Research Article Fault Feature Analysis of a Cracked Gear...

Research ArticleFault Feature Analysis of a Cracked Gear Coupled Rotor System

Hui Ma Rongze Song Xu Pang and Bangchun Wen

School of Mechanical Engineering and Automation Northeastern University Shenyang Liaoning 110819 China

Correspondence should be addressed to Hui Ma mahui 2007163com

Received 19 March 2014 Accepted 22 May 2014 Published 24 June 2014

Academic Editor Minping Jia

Copyright copy 2014 Hui Ma et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Considering themisalignment of gear root circle and base circle and accurate transition curve an improvedmesh stiffnessmodel forhealthy gear is proposed and it is validated by comparison with the finite element method On the basis of the improved methoda mesh stiffness model for a cracked gear pair is built Then a finite element model of a cracked gear coupled rotor system in aone-stage reduction gear box is established The effects of crack depth width initial position and crack propagation directionon gear mesh stiffness fault features in time domain and frequency domain and statistical indicators are investigated Moreoverfault features are also validated by experiment The results show that the improved mesh stiffness model is more accurate thanthe traditional mesh stiffness model When the tooth root crack appears distinct impulses are found in time domain vibrationresponses and sidebands appear in frequency domain Amplitudes of all the statistical indicators ascend gradually with the growthof crack depth and width decrease with the increasing crack initial position angle and firstly increase and then decrease with thegrowth of propagation direction angle

1 Introduction

A crack may initiate under the alternative load which willreduce the structural strength and may lead to tooth fracturewith crack propagation Tooth fracture is the most seriousfault in gear box and it may cause complete failure of the gearIf the tooth crack can be detected early and crack propagationcan be monitored a disastrous breakdown can be avoided byreplacing damaged gear The status of the tooth crack can beevaluated by the vibration response of the gear pair or gearcoupled rotor system for condition monitoring purposes

Vibration response of the gear pair is closely related to thetime-varying mesh stiffness (TVMS) of the gear pair Manyresearchers have developedmany analyticalmethods to studythe TVMS of healthy gears [1ndash9] Considering Hertzianenergy bending energy and axial compressive energy Yangand Lin [1] calculated theTVMSof a gear pair by the potentialenergy principle And this model was further refined by Tian[2] andWu [3] by taking the shear energy into considerationAssuming the gear body as a cantilever beam in the halfplane Zhou et al [4] took the effect of the fillet foundationdeflection into account and improved the method to cal-culate the TVMS Based on the corrected tooth foundationdeformation proposed by Sainsot et al [5] Chaari et al [6]

developed a model to calculate the TVMS Considering thatroot circle and base circle aremisaligned exactlyWan et al [7]developed amodifiedmethod to obtain the TVMS accuratelyby judging the number of gear teeth Including the effect ofthe gear tooth errors Chen and Shao [8] proposed a generalanalytical mesh stiffness model Considering the friction andthe possibility of contacts on both flanks Fernandez DelRincon et al [9] presented a procedure for determiningthe TVMS In addition to the analytical methods manyresearchers also applied finite element (FE)model to calculateTVMS [10ndash14]

Investigations on cracked gears have also been carriedout by many researchers A lot of researches focused on thecrack propagation and the effect of the rim thickness on thecrack propagation paths [15ndash18] In order to study the effect ofcrack propagation on the TVMS different models of crackedgears were also developed based on analytical method and FEmethod Based on the potential energy method proposed byYang and Lin [1] Tian [2] discussed the effects of a chippedtooth a cracked tooth and a broken tooth on TVMS byconsidering the shear energy and compared the vibrationacceleration responses of the gear system under different faulttypes On the basis of [2] Wu et al [19] studied the effects oftooth crack growth on the vibration response of a one-stage

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 832192 22 pageshttpdxdoiorg1011552014832192

2 Mathematical Problems in Engineering

gearbox with spur gears by computer simulation and devel-oped an analytical model for calculating the TVMS undera cracked tooth with different crack levels Chaari et al [6]presented an original analytical model of tooth crack quan-tified the gear mesh stiffness reduction due to spur gear toothcrack and verified the results obtained analytically by com-paring with the results of FE model Considering the toothroot crack propagations along both tooth width and crackdepth Chen and Shao [20] proposed an analytical model toinvestigate the effect of gear tooth crack on the TVMS Bythis analytical formulation the mesh stiffness of a spur gearpair with different crack lengths and depths can be obtainedBased on an improved potential energy method Zhou et al[4] developed a modified mathematical model for simulatinggear crack from root with linear growth path in a pinionby considering the deformation of gear body By assuminga parabolic curve instead of a straight line to deal with thetooth thickness reduction Mohammed et al [21] presented anew method to calculate the TVMS of the gear pair for apropagating crack in the tooth root and investigated theinfluence of gear mesh stiffness on the vibration-based faultdetection indicators the RMS Kurtosis and the crest factor

As the above literatures show extensive efforts have beendevoted to study the TVMS of meshing gear pairs with orwithout tooth crack However previous studies mostlyfocused on gears themselves [20ndash22] and in fact gear trans-mission systems are only one part of the whole rotor systemTo study the dynamic characteristics of the whole rotorsystem better the flexibility of shafts needs to be considered[23ndash31] Kahraman et al [23] developed a FE model of ageared rotor system on flexible bearings including rotaryinertia axial loading stiffness and damping of the gearmesh Kubur et al [25] established a dynamic FE model of ahelical gear transmission multi-parallel-shaft rotor systemand analyzed its dynamic characteristics Taking the flexuralrotary and torsional degrees of freedom into account Leeet al [26 27] established a FE model of a turbo-chillerrotor-bearing systemwith a bull-pinion speed increasing gearand analyzed the coupled natural frequencies and unbalanceresponses of this system By using an extended FE model of atest gear-shaft-bearing systemVelex andAjmi [30] comparedthe dynamic results from the formulations based on transmis-sion error with the reference solutions Ma et al [31] estab-lished a FE model of a geared rotor system and investigatedthe effects of tip relief on vibration responses of this systemOmar et al [32] presented a nine degree-of-freedom (DOF)model of a gear transmission system in which the gearboxstructure is coupled with the gear shafts Jia et al [33] pre-sented a dynamic model of three shafts and two pairs of gearsin mesh with 26 DOF including the effects of TVMS pitchand profile errors friction and a localized tooth crack on oneof the gears

In the abovementioned researches the gear tooth is gen-erally modeled as a nonuniform cantilever beam on base cir-cle when the TVMS of spur gear is solved based on the energymethod However root circle and base circle are misalignedexactly so the solution error of TVMS will appear Aiming atthis situation an improved mesh stiffness model is proposedin which the gear tooth is simplified as a cantilever beam on

the root circle In addition the accurate transition curve isalso considered in the improved method This study focuseson the computation of the TVMS of a cracked gear pairbased on the improved mesh stiffness model In addition thedynamic model of cracked gear coupled rotor system is alsoestablished in which the meshing model of the gear pairwith tooth crack is acquired by a lumpedmass model consid-ering TVMS and constant load torque [31] Finally vibrationresponses are obtained by Newmark-120573 method and the timedomain frequency domain and statistical indicators featuresare analyzed under different crack parameters

The structure of the paper is as follows After this intro-duction mesh stiffness calculation of a spur gear pair withcrack is presented in Section 2 An improved mesh stiffnessmodel for a healthy gear pair is established and verified inSections 21 and 22 respectively an improved mesh stiffnessmodel for a cracked gear pair is built in Section 23 on thebasis of Section 23 TVMS for crack propagation along thetooth width is introduced in Section 24 In Section 3 a FEmodel of a cracked gear coupled rotor system is developedVibration responses of the cracked gear coupled rotor systemare analyzed in Section 4 Effects of crack depth width initialposition and propagation direction on the system vibrationresponses are discussed in Sections 41 42 43 and 44respectively Measured vibration responses of the crackedgear coupled rotor system are performed in Section 5 Finallyconclusions are drawn in Section 6

2 Mesh Stiffness Calculation ofa Spur Gear Pair with Crack

21 An ImprovedMesh Stiffness Model for a Healthy Gear PairIn many published literatures the gear tooth is generallymodeled as a nonuniform cantilever beam on base circleHowever root circle and base circle are misaligned exactlyespecially when the number of teeth is larger or smaller than41 In this section an improved mesh stiffness model ispresented in which the misalignment effect between the rootcircle and base circle is considered and the gear tooth issimplified as a cantilever beam on the root circle In additionthe accurate transition curve is also considered in theimproved method

Before tooth profile curve equations are presented somebasic variable symbols have to be introduced 119898 is themodule 119873 is the number of teeth 120572 is the pressure angle ofthe gear pitch circle 119903 119903

119886 and 119903

119887are the radiuses of the pitch

circle addendum circle and base circle of the gear ℎlowast119886is the

addendum coefficient and 119888lowast is the tip clearance coefficientTooth profile curve can be divided into four parts addendumcurve 119860119861 involute curve 119861119862 transition curve 119862119863 anddedendum curve 119863119864 (see Figure 1) Equations of involutecurve are expressed as follows

119909 = 119903119894sin120593

119910 = 119903119894cos120593

120593 =120587

2119873minus (inv120572

119894minus inv120572)

(1)

Mathematical Problems in Engineering 3

F

Oy

x

C

A

B

D

E

Root circle

Base circle

Addendum circle

x

120573

120591

120591

Fa

Fb

xC120579c

120591c

rc

rb

y120573

y

yC

yD

x120573

Figure 1 Geometric model of the gear profile

where 119903119894= 119903

119887 cos120572

119894 120572119894(120572119888le 120572

119894le 120572

119886) is the pressure angle

of the arbitrary point at the involute curve in which 120572119888

=arccos(119903

119887119903119888) and 120572

119886= arccos(119903

119887119903119886) are the pressure angle

of the involute starting point and the addendum circlerespectively 119903

119888= radic(119903

119887tan120572 minus ℎlowast

119886119898 sin120572)2 + 1199032

119887is the radius

of the involute starting point circle inv120572119894is the involute

function inv120572119894= tan120572

119894minus 120572

119894 inv120572 = tan120572 minus 120572

Transition curve is formed in the process ofmachining byrack which is not engaged in meshing but is of greatimportance Its equations are defined as [34]

119909 = 119903 times sin (Φ) minus (1198861

sin 120574 + 119903120588

) times cos (120574 minus Φ)

119910 = 119903 times cos (Φ) minus (1198861

sin 120574 + 119903120588

) times sin (120574 minus Φ)

120572 le 120574 le120587

2

(2)

where Φ = (1198861 tan 120574 + 119887

1)119903 119903

120588= 119888lowast119898(1 minus sin120572) 119886

1=

(ℎlowast119886+ 119888lowast) times 119898 minus 119903

120588 and 119887

1= 1205871198984 + ℎlowast

119886119898 tan120572 + 119903

120588cos120572

Based on elastic mechanics the relationships betweenHertzian contact stiffness 119896

ℎ bending stiffness 119896

119887 shear

stiffness 119896119904 axial compressive stiffness 119896

119886 fillet foundation

stiffness 119896119891andHertzian energy119880

ℎ bending energy119880

119887 shear

energy119880119904 axial compressive energy119880

119886 and fillet foundation

energy 119880119891are obtained which are written as [1 2 20]

119880ℎ=

1198652

2119896ℎ

119880119887=

1198652

2119896119887

119880119904=

1198652

2119896119904

119880119886=

1198652

2119896119886

119880119891=

1198652

2119896119891

(3)

where 119865 is the total acting force on the contact teethThe summation of Hertzian bending shear and axial

compressive and fillet foundation energies constitutes the

total potential energy 119880 stored in a single pair of meshingteeth which can be expressed as

119880 =1198652

2119896

= 119880ℎ+ 119880

1198871+ 119880

1199041+ 119880

1198861+ 119880

1198911+ 119880

1198872+ 119880

1199042+ 119880

1198862+ 119880

1198912

=1198652

2(

1

119896ℎ

+1

1198961198871

+1

1198961199041

+1

1198961198861

+1

1198961198911

+1

1198961198872

+1

1198961199042

+1

1198961198862

+1

1198961198912

)

(4)

where 119896 represents the total effective mesh stiffness of thepair of meshing teeth and subscripts 1 and 2 denote pinionand gear respectively

According to (4) single-tooth-pair mesh stiffness can begiven as

119896 = 1 times (1

119896ℎ

+1

1198961198871

+1

1198961199041

+1

1198961198861

+1

1198961198911

+1

1198961198872

+1

1198961199042

+1

1198961198862

+1

1198961198912

)

minus1

(5)

Consequently the total TVMSof the gear pair in themeshcycle with contact ratio between 1 and 2 can be calculated as

11989612

=

119895

sum119894=1

119896119894 (119895 = 1 2) (6)

where 11989612is the total TVMS of the gear pair in themesh cycle

and 119895 denotes thenumber of meshing tooth pair at the sametime

Hertzian contact stiffness and fillet foundation stiffness[6 20] can be found in (7) and (8)

119896ℎ=

120587119864119871

4 (1 minus V2) (7)

where 119864 119871 and V represent Youngrsquos modulus tooth widthand Poissonrsquos ratio respectively

1

119896119891

=cos2120573119864119871

119871lowast(119906119891

119878119891

)

2

+ 119872lowast (119906119891

119878119891

) + 119875lowast (1 + 119876lowasttan2120573)

(8)

where 120573 is the operating pressure angle (see Figure 1) andparameters 119906

119891 119878119891 119871lowast 119872lowast 119875lowast and 119876lowast can be found in [5]


F

y

xO

F

120573

120573

120575x

120575120575y

Figure 2 FE model of a healthy gear

Based on the beam theory axial compressive bendingand shear energy of gear tooth comprised of involute andtransitional part can be calculated by

119880119886=

1198652

2119896119886

= int119910119862

119910119863

1198652119886

21198641198601199101

d1199101+ int

119910120573

119910119862

1198652119886

21198641198601199102

d1199102

119880119887=

1198652

2119896119887

= int119910119862

119910119863

1198722

1

21198641198681199101

d1199101+ int

119910120573

119910119862

1198722

2

21198641198681199102

d1199102

119880119904=

1198652

2119896119904

= int119910119862

119910119863

121198652119887

21198661198601199101

d1199101+ int

119910120573

119910119862

121198652119887

21198661198601199102

d1199102

(9)

where 119865119887= 119865 cos120573 119865

119886= 119865 sin120573 119909

120573is the distance between

the contact point and the central line of the tooth 119910120573is the

distance between the contact point and original point in thehorizontal direction 119866 = 1198642(1 + V) is the shear modulus1199101 1199102represent the horizontal coordinates of arbitrary point

at transition curve and involute curve 119910119862 119910119863denote the

horizontal coordinates of the starting point and ending pointof the transition curve 119868

1199101 119860

1199101 1198681199102 and 119860

1199102represent

the area moment of inertia and the cross-sectional arearespectively and torques119872

1= 119865

119887(119910120573minus119910

1) minus 119865

119886119909120573and 119872

2=

119865119887(119910120573minus 119910

2) minus 119865

119886119909120573denote the bending effect of 119865

119887and 119865

119886on

the part of transition and involute curvesFor convenience angular displacement is adopted in the

calculation According to the characteristics of involute andtransition curve 119909

120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 119860

1199101 and 119860

1199102

can be expressed as the functions of angular 120574 120573 or 120591 (see theappendix) and then axial compressive bending and shearstiffness can be written as follows based on (9)

1

119896119886

= int120572

1205872

sin2120573119864119860

1199101

d1199101

d120574d120574 + int

120573

120591119888

sin2120573119864119860

1199102

d1199102

d120591d120591 (10)

1

119896119887

= int120572

1205872

[cos120573 (119910120573minus 119910

1) minus 119909

120573sin120573]

2

1198641198681199101

d1199101

d120574d120574

+ int120573

120591119888

(cos120573 (119910120573minus 119910

2) minus 119909

120573sin120573)

2

1198641198681199102

d1199102

d120591d120591

(11)

1

119896119904

= int120572

1205872

12cos2120573119866119860

1199101

d1199101

d120574d120574 + int

120573

120591119888

12cos2120573119866119860

1199102

d1199102

d120591d120591 (12)

where d1199101d120574 and d119910

2d120591 are shown in the appendix

22 Verification for Improved Mesh Stiffness Model by FEModel FE method is considered as an efficient and reliabletool for solving problems and it is widely applied to calculateTVMS [10ndash14] To reduce the computational time analysis isperformed by a 2Dmodel with only one tooth (see Figure 2)which is widely used and accepted in [6 13] The load isconsidered to be in the plane of the gear body and uniformlydistributed along the tooth width No load is applied in theaxial direction and the stress is negligible Without consid-ering the contact between two gear teeth meshing forces areapplied on the tooth profile under the plane strain assump-tion The inner ring nodes of the gear are coupled withthe master node (the geometric centre of the gear) and themaster node is restrained from all degrees of freedom Subse-quently according to [21] the deformation 120575 in the directionof action line can be acquired by deformations 120575

119909and 120575

119910

which are taken from FE model and given out as

120575 = 120575119909cos120573 + 120575

119910sin120573 (13)

The single-tooth stiffness without considering the contactbetween two teeth is calculated as

119896119905=

119865

120575 (14)


0 10 20 302

4

6

TMIM

FEM

Angular position (∘)

TVM

S (times10

8N

m)

(a)

5

4

3

20 4 8 12

TMIM

FEM


TVM

S (times10

8N

m)

(b)

Figure 3 TVMS based on different methods (a) gear pair 1 (b) gear pair 2 (TM stands for traditional method IM improved method andFEM finite element method)

G

Q

PB

x

yo

C

D

C

Dq = q1

hqhB

q1

120595υ

(a)

υ

G

Q

B

υ

Px

yo

C

D

C

D

120595

q = q1max + q2

hqq2

hB

hBymax

q1max

(b)

Figure 4 Schematic of crack propagation along the depth direction (a) 119902 le 1199021max (b) 119902 gt 119902

1max

Considering the influence ofHertzian contact 119896ℎis intro-

duced as is shown in (7) and the single-tooth-pair meshstiffness can be given as

119896 =1

(1119896ℎ) + (1119896

1199051) + (1119896

1199052) (15)

where subscripts 1 and 2 denote the pinion and gear respec-tively

Two gear pairs are selected to verify the validity of theimproved mesh stiffness model and their parameters arelisted in Table 1 TVMS based on traditional method (TM) inwhich the gear tooth is modeled as a nonuniform cantileverbeam on the base circle improved method (IM) in which thegear tooth is simplified as a cantilever beam on the root circleand FE method (FEM) with precise profile curve are dis-played in Figure 3 Obviously the results from IM and FEMshow good agreement while TM shows great difference fromFEMThus it can be seen that IM ismore accurate to calculatethe TVMS than TM compared with the result of FEM

23 An Improved Mesh Stiffness Model for a Cracked GearPair In order to simplify the crack model the crack path isassumed to be a straight The crack starts at the root of thedriven gear and then propagates as is shown in Figure 4The

Table 1 Parameters of spur gear pairs

Parameters Gear pair 1 Gear pair 2Pinion Gear Pinion Gear

Number of teeth N 22 22 62 62Youngrsquos modulus E (GPa) 206 206 206 206Poissonrsquos ratio v 03 03 03 03Modulem (mm) 3 3 3 3Addendum coefficient ℎ

119886

lowast 1 1 1 1Tip clearance coefficient clowast 025 025 025 025Face width L (mm) 20 20 20 20Pressure angle 120572 (∘) 20 20 20 20Hub bore radius rint (mm) 117 117 357 357

geometrical parameters of the crack (119902 120592 120595) are presentedin Figure 4 in which 119902 denotes the crack depth 120592 the crackpropagation direction and 120595 the crack initial position

Assume that the crack extends along the whole toothwidth with a uniform crack depth distribution and the toothwith this crack is still considered as a cantilevered beamThecurve of the tooth profile remains perfect and the foundationstiffness is not affected In this case based on (7) theHertzian


contact stiffness will still be a constant since the work surfaceof the tooth has no defect and the width of the effective worksurface will always be a constant 119871 For the axial compressivestiffness it will be considered the same with that under theperfect condition in that the crack part can still bear theaxial compressive force as if no crack exists Therefore forthe gear with a cracked tooth Hertzian fillet foundation andaxial compressive stiffness can still be calculated according to(7) (8) and (10) respectively However the bending andshear stiffnesses will change due to the influence of the crackTherefore based on (5) single-tooth-pair mesh stiffness for acracked gear pair can be calculated as

119896 = 1 times (1

119896ℎ

+1

1198961198871

+1

1198961199041

+1

1198961198861

+1

1198961198911

+1

119896119887 crack

+1

119896119904 crack

+1

1198961198862

+1

1198961198912

)

minus1

(16)

where 119896119887 crack and 119896

119904 crack denote bending and shear stiffnesseswhen a crack is introduced and subscripts 1 and 2 representpinion and gear respectively

When the tooth crack is present and 119902 le 1199021max (see

Figure 4(a)) the effective area moment of inertia and area ofthe cross section at the position of 119910 can be calculated as

119868 =

1

12(ℎ119902+ 119909)

3

119871 119910119875le 119910 le 119910

119866

1

12(2119909)3119871 =

2

31199093119871 119910 lt 119910

119875or 119910 gt 119910

119866

119860 =

(ℎ119902+ 119909) 119871 119910

119875le 119910 le 119910

119866

2119909119871 119910 lt 119910119875or 119910 gt 119910

119866

(17)

where ℎ119902is the distance from the root of the crack to the

central line of the tooth which corresponds to point119866 on thetooth profile and ℎ

119902can be calculated by

ℎ119902= 119909

119876minus 119902

1sin 120592 (18)

When the crack propagates to the central line of the tooth1199021reaches its maximum value 119902

1max = 119909119876 sin 120592 Then the

crack will change direction to 1199022 which is assumed to be

exactly symmetricwith 1199021 and in this stage 119902 = 119902

1max+1199022 (seeFigure 4(b)) In theory maximum value of 119902

2is 119902

1max how-ever the tooth is expected to suffer sudden breakage beforecrack runs through the whole tooth therefore the maximumvalue of 119902

2is smaller than 119902

1max When the crack propagates

along 1199022 the effective area moment of inertia and area of the

cross section at the position of 119910 can be calculated as

119868 =

2

31199093119871 119910 lt 119910max

1

12[119909 minus

(119910 minus 119910max) ℎ119902

119910119875minus 119910max

]

3

119871 119910max le 119910 lt 119910119875

1

12(119909 minus ℎ

119902)3

119871 119910119875le 119910 le 119910

119866

0 119910 gt 119910119866

119860 =

2119909119871 119910 lt 119910max

[119909 minus(119910 minus 119910max) ℎ119902

119910119875minus 119910max

]119871 119910max le 119910 lt 119910119875

(119909 minus ℎ119902) 119871 119910

119875le 119910 le 119910

119866

0 119910 gt 119910119866

(19)

where 119910max denotes the leftmost position of crack propaga-tion (shown in Figure 4(b)) and ℎ

119902= 119902

2sin 120592

Based on (11) and (12) the bending and shear stiffness ofthe cracked gear pair can be calculated The schematic of thecalculation process is presented in Figure 5

24 TVMS for Crack Propagation along the ToothWidth Themesh stiffness model proposed in [20] divided the tooth intosome independent thin slices to represent the crack propa-gation along the tooth width as is shown in Figure 6 Whend119911 is small the crack depth can be assumed to be a constantthrough the width for each slice and stiffness of each slicedenoted as 119896(119911) can be calculated by (16) Then the stiffnessof the whole tooth can be obtained by integration of 119896(119911)along the tooth width

119896 = int119871

0

119896 (119911) (20)

Assume that the crack propagation is in the plane (seeA-A in Figure 6) The crack depth along tooth width can bedescribed as a function of 119911 in the coordinate system 11991010158401199001015840119911which can be written as

119902 (119911) = 119891 (119911) (21)The distribution of the crack depth is assumed to be a

parabolic function along the tooth width to investigate theinfluence of crack width on TVMS [20] When the crackwidth is less than the whole tooth width (see the solid curvein Figure 7)

119902 (119911) = 1199020radic

119911 + 119871119888minus 119871

119871119888

119911 isin [119871 minus 119871119888 119871]

119902 (119911) = 0 119911 isin [0 119871 minus 119871119888]

(22)

where 119871119888is the crack width and 119902

0is the crack depth on crack

starting surfaceWhen the crack width extends through the whole tooth

width (see the dashed curve in Figure 7)

119902 (119911) = radic 11990220minus 1199022

119890

119871119911 + 1199022

119890 (23)

where 119902119890is the crack depth on crack ending surface


Involute

Involute

Yes

Yes

Yes

No

No

No

No

Yes

Transition curve

Involute

Involute

Yes

No

No

No

Yes

Yes

No

Yes

Yes

No

Yes

Yes

Yes

No

No

No

Yes

No

Yes

No

Yes

No

No

Yes

Crack parameters

Transition curve

Transition curve

Transition curve

(q 120592 120595)

q le q1max

yP le y le yG y lt yPyG le yB

y120573 le yG

y le yG

y lt ymax

ymax le y le yP

yP le y le yG

yG le yB

y120573 le yG

y le yG

B2 =2

3x3L

B3 =1

12[x minus

(y minus ymax)hq

yP minus ymax]3

L

B4 =1

12(x minus hq)

3L

C1 = (hq + x)L

C2 = 2xL

C3 = [x minus(y minus ymax)hq

yP minus ymax]L

C4 = (x minus hq)L

ymax le y lt yP

yltymax

Equations (14) and (15)

kb crack ks crack

G isin CD

G isin CD

Iy1 = B3

Ay1 = C3

Iy1 = B4

Ay1 = C4

Iy1 = 0

Ay1 = 0

Iy2 = 0

Ay2 = 0

Iy1 = B2

Ay1 = C2

Iy1 = B3

Ay1 = C3

Iy1 = B4

Ay1 = C4

Iy2 = 0

Ay2 = 0Iy2 = B4

Ay2 = C4

Iy1 = B2

Ay1 = C2

Iy1 = B1

Ay1 = C1

Iy1 = B1

Ay1 = C1

Iy1 = B2Ay1 = C2

Iy1 = B2

Ay1 = C2

Iy2 = B2

Ay2 = C2

Iy2 = B2

Ay2 = C2

Iy2 = B1

Ay1 = C1

B1 =1

12(hq + x)3L

Figure 5 Schematic to obtain bending and shear stiffness for a cracked tooth

3 FE Model of a Cracked GearCoupled Rotor System

In order to research on fault features of a cracked gear itis necessary to consider the dynamic characteristics of thegeared rotor system which owns realistic significance Basedon [31] a geared rotor system model will be established

Dynamic model of a spur gear pair formed by gears 1 and2 is shown in Figure 8 where the typical spur gear meshingis represented by a pair of rigid disks connected by a spring-damper set along the plane of action which is tangent to thebase circles of the gears119874

1and119874

2are their geometrical cen-

tersΩ1andΩ

2are the rotating speeds and 119903

1198871and 119903

1198872are the

radiuses of the gear base circles respectively 11989012(119905) denotes

nonloaded static transmission error which is formed solelyby the geometric deviation component Assuming that thespur gear pair studied in the paper is perfect nonloaded statictransmission error will be zero that is 119890

12(119905) = 0 119888

12(119905) rep-

resents the mesh damping which is assumed to be zero Theangle between the plane of action and the positive 119910-axis isrepresented by 120595

12defined as

12059512

= minus120572 + 120572

12Ω1 Counterclosewise

120572 + 12057212

minus 120587 Ω1 Clockwise

(24)


A

Az

L

z

q0

120592

y998400

x998400

o998400

dz

(a)

q(z)

k(z)

120592

dz

(b)

z

L

Lc

q(z)q0

z

f(z)

y998400

o998400

A-A

dz

(c)

Figure 6 Crack model at gear tooth root

qe

y998400

o998400

L

z

q0q(z)

Lc

Figure 7 Crack depth along tooth width

where 12057212

(0 le 12057212

le 2120587) represents the angle between theline connecting the gear centers and the positive 119909-axis of thepinion and in this work 120572

12= 0

In order to consider the influence of the rotating directionof the pinion the function 120590 is introduced as

120590 = 1 Ω

1 Counterclockwise

minus1 Ω1 Clockwise

(25)

y1

120579y1

rb1

120579z112057212

O1

k12(t)

e12(t)

c12(t)

x1

120579x1

12059512

y2

120579y2

rb2

120579z2 x2

120579x2

Gear 2 (gear)

Gear 1 (pinion)

O2

Ω2

Ω1

Figure 8 Dynamic model of a spur gear pair

Each gear owns six degrees of freedom including transla-tion in 119909 119910 and 119911 directions and rotation about these threeaxesWith these six degrees of freedom for each gear the gearpair has a total of 12 degrees of freedom that defines thecoupling between the two shafts holding the gears Without


considering the tooth separation backlash friction forces atthe mesh point and the position change of the pressure linethe motion equations of the gear pair are written as

11989811minus 119896

1211990112

(119905) sin12059512

minus 11988812

(119905) 12

(119905) sin12059512

= 0

1198981

1199101+ 119896

1211990112

(119905) cos12059512

+ 11988812

(119905) 12

(119905) cos12059512

= 0

11989811= 0

1198681199091

1205791199091

+ 1198681199111Ω1

1205791199101

= 0

1198681199101

1205791199101

minus 1198681199111Ω1

1205791199091

= 0

1198681199111

1205791199111

+ 120590 times 11990311988711198961211990112

(119905) + 120590 times 119903119887111988812

(119905) 12

(119905) = 1198791

11989822+ 119896

1211990112

(119905) sin12059512

+ 11988812

(119905) 12

(119905) sin12059512

= 0

1198982

1199102minus 119896

1211990112

(119905) cos12059512

minus 11988812

(119905) 12

(119905) cos12059512

= 0

11989822= 0

1198681199092

1205791199092

+ 1198681199112Ω2

1205791199102

= 0

1198681199102

1205791199102

minus 1198681199112Ω2

1205791199092

= 0

1198681199112

1205791199112

+ 120590 times 11990311988721198961211990112

(119905) + 120590 times 119903119887211988812

(119905) 12

(119905) = 1198792

(26)

where 1198981and 119898

2are the mass of gears 1 and 2 and 119868

1199091 1198681199101

1198681199111 1198681199092 1198681199102 and 119868

1199112 respectively represent the moments of

inertia about the 119909- 119910- and 119911-axis of gears 1 and 2 1198791and 119879

2

are constant load torques The term 11990112(119905) represents the

relative displacement of the gears in the direction of the actionplane and is defined as

11990112

(119905) = (minus1199091sin120595

12+ 119909

2sin120595

12+ 119910

1cos120595

12minus 119910

2cos120595

12

+120590 times 11990311988711205791199111

+ 120590 times 11990311988721205791199112) minus 119890

12(119905)

(27)

Equation (26) can be written in matrix form as follows

M12X12

+ (C12

+ G12) X

12+ K

12X12

= F12 (28)

where X12is the displacement vector of a gear pair M

12K12

C12 and G

12represent the mass matrix mesh stiffness

matrix mesh damping matrix and gyroscopic matrix of thespur gear pair respectively F

12is the exciting force vector of

the spur gear pair These matrices and vectors are definedthoroughly in [31]

Motion equations of the whole geared rotor system can bewritten in matrix form as

Mu + (C + G) u + Ku = Fu (29)

where M K C and G are the mass stiffness damping andgyroscopic matrix of the global system u and Fu denote thedisplacement and external force vectors of the global systemAll these matrixes and vectors can be obtained in [31]

In this paper the cracked gear coupled rotor systemincludes two shafts a gear pairwith a tooth crack in the driven

Shaft 2

Shaft 1 z

x

y

Bearing 1 Bearing 2

Bearing 3 Bearing 4

Gear 1 (node 8)

Gear 2 (node 22)

Figure 9 FE model of a cracked gear coupled rotor system

gear (gear 2) and four ball bearingsTheparameters of the sys-tem are listed in Table 2 and the FEmodel of the geared rotorsystem is shown in Figure 9Material parameters of the shaftsare as follows Youngrsquos modulus 119864 = 210GPa Poissonrsquos ratioV = 03 and density 120588 = 7850 kgm3 Shafts 1 and 2 are bothdivided into 13 elements and gears 1 and 2 are located at nodes8 and 22 respectively (see Figure 9)

4 Vibration Response Analysis ofa Cracked Gear Coupled Rotor System

Based on the cracked gear coupled rotor system proposed inSection 3 the system dynamic responses can be calculated byNewmark-120573 numerical integrationmethod Firstly a crackedgear with a constant depth 3mm through the whole toothwidth is chosen as an example to analyze the influence of thecrack at the root of the gear on the systemvibration responsesThe parameters are as follows rotating speed of gear 1 Ω

1=

1000 revmin 120595 = 35∘ 120592 = 45∘ and crack depth 119902 = 3mm(1199021max = 32mm under these parameters)In this paper vibration responses at right journal of

the driving shaft in 119909 direction are selected to analyze thefault features under tooth crack condition The accelerationresponses of the healthy and cracked gears under steady stateare compared and the results are shown in Figure 10 In thefigure it is shown that three distinct impulses representingthe mating of the cracked tooth appear compared with thehealthy condition And the time interval between every twoadjacent impulses is exactly equal to the rotating period of thedriven gear (00818 s) because the influence caused by crackedtooth repeats only once in a revolution of the driven gear Inorder to clearly reflect the vibration features when the tooth


Table 2 Parameters of the cracked gear coupled rotor system

SegmentsShaft dimensions

Shaft 1 Shaft 2Diameters (mm) Length (mm) Diameters (mm) Length (mm)

1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20

Gear parametersGears 119868

119909= 119868119910(kgsdotmm2) 119868

119911(kgsdotmm2) m (kg) Pressure angle (∘) Pitch diameter (mm) Number of teeth N Tooth width (mm)

1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20

Bearing parametersBearings 119896

119909119909(MNm) 119896

119910119910(MNm) 119896

119911119911(MNm) 119896

120579119909120579119909(MNsdotmrad) 119896

120579119910120579119910(MNsdotmrad)

All 200 200 200 01 01

100

0

minus100

002 006 01 014 018 022

Time (s)

a(m

s2)

(a)

002 006 01 014 018 022

Time (s)

200

0

minus200a(m

s2) Δt = 00818 s

(b)

200

100

0

minus100

0032 0034 0036

Time (s)

a(m

s2)

Healthy gearCracked gear

(c)

Figure 10 Accelerations in time domain (a) healthy gear (b) cracked gear and (c) enlarged view

with crack is in mesh the enlarged views of vibration acceler-ation are displayed in Figure 10(c) in which the gray regionrepresents the double-tooth meshing area and the whiteregion denotes the single-toothmeshing area From the figureit can be seen that the vibration level increases apparentlycompared with that of the healthy gear

Amplitude spectra of the acceleration are shown inFigure 11 In the figure 119891

119890denotes the meshing frequency

(91667Hz) The figure shows that the amplitudes of 119891119890and

those of its harmonics (2119891119890 3119891

119890 etc) for the cracked gear

hardly change compared with the healthy gear but the side-bands appear under the tooth crack condition which can beobserved clearly in Figure 11(b) The interval Δ119891 betweenevery two sideband components is exactly equal to therotating frequency of the cracked gear that is 119891

2= 12Hz So

a conclusion can be drawn that the sideband components canbe used to diagnose the crack fault of gear


Table 3 Crack case data under different crack depths

Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()

Crack size(mm) Condition

1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

119871119888= 119871

119902119890= 119902

0= q

1199021max= 321mm

2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295

3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308

4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321

5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334

6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347

7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360

8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372

9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385

10 18 119902 = 116 21 (C) 40 119902 = 257

11 (B) 20 119902 = 128 22 42 119902 = 270

fe

2fe3fe

4fe

5fe

6fe

7fe

8fe

9fe

10fe

11fe

12fe

13fe

14fe15fe

16fe

0 5 10 15

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

30

20

10

0


times103

(a)

04

02

04500 4600 4700

5fe

Δf = 12Hz

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)


(b)

Figure 11 Amplitude spectra of the acceleration (a) amplitude spectra (b) local enlarged view (119891119890 2119891

119890 16119891

119890stand for mesh frequency and

its harmonic components)

41 Effects of Crack Depth In this section the angle 120592 standsfor crack propagation direction and is set at a constant 45∘and the angle 120595 stands for crack initial position and is set at35∘ In order to analyze the influence of crack depth on thegeared rotor system 31 crack depth cases are consideredwhich are shown in Table 3 The crack depth percentage isdefined as the ratio of actual crack depth with theoreticalthrough-tooth crack size The rotating speed of gear 1 Ω

1=

1000 revmin (in the following study the same speed is cho-sen) Four cases each of which concerns a different crack sizeare selected to study the influences of different crack depthsThe selected cases are case 1 which is the healthy case case 11which represents a smaller crack case 21 which represents amoderate crack finally case 31 which is a deep crack Thesefour cases selected for stiffness and vibration responsescomparison are referred to as A B C and D respectively

(see Table 3) In order to verify the validity of the improvedmesh stiffness model for a cracked gear pair a FE model isalso established in which the crack is modeled using 2Dsingularity elements (see Figure 12)

TVMS under the four crack sizes are calculated as a func-tion of the shaft rotation angle by the improvedmesh stiffnessmodel and the FE model And the results are shown inFigure 13 There will be eight lines when four crack cases oftwomodels are considered In order to clearly show the resultdifference between the improved method (IM) and finiteelement method (FEM) under different crack cases the fourcases are shown in two figures (see Figures 13(a) and 13(b))In the figures 1 and 2 stand for IM and FEM respectively AB C and D stand for different crack cases It can be seenfrom the figure that distinct reduction of stiffness appearswhen tooth crack is introduced and the maximum stiffness


Table 4 Comparisons of mean TVMS under different crack depths

Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896

119898(times108 Nm) Reduction percentage

1A 3582 0 2A 3323 01B 3511 minus198 2B 3194 minus3881C 3327 minus712 2C 3031 minus8791D 2959 minus1739 2D 2869 minus1366

Singularity elements

F

Crack

120573

Figure 12 FE model of a tooth with crack

reduction during the mesh process for a fixed crack appearswhere the cracked tooth is just going to engage This isbecause the relative bigger flexibility of the tooth at the adden-dum circle compared with that at the dedendum circle IMshows a better agreement with the FEM under small crackdepth conditions such as cases A B andCThe result of FEMis slightly larger than that of the IM under the large crackcondition such as case D In order to clearly display the vari-ation of the stiffness with crack depths the comparisons ofthe mean stiffness 119896

119898about the four cases are shown in

Table 4 In the table reduction percentages refer to thereduction percentage of the stiffness in case B (or C D)relative to case A It can be seen that the stiffness decreaseswith the growth of crack depth and the FEM shows the samechanging trends as the IM In the analysis of vibrationresponses the TVMS acquired from IM will be used

Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 14 For visual convenience only the local enlargedview is displayed It is very hard to observe the change inacceleration waveform produced by the gear tooth root crackwhen it is at early stage such as case B However with thegrowth of gear tooth crack depth the amplitudes of theimpulsive vibration signals increase The impulses causedby tooth crack are very obvious when the crack propagatesto 60 (case D) In the acceleration amplitude spectra themagnitudes of sidebands ascend with the crack propagationalong crack depth

Statistical features which are commonly used to provide ameasurement of the vibration level are widely used inmechanical fault detection [3 20 21] RMS andKurtosis indi-cators are used to explore the effect of the crack propagationalong depth [3] The RMS value is defined as

RMS = radic1

119873

119873

sum119899=1

(119909 (119899) minus 119909)2 119909 =1

119873

119873

sum119899=1

119909 (119899) (30)

And Kurtosis is calculated by

Kurtosis =1119873sum

119873

119899=1(119909 (119899) minus 119909)4

[1119873sum119873

119899=1(119909 (119899) minus 119909)2]

2 (31)

Variation of statistical indicators of vibration accelerationwith the increase of crack depth is shown in Figure 15 Andindicators are acquired from two types of signals (original sig-nal and residual signal [3]) which are shown in Figures 15(a)and 15(b) respectively For original signals in order to reflectthe vibration level compared with the healthy condition theindicators are expressed as percentages which are defined as

119903Kurtosis =Kurtosiscrack minus Kurtosishealth

Kurtosishealthtimes 100

119903RMS =RMScrack minus RMShealth

RMShealthtimes 100

(32)

where 119903 shows the relative change of cracked signal indicatorsubscripts Kurtosis and RMS denote different statisticalindicators and subscripts crack and health are cracked andhealthy gear

Relative change ratios in Figure 15(b) refer to the ratiosof RMS and Kurtosis with those under the maximum crackdepth (60 under this condition) It can be seen from thefigures that for original signals RMS and Kurtosis changelittle when the crack depth is smaller but when the crackdepth becomes higher than 40 Kurtosis exhibits an obviousincreasing trend while RMS increases slightly For residualsignals unlike the results obtained fromoriginal signals RMSshows obvious increasing pattern when the crack depthexceeds 20 Compared with RMS however Kurtosischanges less obviously SoRMSacquired from residual signalsis a robust indicator for early fault identification which issimilar to the results shown in [3]

42 Effects of Crack Width In order to investigate theinfluences of crack width on the geared rotor system 22 crackwidth cases are selectedwhich are shown inTable 5The crackwidth percentage is defined as the ratio of actual crack width119871119888to the whole tooth width (119871 = 20mm) It is noted that

120 of crack width represents the case when 119871119888

= 119871 =20mm and 119902

119890= 119902

0= 3mm Five cases corresponding to

cases 1 6 11 21 and 22 in Table 5 are referred to as A B Cand D and E TVMS under the five cases are analyzed toillustrate the influences of crack width by IM (see Figure 16)Since the FE model used in the paper is a 2D model it ispowerless to simulate the crack propagation along the toothwidth And because the crack depth is changing along the


4

3

20 4 8 12

TVM

S (times10

8N

m)

1A1B

2A2B


(a)

4

3

2

10 4 8 12

TVM

S (times10

8N

m)


1C1D

2C2D

(b)

Figure 13 TVMS under different crack depths (a) crack cases A and B and (b) crack cases C and D

AB

CD

300

200

100

0

minus100

minus200

a(m

s2)

0032 0034 0036

Time (s)

(a)

AB

CD

1

05

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)

Figure 14 Acceleration responses under different crack depths (a) time domain waveform and (b) amplitude spectra

Kurtosis

25

15

5

minus50 20 40 60

Crack depth ()

r Kur

tosi

sr R

MS

()

RMS

(a)

100

50

0

Rela

tive c

hang

e rat

io (

)

Kurtosis

0 20 40 60

Crack depth ()

RMS

(b)

Figure 15 Acceleration statistical indicators under different crack depths (a) original signal and (b) residual signal


Table 5 Crack propagation case data under different crack widths

Crackcase

Crack width()

Crack size(mm)

Crackcase

Crack width() Crack size (mm) Crack

caseCrack width

() Crack size (mm) Condition

1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902

119890= 0 119871

119888= 8 17 80 119902

119890= 0 119871

119888= 16

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

1199020= 3mm

2 5 119902119890= 0 119871119888 = 1 10 45 119902

119890= 0 119871

119888= 9 18 85 119902

119890= 0 119871

119888= 17

3 10 119902119890= 0 119871

119888= 2 11 (C) 50 119902

119890= 0 119871

119888= 10 19 90 119902

119890= 0 119871

119888= 18

4 15 119902119890= 0 119871

119888= 3 12 55 119902

119890= 0 119871119888 = 11 20 95 119902

119890= 0 119871

119888= 19

5 20 119902119890= 0 119871

119888= 4 13 60 119902

119890= 0 119871

119888= 12 21 (D) 100 119902

119890= 0 119871

119888= 20

6 (B) 25 119902119890= 0 119871

119888= 5 14 65 119902

119890= 0 119871

119888= 13 22 (E) 120 119902

119890= 3 119871

119888= 20

7 30 119902119890= 0 119871

119888= 6 15 70 119902

119890= 0 119871

119888= 14

8 35 119902119890= 0 119871

119888= 7 16 75 119902

119890= 0 119871

119888= 15

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

ABC

DE

Figure 16 TVMS under different crack widths by IM

tooth width even if a 3D model is adopted some difficultieswill also occur So in this section the comparison of themesh stiffness between the improved model and the FEmodel under different crack widths is not carried out Thecomparisons of the mean stiffness 119896

119898about the five cases are

shown in Table 6 It can be seen from Figure 16 and Table 6that the stiffness decreases with the growth of crack width

Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 17 The amplitudesof the impulsive vibration and the magnitudes of sidebandsincrease with the growth of crack width For statisticalfeatures (see Figure 18) RMS and Kurtosis of original signalincrease slightly when the crack width is smaller When thecrack width becomes higher than 80 Kurtosis exhibits anobvious increasing trend Reversely RMS increases slightlyFor residual signals unlike the results obtained from theoriginal signals RMS shows obvious increasing pattern Inaddition there is a prompt increase for the two indicatorswhen the crack propagates along the whole tooth width andcontinues to propagate along the crack depth

Table 6 Comparisons ofmean TVMS under different crack widths

Case (IM) 119896119898(times108 Nm) Reduction percentage

A 3582 0B 3534 minus134C 3490 minus257D 3401 minus505E 3231 minus980

43 Effects of Crack Initial Position In this section 12 crackinitial position cases are considered which are shown inTable 7 Four cases marked as A (stands for healthy gear) BC and D (see Table 7) are selected to study the influencesof different crack initial positions on TVMS and vibrationresponses The mesh stiffness curves obtained by IM andFEM under the above four cases are displayed in Figure 19The comparisons of themean stiffness 119896

119898about the four cases

are shown inTable 8 It can be seen fromFigure 19 andTable 8that the stiffness ascends with the increase of the crack initialposition angle 120595 and the FEM shows the same changingtrends as IM

Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 20 With the growth of initial position angle 120595 theamplitudes of the impulsive vibration signals decrease Simi-larly in the acceleration amplitude spectra themagnitudes ofsidebands also decrease with the growth of initial positionangle 120595

The variation of statistical indicators acquired from orig-inal signals and residual signals with the increase of initialposition angle 120595 is shown in Figure 21 For residual signalsthe relative change ratio of120595=65∘ crack is defined as 100Asseen from the figures for original signals Kurtosis exhibits anobvious decreasing trend while RMS decreases slightly Forresidual signals just the opposite RMS shows obviousdecreasing pattern with the growth of initial position angle120595In addition compared with original signals RMS acquiredfrom residual signals changes more obviously

44 Effects of Crack Propagation Direction In this section 19crack propagation direction cases are considered as shown inTable 9 Seven cases marked as A (stands for healthy gear)


200

100

0

minus100

a(m

s2)

0032 0034 0036

Time (s)

ABC

DE

(a)

4500 4600 4700

04

03

02

01

0

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

ABC

DE

(b)

Figure 17 Acceleration responses under different crack widths (a) time domain waveform and (b) amplitude spectra

3

2

1

0

0 40 80 120

Crack width ()

r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

100

50

0

Relat

ive c

hang

e rat

io (

)

0 40 80 120

Crack width ()

Kurtosis

RMS

(b)

Figure 18 Acceleration statistical indicators under different crack widths (a) original signal and (b) residual signal (120 represents the case119871119888= 119871 = 20mm and 119902

119890= 119902

0= 3mm)

Table 7 Crack case data under different crack initial positions

Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Condition1 15 5 35 9 55 Ω1 = 1000 revmin

120592 = 45∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65

4 30 8 50 12

Table 8 Comparisons of mean TVMS under different crack initial positions





45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1A1B

2A2B

(a)

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1C1D

2C2D

(b)

Figure 19 TVMS under different crack initial positions (a) crack cases A and B and (b) crack cases C and D

200

100

0

minus100

a(m

s2 )

0032 0034 0036

Time (s)

AB

CD

(a)

AB

CD

04

02

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)

Figure 20 Acceleration responses under different crack initial positions (a) time domain waveform and (b) amplitude spectra45

3

15

015 25 35 45 55 65

Crack position 120595 (∘)

r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

180

140

100

Rela

tive c

hang

e rat

io (

)

15 25 35 45 55 65


Kurtosis

RMS

(b)

Figure 21 Acceleration statistical indicators under different crack initial positions (a) original signal and (b) residual signal


4

3

2

10 4 8 12


TVM

S (times10

8N

m)

1A1B1C1D

1E1F1G

(a)

4

3

2

10 4 8 12


TVM

S (times10

8N

m)

2A2B2C2D

2E2F2G

(b)

Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM

Table 9 Crack case data under different crack propagation directions

Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation

direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition

1 (B) 0 8 35 15 (E) 70

Ω1 = 1000 revmin120595 = 35∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 5 9 40 16 753 10 10 45 17 (F) 80

4 15 11 50 18 855 20 12 55 19 (G) 90

6 25 13 (D) 60

7 (C) 30 14 65

B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses

Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896

119898about the seven cases are

shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM

Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592

The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of

propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously

5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System

The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed


400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)

Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra

Table 10 Comparisons of mean TVMS under different crack propagation directions



1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583


80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90

Crack propagation direction 120592 (∘)

RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)

Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal

in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891

1and 119891

2 respectively and meshing frequency 119891

119890is equal

to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively

Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891

1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891

1 For cracked geared

rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891

119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2

In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results

Electromotor

One-stage reduction gear box

Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position

Figure 25 Test rig of a cracked gear coupled rotor system

6 Conclusions

Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows

(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear


40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000

Frequency (Hz) Frequency (Hz) Frequency (Hz)

fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)

Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra

60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe

2fe minus 2f22fe + f2

2fe + 2f22fe minus f2

fe + 2f2

(b)

Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra

tooth is modeled as a cantilever beam on the basecircle

(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases

with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592

(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and


the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults

(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously

It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus

Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860

1199102can be expressed

as the function of angular 120574 120573 or 120591 which can be expressedas

119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910

2d120591 can be calculated as follows

d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574

sin2120574sin(120574 minus

1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)

times cos(120574 minus1198861 tan 120574 + 119887

1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)

where 120579119887is the half tooth angle on the base circle of the gear

120579119887= 1205872119873 + inv120572

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work

References

[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987

[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004

[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007

[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic


model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012

[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004

[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009

[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013

[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013

[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013

[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006

[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005

[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999

[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003

[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004

[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997

[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013

[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013

[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010

[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008

[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011

[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013

[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012

[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992

[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998

[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004

[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003

[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005

[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012

[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012

[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006

[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014

[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012

[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003

[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


gearbox with spur gears by computer simulation and devel-oped an analytical model for calculating the TVMS undera cracked tooth with different crack levels Chaari et al [6]presented an original analytical model of tooth crack quan-tified the gear mesh stiffness reduction due to spur gear toothcrack and verified the results obtained analytically by com-paring with the results of FE model Considering the toothroot crack propagations along both tooth width and crackdepth Chen and Shao [20] proposed an analytical model toinvestigate the effect of gear tooth crack on the TVMS Bythis analytical formulation the mesh stiffness of a spur gearpair with different crack lengths and depths can be obtainedBased on an improved potential energy method Zhou et al[4] developed a modified mathematical model for simulatinggear crack from root with linear growth path in a pinionby considering the deformation of gear body By assuminga parabolic curve instead of a straight line to deal with thetooth thickness reduction Mohammed et al [21] presented anew method to calculate the TVMS of the gear pair for apropagating crack in the tooth root and investigated theinfluence of gear mesh stiffness on the vibration-based faultdetection indicators the RMS Kurtosis and the crest factor

As the above literatures show extensive efforts have beendevoted to study the TVMS of meshing gear pairs with orwithout tooth crack However previous studies mostlyfocused on gears themselves [20ndash22] and in fact gear trans-mission systems are only one part of the whole rotor systemTo study the dynamic characteristics of the whole rotorsystem better the flexibility of shafts needs to be considered[23ndash31] Kahraman et al [23] developed a FE model of ageared rotor system on flexible bearings including rotaryinertia axial loading stiffness and damping of the gearmesh Kubur et al [25] established a dynamic FE model of ahelical gear transmission multi-parallel-shaft rotor systemand analyzed its dynamic characteristics Taking the flexuralrotary and torsional degrees of freedom into account Leeet al [26 27] established a FE model of a turbo-chillerrotor-bearing systemwith a bull-pinion speed increasing gearand analyzed the coupled natural frequencies and unbalanceresponses of this system By using an extended FE model of atest gear-shaft-bearing systemVelex andAjmi [30] comparedthe dynamic results from the formulations based on transmis-sion error with the reference solutions Ma et al [31] estab-lished a FE model of a geared rotor system and investigatedthe effects of tip relief on vibration responses of this systemOmar et al [32] presented a nine degree-of-freedom (DOF)model of a gear transmission system in which the gearboxstructure is coupled with the gear shafts Jia et al [33] pre-sented a dynamic model of three shafts and two pairs of gearsin mesh with 26 DOF including the effects of TVMS pitchand profile errors friction and a localized tooth crack on oneof the gears

In the abovementioned researches the gear tooth is gen-erally modeled as a nonuniform cantilever beam on base cir-cle when the TVMS of spur gear is solved based on the energymethod However root circle and base circle are misalignedexactly so the solution error of TVMS will appear Aiming atthis situation an improved mesh stiffness model is proposedin which the gear tooth is simplified as a cantilever beam on

the root circle In addition the accurate transition curve isalso considered in the improved method This study focuseson the computation of the TVMS of a cracked gear pairbased on the improved mesh stiffness model In addition thedynamic model of cracked gear coupled rotor system is alsoestablished in which the meshing model of the gear pairwith tooth crack is acquired by a lumpedmass model consid-ering TVMS and constant load torque [31] Finally vibrationresponses are obtained by Newmark-120573 method and the timedomain frequency domain and statistical indicators featuresare analyzed under different crack parameters

The structure of the paper is as follows After this intro-duction mesh stiffness calculation of a spur gear pair withcrack is presented in Section 2 An improved mesh stiffnessmodel for a healthy gear pair is established and verified inSections 21 and 22 respectively an improved mesh stiffnessmodel for a cracked gear pair is built in Section 23 on thebasis of Section 23 TVMS for crack propagation along thetooth width is introduced in Section 24 In Section 3 a FEmodel of a cracked gear coupled rotor system is developedVibration responses of the cracked gear coupled rotor systemare analyzed in Section 4 Effects of crack depth width initialposition and propagation direction on the system vibrationresponses are discussed in Sections 41 42 43 and 44respectively Measured vibration responses of the crackedgear coupled rotor system are performed in Section 5 Finallyconclusions are drawn in Section 6

2 Mesh Stiffness Calculation ofa Spur Gear Pair with Crack

21 An ImprovedMesh Stiffness Model for a Healthy Gear PairIn many published literatures the gear tooth is generallymodeled as a nonuniform cantilever beam on base circleHowever root circle and base circle are misaligned exactlyespecially when the number of teeth is larger or smaller than41 In this section an improved mesh stiffness model ispresented in which the misalignment effect between the rootcircle and base circle is considered and the gear tooth issimplified as a cantilever beam on the root circle In additionthe accurate transition curve is also considered in theimproved method

Before tooth profile curve equations are presented somebasic variable symbols have to be introduced 119898 is themodule 119873 is the number of teeth 120572 is the pressure angle ofthe gear pitch circle 119903 119903

119886 and 119903

119887are the radiuses of the pitch

circle addendum circle and base circle of the gear ℎlowast119886is the

addendum coefficient and 119888lowast is the tip clearance coefficientTooth profile curve can be divided into four parts addendumcurve 119860119861 involute curve 119861119862 transition curve 119862119863 anddedendum curve 119863119864 (see Figure 1) Equations of involutecurve are expressed as follows

119909 = 119903119894sin120593

119910 = 119903119894cos120593

120593 =120587

2119873minus (inv120572

119894minus inv120572)

(1)


F

Oy

x

C

A

B

D

E

Root circle

Base circle

Addendum circle

x

120573

120591

120591

Fa

Fb

xC120579c

120591c

rc

rb

y120573

y

yC

yD

x120573


where 119903119894= 119903

119887 cos120572

119894 120572119894(120572119888le 120572

119894le 120572



=arccos(119903

119887119903119888) and 120572

119886= arccos(119903



119888= radic(119903


119886119898 sin120572)2 + 1199032

119887is the radius



119894minus 120572

119894 inv120572 = tan120572 minus 120572


119909 = 119903 times sin (Φ) minus (1198861

sin 120574 + 119903120588


119910 = 119903 times cos (Φ) minus (1198861

sin 120574 + 119903120588


120572 le 120574 le120587

2

(2)

where Φ = (1198861 tan 120574 + 119887

1)119903 119903

120588= 119888lowast119898(1 minus sin120572) 119886

1=


120588 and 119887

1= 1205871198984 + ℎlowast

119886119898 tan120572 + 119903

120588cos120572



119887 shear





119887 shear




119880ℎ=

1198652

2119896ℎ

119880119887=

1198652

2119896119887

119880119904=

1198652

2119896119904

119880119886=

1198652

2119896119886

119880119891=

1198652

2119896119891

(3)




119880 =1198652

2119896

= 119880ℎ+ 119880

1198871+ 119880

1199041+ 119880

1198861+ 119880

1198911+ 119880

1198872+ 119880

1199042+ 119880

1198862+ 119880

1198912

=1198652

2(

1

119896ℎ

+1

1198961198871

+1

1198961199041

+1

1198961198861

+1

1198961198911

+1

1198961198872

+1

1198961199042

+1

1198961198862

+1

1198961198912

)

(4)



119896 = 1 times (1

119896ℎ

+1

1198961198871

+1

1198961199041

+1

1198961198861

+1

1198961198911

+1

1198961198872

+1

1198961199042

+1

1198961198862

+1

1198961198912

)

minus1

(5)


11989612

=

119895

sum119894=1

119896119894 (119895 = 1 2) (6)




119896ℎ=

120587119864119871

4 (1 minus V2) (7)


1

119896119891

=cos2120573119864119871

119871lowast(119906119891

119878119891

)

2

+ 119872lowast (119906119891

119878119891


(8)




F

y

xO

F

120573

120573

120575x

120575120575y



119880119886=

1198652

2119896119886

= int119910119862

119910119863

1198652119886

21198641198601199101

d1199101+ int

119910120573

119910119862

1198652119886

21198641198601199102

d1199102

119880119887=

1198652

2119896119887

= int119910119862

119910119863

1198722

1

21198641198681199101

d1199101+ int

119910120573

119910119862

1198722

2

21198641198681199102

d1199102

119880119904=

1198652

2119896119904

= int119910119862

119910119863

121198652119887

21198661198601199101

d1199101+ int

119910120573

119910119862

121198652119887

21198661198601199102

d1199102

(9)

where 119865119887= 119865 cos120573 119865

119886= 119865 sin120573 119909






1199101 119860

1199101 1198681199102 and 119860

1199102represent


1= 119865

119887(119910120573minus119910

1) minus 119865

119886119909120573and 119872

2=

119865119887(119910120573minus 119910

2) minus 119865


119887and 119865

119886on



120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 119860

1199101 and 119860

1199102


1

119896119886

= int120572

1205872

sin2120573119864119860

1199101

d1199101

d120574d120574 + int

120573

120591119888

sin2120573119864119860

1199102

d1199102

d120591d120591 (10)

1

119896119887

= int120572

1205872

[cos120573 (119910120573minus 119910

1) minus 119909

120573sin120573]

2

1198641198681199101

d1199101

d120574d120574

+ int120573

120591119888

(cos120573 (119910120573minus 119910

2) minus 119909

120573sin120573)

2

1198641198681199102

d1199102

d120591d120591

(11)

1

119896119904

= int120572

1205872

12cos2120573119866119860

1199101

d1199101

d120574d120574 + int

120573

120591119888

12cos2120573119866119860

1199102

d1199102

d120591d120591 (12)

where d1199101d120574 and d119910



119909and 120575

119910


120575 = 120575119909cos120573 + 120575

119910sin120573 (13)


119896119905=

119865

120575 (14)


0 10 20 302

4

6

TMIM

FEM


TVM

S (times10

8N

m)

(a)

5

4

3

20 4 8 12

TMIM

FEM


TVM

S (times10

8N

m)

(b)


G

Q

PB

x

yo

C

D

C

Dq = q1

hqhB

q1

120595υ

(a)

υ

G

Q

B

υ

Px

yo

C

D

C

D

120595

q = q1max + q2

hqq2

hB

hBymax

q1max

(b)


1max



119896 =1

(1119896ℎ) + (1119896

1199051) + (1119896

1199052) (15)







119886






119896 = 1 times (1

119896ℎ

+1

1198961198871

+1

1198961199041

+1

1198961198861

+1

1198961198911

+1

119896119887 crack

+1

119896119904 crack

+1

1198961198862

+1

1198961198912

)

minus1

(16)

where 119896119887 crack and 119896




119868 =

1

12(ℎ119902+ 119909)

3

119871 119910119875le 119910 le 119910

119866

1

12(2119909)3119871 =

2

31199093119871 119910 lt 119910

119875or 119910 gt 119910

119866

119860 =

(ℎ119902+ 119909) 119871 119910

119875le 119910 le 119910

119866

2119909119871 119910 lt 119910119875or 119910 gt 119910

119866

(17)




ℎ119902= 119909

119876minus 119902

1sin 120592 (18)


1max = 119909119876 sin 120592 Then the




2is 119902






119868 =

2

31199093119871 119910 lt 119910max

1

12[119909 minus

(119910 minus 119910max) ℎ119902

119910119875minus 119910max

]

3

119871 119910max le 119910 lt 119910119875

1

12(119909 minus ℎ

119902)3

119871 119910119875le 119910 le 119910

119866

0 119910 gt 119910119866

119860 =

2119909119871 119910 lt 119910max

[119909 minus(119910 minus 119910max) ℎ119902

119910119875minus 119910max

]119871 119910max le 119910 lt 119910119875

(119909 minus ℎ119902) 119871 119910

119875le 119910 le 119910

119866

0 119910 gt 119910119866

(19)


119902= 119902

2sin 120592



119896 = int119871

0

119896 (119911) (20)




119902 (119911) = 1199020radic

119911 + 119871119888minus 119871

119871119888

119911 isin [119871 minus 119871119888 119871]

119902 (119911) = 0 119911 isin [0 119871 minus 119871119888]

(22)





119902 (119911) = radic 11990220minus 1199022

119890

119871119911 + 1199022

119890 (23)



Involute

Involute

Yes

Yes

Yes

No

No

No

No

Yes

Transition curve

Involute

Involute

Yes

No

No

No

Yes

Yes

No

Yes

Yes

No

Yes

Yes

Yes

No

No

No

Yes

No

Yes

No

Yes

No

No

Yes

Crack parameters

Transition curve

Transition curve

Transition curve

(q 120592 120595)

q le q1max


y120573 le yG

y le yG

y lt ymax

ymax le y le yP

yP le y le yG

yG le yB

y120573 le yG

y le yG

B2 =2

3x3L

B3 =1

12[x minus

(y minus ymax)hq

yP minus ymax]3

L

B4 =1

12(x minus hq)

3L

C1 = (hq + x)L

C2 = 2xL


yP minus ymax]L

C4 = (x minus hq)L

ymax le y lt yP

yltymax


kb crack ks crack

G isin CD

G isin CD

Iy1 = B3

Ay1 = C3

Iy1 = B4

Ay1 = C4

Iy1 = 0

Ay1 = 0

Iy2 = 0

Ay2 = 0

Iy1 = B2

Ay1 = C2

Iy1 = B3

Ay1 = C3

Iy1 = B4

Ay1 = C4

Iy2 = 0

Ay2 = 0Iy2 = B4

Ay2 = C4

Iy1 = B2

Ay1 = C2

Iy1 = B1

Ay1 = C1

Iy1 = B1

Ay1 = C1

Iy1 = B2Ay1 = C2

Iy1 = B2

Ay1 = C2

Iy2 = B2

Ay2 = C2

Iy2 = B2

Ay2 = C2

Iy2 = B1

Ay1 = C1

B1 =1

12(hq + x)3L





1and119874


tersΩ1andΩ


1198871and 119903

1198872are the



12(119905) = 0 119888

12(119905) rep-


12defined as

12059512

= minus120572 + 120572


120572 + 12057212


(24)


A

Az

L

z

q0

120592

y998400

x998400

o998400

dz

(a)

q(z)

k(z)

120592

dz

(b)

z

L

Lc

q(z)q0

z

f(z)

y998400

o998400

A-A

dz

(c)


qe

y998400

o998400

L

z

q0q(z)

Lc


where 12057212

(0 le 12057212


12= 0


120590 = 1 Ω

1 Counterclockwise


(25)

y1

120579y1

rb1

120579z112057212

O1

k12(t)

e12(t)

c12(t)

x1

120579x1

12059512

y2

120579y2

rb2

120579z2 x2

120579x2

Gear 2 (gear)

Gear 1 (pinion)

O2

Ω2

Ω1





11989811minus 119896

1211990112

(119905) sin12059512

minus 11988812

(119905) 12

(119905) sin12059512

= 0

1198981

1199101+ 119896

1211990112

(119905) cos12059512

+ 11988812

(119905) 12

(119905) cos12059512

= 0

11989811= 0

1198681199091

1205791199091

+ 1198681199111Ω1

1205791199101

= 0

1198681199101

1205791199101

minus 1198681199111Ω1

1205791199091

= 0

1198681199111

1205791199111

+ 120590 times 11990311988711198961211990112

(119905) + 120590 times 119903119887111988812

(119905) 12

(119905) = 1198791

11989822+ 119896

1211990112

(119905) sin12059512

+ 11988812

(119905) 12

(119905) sin12059512

= 0

1198982

1199102minus 119896

1211990112

(119905) cos12059512

minus 11988812

(119905) 12

(119905) cos12059512

= 0

11989822= 0

1198681199092

1205791199092

+ 1198681199112Ω2

1205791199102

= 0

1198681199102

1205791199102

minus 1198681199112Ω2

1205791199092

= 0

1198681199112

1205791199112

+ 120590 times 11990311988721198961211990112

(119905) + 120590 times 119903119887211988812

(119905) 12

(119905) = 1198792

(26)

where 1198981and 119898


1199091 1198681199101

1198681199111 1198681199092 1198681199102 and 119868



2



11990112

(119905) = (minus1199091sin120595

12+ 119909

2sin120595

12+ 119910

1cos120595

12minus 119910

2cos120595

12

+120590 times 11990311988711205791199111

+ 120590 times 11990311988721205791199112) minus 119890

12(119905)

(27)


M12X12

+ (C12

+ G12) X

12+ K

12X12

= F12 (28)


12K12

C12 and G






Mu + (C + G) u + Ku = Fu (29)



Shaft 2

Shaft 1 z

x

y

Bearing 1 Bearing 2

Bearing 3 Bearing 4

Gear 1 (node 8)

Gear 2 (node 22)





1=







1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20


119909= 119868119910(kgsdotmm2) 119868


1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20


119909119909(MNm) 119896

119910119910(MNm) 119896

119911119911(MNm) 119896

120579119909120579119909(MNsdotmrad) 119896

120579119910120579119910(MNsdotmrad)

All 200 200 200 01 01

100

0

minus100

002 006 01 014 018 022

Time (s)

a(m

s2)

(a)

002 006 01 014 018 022

Time (s)

200

0

minus200a(m

s2) Δt = 00818 s

(b)

200

100

0

minus100

0032 0034 0036

Time (s)

a(m

s2)


(c)









2= 12Hz So




Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()


1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

119871119888= 119871

119902119890= 119902

0= q

1199021max= 321mm

2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295

3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308

4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321

5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334

6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347

7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360

8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372

9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385

10 18 119902 = 116 21 (C) 40 119902 = 257

11 (B) 20 119902 = 128 22 42 119902 = 270

fe

2fe3fe

4fe

5fe

6fe

7fe

8fe

9fe

10fe

11fe

12fe

13fe

14fe15fe

16fe

0 5 10 15

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

30

20

10

0


times103

(a)

04

02

04500 4600 4700

5fe

Δf = 12Hz

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)


(b)


119890 16119891




1=










F

Crack

120573







RMS = radic1

119873

119873

sum119899=1

(119909 (119899) minus 119909)2 119909 =1

119873

119873

sum119899=1

119909 (119899) (30)



119873

119899=1(119909 (119899) minus 119909)4

[1119873sum119873

119899=1(119909 (119899) minus 119909)2]

2 (31)





RMShealthtimes 100

(32)





= 119871 =20mm and 119902

119890= 119902




4

3

20 4 8 12

TVM

S (times10

8N

m)

1A1B

2A2B


(a)

4

3

2

10 4 8 12

TVM

S (times10

8N

m)


1C1D

2C2D

(b)


AB

CD

300

200

100

0

minus100

minus200

a(m

s2)

0032 0034 0036

Time (s)

(a)

AB

CD

1

05

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


Kurtosis

25

15

5

minus50 20 40 60

Crack depth ()

r Kur

tosi

sr R

MS

()

RMS

(a)

100

50

0

Rela

tive c

hang

e rat

io (

)

Kurtosis

0 20 40 60

Crack depth ()

RMS

(b)




Crackcase

Crack width()

Crack size(mm)

Crackcase


caseCrack width


1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902

119890= 0 119871

119888= 8 17 80 119902

119890= 0 119871

119888= 16

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

1199020= 3mm

2 5 119902119890= 0 119871119888 = 1 10 45 119902

119890= 0 119871

119888= 9 18 85 119902

119890= 0 119871

119888= 17

3 10 119902119890= 0 119871

119888= 2 11 (C) 50 119902

119890= 0 119871

119888= 10 19 90 119902

119890= 0 119871

119888= 18

4 15 119902119890= 0 119871

119888= 3 12 55 119902

119890= 0 119871119888 = 11 20 95 119902

119890= 0 119871

119888= 19

5 20 119902119890= 0 119871

119888= 4 13 60 119902

119890= 0 119871

119888= 12 21 (D) 100 119902

119890= 0 119871

119888= 20

6 (B) 25 119902119890= 0 119871

119888= 5 14 65 119902

119890= 0 119871

119888= 13 22 (E) 120 119902

119890= 3 119871

119888= 20

7 30 119902119890= 0 119871

119888= 6 15 70 119902

119890= 0 119871

119888= 14

8 35 119902119890= 0 119871

119888= 7 16 75 119902

119890= 0 119871

119888= 15

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

ABC

DE
















200

100

0

minus100

a(m

s2)

0032 0034 0036

Time (s)

ABC

DE

(a)

4500 4600 4700

04

03

02

01

0

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

ABC

DE

(b)


3

2

1

0

0 40 80 120

Crack width ()

r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

100

50

0

Relat

ive c

hang

e rat

io (

)

0 40 80 120

Crack width ()

Kurtosis

RMS

(b)


119890= 119902

0= 3mm)



120592 = 45∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65

4 30 8 50 12






45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1A1B

2A2B

(a)

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1C1D

2C2D

(b)


200

100

0

minus100

a(m

s2 )

0032 0034 0036

Time (s)

AB

CD

(a)

AB

CD

04

02

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


3

15

015 25 35 45 55 65


r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

180

140

100

Rela

tive c

hang

e rat

io (

)

15 25 35 45 55 65


Kurtosis

RMS

(b)



4

3

2

10 4 8 12


TVM

S (times10

8N

m)

1A1B1C1D

1E1F1G

(a)

4

3

2

10 4 8 12


TVM

S (times10

8N

m)

2A2B2C2D

2E2F2G

(b)





1 (B) 0 8 35 15 (E) 70

Ω1 = 1000 revmin120595 = 35∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 5 9 40 16 753 10 10 45 17 (F) 80

4 15 11 50 18 855 20 12 55 19 (G) 90

6 25 13 (D) 60

7 (C) 30 14 65











400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)







80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










F

Oy

x

C

A

B

D

E

Root circle

Base circle

Addendum circle

x

120573

120591

120591

Fa

Fb

xC120579c

120591c

rc

rb

y120573

y

yC

yD

x120573


where 119903119894= 119903

119887 cos120572

119894 120572119894(120572119888le 120572

119894le 120572



=arccos(119903

119887119903119888) and 120572

119886= arccos(119903



119888= radic(119903


119886119898 sin120572)2 + 1199032

119887is the radius



119894minus 120572

119894 inv120572 = tan120572 minus 120572


119909 = 119903 times sin (Φ) minus (1198861

sin 120574 + 119903120588


119910 = 119903 times cos (Φ) minus (1198861

sin 120574 + 119903120588


120572 le 120574 le120587

2

(2)

where Φ = (1198861 tan 120574 + 119887

1)119903 119903

120588= 119888lowast119898(1 minus sin120572) 119886

1=


120588 and 119887

1= 1205871198984 + ℎlowast

119886119898 tan120572 + 119903

120588cos120572



119887 shear





119887 shear




119880ℎ=

1198652

2119896ℎ

119880119887=

1198652

2119896119887

119880119904=

1198652

2119896119904

119880119886=

1198652

2119896119886

119880119891=

1198652

2119896119891

(3)




119880 =1198652

2119896

= 119880ℎ+ 119880

1198871+ 119880

1199041+ 119880

1198861+ 119880

1198911+ 119880

1198872+ 119880

1199042+ 119880

1198862+ 119880

1198912

=1198652

2(

1

119896ℎ

+1

1198961198871

+1

1198961199041

+1

1198961198861

+1

1198961198911

+1

1198961198872

+1

1198961199042

+1

1198961198862

+1

1198961198912

)

(4)



119896 = 1 times (1

119896ℎ

+1

1198961198871

+1

1198961199041

+1

1198961198861

+1

1198961198911

+1

1198961198872

+1

1198961199042

+1

1198961198862

+1

1198961198912

)

minus1

(5)


11989612

=

119895

sum119894=1

119896119894 (119895 = 1 2) (6)




119896ℎ=

120587119864119871

4 (1 minus V2) (7)


1

119896119891

=cos2120573119864119871

119871lowast(119906119891

119878119891

)

2

+ 119872lowast (119906119891

119878119891


(8)




F

y

xO

F

120573

120573

120575x

120575120575y



119880119886=

1198652

2119896119886

= int119910119862

119910119863

1198652119886

21198641198601199101

d1199101+ int

119910120573

119910119862

1198652119886

21198641198601199102

d1199102

119880119887=

1198652

2119896119887

= int119910119862

119910119863

1198722

1

21198641198681199101

d1199101+ int

119910120573

119910119862

1198722

2

21198641198681199102

d1199102

119880119904=

1198652

2119896119904

= int119910119862

119910119863

121198652119887

21198661198601199101

d1199101+ int

119910120573

119910119862

121198652119887

21198661198601199102

d1199102

(9)

where 119865119887= 119865 cos120573 119865

119886= 119865 sin120573 119909






1199101 119860

1199101 1198681199102 and 119860

1199102represent


1= 119865

119887(119910120573minus119910

1) minus 119865

119886119909120573and 119872

2=

119865119887(119910120573minus 119910

2) minus 119865


119887and 119865

119886on



120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 119860

1199101 and 119860

1199102


1

119896119886

= int120572

1205872

sin2120573119864119860

1199101

d1199101

d120574d120574 + int

120573

120591119888

sin2120573119864119860

1199102

d1199102

d120591d120591 (10)

1

119896119887

= int120572

1205872

[cos120573 (119910120573minus 119910

1) minus 119909

120573sin120573]

2

1198641198681199101

d1199101

d120574d120574

+ int120573

120591119888

(cos120573 (119910120573minus 119910

2) minus 119909

120573sin120573)

2

1198641198681199102

d1199102

d120591d120591

(11)

1

119896119904

= int120572

1205872

12cos2120573119866119860

1199101

d1199101

d120574d120574 + int

120573

120591119888

12cos2120573119866119860

1199102

d1199102

d120591d120591 (12)

where d1199101d120574 and d119910



119909and 120575

119910


120575 = 120575119909cos120573 + 120575

119910sin120573 (13)


119896119905=

119865

120575 (14)


0 10 20 302

4

6

TMIM

FEM


TVM

S (times10

8N

m)

(a)

5

4

3

20 4 8 12

TMIM

FEM


TVM

S (times10

8N

m)

(b)


G

Q

PB

x

yo

C

D

C

Dq = q1

hqhB

q1

120595υ

(a)

υ

G

Q

B

υ

Px

yo

C

D

C

D

120595

q = q1max + q2

hqq2

hB

hBymax

q1max

(b)


1max



119896 =1

(1119896ℎ) + (1119896

1199051) + (1119896

1199052) (15)







119886






119896 = 1 times (1

119896ℎ

+1

1198961198871

+1

1198961199041

+1

1198961198861

+1

1198961198911

+1

119896119887 crack

+1

119896119904 crack

+1

1198961198862

+1

1198961198912

)

minus1

(16)

where 119896119887 crack and 119896




119868 =

1

12(ℎ119902+ 119909)

3

119871 119910119875le 119910 le 119910

119866

1

12(2119909)3119871 =

2

31199093119871 119910 lt 119910

119875or 119910 gt 119910

119866

119860 =

(ℎ119902+ 119909) 119871 119910

119875le 119910 le 119910

119866

2119909119871 119910 lt 119910119875or 119910 gt 119910

119866

(17)




ℎ119902= 119909

119876minus 119902

1sin 120592 (18)


1max = 119909119876 sin 120592 Then the




2is 119902






119868 =

2

31199093119871 119910 lt 119910max

1

12[119909 minus

(119910 minus 119910max) ℎ119902

119910119875minus 119910max

]

3

119871 119910max le 119910 lt 119910119875

1

12(119909 minus ℎ

119902)3

119871 119910119875le 119910 le 119910

119866

0 119910 gt 119910119866

119860 =

2119909119871 119910 lt 119910max

[119909 minus(119910 minus 119910max) ℎ119902

119910119875minus 119910max

]119871 119910max le 119910 lt 119910119875

(119909 minus ℎ119902) 119871 119910

119875le 119910 le 119910

119866

0 119910 gt 119910119866

(19)


119902= 119902

2sin 120592



119896 = int119871

0

119896 (119911) (20)




119902 (119911) = 1199020radic

119911 + 119871119888minus 119871

119871119888

119911 isin [119871 minus 119871119888 119871]

119902 (119911) = 0 119911 isin [0 119871 minus 119871119888]

(22)





119902 (119911) = radic 11990220minus 1199022

119890

119871119911 + 1199022

119890 (23)



Involute

Involute

Yes

Yes

Yes

No

No

No

No

Yes

Transition curve

Involute

Involute

Yes

No

No

No

Yes

Yes

No

Yes

Yes

No

Yes

Yes

Yes

No

No

No

Yes

No

Yes

No

Yes

No

No

Yes

Crack parameters

Transition curve

Transition curve

Transition curve

(q 120592 120595)

q le q1max


y120573 le yG

y le yG

y lt ymax

ymax le y le yP

yP le y le yG

yG le yB

y120573 le yG

y le yG

B2 =2

3x3L

B3 =1

12[x minus

(y minus ymax)hq

yP minus ymax]3

L

B4 =1

12(x minus hq)

3L

C1 = (hq + x)L

C2 = 2xL


yP minus ymax]L

C4 = (x minus hq)L

ymax le y lt yP

yltymax


kb crack ks crack

G isin CD

G isin CD

Iy1 = B3

Ay1 = C3

Iy1 = B4

Ay1 = C4

Iy1 = 0

Ay1 = 0

Iy2 = 0

Ay2 = 0

Iy1 = B2

Ay1 = C2

Iy1 = B3

Ay1 = C3

Iy1 = B4

Ay1 = C4

Iy2 = 0

Ay2 = 0Iy2 = B4

Ay2 = C4

Iy1 = B2

Ay1 = C2

Iy1 = B1

Ay1 = C1

Iy1 = B1

Ay1 = C1

Iy1 = B2Ay1 = C2

Iy1 = B2

Ay1 = C2

Iy2 = B2

Ay2 = C2

Iy2 = B2

Ay2 = C2

Iy2 = B1

Ay1 = C1

B1 =1

12(hq + x)3L





1and119874


tersΩ1andΩ


1198871and 119903

1198872are the



12(119905) = 0 119888

12(119905) rep-


12defined as

12059512

= minus120572 + 120572


120572 + 12057212


(24)


A

Az

L

z

q0

120592

y998400

x998400

o998400

dz

(a)

q(z)

k(z)

120592

dz

(b)

z

L

Lc

q(z)q0

z

f(z)

y998400

o998400

A-A

dz

(c)


qe

y998400

o998400

L

z

q0q(z)

Lc


where 12057212

(0 le 12057212


12= 0


120590 = 1 Ω

1 Counterclockwise


(25)

y1

120579y1

rb1

120579z112057212

O1

k12(t)

e12(t)

c12(t)

x1

120579x1

12059512

y2

120579y2

rb2

120579z2 x2

120579x2

Gear 2 (gear)

Gear 1 (pinion)

O2

Ω2

Ω1





11989811minus 119896

1211990112

(119905) sin12059512

minus 11988812

(119905) 12

(119905) sin12059512

= 0

1198981

1199101+ 119896

1211990112

(119905) cos12059512

+ 11988812

(119905) 12

(119905) cos12059512

= 0

11989811= 0

1198681199091

1205791199091

+ 1198681199111Ω1

1205791199101

= 0

1198681199101

1205791199101

minus 1198681199111Ω1

1205791199091

= 0

1198681199111

1205791199111

+ 120590 times 11990311988711198961211990112

(119905) + 120590 times 119903119887111988812

(119905) 12

(119905) = 1198791

11989822+ 119896

1211990112

(119905) sin12059512

+ 11988812

(119905) 12

(119905) sin12059512

= 0

1198982

1199102minus 119896

1211990112

(119905) cos12059512

minus 11988812

(119905) 12

(119905) cos12059512

= 0

11989822= 0

1198681199092

1205791199092

+ 1198681199112Ω2

1205791199102

= 0

1198681199102

1205791199102

minus 1198681199112Ω2

1205791199092

= 0

1198681199112

1205791199112

+ 120590 times 11990311988721198961211990112

(119905) + 120590 times 119903119887211988812

(119905) 12

(119905) = 1198792

(26)

where 1198981and 119898


1199091 1198681199101

1198681199111 1198681199092 1198681199102 and 119868



2



11990112

(119905) = (minus1199091sin120595

12+ 119909

2sin120595

12+ 119910

1cos120595

12minus 119910

2cos120595

12

+120590 times 11990311988711205791199111

+ 120590 times 11990311988721205791199112) minus 119890

12(119905)

(27)


M12X12

+ (C12

+ G12) X

12+ K

12X12

= F12 (28)


12K12

C12 and G






Mu + (C + G) u + Ku = Fu (29)



Shaft 2

Shaft 1 z

x

y

Bearing 1 Bearing 2

Bearing 3 Bearing 4

Gear 1 (node 8)

Gear 2 (node 22)





1=







1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20


119909= 119868119910(kgsdotmm2) 119868


1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20


119909119909(MNm) 119896

119910119910(MNm) 119896

119911119911(MNm) 119896

120579119909120579119909(MNsdotmrad) 119896

120579119910120579119910(MNsdotmrad)

All 200 200 200 01 01

100

0

minus100

002 006 01 014 018 022

Time (s)

a(m

s2)

(a)

002 006 01 014 018 022

Time (s)

200

0

minus200a(m

s2) Δt = 00818 s

(b)

200

100

0

minus100

0032 0034 0036

Time (s)

a(m

s2)


(c)









2= 12Hz So




Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()


1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

119871119888= 119871

119902119890= 119902

0= q

1199021max= 321mm

2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295

3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308

4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321

5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334

6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347

7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360

8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372

9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385

10 18 119902 = 116 21 (C) 40 119902 = 257

11 (B) 20 119902 = 128 22 42 119902 = 270

fe

2fe3fe

4fe

5fe

6fe

7fe

8fe

9fe

10fe

11fe

12fe

13fe

14fe15fe

16fe

0 5 10 15

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

30

20

10

0


times103

(a)

04

02

04500 4600 4700

5fe

Δf = 12Hz

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)


(b)


119890 16119891




1=










F

Crack

120573







RMS = radic1

119873

119873

sum119899=1

(119909 (119899) minus 119909)2 119909 =1

119873

119873

sum119899=1

119909 (119899) (30)



119873

119899=1(119909 (119899) minus 119909)4

[1119873sum119873

119899=1(119909 (119899) minus 119909)2]

2 (31)





RMShealthtimes 100

(32)





= 119871 =20mm and 119902

119890= 119902




4

3

20 4 8 12

TVM

S (times10

8N

m)

1A1B

2A2B


(a)

4

3

2

10 4 8 12

TVM

S (times10

8N

m)


1C1D

2C2D

(b)


AB

CD

300

200

100

0

minus100

minus200

a(m

s2)

0032 0034 0036

Time (s)

(a)

AB

CD

1

05

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


Kurtosis

25

15

5

minus50 20 40 60

Crack depth ()

r Kur

tosi

sr R

MS

()

RMS

(a)

100

50

0

Rela

tive c

hang

e rat

io (

)

Kurtosis

0 20 40 60

Crack depth ()

RMS

(b)




Crackcase

Crack width()

Crack size(mm)

Crackcase


caseCrack width


1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902

119890= 0 119871

119888= 8 17 80 119902

119890= 0 119871

119888= 16

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

1199020= 3mm

2 5 119902119890= 0 119871119888 = 1 10 45 119902

119890= 0 119871

119888= 9 18 85 119902

119890= 0 119871

119888= 17

3 10 119902119890= 0 119871

119888= 2 11 (C) 50 119902

119890= 0 119871

119888= 10 19 90 119902

119890= 0 119871

119888= 18

4 15 119902119890= 0 119871

119888= 3 12 55 119902

119890= 0 119871119888 = 11 20 95 119902

119890= 0 119871

119888= 19

5 20 119902119890= 0 119871

119888= 4 13 60 119902

119890= 0 119871

119888= 12 21 (D) 100 119902

119890= 0 119871

119888= 20

6 (B) 25 119902119890= 0 119871

119888= 5 14 65 119902

119890= 0 119871

119888= 13 22 (E) 120 119902

119890= 3 119871

119888= 20

7 30 119902119890= 0 119871

119888= 6 15 70 119902

119890= 0 119871

119888= 14

8 35 119902119890= 0 119871

119888= 7 16 75 119902

119890= 0 119871

119888= 15

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

ABC

DE
















200

100

0

minus100

a(m

s2)

0032 0034 0036

Time (s)

ABC

DE

(a)

4500 4600 4700

04

03

02

01

0

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

ABC

DE

(b)


3

2

1

0

0 40 80 120

Crack width ()

r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

100

50

0

Relat

ive c

hang

e rat

io (

)

0 40 80 120

Crack width ()

Kurtosis

RMS

(b)


119890= 119902

0= 3mm)



120592 = 45∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65

4 30 8 50 12






45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1A1B

2A2B

(a)

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1C1D

2C2D

(b)


200

100

0

minus100

a(m

s2 )

0032 0034 0036

Time (s)

AB

CD

(a)

AB

CD

04

02

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


3

15

015 25 35 45 55 65


r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

180

140

100

Rela

tive c

hang

e rat

io (

)

15 25 35 45 55 65


Kurtosis

RMS

(b)



4

3

2

10 4 8 12


TVM

S (times10

8N

m)

1A1B1C1D

1E1F1G

(a)

4

3

2

10 4 8 12


TVM

S (times10

8N

m)

2A2B2C2D

2E2F2G

(b)





1 (B) 0 8 35 15 (E) 70

Ω1 = 1000 revmin120595 = 35∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 5 9 40 16 753 10 10 45 17 (F) 80

4 15 11 50 18 855 20 12 55 19 (G) 90

6 25 13 (D) 60

7 (C) 30 14 65











400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)







80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










F

y

xO

F

120573

120573

120575x

120575120575y



119880119886=

1198652

2119896119886

= int119910119862

119910119863

1198652119886

21198641198601199101

d1199101+ int

119910120573

119910119862

1198652119886

21198641198601199102

d1199102

119880119887=

1198652

2119896119887

= int119910119862

119910119863

1198722

1

21198641198681199101

d1199101+ int

119910120573

119910119862

1198722

2

21198641198681199102

d1199102

119880119904=

1198652

2119896119904

= int119910119862

119910119863

121198652119887

21198661198601199101

d1199101+ int

119910120573

119910119862

121198652119887

21198661198601199102

d1199102

(9)

where 119865119887= 119865 cos120573 119865

119886= 119865 sin120573 119909






1199101 119860

1199101 1198681199102 and 119860

1199102represent


1= 119865

119887(119910120573minus119910

1) minus 119865

119886119909120573and 119872

2=

119865119887(119910120573minus 119910

2) minus 119865


119887and 119865

119886on



120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 119860

1199101 and 119860

1199102


1

119896119886

= int120572

1205872

sin2120573119864119860

1199101

d1199101

d120574d120574 + int

120573

120591119888

sin2120573119864119860

1199102

d1199102

d120591d120591 (10)

1

119896119887

= int120572

1205872

[cos120573 (119910120573minus 119910

1) minus 119909

120573sin120573]

2

1198641198681199101

d1199101

d120574d120574

+ int120573

120591119888

(cos120573 (119910120573minus 119910

2) minus 119909

120573sin120573)

2

1198641198681199102

d1199102

d120591d120591

(11)

1

119896119904

= int120572

1205872

12cos2120573119866119860

1199101

d1199101

d120574d120574 + int

120573

120591119888

12cos2120573119866119860

1199102

d1199102

d120591d120591 (12)

where d1199101d120574 and d119910



119909and 120575

119910


120575 = 120575119909cos120573 + 120575

119910sin120573 (13)


119896119905=

119865

120575 (14)


0 10 20 302

4

6

TMIM

FEM


TVM

S (times10

8N

m)

(a)

5

4

3

20 4 8 12

TMIM

FEM


TVM

S (times10

8N

m)

(b)


G

Q

PB

x

yo

C

D

C

Dq = q1

hqhB

q1

120595υ

(a)

υ

G

Q

B

υ

Px

yo

C

D

C

D

120595

q = q1max + q2

hqq2

hB

hBymax

q1max

(b)


1max



119896 =1

(1119896ℎ) + (1119896

1199051) + (1119896

1199052) (15)







119886






119896 = 1 times (1

119896ℎ

+1

1198961198871

+1

1198961199041

+1

1198961198861

+1

1198961198911

+1

119896119887 crack

+1

119896119904 crack

+1

1198961198862

+1

1198961198912

)

minus1

(16)

where 119896119887 crack and 119896




119868 =

1

12(ℎ119902+ 119909)

3

119871 119910119875le 119910 le 119910

119866

1

12(2119909)3119871 =

2

31199093119871 119910 lt 119910

119875or 119910 gt 119910

119866

119860 =

(ℎ119902+ 119909) 119871 119910

119875le 119910 le 119910

119866

2119909119871 119910 lt 119910119875or 119910 gt 119910

119866

(17)




ℎ119902= 119909

119876minus 119902

1sin 120592 (18)


1max = 119909119876 sin 120592 Then the




2is 119902






119868 =

2

31199093119871 119910 lt 119910max

1

12[119909 minus

(119910 minus 119910max) ℎ119902

119910119875minus 119910max

]

3

119871 119910max le 119910 lt 119910119875

1

12(119909 minus ℎ

119902)3

119871 119910119875le 119910 le 119910

119866

0 119910 gt 119910119866

119860 =

2119909119871 119910 lt 119910max

[119909 minus(119910 minus 119910max) ℎ119902

119910119875minus 119910max

]119871 119910max le 119910 lt 119910119875

(119909 minus ℎ119902) 119871 119910

119875le 119910 le 119910

119866

0 119910 gt 119910119866

(19)


119902= 119902

2sin 120592



119896 = int119871

0

119896 (119911) (20)




119902 (119911) = 1199020radic

119911 + 119871119888minus 119871

119871119888

119911 isin [119871 minus 119871119888 119871]

119902 (119911) = 0 119911 isin [0 119871 minus 119871119888]

(22)





119902 (119911) = radic 11990220minus 1199022

119890

119871119911 + 1199022

119890 (23)



Involute

Involute

Yes

Yes

Yes

No

No

No

No

Yes

Transition curve

Involute

Involute

Yes

No

No

No

Yes

Yes

No

Yes

Yes

No

Yes

Yes

Yes

No

No

No

Yes

No

Yes

No

Yes

No

No

Yes

Crack parameters

Transition curve

Transition curve

Transition curve

(q 120592 120595)

q le q1max


y120573 le yG

y le yG

y lt ymax

ymax le y le yP

yP le y le yG

yG le yB

y120573 le yG

y le yG

B2 =2

3x3L

B3 =1

12[x minus

(y minus ymax)hq

yP minus ymax]3

L

B4 =1

12(x minus hq)

3L

C1 = (hq + x)L

C2 = 2xL


yP minus ymax]L

C4 = (x minus hq)L

ymax le y lt yP

yltymax


kb crack ks crack

G isin CD

G isin CD

Iy1 = B3

Ay1 = C3

Iy1 = B4

Ay1 = C4

Iy1 = 0

Ay1 = 0

Iy2 = 0

Ay2 = 0

Iy1 = B2

Ay1 = C2

Iy1 = B3

Ay1 = C3

Iy1 = B4

Ay1 = C4

Iy2 = 0

Ay2 = 0Iy2 = B4

Ay2 = C4

Iy1 = B2

Ay1 = C2

Iy1 = B1

Ay1 = C1

Iy1 = B1

Ay1 = C1

Iy1 = B2Ay1 = C2

Iy1 = B2

Ay1 = C2

Iy2 = B2

Ay2 = C2

Iy2 = B2

Ay2 = C2

Iy2 = B1

Ay1 = C1

B1 =1

12(hq + x)3L





1and119874


tersΩ1andΩ


1198871and 119903

1198872are the



12(119905) = 0 119888

12(119905) rep-


12defined as

12059512

= minus120572 + 120572


120572 + 12057212


(24)


A

Az

L

z

q0

120592

y998400

x998400

o998400

dz

(a)

q(z)

k(z)

120592

dz

(b)

z

L

Lc

q(z)q0

z

f(z)

y998400

o998400

A-A

dz

(c)


qe

y998400

o998400

L

z

q0q(z)

Lc


where 12057212

(0 le 12057212


12= 0


120590 = 1 Ω

1 Counterclockwise


(25)

y1

120579y1

rb1

120579z112057212

O1

k12(t)

e12(t)

c12(t)

x1

120579x1

12059512

y2

120579y2

rb2

120579z2 x2

120579x2

Gear 2 (gear)

Gear 1 (pinion)

O2

Ω2

Ω1





11989811minus 119896

1211990112

(119905) sin12059512

minus 11988812

(119905) 12

(119905) sin12059512

= 0

1198981

1199101+ 119896

1211990112

(119905) cos12059512

+ 11988812

(119905) 12

(119905) cos12059512

= 0

11989811= 0

1198681199091

1205791199091

+ 1198681199111Ω1

1205791199101

= 0

1198681199101

1205791199101

minus 1198681199111Ω1

1205791199091

= 0

1198681199111

1205791199111

+ 120590 times 11990311988711198961211990112

(119905) + 120590 times 119903119887111988812

(119905) 12

(119905) = 1198791

11989822+ 119896

1211990112

(119905) sin12059512

+ 11988812

(119905) 12

(119905) sin12059512

= 0

1198982

1199102minus 119896

1211990112

(119905) cos12059512

minus 11988812

(119905) 12

(119905) cos12059512

= 0

11989822= 0

1198681199092

1205791199092

+ 1198681199112Ω2

1205791199102

= 0

1198681199102

1205791199102

minus 1198681199112Ω2

1205791199092

= 0

1198681199112

1205791199112

+ 120590 times 11990311988721198961211990112

(119905) + 120590 times 119903119887211988812

(119905) 12

(119905) = 1198792

(26)

where 1198981and 119898


1199091 1198681199101

1198681199111 1198681199092 1198681199102 and 119868



2



11990112

(119905) = (minus1199091sin120595

12+ 119909

2sin120595

12+ 119910

1cos120595

12minus 119910

2cos120595

12

+120590 times 11990311988711205791199111

+ 120590 times 11990311988721205791199112) minus 119890

12(119905)

(27)


M12X12

+ (C12

+ G12) X

12+ K

12X12

= F12 (28)


12K12

C12 and G






Mu + (C + G) u + Ku = Fu (29)



Shaft 2

Shaft 1 z

x

y

Bearing 1 Bearing 2

Bearing 3 Bearing 4

Gear 1 (node 8)

Gear 2 (node 22)





1=







1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20


119909= 119868119910(kgsdotmm2) 119868


1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20


119909119909(MNm) 119896

119910119910(MNm) 119896

119911119911(MNm) 119896

120579119909120579119909(MNsdotmrad) 119896

120579119910120579119910(MNsdotmrad)

All 200 200 200 01 01

100

0

minus100

002 006 01 014 018 022

Time (s)

a(m

s2)

(a)

002 006 01 014 018 022

Time (s)

200

0

minus200a(m

s2) Δt = 00818 s

(b)

200

100

0

minus100

0032 0034 0036

Time (s)

a(m

s2)


(c)









2= 12Hz So




Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()


1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

119871119888= 119871

119902119890= 119902

0= q

1199021max= 321mm

2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295

3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308

4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321

5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334

6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347

7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360

8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372

9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385

10 18 119902 = 116 21 (C) 40 119902 = 257

11 (B) 20 119902 = 128 22 42 119902 = 270

fe

2fe3fe

4fe

5fe

6fe

7fe

8fe

9fe

10fe

11fe

12fe

13fe

14fe15fe

16fe

0 5 10 15

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

30

20

10

0


times103

(a)

04

02

04500 4600 4700

5fe

Δf = 12Hz

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)


(b)


119890 16119891




1=










F

Crack

120573







RMS = radic1

119873

119873

sum119899=1

(119909 (119899) minus 119909)2 119909 =1

119873

119873

sum119899=1

119909 (119899) (30)



119873

119899=1(119909 (119899) minus 119909)4

[1119873sum119873

119899=1(119909 (119899) minus 119909)2]

2 (31)





RMShealthtimes 100

(32)





= 119871 =20mm and 119902

119890= 119902




4

3

20 4 8 12

TVM

S (times10

8N

m)

1A1B

2A2B


(a)

4

3

2

10 4 8 12

TVM

S (times10

8N

m)


1C1D

2C2D

(b)


AB

CD

300

200

100

0

minus100

minus200

a(m

s2)

0032 0034 0036

Time (s)

(a)

AB

CD

1

05

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


Kurtosis

25

15

5

minus50 20 40 60

Crack depth ()

r Kur

tosi

sr R

MS

()

RMS

(a)

100

50

0

Rela

tive c

hang

e rat

io (

)

Kurtosis

0 20 40 60

Crack depth ()

RMS

(b)




Crackcase

Crack width()

Crack size(mm)

Crackcase


caseCrack width


1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902

119890= 0 119871

119888= 8 17 80 119902

119890= 0 119871

119888= 16

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

1199020= 3mm

2 5 119902119890= 0 119871119888 = 1 10 45 119902

119890= 0 119871

119888= 9 18 85 119902

119890= 0 119871

119888= 17

3 10 119902119890= 0 119871

119888= 2 11 (C) 50 119902

119890= 0 119871

119888= 10 19 90 119902

119890= 0 119871

119888= 18

4 15 119902119890= 0 119871

119888= 3 12 55 119902

119890= 0 119871119888 = 11 20 95 119902

119890= 0 119871

119888= 19

5 20 119902119890= 0 119871

119888= 4 13 60 119902

119890= 0 119871

119888= 12 21 (D) 100 119902

119890= 0 119871

119888= 20

6 (B) 25 119902119890= 0 119871

119888= 5 14 65 119902

119890= 0 119871

119888= 13 22 (E) 120 119902

119890= 3 119871

119888= 20

7 30 119902119890= 0 119871

119888= 6 15 70 119902

119890= 0 119871

119888= 14

8 35 119902119890= 0 119871

119888= 7 16 75 119902

119890= 0 119871

119888= 15

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

ABC

DE
















200

100

0

minus100

a(m

s2)

0032 0034 0036

Time (s)

ABC

DE

(a)

4500 4600 4700

04

03

02

01

0

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

ABC

DE

(b)


3

2

1

0

0 40 80 120

Crack width ()

r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

100

50

0

Relat

ive c

hang

e rat

io (

)

0 40 80 120

Crack width ()

Kurtosis

RMS

(b)


119890= 119902

0= 3mm)



120592 = 45∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65

4 30 8 50 12






45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1A1B

2A2B

(a)

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1C1D

2C2D

(b)


200

100

0

minus100

a(m

s2 )

0032 0034 0036

Time (s)

AB

CD

(a)

AB

CD

04

02

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


3

15

015 25 35 45 55 65


r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

180

140

100

Rela

tive c

hang

e rat

io (

)

15 25 35 45 55 65


Kurtosis

RMS

(b)



4

3

2

10 4 8 12


TVM

S (times10

8N

m)

1A1B1C1D

1E1F1G

(a)

4

3

2

10 4 8 12


TVM

S (times10

8N

m)

2A2B2C2D

2E2F2G

(b)





1 (B) 0 8 35 15 (E) 70

Ω1 = 1000 revmin120595 = 35∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 5 9 40 16 753 10 10 45 17 (F) 80

4 15 11 50 18 855 20 12 55 19 (G) 90

6 25 13 (D) 60

7 (C) 30 14 65











400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)







80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 302

4

6

TMIM

FEM


TVM

S (times10

8N

m)

(a)

5

4

3

20 4 8 12

TMIM

FEM


TVM

S (times10

8N

m)

(b)


G

Q

PB

x

yo

C

D

C

Dq = q1

hqhB

q1

120595υ

(a)

υ

G

Q

B

υ

Px

yo

C

D

C

D

120595

q = q1max + q2

hqq2

hB

hBymax

q1max

(b)


1max



119896 =1

(1119896ℎ) + (1119896

1199051) + (1119896

1199052) (15)







119886






119896 = 1 times (1

119896ℎ

+1

1198961198871

+1

1198961199041

+1

1198961198861

+1

1198961198911

+1

119896119887 crack

+1

119896119904 crack

+1

1198961198862

+1

1198961198912

)

minus1

(16)

where 119896119887 crack and 119896




119868 =

1

12(ℎ119902+ 119909)

3

119871 119910119875le 119910 le 119910

119866

1

12(2119909)3119871 =

2

31199093119871 119910 lt 119910

119875or 119910 gt 119910

119866

119860 =

(ℎ119902+ 119909) 119871 119910

119875le 119910 le 119910

119866

2119909119871 119910 lt 119910119875or 119910 gt 119910

119866

(17)




ℎ119902= 119909

119876minus 119902

1sin 120592 (18)


1max = 119909119876 sin 120592 Then the




2is 119902






119868 =

2

31199093119871 119910 lt 119910max

1

12[119909 minus

(119910 minus 119910max) ℎ119902

119910119875minus 119910max

]

3

119871 119910max le 119910 lt 119910119875

1

12(119909 minus ℎ

119902)3

119871 119910119875le 119910 le 119910

119866

0 119910 gt 119910119866

119860 =

2119909119871 119910 lt 119910max

[119909 minus(119910 minus 119910max) ℎ119902

119910119875minus 119910max

]119871 119910max le 119910 lt 119910119875

(119909 minus ℎ119902) 119871 119910

119875le 119910 le 119910

119866

0 119910 gt 119910119866

(19)


119902= 119902

2sin 120592



119896 = int119871

0

119896 (119911) (20)




119902 (119911) = 1199020radic

119911 + 119871119888minus 119871

119871119888

119911 isin [119871 minus 119871119888 119871]

119902 (119911) = 0 119911 isin [0 119871 minus 119871119888]

(22)





119902 (119911) = radic 11990220minus 1199022

119890

119871119911 + 1199022

119890 (23)



Involute

Involute

Yes

Yes

Yes

No

No

No

No

Yes

Transition curve

Involute

Involute

Yes

No

No

No

Yes

Yes

No

Yes

Yes

No

Yes

Yes

Yes

No

No

No

Yes

No

Yes

No

Yes

No

No

Yes

Crack parameters

Transition curve

Transition curve

Transition curve

(q 120592 120595)

q le q1max


y120573 le yG

y le yG

y lt ymax

ymax le y le yP

yP le y le yG

yG le yB

y120573 le yG

y le yG

B2 =2

3x3L

B3 =1

12[x minus

(y minus ymax)hq

yP minus ymax]3

L

B4 =1

12(x minus hq)

3L

C1 = (hq + x)L

C2 = 2xL


yP minus ymax]L

C4 = (x minus hq)L

ymax le y lt yP

yltymax


kb crack ks crack

G isin CD

G isin CD

Iy1 = B3

Ay1 = C3

Iy1 = B4

Ay1 = C4

Iy1 = 0

Ay1 = 0

Iy2 = 0

Ay2 = 0

Iy1 = B2

Ay1 = C2

Iy1 = B3

Ay1 = C3

Iy1 = B4

Ay1 = C4

Iy2 = 0

Ay2 = 0Iy2 = B4

Ay2 = C4

Iy1 = B2

Ay1 = C2

Iy1 = B1

Ay1 = C1

Iy1 = B1

Ay1 = C1

Iy1 = B2Ay1 = C2

Iy1 = B2

Ay1 = C2

Iy2 = B2

Ay2 = C2

Iy2 = B2

Ay2 = C2

Iy2 = B1

Ay1 = C1

B1 =1

12(hq + x)3L





1and119874


tersΩ1andΩ


1198871and 119903

1198872are the



12(119905) = 0 119888

12(119905) rep-


12defined as

12059512

= minus120572 + 120572


120572 + 12057212


(24)


A

Az

L

z

q0

120592

y998400

x998400

o998400

dz

(a)

q(z)

k(z)

120592

dz

(b)

z

L

Lc

q(z)q0

z

f(z)

y998400

o998400

A-A

dz

(c)


qe

y998400

o998400

L

z

q0q(z)

Lc


where 12057212

(0 le 12057212


12= 0


120590 = 1 Ω

1 Counterclockwise


(25)

y1

120579y1

rb1

120579z112057212

O1

k12(t)

e12(t)

c12(t)

x1

120579x1

12059512

y2

120579y2

rb2

120579z2 x2

120579x2

Gear 2 (gear)

Gear 1 (pinion)

O2

Ω2

Ω1





11989811minus 119896

1211990112

(119905) sin12059512

minus 11988812

(119905) 12

(119905) sin12059512

= 0

1198981

1199101+ 119896

1211990112

(119905) cos12059512

+ 11988812

(119905) 12

(119905) cos12059512

= 0

11989811= 0

1198681199091

1205791199091

+ 1198681199111Ω1

1205791199101

= 0

1198681199101

1205791199101

minus 1198681199111Ω1

1205791199091

= 0

1198681199111

1205791199111

+ 120590 times 11990311988711198961211990112

(119905) + 120590 times 119903119887111988812

(119905) 12

(119905) = 1198791

11989822+ 119896

1211990112

(119905) sin12059512

+ 11988812

(119905) 12

(119905) sin12059512

= 0

1198982

1199102minus 119896

1211990112

(119905) cos12059512

minus 11988812

(119905) 12

(119905) cos12059512

= 0

11989822= 0

1198681199092

1205791199092

+ 1198681199112Ω2

1205791199102

= 0

1198681199102

1205791199102

minus 1198681199112Ω2

1205791199092

= 0

1198681199112

1205791199112

+ 120590 times 11990311988721198961211990112

(119905) + 120590 times 119903119887211988812

(119905) 12

(119905) = 1198792

(26)

where 1198981and 119898


1199091 1198681199101

1198681199111 1198681199092 1198681199102 and 119868



2



11990112

(119905) = (minus1199091sin120595

12+ 119909

2sin120595

12+ 119910

1cos120595

12minus 119910

2cos120595

12

+120590 times 11990311988711205791199111

+ 120590 times 11990311988721205791199112) minus 119890

12(119905)

(27)


M12X12

+ (C12

+ G12) X

12+ K

12X12

= F12 (28)


12K12

C12 and G






Mu + (C + G) u + Ku = Fu (29)



Shaft 2

Shaft 1 z

x

y

Bearing 1 Bearing 2

Bearing 3 Bearing 4

Gear 1 (node 8)

Gear 2 (node 22)





1=







1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20


119909= 119868119910(kgsdotmm2) 119868


1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20


119909119909(MNm) 119896

119910119910(MNm) 119896

119911119911(MNm) 119896

120579119909120579119909(MNsdotmrad) 119896

120579119910120579119910(MNsdotmrad)

All 200 200 200 01 01

100

0

minus100

002 006 01 014 018 022

Time (s)

a(m

s2)

(a)

002 006 01 014 018 022

Time (s)

200

0

minus200a(m

s2) Δt = 00818 s

(b)

200

100

0

minus100

0032 0034 0036

Time (s)

a(m

s2)


(c)









2= 12Hz So




Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()


1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

119871119888= 119871

119902119890= 119902

0= q

1199021max= 321mm

2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295

3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308

4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321

5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334

6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347

7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360

8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372

9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385

10 18 119902 = 116 21 (C) 40 119902 = 257

11 (B) 20 119902 = 128 22 42 119902 = 270

fe

2fe3fe

4fe

5fe

6fe

7fe

8fe

9fe

10fe

11fe

12fe

13fe

14fe15fe

16fe

0 5 10 15

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

30

20

10

0


times103

(a)

04

02

04500 4600 4700

5fe

Δf = 12Hz

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)


(b)


119890 16119891




1=










F

Crack

120573







RMS = radic1

119873

119873

sum119899=1

(119909 (119899) minus 119909)2 119909 =1

119873

119873

sum119899=1

119909 (119899) (30)



119873

119899=1(119909 (119899) minus 119909)4

[1119873sum119873

119899=1(119909 (119899) minus 119909)2]

2 (31)





RMShealthtimes 100

(32)





= 119871 =20mm and 119902

119890= 119902




4

3

20 4 8 12

TVM

S (times10

8N

m)

1A1B

2A2B


(a)

4

3

2

10 4 8 12

TVM

S (times10

8N

m)


1C1D

2C2D

(b)


AB

CD

300

200

100

0

minus100

minus200

a(m

s2)

0032 0034 0036

Time (s)

(a)

AB

CD

1

05

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


Kurtosis

25

15

5

minus50 20 40 60

Crack depth ()

r Kur

tosi

sr R

MS

()

RMS

(a)

100

50

0

Rela

tive c

hang

e rat

io (

)

Kurtosis

0 20 40 60

Crack depth ()

RMS

(b)




Crackcase

Crack width()

Crack size(mm)

Crackcase


caseCrack width


1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902

119890= 0 119871

119888= 8 17 80 119902

119890= 0 119871

119888= 16

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

1199020= 3mm

2 5 119902119890= 0 119871119888 = 1 10 45 119902

119890= 0 119871

119888= 9 18 85 119902

119890= 0 119871

119888= 17

3 10 119902119890= 0 119871

119888= 2 11 (C) 50 119902

119890= 0 119871

119888= 10 19 90 119902

119890= 0 119871

119888= 18

4 15 119902119890= 0 119871

119888= 3 12 55 119902

119890= 0 119871119888 = 11 20 95 119902

119890= 0 119871

119888= 19

5 20 119902119890= 0 119871

119888= 4 13 60 119902

119890= 0 119871

119888= 12 21 (D) 100 119902

119890= 0 119871

119888= 20

6 (B) 25 119902119890= 0 119871

119888= 5 14 65 119902

119890= 0 119871

119888= 13 22 (E) 120 119902

119890= 3 119871

119888= 20

7 30 119902119890= 0 119871

119888= 6 15 70 119902

119890= 0 119871

119888= 14

8 35 119902119890= 0 119871

119888= 7 16 75 119902

119890= 0 119871

119888= 15

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

ABC

DE
















200

100

0

minus100

a(m

s2)

0032 0034 0036

Time (s)

ABC

DE

(a)

4500 4600 4700

04

03

02

01

0

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

ABC

DE

(b)


3

2

1

0

0 40 80 120

Crack width ()

r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

100

50

0

Relat

ive c

hang

e rat

io (

)

0 40 80 120

Crack width ()

Kurtosis

RMS

(b)


119890= 119902

0= 3mm)



120592 = 45∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65

4 30 8 50 12






45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1A1B

2A2B

(a)

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1C1D

2C2D

(b)


200

100

0

minus100

a(m

s2 )

0032 0034 0036

Time (s)

AB

CD

(a)

AB

CD

04

02

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


3

15

015 25 35 45 55 65


r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

180

140

100

Rela

tive c

hang

e rat

io (

)

15 25 35 45 55 65


Kurtosis

RMS

(b)



4

3

2

10 4 8 12


TVM

S (times10

8N

m)

1A1B1C1D

1E1F1G

(a)

4

3

2

10 4 8 12


TVM

S (times10

8N

m)

2A2B2C2D

2E2F2G

(b)





1 (B) 0 8 35 15 (E) 70

Ω1 = 1000 revmin120595 = 35∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 5 9 40 16 753 10 10 45 17 (F) 80

4 15 11 50 18 855 20 12 55 19 (G) 90

6 25 13 (D) 60

7 (C) 30 14 65











400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)







80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896 = 1 times (1

119896ℎ

+1

1198961198871

+1

1198961199041

+1

1198961198861

+1

1198961198911

+1

119896119887 crack

+1

119896119904 crack

+1

1198961198862

+1

1198961198912

)

minus1

(16)

where 119896119887 crack and 119896




119868 =

1

12(ℎ119902+ 119909)

3

119871 119910119875le 119910 le 119910

119866

1

12(2119909)3119871 =

2

31199093119871 119910 lt 119910

119875or 119910 gt 119910

119866

119860 =

(ℎ119902+ 119909) 119871 119910

119875le 119910 le 119910

119866

2119909119871 119910 lt 119910119875or 119910 gt 119910

119866

(17)




ℎ119902= 119909

119876minus 119902

1sin 120592 (18)


1max = 119909119876 sin 120592 Then the




2is 119902






119868 =

2

31199093119871 119910 lt 119910max

1

12[119909 minus

(119910 minus 119910max) ℎ119902

119910119875minus 119910max

]

3

119871 119910max le 119910 lt 119910119875

1

12(119909 minus ℎ

119902)3

119871 119910119875le 119910 le 119910

119866

0 119910 gt 119910119866

119860 =

2119909119871 119910 lt 119910max

[119909 minus(119910 minus 119910max) ℎ119902

119910119875minus 119910max

]119871 119910max le 119910 lt 119910119875

(119909 minus ℎ119902) 119871 119910

119875le 119910 le 119910

119866

0 119910 gt 119910119866

(19)


119902= 119902

2sin 120592



119896 = int119871

0

119896 (119911) (20)




119902 (119911) = 1199020radic

119911 + 119871119888minus 119871

119871119888

119911 isin [119871 minus 119871119888 119871]

119902 (119911) = 0 119911 isin [0 119871 minus 119871119888]

(22)





119902 (119911) = radic 11990220minus 1199022

119890

119871119911 + 1199022

119890 (23)



Involute

Involute

Yes

Yes

Yes

No

No

No

No

Yes

Transition curve

Involute

Involute

Yes

No

No

No

Yes

Yes

No

Yes

Yes

No

Yes

Yes

Yes

No

No

No

Yes

No

Yes

No

Yes

No

No

Yes

Crack parameters

Transition curve

Transition curve

Transition curve

(q 120592 120595)

q le q1max


y120573 le yG

y le yG

y lt ymax

ymax le y le yP

yP le y le yG

yG le yB

y120573 le yG

y le yG

B2 =2

3x3L

B3 =1

12[x minus

(y minus ymax)hq

yP minus ymax]3

L

B4 =1

12(x minus hq)

3L

C1 = (hq + x)L

C2 = 2xL


yP minus ymax]L

C4 = (x minus hq)L

ymax le y lt yP

yltymax


kb crack ks crack

G isin CD

G isin CD

Iy1 = B3

Ay1 = C3

Iy1 = B4

Ay1 = C4

Iy1 = 0

Ay1 = 0

Iy2 = 0

Ay2 = 0

Iy1 = B2

Ay1 = C2

Iy1 = B3

Ay1 = C3

Iy1 = B4

Ay1 = C4

Iy2 = 0

Ay2 = 0Iy2 = B4

Ay2 = C4

Iy1 = B2

Ay1 = C2

Iy1 = B1

Ay1 = C1

Iy1 = B1

Ay1 = C1

Iy1 = B2Ay1 = C2

Iy1 = B2

Ay1 = C2

Iy2 = B2

Ay2 = C2

Iy2 = B2

Ay2 = C2

Iy2 = B1

Ay1 = C1

B1 =1

12(hq + x)3L





1and119874


tersΩ1andΩ


1198871and 119903

1198872are the



12(119905) = 0 119888

12(119905) rep-


12defined as

12059512

= minus120572 + 120572


120572 + 12057212


(24)


A

Az

L

z

q0

120592

y998400

x998400

o998400

dz

(a)

q(z)

k(z)

120592

dz

(b)

z

L

Lc

q(z)q0

z

f(z)

y998400

o998400

A-A

dz

(c)


qe

y998400

o998400

L

z

q0q(z)

Lc


where 12057212

(0 le 12057212


12= 0


120590 = 1 Ω

1 Counterclockwise


(25)

y1

120579y1

rb1

120579z112057212

O1

k12(t)

e12(t)

c12(t)

x1

120579x1

12059512

y2

120579y2

rb2

120579z2 x2

120579x2

Gear 2 (gear)

Gear 1 (pinion)

O2

Ω2

Ω1





11989811minus 119896

1211990112

(119905) sin12059512

minus 11988812

(119905) 12

(119905) sin12059512

= 0

1198981

1199101+ 119896

1211990112

(119905) cos12059512

+ 11988812

(119905) 12

(119905) cos12059512

= 0

11989811= 0

1198681199091

1205791199091

+ 1198681199111Ω1

1205791199101

= 0

1198681199101

1205791199101

minus 1198681199111Ω1

1205791199091

= 0

1198681199111

1205791199111

+ 120590 times 11990311988711198961211990112

(119905) + 120590 times 119903119887111988812

(119905) 12

(119905) = 1198791

11989822+ 119896

1211990112

(119905) sin12059512

+ 11988812

(119905) 12

(119905) sin12059512

= 0

1198982

1199102minus 119896

1211990112

(119905) cos12059512

minus 11988812

(119905) 12

(119905) cos12059512

= 0

11989822= 0

1198681199092

1205791199092

+ 1198681199112Ω2

1205791199102

= 0

1198681199102

1205791199102

minus 1198681199112Ω2

1205791199092

= 0

1198681199112

1205791199112

+ 120590 times 11990311988721198961211990112

(119905) + 120590 times 119903119887211988812

(119905) 12

(119905) = 1198792

(26)

where 1198981and 119898


1199091 1198681199101

1198681199111 1198681199092 1198681199102 and 119868



2



11990112

(119905) = (minus1199091sin120595

12+ 119909

2sin120595

12+ 119910

1cos120595

12minus 119910

2cos120595

12

+120590 times 11990311988711205791199111

+ 120590 times 11990311988721205791199112) minus 119890

12(119905)

(27)


M12X12

+ (C12

+ G12) X

12+ K

12X12

= F12 (28)


12K12

C12 and G






Mu + (C + G) u + Ku = Fu (29)



Shaft 2

Shaft 1 z

x

y

Bearing 1 Bearing 2

Bearing 3 Bearing 4

Gear 1 (node 8)

Gear 2 (node 22)





1=







1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20


119909= 119868119910(kgsdotmm2) 119868


1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20


119909119909(MNm) 119896

119910119910(MNm) 119896

119911119911(MNm) 119896

120579119909120579119909(MNsdotmrad) 119896

120579119910120579119910(MNsdotmrad)

All 200 200 200 01 01

100

0

minus100

002 006 01 014 018 022

Time (s)

a(m

s2)

(a)

002 006 01 014 018 022

Time (s)

200

0

minus200a(m

s2) Δt = 00818 s

(b)

200

100

0

minus100

0032 0034 0036

Time (s)

a(m

s2)


(c)









2= 12Hz So




Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()


1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

119871119888= 119871

119902119890= 119902

0= q

1199021max= 321mm

2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295

3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308

4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321

5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334

6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347

7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360

8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372

9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385

10 18 119902 = 116 21 (C) 40 119902 = 257

11 (B) 20 119902 = 128 22 42 119902 = 270

fe

2fe3fe

4fe

5fe

6fe

7fe

8fe

9fe

10fe

11fe

12fe

13fe

14fe15fe

16fe

0 5 10 15

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

30

20

10

0


times103

(a)

04

02

04500 4600 4700

5fe

Δf = 12Hz

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)


(b)


119890 16119891




1=










F

Crack

120573







RMS = radic1

119873

119873

sum119899=1

(119909 (119899) minus 119909)2 119909 =1

119873

119873

sum119899=1

119909 (119899) (30)



119873

119899=1(119909 (119899) minus 119909)4

[1119873sum119873

119899=1(119909 (119899) minus 119909)2]

2 (31)





RMShealthtimes 100

(32)





= 119871 =20mm and 119902

119890= 119902




4

3

20 4 8 12

TVM

S (times10

8N

m)

1A1B

2A2B


(a)

4

3

2

10 4 8 12

TVM

S (times10

8N

m)


1C1D

2C2D

(b)


AB

CD

300

200

100

0

minus100

minus200

a(m

s2)

0032 0034 0036

Time (s)

(a)

AB

CD

1

05

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


Kurtosis

25

15

5

minus50 20 40 60

Crack depth ()

r Kur

tosi

sr R

MS

()

RMS

(a)

100

50

0

Rela

tive c

hang

e rat

io (

)

Kurtosis

0 20 40 60

Crack depth ()

RMS

(b)




Crackcase

Crack width()

Crack size(mm)

Crackcase


caseCrack width


1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902

119890= 0 119871

119888= 8 17 80 119902

119890= 0 119871

119888= 16

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

1199020= 3mm

2 5 119902119890= 0 119871119888 = 1 10 45 119902

119890= 0 119871

119888= 9 18 85 119902

119890= 0 119871

119888= 17

3 10 119902119890= 0 119871

119888= 2 11 (C) 50 119902

119890= 0 119871

119888= 10 19 90 119902

119890= 0 119871

119888= 18

4 15 119902119890= 0 119871

119888= 3 12 55 119902

119890= 0 119871119888 = 11 20 95 119902

119890= 0 119871

119888= 19

5 20 119902119890= 0 119871

119888= 4 13 60 119902

119890= 0 119871

119888= 12 21 (D) 100 119902

119890= 0 119871

119888= 20

6 (B) 25 119902119890= 0 119871

119888= 5 14 65 119902

119890= 0 119871

119888= 13 22 (E) 120 119902

119890= 3 119871

119888= 20

7 30 119902119890= 0 119871

119888= 6 15 70 119902

119890= 0 119871

119888= 14

8 35 119902119890= 0 119871

119888= 7 16 75 119902

119890= 0 119871

119888= 15

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

ABC

DE
















200

100

0

minus100

a(m

s2)

0032 0034 0036

Time (s)

ABC

DE

(a)

4500 4600 4700

04

03

02

01

0

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

ABC

DE

(b)


3

2

1

0

0 40 80 120

Crack width ()

r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

100

50

0

Relat

ive c

hang

e rat

io (

)

0 40 80 120

Crack width ()

Kurtosis

RMS

(b)


119890= 119902

0= 3mm)



120592 = 45∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65

4 30 8 50 12






45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1A1B

2A2B

(a)

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1C1D

2C2D

(b)


200

100

0

minus100

a(m

s2 )

0032 0034 0036

Time (s)

AB

CD

(a)

AB

CD

04

02

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


3

15

015 25 35 45 55 65


r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

180

140

100

Rela

tive c

hang

e rat

io (

)

15 25 35 45 55 65


Kurtosis

RMS

(b)



4

3

2

10 4 8 12


TVM

S (times10

8N

m)

1A1B1C1D

1E1F1G

(a)

4

3

2

10 4 8 12


TVM

S (times10

8N

m)

2A2B2C2D

2E2F2G

(b)





1 (B) 0 8 35 15 (E) 70

Ω1 = 1000 revmin120595 = 35∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 5 9 40 16 753 10 10 45 17 (F) 80

4 15 11 50 18 855 20 12 55 19 (G) 90

6 25 13 (D) 60

7 (C) 30 14 65











400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)







80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Involute

Involute

Yes

Yes

Yes

No

No

No

No

Yes

Transition curve

Involute

Involute

Yes

No

No

No

Yes

Yes

No

Yes

Yes

No

Yes

Yes

Yes

No

No

No

Yes

No

Yes

No

Yes

No

No

Yes

Crack parameters

Transition curve

Transition curve

Transition curve

(q 120592 120595)

q le q1max


y120573 le yG

y le yG

y lt ymax

ymax le y le yP

yP le y le yG

yG le yB

y120573 le yG

y le yG

B2 =2

3x3L

B3 =1

12[x minus

(y minus ymax)hq

yP minus ymax]3

L

B4 =1

12(x minus hq)

3L

C1 = (hq + x)L

C2 = 2xL


yP minus ymax]L

C4 = (x minus hq)L

ymax le y lt yP

yltymax


kb crack ks crack

G isin CD

G isin CD

Iy1 = B3

Ay1 = C3

Iy1 = B4

Ay1 = C4

Iy1 = 0

Ay1 = 0

Iy2 = 0

Ay2 = 0

Iy1 = B2

Ay1 = C2

Iy1 = B3

Ay1 = C3

Iy1 = B4

Ay1 = C4

Iy2 = 0

Ay2 = 0Iy2 = B4

Ay2 = C4

Iy1 = B2

Ay1 = C2

Iy1 = B1

Ay1 = C1

Iy1 = B1

Ay1 = C1

Iy1 = B2Ay1 = C2

Iy1 = B2

Ay1 = C2

Iy2 = B2

Ay2 = C2

Iy2 = B2

Ay2 = C2

Iy2 = B1

Ay1 = C1

B1 =1

12(hq + x)3L





1and119874


tersΩ1andΩ


1198871and 119903

1198872are the



12(119905) = 0 119888

12(119905) rep-


12defined as

12059512

= minus120572 + 120572


120572 + 12057212


(24)


A

Az

L

z

q0

120592

y998400

x998400

o998400

dz

(a)

q(z)

k(z)

120592

dz

(b)

z

L

Lc

q(z)q0

z

f(z)

y998400

o998400

A-A

dz

(c)


qe

y998400

o998400

L

z

q0q(z)

Lc


where 12057212

(0 le 12057212


12= 0


120590 = 1 Ω

1 Counterclockwise


(25)

y1

120579y1

rb1

120579z112057212

O1

k12(t)

e12(t)

c12(t)

x1

120579x1

12059512

y2

120579y2

rb2

120579z2 x2

120579x2

Gear 2 (gear)

Gear 1 (pinion)

O2

Ω2

Ω1





11989811minus 119896

1211990112

(119905) sin12059512

minus 11988812

(119905) 12

(119905) sin12059512

= 0

1198981

1199101+ 119896

1211990112

(119905) cos12059512

+ 11988812

(119905) 12

(119905) cos12059512

= 0

11989811= 0

1198681199091

1205791199091

+ 1198681199111Ω1

1205791199101

= 0

1198681199101

1205791199101

minus 1198681199111Ω1

1205791199091

= 0

1198681199111

1205791199111

+ 120590 times 11990311988711198961211990112

(119905) + 120590 times 119903119887111988812

(119905) 12

(119905) = 1198791

11989822+ 119896

1211990112

(119905) sin12059512

+ 11988812

(119905) 12

(119905) sin12059512

= 0

1198982

1199102minus 119896

1211990112

(119905) cos12059512

minus 11988812

(119905) 12

(119905) cos12059512

= 0

11989822= 0

1198681199092

1205791199092

+ 1198681199112Ω2

1205791199102

= 0

1198681199102

1205791199102

minus 1198681199112Ω2

1205791199092

= 0

1198681199112

1205791199112

+ 120590 times 11990311988721198961211990112

(119905) + 120590 times 119903119887211988812

(119905) 12

(119905) = 1198792

(26)

where 1198981and 119898


1199091 1198681199101

1198681199111 1198681199092 1198681199102 and 119868



2



11990112

(119905) = (minus1199091sin120595

12+ 119909

2sin120595

12+ 119910

1cos120595

12minus 119910

2cos120595

12

+120590 times 11990311988711205791199111

+ 120590 times 11990311988721205791199112) minus 119890

12(119905)

(27)


M12X12

+ (C12

+ G12) X

12+ K

12X12

= F12 (28)


12K12

C12 and G






Mu + (C + G) u + Ku = Fu (29)



Shaft 2

Shaft 1 z

x

y

Bearing 1 Bearing 2

Bearing 3 Bearing 4

Gear 1 (node 8)

Gear 2 (node 22)





1=







1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20


119909= 119868119910(kgsdotmm2) 119868


1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20


119909119909(MNm) 119896

119910119910(MNm) 119896

119911119911(MNm) 119896

120579119909120579119909(MNsdotmrad) 119896

120579119910120579119910(MNsdotmrad)

All 200 200 200 01 01

100

0

minus100

002 006 01 014 018 022

Time (s)

a(m

s2)

(a)

002 006 01 014 018 022

Time (s)

200

0

minus200a(m

s2) Δt = 00818 s

(b)

200

100

0

minus100

0032 0034 0036

Time (s)

a(m

s2)


(c)









2= 12Hz So




Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()


1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

119871119888= 119871

119902119890= 119902

0= q

1199021max= 321mm

2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295

3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308

4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321

5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334

6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347

7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360

8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372

9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385

10 18 119902 = 116 21 (C) 40 119902 = 257

11 (B) 20 119902 = 128 22 42 119902 = 270

fe

2fe3fe

4fe

5fe

6fe

7fe

8fe

9fe

10fe

11fe

12fe

13fe

14fe15fe

16fe

0 5 10 15

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

30

20

10

0


times103

(a)

04

02

04500 4600 4700

5fe

Δf = 12Hz

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)


(b)


119890 16119891




1=










F

Crack

120573







RMS = radic1

119873

119873

sum119899=1

(119909 (119899) minus 119909)2 119909 =1

119873

119873

sum119899=1

119909 (119899) (30)



119873

119899=1(119909 (119899) minus 119909)4

[1119873sum119873

119899=1(119909 (119899) minus 119909)2]

2 (31)





RMShealthtimes 100

(32)





= 119871 =20mm and 119902

119890= 119902




4

3

20 4 8 12

TVM

S (times10

8N

m)

1A1B

2A2B


(a)

4

3

2

10 4 8 12

TVM

S (times10

8N

m)


1C1D

2C2D

(b)


AB

CD

300

200

100

0

minus100

minus200

a(m

s2)

0032 0034 0036

Time (s)

(a)

AB

CD

1

05

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


Kurtosis

25

15

5

minus50 20 40 60

Crack depth ()

r Kur

tosi

sr R

MS

()

RMS

(a)

100

50

0

Rela

tive c

hang

e rat

io (

)

Kurtosis

0 20 40 60

Crack depth ()

RMS

(b)




Crackcase

Crack width()

Crack size(mm)

Crackcase


caseCrack width


1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902

119890= 0 119871

119888= 8 17 80 119902

119890= 0 119871

119888= 16

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

1199020= 3mm

2 5 119902119890= 0 119871119888 = 1 10 45 119902

119890= 0 119871

119888= 9 18 85 119902

119890= 0 119871

119888= 17

3 10 119902119890= 0 119871

119888= 2 11 (C) 50 119902

119890= 0 119871

119888= 10 19 90 119902

119890= 0 119871

119888= 18

4 15 119902119890= 0 119871

119888= 3 12 55 119902

119890= 0 119871119888 = 11 20 95 119902

119890= 0 119871

119888= 19

5 20 119902119890= 0 119871

119888= 4 13 60 119902

119890= 0 119871

119888= 12 21 (D) 100 119902

119890= 0 119871

119888= 20

6 (B) 25 119902119890= 0 119871

119888= 5 14 65 119902

119890= 0 119871

119888= 13 22 (E) 120 119902

119890= 3 119871

119888= 20

7 30 119902119890= 0 119871

119888= 6 15 70 119902

119890= 0 119871

119888= 14

8 35 119902119890= 0 119871

119888= 7 16 75 119902

119890= 0 119871

119888= 15

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

ABC

DE
















200

100

0

minus100

a(m

s2)

0032 0034 0036

Time (s)

ABC

DE

(a)

4500 4600 4700

04

03

02

01

0

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

ABC

DE

(b)


3

2

1

0

0 40 80 120

Crack width ()

r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

100

50

0

Relat

ive c

hang

e rat

io (

)

0 40 80 120

Crack width ()

Kurtosis

RMS

(b)


119890= 119902

0= 3mm)



120592 = 45∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65

4 30 8 50 12






45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1A1B

2A2B

(a)

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1C1D

2C2D

(b)


200

100

0

minus100

a(m

s2 )

0032 0034 0036

Time (s)

AB

CD

(a)

AB

CD

04

02

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


3

15

015 25 35 45 55 65


r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

180

140

100

Rela

tive c

hang

e rat

io (

)

15 25 35 45 55 65


Kurtosis

RMS

(b)



4

3

2

10 4 8 12


TVM

S (times10

8N

m)

1A1B1C1D

1E1F1G

(a)

4

3

2

10 4 8 12


TVM

S (times10

8N

m)

2A2B2C2D

2E2F2G

(b)





1 (B) 0 8 35 15 (E) 70

Ω1 = 1000 revmin120595 = 35∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 5 9 40 16 753 10 10 45 17 (F) 80

4 15 11 50 18 855 20 12 55 19 (G) 90

6 25 13 (D) 60

7 (C) 30 14 65











400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)







80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A

Az

L

z

q0

120592

y998400

x998400

o998400

dz

(a)

q(z)

k(z)

120592

dz

(b)

z

L

Lc

q(z)q0

z

f(z)

y998400

o998400

A-A

dz

(c)


qe

y998400

o998400

L

z

q0q(z)

Lc


where 12057212

(0 le 12057212


12= 0


120590 = 1 Ω

1 Counterclockwise


(25)

y1

120579y1

rb1

120579z112057212

O1

k12(t)

e12(t)

c12(t)

x1

120579x1

12059512

y2

120579y2

rb2

120579z2 x2

120579x2

Gear 2 (gear)

Gear 1 (pinion)

O2

Ω2

Ω1





11989811minus 119896

1211990112

(119905) sin12059512

minus 11988812

(119905) 12

(119905) sin12059512

= 0

1198981

1199101+ 119896

1211990112

(119905) cos12059512

+ 11988812

(119905) 12

(119905) cos12059512

= 0

11989811= 0

1198681199091

1205791199091

+ 1198681199111Ω1

1205791199101

= 0

1198681199101

1205791199101

minus 1198681199111Ω1

1205791199091

= 0

1198681199111

1205791199111

+ 120590 times 11990311988711198961211990112

(119905) + 120590 times 119903119887111988812

(119905) 12

(119905) = 1198791

11989822+ 119896

1211990112

(119905) sin12059512

+ 11988812

(119905) 12

(119905) sin12059512

= 0

1198982

1199102minus 119896

1211990112

(119905) cos12059512

minus 11988812

(119905) 12

(119905) cos12059512

= 0

11989822= 0

1198681199092

1205791199092

+ 1198681199112Ω2

1205791199102

= 0

1198681199102

1205791199102

minus 1198681199112Ω2

1205791199092

= 0

1198681199112

1205791199112

+ 120590 times 11990311988721198961211990112

(119905) + 120590 times 119903119887211988812

(119905) 12

(119905) = 1198792

(26)

where 1198981and 119898


1199091 1198681199101

1198681199111 1198681199092 1198681199102 and 119868



2



11990112

(119905) = (minus1199091sin120595

12+ 119909

2sin120595

12+ 119910

1cos120595

12minus 119910

2cos120595

12

+120590 times 11990311988711205791199111

+ 120590 times 11990311988721205791199112) minus 119890

12(119905)

(27)


M12X12

+ (C12

+ G12) X

12+ K

12X12

= F12 (28)


12K12

C12 and G






Mu + (C + G) u + Ku = Fu (29)



Shaft 2

Shaft 1 z

x

y

Bearing 1 Bearing 2

Bearing 3 Bearing 4

Gear 1 (node 8)

Gear 2 (node 22)





1=







1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20


119909= 119868119910(kgsdotmm2) 119868


1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20


119909119909(MNm) 119896

119910119910(MNm) 119896

119911119911(MNm) 119896

120579119909120579119909(MNsdotmrad) 119896

120579119910120579119910(MNsdotmrad)

All 200 200 200 01 01

100

0

minus100

002 006 01 014 018 022

Time (s)

a(m

s2)

(a)

002 006 01 014 018 022

Time (s)

200

0

minus200a(m

s2) Δt = 00818 s

(b)

200

100

0

minus100

0032 0034 0036

Time (s)

a(m

s2)


(c)









2= 12Hz So




Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()


1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

119871119888= 119871

119902119890= 119902

0= q

1199021max= 321mm

2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295

3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308

4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321

5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334

6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347

7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360

8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372

9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385

10 18 119902 = 116 21 (C) 40 119902 = 257

11 (B) 20 119902 = 128 22 42 119902 = 270

fe

2fe3fe

4fe

5fe

6fe

7fe

8fe

9fe

10fe

11fe

12fe

13fe

14fe15fe

16fe

0 5 10 15

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

30

20

10

0


times103

(a)

04

02

04500 4600 4700

5fe

Δf = 12Hz

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)


(b)


119890 16119891




1=










F

Crack

120573







RMS = radic1

119873

119873

sum119899=1

(119909 (119899) minus 119909)2 119909 =1

119873

119873

sum119899=1

119909 (119899) (30)



119873

119899=1(119909 (119899) minus 119909)4

[1119873sum119873

119899=1(119909 (119899) minus 119909)2]

2 (31)





RMShealthtimes 100

(32)





= 119871 =20mm and 119902

119890= 119902




4

3

20 4 8 12

TVM

S (times10

8N

m)

1A1B

2A2B


(a)

4

3

2

10 4 8 12

TVM

S (times10

8N

m)


1C1D

2C2D

(b)


AB

CD

300

200

100

0

minus100

minus200

a(m

s2)

0032 0034 0036

Time (s)

(a)

AB

CD

1

05

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


Kurtosis

25

15

5

minus50 20 40 60

Crack depth ()

r Kur

tosi

sr R

MS

()

RMS

(a)

100

50

0

Rela

tive c

hang

e rat

io (

)

Kurtosis

0 20 40 60

Crack depth ()

RMS

(b)




Crackcase

Crack width()

Crack size(mm)

Crackcase


caseCrack width


1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902

119890= 0 119871

119888= 8 17 80 119902

119890= 0 119871

119888= 16

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

1199020= 3mm

2 5 119902119890= 0 119871119888 = 1 10 45 119902

119890= 0 119871

119888= 9 18 85 119902

119890= 0 119871

119888= 17

3 10 119902119890= 0 119871

119888= 2 11 (C) 50 119902

119890= 0 119871

119888= 10 19 90 119902

119890= 0 119871

119888= 18

4 15 119902119890= 0 119871

119888= 3 12 55 119902

119890= 0 119871119888 = 11 20 95 119902

119890= 0 119871

119888= 19

5 20 119902119890= 0 119871

119888= 4 13 60 119902

119890= 0 119871

119888= 12 21 (D) 100 119902

119890= 0 119871

119888= 20

6 (B) 25 119902119890= 0 119871

119888= 5 14 65 119902

119890= 0 119871

119888= 13 22 (E) 120 119902

119890= 3 119871

119888= 20

7 30 119902119890= 0 119871

119888= 6 15 70 119902

119890= 0 119871

119888= 14

8 35 119902119890= 0 119871

119888= 7 16 75 119902

119890= 0 119871

119888= 15

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

ABC

DE
















200

100

0

minus100

a(m

s2)

0032 0034 0036

Time (s)

ABC

DE

(a)

4500 4600 4700

04

03

02

01

0

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

ABC

DE

(b)


3

2

1

0

0 40 80 120

Crack width ()

r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

100

50

0

Relat

ive c

hang

e rat

io (

)

0 40 80 120

Crack width ()

Kurtosis

RMS

(b)


119890= 119902

0= 3mm)



120592 = 45∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65

4 30 8 50 12






45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1A1B

2A2B

(a)

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1C1D

2C2D

(b)


200

100

0

minus100

a(m

s2 )

0032 0034 0036

Time (s)

AB

CD

(a)

AB

CD

04

02

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


3

15

015 25 35 45 55 65


r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

180

140

100

Rela

tive c

hang

e rat

io (

)

15 25 35 45 55 65


Kurtosis

RMS

(b)



4

3

2

10 4 8 12


TVM

S (times10

8N

m)

1A1B1C1D

1E1F1G

(a)

4

3

2

10 4 8 12


TVM

S (times10

8N

m)

2A2B2C2D

2E2F2G

(b)





1 (B) 0 8 35 15 (E) 70

Ω1 = 1000 revmin120595 = 35∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 5 9 40 16 753 10 10 45 17 (F) 80

4 15 11 50 18 855 20 12 55 19 (G) 90

6 25 13 (D) 60

7 (C) 30 14 65











400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)







80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











11989811minus 119896

1211990112

(119905) sin12059512

minus 11988812

(119905) 12

(119905) sin12059512

= 0

1198981

1199101+ 119896

1211990112

(119905) cos12059512

+ 11988812

(119905) 12

(119905) cos12059512

= 0

11989811= 0

1198681199091

1205791199091

+ 1198681199111Ω1

1205791199101

= 0

1198681199101

1205791199101

minus 1198681199111Ω1

1205791199091

= 0

1198681199111

1205791199111

+ 120590 times 11990311988711198961211990112

(119905) + 120590 times 119903119887111988812

(119905) 12

(119905) = 1198791

11989822+ 119896

1211990112

(119905) sin12059512

+ 11988812

(119905) 12

(119905) sin12059512

= 0

1198982

1199102minus 119896

1211990112

(119905) cos12059512

minus 11988812

(119905) 12

(119905) cos12059512

= 0

11989822= 0

1198681199092

1205791199092

+ 1198681199112Ω2

1205791199102

= 0

1198681199102

1205791199102

minus 1198681199112Ω2

1205791199092

= 0

1198681199112

1205791199112

+ 120590 times 11990311988721198961211990112

(119905) + 120590 times 119903119887211988812

(119905) 12

(119905) = 1198792

(26)

where 1198981and 119898


1199091 1198681199101

1198681199111 1198681199092 1198681199102 and 119868



2



11990112

(119905) = (minus1199091sin120595

12+ 119909

2sin120595

12+ 119910

1cos120595

12minus 119910

2cos120595

12

+120590 times 11990311988711205791199111

+ 120590 times 11990311988721205791199112) minus 119890

12(119905)

(27)


M12X12

+ (C12

+ G12) X

12+ K

12X12

= F12 (28)


12K12

C12 and G






Mu + (C + G) u + Ku = Fu (29)



Shaft 2

Shaft 1 z

x

y

Bearing 1 Bearing 2

Bearing 3 Bearing 4

Gear 1 (node 8)

Gear 2 (node 22)





1=







1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20


119909= 119868119910(kgsdotmm2) 119868


1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20


119909119909(MNm) 119896

119910119910(MNm) 119896

119911119911(MNm) 119896

120579119909120579119909(MNsdotmrad) 119896

120579119910120579119910(MNsdotmrad)

All 200 200 200 01 01

100

0

minus100

002 006 01 014 018 022

Time (s)

a(m

s2)

(a)

002 006 01 014 018 022

Time (s)

200

0

minus200a(m

s2) Δt = 00818 s

(b)

200

100

0

minus100

0032 0034 0036

Time (s)

a(m

s2)


(c)









2= 12Hz So




Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()


1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

119871119888= 119871

119902119890= 119902

0= q

1199021max= 321mm

2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295

3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308

4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321

5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334

6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347

7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360

8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372

9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385

10 18 119902 = 116 21 (C) 40 119902 = 257

11 (B) 20 119902 = 128 22 42 119902 = 270

fe

2fe3fe

4fe

5fe

6fe

7fe

8fe

9fe

10fe

11fe

12fe

13fe

14fe15fe

16fe

0 5 10 15

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

30

20

10

0


times103

(a)

04

02

04500 4600 4700

5fe

Δf = 12Hz

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)


(b)


119890 16119891




1=










F

Crack

120573







RMS = radic1

119873

119873

sum119899=1

(119909 (119899) minus 119909)2 119909 =1

119873

119873

sum119899=1

119909 (119899) (30)



119873

119899=1(119909 (119899) minus 119909)4

[1119873sum119873

119899=1(119909 (119899) minus 119909)2]

2 (31)





RMShealthtimes 100

(32)





= 119871 =20mm and 119902

119890= 119902




4

3

20 4 8 12

TVM

S (times10

8N

m)

1A1B

2A2B


(a)

4

3

2

10 4 8 12

TVM

S (times10

8N

m)


1C1D

2C2D

(b)


AB

CD

300

200

100

0

minus100

minus200

a(m

s2)

0032 0034 0036

Time (s)

(a)

AB

CD

1

05

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


Kurtosis

25

15

5

minus50 20 40 60

Crack depth ()

r Kur

tosi

sr R

MS

()

RMS

(a)

100

50

0

Rela

tive c

hang

e rat

io (

)

Kurtosis

0 20 40 60

Crack depth ()

RMS

(b)




Crackcase

Crack width()

Crack size(mm)

Crackcase


caseCrack width


1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902

119890= 0 119871

119888= 8 17 80 119902

119890= 0 119871

119888= 16

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

1199020= 3mm

2 5 119902119890= 0 119871119888 = 1 10 45 119902

119890= 0 119871

119888= 9 18 85 119902

119890= 0 119871

119888= 17

3 10 119902119890= 0 119871

119888= 2 11 (C) 50 119902

119890= 0 119871

119888= 10 19 90 119902

119890= 0 119871

119888= 18

4 15 119902119890= 0 119871

119888= 3 12 55 119902

119890= 0 119871119888 = 11 20 95 119902

119890= 0 119871

119888= 19

5 20 119902119890= 0 119871

119888= 4 13 60 119902

119890= 0 119871

119888= 12 21 (D) 100 119902

119890= 0 119871

119888= 20

6 (B) 25 119902119890= 0 119871

119888= 5 14 65 119902

119890= 0 119871

119888= 13 22 (E) 120 119902

119890= 3 119871

119888= 20

7 30 119902119890= 0 119871

119888= 6 15 70 119902

119890= 0 119871

119888= 14

8 35 119902119890= 0 119871

119888= 7 16 75 119902

119890= 0 119871

119888= 15

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

ABC

DE
















200

100

0

minus100

a(m

s2)

0032 0034 0036

Time (s)

ABC

DE

(a)

4500 4600 4700

04

03

02

01

0

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

ABC

DE

(b)


3

2

1

0

0 40 80 120

Crack width ()

r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

100

50

0

Relat

ive c

hang

e rat

io (

)

0 40 80 120

Crack width ()

Kurtosis

RMS

(b)


119890= 119902

0= 3mm)



120592 = 45∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65

4 30 8 50 12






45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1A1B

2A2B

(a)

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1C1D

2C2D

(b)


200

100

0

minus100

a(m

s2 )

0032 0034 0036

Time (s)

AB

CD

(a)

AB

CD

04

02

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


3

15

015 25 35 45 55 65


r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

180

140

100

Rela

tive c

hang

e rat

io (

)

15 25 35 45 55 65


Kurtosis

RMS

(b)



4

3

2

10 4 8 12


TVM

S (times10

8N

m)

1A1B1C1D

1E1F1G

(a)

4

3

2

10 4 8 12


TVM

S (times10

8N

m)

2A2B2C2D

2E2F2G

(b)





1 (B) 0 8 35 15 (E) 70

Ω1 = 1000 revmin120595 = 35∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 5 9 40 16 753 10 10 45 17 (F) 80

4 15 11 50 18 855 20 12 55 19 (G) 90

6 25 13 (D) 60

7 (C) 30 14 65











400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)







80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20


119909= 119868119910(kgsdotmm2) 119868


1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20


119909119909(MNm) 119896

119910119910(MNm) 119896

119911119911(MNm) 119896

120579119909120579119909(MNsdotmrad) 119896

120579119910120579119910(MNsdotmrad)

All 200 200 200 01 01

100

0

minus100

002 006 01 014 018 022

Time (s)

a(m

s2)

(a)

002 006 01 014 018 022

Time (s)

200

0

minus200a(m

s2) Δt = 00818 s

(b)

200

100

0

minus100

0032 0034 0036

Time (s)

a(m

s2)


(c)









2= 12Hz So




Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()


1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

119871119888= 119871

119902119890= 119902

0= q

1199021max= 321mm

2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295

3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308

4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321

5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334

6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347

7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360

8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372

9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385

10 18 119902 = 116 21 (C) 40 119902 = 257

11 (B) 20 119902 = 128 22 42 119902 = 270

fe

2fe3fe

4fe

5fe

6fe

7fe

8fe

9fe

10fe

11fe

12fe

13fe

14fe15fe

16fe

0 5 10 15

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

30

20

10

0


times103

(a)

04

02

04500 4600 4700

5fe

Δf = 12Hz

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)


(b)


119890 16119891




1=










F

Crack

120573







RMS = radic1

119873

119873

sum119899=1

(119909 (119899) minus 119909)2 119909 =1

119873

119873

sum119899=1

119909 (119899) (30)



119873

119899=1(119909 (119899) minus 119909)4

[1119873sum119873

119899=1(119909 (119899) minus 119909)2]

2 (31)





RMShealthtimes 100

(32)





= 119871 =20mm and 119902

119890= 119902




4

3

20 4 8 12

TVM

S (times10

8N

m)

1A1B

2A2B


(a)

4

3

2

10 4 8 12

TVM

S (times10

8N

m)


1C1D

2C2D

(b)


AB

CD

300

200

100

0

minus100

minus200

a(m

s2)

0032 0034 0036

Time (s)

(a)

AB

CD

1

05

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


Kurtosis

25

15

5

minus50 20 40 60

Crack depth ()

r Kur

tosi

sr R

MS

()

RMS

(a)

100

50

0

Rela

tive c

hang

e rat

io (

)

Kurtosis

0 20 40 60

Crack depth ()

RMS

(b)




Crackcase

Crack width()

Crack size(mm)

Crackcase


caseCrack width


1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902

119890= 0 119871

119888= 8 17 80 119902

119890= 0 119871

119888= 16

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

1199020= 3mm

2 5 119902119890= 0 119871119888 = 1 10 45 119902

119890= 0 119871

119888= 9 18 85 119902

119890= 0 119871

119888= 17

3 10 119902119890= 0 119871

119888= 2 11 (C) 50 119902

119890= 0 119871

119888= 10 19 90 119902

119890= 0 119871

119888= 18

4 15 119902119890= 0 119871

119888= 3 12 55 119902

119890= 0 119871119888 = 11 20 95 119902

119890= 0 119871

119888= 19

5 20 119902119890= 0 119871

119888= 4 13 60 119902

119890= 0 119871

119888= 12 21 (D) 100 119902

119890= 0 119871

119888= 20

6 (B) 25 119902119890= 0 119871

119888= 5 14 65 119902

119890= 0 119871

119888= 13 22 (E) 120 119902

119890= 3 119871

119888= 20

7 30 119902119890= 0 119871

119888= 6 15 70 119902

119890= 0 119871

119888= 14

8 35 119902119890= 0 119871

119888= 7 16 75 119902

119890= 0 119871

119888= 15

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

ABC

DE
















200

100

0

minus100

a(m

s2)

0032 0034 0036

Time (s)

ABC

DE

(a)

4500 4600 4700

04

03

02

01

0

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

ABC

DE

(b)


3

2

1

0

0 40 80 120

Crack width ()

r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

100

50

0

Relat

ive c

hang

e rat

io (

)

0 40 80 120

Crack width ()

Kurtosis

RMS

(b)


119890= 119902

0= 3mm)



120592 = 45∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65

4 30 8 50 12






45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1A1B

2A2B

(a)

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1C1D

2C2D

(b)


200

100

0

minus100

a(m

s2 )

0032 0034 0036

Time (s)

AB

CD

(a)

AB

CD

04

02

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


3

15

015 25 35 45 55 65


r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

180

140

100

Rela

tive c

hang

e rat

io (

)

15 25 35 45 55 65


Kurtosis

RMS

(b)



4

3

2

10 4 8 12


TVM

S (times10

8N

m)

1A1B1C1D

1E1F1G

(a)

4

3

2

10 4 8 12


TVM

S (times10

8N

m)

2A2B2C2D

2E2F2G

(b)





1 (B) 0 8 35 15 (E) 70

Ω1 = 1000 revmin120595 = 35∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 5 9 40 16 753 10 10 45 17 (F) 80

4 15 11 50 18 855 20 12 55 19 (G) 90

6 25 13 (D) 60

7 (C) 30 14 65











400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)







80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()

Crack size(mm)

Crackcase

Crackdepth ()


1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

119871119888= 119871

119902119890= 119902

0= q

1199021max= 321mm

2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295

3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308

4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321

5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334

6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347

7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360

8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372

9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385

10 18 119902 = 116 21 (C) 40 119902 = 257

11 (B) 20 119902 = 128 22 42 119902 = 270

fe

2fe3fe

4fe

5fe

6fe

7fe

8fe

9fe

10fe

11fe

12fe

13fe

14fe15fe

16fe

0 5 10 15

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

30

20

10

0


times103

(a)

04

02

04500 4600 4700

5fe

Δf = 12Hz

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)


(b)


119890 16119891




1=










F

Crack

120573







RMS = radic1

119873

119873

sum119899=1

(119909 (119899) minus 119909)2 119909 =1

119873

119873

sum119899=1

119909 (119899) (30)



119873

119899=1(119909 (119899) minus 119909)4

[1119873sum119873

119899=1(119909 (119899) minus 119909)2]

2 (31)





RMShealthtimes 100

(32)





= 119871 =20mm and 119902

119890= 119902




4

3

20 4 8 12

TVM

S (times10

8N

m)

1A1B

2A2B


(a)

4

3

2

10 4 8 12

TVM

S (times10

8N

m)


1C1D

2C2D

(b)


AB

CD

300

200

100

0

minus100

minus200

a(m

s2)

0032 0034 0036

Time (s)

(a)

AB

CD

1

05

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


Kurtosis

25

15

5

minus50 20 40 60

Crack depth ()

r Kur

tosi

sr R

MS

()

RMS

(a)

100

50

0

Rela

tive c

hang

e rat

io (

)

Kurtosis

0 20 40 60

Crack depth ()

RMS

(b)




Crackcase

Crack width()

Crack size(mm)

Crackcase


caseCrack width


1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902

119890= 0 119871

119888= 8 17 80 119902

119890= 0 119871

119888= 16

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

1199020= 3mm

2 5 119902119890= 0 119871119888 = 1 10 45 119902

119890= 0 119871

119888= 9 18 85 119902

119890= 0 119871

119888= 17

3 10 119902119890= 0 119871

119888= 2 11 (C) 50 119902

119890= 0 119871

119888= 10 19 90 119902

119890= 0 119871

119888= 18

4 15 119902119890= 0 119871

119888= 3 12 55 119902

119890= 0 119871119888 = 11 20 95 119902

119890= 0 119871

119888= 19

5 20 119902119890= 0 119871

119888= 4 13 60 119902

119890= 0 119871

119888= 12 21 (D) 100 119902

119890= 0 119871

119888= 20

6 (B) 25 119902119890= 0 119871

119888= 5 14 65 119902

119890= 0 119871

119888= 13 22 (E) 120 119902

119890= 3 119871

119888= 20

7 30 119902119890= 0 119871

119888= 6 15 70 119902

119890= 0 119871

119888= 14

8 35 119902119890= 0 119871

119888= 7 16 75 119902

119890= 0 119871

119888= 15

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

ABC

DE
















200

100

0

minus100

a(m

s2)

0032 0034 0036

Time (s)

ABC

DE

(a)

4500 4600 4700

04

03

02

01

0

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

ABC

DE

(b)


3

2

1

0

0 40 80 120

Crack width ()

r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

100

50

0

Relat

ive c

hang

e rat

io (

)

0 40 80 120

Crack width ()

Kurtosis

RMS

(b)


119890= 119902

0= 3mm)



120592 = 45∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65

4 30 8 50 12






45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1A1B

2A2B

(a)

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1C1D

2C2D

(b)


200

100

0

minus100

a(m

s2 )

0032 0034 0036

Time (s)

AB

CD

(a)

AB

CD

04

02

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


3

15

015 25 35 45 55 65


r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

180

140

100

Rela

tive c

hang

e rat

io (

)

15 25 35 45 55 65


Kurtosis

RMS

(b)



4

3

2

10 4 8 12


TVM

S (times10

8N

m)

1A1B1C1D

1E1F1G

(a)

4

3

2

10 4 8 12


TVM

S (times10

8N

m)

2A2B2C2D

2E2F2G

(b)





1 (B) 0 8 35 15 (E) 70

Ω1 = 1000 revmin120595 = 35∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 5 9 40 16 753 10 10 45 17 (F) 80

4 15 11 50 18 855 20 12 55 19 (G) 90

6 25 13 (D) 60

7 (C) 30 14 65











400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)







80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















F

Crack

120573







RMS = radic1

119873

119873

sum119899=1

(119909 (119899) minus 119909)2 119909 =1

119873

119873

sum119899=1

119909 (119899) (30)



119873

119899=1(119909 (119899) minus 119909)4

[1119873sum119873

119899=1(119909 (119899) minus 119909)2]

2 (31)





RMShealthtimes 100

(32)





= 119871 =20mm and 119902

119890= 119902




4

3

20 4 8 12

TVM

S (times10

8N

m)

1A1B

2A2B


(a)

4

3

2

10 4 8 12

TVM

S (times10

8N

m)


1C1D

2C2D

(b)


AB

CD

300

200

100

0

minus100

minus200

a(m

s2)

0032 0034 0036

Time (s)

(a)

AB

CD

1

05

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


Kurtosis

25

15

5

minus50 20 40 60

Crack depth ()

r Kur

tosi

sr R

MS

()

RMS

(a)

100

50

0

Rela

tive c

hang

e rat

io (

)

Kurtosis

0 20 40 60

Crack depth ()

RMS

(b)




Crackcase

Crack width()

Crack size(mm)

Crackcase


caseCrack width


1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902

119890= 0 119871

119888= 8 17 80 119902

119890= 0 119871

119888= 16

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

1199020= 3mm

2 5 119902119890= 0 119871119888 = 1 10 45 119902

119890= 0 119871

119888= 9 18 85 119902

119890= 0 119871

119888= 17

3 10 119902119890= 0 119871

119888= 2 11 (C) 50 119902

119890= 0 119871

119888= 10 19 90 119902

119890= 0 119871

119888= 18

4 15 119902119890= 0 119871

119888= 3 12 55 119902

119890= 0 119871119888 = 11 20 95 119902

119890= 0 119871

119888= 19

5 20 119902119890= 0 119871

119888= 4 13 60 119902

119890= 0 119871

119888= 12 21 (D) 100 119902

119890= 0 119871

119888= 20

6 (B) 25 119902119890= 0 119871

119888= 5 14 65 119902

119890= 0 119871

119888= 13 22 (E) 120 119902

119890= 3 119871

119888= 20

7 30 119902119890= 0 119871

119888= 6 15 70 119902

119890= 0 119871

119888= 14

8 35 119902119890= 0 119871

119888= 7 16 75 119902

119890= 0 119871

119888= 15

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

ABC

DE
















200

100

0

minus100

a(m

s2)

0032 0034 0036

Time (s)

ABC

DE

(a)

4500 4600 4700

04

03

02

01

0

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

ABC

DE

(b)


3

2

1

0

0 40 80 120

Crack width ()

r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

100

50

0

Relat

ive c

hang

e rat

io (

)

0 40 80 120

Crack width ()

Kurtosis

RMS

(b)


119890= 119902

0= 3mm)



120592 = 45∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65

4 30 8 50 12






45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1A1B

2A2B

(a)

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1C1D

2C2D

(b)


200

100

0

minus100

a(m

s2 )

0032 0034 0036

Time (s)

AB

CD

(a)

AB

CD

04

02

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


3

15

015 25 35 45 55 65


r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

180

140

100

Rela

tive c

hang

e rat

io (

)

15 25 35 45 55 65


Kurtosis

RMS

(b)



4

3

2

10 4 8 12


TVM

S (times10

8N

m)

1A1B1C1D

1E1F1G

(a)

4

3

2

10 4 8 12


TVM

S (times10

8N

m)

2A2B2C2D

2E2F2G

(b)





1 (B) 0 8 35 15 (E) 70

Ω1 = 1000 revmin120595 = 35∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 5 9 40 16 753 10 10 45 17 (F) 80

4 15 11 50 18 855 20 12 55 19 (G) 90

6 25 13 (D) 60

7 (C) 30 14 65











400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)







80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4

3

20 4 8 12

TVM

S (times10

8N

m)

1A1B

2A2B


(a)

4

3

2

10 4 8 12

TVM

S (times10

8N

m)


1C1D

2C2D

(b)


AB

CD

300

200

100

0

minus100

minus200

a(m

s2)

0032 0034 0036

Time (s)

(a)

AB

CD

1

05

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


Kurtosis

25

15

5

minus50 20 40 60

Crack depth ()

r Kur

tosi

sr R

MS

()

RMS

(a)

100

50

0

Rela

tive c

hang

e rat

io (

)

Kurtosis

0 20 40 60

Crack depth ()

RMS

(b)




Crackcase

Crack width()

Crack size(mm)

Crackcase


caseCrack width


1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902

119890= 0 119871

119888= 8 17 80 119902

119890= 0 119871

119888= 16

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

1199020= 3mm

2 5 119902119890= 0 119871119888 = 1 10 45 119902

119890= 0 119871

119888= 9 18 85 119902

119890= 0 119871

119888= 17

3 10 119902119890= 0 119871

119888= 2 11 (C) 50 119902

119890= 0 119871

119888= 10 19 90 119902

119890= 0 119871

119888= 18

4 15 119902119890= 0 119871

119888= 3 12 55 119902

119890= 0 119871119888 = 11 20 95 119902

119890= 0 119871

119888= 19

5 20 119902119890= 0 119871

119888= 4 13 60 119902

119890= 0 119871

119888= 12 21 (D) 100 119902

119890= 0 119871

119888= 20

6 (B) 25 119902119890= 0 119871

119888= 5 14 65 119902

119890= 0 119871

119888= 13 22 (E) 120 119902

119890= 3 119871

119888= 20

7 30 119902119890= 0 119871

119888= 6 15 70 119902

119890= 0 119871

119888= 14

8 35 119902119890= 0 119871

119888= 7 16 75 119902

119890= 0 119871

119888= 15

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

ABC

DE
















200

100

0

minus100

a(m

s2)

0032 0034 0036

Time (s)

ABC

DE

(a)

4500 4600 4700

04

03

02

01

0

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

ABC

DE

(b)


3

2

1

0

0 40 80 120

Crack width ()

r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

100

50

0

Relat

ive c

hang

e rat

io (

)

0 40 80 120

Crack width ()

Kurtosis

RMS

(b)


119890= 119902

0= 3mm)



120592 = 45∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65

4 30 8 50 12






45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1A1B

2A2B

(a)

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1C1D

2C2D

(b)


200

100

0

minus100

a(m

s2 )

0032 0034 0036

Time (s)

AB

CD

(a)

AB

CD

04

02

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


3

15

015 25 35 45 55 65


r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

180

140

100

Rela

tive c

hang

e rat

io (

)

15 25 35 45 55 65


Kurtosis

RMS

(b)



4

3

2

10 4 8 12


TVM

S (times10

8N

m)

1A1B1C1D

1E1F1G

(a)

4

3

2

10 4 8 12


TVM

S (times10

8N

m)

2A2B2C2D

2E2F2G

(b)





1 (B) 0 8 35 15 (E) 70

Ω1 = 1000 revmin120595 = 35∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 5 9 40 16 753 10 10 45 17 (F) 80

4 15 11 50 18 855 20 12 55 19 (G) 90

6 25 13 (D) 60

7 (C) 30 14 65











400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)







80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Crackcase

Crack width()

Crack size(mm)

Crackcase


caseCrack width


1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902

119890= 0 119871

119888= 8 17 80 119902

119890= 0 119871

119888= 16

Ω1 = 1000 revmin120592 = 45∘

120595 = 35∘

1199020= 3mm

2 5 119902119890= 0 119871119888 = 1 10 45 119902

119890= 0 119871

119888= 9 18 85 119902

119890= 0 119871

119888= 17

3 10 119902119890= 0 119871

119888= 2 11 (C) 50 119902

119890= 0 119871

119888= 10 19 90 119902

119890= 0 119871

119888= 18

4 15 119902119890= 0 119871

119888= 3 12 55 119902

119890= 0 119871119888 = 11 20 95 119902

119890= 0 119871

119888= 19

5 20 119902119890= 0 119871

119888= 4 13 60 119902

119890= 0 119871

119888= 12 21 (D) 100 119902

119890= 0 119871

119888= 20

6 (B) 25 119902119890= 0 119871

119888= 5 14 65 119902

119890= 0 119871

119888= 13 22 (E) 120 119902

119890= 3 119871

119888= 20

7 30 119902119890= 0 119871

119888= 6 15 70 119902

119890= 0 119871

119888= 14

8 35 119902119890= 0 119871

119888= 7 16 75 119902

119890= 0 119871

119888= 15

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

ABC

DE
















200

100

0

minus100

a(m

s2)

0032 0034 0036

Time (s)

ABC

DE

(a)

4500 4600 4700

04

03

02

01

0

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

ABC

DE

(b)


3

2

1

0

0 40 80 120

Crack width ()

r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

100

50

0

Relat

ive c

hang

e rat

io (

)

0 40 80 120

Crack width ()

Kurtosis

RMS

(b)


119890= 119902

0= 3mm)



120592 = 45∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65

4 30 8 50 12






45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1A1B

2A2B

(a)

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1C1D

2C2D

(b)


200

100

0

minus100

a(m

s2 )

0032 0034 0036

Time (s)

AB

CD

(a)

AB

CD

04

02

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


3

15

015 25 35 45 55 65


r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

180

140

100

Rela

tive c

hang

e rat

io (

)

15 25 35 45 55 65


Kurtosis

RMS

(b)



4

3

2

10 4 8 12


TVM

S (times10

8N

m)

1A1B1C1D

1E1F1G

(a)

4

3

2

10 4 8 12


TVM

S (times10

8N

m)

2A2B2C2D

2E2F2G

(b)





1 (B) 0 8 35 15 (E) 70

Ω1 = 1000 revmin120595 = 35∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 5 9 40 16 753 10 10 45 17 (F) 80

4 15 11 50 18 855 20 12 55 19 (G) 90

6 25 13 (D) 60

7 (C) 30 14 65











400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)







80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










200

100

0

minus100

a(m

s2)

0032 0034 0036

Time (s)

ABC

DE

(a)

4500 4600 4700

04

03

02

01

0

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

ABC

DE

(b)


3

2

1

0

0 40 80 120

Crack width ()

r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

100

50

0

Relat

ive c

hang

e rat

io (

)

0 40 80 120

Crack width ()

Kurtosis

RMS

(b)


119890= 119902

0= 3mm)



120592 = 45∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65

4 30 8 50 12






45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1A1B

2A2B

(a)

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1C1D

2C2D

(b)


200

100

0

minus100

a(m

s2 )

0032 0034 0036

Time (s)

AB

CD

(a)

AB

CD

04

02

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


3

15

015 25 35 45 55 65


r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

180

140

100

Rela

tive c

hang

e rat

io (

)

15 25 35 45 55 65


Kurtosis

RMS

(b)



4

3

2

10 4 8 12


TVM

S (times10

8N

m)

1A1B1C1D

1E1F1G

(a)

4

3

2

10 4 8 12


TVM

S (times10

8N

m)

2A2B2C2D

2E2F2G

(b)





1 (B) 0 8 35 15 (E) 70

Ω1 = 1000 revmin120595 = 35∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 5 9 40 16 753 10 10 45 17 (F) 80

4 15 11 50 18 855 20 12 55 19 (G) 90

6 25 13 (D) 60

7 (C) 30 14 65











400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)







80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1A1B

2A2B

(a)

45

35

25

150 4 8 12


TVM

S (times10

8N

m)

1C1D

2C2D

(b)


200

100

0

minus100

a(m

s2 )

0032 0034 0036

Time (s)

AB

CD

(a)

AB

CD

04

02

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

Δf = 12Hz

5fe

(b)


3

15

015 25 35 45 55 65


r Kur

tosi

sr R

MS

()

Kurtosis

RMS

(a)

180

140

100

Rela

tive c

hang

e rat

io (

)

15 25 35 45 55 65


Kurtosis

RMS

(b)



4

3

2

10 4 8 12


TVM

S (times10

8N

m)

1A1B1C1D

1E1F1G

(a)

4

3

2

10 4 8 12


TVM

S (times10

8N

m)

2A2B2C2D

2E2F2G

(b)





1 (B) 0 8 35 15 (E) 70

Ω1 = 1000 revmin120595 = 35∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 5 9 40 16 753 10 10 45 17 (F) 80

4 15 11 50 18 855 20 12 55 19 (G) 90

6 25 13 (D) 60

7 (C) 30 14 65











400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)







80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4

3

2

10 4 8 12


TVM

S (times10

8N

m)

1A1B1C1D

1E1F1G

(a)

4

3

2

10 4 8 12


TVM

S (times10

8N

m)

2A2B2C2D

2E2F2G

(b)





1 (B) 0 8 35 15 (E) 70

Ω1 = 1000 revmin120595 = 35∘

119871119888= 119871 = 20mm

119902119890= 119902

0= 3mm

2 5 9 40 16 753 10 10 45 17 (F) 80

4 15 11 50 18 855 20 12 55 19 (G) 90

6 25 13 (D) 60

7 (C) 30 14 65











400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)







80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










400

200

0

minus200

a(m

s2)

0032 0034 0036

Time (s)

ABCD

EFG

(a)

16

08

04500 4600 4700

Frequency (Hz)

Acce

lera

tion

ampl

itude

(ms2)

5fe

Δf = 12Hz

ABCD

EFG

(b)

4595 4605 4615

Frequency (Hz)

12

06

0

Δf = 12Hz

Acce

lera

tion

ampl

itude

(ms2)

ABCD

EFG

(c)







80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










80

50

20

minus10

r Kur

tosi

sr R

MS

()

0 30 60 90


RMS

Kurtosis

(a)

120

80

40

0

Rela

tive c

hang

e rat

io (

)

0 30 60 90


RMS

Kurtosis

(b)



1and 119891


119890is equal



1 for example

9328Hz asymp 119891119890+ 119891

1 1815Hz asymp 2119891

119890minus 119891



119890minus 119891

2 1844Hz asymp

2119891119890+ 119891

2


Electromotor


Detent

Cracked gear

Coupling

Foundation

Acceleration sensor

Crack position


6 Conclusions




40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










40

20

0

minus20

minus4011 12 13 14 15

Time (s)

a(m

s2)

(a)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

0 2500 5000


fe

2fe

3fe9160

18320

27510

1800 1820 1840 1860 1880880 900 920 940 960

8826

8994

9160

9328

9492

17990

18320

18150 18470

18650

fe minus 2f1

fe minus f1

fe

fe + f1

fe + 2f1

2fe

2fe minus 2f1

2fe minus f12fe + 1

2fe + 2f1

f1

1

08

06

04

02

0

04

03

02

01

0

04

03

02

01

0

(b)


60

0

minus60

Δt = 00818 s

Time (s)45 46 47 48

a(m

s2)

(a)

0 2500 5000 880 900 920 940 1800 1820 1840 1860


08

04

0

08

04

0

025

020

015

010

005

0

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

Am

plitu

de (m

s2)

27470 8913

90359158

91589279

9406

18090

18200

18440

1856018310

fe 2fe

3fe

18310

fe minus 2f2

fe minus f2

fe fe + f2

2fe



fe + 2f2

(b)










Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Appendix

119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860



119909120573= 119903119887[ (120573 + 120579

119887) cos120573 minus sin120573]

119910120573= 119903119887[ (120573 + 120579

119887) sin120573 + cos120573]

1199091= 119903 times sin (Φ) minus (

1198861

sin 120574 + 119903120588


1199101= 119903 times cos (Φ) minus (

1198861

sin 120574 + 119903120588


1199092= 119903119887[ (120591 + 120579

119887) cos 120591 minus sin 120591]

1199102= 119903119887[ (120591 + 120579

119887) sin 120591 + cos 120591]

1198601199101

= 21199091119871

1198601199102

= 21199092119871

1198681199101

=2

311990931119871

1198681199102

=2

311990932119871

(A1)

So d1199101d120574 and d119910


d1199101

d120574=

1198861sin ((119886

1 tan 120574 + 119887

1) 119903) (1 + tan2120574)

tan2120574

+1198861cos 120574


1198861 tan 120574 + 119887

1

119903) minus (

1198861

sin 120574+ 119903120588)


1

119903)(1 +

1198861(1 + tan2120574)119903tan2120574

)

d1199102

d120591= 119903119887(120591 + 120579

119887) cos 120591

(A2)


120579119887= 1205872119873 + inv120572



Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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