2011- on the finite element modeling of the asymmetric cracked rotor

7/25/2019 2011- On the Finite Element Modeling of the Asymmetric Cracked Rotor

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On the finite element modeling of the asymmetriccracked rotor

Mohammad A. AL-Shudeifat n

Aerospace Engineering, Khalifa University of Science, Technology and Research, P.O. Box 127788, Abu Dhabi, UAE

a r t i c l e i n f o

Article history:

Received 1 October 2011

Received in revised form

8 November 2012

Accepted 24 December 2012

Handling Editor: H. OuyangAvailable online 24 January 2013

Keywords:

Structural health monitoring

Cracked rotor modeling

Damage detection

Rotor damage

a b s t r a c t

The advanced phase of the breathing crack in the heavy duty horizontal rotor system is

expected to be dominated by the open crack state rather than the breathing state after a

short period of operation. The reason for this scenario is the expected plastic deformation

in crack location due to a large compression stress field appears during the continuous

shaft rotation. Based on that, the finite element modeling of a cracked rotor system with

a transverse open crack is addressed here. The cracked rotor with the open crack model

behaves as an asymmetric shaft due to the presence of the transverse edge crack. Hence,

the time-varying area moments of inertia of the cracked section are employed in

formulating the periodic finite element stiffness matrix which yields a linear time-

periodic system. The harmonic balance method (HB) is used for solving the finite element

(FE) equations of motion for studying the dynamic behavior of the system. The behavior

of the whirl orbits during the passage through the subcritical rotational speeds of the

open crack model is compared to that for the breathing crack model. The presence of the

open crack with the unbalance force was found only to excite the 1/2 and 1/3 of the

backward critical whirling speed. The whirl orbits in the neighborhood of these

subcritical speeds were found to have nearly similar behavior for both open and

breathing crack models. While unlike the breathing crack model, the subcritical forward

whirling speeds have not been observed for the open crack model in the response to the

unbalance force. As a result, the behavior of the whirl orbits during the passage through

the forward subcritical rotational speeds is found to be enough to distinguish the

breathing crack from the open crack model. These whirl orbits with inner loops that

appear in the neighborhood of the forward subcritical speeds are then a unique property

for the breathing crack model.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Rotordynamic systems have wide application in power generation, aircraft engines, compressors, pumps and in many

other industrial fields. They mostly operate in a heavy loading environment where damage due to fatigue cracks or failure

is always expected to occur. Finding an efficient technique that helps in detecting damage in its early phase is a promising

area of research. This may help in preventing further damage to occur and in saving life and equipment. In the literature,

damage has mostly been characterized by propagating cracks. The breathing and the open transverse crack models are

considered as the main causes of the damage in rotor systems. Identifying the vibration signature of the damaged rotor

Contents lists available at SciVerse ScienceDirect

journal homepage: www.elsevier.com/locate/jsvi

Journal of Sound and Vibration

0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jsv.2012.12.026

n Tel.: 971 2 5018564; fax: 971 2 4472442.

E-mail address: [email protected]

Journal of Sound and Vibration 332 (2013) 27952807
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system requires finding efficient tools for modeling the different types of the cracks which may help in studying the actual

dynamic behavior of the cracked system. If the dynamic behavior of the cracked rotor system can be identified via a correct

theoretical model that can be experimentally verified then the vibration signature in the real life applications can be

approximated. In this study, efforts have been made to add improvements for the previous models to achieve the common

goal for finding more efficient techniques in damage detection.

The common technique for formulating the finite element stiffness of the cracked rotor is the flexibility matrix method

[19]. This technique was used in [1] for studying the effect of the coupling between the longitudinal and bending

vibration of a cracked rotor system with an open transverse crack. It was also used in [2]for studying the same issue of acracked rotor system with two breathing cracks. Similarly, the finite element stiffness matrix of the cracked element of a

cracked rotor was determined in[39]using the flexibility matrix method. As a result, the finite element model (FEM) was

formulated and solved for studying the dynamic behavior of the cracked rotor system.

Another form for the finite element stiffness matrix, which is similar to that of an asymmetric rod in space in[10], was

used in modeling the cracked rotor with a breathing crack model in [1116]. As a result, the time-varying stiffness matrix

of the cracked rotor was formulated based on the use of the classical breathing function that proposed in [17]. The Finite

element model was solved by the harmonic balance method for the vibration amplitudes, whirl orbits and the shift in the

critical and the subcritical rotational speeds.

The dynamic behavior of the whirl orbits in the neighborhood of the subcritical rotational speeds or during the passage

through these speeds was studied in [16,1826]for a cracked rotor with a breathing crack model. The transfer matrix

method was used in [18,19] for deriving the stiffness matrix of the cracked rotor. The whirl orbit reversal during the

passage through the critical and the subcritical rotational speeds was noticed in [19]. The whirl orbit with one inner loop

appears during the passage through 1/2 of the critical rotational speed and it was found to be sensitive to the unbalanceforce direction[16,2426]. A review of some previous techniques for modeling the open and the breathing crack models

and the different methods of solution were introduced in[27].

The finite element modeling of the open crack is addressed here. This crack model could be a cut or an advanced phase

of a breathing crack. The crack starts to propagate at a location where the elastic properties of the shaft at that location are

subjected to change. This leads to a reduction in the stiffness where the synchronous breathing of the crack between

compression and tension stress fields on the crack faces of contact may lead to a permanent plastic deformation by which

the breathing mechanism becomes dominated by the permanently open crack state. This change in the crack state to a

permanently open state changes the crack signature which is expected to be distinguished from the breathing crack

signature.

The time-varying finite element stiffness matrix of the cracked element is derived here based on the transformation of

the constant stiffness matrix of the rod in space in[10]from the rotating coordinates into the fixed coordinates. The time-

varying transformation matrix is also derived for performing this transformation. The resulting time-varying stiffness

matrix of the cracked element was found to include the effect of the time-varying cross-coupling stiffness. The results ofincluding the time-varying cross-coupling stiffness are compared with the results when the cross-coupling stiffness is

ignored where it is found that ignoring this cross-coupling stiffness significantly affects the whirl orbits behavior during

the passage through the subcritical speeds.

The open crack model of the cracked rotor is compared in this study with the breathing crack model in the literature

based on the whirl orbits behavior at the neighborhood of the subcritical rotational speeds. It is found that the whirl orbit

behavior for the breathing crack model is necessarily enough to distinguish between the open crack and the breathing

crack models during the passage through the forward subcritical rotational speeds.

2. Modeling of the rotor-bearing-disk system

The finite element method is employed in formulating the equations of motion of the rotor system shown in Fig. 1.

The finite element equations of motion of the N-elements rotor with N1 nodes are given in matrix form as[16,2830]M qt C G _qt Kqt Fut Fg (1)

where qt qT1qT2. . .q

Ti . . .q

TN 1

T is the 4(N1) 1 dimensional nodal displacement vector, qTi t ui vi jxi j

yi is the

single node displacement vector of the translational and rotational displacements about the stationary axes in the bearing

center for i 1,2,y,N1, Fu(t) is the 4(N1) 1 unbalance force vector, and Fg is the 4(N1) 1 gravity force vector

[15,28,29]. TheM, K, C and Gare the global mass, stiffness, damping, and gyroscopic matrices, respectively, of the intact

shaft where each is of dimension 4(N 1) 4(N1) as in[15,28,29]. Each rigid disk is placed at the selected node and its

1 X2 ... N1

Y

ZN

Y

3

Fig. 1. Finite element model of the rotor.

M.A. AL-Shudeifat / Journal of Sound and Vibration 332 (2013) 279528072796


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center has the same nodal displacement vector of the node. The bearings can be either journal or ball bearings. The nodal

equations of motion of the disk and the equations of forces for the bearing nodes are also found in[15,28,29].

3. Finite element model of a cracked rotor system with an open crack

The open transverse crack is modeled as shown inFig. 2where the dashed segment represents the crack segment. The

crack is assumed at anglef relative to the fixed negativeY-axis att 0 as shown inFig. 2a. As the shaft rotates the crackangle with the negativeY-axis changes with time tofOtas shown inFig. 2b, whereOis the rotational speed of the rotor.

The anglef is selected to be equal to zero in the following analysis.

The area moments of inertia of the cracked element about its centroidal x- andy-axes are constant quantities during the

rotation of the shaft while the area moments of inertia of the cracked element about its fixed centroidal X- andY-axes are

time-varying quantities during the rotation of the shaft. The axes Xand Yremain parallel to the stationaryX- andY-axes

during rotation. The cracked element stiffness matrix in the rotatingx- andy-axes can be written in a form similar to that

of the asymmetric rod in space in[10].

The centroidal area moments of inertia of the cracked element about theX- andY-axes areIX

t andIY

t, respectively.

From[10], the time-varying area moments of inertia IXt, IYt and IX Yt about the centroidal X- and Y-axes during the

shaft rotation are given in terms of centroidal area moments of inertias Ix andIy in the rotating x- and y-axes as[10]

IX t

Ix Iy

2

IxIy

2 cos 2Ot Ix ysin 2Ot (2a)

IY

t Ix Iy

2

IxIy2

cos 2Ot Ix ysin 2Ot (2b)

IX Y

t IxIy

2 sin 2Ot Ix ycos 2Ot (2c)

whereIx IxAcee2,Iy Iy,IxandIyare the area moments of inertia of the cracked element cross-section about the rotating

x- andy-axes,Aceis the area of the cracked element cross-section and e is its centroid location on they-axis. Sincey is the

axis of symmetry of the cracked element cross-sectional area during rotation, then Ixy 0. The quantities Ace and e have

been derived in[16]as

Ace R2pcos11m 1mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2m

p (3a)

e 2R3

3Acem2m3=2 (3b)

Fig. 2. Schematic diagrams of the cracked element cross-section (a) before rotation, (b) after shaft rotates. The dashed area represents the crack segment.

M.A. AL-Shudeifat / Journal of Sound and Vibration 332 (2013) 27952807 2797


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The area moments of inertiaIxandIyof the cracked element cross-section about the rotating x- andy-axes, respectively,

have also been derived in[16]for 0rmr1 as

Ix pR4

4

R4

4 1m2m24m 1gsin11m (4a)

Iy pR4

4

R4

12 1m

2m24m3

g3sin1 g

(4b)

where gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim 2m q

, mh/R is the non-dimensional crack depth and h is the crack depth in the radial direction ofthe shaft.

The finite element stiffness matrix of the rod in space in[10]is used to express the jth cracked element stiffness matrix

in the rotating coordinates as

kjR

E

l3

12Ix 0 0 6lIx 12lIx 0 0 6lIx0 12Iy 6lIy 0 0 12Iy 6lIy 0

0 6lIy 4l2

Iy 0 0 6lIy 2l2

Iy 0

6lIx 0 0 4l2

Ix 6lIx 0 0 2l2

Ix

12lIx 0 0 6lIx 12Ix 0 0 6lIx0 12Iy 6lIy 0 0 12Iy 6lIy 0

0 6lIy 2l2

Iy 0 0 6lIy 4l2

Iy 0

6lIx 0 0 2l2

Ix 6lIx 0 0 4l2

Ix

266666666666666664

377777777777777775

(5)

the cracked element stiffness matrix kjFin the fixed coordinates is found via the transformation

kjF W

TkjRW (6)

where W is the transformation matrix of dimension 8 8. This transformation matrix is derived based on the coordinate

transformation shown inFig. 3.

For the nodal displacement vectorqT(t) [u vjxjy] of the left node of the elementj in the rotating coordinate systemOxyz,jx andjy have the following relationships[28,29]:

jx @vy@w

, jy @ux@w

(7)

the coordinate systemOXYZis rotated byOtaboutz-axis which yields the rotating coordinate system Oxyz. Hence,u andv

in the fixed coordinate systemOXYZcan be written asu uxcosOtvysinOt (8a)

v uxsinOt vycosOt (8b)

hence,jX andjY in the fixed coordinate system have the following relationships:

jX @v

@w

@ux@w

sin Ot @vy@w

cos Ot jxcosOt jysinOt (9a)

jY @u

@w

@ux@w

cos Otj

@vy@w

sin Ot jxsinOt jycosOt (9b)

Fig. 3. The fixed and rotating coordinate systems at jth rotor element (left node).



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as a result, the transformation matrix that transforms the stiffness from the rotating x- andy-axes to the fixed X- and Y-

axes inFig. 3or the rotating x- and y-axes to the fixed X- andY-axes inFig. 2is given as

W

cosOt sinOt 0 0 0 0 0 0

sinOt cosOt 0 0 0 0 0 0

0 0 cosOt sinOt 0 0 0 0

0 0 sinOt cosOt 0 0 0 0

0 0 0 0 cosOt sinOt 0 00 0 0 0 sinOt cosOt 0 0

0 0 0 0 0 0 cosOt sinOt

0 0 0 0 0 0 sinOt cosOt

2666666666666664

3777777777777775

(10)

the application of Eq.(6)with Eq.(10)yields the time-varying stiffness matrix of the cracked element in the fixed X- and

Y-axes which is given as

kjF k

ja k

jb (11)

where

kja

E

l3

12IXt 0 0 6lIXt 12lIXt 0 0 6lIXt0 12I

Yt 6lI

Yt 0 0 12I

Yt 6lI

Yt 0

0 6lIY

t 4l2

IY

t 0 0 6lIY

t 2l2

IY

t 0

6lIX

t 0 0 4l2

IX

t 6lIX

t 0 0 2l2

IX

t

12lIX

t 0 0 6lIX

t 12IX

t 0 0 6lIX

t

0 12IYt 6lIYt 0 0 12IYt 6lIYt 0

0 6lIY

t 2l2

IY

t 0 0 6lIY

t 4l2

IY

t 0

6lIXt 0 0 2l2

IXt 6lIXt 0 0 4l

2IXt

266666666666666664

377777777777777775

(12)

kjb

E

l3

0 12IXY

t 6lIXY

t 0 0 12IXY

t 6lIXY

t 0

12IXYt 0 0 6lIXYt 12IXYt 0 0 6lIXYt

6lIXY

t 0 0 4l2

IXY

t 6lIXY

t 0 0 2l2

IXY

t

0 6lIXY

t 4l2

IXY

t 0 0 6lIXY

t 2l2

IXY

t 0

0 12IXY

t 6lIXY

t 0 0 12IXY

t 6lIXY

t 0

12IXY

t 0 0 6lIXY

t 12IXY

t 0 0 6lIXY

t

6lIXYt 0 0 2l2

IXY

t 6lIXYt 0 0 4l2

IXY

t

0 6lIXYt 2l2

IXY

t 0 0 6lIXYt 4l2

IXY

t 0

266666666666666664

377777777777777775

(13)

whereIX

t, IY

t and IX Y

thave the same formulas in Eqs. (2a)(2b). In addition, the cracked element stiffness matrix in

the fixedXandYcoordinates in Eq. (11)can be rewritten as

kjF kj1 kj2cos2Ot kj3sin2Ot (14)

the stiffness matriceskj1 , k

j2 andk

j3 are given as

kj1

E

l3

12I1 0 0 6lI1 12lI1 0 0 6lI1

0 12I1 6lI1 0 0 12I1 6lI1 0

0 6lI1 4l2

I1 0 0 6lI1 2l2

I1 0

6lI1 0 0 4l2

I1 6lI1 0 0 2l2

I1

12lI1 0 0 6lI1 12I1 0 0 6lI1

0 12I1 6lI1 0 0 12I1 6lI1 0

0 6lI1 2l2

I1 0 0 6lI1 4l2

I1 0

6lI1 0 0 2l2

I1 6lI1 0 0 4l2

I1

2666666666666664

3777777777777775

(15a)

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6/13

kj2

E

l3

12I2 0 0 6lI2 12lI2 0 0 6lI2

0 12I3 6lI3 0 0 12I3 6lI3 0

0 6lI3 4l2

I3 0 0 6lI3 2l2

I3 0

6lI2 0 0 4l2

I2 6lI2 0 0 2l2

I2

12lI2 0 0 6lI2 12I2 0 0 6lI2

0 12I3 6lI3 0 0 12I3 6lI3 0

0 6lI3 2l

2

I3 0 0 6lI3 4l

2

I3 06lI2 0 0 2l

2I2 6lI2 0 0 4l

2I2

266666666666

6664

377777777777

7775

(15b)

kj3

E

l3

0 12I2 6lI2 0 0 12I2 6lI2 0

12I2 0 0 6lI2 12I2 0 0 6lI2

6lI2 0 0 4l2

I2 6lI2 0 0 2l2

I2

0 6lI2 4l2

I2 0 0 6lI2 2l2

I2 0

0 12I2 6lI2 0 0 12I2 6lI2 0

12I2 0 0 6lI2 12I2 0 0 6lI2

6lI2 0 0 2l2

I2 6lI2 0 0 4l2

I2

0 6lI2 2l2

I2 0 0 6lI2 4l2

I2 0

2666666666666664

3777777777777775

(15c)

where I1

1=

2 Ix

Iy

, I2

1=

2 Ix

Iy

, I3

I2

, Ix

and Iy

are all constant quantities during the rotation of the cracked

shaft. The matrix kj3 includes the effect of the cross-coupling area moment of inertiaIX Yt on the cracked rotor.

As a result, the FEM equations of motion of the rotor-bearing-system with an open crack model are rewritten as

M qt C _qt K1 K2cos2Ot K3sin2Otqt F1cosOtF2sinOt Fg (16)

where K1 is the 4(N1) 4(N 1) stiffness matrix of which the entries of the cracked element stiffness matrix kj1 are

merged instead of the uncracked element entries of element j in K [15,28,29], K2 and K3 are 4(N1) 4(N1) stiffness

matrices of zero entries except at the cracked element where the entries are equal to kj2 inK2and k

j3 in K3,F1and F2are

the vectors of the unbalance force amplitudes[15], and C G Cis a combination of the gyroscopic and damping matrices.

The solution of the system in Eq.(16)is expressed as a finite Fourier series as

qt A0 Xn

k 1

AkcoskOt BksinkOt (17)

inserting this solution into Eq. (16)yieldsH m,O

K1 O O 0:5K2 0:5K3 O O O O O O

O C1 C2 C11 C3 O O C2 C3 O O O O

O C11 C3 C1

C2 O O C3 C2 O O O O

K2 O O C2 C

21 O O C2 C3 & O O

K3 O O C21 C

2 O O C3 C2 & ^ ^

O C2 C3 O O C3 C

31 O O & O O

O C3 C2 O O C31 C

3 O O & C2 C3

O O O C2 C3 O O C4 C41 & C3 C2

O O O C3 C2 O O C41 C4 & O O^ ^ ^ & & & & & & & O O

O O O O O C2 C3 O O Cn Cn1

O O O O O C3 C2 O O Cn1 C

n

26666666666666666666666666664

37777777777777777777777777775

A0

A1

B1

A2

B2

A3

B3

A4

B4^

An

Bn

26666666666666666666666664

37777777777777777777777775

Fg

F1

F2

0

0

0

0

0

0^

0

0

26666666666666666666666664

37777777777777777777777775

(18)

where C2(1/2)K2, C3(1/2)K3, C(j)K1 (jO)

2Mand Cj1 jOC for j 1,2,. . .,n, and n is the number of harmonics used.

Eq.(17)is solved for A0, Ak and Bk for k 1,2,. . .,n .

4. Theoretical results for the open crack model

The rotor-bearing-disk system is divided into 12 elements as shown inFig. 4where the crack location is assumed to be

in element 3 where the length of this cracked element is equal to 60.3 mm. The damping matrix of the shaft is assumed to

be equal to zero while the damping of the bearings in Table 1is the only damping included for simulation. Two disks areattached to nodes 3 and 11. The mass unbalance meis attached to the left disk at an angleb relative to the positiveX-axis

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


7/13

at distanced from the shaft centerline. The values of the physical parameters of the system are given in Table 1. The HB

solution is used to generate the results for the whirl orbits and the shift in the critical and subcritical speeds.

To distinguish between the critical forward and backward whirling rotational speeds, anisotropic bearings are used

for the system inFig. 4with kxx5 107 N/m and kyy7 10

7 N/m. The anisotropic bearings are only used here for the

results inFig. 5while isotropic bearings are used for all following cases. It is clear fromFig. 5that the critical forward whirl

speed of12884.5 rev/min appears before the critical backward whirl speed ob1ffi 3043 rev/min when anisotropicbearings are used. Similarly, as the crack starts to appear for the asymmetric shaft with a transverse open crack and

isotropic bearings ofkxxkyy7 107

N/m the critical backward speed is also excited at ob1ffi 3043 rev/min when m-

0 asshown in the waterfall plot inFig. 6. For the asymmetric cracked shaft with isotropic bearing or the symmetric intact shaft

with anisotropic bearings, the first forward and backward whirl speeds have been excited by the unbalance force.

The appearance of the open transverse crack is found to only excite the 1/2 and 1/3 of the first backward critical speed

and as shown in Figs. 79. The first pair of the critical forward and backward whirling rotational speeds and their

corresponding subcritical rotational speeds are shown in Fig. 8for m0.3 where the significance of including K3 whichinclude the effect of IX Y in the cracked element stiffness matrix is shown. IfK3 is not included in the cracked element

stiffness matrix, extra sub-harmonics appear as shown in Fig. 8which are more familiar with the breathing crack model

than the open crack model[16]. These extra harmonics do not appear when IX Y

is included in the crack element stiffness

matrix even at high crack depth as shown inFig. 9.

The whirl orbits during the passage through the 1/2 and 1/3 of the first backward critical speed which are excited by the

open transverse crack are plotted inFigs. 10 and 11. The whirl orbit with three outer loops appears in the neighborhood of

the 1/2 of the first backward whirl speeds reverses its direction during the passage through this subcritical whirl speed.

The whirl orbit with four outer loops that appears during the passage through 1/3 of the backward critical speed is found

Fig. 4. Finite element model of the rotor-bearing-disk system.

Table 1

Physical parameters of the rotor-bearing-disk system.

Description Value Description Value

Length of the rotor, L 0.724 m Disk outer radius, Ro 76.2 mm

Radius of the rotor, R 7.9 mm Disk inner radius, R i 7.9 mm

Density of rotor, r 7800 kg/m3 Density of disk, r 2700 kg/m3

Modulus of elasticity, E 2.1 1011 N/m2 Mass of the disk, md 0.571 kg

Bearing stiffness (kxxkyy) 7 107 N/m Mass unbalance, med

2 103 kg m2

Bearing damping (cxxcyy) 5 102 N s/m Mass unbalance angle,b p/2 rad

Fig. 5. The first critical forward and backward whirling rotational speeds of the bearing-disk-system with anisotropic bearings ofkxx5 107 N/m andkyy7 10

7 N/m based on the vibration amplitudes at node 7.



8/13

to be rotated by 451during the passage through this subcritical speed. These orbits have nearly similar behavior and shape

to those of the breathing crack model in [16,19]during the passage through these subcritical backward whirl speeds.

Unlike the breathing crack model in[16,19,2226], the open crack model is found to be not exciting the subcritical forward

whirl speeds. Hence, the whirl orbits with inner loops which appear during the passage through the forward subcritical

speeds are unique signature for the breathing crack model compared to the open crack model.

b1

f1

Fig. 6. Waterfall of the shift in the forward and backward critical whirling speeds of the rotor-bearing-disk-system versus crack depth for

med25 104 kg m2 based on the vibration amplitudes at node 7.

2

1

3

1b1

b1

Fig. 7. Waterfall of the shift in the 1/2 of the first backward critical whirling speed and 1/3 of the first forward critical whirl speed of the rotor-bearing-

disk-system versus crack depth for med2

5 104 kg m2 based on the vibration amplitudes at node 7.

212

1

314

1

b1

f1 b1

f1b1

b1

Fig. 8. The vibration amplitudes of node 2 versus the rotational speed form0.3,bp/2 andmed2

5 104 kg m2 based on the vibration amplitudes at

node 7 where 6 harmonics are used in the HB solution.

2

1

3

1

b1

b1

f1

b1

Fig. 9. The vibration amplitudes of node 2 versus the rotational speed for m 0.75,bp/2 andmed2 5 104 kg m2 based on the vibration amplitudes

at node 7 where 6 harmonics are used in the HB solution.



9/13

The whirl orbits during the passage through the first backward critical whirl speed are plotted in Fig. 12. It is interesting

to notice that these orbits have the same shape and behavior as the whirl orbits at the 1/3 of this critical backward whirl

speed as previously shown in Fig. 11.

The effect of the unbalance mass on the three outer loops whirl orbit that appear during the passage through the 1/2 of

the first critical backward whirl speed is shown in Fig. 13. It is shown that these loops disappear at relatively high

unbalance force as shown in Fig. 13a and b where the whirl orbit has nearly an egg-shaped. In addition, for small

unbalance mass, the centers of these loops converge to main center of the whole whirl orbit as shown in Fig. 13e.

The initial crack angle f at t 0 has no effect on the whirl orbit as long as the relative angle Dybf between the

crack orientation and the unbalance force direction remains constant during the rotation. For a fixed crack angle f thewhirl orbits are sensitive to the change in the unbalance force direction as shown inFig. 14. If the unbalance force angleb

=1494 rpm =1491 rpm =1488 rpm =1485 rpm =1482 rpm

Fig. 10. The whirl orbits of node 2 during the passage through the subcritical backward whirl speed 1/2ob1ffi1488 rev/min for med25 103 kg m2,

m0.3 and b0.

=978 rpm =984 rpm =988 rpm =992 rpm =998 rpm

Fig. 11. The whirl orbits of node 2 during the passage through the subcritical backward whirl speed (1/3ob1ffi988 rev/min) for med25 103 kg m2,

m0.3 and b0.

=2975 rpm =2985 rpm =2990rpm =2995 rpm =3005 rpm

Fig. 12. The whirl orbits of node 2 during the passage through the first critical backward whirl speed ob1ffi2990 rev/min for med25 103 kg m2,m0.3 and b0.

med2=104 kg.m2med

2=104 kg.m2med

2=0.001 kg.m2med

2=0.005 kg.m2med

2=0.01 kg.m2

Fig. 13. The effect of the unbalance mass on the whirl orbit of node 2 during the passage through the 1/2 first critical backward whirl speed at

Offi1482 rev/min for m 0.3 andb 0.



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is changed tob7xthe whirl orbit rotates by 72xfrom its original position as shown in the figure. This behavior is also

observed in[26]for the whirl orbit of a cracked system with two open transverse cracks. This observation may help in

detecting the crack based on the sensitivity of the orbits in the neighborhood of the subcritical rotational speeds to the

unbalance force direction.

5. Experimental results for the open crack model

The Spectra-Quest MFS-RDS rotordynamic simulator, shown in Fig. 15, was used for finding the experimental time

histories for the whirl amplitudes and orbits near to node 2 and node 11 of the finite element model shown in Fig. 4in the

neighborhood of the critical and subcritical whirl speeds. The physical parameters of the MFS-RDS rotordynamic simulator

have been previously given inTable 1. Two sets of proximity probes have been installed. The first set of two perpendicular

proximity probes is installed at the left side of the shaft near to the left bearing (close to node 2) while the other set isinstalled at the right side of the shaft near to the right bearing (close to node 11) as shown in the picture in Fig. 15. These

sets measure the horizontal and vertical displacements at each side of the shaft. The whirl amplitude is calculated as

Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu

2 v

2q

where uis the reading of the first probe while vis the reading of the second probe which is perpendicular to

the first one in the same set. The readings of the proximity probes were collected at a frequency of 10 kH for 10 s of shaft

rotation at a fixed rotational speed.

The wavelet transform [31] is used to explore the frequency content in the signal in the neighborhood of the first

critical speed and its corresponding subcritical speed ( 1/2 of this critical speed). For a rotor speed O2906 rev/min in

the neighborhood of the first critical speed form 0.36, the experimental data of the whirl amplitudes obtained by the leftset of the proximity probes are plotted inFig. 16a where no filtration was used for these data. The corresponding frequency

contents inFig. 16b are normalized to the rotor speed itself (O 2906 rev/min). It is clear from this figure that the sub and

super harmonic frequency components of 1/2, 2/2, 3/2 and 4/2 ofOappear in the signal due to the appearance of

the open crack.For a rotor speed O 1537 rev/min in the neighborhood of the 1/2 of the first critical speed for m0.36, the

experimental data of the whirl amplitudes obtained by the right set of the proximity probes are plotted in Fig. 17a where

no filtration was used for these data. The corresponding frequency contents inFig. 17b are also normalized to the rotor

speed itself (O1537 rev/min). It is clear from this figure that the sub and super harmonic frequency components of 1/2,

2/2, 3/2, 4/2 and 5/2 ofOappear in the signal due to the appearance of the open crack. Even though the data of the

right set of the proximity probes at O1537 rev/min has a big noise as shown in Fig. 18a, the corresponding wavelet

transform inFig. 18b still capable to capture similar frequency content to that inFig. 17b which is for the data with a very

small noise. As a result, the wavelet transform is found to perform well even with data of big noise.

The experimental whirl orbits in Fig. 19 were found to be very close to the theoretical egg-shape whirl orbit that

previously shown inFig. 13a and b in the neighborhood of 1/2 of the critical backward whirl speed. In addition, the whirl

orbit reversal during the passage through the 1/2 of the first critical whirl speed is clearly noticed inFig. 19f and g. Hence,

the 1/2 of the first critical backward whirl speed is experimentally expected to be between O1444 rev/min and

O1500 rev/min form0.48. As a result, the open crack model is found theoretically and experimentally to excite the 1/2of the subcritical backward whirl speed.

= /2 = /2 = /4 = /4 = 0

Fig. 14. The effect of the unbalance force angle b on the whirl orbit of node 2 at Offi1482 rev/min for m0.6 and med2

5 103 kg m.

Fig. 15. The MFS-RDS Spectra-Quest rotordynamic simulator used for experimental analysis.



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6. Conclusions

In this study the finite element model of the time-varying stiffness matrix is introduced for a cracked rotor with an

open transverse crack. The harmonic balance method is used here for solving the time-varying finite element equations of

motion of the cracked rotor for critical and subcritical harmonic analysis. The well-known whirl orbits with inner loops

that appear in the neighborhood of the subcritical forward whirl speeds with the breathing crack model are not observed

here for the open transverse crack model. The appearance of the open crack is found to excite the backward critical and

subcritical whirl speeds where the whirl orbits with the outer loops appear during the passage through these subcriticalspeeds. In addition, the whirl orbit reversal is observed theoretically and experimentally during the passage through the 1/2

2906 rpm (48.438 Hz)=

Fig. 16. (a) Time histories of the vibration amplitudes of the signal in the neighborhood of the first critical whirl speeds measured by the left side

proximity probes and (b) the corresponding wavelet transform of the normalized frequency contents for m 0.36 and med21 103 kg m2.

1537 rpm (25.6 Hz)=

Fig. 17. (a) Time histories of the vibration amplitudes in the neighborhood of 1/2 of the first critical whirl speeds measured by the left side proximity

probes and (b) the corresponding wavelet transform of the normalized frequency contents for m 0.36 and med21 103 kg m2.

1537 rpm (25.6 Hz)=

Fig. 18. (a) Time histories of the vibration amplitudes in the neighborhood of 1/2 of the first critical whirl speeds measured by the right side proximity

probes and (b) the corresponding wavelet transform of the normalized frequency contents for m 0.36 and at med2 1 103 kg m2.



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of the critical backward whirl speed for the open crack model. Hence, the experimental results have shown that the open

crack model excites the backward subcritical whirl.Furthermore, the whirl orbits in the neighborhood of the critical backward whirl speed are also found to have the same

shape and behavior of those appear in the neighborhood of the 1/3 of this critical backward whirl speed. However, it is

verified here that the whirl orbits with inner loops are a unique signature for the breathing crack rather than the

open crack.

In addition, the whirl orbits in the neighborhood of the subcritical backward whirl have the same shape and behavior

for both breathing and open crack models. These whirl orbits are found to be sensitive to the unbalance force direction. For

a constant relative angle between the crack orientation and the unbalance force direction, the initial orientation of the

crack has no effect on the orientation of the whirl orbits, the shift in the critical and subcritical speeds and the vibration

amplitudes during rotation.

Acknowledgment

Prof. Alexander Vakakis group at the University of Illinois at Urban-Champaign is highly acknowledged for the help in

using the wavelet Matlab code in this paper.

Appendix A

For the 12 elements model, previously shown in Fig. 3,the cracked element length was 60.3 mm. For the 24 elements

model the cracked element length is maintained to be 60.3 mm which reduces the total number of elements to 23 as

shown inFig. A1since this crack length spans through two elements of the 24 elements model. Fixing the crack element

length while increasing the number of elements to greater than 12 elements has found to have slight effect on the

convergence as shown inFig. A2where the critical forward and backward whirling amplitudes are plotted versus the shaftrotating speed for m 0.3.

=1444 rpm =1481 rpm =1500 rpm =1519 rpm

=1538 rpm=1500 rpm=1444 rpm=1406 rpm

Fig. 19. Experimental whirl orbits of node 2 in the neighborhood of the 1/2 backward critical whirl speed for unbalancemed21 103 kg m2,f0 rad,

bp/2 rad, (a)(d) for m0.36, (e)(h) for m0.48.

Fig. A1. The 23 elements model of the cracked shaft.



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Fig. A2. The vibration amplitudes of node 2 versus the rotational speed for m0.3, bp/2 and at med21 103 kg m2 for the 12 and 13 elements

models.


2011- on the finite element modeling of the asymmetric cracked rotor

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