Experimental Dynamic Analysis of a Breathing Cracked Rotor
Experimental Dynamic Analysis of a Breathing Cracked Rotor
Chao-Zhong Guo 0 1
Ji-Hong Yan 0 1
Lawrence A. Bergman 0 1
0 Department of Aerospace Engineering, University of Illinois at Urbana-Champaign , Urbana 61801 , USA
1 Supported by National Natural Science Foundation of China , Grant No. 51505099
Crack fault diagnostics plays a critical role for rotating machinery in the traditional and Industry 4.0 factory. In this paper, an experiment is set up to study the dynamic response of a rotor with a breathing crack as it passes through its 1/2, 1/3, 1/4 and 1/5 subcritical speeds. A cracked shaft is made by applying fatigue loads through a three-point bending apparatus and then placed in a rotor testbed. The vibration signals of the testbed during the coasting-up process are collected. Whirl orbit evolution at these subcritical speed zones is analyzed. The Fourier spectra obtained by FFT are used to investigate the internal frequencies corresponding to the typical orbit characteristics. The results show that the appearance of the inner loops and orientation change of whirl orbits in the experiment are agreed well with the theoretical results obtained previously. The presence of higher frequencies 2X, 3X, 4X and 5X in Fourier spectra reveals the causes of subharmonic resonances at these subcritical speed zones. The experimental investigation is more systematic and thorough than previously reported in the literature. The unique dynamic behavior of the orbits and frequency spectra are feasible features for practical crack diagnosis. This paper provides a critical technology support for the self-aware health management of rotating machinery in the Industry 4.0 factory.
Industry 4; 0 Fault diagnosis Cracked rotor FFT spectra
School of Mechatronics Engineering, Harbin Institute of
Technology, Harbin 150001, China
The recently emerged conception ‘‘Industry 4.0’’ is one of
the most popular manufacturing topics among the industry
and academia in the world which was first announced at the
2013 Hannover Fair [
]. Similar strategies have also been
proposed by other main industrial countries, such as
‘‘Industrial Internet’’ by US and ‘‘Made in China 2025’’ by
]. Focusing on cyber-physical systems (CPS),
Industry 4.0 is regarded as the next-generation production
framework for the fourth industrial revolution [
promises to create the smart factory [
]. As one of the main
components of Industry 4.0, the smart factory will involve
a new integrative system, where not only all manufacturing
resources (sensors, actuators, machines, robots, etc.) are
connected and exchange information automatically, but
also the factory will become conscious and intelligent
enough to predict and maintain the machines; and
intelligent enough to predict and maintain the machines [
Many new technologies and methodologies have been
developed in the related fields to promote the revolution,
such as the crowsdsourcing based new production
development in manufacturing SMEs , Quality assurance
], standardization towards Industry 4.0 [
wireless device connection [
]. Because of high
connection, the health condition of a single machine has greater
influence on the factory operation than ever before. Online
fault diagnostics and prognostics provide critical health
information for the self-aware building and decision
making which will play an important role in the Industry 4.0
Rotating machines are extensively used in industry, such
as the compressors, rotors in manufacturing machines,
steam and gas turbines, generators, and pumps [
Fatigue cracking of rotor shafts is an important
phenomenon that can lead to severe damage and great
economic loss if not detected in time, especially for the highly
connected and automatic production system in the Industry
4.0 factory. The CPS provides great opportunity to perform
online crack detection based on the broad implement of
sensors, data acquisition systems, computer networks and
cloud computing systems together with the rotor dynamic
theory so that the sudden breakdown of production lines
due to the cracked shafts can be avoided.
It has been shown in the literature that the presence of a
crack introduces additional flexibility to the shaft, which
reduces its overall stiffness and generates complex orbits
and super-harmonic frequency components [
dynamic analysis of cracked rotor systems has been
intensively studied by many researchers [
et al.  evaluated the dynamic response of the rotor with
a breathing crack by expanding the changing stiffness of
the crack in a truncated Fourier series and using the
Harmonic Balance Method. The orbits during transient
operation at the critical speed and at half of the critical speed
were considered to be the unique characteristics of the
cracked rotor system. Babu, et al. [
] used the
HilbertHuang transform to study the transient response of a
cracked rotor passing through its critical speed. A
frequency fluctuation was observed at the sub-critical speed.
] and Silani, et al. [
] used finite element
models to establish the dynamic equations of the cracked
rotor system. The whirl orbit and the shifts in the critical
and subcritical speeds were studied. Shravankumar, et al.
] utilized a full-spectrum method obtained by complex
Fast Fourier transform to estimate the force and
displacement coefficients for crack identification. The typical orbits
and the frequency spectrum at the sub-critical speed were
investigated. Lu, et al. [
] studied the nonlinear response
characteristics of a breathing transverse crack rotor. The
results indicate that the transverse crack causes
super-harmonic resonance peaks at the second, third and fourth
subcritical speeds. In an earlier paper by a subset of the
], a new breathing function was proposed and
the empirical mode decomposition was used to study the
high-order frequency variation of a Jeffcott rotor with a
transvers breathing crack. The typical whirl orbits during
passage through the 1/3 and 1/2 subcritical speeds were
observed. Based on aforementioned literature and other
references therein, it can be concluded that analyzing
unique characteristics of the dynamic response is a feasible
and widely used method for crack detection.
However, although many papers have been published in
this area, only a few employ actual results from laboratory
]. Darpe, et al. [
] verified theoretical
findings through experiments with a fatigue crack rotor.
The orbits and FFT spectra during the rotor passage
through the 1/3 and 1/2 critical speeds consistently
matched the theoretical findings. Later, Zhou, et al. [
Ren, et al. [
] adopted a similar method to further
examine fatigue in cracked rotors. Compared with real
fatigue cracks, the open crack is easier to implement in the
experiment. Dong, et al. [
], Lin, et al. [
Mohammed, et al. [
] used a wire cutter machine to
generate slits with different depths in the rotor which were
generally viewed as open cracks.
In this research, the dynamic response of a breathing
cracked rotor was studied experimentally. A real fatigue
crack was induced using a three-point bending machine.
The unique orbit evolution was compared with the
theoretical findings given in Ref. [
], and the frequency
spectra obtained by FFT method were analyzed as well.
This work presents an effective crack detection method
based on the dynamic response for the online diagnosis of
rotor systems in the future smart factory.
2 Theoretical Analysis of the Cracked Rotor
In Ref. [
], a Jeffcott rotor model was established, and
a breathing function was synthesized using Fourier series
to approximate the actual breathing process. The
breathing of the crack during the shaft rotation can be
described as shown Figure 1. When open, the crack
causes a reduction in bending stiffness of the shaft,
while when fully closed (see Figure 1(f), the bending
stiffness is equal to that of the uncracked rotor.
Therefore, the breathing phenomenon leads to a time-varying
stiffness matrix in the governing equations of the
cracked rotor system. The governing equations of the
cracked rotor system are given by
( mu€ þ cu_ þ k1ðtÞu þ k12ðtÞv ¼ medX2 sinðXt þ bÞ;
mv€ þ cv_ þ k21ðtÞu þ k2ðtÞv ¼ medX2 cosðXt þ bÞ
where k1(t), k2(t) are the instantaneous stiffnesses
respectively in the horizontal and vertical directions, and k12(t),
k21(t) are the cross-coupling stiffnesses. The definitions of
other parameters can be found in Ref. [
The governing equations are solved for relative crack
depth of l = 0.2, and the whirl orbits during passage
through the 1/3 and 1/2 subcritical speeds are shown in
Figure 2. It can be seen that near the 1/3 subcritical speed
two inner loops appear (see Figure 2(a)) and get larger as
the rotating speed approaches the 1/3 subcritical speed in
Y , Y
], which leads us to consider the orbits as a
reasonable feature for crack identification.
3 Experimental Validation
An experiment is set up on a rotor test rig which consisted
of a real fatigue shaft supported by a pair of identical ball
bearings, a disk, two eddy current sensors and a DC motor
as shown in Figure 3. The mass of the disk is 500 g. The
diameter of the shaft is 10 mm, and the span between the
two bearings is 400 mm.
A fatigue crack is generated on the shaft transversely by
using a three-point-bending machine. Firstly, a slot is made
by a wire-electrode cutting machine near the middle of the
effective supported span as the initial fault. Then the shaft
is placed in the three-point-bending machine and subjected
to cyclic loading in a sinusoidal form at the nearby of the
precut slot. After about 5000 cycles, the crack propagates
to a depth of about 3.2 mm. The displacement of the center
of the cracked section is measured by the eddy current
sensors implemented in both the horizontal and vertical
directions. A speed controller is used to adjust the rotation
speed. The vibration signal is collected by a PXI data
acquisition box produced by the National Instruments
In order to verify the typical whirl orbits during passage
through the subcritical speeds, a coast up and rundown
process was performed. The first critical speed of the
cracked rotor system was found at about 3300 r/min. The
Eddy current sensors Disk
orbits around 1660 r/min, which is the 1/2 sub-critical
speed zone, are plotted in Figure 4. It can be seen that the
experimental results agree well with the theoretical
analysis. The evolution process of the typical loop shown here
is clearer and more complete than similar results shown in
the prior Refs. [
]. In Ref. [
], only two whirl orbits
were shown which was incomplete to display the
evaluation process. In Ref. [
] only the frequency spectra were
studied while orbits were not examined. In Ref. [
whirl orbits at the subcritical speeds were not quite
consistent with the theoretical results. The reason may lie in
that in Ref. [
] the oil film force generated by the journal
bearings affected the rotor orbits, but in this research the
high precision ball bearings are used which can help to
reduce the additional force.
The orientation change of the loop during passage
through the 1/2 subcritical speed zone is about p rad which
verifies the modelling and simulation findings in our
previous research [
]. The corresponding frequency spectra
are shown in Figure 5. They indicate that when the rotation
speed approaches the 1/2 sub-critical speed zone, the 2X
component dominates, as shown in Figures 5(b) and (c). As
the speed increases past the center of the subcritical zone,
the 2X component weakens, as expected.
Similarly, the whirl orbits during passage through the
1/3 subcritical speed zone are shown in Figure 6. The
evolution of the inner loops and their orientation agrees
well with simulation results. The frequency spectra in
Figure 7 show that in this region, in addition to the basic
frequency a high order (3X) component exists, indicated by
the two inner loops in Figure 6, and we observe that the 2X
component is relatively smaller in the 1/3 subcritical speed
During the experiment, the unique orbits near the 1/4
and 1/5 subcritical speeds were also observed as shown in
Figure 8 and Figure 10. To our knowledge, these
subcritical speed zones have not been observed experimentally
and reported in the literature. The whirl orbits including the
three inner loops shown in Figure 8 agreed well with
simulation results given in Ref. [
]. The frequency spectra
of the horizontal response in the 1/4 subcritical speed zone
are shown in Figure 9. It indicates that in this zone the 4X
component exists and shows a similar variation pattern as
the high-order frequency components in the 1/2 and 1/3
subcritical speed zones.
In Figure 10, four inner loops appear in the orbit when
the rotating speed passes through the 1/5 subcritical speed
zone. However, compared to Figure 4, Figure 6 and
basic frequency, the high frequency harmonic of the first
critical speed is obvious which leads to the appearance
of the typical orbits. When the rotating speed is in the
lower subcritical speed zone, the corresponding high
resonance frequency component is weaker. From
Figures 4–11, we believe that the unique dynamic
responses in the sub-critical speed zones have been
systematically investigated and documented by the
current set of experiments.
In the experiment, typical inner loops appear when
the cracked rotor passes the 1/5, 1/4, 1/3 and 1/2
subcritical speed zones, which well proves the
theoretical findings in previous research.
The FFT spectra indicates that in each subcritical
speed region, higher frequency components always
exist. In addition, during the passage, the closer the
In future work, the experiment can be designed with
multiple faults, such as the crack with bearing fatigue,
oilfilm force, or rub-impact fault to study the influence of
other faults on the dynamics of cracked rotor systems and
the crack detection method in multi-fault cases.
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Chao-Zhong Guo , born in 1982, is currently an assistant professor at Harbin Institute of Technology, China. He received his PhD degree from Harbin Institute of Technology, China , in 2014 . During 2010 to 2012, he studied in University of Illinois at Urbana-Champaign as a joint PhD candidate . His research interests include prognostics, rotor dynamics . Tel: ? 86 - 451 -86414374; E-mail: Ji-Hong Yan, born in 1972, is currently a professor at Harbin Institute of Technology, China. She received her PhD degree from Harbin Institute of Technology, China , in 1999 . Her research interests include prognostics, sustainable manufacturing, and intelligent manufacturing . Tel: ? 86 - 451 -86402972; E-mail: Lawrence A. Bergman, is currently a professor at University of Illinois at Urbana-Champaign, USA. He received his PhD degree from Case Western Reserve University, USA, in 1980 . His research interests include stochastic dynamics, linear and nonlinear structural dynamics and control, nonlinear system identification . Tel: ? 1 - 217 - 333-4970; E-mail: