Dynamic modelling of a one-stage spur gear systemand
vibration-based tooth crack detection analysisOmar D.
Mohammeda,b,n, Matti Rantataloa, Jan-Olov AidanpcaDivision of
Operation, Maintenance and Acoustics, Lulea University of
Technology, SwedenbMechanical Engineering Department, College of
Engineering, University of Mosul, IraqcDivision of Product and
Production Development, Lulea University of Technology,
Swedenarticle infoArticle history:Received 29 December 2013Received
in revised form4 June 2014Accepted 2 September 2014Available online
22 September 2014Keywords:Tooth crackMesh stiffnessGear dynamic
modelGyroscopic DOFVibration analysisCrack detectionabstractFor
thepurposeof simulationandvibration-basedconditionmonitoringof
agearedsystem, itisimportanttomodelthesystem
withanappropriatenumberofdegreesoffreedom(DOF).
Inearlierpapersseveral
modelsweresuggestedanditisthereforeofinterest to evaluate their
limitations. In the present study a 12 DOF gear dynamic
modelincluding a gyroscopic effect was developed and the equations
of motions were derived.A one-stage reduction gear was modelled
using three different dynamic models (with 6, 8and 8 reduced to 6
DOF), as well as the developed model (with 12 DOF), which is
referredasthefourthmodel inthispaper.
Thetime-varyingmeshstiffnesswascalculated,
anddynamicsimulationwasthenperformedfordifferent cracksizes.
Timedomainscalarindicators (the RMS, kurtosis andthe crest factor)
were appliedfor fault detectionanalysis. Theresultsof thefirstmodel
showaclearlyvisibledifferencefromthoseofthe other studied models,
which were made more realistic by including two more DOF
todescribethemotorandload.
Boththesymmetricandtheasymmetricdisccaseswerestudied using the
fourth model. In the case of disc symmetry, the results of the
obtainedresponse are close to those obtained from both the second
and third models. Furthermore,the second model showed a slight
influence from inter-tooth friction, and therefore thethirdmodel
isadequateforsimulatingthepinion'sy-displacementinthecaseof
thesymmetric disc. Inthe case of the asymmetric disc, the results
deviate fromthoseobtained in the symmetric case. Therefore, for
simulating the pinion's y-displacement, thefourth model can be
considered for more accurate modelling in the case of theasymmetric
disc.& 2014 Elsevier Ltd. All rights reserved.1.
IntroductionThe vibration-basedconditionmonitoring technique has
gaineda great deal of importance inthe maintenanceengineeringof
industrial geartransmissions. Theroleof thistechniqueistodetect
deterioration, onthebasisof theobtained vibration signal, before
the occurrence of sudden breakage. Any undetected fault can result
in a malfunction andContentslistsavailableatScienceDirectjournal
homepage: www.elsevier.com/locate/ymsspMechanical Systems and
Signal
Processinghttp://dx.doi.org/10.1016/j.ymssp.2014.09.0010888-3270/&
2014 Elsevier Ltd. All rights reserved.nCorresponding author at:
Division of Operation, Maintenance and Acoustics, Lulea University
of Technology, Sweden. Tel.: 46 920 49 1840;fax: 46 920 49
2818.E-mail addresses: [email protected],
[email protected] (O.D. Mohammed).Mechanical Systems and
Signal Processing 54-55 (2015) 293305thenaffecttheavailabilityof
thewholesystem. Therefore,
earlyfaultdetectionisrequiredtoallowproperscheduledmaintenance to
prevent catastrophic failure and consequently provide safer
operation and higher cost savings.Vibrationresponse can be measured
experimentally or modelled theoretically. The experimental approach
is usually associated withhigher costs and problems in accessing
the measurement nodes, and is often time-consuming. Furthermore,
experimentalwork is usually restricted in terms of producing enough
real faults of desired dimensions. In many cases, therefore,
dynamiclumped-parametersmodellingcan
provideuswithaclearunderstandingofthedynamicbehaviourofthestudiedgearsystem.Gear
modelling can be considered as a fundamental problem which is still
the object of much on-going research. A greatdeal of research has
been conducted to study different dynamic models of gear systems
[1,2]. Different mathematical gearmodelswereexaminedin[2],
andgearmodellingwithbothtorsionalandtranslationalvibration
wasadoptedin[3,4].The one-stage 8 DOF gear dynamic model was
applied in [3], while the 6 DOF model was investigated in [4],
ignoring theinter-tooth friction. A different 6 DOF gear dynamic
model was applied in [510]; in this model the friction was
consideredby simulating 3 DOF for each disc (one torsional and two
translational degrees). A one-stage 16 DOF gear dynamic modelwas
developed in [11] and then adopted in [12] for simulating the
system dynamic behaviour.Among the above-mentioned research
studies, different dynamic models have been presented for different
gear systems.However, there is no study that has examined the
influence of adding more DOF to describe the gyroscopic effect of
the geardisc. Inthepresent study, a one-stage12DOF spur gear model
was developedfor describingthegyroscopic DOF.This developed model
was used to simulate the studied gear system to examine, from a
fault detection perspective, if it isnecessary to consider the disc
asymmetry effect for the studied system. This presented model and
three other models wereused to simulate the same gear system for
different crack sizes. In addition, the present paper explains gear
mesh stiffnesscalculation with a cracked tooth and presents the
results of fault detection analysis applied on the dynamic response
of thefour studied models.2. Gear dynamic modellingThe modelling of
a one-stage reduction gear system is presented in this paper. The
main gear parameters were obtainedfrom a real spur gear
transmission and are explained in Table 1. This gear transmission
is a part of a machine which is
usedasatest-rigintheConditionBasedMaintenance(CBM) Laboratoryat
LuleaUniversityof Technology. Toperformthedynamic simulation, some
more parameters need to be introduced in the studied dynamic
models, and these parametersare explained in Table 2.The term
pinion refers here to the smaller gear, which is a driver gear
connected to the input shaft, and the term gearrefers to the larger
gear, which is a driven gear connected to the output shaft. The
following notation is used:mp/mg: mass of the pinion/gear;Table
1Parameters of the gearpinion set.Parameter Pinion GearMass (kg)
0.289 1.789Number of teeth 36 90Module (mm) 1.5Teeth width (mm)
15Pressure angle (deg) 20Contact ratio 1.76Gear ratio 2.5Young's
modulus, E (N/mm2) 2105Poisson's ratio 0.3Table 2Parameters of the
dynamic modelling.Parameter InputshaftOutputshaftRadial stiffness
of the bearings in x and y direction (N/m) 6.01086.0108Radial
damping of the bearings in x and y direction (N s/m) 1.81031.8
103Applied torque (N m) 50 125Torsional stiffness (N m/rad) 11041
104Torsional damping (N m s/rad) 10 10Rotational speed (Hz) 55.55
22.22Gear mesh frequency (Hz) 2000Coefficient of friction 0.06O.D.
Mohammed et al. / Mechanical Systems and Signal Processing 54-55
(2015) 293305 294Ip/Ig: mass moment of inertia of the
pinion/gear;Kxp/Kyp: radial stiffness in the x/y directions of the
pinion;Kxg/Kyg: radial stiffness in the x/y directions of the
gear;Cxp/Cyp: radial damping in the x/y directions of the
pinion;Cxg/Cyg: radial damping in the x/y directions of the
gear;Km: equivalent mesh stiffness;Cm: mesh damping
coefficient;rbp/rbg: base circle radius of the pinion/gear;Tp/Tg:
torque applied on the pinion/gear;Tm/Tb: torque applied on the
motor/load;kt/ct: torsional stiffness/damping of the input and
output shaft; andwp/wg: constant speed of the
pinion/gear.Inthepresent work, four dynamicmodels
wereusedtosimulatethedynamicresponse.
Inter-toothfrictionwasintroduced in three of the studied models,
and the effect of ignoring it was examined, as stated in Section
2.3.2.1. First model (6 DOF)For simplicity the one-stage gear
system can be modelled without considering the motor and load. This
model consists of6 DOF, is currently applied and was adopted in
[510]. A schematic diagram of the 6 DOF model, which has 3 DOF
(onerotational andtwotranslational)foreachgeardisc, isshowninFig.
1. Theequationsof motionforthismodel canbeexplained as follows.The
equations of motion in thex direction for the pinion and gear
arempxp KxpxpCxp_xpFp1mg xg KxgxgCxg _ xgFg2The equations of motion
in they direction for the pinion and gear aremp yp NKypypCyp _
yp3mg ygNKygygCyg _ yg4The equations of motion in the direction for
the pinion and gear areIp prpNTpMp5Ig g rgNTgMg6Fig. 1. Dynamic
model of a reduction gear system with 6 DOF.O.D. Mohammed et al. /
Mechanical Systems and Signal Processing 54-55 (2015) 293305 295In
these equations,N Kmypyg_ _rpprgg_ __ _Cm_ yp_ yg_ _rp _prg _g_ _ _
_N N1N2Nn(n is the number of teeth in contact) Mp/Mg: the moments
due to the friction forces Fp/FgFpFp1Fp2Fn; Fp1m N1and Fp2 m
N2FgFg1Fg2Fn; Fg1 m N1and Fg2m N22.2. Second model (8 DOF)For more
reality the one-stage gear system can be modelled taking the motor
and load into consideration. This modelconsists of 8 DOF and was
applied in [2,3]; it has 3 DOF (one rotational and two
translational) for each gear disc, as well as 1DOF for each motor
disc and load disc to describe the rotation. A schematic diagram of
the 8 DOF model is shown in Fig. 2.The equations of motion for this
model can be explained as follows.The equations of motion in thex
direction for the pinion and gear arempxp KxpxpCxp_xpFp7mg xg
KxgxgCxg _xgFg8The equations of motion in they direction for the
pinion and gear aremp yp NKypypCyp _ yp9mg ygNKygygCyg _yg10The
equations of motion in the direction for the pinion and gear
areIpprp NMpktpm ct_p_m 11Ig g rgNMgktgbct_g_b 12The equations of
motion in the direction for the motor and load areImm ktmp
ct_m_pTm13Ibb ktbg ct_b_g Tb14Fig. 2. Dynamic model of a reduction
gear system with 8 DOF.O.D. Mohammed et al. / Mechanical Systems
and Signal Processing 54-55 (2015) 293305 2962.3. Third model (8
DOF reduced to 6 DOF)To examine the effect of including inter-tooth
friction, the second model (with 8 DOF) can be used with the
friction effectignored. Accordingly, the xp and xg displacements
are then excluded. The 8 DOF are then reduced to 6 DOF, and this
model isreferred to as having8 DOF reduced to 6 DOF. This model
(with 6 DOF) was used previously in [2,4].2.4. Fourth model (12
DOF), the developed modelA one-stage gear dynamic model including a
gyroscopic effect has been developed in this study. This model
consists of 12DOF and has 5 DOF (three rotational and two
translational) for each gear disc, as well as 1 DOF for each motor
disc and loaddisc to describe the rotation. A schematic diagram of
the 12 DOF model is shown in Fig. 3. The equations of motion for
thismodel can be explained as follows.The equations of motion in
thex direction for the pinion and gear aremp xp KxpT xpCxpT_
xpFpKxpCpCxpC_ p15mg xg KxgT xgCxgT_ xgFgKxgCgCxgC_ g16The
equations of motion in they direction for the pinion and gear
arempyp KypT ypCypT_ypNKypCpCypC_ p17mg yg KygT ygCygT_
ygNKygCgCygC_ g18The equations of motion in the direction for the
pinion and gear areIpprp NMpktpm_ _ct_p_m_ _19Igg rg NMgktgb_
_ct_g_b_ _20The equations of motion in the direction for the pinion
and gear areIdp pIpwp_ pKypC ypKypR pCypC _ ypCypR_ p21Idg g Igwg_
gKygC ygKygR gCygC _ygCygR_ g22The equations of motion in the
direction for the pinion and gear areIdp p Ipwp_ pKxpC xpKxpR pCxpC
_xpCxpR_ p23Idg gIgwg_ gKxgC xgKxgR gCxgC _ xgCxgR_ g24The
equations of motion in the direction for the motor and load areImm
ktmp_ _ct_m_p_ _Tm25Fig. 3. Dynamic model of a reduction gear
system with 12 DOF.O.D. Mohammed et al. / Mechanical Systems and
Signal Processing 54-55 (2015) 293305 297Ibb ktbg_ _ct_b_g_
_Tb26The subscript T in the stiffness and damping terms is used to
distinguish the radial or translational terms from the
otherrotational terms, which are denoted by R, and the coupling
terms, which are denoted by C. These terms are explained inAppendix
A.3. Vibration-based tooth fault
detectionForgearfaultdetectionpurposes, thestatusof
thetoothdeteriorationcanbeevaluatedmainlybyintroducingthereductioninthetime-varyinggear
meshstiffness. Several researchstudies [35,913] haveintroducedthis
stiffnessreductionindynamic modellingfor fault detectionpurposes.
Inthepresent paper, themeshstiffness parameter isconsidered to
assess the fault status, as the mesh stiffness affects the output
dynamic response.3.1. Modelling of gear mesh stiffness with a crack
in one toothTo describe how deep a crack in the tooth root is, the
crack level (CL) can be defined as the crack depth percentage of
thetotal tooth root thickness measured at the crack initiation
point, which is 2.96 mm in the studied model. Different
crackpropagationscenarios were discussedin[5,9], where their
performances, froma fault detectionperspective, werecompared.
However, for simplicity the crack is assumed in the present study
to extend along the whole tooth width
withauniformcrackdepthdistribution. Thirteencrackcasesof
differentcrackdepthswereintroduced, withastepsizeof0.1 mm, as
stated in Table 3.The mesh stiffness calculation presented in [5]
was adopted in [9], as it is a more comprehensive approach and
offers thepossibility of simulating a parabolic crack distribution.
A modified method for stiffness evaluation was discussed in
[10].One conclusion drawn was that the method presented in [5] had
been proved valid for stiffness evaluation for crack levelsless
than 30% based on the studied model. Another conclusion was that
the modified method could be considered as
analternativeforlargecracksizes, and, basedonthestudiedmodel,
abetterresultagreementhasbeenshownwiththemodified method for crack
sizes larger than 30%. In the present study the mesh stiffness
evaluation was investigated usingboth methods, and, based on the
present studied model, the difference was found to be insignificant
(less than 2%) for thelargest studied crack size, which can be
shown in Table 3. Therefore, the method presented in [5] was
applied in the presentstudy and the equivalent mesh stiffness could
be obtained as explained in the following sub-sections.3.1.1. Tooth
stiffness with a constant crack depth along the tooth widthThe
deflections under the action of the force can be determined, and
then the stiffness can be calculated by consideringthe tooth as a
non-uniform cantilever beam with an effective length of d, see Fig.
4a and b. The bending, shear, and axialcompressive stiffnesses act
in the direction of the applied load and can be obtained as follows
[5]:1Kb_d0ycos 1 hsin1 2EIxdy 271Ks_d01:2cos21 GAxdy 281Ka_d0sin21
EAxdy 29Kbis the bending stiffness, Ksis the shear stiffness and
Kais the axial compressive stiffness.h, hq, hc, hx, y, dy, d, and
1are illustrated in Fig. 4b. 1varies with the gear tooth
position.Table 3Crack propagation case data.Crack case qo (mm) CL
(%) Crack case qo (mm) CL (%)1 0 0.00 8 0.7 23.642 0.1 3.37 9 0.8
27.023 0.2 6.75 10 0.9 30.404 0.3 10.13 11 1.0 33.785 0.4 13.51 12
1.1 37.166 0.5 16.89 13 1.2 40.547 0.6 20.27O.D. Mohammed et al. /
Mechanical Systems and Signal Processing 54-55 (2015) 293305
298Moreover, the following notation is used: G: shear modulus, G
E=21; Ix: area moment of inertia,Ix1=12hxhx3W; hxrhq1=12hxhq3W;
hx4hq_Ax: area of the section of distancey measured from the load
application point,Axhxhx W; hxrhqhxhq W; hx4hq_hqhcq0
sin(c);andq0andcarethecrackdepthandcrackangle, respectively,
(seeFig. 4b). Thecrackanglecisconsidered to be 703.The total tooth
stiffness resulting from the effect of all the stiffnesses
calculated previously can be obtained as follows:Ktp1=1Kb 1Ks 1Ka_
_303.1.2. The effect of the fillet foundation deflection on the
mesh stiffnessThe fillet foundation deflection can be calculated as
follows [14]:f F cos2m WELnufSf_ _2MnufSf_ __Pn1Qntan2m_31mis the
pressure angle, and uf and Sf are illustrated in Fig. 5.Ln; Mn; Pn;
and Qncan be approximated using polynomial functions as follows
[14]:Xnihf i; f_ _Ai=2f Bih2f iCihf i=fDi=fEihf iFi32Xnirepresents
the coefficients Ln; Mn; Pn; and Qn. The coefficients Ai; Bi; Ci;
Di; Ei and Fiare given in Table 4.hf irf =rint; rf ; rint; and fare
illustrated in Fig. 5.Then the stiffness due to the fillet
foundation deflection can be obtained as1KffF33For a pinion it can
be denoted by Kfp.Fig. 4. Modelling of a gear tooth crack: (a)
modelling of a cracked tooth, (b) tooth notation, and (c) uniform
crack distribution.O.D. Mohammed et al. / Mechanical Systems and
Signal Processing 54-55 (2015) 293305 2993.1.3. The effect of
Hertzian contact on the mesh stiffnessThe Hertzian contact
stiffness Kh can be calculated as stated in [15] as follows:1Kh412
E W343.1.4. Total mesh
stiffnessAftercalculatingthestiffnessofacrackedpiniontooth, Ktp,
duetobending, shear, andaxial compression, andthencalculating the
stiffness due to the fillet foundation deflection, Kfp, the same
calculations can be performed for an uncrackedmating gear tooth to
find Ktg and Kfg. Thus, the total mesh stiffness for one meshing
tooth pair isKt11=Ktp_ _1=Kf p_ _1=Ktg_ _1=Kf g_ _1=Kh_ _
Km135where Km1 is the total mesh stiffness for the first tooth
pair.In cases where there are two tooth pairs in contact, the same
calculations are repeated for the second tooth pair to findKm2.
Then we can obtain the equivalent mesh stiffness as
follows:KmKm1Km236Fig. 6 shows the varying mesh stiffness Km
obtained for the studied crack sizes. Actually, the damping between
meshingteeth is proportional to the equivalent mesh stiffness Km,
and can be evaluated approximately using the following equation
[16]:Cm2Km1=mp1=mg37In this study the mesh damping value Cm is
considered to be 1147 N s/m for the case of two teeth in contact,
and 869 N s/mfor the case of one tooth in contact.3.2. Dynamic
simulation using the three studied dynamic modelsA dynamic
simulation was performed for the healthy case, after which the
simulation was repeated for the faulty cases.To introduce the tooth
fault in dynamic simulation, the equivalent mesh stiffness
corresponding to the crack sizes can
beinputinthestudieddynamicmodel. Fourgeardynamicmodelswerestudied,
andthesimulation wasrepeatedforthethirteen studied crack cases for
each model.Fig. 5. Geometrical parameters for fillet foundation
deflection [14].Table 4Values of the coefficients of Eq. (32).Ai (
105) Bi ( 103) Ci ( 104) Di ( 103) EiFiLn(hfi,f) 5.574 1.9986
2.3015 4.7702 0.0271 6.8045Mn(hfi,f) 60.111 28.100 83.431 9.9256
0.1624 0.9086Pn(hfi,f) 50.952 185.50 0.0538 53.300 0.2895
0.9236Qn(hfi,f) 6.2042 9.0889 4.0964 7.8297 0.1472 0.6904O.D.
Mohammed et al. / Mechanical Systems and Signal Processing 54-55
(2015) 293305
300AMatlabcomputersimulationusingtheODE45functionwascarriedouttomodel
theequationsof motionwithasamplingfrequencyof 200 kHz toprevent
aliasingof the highest detectable harmonics around20 kHz.
Adynamicsimulation with timedomain analysis was performed to obtain
thevibrationalsignals ofall thestudied cases. Normallydistributed
noise was added with an SNR value of 30 dB to include the influence
of measurement noise. In reality a
randommanufacturingerrororanyothercontribution
whichissynchronouswiththerotationalspeedwilladdenergiesinthespectra
at multiple integers of the rotational speed. These signal types
are theoretically removed from future signal
contentbytheresidualsignalprocess. Thisworkwascarriedoutby
studyingthepinionsy-displacement, whichwasthemostsensitive movement
for crack propagation in the pinion tooth root.3.3. Residual signal
and time domain statistical indicatorsThe performance of some of
the statistical indicators was studied in [4]. When the second
proposed method of generatingthe residual signal is applied, the
RMS indicator shows the best performance. Kurtosis is the most
robust indicator for all thesignalsused. Thesecond
methodrecommendedbyWuetal.
[4]forgeneratingtheresidualsignalinvolves removingthewhole original
signal of the healthy case from the original signal of the faulty
cases. The original signal obtained for the
healthycaseisconsideredasaregularsignal,
andbyremovingthisregularsignalfromtheoriginalsignalofthefaultycase,
theinfluence of regular vibration can be removed and then the
signal components generated due to crack propagation can
behighlighted. This method for generating the residual signal was
applied in the present study, after which the RMS, kurtosis andthe
crest factor were applied to the obtained residual signal. These
indicators can be explained as follows.The RMS is considered as one
of the basic statistical indicators that measure the energy level
of a signal. The RMS can bedefined as follows [4]:RMS 1NNn 1xn x 2;
where x 1NNn 1xn 38Kurtosisis anindicator whichmeasures thedegreeof
peakinessof adistributionanddescribes thesignal shapeascompared to
the normal distribution. The kurtosis value depends on the
distribution tail length, so that the kurtosis value ofthe residual
signal is much higher than that of the original signal. The
kurtosis indicator can be defined as follows [4,17]:Kurtosis 1=NNn
1xnx41=NNn 1xn x2
2_ 39The crest factor is the ratio between the maximum absolute
value reached by the signal and the RMS of the signal.
Thisindicator gives one an idea as to whether any impacting can
exist in the signal [17].CF max xnRMS40Fig. 6. Gear mesh stiffness
cycle with different crack sizes.O.D. Mohammed et al. / Mechanical
Systems and Signal Processing 54-55 (2015) 293305 3014. Results and
discussionDynamic simulations were performed for the four gear
dynamic models studied. The fourth model (with 12 DOF)
wasappliedtothesymmetricdisccaseinadditiontotheasymmetricdisccase,
whichwas simulatedtostudythediscasymmetry effect on the obtained
dynamic response. The response of the pinion's y-displacement was
analysed, because itwas the most sensitive movement for crack
propagation in the pinion tooth root. Some of the time signals
obtained usingthe fourth model with the disc symmetry case are
shown in Fig. 7. Moreover, Fig. 8 shows the percentage change in
theperformance of the statistical indicators applied on the time
signals obtained from the different dynamic models used inthis
study.The results of the first model (with 6 DOF) show a clearly
visible difference from those of the other studied models. It canbe
observed in Fig. 8 that, with the higher studied crack level, the
RMS change obtained from the first model was about 160%higher than
the RMS change obtained from the other models. In the first model
the input and output torques were
applieddirectlyonthegearandpinion, andthereforethedisplacement
responseshowsahighersensitiveresponsethantheresponseobtainedfromtheothermodels,
whichweremademorerealisticbyincludingtwomoreDOFtodescribethemotor
and load. The results of the third model (8 DOF reduced to 6 DOF),
where inter-tooth friction is ignored, are very closeto those
obtained from the second model (with 8 DOF), and it is shown that
friction exerts a very slight influence. Moreover,thefourthmodel
appliedto thesymmetric discshows aresponsethatisvery closeto
theresponsesobtainedfromthesecond and third models, because the
coupling terms which affect the obtained response yp are zero when
the lengths a andb are equal. The slight difference between their
results is due to a re-generation of the random signal accompanying
theobtained simulated signal.There is a difference between the
results for the 12 DOF model applied to the asymmetric disc case
and the results for thesymmetric disc case, since the coupling
terms start to contribute to the obtained dynamic response. These
coupling termsdescribe the influence of the gyroscopic DOF on the
dynamic response. For fault detection purposes, the disc
asymmetryeffect is important to include, as the obtained response
yp was affected by the asymmetry of the gear
disc.Todemonstratethegyroscopiceffect, aCampbell diagramof
thefourthmodel withthesymmetricdisccasewasobtainedfor
constantaverage meshstiffness, seeFig. 9. Despitethesmallinertia
valuesofthestudiedgearsystem, theinfluence of the gyroscopic effect
can be seen as the fifth eigenfrequency varies with the running
speed. This eigenfrequencyis much greater than the operating speed,
but it is useful to examine the influence of the gyroscopic effect
on the system'seigenfrequencies.Fig. 7. Original and residual
signals of three selected crack cases obtained using the 12 DOF
model with the disc symmetry case.O.D. Mohammed et al. / Mechanical
Systems and Signal Processing 54-55 (2015) 293305 302Fig. 8.
Performance of the statistical indicators applied on the residual
signals obtained from the studied models.Fig. 9. Campbell diagram
of the 12 DOF model with the symmetric disc case.O.D. Mohammed et
al. / Mechanical Systems and Signal Processing 54-55 (2015) 293305
3035. ConclusionsIn this paper,a gear dynamic model (with 12 DOF)
including a gyroscopic effect was developed and the equations
ofmotionswerederived.
Thestudiedgearsystemwasmodelledusingthreedifferentdynamicmodels(with6,
8and8reducedto6DOF), aswellasthedevelopedmodel(with12DOF),
whichisreferredasthefourthmodelinthispaper.Dynamic simulation was
performed for different crack sizes, and the pinion's
y-displacement was analysed as it is the mostsensitive movement for
crack propagation in the pinion tooth root. Time domain scalar
indicators (the RMS, kurtosis and thecrest factor) were applied for
fault detection analysis.The results of the first model show a
clearly visible difference from those of the other studied models,
which were mademore realistic by including two more DOF to describe
the motor and load.Both the symmetric and the asymmetric disc cases
were studied using the fourth model. In the case of disc
symmetry,the results for the obtained response yp are close to
those obtained from both the second and third models. Furthermore,
thesecond model showed a slight influence from inter-tooth
friction, and therefore the third model is adequate for
simulatingthe pinion's y-displacement in the case of the symmetric
disc.In the case of the asymmetric disc using the fourth model, the
results deviate from those obtained in the symmetric
case.Therefore, forsimulatingthepinion'sy-displacement, thismodel
(with12DOF) canbeconsideredformoreaccuratemodelling in the case of
the asymmetric disc.Appendix A. Terms used in the derived equations
of motionIn the equations of motion derived in Section 2, the
subscript T in the stiffness and damping terms is used to
distinguishthe radial or translational terms from the other
rotational terms, which are denoted by R, and the coupling terms,
which aredenoted by C. These terms are explained as follows, see
Fig. A1.The translational terms represent the total effect of the
supports on both sides.KxpT K1xpK2xp ; KypT K1ypK2yp; KxgT
K1xgK2xg; KygT K1ygK2ygCxpT C1xpC2xp; CypT C1ypC2yp ; CxgT
C1xgC2xg; CygT C1ygC2ygTherotational
termsareaffectedbythediscsymmetryandthenbytheparametersaandb.
Inthisstudy, forthesymmetric disc case the gear shaft lengths are
ab0.05 m, and for the asymmetric case a0.07 m and b0.03
m.KxpRa2K1xpb2K2xp; KypRa2K1ypb2K2yp; KxgRa2K1xgb2K2xg;
KygRa2K1ygb2K2ygCxpRa2C1xpb2C2xp; CypRa2C1ypb2C2yp;
CxgRa2C1xgb2C2xg; CygRa2C1ygb2C2ygThe coupling terms are also
affected by the parameters a and b; note that these terms are zero
in the case of the symmetricdisc.KxpCa K1xpb K2xp; KypCa K1ypb
K2yp; KxgC a K1xgb K2xg; KygC a K1ygb K2ygCxpCa C1xpb C2xp; CypCa
C1ypb C2yp; CxgC a C1xgb C2xg; CygC a C1ygb C2ygFig. A1. Definition
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