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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
http://www.elsevier.com/locate/jsvihttp://www.elsevier.com/locate/jsvihttp://dx.doi.org/10.1016/j.jsv.2012.12.026mailto:[email protected]://dx.doi.org/10.1016/j.jsv.2012.12.026http://dx.doi.org/10.1016/j.jsv.2012.12.026mailto:[email protected]://dx.doi.org/10.1016/j.jsv.2012.12.026http://dx.doi.org/10.1016/j.jsv.2012.12.026http://dx.doi.org/10.1016/j.jsv.2012.12.026http://www.elsevier.com/locate/jsvihttp://www.elsevier.com/locate/jsvi7/25/2019 2011- On the Finite Element Modeling of the Asymmetric Cracked Rotor
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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.
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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.
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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)
M.A. AL-Shudeifat / Journal of Sound and Vibration 332 (2013) 27952807 2799
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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
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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.
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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.
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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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References
[1] C.A. Papadopoulos, A.D. Dimarogonas, Coupled longitudinal and vertical vibrations of a rotating shaft with an open crack, Journal of Sound andVibration 117 (1) (1987) 8193.
[2] A.C. Chasalevris, C.A. Papadopoulos, Coupled horizontal and vertical bending vibrations of stationary shaft with two cracks, Journal of Sound andVibration 309 (2007) 507528.
[3] A.S. Sekhar, B.S. Prabhu, Transient analysis of a cracked rotor passing through critical speeds,Journal of Sound and Vibration173 (3) (1994) 415421.[4] A.S. Sekhar, Crack identification in a rotor system: a model-based approach,Journal of Sound and Vibration 270 (2004) 887902.[5] A.S. Sekhar, A.R. Monhanty, S. Prabhakar, Vibration of cracked rotor system: transverse crack versus slant crack, Journal of Sound and Vibration 279
(2005) 12031217.[6] A.K. Darpe, A novel way to detect transverse surface crack in a rotating shaft, Journal of Sound and Vibration 305 (2007) 151171.[7] A.S. Sekhar, B.S. Prabhu, Transient analysis of a cracked rotor passing through critical speed, Journal of Sound and Vibration173 (3) (1994) 415421.[8] A.S. Sekhar, B.S. Prabhu, Vibration and stress fluctuation in cracked shafts,Journal of Sound and Vibration 169 (5) (1994) 655667.[9] A.S. Sekhar, Vibration characteristics of a cracked rotor with two open cracks,Journal of Sound and Vibration 223 (4) (1999) 497512.
[10] W.D. Pilkey,Analysis and Design of Elastic Beams , First edition, John Wiley and Sons, New York, 2002, pp. 2429.[11] J. Sinou, Detection of cracks in rotor based on the 2 and 3 super-harmonics frequency components and the crack-unbalance interactions,
Communications in Nonlinear Science and Numerical Simulation 13 (2008) 20242040.[12] J. Sinou, A.W. Lees, A non-linear study of a cracked rotor, European Journal of Mechanics 26 (2007) 152170.[13] J. Sinou, A.W. Lees, The influence of cracks in rotating shafts,Journal of Sound and Vibration 285 (2005) 10151037.[14] J. Sinou, Effects of a crack on the stability of a non-linear rotor system, International Journal of Non-Linear Mechanics 42 (7) (2007) 959972.[15] M.A. AL-shudeifat, E.A. Butcher, C.R. Stern, General harmonic balance solution of a cracked rotor-bearing-disk system for harmonic and sub-
harmonic analysis: analytical and experimental approach, International Journal of Engineering Science 48 (10) (2010) 921935.[16] M.A. AL-Shudeifat, E.A. Butcher, New breathing functions for the transverse breathing crack of the cracked rotor system: approach for critical and
subcritical harmonic analysis, Journal of Sound and Vibration 330 (3) (2011) 526544.[17] I.W. Mayes, W.G.R. Davies, Analysis of the response of a multi-rotor-bearing system containing a transverse crack in a rotor, Journal of Vibration,
Acoustics, Stress, and Reliability in Design 106 (1984) 139145.[18] I. Green, C. Casy, Crack detection in a rotor dynamic system by vibration monitoringpart I: analysis,Journal of Engineering for Gas Turbine and Power
127 (2005) 425436.[19] O.S. Jun, M.S. Gadala, Dynamic behavior analysis of cracked rotor,Journal of Sound and Vibration 309 (2008) 210245.[20] L. Xiao-feng, X. Ping-yong, S. Tie-lin, Y. Shu-zi, Nonlinear analysis of a cracked rotor with whirling, Applied Mathematics and Mechanics 23 (2002)
721731.[21] C.M. Stoisser, S. Audebert, A comprehensive theoretical, numerical and experimental approach for crack detection in power plant rotating
machinery, Mechanical Systems and Signal Processing 22 (2008) 818844.[22] T.H. Patel, A.K. Darpe, Influence of crack breathing model on nonlinear dynamics of a cracked rotor, Journal of Sound and Vibration 311 (2008)
953972.[23] T. Zhou, Z. Sun, J. Xu, W. Han, Experimental analysis of a cracked rotor,Journal of Dynamic Systems, Measurements, and Control 127 (2005) 313320.[24] A.K. Darpe, K. Gupta, A. Chawla, Transient response and breathing behavior of a cracked Jeffcott rotor,Journal of Sound and Vibration 272 (2004)
207243.[25] A.K. Darpe, K. Gupta, A. Chawla, Dynamics of bowed rotor with transient surface crack, Journal of Sound and Vibration 296 (2006) 888907.[26] A.K. Darpe, K. Gupta, A. Chawla, Dynamics of a two-crack rotor,Journal of Sound and Vibration 259 (3) (2003) 649675.[27] C.A. Papadopoulos, The strain energy release approach for modeling cracks in rotors: a state of the art review, Mechanical Systems and Signal
Processing 22 (2008) 763789.[28] M. Lalanne, G. Ferraris, Rotor Dynamics Prediction in Engineering, Second edition, John Wiley and Sons, New York, 1998.[29] T. Yamamoto, Y. Ishida, Linear and Nonlinear Rotordynamics, Second edition, John Wiley and Sons, New York, 2001.[30] M.A. AL-Shudeifat, E.A. Butcher, On the modeling of open and breathing cracks of a cracked rotor system,Proceedings of the ASME 2010 International
Design Engineering Technical Conferences & Computers and Information in Engineering Conference, DETC2010-28289, Vol. 5, pp. 919928.[31] A.F. Vakakis, O. Gendelman, L.A. Bergman, D.M. McFarland, G. Kerschen, Y.S. Lee, Passive Nonlinear Targeted Energy Transfer in Mechanical and
Structural Systems: I and II, Springer Verlag, Berlin and New York, 2008.
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.
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