10th international conference on vibrations in rotating machinery

10th International Conference on

Vibrations in Rotating Machinery

Tribology Group and Mechatronics, Informatics & Control Group

Organising Committee

Professor David Ewins University of Bristol, UK

Dr Philip Bonello University of Manchester, UK

Professor Izhak Bucher Technion, Israel

Professor Seamus Garvey University of Nottingham, UK

Dr Jeffrey Green Rolls-Royce, UK

Dr Robert Herbert RWE npower, UK

Professor Georges Jacquet-Ricardet INSA Lyon, France

Professor Patrick Keogh University of Bath, UK

Professor Arthur Lees University of Swansea, UK

Professor Asoke Nandi University of Liverpool, UK

Professor Robert Wood University of Southampton, UK

International Scientific Committee

Professor Nicolò Bachschmid Politecnico di Milano, Italy

Professor Katia Lucchesi Cavalca Universidade Estadual de Campinas, Brazil

Professor Aly El-Shafei RITEC, Egypt

Professor Eric Hahn University of New South Wales, Australia

Professor Timo Holopainen ABB, Finland

Professor Gordon Kirk Virginia Tech, USA

Professor Chong-Won Lee Korea Advanced Institute of Science and

Technology, Republic of Korea

Dr Rainer Nordmann Alstom, Germany

Professor Nevzat Ozgüven Middle East Technical University, Turkey

Professor Paolo Pennacchi Politecnico di Milano, Italy

Professor J S Rao Altair Engineering, India

Professor Damian Vogt KTH, Sweden

The Committee would like to thank the following supporters:

Aerospace Division

Structural Technology and Materials Group

Power Industries Division

Pressure Systems Group

Process Industries Division

Automobile Division

Railway Division

10th International Conference on

Vibrations in Rotating Machinery

11–13 SEPTEMBER 2012

IMECHE, LONDON

Oxford Cambridge Philadelphia New Delhi

Co-sponsored by:

British Society for Strain Measurement

Engineering Integrity Society

Forum for Applied Mechanics

Forum for Engineering Structural Integrity

The Society of Environmental Engineers

South African Institute of Tribology

Association Française de Mécanique

Royal Aeronautical Society

Brazilian Society of Mechanical Engineering and Sciences

Japan Society of Mechanical Engineers

British Gear Association

Institute of Materials, Minerals and Mining

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© Paolo Pennacchi, 2012

Modelling, dynamic behaviour and

diagnostics of cracked rotors

P Pennacchi

Politecnico di Milano, Dept. of Mechanical Engineering, Italy

ABSTRACT

One of the most common incipient losses of structural integrity in mechanical

structures is the development and propagation of a crack. This happens also in

rotating machinery and a very rich, but also in some way confusing, literature

about cracked rotors has appeared in the last 30 years.

In the paper, a general and wide overview about the behaviour of cracked rotors

will be presented, covering several aspects of this topic. In particular modelling,

dynamic behaviour and diagnostics of cracks will be analyzed in detail, introducing

also practical examples and cases of industrial machinery, including some topics

barely documented in literature such as helical and annular crack development.

1 INTRODUCTION

The propagation of shaft cracks is one of the most dangerous faults that can occur

in rotating machinery [1]. If this fault is not early detected, it can cause serious

damage as well as long outages and expensive maintenance actions.

Transverse cracks that propagate in the shafts of rotating machines cause a change

of the local flexural stiffness of the rotor. Moreover, in horizontal machine-trains,

the shaft weight causes rotor bending that gives rise to tensile axial stresses, acting

in the lower area of the faulty cross-section, which tend to open the crack.

Conversely, the axial compressive stresses generated in upper area of the cracked

cross-section tend to cause the contact between the crack surfaces. Therefore,

during a complete revolution of the shaft, a transverse crack can be subjected to a

breathing phenomenon, that is a periodic closure and opening of the crack.

Transverse cracks can propagate in the cross-section of the shafts of rotating

machines as a consequence of very high fatigue stresses. In general, the stress

distribution in a rotor is rather complex, being the consequence of many

concomitant static and dynamic loads. Owing to some of the most important of

them, the cracked section is subjected to time-varying axial stresses generated by

different causes, in operating condition, both at low and high rotational speeds.

Some of these stresses are periodic as they depend on mechanical phenomena that

are influenced by the shaft angular position. Other important axial stresses are

aperiodic as they are caused by quasi-static loads. With regard to this, the stresses

caused, for instance, by changes of the machine thermal condition can be very

important. In general, these thermal transients occur within time intervals that are

considerably longer than the revolution period of the shaft. Therefore, the

corresponding axial stresses gradually increase, or decrease, during sufficiently

long time intervals. Detailed analyses of thermal effects on breathing mechanism

are presented in [1][2].

3

In the case of very thin cracks, having planar surfaces, the time-varying axial

stresses acting on the cracked section of the shaft can cause a temporary opening,

or closure, of the crack. Negative compressive stresses can cause contact between

the fracture surfaces, whereas positive tensile stresses tend to open the crack. This

phenomenon is often called crack breathing.

2 CRACK BREATHING MECHANISM

The breathing mechanism has been analysed in literature by several authors:

Mayes and Davies [3] proposed the so called “switching crack”, in which the crack

is either completely closed or open and passes abruptly from one state to the other.

Nelson and Nataraj [4] proposed a switching criterion based on the curvature of the

shaft, while Wauer [5] based his criterion on the sign of the total axial strain at

extreme fiber of the cross section of the shaft. Mayes and Davies [6] improved their

model by introducing a smoothing trigonometric function of the rotation angle of

the section. However, all these approaches, along with similar ones [7][8], suppose

that the area of the cracks that is in contact for various rotation angles is a priori

known as observed by Andrieux and Varé [9]. Functions that rule the breathing of

the crack are still used in the literature, as in Sinou and Lees [10], Patel and Darpe

[11], Sawicki et al. [12], Ishida et al. [13][14] and Sinou and Faverjon [15].

A different approach was proposed by Dimarogonas and Papadopoulos [16] by

considering shaft elasticity and stress intensity factors, with a typical approach of

Solid Mechanics. Starting from that and by the introduction of the concept of crack

closure line by Darpe et al. [17][18] and of other improvements by Papadopoulos

[19]. This method has been widely used (see for instance Papadopoulos [20] and

Sekhar [21][22][23]). However, some limitations of this approach have been

highlighted, such as the limitation to consider cracks deeper than 50% of the

diameter [19] or thermal stresses or crack closure effects [1].

Other approaches, able to manage the previously mentioned limitations have been

proposed in [9][24] and [25].

However, owing to the interest in defining the true breathing behaviour of cracked

shafts, some tests have been performed in the laboratory of the Dept. of

Mechanical Engineering of Politecnico di Milano, as described in [1], to measure the

strains in different points of a cracked shaft, with 70 mm diameter, under different

load conditions. To this aim, a series of strain-gauges have been applied close to

the crack and also directly across the crack lips (see figure 1).

Figure 1. Detail of strain-gauge positions close

and across the crack, from [1].

4

The horizontal cracked specimen was subjected to different stationary loads and

rotated in different angular positions in order to excite the breathing of the crack.

Two different non trivial effects have been observed: the crack closure effect and

the local contact conditions of the crack lips in closed crack configuration, which are

now discussed:

a) Crack closure effect: Small loads, generating small bending moments in the

locality of the crack, were not able to open the crack. The crack closure effect

generates an internal bending moment that holds the crack closed. Only when

the external bending moment overcomes the internal bending moment, then the

lips of the crack start to open.

b) Local contact conditions of the crack lips: When the crack is closed, with an

external bending moment that sums up to the internal bending moment, the

measured compressive strain is much higher than the theoretical strain

calculated assuming a linear compressive stress distribution over the cracked

section. This can be explained by assuming that the contact is not spread over

all the cracked area when the crack is closed, but it occurs only on a smaller

portion of the cracked surface, or on the crack lips only, determining higher

strains associated also to stress intensity factors. This aspect is also related to

the crack closure effect.

Despite the fact that crack closure effects have been studied by several researchers

(see for instance [26]), their influence, on the breathing behaviour of rotating

shafts, has never been modelled suitably, to the author’s knowledge.

Despite the highly non-linear stress and strain distribution in the cracked area and

the non-linear breathing behaviour, the overall load-strain behaviour results are

quite linear. The overall load vs. deflection law can therefore easily be represented

by a linear model like the FLEX model presented in [1].

Cracked shaft diameter was larger than 70 mm when the crack had been initiated

by means of a small notch generated by electro-erosion and had propagated

roughly up to half way the shaft cross section by applying a constant bending load

to the rotating shaft. The specimen was then machined and the diameter was

reduced to the final one by turning, so that the initial slot was removed.

The final cracked section has the shape shown in figure 2, as determined from

ultrasonic test measurements. The cracked shaft has been clamped at one end but

can be rotated around his axis by steps of 15° each. A vertical load has been

applied at the other end and has been increased by steps. Theoretical stresses and

strains are calculated assuming no crack.

Strain-gauges from A1 to A11 were applied each 15°, as close as possible to the

crack. Strain-gauges from A12 to A15 were applied to an integer section, which

was sufficiently far away from the crack to be not influenced by its breathing

behaviour and their measurements are used as reference signals. Strain-gauges

from A16 to A28 were applied at 15° in correspondence of the crack, but on the

integer part opposite to the crack. Strain-gauges from B31 to B39 were applied

across the lips, as it is shown in figure 2.

5

Figure 2. Shape of the crack and strain-gauge positions.

Typical results are shown in figure 3 for measuring point A6 , in which the

maximum effect of the crack is expected. Angle 0° indicates that crack axis (which

passes close to measuring point A6 ) is directed vertically downwards (crack open),

angle 180° indicates that crack axis is directed vertically upwards (crack closed). A

flat zone in strain level (about 140 µε) is clearly recognisable between 0° and 60°

and between 300° and 360°, for a total range of angular positions of 120°. In this

range of angular positions where the strain does not change, provided that

sufficient load is applied, the crack is supposed to be definitely open. The strain for

the open crack is not zero but a tensile strain of 140 µε is measured on the open

crack lips.

Figure 3. Strain vs. rotation measured by strain-gauge A6 .

This can be explained by the presence of an internal pre-stress on the crack lips

due to the crack closure effect, generating a compressive strain of 140 µε, when

the strain-gauges have been applied to the shaft with no external loads. Only the

loads that generate a similar value of tensile stresses are they able to open the

crack. Similar results, but generally with higher values (up to 240 µε) of

16

Rotation angle [degrees]

B B

6

compressive strains due to the crack closure effect, have been found in the other

measuring points A3 , A4 , A5 and A7 . The behaviour is different in positions

that are closer to the crack ends, i.e. also closer to the crack tip, as in measuring

point A10 .

When the crack is closed (at the angular position of 180°), the compressive stress

of 350 µε in correspondence of maximum load, is much higher with respect to the

theoretical value of 190 µε, calculated considering the load applied on a un-cracked

specimen. This can be attributed to the fact that only a smaller part of the cracked

surface is in contact, generating locally higher stresses and also stress intensity

factors which would be absent if the crack faces were completely in contact each

other over the complete crack area. This strain magnification is present for all loads

in proportion with the same value, but its value is different for different

measurement points and tends to unity (no strain magnification) close to the crack

ends (in measuring points A1 and A10 ), as expected.

At the measuring points close to the crack ends, when the loads generate tensile

stresses, during the gradually opening of the crack, rather high stress intensity

factors have been measured that are due to the closeness of the crack tip to the

measuring points. At the measuring point A16 (opposite to A6 ), the maximum

tensile strain is 240 µε (opposite to the closed crack), which is somewhat higher

than the theoretical value of 190 µε: this might be caused by the fact that not all

the cracked area is in contact with the opposite face.

The measured behaviour has been also simulated, with the aid of a model of the

crack. The FLEX model used for calculating the breathing behaviour and the

reduced stiffness of the cracked shaft is described in [1] and the details are not

reported here for the sake of brevity. The crack closure effect has been simulated

by an external bending moment that tends to hold the crack closed, generating a

maximum of 140 µε of compressive strain. Figure 4 shows the comparison of

calculated results obtained with the above specified model, with the experimental

results in measuring point A6 for the maximum load. It is surprising how good the

simplified model is able to reproduce the experimental behaviour. This occurs at

almost all measuring points. The quasi-linear breathing behaviour model can be

considered completely validated with these experimental results.

Figure 4. Comparison of calculated and experimental results at point A6 .

0 60 120 180 240 300 360 -400

-300

-200

-100

0

100

200

Rotation angle [degrees]

Strain []

Experimental

Calculated

7

3 CRACK PROPAGATION AND SHAPE

A crack may propagate from some small imperfections on the surface of the body

or inside of the material and it is most likely to develop in regions of high stress

concentration. Cracks in rotating shafts are most likely to appear due to sharp

changes of the diameter or of the geometry of the shaft (for instance due to the

presence of holes, slots for keys, threads and so on) in regions of high stress

concentration.

Also thermal stresses that develop as a consequence of the working fluid in thermal

machines, such as steam and gas turbines, and thermal shocks are responsible for

generating high local stress intensity factors, which can cause the initiation of a

crack and its propagation.

In rotating shafts, the cracks propagate generally in a plane perpendicular to the

shaft axis, when axial bending stresses are prevailing, generating a transverse

crack. A typical example is shown in figure 5, where the actual shape of the crack in

a generator was found definitively after dismantling the rotor and breaking it. The

crack had propagated to nearly 50% of the cross section area. The starting point

was in correspondence to a defect in a slot for the windings (see figure 5).

However, conical crack surfaces and helical crack surfaces, as well as annular and

longitudinal cracks, have also been reported in the literature.

Figure 5. Transverse crack in a generator (left);

starting point (right), from [1].

The crack may have any orientation at its starting point, depending on local

conditions, but when the crack propagates more deeply, its direction is mainly

radial and the cracked surface, although not exactly, is roughly planar, with

vanishingly small curvatures.

8

The propagation velocity may change from case to case in rotating shafts.

Propagation times of 74,000 up to 101,000 hours of operation are reported from

detection, or suspected presence, to dangerous crack depth, but only 2,500 hours

of operation has been enough to deepen consistently the crack in other cases. Very

frequently the crack moves in steps, alternating in growth to stop: both can be

seen on the cracked surface pattern where the rest lines called beach marks are

recognizable. These are evident in figure 5.

Generally, when the crack approaches a dangerous depth, it propagates more

quickly, with a propagation velocity that increases almost exponentially. The final

growth, up to a critical dangerous depth, takes sometimes only few days of

operation.

3.1 Helical cracks

Generally, cracks propagate in surfaces which are roughly planar, even if not

exactly as analyzed in detail in [27][28], and perpendicular to the rotation axis of

the shaft. However, if a high torque combines with high bending loads, the crack

may also propagate along a helical path, therefore these cracks are called helical or

slant cracks.

Cracks caused by torsional stresses develop mostly along helical surfaces rather

than along planar inclined surfaces. Slant cracks (with helicoid angles up to 45°)

can develop only when torque is alternating, which is usually not the case in

turbomachinery.

The crack studied in [1][25], see figure 6, developed along a helical path with an

angle of 6° only on the outer surface of the shaft, owing to the combined action of

bending and torsion. Such a crack could develop at mid-span of double flow steam

turbines in high power turbogenerators, where the maximum bending moment due

to weight combines to the transmitted torque.

The results of the theoretical and experimental analyses showed that at full load

the differences between helical and flat crack are so small that they could be

neglected.

Figure 6. Slightly helical crack highlighted by dye penetrant, from [1].

9

When torsional load is removed, then higher differences arise, but mainly for the

torsional degree of freedom, which is excited by the bending load by means of a

coupling effect.

In this condition torsional deflections are generated by bending moments due to

coupling effects and torsional vibrations are excited in rotating shafts. This

constitutes a clear symptom of the presence of a helical crack in a shaft line: when

during a run-down transient (at no torsional load) the torsional natural frequencies

are strongly excited at the corresponding rotating speed, this could be due to the

relevant coupling effect of the helical crack.

3.2 Annular cracks

Occasionally, multiple crack initiations can occur along the circumference of the

external surface of the shaft. This phenomenon can give rise to the propagation of

a full-annular transverse crack [2][29]. In general, the contour of the

corresponding residual section is nearly a circle, the centre of which can be affected

by an eccentricity with respect to the shaft axis (see figure 7).

The analysis of the breathing mechanism of annular cracks shows that the highest

values of the eccentricity, as well as the lowest values of the radius, cause the

highest values of the crack depth. The highest changes of the angle of the neutral

axis, with respect to the reference, occur during a complete revolution. However,

owing to the crack shape, even the highest values of the inclination angle of the

neutral axis are limited. This reduces the amount of the complex flexion of the

shaft, caused by its weight, and the levels of the super-synchronous vibrations.

Therefore, this type of crack, the propagation of which has been occasionally

detected near high-pressure stages of steam turbines, can cause only small levels

of the twice per revolution vibrations of the shaft that, in general, are considered

the most common and characteristic symptom of the presence of a shaft crack.

Figure 7. Cross-section of shaft showing an annular crack, from [2].

10

4 IDENTIFICATION OF CRACKS IN ROTORS

Model based techniques are able to identify the presence, the position and even the

depth of cracks in rotors. Crack diagnosis is performed by means of parity

equations, by introducing external forces that are equivalent to the effects of the

crack.

The starting point for the definition of equivalent external forces to cracks is the

system of dynamic equations of the machine, obtained by considering the rotor

modelled by means of finite beam elements, the bearings by equivalent damping

and stiffness and the foundation by a suitable representation (pedestals, modal,

transfer matrix or rigid).

The system of dynamic equations of a large rotating machine, with several d.o.f.s,

is typically:

      ( ) total total total Mx Cx Kx F     t (1)

where the mass matrix M takes also into account the secondary effect of the

rotatory inertia, the damping matrix C includes also the speed depending

gyroscopic matrix and the stiffness matrix K takes also into account the shear

effect.

Considering eq. (1), it is difficult to identify the changes due to the developing fault

in the matrices M , C and K , which are of high order, from measurement of

the absolute vibration total x in only few measuring planes along the shaft.

In real rotating machines, the measurement points of the vibration along the shaft

during operation are few and the transducers are normally in correspondence with

the bearings. Only lateral vibrations are normally monitored, hence in the following

only lateral behaviour of the rotor will be considered.

The consideration of further measuring planes along the rotor span, due to the

presence of the casings and of the possible working fluids, is practically impossible,

therefore methods that reconstruct modal shapes of the rotor cannot be used.

The right hand side (r.h.s.) external forces F( )t of eq. (1) are composed of the

weight (which is known) and by the original unbalance and bow (which are

unknown). The system parameter changes due to the fault are indicated as dM ,

dC and dK and eq. (1) becomes:

                i e t

total total total u d dd  M Mx C Cx K Kx W UM         (2)

If the system behaviour is considered to be linear, which is acceptable for a wide

class of faults [30][31], then the total vibration total x can be considered as due to

two superposed effects:

x xx total ref a  .  (3)

It can be shown that the overall behaviour of a horizontal axis heavy cracked shaft

is linear, under the conditions shown in [1]. Only in extreme operating conditions,

the non-linear effect of the breathing crack, which is weak in normal conditions, will

influence its behaviour.

The first vibration vector ref . x is the pre-fault vibration, which is due to the weight

W and the unknown unbalance force i e t U and unbalance moment i e t

u M  . The

11

second vibration a x is due to the developing fault. The last is also called additional

vibration. The vibration component a x may be obtained by calculating the vector

differences of the actual vibrations (due to weight, original unbalance, bow and

fault) and the original vibrations measured, in the same operating conditions in a

reference case (rotation speed, flow rate, power, temperature, etc.) before the fault

was developing. A discussion about the possible errors introduced and their

tracking is presented in [32]. Recalling the definition of the pre-fault vibration ref . x ,

the following equation holds:

        i

.. . e t

ref ref ref u

 Mx Cx Kx W U M      (4)

which substituted in eq. (2) with eq. (3) gives:

  a   a   a   total   total   total Mx Cx Kx Mx Cx Kx        d dd   (5)

The r.h.s. of eq. (5) can be considered as a system of equivalent external forces

which force the fault-free system to have the change in the additional vibration a x

that is due to the developing fault only:

      ( ) a a af Mx Cx Kx F     t (6)

A rather complete overview of the equivalent forcing systems to the most common

faults in rotating machinery is presented in [30][31].

Note that in eq. (6) system parameters M , C and K are time invariant and

known, but normally C and K are functions of the operating speed with

regard to the gyroscopic effect and the bearing coefficients. Using this approach,

the problem of fault identification is reduced to an external force identification by

means of parity equations. The effect of a crack on the statical and dynamical

behaviour of the rotor can be simulated in the frequency domain, by applying

different sets of equivalent forces to the rotor in correspondence of the cracked

beam element, one set for each of the harmonic components considered.

The stiffness of a cracked shaft is periodic due to the breathing and the rotation of

the crack, see [1] where it is shown that the equivalent crack forces are given by

the following expressions:

   

  0

** *

11 1 1 2 2

* **

2 2 33 3 3

11 1

44 4

1 11

4 44

f

              

             

Kx K x K x

F

K x Kx K x

 (7)

     

1

* ** *

1 2 1 3 2 1 2 2 3

i i

* *

12 23

111 1

222 2

e e

1 1

2 2

s

t t

f

 

               

       

Kx K x Kx K x K x

F

Kx Kx

 (8)

      2

i 2 * * i2

2 3 1 1 1 1 3

1 11

e e

222

t t

f s

                   F K x Kx Kx K x

 (9)

12

      3

i3 i3

3 1 2 2 1

1 1

e e

2 2

t t

f s

              F Kx Kx K x

 (10)

of which eqs. (8), (9) and (10) are respectively the first, the second and the third

harmonic components of the crack force system. Usually higher order harmonic

components are not considered for a practical reason: actual condition monitoring

systems installed on real machines that performs order analysis normally store only

1X, 2X and sometimes 3X components. Note that the projections along reference

axes of the harmonic component of the force system are not necessarily equal, thus

the equivalent force system is not in general represented by rotating forces.

From the experimental point of view and under the previously exposed hypothesis

of linearity, the difference a total ref   . xx x , between the measured vibration total x

of a rotor system that has a fault and the reference case ref . x , represents the

vibrational behaviour due to the fault, i.e. the additional vibrations. These

vibrations are used in the identification procedure, since they are due to the

impending fault only. In fact, the reference case vibrations ref . x are given by eq.

(11):

Mx Cx Kx F  ref .. .   ref     ref e   (11)

while those caused by the developing crack are given by:

      01 2 3

i i 2 i 3 ee e ttt

total total m total e f f f f

   Mx Cx K x F F F F F            (12)

If eq. (12) is considered for an unknown crack, also Km  is unknown. Anyhow it

can be approximated by K of the un-cracked shaft, from which it differs only

very little: the crack affects the stiffness of one element only. Therefore the

additional vibrations are given by:

      01 2 3

i i 2 i3 ee e ttt

a a af f f f

   Mx Cx Kx F F F F           (13)

By applying the harmonic balance criteria in the frequency domain to eq. (13) and

considering the harmonic components of the additional vibrations Xn , the following

equations are obtained for each one of the considered harmonic components:

2 () i 1, 2,3 n n f        nn n   M C KX F (14)

The force vectors n Ff in eq. (14) are those to be identified. For the sake of clarity

and simplicity, a single fault is considered, while multiple faults can be handled

using the method fully described in [30][31].

In the r.h.s. of eq. (14), the equivalent force system is applied to the two nodes of

the element that contains the crack. Since lateral vibrations only are considered,

the model has 4 d.o.f.s per node (see figure 8) and the equivalent force is a vector

of eight generalized forces.

Owing to energy considerations, the most important among these forces are the

bending moments that are roughly equal and opposite on the two nodes of the

cracked element (see figure 8).

13