ofthe rotor is analyzed in two separate time intervals �t1and�t2 (Fig. 8). These time intervals are applied in thefollowing manner: �t1 is located to the left from a rec-ognized peak value, and�t2—to the right. The numbern of signal samples in each time interval is constant andis selected arbitrarily to cover the most of the increas-ing/decreasing parts of the vibration response duringone revolution of the rotor. For the rotating speed of800 rpm, the number of samples is n = 299, while thesampling frequency f = 8000 Hz.

Shaft vibrations yk1(n) and yk2(n) in each time inter-val �t1, �t2 are scaled by Hanning window (Eq. 11),and then auto-correlation functions Ryy1(m), Ryy2(m)

of the obtained signals yHk1(n), yHk2(n) are calcu-lated (Eq. 13). Example functions Ryy1(m), Ryy2(m)

obtained for both time intervals of a selected vibra-tion peak are shown in Fig. 9. It is obvious that the

123

Method of shaft crack detection based on squared gain 681

Fig. 8 Time intervals for auto-correlation calculations–simulated vertical positions of the shaft rotating at Ω = 800rpm with no noise

direct comparison of the auto-correlation functionsis difficult. Therefore, their analytical representationsRyy1(τ ), Ryy2(τ ) are calculated (Eq. 14) and relativechanges �A j , �Bj in model parameters (Eq. 17) areused to assess the appearance of a probable shaft crack.After several initial calculations, the order r of theapproximating polynomials (Eq. 14) has been chosenas r = 6 resulting in the coefficient of determinationR2 > 0.997. Next, threshold ranges (Eq. 20) for indi-vidual parameter changes�A j ,�Bj are calculated anddamage maps are created. Auto-correlation analysis isconducted in a series of K = 30 subsequent peaks.The results obtained for each peak in a series are aver-aged over the number of peaks. The number of K = 30peaks is selected to ensure a statistically representativedata sample.

Fig. 9 Simulation auto-correlation functions of a selected vibration peak: a at time interval �t1, no noise, b at time interval �t2, nonoise, c at time interval �t1, σ = 10−7 m noise, b at time interval �t2, σ = 10−7 m noise

123

682 R. Gradzki et al.

Fig. 10 Simulation damage map of the uncracked rotor (Cnf1);Ω = 800 rpm

Damagemaps obtained for the three considered con-figurations of the rotor are presented in Figs. 10, 11and 12. Figure 10 explains the structure of the dam-age map for the uncracked shaft (Cnf1). The mapconsists of 30 rows indicating the location of relativechanges �A j , �Bj within respective threshold ranges�A j ±σA j ,�A j ±2σA j ,�A j ±3σA j and�B j ±σBj ,�B j ±2σBj ,�B j ±3σBj at 30 vibration peaks. Dam-age maps for other rotor configurations are created ina similar manner (Figs. 11, 12). Figures 11 (with nonoise) and 12 (with σ = 10−7 m noise) contain twosets of 30 peak series registered for each rotor configu-ration (Cnf1–Cnf3). Rotation numbers at which vibra-tion peaks are analyzed have been chosen freely (in arandomized way).

The indications of a possible shaft crack in the rotorshown in Fig. 11 (with no noise) are clearly visible. Thedamage maps of the healthy rotor (Cnf1) are predomi-nantly green,while themaps of the cracked rotor (Cnf2,Cnf3) are predominantly black. For a noisy case, sim-ilar indications are observed (Fig. 12)—damage mapsof the cracked rotor are completely different than themaps of the cracked rotor. These changes in damagemaps reflect the technical condition of the rotor andconfirm the effectiveness of the proposed method inthe simulated case.

Fig. 11 Simulation damage maps (no noise) at Ω = 800 rpm: a uncracked rotor (Cnf1), b, c cracked rotor (Cnf2 and Cnf3)

123

Method of shaft crack detection based on squared gain 683

Fig. 12 Simulation damage maps (σ = 10−7 m noise) at Ω = 800 rpm: a uncracked rotor (Cnf1), b, c cracked rotor (Cnf2 and Cnf3)

6 Experimental results

6.1 Experimental frequency response

Frequency responses of the rotor obtained experimen-tally are presented in Fig. 13a–c. As can be seen exper-imental vibration response of the uncracked rotor con-tains not only the 1× synchronous frequency of therotor located at f = 13.3 Hz (Ω = 800 rpm) but alsoits multiples at 2×, 3×, 4×. These additional 2×, 3×,4× components are the main differences in frequencyresponses of the uncracked rotor obtained experimen-tally and by simulations (compare Figs. 4 and 13). Inreal rotor systems, these additional components appearas a result of inevitable nonlinearities that are alwayspresent even in healthy rotors [17,26].

As can be seen in Fig. 13, frequency responses ofthe uncracked as well as the cracked rotor are almostthe same. Therefore, it becomes clear that the changesin the frequency response of the rotor (appearance ofadditional frequency components or increase in theiramplitudes) cannot be used as reliable shaft crack indi-cations.

6.2 Experimental damage maps

Example time histories of vertical shaft positionsobtained for the rotor rotating with a constant speed of800 rpm are shown in Fig. 14. The evident differencebetween the responses in Fig. 14 is in the amplitudesof vibration. However, there is no clear trend in thesechanges—the amplitude for the cracked shaft may belarger (Cnf3) or smaller (Cnf2) than for the uncracked(Cnf1) shaft. This observation is true for other exper-imentally registered cases, not shown in Fig. 14, andconfirms, what has already been noticed in Sect. 5.2(and in literature), that the amplitude changes cannotbe considered as reliable shaft crack indications. Notealso that the obtained vibration responses are disturbedwith somemeasurement noise. Therefore, the proposedmethod may be applied, expecting that due to its abil-ity to eliminate the influence of external disturbances,and independence on absolute amplitude values, theobtained crack indications will be more robust.

According to the method, also in the experimentalcase the vibration response of the rotor is analyzed intwo separate time intervals �t1 and �t2 (Fig. 15). Todetect the peaks, a simple procedure is applied inwhich

123

684 R. Gradzki et al.

Fig. 13 Experimental frequency responses of the rotor, a uncracked rotor (Cnf1), b, c cracked rotor (Cnf2 and Cnf3)

Fig. 14 Experimental shaft positions at Ω = 800 rpm

the measured vibration response is approximated witha sine function.

Example auto-correlation functions Ryy1(m),Ryy2(m) are shown in Fig. 16. The differences betweenthese functions obtained for the uncracked and crackedrotor are larger than in the simulated case (compareFigs. 9 and 16), yet again their analytical representa-tions Ryy1(τ ), Ryy2(τ ) are calculated (Eq. 14) and rel-ative changes�A j ,�Bj in model parameters (Eq. 17)are used to create damage maps and assess the possibleappearance of a shaft crack.

Experimental damage maps obtained for the threeconsidered configurations of the rotor are presented inFig. 17. The indications of a possible shaft crack in therotor shown in Fig. 17 are clearly visible. The damagemaps of the healthy rotor (Cnf1) are predominantly

123

Method of shaft crack detection based on squared gain 685

Fig. 15 Time intervals for auto-correlation calculations–experimental vertical positions of the shaft rotating at Ω = 800rpm

green. The maps of the cracked rotor (Cnf2, Cnf3) arepredominantly black. Again the predominant colors ineachmap reflect the technical condition of the rotor andconfirm the effectiveness of the proposed method alsoin the experimental case.

When comparing the simulation and experimentalresults, it appears that the proposed method is evenmore sensitive to shaft crackswhen applied experimen-tally than when introduced in a simulation procedure.As can be seen in experimentally obtained Fig. 17b, cdamage maps are almost black. Similar damage mapswith mostly black areas are obtained in an undisturbedsimulation case in Fig. 11. For noisy simulation resultsin Fig. 12, the damage maps are only red/blue for thecase of the cracked shaft.

This suggests that the white noise added to the sim-ulation data does not accurately regenerate the realdisturbances that are present in the tested rotatingmachine. Furthermore, the practical sensitivity of themethod may be even higher than predicted by simu-lation calculations. This is because the mathematicalmodel does not include all physical phenomena thatexist in a real rotor.

Therefore, it is expected that the practical applica-bility of the method will by high enough to detect notonly large (μ > 25%) but also small cracks. However,to check and confirm this expectations, experimentaltests for rotors of different shaft crack depths and loca-tions are required and are underway.

7 Conclusions

The rotor fault detection method presented in the paperis based on auto-correlation and power spectral densityfunctions of the rotor vibration response measured atthe bearings. The response is analyzed in two separatetime intervals.When the intervals are in a close vicinityto each other, the influence of external disturbances iseliminated. Therefore, each change in the parameters ofthe diagnostic model shall be interpreted as the changein the technical condition of the machine.

Shaft crack indications are clearly readable and pre-sented in a form of distinctive color maps, where thepredominant green implies the healthy shaft, and thepredominant black—the cracked shaft. The exact loca-tions of the colors at the damage maps may be differ-

Fig. 16 Experimental auto-correlation functions of a selected vibration peak: a at time interval �t1, b at time interval �t2

123

686 R. Gradzki et al.

Fig. 17 Experimental damage maps at Ω = 800 rpm: a uncracked rotor (Cnf1), b, c cracked rotor (Cnf2 and Cnf3)

123

Method of shaft crack detection based on squared gain 687

ent, as the diagnostic thresholds are defined with meanand standard deviation, i.e., with statistic parameters.Therefore, not the exact locations but predominanceof a given color are important. The rotations at whichvibration peaks are analyzed can be chosen freely, yetstill the obtained damage maps clearly reflect the tech-nical condition of the rotor.

Provided experimental and simulation results con-firm that the method is able to reliably indicate the faulteven in the presence of variable amplitude values andrelatively high measurement disturbances. The methodis simple and uses only themeasured vibration data. Noinitial preparation of the rotor for tests is required. Themachine can be monitored continuously online. Thisgives chance for future practical implementation of themethod.

However, at the current state of its development, themethod cannot indicate the depth or location of theshaft crack. It can only warn about a possible malfunc-tion of the rotor, i.e., it can inform a diagnostic staff,that the overhaul of the machine is required (when thedamage map is predominantly blue and red), or that thereplacement of the rotor can be considered (when thedamage map is predominantly red and black).

The efficiency of the method has been confirmedonly for shaft crack detection problems. Based on thepositive results obtained, it is expected that the methodcan also accurately predict the appearance of othermalfunctions, e.g., rubbing or misalignment. However,experimental verification is needed,which is underway.

Acknowledgements This work was supported by theMinistryof Science and Higher Education in Poland (Research ProjectsNo. S/WM/1/2016, WZ/WM/1/2019, MB/WM/2/2018).

Compliance with ethical standards

Conflict of interest The authors declare that they have no con-flict of interest.

Open Access This article is distributed under the terms ofthe Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permitsunrestricted use, distribution, and reproduction in any medium,provided you give appropriate credit to the original author(s) andthe source, provide a link to the Creative Commons license, andindicate if changes were made.

Appendix 1: Matrices of the rotor model

The forms of the matrices introduced in the motion Eq.(23) are explained below.

Symmetrical inertia matrix of the shaft finite ele-ment:

M = ρl

⎡

⎢⎢⎢⎢⎢⎢⎣

m1,1 m1,2 m1,3 m1,4 · · · m1,11 m1,12

m2,2 m2,3 m2,4 · · · m2,11 m2,12

m3,3 m3,4 · · · m3,11 m3,12

· · · · · · · · · · · ·m11,11 m11,12

sym. m12,12

⎤

⎥⎥⎥⎥⎥⎥⎦

,

(A.1)

where the nonzero elements lying on themain diagonaland above it are as follows:

m1,1 = A

3, m1,2 = A

6,

m2,2 = 13A

35+ 6I3

5l2, m2,6 = 11Al

210+ I3

10l,

m2,8 = 9A

70+ I3

5l2, m2,12 = −13Al

420+ I3

10l,

m3,3 = 13A

35+ 6I2

5l2, m3,5 = −11Al

210− I2

10l,

m3,9 = 9A

70− 6I2

5l2, m3,11 = 13Al

420− I2

10l,

m4,4 = I13

, m4,10 = I16

,

m5,5 = Al2

105+ 2I2

15, m5,9 = −m3,11,

m5,11 = −Al2

140− I2

30,

m6,6 = Al2

105+ 2I3

15, m6,8 = −m2,12,

m6,12 = −Al2

140− I3

30,

m7,7 = m1,1,

m8,8 = m2,2, m8,12 = −m2,6,

m9,9 = m3,3, m9,11 = −m3,5,

m10,10 = m4,4,

m11,11 = m5,5,

m12,12 = m6,6.

Symmetrical stiffnessmatrix of the shaft finite element:

K = E

l

⎡

⎢⎢⎢⎢⎢⎢⎣

k1,1 k1,2 k1,3 k1,4 · · · k1,11 k1,12k2,2 k2,3 k2,4 · · · k2,11 k2,12

k3,3 k3,4 · · · k3,11 k3,12· · · · · · · · · · · ·

k11,11 k11,12sym. k12,12

⎤

⎥⎥⎥⎥⎥⎥⎦

,

(A.2)

where the nonzero elements lying on themain diagonaland above it are as follows:

123

688 R. Gradzki et al.

k1,1 = A, k1,7 = − k1,1,

k2,2 = 12I3l2

, k2,6 = 6I3l

,

k2,8 = − k2,2, k2,12 = k2,6,

k3,3 = 12I2l2

, k3,3 = −6I2l

,

k3,9 = − k3,3, k3,11 = k3,5,

k4,4 = I12 (1 + ν)

, k4,10 = − k4,4,

k5,5 = 4I2, k5,9 = − k3,5, k5,11 = 2I2,

k6,6 = 4I3, k6,8 = − k2,6, k6,12 = 2I3,

k7,7 = k1,1,

k8,8 = k2,2, k8,12 = − k2,6,

k9,9 = k3,3, k9,11 = k3,5,

k10,10 = k4,4,

k11,11 = k5,5,

k12,12 = k6,6.

Anti-symmetrical gyroscopic matrix of the shaft finiteelement:

DG = 2ρ

⎡

⎢⎢⎢⎢⎢⎢⎣

d1,1 d1,2 d1,3 d1,4 · · · d1,11 d1,12d2,2 d2,3 d2,4 · · · d2,11 d2,12

d3,3 d3,4 · · · d3,11 d3,12· · · · · · · · · · · ·

d11,11 d11,12antisym. d12,12

⎤

⎥⎥⎥⎥⎥⎥⎦

,

(A.3)

where the nonzero elements lying above diagonal areas follows:

d2,3 = −13Al

35, d2,5 = −11Al2

210, d2,9 = −9Al

70,

d2,11 = −13Al2

420, d3,6 = d2,5, d3,8 = − d2,9,

d3,12 = d2,11, d5,6 = − Al3

105, d5,8 = d2,11,

d5,12 = − Al3

140, d6,9 = d2,11, d6,11 = − d5,12,

d8,9 = d2,3, d8,11 = − d2,5,

d9,12 = − d2,5, d11,12 = d5,6.

In the above formulas A is the cross-sectional area ofthe shaft element, l is the length of the shaft element,I1, I2 and I3 are the moments of inertia of the crosssection of the shaft element with respect to the x1, x2,x3 axis, ρ is the density, E the Young’s modulus and ν

the Poisson’s ratio of the rotor material.

References

1. Bently, D.E., Muszynska, A.: Early detection of shaft crackson fluid-handling machines, In: Proceedings of the ASMEInternational Symposium on Fluid Machinery TroubleShooting, Anaheim, California, December 7–12 (1986)

2. Kulesza, Z., Sawicki, J.T.: New finite element modelingapproach of a propagating cracked shaft. ASME J. Appl.Mech. 80(2), 021025 (2012)

3. Kulesza, Z.: Dynamic behavior of cracked rotor subjected tomultisine excitation. J. Sound Vib. 333, 1369–1378 (2014)

4. Ma, H., Zhao, Q., Han, Q.,Wen, B.: Dynamic characteristicsanalysis of a rotor–stator system under different rubbingforms. Appl. Math. Model. 39, 2392–2408 (2015)

5. Patel, T.H., Darpe, A.K.: Vibration response of misalignedrotors. J. Sound Vib. 325, 609–628 (2009)

6. Saavedra, P.N., Cuitino, L.A.: Vibration analysis of rotorfor crack identification. J. Vib. Control 8(1), 51–67 (2002)

7. Torkhani, M., May, L., Voinis, P.: Light, medium and heavypartial rubs during speed transients of rotating machines:numerical simulation and experimental observation. Mech.Syst. Signal Process. 29, 45–66 (2012)

8. Werner, F.: The ratio of 2× to 1× vibration—a shaft crackdetection myth. Orbit 14(3), 11 (1993)

9. Allen, J.W., Bohanick, J.S.: Cracked shaft diagnosisand detection on reactor recirculation pumps at grandgulf nuclear station. In: Proceedings of the InternationalExhibition and Conference for the Power GenerationIndustries-Power-Gen, Houston, Texas, May–June 5–6, pp.1021–1034 (1990)

10. Bachschmid, N., Pennacchi, P., Tanzi, E., Vania, A.:Identification of transverse crack position and depth in rotorsystems. Meccanica 35, 563–582 (2000)

11. Bently, D.E., Muszynska, A.: Detection of rotor cracks. In:Proceedings of Texas A&M University 15th Turbomachin-ery Symposium and Short Courses, Corpus Christi, Texas,November (1986), pp. 129–139 (1986)

12. Lazzeri, L., Cecconi, S., Faravelli, M., Scala, M., Tolle, E.:Second harmonic vibration monitoring of a cracked shaft ina turbo-generator. In: Proceedings of the American PowerConference, Chicago, Illinois, September 2 (1992), vol. 54,pp. 1337–1342 (1992)

13. Babu, T.R., Srikanth, S., Sekhar, A.S.: Hilbert–Huang trans-form for detection and monitoring of crack in a transientrotor. Mech. Syst. Signal Process. 22, 905–914 (2008)

14. Chandra, N.H., Sekhar, A.S.: Fault detection in rotorbearing systems using time frequency techniques. Mech.Syst. Signal Process. 72–73, 105–133 (2016)

15. Patel, T.H., Darpe, A.K.: Study of coast-up vibrationresponse for rub detection. Mech. Mach. Theory 44(8),1570–1579 (2009)

16. Plaut, R.H., Andruet, R.H., Suherman, S.: Behavior of acracked rotating shaft during passage through a criticalspeed. J. Sound Vib. 173(5), 577–589 (1994)

17. Sawicki, J.T., Wu, X., Baaklini, G., Gyekenyesi, A.L.:Vibration-based crack diagnosis in rotating shafts duringacceleration through resonance. In: Proceedings of SPIE10th Annual International Symposium on Smart Structuresand Materials, San Diego, California, August 4 (2003)

123

Method of shaft crack detection based on squared gain 689

18. Sekhar, A.S., Prabhu, B.S.: Condition monitoring ofcracked rotors through transient response. Mech. Mach.Theory 33(8), 1167–1175 (1998)

19. Söffker, D., Wei, C., Wolff, S., Saadawia, M.S.: Detectionof rotor cracks: comparison of an old model-based approachwith a new signal-based approach. Nonlinear Dyn. 83(3),1153–1170 (2016)

20. Isermann, R.: Model-based fault detection and diagnosis-status and applications. Ann. Rev. Control 29, 71–85 (2005)

21. Kulesza, Z., Sawicki, J.T.: Auxiliary state variables forrotor crack detection. J. Vib. Control 17(6), 857–872 (2010)

22. Kulesza, Z., Sawicki, J.T., Gyekenyesi, A.L.: Robust faultdetection filter using linear matrix inequalities’ approachfor shaft crack diagnosis. J. Vib. Control 19(9), 1421–1440(2013)

23. Pennacchi, P., Bachschmid, N., Vania, A.: A model-basedidentification method of transverse cracks in rotatingshafts suitable for industrial machines. Mech. Syst. SignalProcess. 20, 2112–2147 (2006)

24. Pennacchi, P., Vania, A.: Diagnosis and model basedidentification of a coupling misalignment. Shock Vib.12(4), 293–308 (2005)

25. Seibold, S., Weinert, K.: A time domain method for thelocalization of cracks in rotors. J. Sound Vib. 195(1), 57–73(1996)

26. Sawicki, J.T., Friswell, M.I., Kulesza, Z., Wroblewski,A., Lekki, J.D.: Detecting cracked rotors using auxiliaryharmonic excitation. J. Sound Vib. 330, 1365–1381 (2011)

27. Sinou, J.J., Lees, A.W.: The influence of cracks in rotatingshafts. J. Sound Vib. 285, 1015–1037 (2005)

28. Guo, D., Peng, Z.K.: Vibration analysis of a crackedrotor using Hilbert–Huang transform. Mech. Syst. SignalProcess. 21, 3030–3041 (2007)

29. He, Y., Guo, D., Chu, F.: Using genetic algorithms to detectand configure shaft crack for rotor-bearing system. Comput.Methods Appl. Mech. Eng. 190, 5895–5906 (2001)

30. Sawicki, J.T., Sen, A.K., Litak, G.: Multiresolution waveletanalysis of the dynamics of a cracked rotor. Int. J. Rotat.Mach. Article ID 265198 (2009). https://doi.org/10.1155/2009/265198

31. Sekhar, A.S.: Identification of a crack in a rotor systemusing a model-based wavelet approach. Struct. HealthMonit. 2(4), 293–308 (2003)

32. Xiang, J., Zhong, Y., Chen, X., He, Z.: Crack detection in ashaft by combination of wavelet-based elements and geneticalgorithm. Int. J. Solids Struct. 45, 4782–4795 (2008)

33. Ishida, Y., Hirokawa, K., Hirose, M.: Vibrations of acracked rotor: 3/2-order super-subharmonic and one half-order sub-harmonic resonances. In: Proceedings of the 15thBiennial Conference on Mechanical Vibration and Noise,Boston, Massachussets, vol. 84(3), pp. 605–612 (1995)

34. Szolc, T., Tauzowski, P., Knabel, J., Stocki, R.: Nonlinearand parametric coupled vibrations of the rotor-shaft systemas fault identification symptom using stochastic methods.Nonlinear Dyn. 57(4), 533–557 (2009)

35. Gasch, R., Liao, M.: Process for the early detection of acrack in a rotating shaft. US Patent No. 5,533,400 (1996)

36. Goldman, P., Muszynska, A., Bently, D.E., Dayton, K.P.,Garcin, M.: Application of perturbation methodologyand directional filtering for early rotor crack detection.In: Proceedings of the ASME International Gas Turbine

and Aeroengine Congress and Exhibition (IGTI 1999),Indianapolis, Indiana, June 7–10 (1999)

37. Wang, S., Zi, Y., Qian, S., Zi, B., Bi, C.: Effects of unbalanceon the nonlinear dynamics of rotors with transverse cracks.Nonlinear Dyn. 91, 2755–2772 (2018)

38. Al-Shudeifat, M.A.A., Butcher, E.A., Stern, C.R.: Generalharmonic balance solution of a cracked rotor-bearing-disksystem for harmonic and sub-harmonic analysis: analyticaland experimental approach. Int. J. Eng. Sci. 48, 921–935(2010)

39. Han, Q., Chu, F.: Dynamic response of cracked rotor-bearing system under time-dependent base movements. J.Sound Vib. 332, 6847–6870 (2013)

40. Al-Shudeifat, M.A.A.: Stability analysis and backwardwhirl investigation of cracked rotors with time-varyingstiffness. J. Sound Vib. 348, 365–380 (2015)

41. Ha, J., Youn, B.D., Oh, H., Han, B., Jung, Y., Park, J.:Autocorrelation-based time synchronous averaging for con-dition monitoring of planetary gearboxes in wind turbines.Mech. Syst. Signal Process. 70–71, 161–175 (2016)

42. Ni, P., Xia, Y., Law, S.S., Zhu, S.: Structural damagedetection using auto/cross-correlation functions undermultiple unknown excitations. Int. J. Struct. Stab. Dyn.14(5), 1440006 (2014)

43. Ramsagar, P.K., Pardue, S.J.: Damage detection methodusing autocorrelation. Proc. SPIE 4359(2), 1622–1627(2001)

44. Zubaydi, A., Haddara, M.R., Swamidas, A.S.J.: On the useof the autocorrelation function to identify the damage in theside shell of a ship’s hull. Mar. Struct. 13, 537–551 (2000)

45. Zhang, M., Schmidt, R.: Sensitivity analysis of anauto-correlation-function-based damage index and itsapplication in structural damage detection. J. Sound Vib.333, 7352–7363 (2014)

46. Gosiewski, Z., Sawicki, J.T.: A new vibroacoustic methodfor shaft crack detection. In: Proceedings of the InternationalSymposium on Stability Control of Rotating MachineryISCORMA-4, Calgary, Alberta, 2007, August 27–30

47. JianPing, J., Guang, M.: A novel method for multi-faultdiagnosis of rotor system.Mech.Mach. Theory 44, 697–709(2009)

48. Kotowski, A., Lindstedt, P.: The using of signals of impulseacoustic response in test of rotor blades in stationaryconditions. In: Proceeding of the International Symposiumon Stability Control of Rotating Machinery ISCORMA-4,Calgary, Alberta, Canada, August 27–30 (2007)

49. Lindstedt, P., Rokicki, E., Borowczyk, H., Majewski, P.:Rotor blades condition monitoring method based on theelimination of the environmental signal. In: ResearchWorksof Air Force Institute of Technology AFIT 25, pp. 15–24(2009). https://doi.org/10.2478/v10041-009-0005-y

50. Lindstedt, P., Gradzki, R.: Parametrical models of workingrotor machine blade diagnostics with its unmeasurableenvironment elimination. Acta Mech. Autom. 4(4), 56–63(2010)

51. Gradzki, R., Lindstedt, P., Kulesza, Z., Bartoszewicz, B.:Rotor blades diagnosis method based on differences inphase shifts. Shock Vib. Article ID 9134607, 13 pages(2018). https://doi.org/10.1155/2018/9134607

52. Gradzki, R., Golak, K., Lindstedt, P.: Parametric andnonparametric diagnostic models for blades in the rotating

123

690 R. Gradzki et al.

machinery with environment elimination. J. KONES 23(2),137–145 (2016)

53. Gradzki, R.: The influence of diagnostic signalmeasurementperiod on blades technical condition images determinedfrom phase shift difference. Solid State Phenom. 199,67–72 (2013)

54. Golak, K., Lindstedt, P., Gradzki, R.: Studies of the jetengine control quality based on its response to the distur-bance inflicted on the object, designated from its responseto the set point inflicted to the controller, safety andreliability: methodology and applications. In: Proceedingsof the European Safety and Reliability Conference ESREL,14–18 September 2014, Wroclaw, pp. 137–140 (2014)

55. Chatfield, C.: The Analysis of Time Series—An Introduc-tion. Chapman and Hall, London (2003)

56. Darpe, A.K., Gupta, K., Chawla, A.: Coupled bending,longitudinal and torsional vibrations of a cracked rotor. J.Sound Vib. 269, 33–60 (2004)

57. Dimarogonas, A.D., Paipetis, S.A.: Analytical Methodsin Rotor Dynamics. Applied Science Publishing, London(1983)

58. Papadopoulos, C.A., Dimarogonas, A.D.: Coupling ofbending and torsional vibration of a cracked Timoshenkoshaft. Ing. Arch. 57, 257–266 (1987)

Publisher’s Note Springer Nature remains neutral with regardto jurisdictional claims in published maps and institutional affil-iations.

123