-
on measured vibration signals
Liu Hong, Jaspreet Singh DhupiSchool of Mechanical and Aerospace
Engineering, Nan
a r t i c l e i n f o
Article history:Received 6 June 2013Received in revised form28
October 2013Accepted 21 November 2013Handling Editor: I.
TrendafilovaAvailable online 3 January 2014
st DTW algorithm.All rights reserved.
Early detection of local gear faults in practical industrial
environments is crucial to optimize the maintenance schedule
operation of the system [4,5]. If there is any local gear fault,
e.g., such as a scuffing, cracking or a fretting corrosive tooth
as
Contents lists available at ScienceDirect
Journal of Sound and Vibration
Journal of Sound and Vibration 333 (2014) 216421800022-460X/$ -
see front matter & 2013 Elsevier Ltd. All rights
reserved.http://dx.doi.org/10.1016/j.jsv.2013.11.033n Corresponding
author at. Division of Mechatronics and Design, School of
Mechanical and Aerospace Engineering, Nanyang Technological
University,N3-02c-78, 50 Nanyang Avenue, Singapore 639798,
Singapore. Tel.: 65 6790 5506; fax: 65 6792-4062.
E-mail address: [email protected] (J.S. Dhupia).and reduce
the financial cost of gearbox damage [13]. Vibration based
diagnosis using a sensor fixed to a gearbox housingis the most
preferred monitoring technique because of the ease of measurement
and no interference with the normalthe planet motion with respect
to the fixed sensor, which is experimand is later employed for the
estimation of reference signal used in Fa
& 2013 Elsevier Ltd.
1. Introductiondetection of gears is validated using
experimental signals from a planetary gearbox test rig.For fault
detection in planetary gear-sets, a window function is introduced
to account for
entally determinedtechnique is beneficial in practical analysis
to highlight sideband patterns in situationswhere data is often
contaminated by process/measurement noises and small fluctuationsa
n
yang Technological University, 50 Nanyang Avenue, Singapore
639798, Singapore
a b s t r a c t
Spectral analysis techniques to process vibration measurements
have been widely studiedto characterize the state of gearboxes.
However, in practice, the modulated sidebandsresulting from the
local gear fault are often difficult to extract accurately from
anambiguous/blurred measured vibration spectrum due to the limited
frequency resolutionand small fluctuations in the operating speed
of the machine that often occurs in anindustrial environment. To
address this issue, a new time-domain diagnostic algorithm
isdeveloped and presented herein for monitoring of gear faults,
which shows an improvedfault extraction capability from such
measured vibration signals. This new time-domainfault detection
method combines the fast dynamic time warping (Fast DTW) as well as
thecorrelated kurtosis (CK) techniques to characterize the local
gear fault, and identify thecorresponding faulty gear and its
position. Fast DTW is employed to extract the periodicimpulse
excitations caused from the faulty gear tooth using an estimated
reference signalthat has the same frequency as the nominal gear
mesh harmonic and is built usingvibration characteristics of the
gearbox operation under presumed healthy conditions. This
in operating speeds that occur even at otherwise presumed
steady-state conditions. Theextracted signal is then resampled for
subsequent diagnostic analysis using CK technique.CK takes
advantages of the periodicity of the geared faults; it is used to
identify the positionof the local gear fault in the gearbox. Based
on simulated gear vibration signals, the FastDTW and CK based
approach is shown to be useful for condition monitoring in both
fixedaxis as well as epicyclic gearboxes. Finally the effectiveness
of the proposed method in faultA time domain approach to diagnose
gearbox fault based
journal homepage: www.elsevier.com/locate/jsvi
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L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014)
21642180 2165shown in Fig. 1, it results in amplitude and frequency
modulations in the original vibration measurements whose
sidebandsare periodic at the shaft rotation frequency and its
harmonics [6,7]. However, gearboxes often operate under some
smallfluctuation around nominal load/speed conditions during their
normal service [8,9]. These fluctuations result in a variation
ofboth the modulations and their carrier frequencies (gear mesh
harmonics) that blurs the sideband components in the spectraof the
vibration measurement, often making it difficult to be recognized
[10]. Such smearing effect can be abated by the ordertracking
technique or the time synchronous averaging (TSA) that acquires the
measurements synchronized at identical angleincrement instead of
the identical sampling period [11,12]. Although TSA is a
well-established technique for analyzing gearboxvibration signal
[1315], its commercial implementation is limited because of the
requirement for additional shaft mountedencoders to provide a
measure of shaft angular position and sophisticated interpolation
algorithms to resample the vibrationdata. Since such equipment and
resources lead to increased cost to applications, they are usually
absent in most industrialapplications [16]. In such cases, the
conventional method is to extract the measurement over a shorter
time duration using asliding window during which the gearbox is
presumed to operate under stationary condition. However, these
shorter lengthvibration signals are usually analyzed using Fourier
transforms that has limitations such as the limited frequency
resolutionand spectral leakage, while the small operational speed
oscillations continue to exist.
To avoid the extra cost incurred in implementation of TSA and
shortcomings of the Fourier transform based analysis,a time domain
method that uses dynamic time warping (DTW) was recently employed
to detect common faults in areciprocating compressor through
current signals of the driving motor in [17], though no
identification or diagnosis of faultsignals was demonstrated. In
most industrial gearboxes, which contain several gear pairs, the
fault identification anddiagnosis is of interest as replacement
gears can be ordered before the actual disassembly and physical
inspection of gears,which in turn reduces the machine downtime
[10]. Further, even though DTW algorithm was indicated to be
effective inrecognition, data mining and signal processing [18], it
has an O(N2) time and space complexity that limits its usefulness
onlyto small time series containing at most a few thousand data
points [19]. This makes the DTW algorithm hard to be applied
tovibration signal monitoring for applications such as gearboxes,
where data is usually measured for several seconds at asampling
rate which is of the order of thousands of Hertz to capture the
characteristic vibration phenomena occurringaround the meshing
frequency and its harmonics.
To address the limitations of DTW time and space complexity, as
well as, characterize the local gear fault by identifyingthe
corresponding faulted gear and its position, a new time-domain
diagnostic algorithm combining the fast dynamic timewarping (Fast
DTW) as well as the correlated kurtosis (CK) techniques is proposed
herein. Fast DTW runs in linear O(N) timeand space complexity [19],
has been applied to fault diagnostic application in geared
transmissions for the first time in thiswork to highlight the
sideband patterns resulting from the local gear fault. The fault
diagnosis algorithm introduces anestimated reference signal that
has the same frequency as the nominal gear mesh frequency and is
built using vibration
Fig. 1. Examples of the local gear faulty problems in a wind
turbine gearbox: (a) scuffing on the ring gear (credit: GEARTECH,
NREL #19855) and (b) frettingcorrosion occurred along a line of
action on the sun pinion (credit: GEARTECH, NREL
#19750).characteristics of the gearbox when operating under
presumed healthy conditions. The effect of shifting and distortion
thatusually occurs between a measured signal and an estimated
reference signal due to errors in estimation as well as smallspeed
fluctuations is minimized by Fast DTW as it eliminates these
distortion effects by allowing an elastic stretching orcompressing
within the two time series to determine the best fit between the
estimated reference and measured signal.Thus, it enables an
improved time-domain fault diagnosis algorithm wherein the residual
signal is more sensitive to faultsthrough filtering out of normal
process disturbances. After processing the vibration signal using
FAST DTW, the resultingresidual signal is resampled for subsequent
identification of damaged gear and its position using correlated
kurtosis. Thisstep is necessary as the correlated kurtosis analysis
takes advantages of the rotating periodicity of the local gear
faults toidentify the position of the damaged gear tooth, which is
restored during the resampling. The applicability of the
techniquehas been demonstrated through both simulations and
experiments. Simulations for monitoring of fixed-axis and
epicyclicgear-sets show that the vibration signals processed using
the proposed technique have residual signal containing rich
faultinformation which is more sensitive to the gear faults. The
effectiveness of this approach is also demonstrated experi-mentally
through measured vibration signals from a 4 kW planetary gearbox
test rig.
The rest of the paper is organized as follows. In Section 2, the
proposed time domain approach for gear fault diagnosisbased on the
Fast DTW and correlated kurtosis using vibration signal is
described. Section 3 investigates the effectiveness of
-
the proposed method using MATLAB/Simulink simulation studies.
Experimental validation using a controlled planetarygearbox test
rig is described in Section 4. Finally, Section 5 concludes the
paper.
2. Proposed algorithm for gearbox fault diagnosis in
time-domain
Dynamic time warping has been applied to process data from motor
currents to detect mechanical faults in a recipro-cating compressor
[17]. But the quadratic time and space complexity of this algorithm
limits DTW's application. To addressthis limitation, an improved
time domain fault diagnosis method, which combines Fast DTW and
correlated kurtosis, basedon vibration measurement from a rotary
machine is proposed in this work. This approach demonstrates good
performancefor extracting fault signature from measured vibration
data, and is able to identify the local gear fault, i.e., the
position of thedamaged gear in the gearbox. The background of
dynamic time warping, fast dynamic time warping and correlated
kurtosistechniques are summarized in the first part of this section
for the ease of the readers. Afterwards, a brief description of
theproposed algorithm is presented.
L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014)
216421802166Y y1; y2;; yj;; yM (2)Construct a warp path W:
W w1;w2;;wk;;wKwhere K is the length of the warp path and the
kth element of the warp path is
wk i; j (3)such that i is an index from time series X, and j is
an index from time series Y. The warp path must satisfy the
followingconditions:
a) w1(1,1) which implies that the warp path must start at the
beginning of each time series,b) wK(N,M) which implies that the
wrap path must finish at the end of both time series, andc) if
wk(i, j) and wk1(i, j); then iA(i, i1), jA(j, j1), which implies
that between the start and end of the wrap path
every index in both of the given time series must be
utilized.
Dynamic time warping finds the optimal warp path that minimizes
the accumulative distance (usually Euclideandistance) between the
two time series by typically using a dynamic programming approach.
For this approach, a twodimensional cost matrix D (also referred as
the accumulative distance matrix) of dimension NM is constructed.
Fig. 3shows an example to determine optimal warp path based on the
cost matrix for the alignment of two time series signalsshown in
Fig. 2. The detailed dynamic programming approach is described in
[18]. This algorithm is quadratic in both timeand space complexity,
as each cell in cost matrix D is required to be filled exactly
once.
Fig. 2. Dynamic time warping of two time series: (a) before and
(b) after processing. (For interpretation of the references to
color in this figure, the readeris referred to the web version of
this article.)X x1; x2;; xi;; xN (1)
respectively.2.1. Dynamic time warping
Dynamic time warping technique finds the optimal alignment
between two time series by allowing the given time seriesto be
warped nonlinearly by stretching or shrinking along its time axis.
Thus, DTW can be used to determine the similaritybetween the two
time series. Fig. 2 illustrates the alignment of two time series
processed by DTW. Note that, DTW algorithmis able to achieve
alignment by non-uniform warping, i.e., while the time series Y is
shrunk in the start, it is stretched laterto align with the time
series X.
The dynamic time warping problem can be stated formally as
follows: Given two time series, X and Y, of length N and M,
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2.2. Fast dynamic time warping
As a result of quadratic time and space complexity of DTW
algorithm, its implementation to monitoring data containing
onlyaround 10K measurement points may require a gigabyte range of
memory space. Therefore, speeding up computational time andreducing
memory requirement is crucial for successful implementation of DTW
algorithm to most vibration based fault diagnosisproblems. A
multi-level approach, called Fast DTW algorithm, has been developed
that runs in linear O(N) time and spacecomplexity [19]. Fast
DTWalgorithm can be implemented using a three step recursive
approach. First, a lower level/resolution timeseries is created
that has half as many points as the input time series. Next, an
optimal path in this low level/resolution is found,which is
projected to the original higher level/resolution input time
series. Finally, the projected path is expanded by a
pre-definedradius to form a search window that is passed to the DTW
algorithm. Thus, the fast implementation the DTW algorithm
onlyevaluates the cells in the search window rather than the
complete cost matrix of O(N2) dimension. Fig. 4 shows this three
stepapproach of Fast DTWalgorithm pictorially using the same time
series as given in Fig. 2. Shaded cells represent the search
window,in which red shaded cells represent the projected path and
green shaded cells represent the pre-defined radius.
The performance of the programmed Fast DTW code to remove small
fluctuations that can be found in the measuredsignals can be
further evaluated using a simulated case. Consider a test signal
x(t) and reference signal y(t):
xt A cos 2f t0:02t2 0rtr1;
A cos 2f t0:01t2 1otr2; 2f 1:02
((4)
yt A cos 2f t (5)where tA 0;2, amplitude A1 and rotational
frequency f10 Hz. For this test, a simulation time step dt of 1104
s andpre-defined radius of 20 cells is chosen. Fig. 5 shows the
test signal and reference signal before and after application of
Fast
L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014)
21642180 2167Fig. 3. Optimal warping path searched by dynamic
programming.
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L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014)
216421802168DTW algorithm. From Figs. 2 and 5, it can be concluded
that DTW/Fast DTW can align two similar signals that contain
smallphase difference or small speed oscillation that is often the
case in practical applications. This is an important propertywhich
is responsible for the improved sensitivity of residual signal
obtained from the fault detection algorithm, which isdescribed in
detail in Section 2.4.
Fig. 5. Fast dynamic time warping of the two exampled time
series: (a) before and (b) after processing.
Fig. 4. Window and optimal path evaluated during different
level/resolutions when running recursive Fast DTW algorithm with
pre-defined radiusof 4 cells.2.3. Correlated kurtosis
A fault in rotating machine such as a local gear fault
introduces periodic impacts that appear as impulse-like peaks in
themeasurements [20]. Kurtosis has been recommended for detection
of such peaks in measurement signal [21]. Although,a high kurtosis
value for a given data set indicates a presence of a distinct peak;
however, kurtosis value decreases when themeasured data set
contains periodic impulses repeating at the period of a fault.
Correlated kurtosis takes advantage of theperiodicity of the
faults, and therefore, is used in this work to detect periodic
impulses introduced by a faulted gear tooth.Correlated kurtosis of
M-shift for measured data set X is defined as [20]
CKMT Nn 1Mm 0xnmT 2
Nn 1x2nM1(6)
where T is the period of interest (the period for fault
signature that needs to be detected). Correlated kurtosis of first
and second-shift,M1 or 2, is studied in this paper. From (6), it
can be seen that the CK value approaches a maximum only when the
periodof interest Tmatches with the period of the impulses. Fig. 6
illustrates the CKM versus Kurtosis for several simple signals.
Fig. 6(a)
shows a signal with distinct peak, while Fig. 6(b) and (c) are
defined as 0:3floorn=100k 0 nk100 and 0:1floorn=100k 0 nk100,
respectively. It can be seen that while Kurtosis value decreases
as the peaks in data set become periodic, the CK value
increasesabout the specified period. However, comparing Fig. 6(b)
and (c), both Kurtosis and CK values are not sensitive to the
amplitudeof the peaks in the data set. Therefore, to investigate
the variations in amplitude of data sets, which indicates the
severity of thefault, the rms value is also calculated for the
presented simulations and experimental results in the later
sections.
2.4. Proposed fault detection algorithm
The key steps of the proposed diagnosis approach in this study
are band-pass filtering, reference signal estimation, FastDTW
implementation and residual signal analysis of processed vibration
measurement to detect gear faults in gear-sets,which are presented
as a flowchart in Fig. 7. The details of this approach are given as
follows:
Step 1: Band-pass filtering. The measured vibration signal is
pre-processed by band-pass filtering around the dominantgear mesh
frequency harmonic fm to remove other gear mesh harmonics as well
as remaining non-fault related frequency
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L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014)
21642180 2169comdriv
Sundbanbotsigninvo
(1)ponents resulting from high frequency noise and other process
disturbances that are commonly injected in an industriales. The
pre-processed vibration signal is denoted as x(t).tep 2: Reference
signal estimation. An estimated signal is built using vibration
characteristics of the gear-set operationer presumed healthy
conditions, which has the same frequency as the nominal gear mesh
harmonic fm after thed-pass filtering. This reference signal is
used as an input in Step 3. The procedure for estimating the
reference signal forh fixed axis as well as the planetary gear-set
is described separately. This is because the planetary gear-set
vibrationals have more spectral components than a fixed axis gear
systems and therefore, the estimation technique is also
morelved.
Fixed axis gear-set: Considering an error-free fixed axis gear
pair meshing under a constant load and speed, its meshvibration
y(t) around a certain gear mesh frequency harmonic fm can be
expressed as a sinusoidal signal [6]:
yt A cos 2f mt (7)
Kurtosis = 98.01, CK1(100) = 0.09, CK2(100)=0.008, CKM(T
Kurtosis = 998, CK1(100) = 0, CK2(100) = 0, CKM(T 100) = 0
Kurtosis = 98.01, CK1(100) = 0.09, CK2(100) = 0.008,CKM(T
Fig. 6. CK versus Kurtosis for different signals containing
distinct peaks.
-
which can be used as the reference signal y(t) in the subsequent
Fast DTW algorithm. The amplitude of reference signalis estimated
by finding the root mean square (rms) value for the raw measured
vibration signal after band-pass filteringstage, x(t) that has a
length of N as
A2
pxrms
2NNx2n
s(8)
The nominal gear mesh frequency fm can be calculated from the
nominal operational speed of the system. Its time axis tis chosen
to have the same length N and time step as x(t) in this study. The
initial phase of the reference signal isestimated by sweeping the
phase angle of the reference signal from 0 to 2 in small increments
and finding the phaseangle that yields the least mean square error
between x(t) and y(t).
(2) Planetary gear-set: When a planet pinion moves with the
carrier toward the vibration sensor mounted on the
stationarygearbox housing, the level of the measured vibration
signal increases, reaches a peak (r1) when the planet is closest
tothe sensor, and then decreases as the planet gear recedes (Z0).
The window function wtA 0;1, models the effect ofthis amplitude
modulation (AM) phenomenon that is periodic with the frequency of
carrier rotation [22]. Therefore, themesh vibration signal y(t)
around a certain gear mesh frequency harmonic fm of a healthy
planetary gear-set can be
Fig. 8. Scheme of determining the pre-defined window
function.
Fig. 7. Flowchart of the proposed time domain fault detection
method.
L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014)
216421802170Fig. 9. Reference signal estimation for planetary
gear-set.
-
expressed as sinusoidal signal with AM effect [23]:
yt wtA cos 2f mt (9)Further, w(t) can be expanded as a Fourier
series consisting of a sum of sinusoidal functions containing the
carrierrotational frequency and its harmonics as
wt J
j 0Wj cos 2jf ct j (10)
where fc is the nominal speed of the carrier that can be
calculated from the nominal operational speed of the gear-set.Since
the actual pattern of the window function w(t) relates to the
structure of the gearbox and the position of thesensor only, the
value of amplitudeWj and initial phase j of jth harmonic in Eq.
(10) should be independent from carrierrotational frequency fc. The
value of amplitudeWj and initial phase j of jth harmonic in Eq.
(10) can be found and storedas pre-defined parameters to describe
the window function for a given planetary gearbox. These values, Wj
and j, aredetermined using the steps described in Fig. 8. First,
the measured vibration signal under healthy condition is
firstpassed through a band pass filter around the mesh harmonic of
interest. Then, the envelope of the measured vibrationsignal can be
extracted by hardware/software envelope detector/demodulation
algorithm. Afterwards, this envelopesignal is normalized within the
[0, 1] range to form the window function w(t). Further, the order
tracking and timesynchronous averaging (TSA) techniques can be
applied to synchronize the window signal based on the
carrierrotational angle to attenuate aperiodic noise. Finally, the
discrete Fourier series transform is employed to determine
thediscrete magnitude spectrum ofWj and discrete phase spectrum of
j. The overall scheme of reference signal estimationfor planetary
gear-set using these evaluated parameters for window function,Wj
and j, is illustrated in Fig. 9. The initialphase of the window
function can be found by sweeping from 0 to 2 in small increments
to find the value of thatyields the least mean square error between
the envelop of x(t) and w(t). Further, the amplitude A and the
initialphase of Eq. (9) can be also estimated by the same methods
presented in fixed-axis case. Afterwards, Eqs. (9) and (10)are used
to generate the reference signal y(t) as shown in Fig. 9 for the
planetary gear-set, which is employed in thesubsequent Fast DTW
algorithm.
L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014)
21642180 2171Step 3: Fast DTW implementation and residual signal
analysis. The two signals, pre-processed measured signal x(t)
andestimated reference signal y(t), are matched in time domain
using the Fast DTW algorithm described in Section 2.2. Theaim of
applying Fast DTW is to reveal the difference between the two
signals x(t) and y(t), which is highlighted byevaluating the
residual signal. The raw residual signal is defined as
|xwarpedywarped|, where xwarped and ywarped areobtained from
signals x and y respectively, after transforming with the wrap path
obtained from Fast DTW. If thegearbox is operating under ideal
healthy condition, the measured vibration signal after band-pass
filtering should besimilar to the estimated reference signal. Small
phase/rotational frequency differences that may occur between x and
ybecause of the typical machine operation characteristics can be
removed by warping them along the time axis (similar
Fig. 10. The time resampling algorithm.
-
3.
Tpermo
3.1.
Ateet
rota
Fig.andto the illustrations in Figs. 2 and 5) using Fast DTW
algorithm. Thus, the residual signal under healthy condition
issmoothed out and has lower rms value when processed by Fast DTW.
If a damaged gear exists, it introduces periodicimpact/impulse-like
response in the measured signal at its characteristic fault period
T. Thus, the residual signal underfaulty condition contains
periodic peak values with high amplitude. The rms value of the
residual signal is employed todetect variation in the amplitude of
the residual. A higher rms value indicates a larger difference
between the measuredsignal and the reference signal and hence
indicates the severity of the fault. Moreover, damaged gear teeth
on differentgears have different characteristic fault period T
related to rotational frequency of the shaft carrying the gear.
Thus, by
11. Performance of time resampling algorithm: (a) test and
reference signal, (b) test and reference signal after Fast DTW, (c)
the raw residual signal,(d) the residual signal after the proposed
time resample algorithm.L. Hong, J.S. Dhupia / Journal of Sound and
Vibration 333 (2014) 216421802172evaluating the CKM(T) for all
possible characteristic fault frequencies arising from possible
tooth damage at differentgears from the residual signal can
identify the position of the fault. The challenge to evaluate the
CKM(T) on the rawresidual signal obtained after the application of
Fast DTW algorithm is that the length of residual signal K is
usuallydifferent from the length N of the original signals. The
reason for this difference in length of data can be deduced
fromFig. 3 and constrain (c) on the wrap path wherein it can be
observed that the index jmay equal to j in the warping pathfunction
wk(i, j), wk1(i, j). Therefore, a resampling algorithm is required
to restore the length of the raw residualsignal back to the
original measured signal before employing CKM(T) to identify the
fault position. This time resamplingalgorithm is presented in Fig.
10. The capability of the time resample algorithm is illustrated in
Fig. 11. A periodicreference signal with period of 275 data points
and a quasi-periodic test signal with a period fluctuating around
thenominal period of 275 data points is shown before and after
application of Fast DTW in Fig. 11(a) and (b) respectively.The
peaks of the raw residual in Fig. 11(c) are also a quasi-periodic
signal due to the quasi-periodic characteristics of thetest signal
and the warp path generated by the Fast DTW algorithm. Fig. 11(d)
gives the residual signal that is resampledusing the index j of the
periodic reference signal along the original time axis of the
reference. The period of this residualsignal is restored back to
275 data points, which is the same as the reference signal.
Simulations based investigation of proposed algorithm's
performance
he performance of the proposed approach is investigated using
both simulations and experiments. In this section, theformance of
algorithm is tested by developing an analytical model for a single
stage of fixed axis gearbox and a dynamicdel for an equally-spaced
planetary gearbox.
Detection and location of a gear defect in single stage fixed
axis gearbox
simulated case is presented herein: a single stage fixed axis
gear-set with a pinion having 10 teeth and a gear having 13h. Let
mesh harmonic order m1, its amplitude A11, number of pinion teeth
Np10, number of gear teeth Ng13 and
tional frequency of pinion is defined as f s 0:08t10; tA
0;10:04t10; tA 1;2
(to simulate small fluctuations around the nominal
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L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014)
21642180 2173rotational speed fns10 Hz. The amplitude and phase
modulation (AM & FM) functions due to a local fault on the
pinion aredefined as at 0:3 cos 2f st
and bt 0:1 cos 2f st. The simulated vibration signals of the
gearbox under healthy and
pinion faulty condition are generated as xh(t) and xpf(t),
respectively:
xht A1 cos 2Npf stnt (11)
Fig. 12. (a) Simulated signals of the gearbox under healthy and
pinion faulty condition xh(t) and xpf(t), (b) spectra of xh(t) and
xpf(t) after band-pass filter, (c)xh(t) and reference signal y(t)
after initial phase estimation but before Fast DTW, and (d) xh(t)
and reference signal y(t) after Fast DTW.xpf t A11at cos 2Npf
stbtnt (12)where n(t) is the Gaussian white noise whose SNR0 dB.
This mathematical signal characteristics (Eqs. (11) and (12))
forhealthy and faulty cases respectively has been widely used for
simulation studies of gear diagnosis by researchers [4,6,10].Fig.
12(a) plots the time domain waveform of simulated healthy signal
xh(t) and faulty signal xpf(t), which indicates that theAMFM
features are mostly concealed by the noise. Afterwards, the
proposed time domain fault diagnosis approachdescribed in last
section is applied to both xh(t) and xpf(t). Due to the
fluctuations in the operational speed, the fault featurecannot be
clearly identified even from the spectra obtained after the
band-pass filtering of raw signals to remove the highfrequency
noise (as seen from Fig. 12(b)). Fig. 12(c) and (d) presents the
simulated vibration signal xh(t) and the referencesignal y(t)
before and after application of the proposed Fast DTW process. It
can be seen that the simulated and referencesignals are aligned
together as a result of Fast DTW application. For calculation of
CKM(T), T corresponding to local faultperiod of pinion is
Tp(1/fns)/dt1000 while T corresponding to the local fault period
for the gear is Tg(1/(Npfns/Ng))/dt1300. Fig. 13(a) and (b) gives
the raw residual signals rz (just after the application of Fast
DTW) and the residual signals z(after subsequent application of the
proposed resampling algorithm). CKM(T) is calculated for both
residual signals forhealthy as well as faulty case. It can be
observed from Fig. 13 that the proposed resampling algorithm for
the raw residualsignal after Fast DTW is necessary to employ CKM(T)
to identify the fault position. For example, the rz based CK1(1000)
underhealthy and local pinion faulty condition remains at small
values 0.6105 and 1.6105, respectively. This variation inCK1 value
is insensitive to the fault at the pinion location when compared to
the z based CK1(1000) under healthy and faultyconditions with
values of 1.3105 and 23.3105 respectively. Further, the z based
CK1(1300) is insensitive to pinionfault and remains at smaller
value of 2.7105, which indicates that the CK1(T) value is sensitive
only for values of Tcorresponding to the time period of the
characteristic fault frequency and its multiples. Similar trend can
be also observed inthe CK2 values in Fig. 13.
3.2. Dynamic model of planetary gear transmission
The dynamic model of planetary gear system used in this study is
given in Fig. 14 and described in detail in [24]. Thegears are
assumed as rigid bodies connected to each other along the line of
action through the corresponding gear meshstiffness and viscous
damping [25,26]. These gears are held by bearings, which allow them
to translate in x and y directions
-
L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014)
216421802174and freely rotate about their centers in the xy
transverse plane of gear. Such model has been widely applied to
study thedynamics of industrial planetary gearbox [27,28]. The
motion of the sun gear is defined with the translational
displacementxs and ys, and the angular coordinate s. Similarly, the
motion of the carrier is defined by xc, yc, and c. is the pressure
angleand i is the initial angle location for planet i. An error
function espi and erpi with 5 m amplitude error and a profile
similarto saw-tooth [9,29,30] are used to describe gear
imperfections such as deviations in the dimensions and shape of the
gearsdue to the manufacturing error as shown in Fig. 15(a). Thus,
the gear mesh deformation along the line of action between thesun
gear and the ith planet gear can be defined as (R is the radius of
the base circle):
spi xsxiRc cos i sin iysyiRc sin i cos iscRspcRpespi (13)
Similarly, the gear mesh deformation between the ring gear and
the ith planet gear can be written as
api xiRc cos i sin iyiRc sin i cos iicRp0cRrerpi (14)
Healthy condition: RMS=0.06, CK1(Tp)=0.610-5, CK1(Tg)=0.710-5;
CK2(Tp)=0.0310-9, CK2(Tg)=0.310-9.Local pinion fault: RMS=0.12,
CK1(Tp)=1.610-5, CK1(Tg)=7.510-5; CK2(Tp)=0.410-9,
CK2(Tg)=7.310-9.
Healthy condition: RMS=0.06, CK1(Tp)=1.310-5, CK1(Tg)=1.910-5;
CK2(Tp)=0.710-9, CK2(Tg)=0.510-9.Local pinion fault: RMS=0.13,
CK1(Tp)=23.310-5, CK1(Tg)=2.710-5; CK2(Tp)=63.910-9,
CK2(Tg)=1.110-9.
Fig. 13. Residual signals after Fast DTW: (a) the raw residual
signals rz after Fast DTW, and (b) the residual signals z after the
proposed time resample.
Fig. 14. Dynamic gear mesh model of planetary gear-set.
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L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014)
21642180 2175Healthy:
Faulty:
Time t
Faulty gear toothmeshing period
KLoss
Kmax
Kmin
Gea
r mes
hing
stiff
ness
K(t)
Fig. 15. (a) Error function and (b) gear meshing stiffness under
healthy and faulty case.where the ith planet pinion has rotation i.
xi and yi are the translational displacements of the ith planet. If
spi and apio0,their values are compulsorily set to 0 as the teeth
lose contact and the resulting spring force will be equal to zero.
The backcollisions of the teeth are usually not taken into account
in planetary gear-set models, because the gear backlash values of
aplanetary gear-set are considerably larger than those of
fixed-axis gear pair in order to ensure easy assembly and
preventcontact on both flanks of the gear teeth [31]. The global
equation of motion for the gearbox that can be expressed in
matrixform as
M Q tCCb _Q tKtKbQ t Ft (15)
where M is the mass matrix, C is the damping matrix, K(t) is the
time-varying gear meshing stiffness matrix, Cb and Kb arethe
bearing damping and stiffness matrices, F(t) is the externally
applied torques vector, and Q(t) is the degrees of freedomvector
that contains two coordinates for translational vibration and a
coordinate for torsional motion for each gear in theplane
containing the gear. Fig. 15(b) illustrates a square waveform used
to describe the time-varying gear meshing stiffnessbetween a gear
pair that form the corresponding element in matrix K(t). The
degradation of a gearbox as a fault progressesresults in a
degradation of the gear mesh stiffness over the life of a gearbox
[32,33]. In this work, the common local gearfaults are modeled in
dynamic simulation study by assuming a local drop a general squared
wave form gear meshingstiffness function (Fig. 15(b)). The
rationale of this approach is based on the investigation on the
variation in the gearmeshing stiffness studied in [34], wherein it
was concluded that such faults are always accompanied by a local
reduction inthe gear meshing stiffness. In this paper, the maximum
and minimum values of gear meshing stiffness are assumed to
beKmax5108 N/m, Kmin3108 N/m, respectively. The 25 and 50 percent
local stiffness loss Kloss is used to simulate the
Healthy case: RMS = 310-7, CK1(Tsunf) = 1.510-5, CK1(Tringf) =
1.510-5, CK1(Tplanetf) = 1.310-5;Moderate fault: RMS= 410-7,
CK1(Tsunf) = 2.3 10-5, CK1(Tringf)=1.3 10-5, CK1(Tplanetf) = 1.3
10-5;Severe fault: RMS= 710-7, CK1(Tsunf) = 2.5 10-5,
CK1(Tringf)=1.5 10-5, CK1(Tplanetf) = 1.6 10-5;White noise:
CK1(Tsunf) = 1.1 10-5, CK1(Tringf)=1.1 10-5, CK1(Tplanetf) = 1.1
10-5.
Fig. 16. Simulated residual vibration signals from the planetary
gear-set under healthy and local sun gear faulted cases after
application of the proposeddiagnosis approach.
-
moderate and severe local sun gear faulted cases respectively.
The dynamic equations of the lumped parameter of gear-setsare
numerically integrated in MATLAB/Simulink environment. A constant
input torque 300 Nm is exerted on the input sideand k_
2output is used as a load torque. The parameter details of the
investigated gear-set are provided in Appendix A.
3.3. Detection and location of a gear defect in planetary
gearbox
In this simulation study, a gear-set having four planets in
equally spaced configuration, with sun gear and planets'
carrieracting as input and output respectively is investigated.
Number of ring gear teeth Nr100, number of sun gear teeth Ns28,and
number of planet gear teeth Np36. The dynamic response under
healthy conditions and a local gear tooth defect in thesun gear of
the planetary gear-set is simulated using the previously described
lumped parameter model. A 15 dB wide bandwhite noise is added to
the dynamic responses to simulate the practical measured signal.
The nominal rotational frequencyfor sun fs, carrier fc and planets
fp are 69.56 Hz, 15.21 Hz and 27.05 Hz under steady operating
conditions, respectively. Thesimulated signal is then pre-processed
using a band-pass filter with the central frequency around the gear
mesh frequencyfm fcNr1521 Hz. For this planetary gear-set at the
given simulated operating conditions, the local sun gear fault
frequencycan be evaluated as fsunf fm/Ns(fsfc)54.35 Hz [23,35].
Therefore, for evaluation of correlated kurtosis, T correspondingto
local sun gear fault is Tsunf(1/fsunf)/dt. Similarly, T
corresponding to local ring gear fault is Tringf(1/fringf)/dt, and
T valuecorresponding to local planet gear fault is
Tplanetf(1/fplanetf)/dt, respectively. Fig. 16 presents the
simulated residualvibration signals under healthy and local sun
gear tooth faulted conditions. The evaluated CK1 with TTsunf,
Tringf,and Tplanetf,and rms values of the residual signals are
shown for different fault severity levels. Additionally, the CK
values of a whitenoise are also given as a reference in Fig. 16. It
can be observed that the rms values of the residual signals
increase when thefault severity increases. CK1(Tsunf) is found to
have significant increase in presence of faults that indicates the
gear fault
L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014)
216421802176Fig. 17. (a) Schematic of the test rig, (b) planetary
gearbox test rig and (c) Seeded gear fault on the internal ring
gear of the planetary gear-set.
-
happens at the sun gear. However, CK1(Tringf) and CK1(Tplanetf)
are still remaining at smaller values comparable to the whitenoise
case, which increase or decrease somewhat under the local sun gear
tooth fault conditions. Comparing these CK1values in Fig. 16, it
implies that strong periodic components resulting from local gear
fault happen and only happen at thelocal sun gear fault frequency
in the residual signal after the proposed fault diagnosis
algorithm. Consequently, a selection ofappropriate thresholds on
the rms and CK1 value can allow for detection of gear faults and
its position.
Thus, from simulations of fixed axis as well as planetary
gear-sets, it can be seen that a selection of appropriatethresholds
on the rms and CKM values can enable detection of gear faults and
its position using the proposed algorithm.
4. Experimental results
The proposed Fast DTW algorithm along with fault identification
using correlated kurtosis was applied on experimentallymeasured
vibration signals from a planetary gearbox test rig shown in Fig.
17. The input shaft of the gearbox is connected toa 3-phase AC
motor (4 kW 4 pole) controlled using a standard industrial drive. A
generator, with a resistive load bank, isdriven by the output shaft
of the gearbox. The two tested gearboxes have back-to-back
planetary gear-sets, each having fourplanets in an equally spaced
configuration with number of ring gear teeth Nr84, number of sun
gear teeth Ns28, andnumber of planet gear teeth Np28. As a result
of the back-to-back scheme, the overall gear ratio of the gearbox
equals toG1/G41/41, where G represents the gear ratio of a single
planetary gear-set. The vibration signals are measured at a
L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014)
21642180 2177Fig. 18. The pre-defined window function for the
planetary gearbox on the test rig as a sum of sinusoidal signals
that are periodic at carrier rotationharmonics (a) discrete
magnitude spectrum of Wj and (b) discrete phase spectrum of
j.sampling rate of 20 kHz using an accelerometer attached to the
gearbox housing outside the ring gear. A seeded spalled geartooth
fault was introduced to one of the ring gear teeth (Fig. 17(c))
using electro-discharge machining (EDM).
The vibration transmission function from the gear mesh to sensor
location described in Step 2 in Section 2.4 can beexperimentally
obtained through demodulation of the measured vibration signal
under healthy condition with controlledconstant speed to determine
the amplitude modulation (AM) function as illustrated in Fig. 8.
The magnitude of this AMfunction is then normalized from 0 to 1.
Fig. 18 shows the pre-defined AM function of the tested planetary
gearboxes that isestimated from the envelope signal of the measured
vibration of the test rig under presumed healthy condition.
Sinceaccording to Eq. (10) the jth component of window function is
located at jth harmonic of carrier frequency, therefore, thecarrier
order instead of frequency is shown on the x-axis of Fig. 18.
Further note that, this pre-defined AM function containsthe
vibration contribution from the manufacturing errors that may exist
under the presumed healthy conditions at thebeginning of the
test.
The vibration signal is measured with motor operating under
open-loop speed control mode with nominal speed of1400 rev/min.
However, small deviations and fluctuations of the operational speed
could be observed as evident from themeasured spectra shown in Fig.
19. Fig. 19 illustrates the measured vibration spectrum around the
nominal gear meshfrequency fm1400/60/(GNr)490 Hz for both healthy
as well as ring gear fault conditions. It can be observed that
themeasured peak of gear mesh frequency deviates from its nominal
value and the spectra are blurred due to the inherentspeed
fluctuations during the test. Afterwards the proposed time domain
fault diagnosis approach described in Section 2.4 isapplied to the
pre-processed data filtered with a FIR band-pass filter with the
central frequency 490 Hz and bandwidth70 Hz used to extract the
signal around the gear mesh frequency. The characteristic local
ring gear fault frequency isfringf fc fm/Nr5.83 Hz, a corresponding
Tringf equals to (1/fringf)/dt is chosen to evaluate the CKM value.
As Ns equals to Npin this planetary gear-set, Tsunf also equals to
Tplanetf. The residual signals obtained after application of Fast
DTW arepresented in Fig. 20(a). This residual signal clearly
highlights the differences between the measured vibration signal
and thereference signal obtained during the healthy and ring gear
fault conditions. The rms, increase from 1.2104 under thehealthy
condition to 4.9104 under the ring gear faulty condition.
CK1(Tringf) and CK2(Tringf) also increases from 7.9105
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L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014)
216421802178and 6.8109 under the healthy condition to compared
large values 19.7105 and 47.7109, which imply that a localpinion
gear fault exists. On the other hand, while CK1(Tsunf)CK1(Tplanetf)
and CK2(Tsunf)CK2(Tplanetf) also increase to8.2105 and 8.9109, they
still remains at comparatively small values. The effectiveness of
proposed algorithm for faultdetection from practical measured
signals is further demonstrated in Fig. 20(b), which presents the
residual signal obtainedfrom measured vibration signal and
reference signal without transforming either signal using the warp
path obtained usingFast DTW. It can be observed that the residual
signal obtained for healthy and a ring gear fault condition does
not containany statistical differences, and it is difficult to
detect gear faults without application of Fast DTW. Thus, it can be
concludedthat Fast DTW processing optimally matches the measured
vibration signal to reference signal that results in more
accuratefeature extraction using the proposed approach.
Fig. 20. The residual signal between the measured vibration
signal and reference signal (a) after the proposed diagnosis
approach processing (b) only afterinitial phase matching.
Fig. 19. Measured vibration spectrum around gear mesh
frequency.
-
include both practical fixed axis as well as epicyclic
gearboxes.
L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014)
21642180 2179Acknowledgment
The authors are pleased to acknowledge the financial support of
Maritime Port Authority of Singapore (Grant number:M4060926).
Appendix A
See Table A1.
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Industrial gearboxes usually exhibit small fluctuations in speed
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A time domain approach to diagnose gearbox fault based on
measured vibration signalsIntroductionProposed algorithm for
gearbox fault diagnosis in time-domainDynamic time warpingFast
dynamic time warpingCorrelated kurtosisProposed fault detection
algorithm
Simulations based investigation of proposed algorithm's
performanceDetection and location of a gear defect in single stage
fixed axis gearboxDynamic model of planetary gear
transmissionDetection and location of a gear defect in planetary
gearbox
Experimental resultsConclusionsAcknowledgmentReferences