Nonlinear Dyn (2009) 57: 533–557DOI 10.1007/s11071-008-9406-7

O R I G I NA L PA P E R

Nonlinear and parametric coupled vibrationsof the rotor-shaft system as fault identification symptomusing stochastic methods

T. Szolc · P. Tauzowski · J. Knabel · R. Stocki

Received: 22 April 2008 / Accepted: 17 July 2008 / Published online: 12 August 2008© Springer Science+Business Media B.V. 2008

Abstract In the paper several stochastic methods fordetection and identification of cracks in the shafts ofrotating machines are proposed. All these methods arebased on the Monte Carlo simulations of the rotor-shaft lateral-torsional-longitudinal vibrations mutuallycoupled by transverse cracks of randomly selecteddepths and locations on the shaft. For this purposethere is applied a structural hybrid model of a realcracked rotor-shaft. This model is characterized by ahigh practical reliability and great computational effi-ciency, so important for hundreds of thousands numer-ical simulations necessary to build databases used insolving the inverse problem, i.e. crack parameter iden-tifications. In order to ensure a good identification ac-curacy, for creating the Monte Carlo samples of datapoints there are proposed special probability densityfunctions for locations and depths of the crack. Suchan approach helps in enhancing databases correspond-ing to the most probable faults of the rotor-shaft sys-tem of the considered rotor machine. In the presentedstudy six different database sizes are considered tocompare identification efficiency and accuracy of con-sidered methods. A sufficiently large database enablesus to estimate almost immediately (usually in less than

T. Szolc · P. Tauzowski · J. Knabel · R. Stocki (�)Institute of Fundamental Technological Research, PolishAcademy of Sciences, ul. Swietokrzyska 21,00-049 Warsaw, Polande-mail: rstocki@ippt.gov.pl

one second) the crack parameters with precision that isin most of the cases acceptable in practice. Then, as anext stage, one of the proposed fast improvement al-gorithms can be applied to refine identification resultsin a reasonable time. The proposed methods seem toprovide very convenient diagnostic tools for industrialapplications.

Keywords Crack rotor dynamics · Nonlinear andparametric vibrations · Hybrid modeling · MonteCarlo simulation · Crack identification methods

1 Introduction

An efficient detection and localization of defects inthe most heavily affected parts of modern turboma-chinery is always a very important task in exploitationof these machines. For this purpose the on-line act-ing monitoring routines during regular operation ofthe turbomachinery are the most advantageous. Ma-jority of the fault identification methods applied untilpresent are based on analyses of vibratory behavior as-sociated with rotational motion of the rotor-machine.Since natural frequencies and mode-shape functionsare usually not very sensitive to local imperfectionsof such mechanical systems caused by relatively smalldefects, e.g. cracks in the rotor-shafts, the correspond-ing changes of these quantities cannot serve as effec-tive fault indicators. Thus, nonlinear effects introduced

534 T. Szolc et al.

by the mentioned defects, e.g. due to crack breath-ing, have been used by some authors as fault diagnos-tic symptoms [1–4]. Fortunately, typical defects ob-served in the rotor-machines, e.g. transverse cracks inthe rotor-shafts, shaft segment misalignments or bear-ing eccentricities, yield disturbances resulting in cou-plings between several kinds of vibrations affectingthe rotor-shaft systems, i.e. coupling of bending vi-brations with torsional and axial vibrations [3, 5, 6].As it follows from numerous theoretical studies andpractical observations, in the cases of the above-mentioned defects of the rotor-machines these cou-plings are usually nonlinear or parametric in charac-ter. Moreover, magnitudes of the additional vibrationcomponents induced by the coupling effects are largeenough to serve efficiently as identification measurefor the given kind of a defect. According to the above,in the case of cracked rotor-shafts there were studied,e.g. in [4, 7, 8], coupling effects between shaft bendingand torsional vibrations regarded as crack occurrencediagnostic symptoms caused by a local anisotropy ofthe faulty shaft cross section. The complete couplingeffects due to anisotropy of the cracked shaft cross sec-tion, i.e. between shaft bending, torsional and axial vi-brations, are taken into consideration in [3, 5, 6]. Thenumerical simulation results of coupled vibrations ofthe cracked rotor-shaft systems obtained in [3, 5] bymeans of the one-dimensional finite element modelof the real object have been qualitatively analyzedin [5] and used for composing the cause–symptomrelationships necessary for crack diagnostics in [3].Then, in order to identify crack depth and positionon the shaft, several methods of inverse mapping ofthe on-line measured response–investigated symptomrelationships have been developed. Most of them em-ploy neural networks or other adaptive systems , seee.g. [3]. Since the fault identification routines based onthe neural networks seem to be quite labor-consumingand not always sufficiently accurate, in the current pa-per, as an alternative approach, stochastic methods oftransverse crack detection and localization in the ro-tating shafts are proposed.

All the proposed methods of fault identificationare based on the Monte Carlo simulation of lateral–torsional–axial vibrations coupled by the crack in therotor-shaft represented by a proper theoretical model.Here, by means of the Monte Carlo simulation, aninput–output database is generated for various crackdepths and locations on the rotor-shaft. The identifica-tion of the most probable crack parameters consists in

analyzing the database points in the neighborhood ofread-outs obtained from on-line monitoring of the realmachine. The methods also provide a measure of con-fidence for the identified values of crack parameters.In order to guarantee that the relationship of interest isproperly investigated for the most probable crack oc-currence events, some assumptions concerning proba-bility distribution functions of the random crack para-meters have to be made. This is one of the major con-cepts of the proposed approach, which is discussed indetail in Sect. 5.

The numerical simulations should be carried out us-ing a theoretical model which guarantees sufficientlyaccurate representation of the real object as well as avery high computational efficiency, so important forthe Monte Carlo simulation involving computationsrepeated thousands or even hundreds of thousandstimes for various crack depths and axial positions onthe shaft. For this purpose, similarly as in [6, 9], thereis applied a hybrid mechanical model built in an analo-gous way as the beam-like finite element models com-monly used for the rotor-shaft systems.

To demonstrate a high identification accuracy ofthis approach and its good applicability for practi-cal engineering requirements, the computational testshave been performed for the large multi-bearing rotor-shaft system of the steam turbogenerator. The resultsof this study are presented in Sect. 6.

2 Assumptions for the hybrid mechanical model

In order to obtain sufficiently reliable results of nu-merical simulations, together with a reasonable com-putational efficiency, the vibrating rotor-shaft sys-tem of a rotor machine is usually modeled by meansof the one-dimensional finite elements of the beam-type. Nevertheless, such models are characterized byrelatively high number of degrees of freedom in therange between hundreds and even thousands. Thecommonly applied cracked shaft models introducenonlinear and parametric effects into the entire dynam-ical system. Thus, for such large finite-element mod-els, proper algorithms reducing number of degrees offreedom have to be employed in order to shorten com-puter simulation times. Moreover, for the Monte Carlosimulation performed for numerous axial fault posi-tions along the entire rotor-shaft line, the discretizationmesh density of the finite element model must be ap-

Nonlinear and parametric coupled vibrations of the rotor-shaft system as fault identification symptom 535

propriately modified in each case. Thus the necessarycorresponding multiple reductions of degrees of free-dom are troublesome and can lead to computationalinaccuracies.

According to the above, in order to avoid the above-mentioned drawbacks of the finite element approachand to maintain the obvious advantages of this method,in this paper dynamic investigations of the entire ro-tating system are performed by means of the one-dimensional hybrid structural model consisting of con-tinuous visco-elastic macro-elements and discrete os-cillators. This model is employed here for eigenvalueanalyses as well as for Monte Carlo numerical sim-ulations of coupled nonlinear lateral–torsional–axialvibrations of the cracked rotor-shaft. Similarly as in[6, 9], in this model successive cylindrical segmentsof the stepped rotor-shaft are substituted by flexurally,axially and torsionally deformable cylindrical macro-elements of continuously distributed inertial visco-elastic properties. Since in the rotor-shaft system of thereal rotor machine the bladed disks and gears are at-tached along many shaft segments by means of shrink-fit connections, the entire inertia of such fragmentsis increased, whereas usually the shaft cross sectionsonly are affected by elastic deformations due to trans-

Fig. 1 Continuous visco-elastic macro-element with distin-guished cross sections responsible for elastic and inertial prop-erties

mitted loadings. Thus, the corresponding visco-elasticmacro-elements in the hybrid model must be char-acterized by geometric cross-sectional parameters re-sponsible for their elastic and inertial properties andby the separate layers responsible for inertial proper-ties only. These are the diametric and polar momentsof inertia as well as the cross-sectional areas. Such ex-emplary ith continuous visco-elastic macro-element ispresented in Fig. 1. In this figure symbols AEi , IEi andI0Ei denote respectively the cross-sectional area, di-ametric and polar moments of inertia responsible forelastic properties, i = 1,2, . . . , n, where n is the totalnumber of macro-elements in the considered hybridmodel. Symbols AIi , IIi and I0Ii denote respectivelythe cross-sectional area, diametric and polar momentsof inertia responsible for inertial properties. Moreover,values of the material constants of each ith macro-element, i.e. Young’s and Kirchhoff’s moduli, can de-pend on the actual operation temperature of the cor-responding rotor-shaft segment of the machine. Thetransverse and torsional external loads continuouslydistributed along the macro-element length li , if any,are described by the two-argument functions pi(x, t)

and qi(x, t), where x is the spatial coordinate and t

denotes time.With an accuracy that is sufficient for practical

purposes, in the proposed hybrid model of the rotor-shaft system, some heavy rotors or coupling diskscan be represented by rigid bodies attached to themacro-element extreme cross sections, as shown inFig. 1. Each journal bearing is represented by theuse of a dynamic oscillator of two degrees of free-dom, where apart from the oil-film interaction alsothe visco-elastic properties of the bearing housing andfoundation are taken into consideration. This bear-ing model makes possible to represent with relativelyhigh accuracy kinetostatic and dynamic anisotropicand anti-symmetric properties of the oil-film in theform of constant stiffness and damping coefficients,see [3]. An example of such a hybrid model of thesteam turbogenerator rotor-shaft system supported onseven journal bearings is presented in Fig. 2.

Fig. 2 Hybrid mechanical model of the steam turbogenerator rotor-shaft system

536 T. Szolc et al.

Fig. 3 Geometry of thecracked shaft segment

In this system in the selected fragment of the rotor-shaft there is considered a transverse crack imple-mented in the model by a proper elastic connection ofthe respective adjacent left- and right-hand-side partsof the faulty shaft segment. An exemplary crackedshaft segment with its geometrical dimensions is pre-sented in Fig. 3. Additional flexural, torsional and ax-ial flexibilities introduced by the crack into the shaftare represented here by means of mass-less springsconnecting the adjacent beam macro-elements substi-tuting cracked shaft segment. Stiffness values of thesesprings are determined according to [3, 5, 7] using thefundamentals of fracture mechanics.

In this study, a hinge model of the transverse crackhas been applied. Thus, the coefficients of the above-mentioned additional flexural, torsional and axial flex-ibilities introduced by the crack are determined usingthe Paris equation based on the Castigliano theorem.In the considered case a double-partial differentiationof the strain energy density function with respect tothe virtual generalized loadings yields the coefficientsof additional flexibility caused by the crack:

fij = ∂2

∂Pi∂Pj

∫ b

−b

∫ a

0J (η)dη dζ

= fij (a,D,E,ν), i, j = 1,2, . . . ,6, (1)

where Pi,Pj are the virtual generalized loadings act-ing on the cracked shaft segment of diameter D, asshown in Fig. 3; a denotes the crack depth, E isYoung’s modulus, ν denotes Poisson’s ratio of theshaft material and b = √

a(D − a) . Symbol J (η)

in (1) is the strain-energy density function

J (η) = 1

E(1 − ν2)

{[6∑

i=1

KIi (Pi, η,D)

]2

+[

6∑i=1

KIIi (Pi, η,D)

]2

+ (1 + ν)

[6∑

i=1

KIIIi (Pi, η,D)

]2}, (2)

where KIi ,KIIi ,KIIIi denote the stress intensity fac-tors of the first, second and third mode, respectively.Exact analytical expressions for the strain densityfunction for the cracked shafts can by found, e.g.,in [3, 5, 7]. As it follows from these papers, since thestress intensity factors KIi ,KIIi ,KIIIi are quadraticfunctions of the virtual generalized loadings Pi , thecoefficients of flexibility obtained using (1) are func-tions of the material constants and cracked shaft cross-sectional parameters a and D. These coefficients cre-ate the 6 × 6 symmetrical local-flexibility matrix.The inverse of this matrix becomes the crack local-stiffness-coefficients matrix of the following form:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

c11 0 c13 0 0 c16

0 c22 0 c24 c25 0

c13 0 c33 0 0 c36

0 c24 0 c44 c45 0

0 c25 0 c45 c55 0

c16 0 c36 0 0 c66

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (3)

The above matrix is not diagonal. If we assume, ac-cording to Fig. 3, that the direction “1” correspondsto translational motion along 0η axis, the direction“2” corresponds to rotational motion around 0ζ axis,the direction “3” corresponds to translational motionalong 0ζ axis, the direction “4” corresponds to rota-tional motion around 0η axis, the direction “5” cor-responds to translational motion along the shaft rota-

Nonlinear and parametric coupled vibrations of the rotor-shaft system as fault identification symptom 537

tion axis 0x and the direction “6” corresponds to rota-tional motion around 0x axis, it follows from (3) thatdue to local cross-sectional anisotropy caused by thecrack the shaft transverse motion is directly coupledwith its rotational motion around the rotation axis, aswell as the rotational motions around diameters are di-rectly coupled with translational motions in the ax-ial direction. According to the above, such a crackmodel makes possible to take into consideration cou-pling effects between the torsional and axial vibrationsof the rotor-shaft with its bending vibrations inducedby residual unbalances of the rotating system.

All coefficients in the local-stiffness matrix (3) re-main constant in the coordinate system {0xηζ } rotat-ing with the shaft. Since the further considerations aregoing to be carried out in an inertial non-rotating co-ordinate system, matrix (3) has to be properly trans-formed and then all stiffness coefficients become time-dependent functions

k∗11(t) = 1

2(c11 + c33) + 1

2(c11 − c33) cos(2α)

− c13 sin(2α),

k∗13(t) = 1

2(c11 − c33) sin(2α) + c13 cos(2α),

k∗16(t) = c16 cos(α) − c36 sin(α),

k∗33(t) = 1

2(c11 + c33) − 1

2(c11 − c33) cos(2α)

+ c13 sin(2α),

k∗36(t) = c16 sin(α) + c36 cos(α),

k∗22(t) = 1

2(c22 + c44) + 1

2(c22 − c44) cos(2α)

− c24 sin(2α),

k∗24(t) = 1

2(c22 − c44) sin(2α) + c24 cos(2α),

k∗25(t) = c25 cos(α) − c45 sin(α),

k∗44(t) = 1

2(c22 + c44) − 1

2(c22 − c44) cos(2α)

+ c24 sin(2α),

k∗45(t) = c25 sin(α) + c45 cos(α),

k∗55(t) = c55,

k∗66(t) = c66,

(4)

where α = �t + αp , � being the shaft constant an-gular speed and αp the angle of crack circumferentialposition on the shaft with respect of the vector of re-sultant excitation caused by the shaft residual unbal-ance.

3 Mathematical formulation of the crackedrotor-shaft vibration problem

In the hybrid model, motion of cross sections of eachvisco-elastic macro-element of the length li is gov-erned by the partial differential equations derived us-ing the Timoshenko and Rayleigh rotating beam the-ory for flexural motion as well as by the hyperbolicequations of the wave type, separately for torsionaland axial motion. Similarly as in [6, 9], mutual con-nections of the successive macro-elements creating thestepped shaft as well as their interactions with the sup-ports and rigid bodies representing the heavy rotors aredescribed by equations of boundary conditions. Theseequations contain geometrical conditions of confor-mity for translational and rotational displacements ofextreme cross sections x = Li = l1 + l2 + · · · + li−1

of the adjacent (i − 1)th and the ith elastic macro-elements:

vi−1(x, t) = vi(x, t),∂vi−1(x, t)

∂x= ∂vi(x, t)

∂x,

θi−1(x, t) = θi(x, t), zi−1(x, t) = zi(x, t),

(5)

where vi(x, t) = ui(x, t) + jwi(x, t), ui(x, t) beingthe lateral displacement in the vertical direction andwi(x, t) the lateral displacement in the horizontal di-rection, j the imaginary number; θi(x, t) is the angulardisplacement with respect to the shaft rotational uni-form motion with the constant velocity �, and zi(x, t)

the translational axial displacement; i = 1,2, . . . , n

and n the total number of macro-elements in the hy-brid model.

The second group of boundary conditions are dy-namic ones, which contain linear, nonlinear and para-metric equations of equilibrium for external forces andtorques, static and dynamic unbalance forces and mo-ments, inertial, elastic and external damping forces,

538 T. Szolc et al.

support reactions and gyroscopic moments. For exam-ple, the dynamic boundary conditions formulated forthe rotating Rayleigh beam, and describing a simpleconnection of the mentioned adjacent (i −1)th and theith elastic macro-elements, have the following form:

EiIEi

∂3vi

∂x3− ρIIi

∂3vi

∂x∂t2− Ei−1IE,i−1

∂3vi−1

∂x3

+ ρII,i−1∂3vi−1

∂x∂t2+ j�ρI0Ii

∂2vi

∂x∂t

− j�ρI0I,i−1∂2vi−1

∂x∂t= 0,

EiIEi

∂2vi

∂x2− Ei−1IE,i−1

∂2vi−1

∂x2= 0,

GiI0Ei

∂θi

∂x− Gi−1I0E,i−1

∂θi−1

∂x= 0,

EiAEi

∂zi

∂x− Ei−1AE,i−1

∂zi−1

∂x= 0,

(6)

where Ei = Ei(Ti),Gi = Gi(Ti) denote respectivelyYoung’s and Kirchhoff’s moduli of the shaft materialof the density ρ, which are expressed as functions ofthe actual operation temperature Ti of the ith macro-element corresponding to the given rotor-shaft seg-ment of the machine.

The transverse crack in the given shaft segmentis also described by boundary conditions formulatedfor extreme cross sections of the corresponding(k − 1)th and kth adjacent macro-elements represent-ing this shaft segment. Then, their dynamic equationsof equilibrium of forces and moments (6) remain valid.However, the geometric equations of displacementconformity (5) must be appropriately substituted byproperly modified relations. If in the shaft the trans-verse crack is assumed, its “breathing” process usuallyhas to be taken into consideration as, e.g., in [2, 5, 6].Then, for the “open” crack in the shaft cross sectionx = Lcr = l1 + l2 + · · · + lk−1, the respective bound-ary conditions have the following form:

Ek−1IE,k−1∂3vk−1

∂x3

= (k∗

11(t) + jk∗13(t)

)Re[vk − vk−1]

+ j(k∗

33(t) − jk∗13(t)

)Im[vk − vk−1]

+ (k∗

16(t) + jk∗36(t)

)[θk − θk−1],

Ek−1IE,k−1∂2vk−1

∂x2

= (k∗

22(t) + jk∗24(t)

)Re

[∂vk

∂x− ∂vk−1

∂x

]

+ j(k∗

44(t) − jk∗24(t)

)Im

[∂vk

∂x− ∂vk−1

∂x

]

+ (k∗

25(t) + jk∗45(t)

)[zk − zk−1],

Gk−1I0E,k−1∂θk−1

∂x

= c66[θk − θk−1] + k∗16(t)Re[vk − vk−1]

+ k∗36(t) Im[vk − vk−1],

(7)

Ek−1AE,k−1∂zk−1

∂x

= c55[zk − zk−1] + k∗25(t)Re

[∂vk

∂x− ∂vk−1

∂x

]

+ k∗45(t) Im

[∂vk

∂x− ∂vk−1

∂x

],

where the local-stiffness coefficients c55, c66, k∗ij (t),

i, j = 1,2, . . . ,6, are defined in (1) and (3).The open/closed-criterion of crack “breathing” can

be described by the following relation formulated inthe coordinate system rotating with the shaft:

ϕk(Lcr , t) > 0, i.e. the crack is open, and

ϕk(Lcr , t) � 0, i.e. the crack is closed,(8)

where:

ϕk(Lcr , t)

=(

∂uk(x, t)

∂x− ∂uk−1(x, t)

∂x

)· cos(α)

+(

∂wk(x, t)

∂x− ∂wk−1(x, t)

∂x

)· sin(α)

for x = Lcr .It is assumed that for the “closed” crack, i.e. for

(82), the geometric boundary conditions (5) are validinstead of relations (7). Moreover, if the crack is closedand the shaft cracked vicinity is compressed due tothe axial vibrations, the boundary condition (54) isvalid. However, for the vicinity of the cracked shaftin tension this relation must be substituted by (74).

Nonlinear and parametric coupled vibrations of the rotor-shaft system as fault identification symptom 539

It should be noticed here that the problem of crackedshaft is formulated as a parametric-nonlinear one, be-cause the considered system changes its configurationdepending on whether the crack is temporarily “open”or “closed” [6]. The above equations of the bound-ary conditions (61,2) and (71,2) are formulated usingthe Rayleigh beam theory. In an analogous way, suchboundary conditions are derived in the framework ofthe Timoshenko beam theory.

It is worth to mention that similar approach forcrack modeling has been applied in [3, 5] for one-dimensional shaft representation using the finite ele-ment method. In those cases the local shaft weaken-ing due to a transverse crack is artificially distributedalong the entire beam finite element corresponding tothe faulty shaft segment by a proper modification of itslocal-stiffness matrix. However, as mentioned above,in the hybrid model used in this work the crack is im-plemented in the form of a discrete mutual connec-tion of two continuous parts of the cracked shaft seg-ment, which is represented by the visco-elastic macro-elements with numbers k − 1 and k. Such an approachseems to represent better a local character of shaftweakening by the crack.

In order to perform simulation of forced vibrationsof the rotor-shafts taking into consideration the hingemodel of crack breathing, it is necessary to determinetwo separate sets of orthogonal eigenfunctions: for thecracked and uncracked system. Relations (7) demon-strate, how the crack couples the rotor-shaft bend-ing vibrations with the torsional and axial vibrations.All coupling terms in (7) are parametric because theycontain the explicit time-variable stiffness coefficientsk∗

11(t), k∗13(t), k∗

33(t), k∗16(t), k∗

36(t), k∗22(t), k∗

24(t),

k∗25(t), k∗

44(t) and k∗45(t) defined by relations (4). As

it follows from (3) and (4), k∗11(t), k

∗13(t), k∗

33(t),

k∗22(t), k∗

24(t) and k∗44(t) are responsible for the mu-

tual bending-to-bending cross-coupling in two perpen-dicular planes in the above-mentioned inertial non-rotating coordinate system. Apart from the terms in(4) oscillating around zero-mean value with single- 1Xand double-synchronous 2X frequencies of the rotat-ing shaft, k∗

11(t), k∗33(t), k∗

22(t) and k∗44(t) contain also

constant average components. However, the variablestiffness coefficients in the terms which couple thebending vibrations with torsional ones, i.e. k∗

16(t) andk∗

36(t), as well as the stiffness coefficients in the termswhich couple the bending vibrations with axial vibra-tions, i.e. k∗

25(t) and k∗45(t), are characterized by the

components fluctuating around zero-mean value withonly single-synchronous 1X frequency of the rotatingshaft. In order to perform an analysis of natural elas-tic vibrations, all the forcing, viscous, parametric andunbalance terms in the boundary conditions have beenomitted. Due to the truncation of these terms, it fol-lows from (7) that the bending, torsional and axial vi-brations of the rotor-shaft system are mutually uncou-pled. According to the above, similarly as in the caseof the uncracked system, the elastic bending, torsionaland axial eigenvalue problems for the rotor-shaft witha crack can be solved separately. Thus, in both casesone obtains separate characteristic equations for theconsidered three eigenvalue problems. These are:

A(ω)B = 0, for the rotor-shaft bending vibrations,

C(ω)D = 0, for the rotor-shaft torsionalvibrations,

E(ω)F = 0, for the rotor-shaft axial vibrations,

(9)

where A(ω) is the complex characteristic matrix andC(ω),E(ω) are the real characteristic matrices, andB,D,F denote the vectors of unknown constant co-efficients in the analytical eigenfuctions defined in[6, 7, 9] for each macro-element in the hybrid model.Thus, the determination of natural frequencies reducesto the search for values of ω, for which the char-acteristic determinants of matrices A,C and E areequal to zero. The bending, torsional and axial eigen-mode functions are then obtained by solving respec-tive equations (9).

According to the described above approach, incomparison with the corresponding uncracked rotor-shaft, the eigenmodes obtained for the cracked systemare characterized by a local weakening caused by thiscrack. This weakening is expressed by the constantstiffness coefficients c55 and c66 in (4) and (7), respec-tively, for the axial and torsional eigenmodes as well asby the average stiffness coefficients 0.5(c11 + c33) and0.5(c22 +c44) in (4) in the non-rotating inertial coordi-nate system for the bending eigenmodes. Moreover, asit follows from (7), the bending eigenmodes becomemutually cross-coupled by the crack in the two per-pendicular planes.

According to the above, the coupling effects be-tween the bending, torsional and axial vibrations are ofa purely oscillatory character in the non-rotating iner-tial coordinates. Thus, all parametric terms in (7) tem-porarily neglected for the eigenvalue problem solution

540 T. Szolc et al.

are going to be used for the forced vibration analysisin order to simulate bending, torsional and axial vibra-tions as mutually coupled by the crack.

The solution for the forced vibration analysis hasbeen obtained using the analytical–computational ap-proach demonstrated in detail in [9]. Solving the dif-ferential eigenvalue problem for the linearized orthog-onal system and an application of the Fourier solutionsin the form of series leads to the set of modal equationsin the Lagrange coordinates

M(�t)r(t) + C(�,�t)r(t)

+ K(ϕ(t),�t

)r(t) = F

(t,�2,�t

), (10)

where:

M(�t) = M0 + Mu(�t),

C(�,�t) = C0 + Cg(�) + Cu(�t),

K(ϕ(t),�t

) = K0 + Kb + Kcr

(ϕk(Lcr , t),�t

).

The symbols M0,K0 denote, respectively, the con-stant diagonal modal mass and stiffness matrices,C0 is the constant symmetrical damping matrix andCg(�) denotes the skew symmetrical matrix of gy-roscopic effects. The terms of the unbalanced ef-fects are contained in the symmetrical matrix Mu(�t)

and in the non-symmetrical matrix Cu(�t). Anti-symmetric elastic properties of the journal bearingsare described by the skew-symmetrical matrix Kb .Nonlinear and parametric properties of the breath-ing crack are described by the symmetrical matrixKcr (ϕk(Lcr , t),�t) of periodically variable coeffi-cients, and the symbol F(t,�2,�t) denotes the exter-nal excitation vector, e.g., due to the unbalance andgravitational forces. The Lagrange coordinate vectorr(t) consists of subvectors of the unknown time func-tions in the Fourier solutions. In order to obtain thesystem’s dynamic response, (10) are solved by meansof a direct integration. The number of equations in(10) corresponds to the number of eigenmodes takeninto consideration, because the forced bending, tor-sional and axial vibrations of the rotor-shaft are mutu-ally coupled and thus, the total number of equationsto solve is a sum of all bending, torsional and ax-ial eigenmodes of the rotor-shaft model in the rangeof frequency of interest. These equations are mutu-ally coupled by the parametric and anti-symmetricalterms regarded as external excitations expanded in se-ries in the base of orthogonal analytical eigenfunc-

tions. A fast convergence of the applied Fourier so-lutions enables us to reduce the appropriate numberof the modal equations to solve, in order to obtaina sufficient accuracy of results in the given range offrequency. Here, it is necessary to solve only 15÷60coupled modal equations (10), even in cases of greatand complex mechanical systems, contrary to the clas-sical one-dimensional beam finite element formula-tion leading usually to large numbers of motion equa-tions corresponding each to more than one hundredor many hundreds degrees of freedom (if the artificialand often error-prone model reduction algorithms arenot applied). However, the proposed hybrid modelingassures at least the same or even better representationof real objects as well as its mathematical description,is formally strict, demonstrates clearly the qualitativesystem properties and is much more convenient for astable and efficient numerical simulation.

In the considered case of nonlinear-parametricproblem, during simulations of forced vibrations, con-dition (8) describing the open/closed-stage of crack“breathing” is computationally predicted after eachintegration step by means of the explicit numericalmethod. Thus, criterion (8) indicates the actual baseof eigenfunctions, in which the coupled dynamic re-sponse should be sought, i.e. the eigenfunctions cor-responding respectively to the cracked or uncrackedrotor-shaft line. Then, at each “switch” from the open-crack-stage into the closed-crack-stage and vice versa,the current system response becomes the initial condi-tion for the “new” stage. Here, for a sufficiently smalldirect integration step, equal to 1/50 of the period cor-responding to the upper limit of the frequency bandconsidered, using the Newmark method almost no cor-rections of the time-step value were required in orderto obtain sufficient accuracy of the computational rou-tine.

4 Numerical example of vibration analysisof the cracked rotor-shaft system

The presented methodology of vibration analysis isshown here in an example of a rotor-shaft system ofthe typical 200 MW steam turbogenerator consistingof the single high- (HP), intermediate- (IP) and low-pressure (LP) turbines as well as of the generator-rotor(GEN). This rotor-shaft system is supported by seven

Nonlinear and parametric coupled vibrations of the rotor-shaft system as fault identification symptom 541

Table 1 The first 12bending natural frequenciesof the steam turbogeneratorrotor-shaft system obtainedby means of Rayleigh’s andTimoshenko’s beam theory

Eigen- Natural frequency Natural frequency Relative

form obtained using obtained using difference [%]

number Rayleigh’s beam theory Timoshenko’s beam theory

[Hz] [Hz]

1 22.742 22.154 −2.65

2 24.516 23.772 −3.13

3 30.924 30.707 −0.71

4 38.976 38.480 −1.29

5 45.886 45.067 −1.82

6 46.486 45.761 −1.58

7 54.183 52.741 −2.73

8 68.251 67.022 −1.83

9 107.794 105.523 −2.15

10 114.205 112.395 −1.61

11 127.162 125.468 −1.35

12 145.211 141.837 −2.38

journal bearings, as shown in Fig. 2. With the aim oftheoretical study, the stepped-rotor shaft of this turbo-generator of the total length of 25.9 m has been mod-eled by means of n = 49 continuous macro-elements,as an initial approximation of its geometry. More ac-curate modeling of such rotor-shaft systems by meansof a greater number n of macro-elements does not in-troduce more detrimental computational efforts. Allgeometrical parameters of the successive real rotor-shaft segments as well as their material constants havebeen determined using the detailed technical docu-mentation of this turbogenerator. The average stiffnessand damping coefficients of the oil-film in the bear-ings as well as the equivalent masses and stiffness anddamping coefficients of the bearing housings are ob-tained by means of measurements and identificationperformed on the real object. The numerical valuesof Young’s and Kirchhoff’s moduli as the actual tem-perature functions Ei(Ti) and Gi(Ti), i = 1,2, . . . ,49,have been determined by the use of the following rela-tionships:

Gi(Ti) = Ei(Ti)

2(1 + ν), ν = 0.32,

where:

Ei(Ti) = (−0.064286(Ti − 20) + 211) × 109 [Pa],

for Ti � 300 [◦C],

Ei(Ti) = (−0.10(Ti − 300) + 193) × 109 [Pa],

for 300 < Ti � 400 [◦C],

Ei(Ti) = (−0.15(Ti − 400) + 183) × 109 [Pa],

for 400 < Ti � 500 [◦C],

Ei(Ti) = (−0.22(Ti − 500) + 168) × 109 [Pa],

for Ti > 500 [◦C].

Table 1 contains first 12 bending eigenfrequenciesof the considered uncracked rotor-shaft system ob-tained using Rayleigh’s and Timoshenko’s beam ro-tating with the nominal rotational speed 3000 rpm. Asit follows from the performed comparison, the sheareffect taken into consideration in the case of Timo-shenko’s beam theory results in a little bit smaller nat-ural frequency values than these determined by meansof Rayleigh’s beam model. Here, in the frequencyrange 0÷150 Hz, which is the most important fromthe engineering viewpoint, the respective differencesslightly exceed 3%. The eigenfunctions correspond-ing to these natural frequencies and determined us-ing both beam theories respectively overlay each other.According to the above, one can conclude that in thisfrequency range an application of Rayleigh’s rotatingbeam theory seems to be sufficiently accurate for fur-ther simulations of forced vibrations.

The first 10 bending eigenforms corresponding tothe above-mentioned natural frequencies are depicted

542 T. Szolc et al.

in Fig. 4, where the vertical projections of the bend-ing eigenfunctions are presented by the solid lines andtheir horizontal projections by the dashed lines. Inthese plots the vertical bars illustrate the modal trans-verse vertical and horizontal displacement compo-nents of the rigid bodies representing the bearing hous-ings. It is to remark that due to strong anisotropy andskew-symmetry of the journal bearing elastic proper-ties the shapes of the eigenform vertical projectionsare different from the shapes of their horizontal pro-jections, which indicates a spatial character of theseeigenfunctions. As it was easy to expect, from anal-ogous computations performed for the cracked tur-bogenerator rotor-shaft it follows that for the consid-ered crack-depth values the respective differences ofnatural frequencies obtained for the cracked and un-cracked system do not exceed 1 Hz for various axialcrack positions within the frequency range 0÷700 Hz.The local “un-smoothnesses” of the correspondingeigenfunctions caused by the cracks are almost unre-markable.

The first five torsional eigenfunctions together withtheir natural frequency values, as well as the first fiveaxial eigenfunctions with the corresponding naturalfrequencies, are presented in Fig. 5. It is to remarkthat the 1st axial eigenform is almost of the typicalrigid mode type. The corresponding natural frequencyvalue f1A = 19.623 Hz depends first of all on the lon-gitudinal stiffness of bearing #5 (see Fig. 2), play-ing here also the thrust-bearing role. Similarly as inthe case of bending eigenvibrations, all the torsionaland axial eigenfunctions determined for the crackedand uncracked rotor-shaft, respectively, almost over-lay each other within the investigated frequency range0÷700 Hz. Torsional natural vibrations are the mostsensitive to the transverse crack, for which the great-est difference of the eigenfrequencies obtained for thecracked and uncracked shaft reach 2.5 Hz. The analo-gous difference of the axial natural frequencies did notexceed 0.1 Hz in the mentioned frequency range. Thisresult confirms the known fact that values of naturalfrequencies and the corresponding eigenfunctions arenot very sensitive to relatively small transverse cracksin fundamental components of majority of mechani-cal systems and structures. Thus, proper identificationmethods must be employed in order to detect and lo-calize them effectively.

For this system the Monte Carlo simulation of cou-pled forced vibrations is performed for various crack-

depth ratios in the range a/D = 0.1÷0.4 and for var-ious axial positions on the entire length of the shaft.The system dynamic responses are obtained for theconstant nominal rotational speed 3000 rpm for var-ious circumferential crack positions αp on the shaftwithin 0◦÷360◦. In addition to the static gravitationalload, the only assumed source of dynamic externalexcitation are static residual unbalances of the high-pressure rotor equal to 90 gm each and mutuallyshifted “in-phase,” of the intermediate-pressure rotorequal to 135 gm each, mutually shifted “in-phase” andby the phase angle = 180◦ with respect to the unbal-ance of the high-pressure rotor, of the low-pressure ro-tor equal to 180 gm each, mutually shifted “in-phase”and “in-phase” with respect to the unbalance of thehigh-pressure rotor as well as of the generator-rotorequal to 270 gm each, mutually shifted “in-phase” andby the phase angle = 180◦ with respect to the un-balance of the high-pressure rotor. This residual un-balance of the considered rotor-shaft system has beensymbolically marked in Fig. 2 by the circles. Thus, thetorsional and axial vibrations can be regarded here asan output effect caused by bending vibrations of therotor-shaft line. For the assumed hybrid model of theinvestigated turbogenerator rotor-shaft system in thefrequency range 0÷650 Hz, 43 bending, 10 torsionaland 7 axial eigenmodes have been considered to solve(10) with very high computational accuracy of the ob-tained results.

The quantities that are of particular interest for thepurpose of crack identification are the ones which inregular exploitation conditions can be relatively easilymeasured on-line. These are lateral vibration displace-ments of the shaft at the bearing locations, fluctuationof shaft rotational speed due to torsional vibrationsalso at the bearing locations as well as the longitudinaldisplacements of the shaft at the shaft’s both free endsand at the thrust-bearing location.

For the assumed sources of external excitations thelateral response of the considered uncracked rotor-shaft system is harmonic in character with the onlyone synchronous frequency component 1X. Here, thevery weak torsional–lateral coupling due to the resid-ual unbalances results in negligible torsional responseand a lack of the lateral–axial coupling yields zeroaxial response. In the framework of the carried outMonte Carlo simulation the transverse cracks of var-ious depths and axial locations along the entire lengthof the considered turbogenerator rotor-shaft line have

Nonlinear and parametric coupled vibrations of the rotor-shaft system as fault identification symptom 543

Fig. 4 Bendingeigenfunctions of the hybridmodel of the steamturbogenerator rotor-shaftsystem

544 T. Szolc et al.

Fig. 5 Torsional and axialeigenfunctions of the hybridmodel of the steamturbogenerator rotor-shaftsystem

Nonlinear and parametric coupled vibrations of the rotor-shaft system as fault identification symptom 545

Fig. 6 Coupled dynamic response for the rotor-shaft system with the crack of the depth ratio a/D = 0.2 located in the intermedi-ate-pressure rotor, close to bearing #3

been assumed. As it follows from results of thesecomputations, the strongest influence of the crack onthe system-coupled dynamic response is usually ob-served in the cases of investigated quantities regis-tered in the vicinity of the given crack, i.e. close tothe adjacent bearings or to the corresponding rotor-shaft ends. Figure 6 shows the plots of the coupledlateral–torsional–axial response of the system with acrack of the depth ratio a/D = 0.2 localized in theintermediate-pressure rotor (IP) in the neighborhoodof bearing #3, see Fig. 2. These plots are given as func-tions of time and relative to synchronous frequencydomain. In this figure the lateral response is depicted

in the form of journal bearing transverse displacementorbits and amplitude spectra of their time histories,where the gray lines correspond to the adjacent left-hand-side bearing #2 and the black lines correspondto the adjacent right-hand-side bearing #3. The tor-sional and axial responses are demonstrated by meansof time histories and their amplitude spectra. Here, bythe gray lines there are depicted, respectively, the re-sponses registered at the bearing #1 and the shaft left-hand-side free end and by the black lines there are il-lustrated the responses registered at bearing #7 and theshaft right-hand-side free end, Fig. 2. In the case ofsuch a crack the lateral response seems to be also har-

546 T. Szolc et al.

Fig. 7 Coupled dynamic response for the rotor-shaft system with the crack of the depth ratio a/D = 0.25 located at the generator-rotormid-span

monic, which follows from the almost elliptical shaftdisplacement orbits plotted in Fig. 6. Here, althoughthe respective bearing journal displacement orbits arestill elliptical, but the amplitude spectra of their dis-placement time-histories are significantly affected bythe double synchronous 2X and triple synchronous3X frequency components. The torsional and axial re-sponses are characterized by relatively severe extremevalues for the monitoring devices operating on the realobject. Their time histories are affected by many fluc-tuation components of relative frequencies 1X, 2X,3X, 4X as well as by 7X÷13X in the case of the tor-sional response, and by almost all X multiplied fre-

quencies (not all are visible in the presented scale)within 1X÷13X in the case of the axial response. Nev-ertheless, the highest peak of the axial response com-ponent corresponds to ∼1.66X approximately equalto the second axial natural frequency 83.044 Hz. Inthis case, the respective axial eigenform (presentedin Fig. 5) is induced in the form of transient statecomponent due to systematic closings and openingsof the crack during each successive revolution of theshaft.

In the same way as in Fig. 6, in Fig. 7 there are pre-sented results of analogous simulation obtained for abigger crack of the depth ratio a/D = 0.25 and local-

Nonlinear and parametric coupled vibrations of the rotor-shaft system as fault identification symptom 547

ized in the generator rotor-shaft mid-span, i.e. in thecross section between bearings #6 and #7. Here, thelateral response is not harmonic. As it follows fromFig. 7, the bearing journal displacement orbits are notelliptical and the corresponding amplitude spectra ofdisplacement time-histories are affected also by the re-markable frequency components 2X and 3X, in addi-tion to the fundamental synchronous component 1X.In the case of the lateral response registered at bearing#6 the frequency component 3X is predominant whichemphasizes the nonlinear-parametric character of thecoupled dynamic response induced by this crack. Thetorsional response is also significant and the axial re-sponse is much more severe than that in the previouscase. Nevertheless, in this case the ∼ 1.66X transientstate component of the axial response is much smallerin comparison with the remaining ones than in theprevious example, which follows from the respectiveFFT-plots in Figs. 6 and 7.

It is noteworthy that torsional and axial responsesobtained for various depth ratios within a/D =0.1÷0.3 for the cracks located in several axial posi-tions along the entire rotor-shaft line are characterizedby similarly “rich” amplitude spectra as these obtainedin the above examples, i.e. in general the same fluctu-ation components are excited with magnitudes rapidlyincreasing together with the crack a/D ratio. But mu-tual relations between the amplitude values of suc-cessive fluctuation components depend on particularcases of crack depths and locations. Similarly, the 2Xand 3X frequency components of the system lateralresponse also significantly increase together with therise of a/D, whereas maximum values of the peak cor-responding to the synchronous 1X component remainalmost unchanged.

According to the observed facts, as diagnostic pa-rameters necessary for crack identification in the shaft,the maximum amplitude values of the 2X and 3X fre-quency components of the system lateral response aswell as the resultant (global) amplitudes of the tor-sional and axial response have been selected. Thementioned resultant (global) amplitudes of the con-sidered torsional and axial time histories are regardedhere as maximum fluctuation values with respect totheir average values of the steady-state dynamic re-sponse. The listed above values of response ampli-tudes corresponding to randomly generated crack pa-rameters will constitute the database of numerical ex-periments used in the identification process.

5 Crack identification methods

The objective of the proposed methods is an efficientassessment of a possible damage of the vibrating rotor-shaft system when an information on system responsesfrom monitoring devices is available. In addition to theidentification of the most probable crack parameters,i.e. the crack location xc and its depth a, the methodsalso provide a confidence measure of the results. Con-trary to the method proposed by the authors in theirprevious paper [10], the methods described here con-sist of two major phases:

1. fast crack parameters identification based on theanalysis of experimental data points generated be-fore by the Monte Carlo sampling,

2. identification improvement strategy, mainly by en-hancing the database in the neighborhood of thefirst phase estimation.

The second phase algorithms are meant to improve anaccuracy of the identified crack parameters in a rel-atively short time, which is the time from the crackdetection moment to the ultimate decision concerningthe exploitation of the rotor machine. Moreover, thesemethods should enable us to use smaller databases re-ducing the costs of database preparation.

As it was mentioned, the foundation of all the pre-sented methods is the database of points (x,y) relatingthe crack parameters x to the responses y of the vi-brating rotor-shaft system (see Sect. 4). The databaseis created by means of the Monte Carlo sampling. Asample of crack parameters is generated according tothe assumed probability distributions of the crack lo-cation xc, the ratio a/D where D is the shaft diam-eter, and the circumferential crack position αp . Thechoice of probability density functions (PDFs) was de-scribed in detail in [10]. In the identification examplepresented in the next section, 30% of the total numberof sample points are generated using the uniform PDFin the range a/D = 0.1÷0.4 and the remaining pointsare generated using the following half-normal densityfunction:

f (a) = 2ϑ

πexp

[− (a − amin)

2ϑ2

π

], (11)

where amin is the minimal considered identifiablecrack depth and ϑ is the parameter selected such thatthe probability of crack depth greater than the maximal

548 T. Szolc et al.

Fig. 8 Stress envelope forthe uncracked rotor-shaftsystem supported on sevenjournal bearings

Fig. 9 Shape of theprobability density functionand the correspondingcumulative distributionfunction for the cracklocation xc along therotor-shaft system of thesteam turbogenerator

depth observable in practice is negligible. In the caseof PDF for the crack location xc, it is taken to be pro-portional to the maximal reduced stress envelope ob-tained for the nominal uncracked rotor-shaft. Such anenvelope for the considered rotor-shaft system shownin Fig. 2 is presented in Fig. 8. The PDF correspondingto the stress envelope can be expressed as

f (xc) = |σ(xc)|∫ x7x1

|σ(x)|dx, xc ∈ [x1, x7] (12)

where σ(x) is the maximal reduced stress function andx1 and x7 are the axial coordinates of the first andthe seventh bearing, respectively (see Fig. 9 for PDFand the corresponding cumulative distribution func-tion (CDF)). Similarly to the case of crack depth, 30%of the total number of points are generated from theuniform PDF in the range [x1, x7] and the remainingsample points are drawn using the PDF (12) to addnumerical experiments at locations where cracks aremost likely to occur.

Nonlinear and parametric coupled vibrations of the rotor-shaft system as fault identification symptom 549

Fig. 10 Illustration of theidentification algorithm

One of the major advantages of one-dimensionalhybrid dynamical models of the rotor-shaft systems istheir high computational efficiency. A single analysisof the parametric-nonlinear coupled vibration processusually takes less than 75 seconds of CPU on mod-ern PCs. Hence, accounting for inherent parallelismof the Monte Carlo method, in most of the cases itis affordable to perform hundreds of thousands oreven millions of simulations. This allows for thoroughexploration of the response space, which facilitatesthe identification process. Nevertheless, still the num-ber of sample points (size of the database) should belarge enough to ensure acceptable identification accu-racy.

Below, there are presented the “reference” identifi-cation algorithm, named here the nearest-point meth-od, and three second-phase algorithms for improve-ment of the results obtained using the nearest pointmethod. A confidence/tolerance measure for the iden-tified parameters is also described.

Nearest-point identification method (NP) Crack pa-rameter identification in the nearest-point identifica-tion method consists in localizing a point in the spaceof responses of the rotor-shaft system that is the clos-est to the read-out from a monitoring device. The ac-curacy of the NP method increases together with thedatabase size. Therefore, the largest possible databaseshave to be used, cf. [10] where this method is de-scribed in detail. Unquestionable advantage of this ap-proach is almost immediate identification of crack pa-rameter values as well as their tolerances.

The NP method can be schematically presented intwo-dimensional space of parameters x1 and x2 andthe corresponding space of two responses y1 and y2,see Fig. 10. In this figure, the read-out (yR

1 , yR2 ) is

marked with the star and the diamond markers depictsample points—the responses for randomly generatedparameters. In Fig. 10, the point in the response space,which minimizes the distance from the read-out, ispoint A. The coordinates (xR

1 , xR2 ) of point A′, corre-

sponding to A in the parameter space, are assumed asidentification results, i.e. the parameters leading to theread-out. Such identification results are biased. Theyvery much depend on the database size as well as on alocal “saturation” of the responses space with the sam-ple points. Therefore, some measure of confidence foridentification results should also be provided. The con-fidence measure introduced in [10] is briefly describedbelow.

In the response space, there are points Bi , i =1, . . . ,NA, falling into the spherical vicinity of thepoint A of the radius equal to the distance betweenA and the read-out. We also identify NA points B ′

i inthe parameter space that are mapped onto Bi points. Ifthere are no points in the spherical vicinity, its radius isgradually enlarged until at least two points are inside.For each point B ′

i the differences xij = |xR

j − xij |

are computed, where xij is the j th coordinate of the

point B ′i , j = 1, . . . , n, n being the number of para-

meters, n = 2 here. A conservative tolerances for theidentified parameters xR

j can be then defined using thecomputed differences as

xRj = max

i=1,...,NA

xij , (13)

and the identification results given as xRj ± xR

j . Thedescribed confidence measure is the same for all theidentification methods presented in this section.

The nearest-point identification approach is ratherstraightforward. It is particularly well suited for rotor-shaft damage identification where, owing to computa-

550 T. Szolc et al.

Fig. 11 The idea oforthogonal projectionidentification method

tionally efficient models of rotor-shaft systems, verylarge samples can be afforded.

Orthogonal projection identification method (OP)The idea of orthogonal projection method is illustratedin Fig. 11. Similarly as in Fig. 10, here the read-outis marked with the star and the diamond markers de-note sample points, i.e. they are the responses for ran-domly generated crack parameters. The OP algorithmstarts by searching for the pair of points in the read-outvicinity, which define the straight line that is closest tothe read-out. In Fig. 11 these are points are B1 andB2 and the line is denoted as l12. Next, the read-outis projected on l12 determining two segments b1 andb2 connecting the projection point with B1 and B2,respectively. In the parameter space the points corre-sponding to B1 and B2 are B ′

1 and B ′2, respectively,

and the straight line passing through these points isdenoted l′12. Accounting for the proportion b1/b2 andthe position of the projection point with respect to B1

and B2, point A′ is established on l′12 as an approxi-mation of the projection point mapping. Then, at thispoint the vibrating rotor-shaft system responses arecomputed yielding point A with coordinates (yA

1 , yA2 ).

This point is added to the database and the procedurerestarts. If point A is localized in the vicinity of theread-out, it will probably be used in the next iterationas one of the points constituting the closest straightline. In turn, it is likely to produce a new A point thatis closer to the read-out than the previous A point.If this is not the case, the procedure is stopped. An-other stop criterion may be set on the number of it-erations. The parameter values corresponding to thebest A point (provided it is closer to the read-out thanany of the original database points) are taken to be theidentification results.

The main advantage of the OP method is a smallnumber of additional simulations of the rotor-shaft dy-namic responses. However, due to strongly nonlinearcharacter of the considered relationship there is noguarantee that the methodology will always improvethe NP identification results.

Nedler–Mead identification method (NM) This non-linear simplex method is a popular direct search op-timization technique introduced by Nedler and Mead,see [11], that has been adopted to the considered iden-tification problem. The simplex graph based on the setof n + 1 points in the parameter space, n being thenumber of parameters, corresponding to n + 1 pointsclosest to the read-out in the response space is em-ployed. Each time during the optimization process anew point (values of crack parameters x) is establishedthe appropriate rotor-shaft responses y are computedand the new point is added to the database. The objec-tive function that is to be minimized here is the dis-tance in the response space between the read-out andthe closest database point. The procedure is stopped ifthe minimized distance is smaller than certain ε toler-ance or the number of calls to the rotor-shaft simula-tion program is exceeded.

Similarly to the OP method, the NM identificationalgorithm requires some additional simulations of therotor-shaft system dynamic responses, which increasethe identification time. However, this additional com-putational effort can be controlled in a quite flexibleway depending on the user preferences. In terms of thedistance from the read-out the method always resultswith a solution, which is not worse than the NP-basedidentification.

Local sampling identification method (LS) The LSmethod seems to be the simplest choice for improv-

Nonlinear and parametric coupled vibrations of the rotor-shaft system as fault identification symptom 551

ing the accuracy of the NP identification results. Themethod consists in finding the two closest points yA

and yB to the read-out yR and computing the systemresponses for a certain number of new points sampledin the crack parameter space using PDFs determinedin some way by yA and yB . These new points aregenerated from the independent uniform distributions,where point xA (related to the closest point yA) is themean vector and the absolute differences between therespective parameter values of xA and xB are the stan-dard deviations of the assumed distributions.

In the numerical example that follows the identi-fication capacity of the proposed methods is testedfor the turbogenerator rotor-shaft system described inSect. 4. For randomly generated cracks (given by theparameters xc, a/D and αp) the system responsesare first computed using the one-dimensional hybridmodel and then, at the second step, the obtained re-sults are introduced as read-outs to the identificationalgorithm indicating the most probable crack parame-ters. A comparison of the identified parameters and theoriginal ones provides a measure of the identificationquality.

6 Numerical example, identification analysisof the rotor-shaft system of the steamturbogenerator

The effectiveness of the crack parameters identifica-tion methods described in Sect. 5 is compared usingthe hybrid mechanical model of the steam turbogen-erator rotor-shaft system supported on seven journalbearings shown in Fig. 2. Selected statistics for allthe methods and for database sizes of 5,000, 10,000,50,000, 100,000, 200,000 and 500,000 points, respec-tively, as well as the average computational effort havebeen collected in one diagram, a so-called box plot.The box plot consists of the following statistics, seeFig. 12: the lower adjacent value (non-outlier simula-tion), the lower quartile Q0.25, the median, the meanvalue, the upper quartile Q0.75 and the upper adjacentvalue (non-outlier simulation). Moreover, the box plotdepicts simulations considered unusual, called out-liers, which are smaller than Q0.25 − 1.5RQ or graterthan Q0.75 + 1.5RQ, where RQ = Q0.75 − Q0.25 isthe interquartile range. The outliers are divided intotwo groups: mild outliers (marked with navy bluecircles)—simulations not smaller than Q0.25 − 3.0RQ

Fig. 12 Box plot notation

or not grater than Q0.75 + 3.0RQ, and extreme outliers(marked with the red crosses)—which are farther thanthe mild ones. For the sake of presentation quality, notalways all the extreme outliers are shown. When this isthe case, the number of hidden extreme outliers (in thebrackets) and the maximal/minimal values are speci-fied.

Each of the box plots presented below displaysidentification statistics for 100 tests. The box plots areprepared for the identification error of two crack pa-rameters: xc and a/D, and their identification toler-ances.

As it can be expected, the identification results forxc strongly depend on the database size. Irrespec-tive of the identification approach, the smallest data-bases of 5,000 or even 10,000 simulations do not pro-vide a good identification quality and the obtained re-sults are far from being reliable, see Fig. 13. All themethods produce many extreme outliers; moreover,the difference between the lower adjacent value andthe upper adjacent value vary in quite a wide rangeof 0.7–1.4 m related to the entire rotor-shaft length of25.9 m. Slightly better identification results are pro-

552 T. Szolc et al.

Fig. 13 Box plots of xc

identification error fordatabase sizes equal to5,000, 10,000 and 50,000,respectively

Fig. 14 Box plots of xc

identification error fordatabase sizes equal to100,000, 200,000 and500,000, respectively

Nonlinear and parametric coupled vibrations of the rotor-shaft system as fault identification symptom 553

duced when the database of 50,000 points is used. Un-fortunately, even in this case, some extreme outlierscan be observed. There are two tests for the NP iden-tification method that completely failed resulting withidentification errors of up to 6 m. Therefore, the fastNP method alone is not sufficiently reliable for sucha medium size database and some improvement algo-rithm needs to be employed.

More promising results deliver identification testscorresponding to large databases of 100,000, 200,000and 500,000 simulations, see Fig. 14. Here, the NPmethod for all the three cases has almost the samescatter which can be considered as acceptable. Theonly extreme outlier, registered in an identification testperformed with 200,000 points database spoils a lit-tle generally very good accuracy and concentration ofthe results. Unquestionably, the best results for all theexamined methods are obtained using the largest data-base (500,000 data points), which only proves the intu-itive relationship between the database size and iden-tification accuracy. However, also with databases of100,000 and 200,000 points one may obtain reliableand accurate results, especially when it is possible touse one of the improving algorithms, the OP, NM orLS. The fastest and the most successful among themseems to be the OP method, which needs only about 7additional simulations.

Identification tolerances of xc for small databasesconfirm a poor quality of the results obtained by allthe methods, see Fig. 15. The only exception is theOP identification method with the database of 50,000points where the largest computed tolerance does notexceed 0.45 m. The quality of OP is not much im-proved for databases of 100,000 and 200,000 points,Fig. 16. However, significant improvement is observedfor the largest database, where the outermost value ofextreme outlier is approximately 0.13 m, and other sta-tistics are better than for the remaining methods.

Similar tolerance statistics are computed only forthe NM method. However, about 16 additional sim-ulations have to be done compared to 7 for OP. Itshould be noticed that the NP method possesses goodtolerance statistics in the case of the two largest data-bases. Nevertheless, in the considered example onlythe largest database enables us to obtain reliable andaccurate results. By this we assume identification ofthe crack position in the monitored rotor-shaft of thetotal length of 25.9 m with the maximum error equal to0.7 m (the most likely maximum error equals to 0.5 m)

and the maximum tolerance equal to 0.45 m (the mostlikely maximum tolerance is 0.3 m) in the worst case.

At the same time, in the case of xc, the consideredidentification improvement methods give comparableresults, the maximum error smaller than 0.2 m and themaximum tolerance smaller than 0.15 m for the NMand OP methods.

The identification process of a/D ratio yields big-ger scatter of identification results, i.e. every applieddatabase and considered method have many mild andextreme outliers, see Figs. 17–20. As in the case ofthe parameter xc, the scatter magnitude decreases to-gether with the increase of database size, which en-ables us to choose an appropriate level of accuracy suf-ficient for monitoring and improvement of identifica-tion results. In the case of the largest database the NPmethod yields the maximum identification error equalto 0.065 m, which is about 15% of the assumed a/D

value. Slightly better results can be observed for theimprovement algorithms NM and LS. The maximumidentification error they produce is 0.04 m, which isnot more than 10% of the maximum assumed a/D.The most efficient in a/D identification improvementseems to be the OP method, both in terms of accuracyand computational time.

7 Conclusions

In the present paper, nonlinear and parametric com-ponents of coupled bending–torsional–axial vibrationshave been used as a transverse crack occurrence symp-toms in rotor-shafts of the rotor machines. For thispurpose, by means of the structural hybrid model ofthe real object, several databases containing results ofdynamic responses corresponding to various possiblecrack depths and locations on the shaft were gener-ated using the Monte Carlo simulation. Special proba-bility density functions of crack parameters were pro-posed in order to ensure good identification quality forthe most probable crack parameters. In order to de-tect and localize the crack, four identification methodshave been applied and mutually compared. These are:the nearest-point method (NP), the orthogonal projec-tion method (OP), the Nedler–Mead method (NM) andthe local sample method (LS). Each of them was testedfrom the viewpoint of computational efficiency andcrack identification error on the example of the rotor-shaft system of large steam turbogenerator.

554 T. Szolc et al.

Fig. 15 Box plots of xc

identification tolerance fordatabase sizes equal to5,000, 10,000 and 50,000,respectively

Fig. 16 Box plots of xc

identification tolerance fordatabase sizes equal to100,000, 200,000 and500,000, respectively

Nonlinear and parametric coupled vibrations of the rotor-shaft system as fault identification symptom 555

Fig. 17 Box plots for a/D

identification error fordatabase sizes equal to5,000, 10,000 and 50,000,respectively

Fig. 18 Box plots of a/D

identification error fordatabase sizes equal to100,000, 200,000 and500,000, respectively

556 T. Szolc et al.

Fig. 19 Box plots of a/D

identification tolerance fordatabase sizes equal to5,000, 10,000 and 50,000,respectively

Fig. 20 Box plots of a/D

identification tolerance fordatabase sizes equal to100,000, 200,000 and500,000, respectively

Nonlinear and parametric coupled vibrations of the rotor-shaft system as fault identification symptom 557

From the carried out considerations it follows thatthe Monte Carlo sampling approach for damage iden-tification in vibrating rotor-shaft systems, togetherwith the proposed methods of improving the accuracyof results, seems to be an appropriate solution for in-dustrial engineering applications. The crucial advan-tage of the proposed procedure is the immediate iden-tification process of the real rotor-shaft system underon-line monitoring. The method is based on the pre-viously prepared database of hundreds of thousandsnumerical experiments, that takes significant but stillacceptable computational effort proportional to the de-manded accuracy of the results. Additionally, the samedatabase can be used to obtain in relatively short time(5 to 20 minutes on a present-day PC) more preciseestimation of the crack parameters. The identificationerror scatter and the scatter of tolerance decrease withthe database size, which allows to obtain required ac-curacy of crack parameter identification. Based on thetest results the identification improvement method thatseems to be the most efficient and accurate is the or-thogonal projection identification method (OP).

Acknowledgements The partial support from the PolishCommittee for Scientific Research grants PBZ-KBN-105/T10/2003, No. N519 010 31/1601 and No. 3T11F00930, is gratefullyacknowledged.

References

1. Zhao, M., Luo, Z.H.: A convenient method for diagnosis ofshafting crack. In: Proc. of the 12th Biennial ASME Con-ference on Mechanical Vibration and Noise. ASME, Rotat-ing Machinery Dynamics, DE-Vol. 18-1, pp. 29–34. Mon-treal, Canada (1989)

2. Ishida, Y., Yamamoto, T., Hirokawa, K.: Vibrations of arotating shaft containing a transverse crack (major criticalspeed of a horizontal shaft). In: Vibration Institute (eds.),Proc. of the 4th Int. IFToMM Conf. on Rotor Dynamics,pp. 47–52. Chicago, Willowbrook, IL, Sept. (1994)

3. Kicinski, J.: Dynamics of rotors and journal bearings. In:Institute of Fluid-Flow Machinery of the Polish Academyof Sciences (eds.), Fluid-Flow Machinery, p. 28. Gdansk (inPolish) (2005)

4. Pennacchi, P., Bachschmid, N.: A model-based identifica-tion method of transverse cracks in rotating shafts suit-able for industrial machines. Mech. Syst. Signal Process 20,2112–2147 (2006)

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

6. Szolc, T., Bednarek, T., Marczewska, I., Marczewski, A.,Sosnowski, W.: Fatigue analysis of the cracked rotor bymeans of the one- and three-dimensional dynamical model.In: Proc. of the 7th International Conference on RotorDynamics. Organized by the IFToMM in Vienna, Sept.,Austria, Paper-ID 241 (2006)

7. Papadopoulos, C.A., Dimarogonas, A.D.: Coupling ofbending and torsional vibration of a cracked Timoshenkoshaft. Ingenieur-Archiv 57, 257–266 (1987)

8. Ostachowicz, W., Krawczuk, M.: Coupled torsional andbending vibrations of a rotor with an open crack. Arch.Appl. Mech. 62, 191–201 (1992)

9. Szolc, T.: On the discrete-continuous modeling of rotor sys-tems for the analysis of coupled lateral-torsional vibrations.Int. J. Rotating Mach. 6(2), 135–149 (2000)

10. Szolc, T., Tauzowski, P., Stocki, R., Knabel, J.: Damageidentification in vibrating rotor-shaft systems by efficientsampling approach. Mech. Syst. Signal Process. (2008,in press)

11. Nedler, J.A., Mead, R.: Simplex method for function mini-mization. Comput. J. 7, 308–313 (1965)