Nonlinear Dynamic Characteristics of Fixed-Axis Gear Wear in Multistage … · 2019. 7. 30. · p4 o s o p3 p2 p1 “ – 1(t) – 2(t) – 4 (t) – 1 t) – sp4 (t) ff 4 ff ff

Research ArticleNonlinear Dynamic Characteristics of Fixed-Axis Gear Wear inMultistage Gear Transmission Systems

Xin Wang

School of Mechanical Engineering Baoji University of Arts and Sciences Baoji 721016 China

Correspondence should be addressed to Xin Wang 46607190qqcom

Received 21 December 2018 Accepted 10 April 2019 Published 20 June 2019

Academic Editor Zhixiong Li

Copyright copy 2019 Xin Wang is 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

With the increasing service life of a gear transmission system the gear teeth become constantly worn and the gear clearance increasese increase in the clearance will produce a series of nonlinear changes that change the stability of the system and can even cause theloss of stability In this paper dimensionless dynamic equations of a multistage gear transmission system that contains a two-stagefixed-axis gear and a one-stage planetary gear were derived e wear fault was simulated by changes in the gear clearance Systembifurcation diagrams with an increase in the clearance were studied e motion state and transition process of the multistage geartransmission system with a wear fault were studied using the Poincare section e change in the spectrum characteristics with anincrease in the clearance was analyzed e frequency spectrum characteristics of the wear fault were discovered e vibrationmechanism and a fault wear diagnosis method in a multistage gear transmission system were obtained

1 Introduction

With the increase in the service life of a gear transmissionsystem the gear teeth are continuously worn resulting in anincrease in the gear clearance Clearance is the main reasonfor collision motion inside this system Clearance will makethe motion state of the system change or even create in-stability erefore it is very important to study the wearmechanism and the change in the system nonlinear state asthese attributes are of great significance for the designmonitoring and diagnosis of the system

Currently many scholars have researched the mecha-nism of wear failure including the friction and wearmechanism of the medium belt [1] theoretical and exper-imental investigations into the teeth wear of spur gears ofheavy-duty machinery [2] measurement of the mean wearcoefficient [3] theoretical study of wear in internal gears [4]prediction of the steady-state wear rate [5] and the wearcharacteristics surface finish and microgeometry [6]

In the field of dynamics Wojnarowski and Onishchenko[7] carried out analytical and experimental investigations ofthe influence of the deformation and wear on spur gear

dynamics Liu et al [8] examined the dynamic response ofthe spur gear pair system and the interactions betweenbearing clearance and the backlash Ding and Kahraman[9] studied the interaction between gear surface wear andgear dynamic response Osman and Velex [10] studied thestatic and dynamic simulations of mild abrasive wearHowever this literature did not consider the nonlinearcharacteristics of the system Shen et al [11] focused on thenonlinear dynamics of a spur gear pair with a slight wearfault where the backlash time-varying stiffness and wearfault were all included ese studies all focused on thesingle pair gear model and neglected the coupling char-acteristics between multistage gears erefore it is nec-essary to further investigate the nonlinear dynamiccharacteristics of a multistage gear transmission systemwear fault

For wear fault diagnosis most of the studies mainly usethe signal processing method to extract fault signals such asthe gear wear monitoring by a modulation signal bispectrumbased onmotor current signal analysis [12] helical gear wearmonitoring [13] and fault diagnosis of gear wearing basedon order spectrum analysis [14] Plint and Alliston-Greiner

HindawiShock and VibrationVolume 2019 Article ID 5641617 11 pageshttpsdoiorg10115520195641617

[15] proposed a new wear criterion the energy pulse He [16]proposed a novel nonlinear time-frequency feature based ona time-frequency manifold (TFM) technique Liu et al [17]proposed a dynamic wear prediction methodology to in-vestigate the coupling effects between surface wear and thedynamics of spur gear systems is paper will also use thenonlinear wear fault time-frequency characteristics topresent a diagnostic method for wear faults

In this paper a nonlinear dynamic model of a multistagegear transmission system consisting of a two-stage fixed-axisgear and a one-stage planetary gear was established esimulation method was used to analyze the bifurcationcharacteristics of the system under different clearances todetermine the wear excitation frequency range and thechanges in the system motion state to understand the fre-quency characteristics of the system e transition processand fault frequency characteristics of each motion state underdifferent excitation frequencies are discussed e vibrationmechanism and a diagnostic method of wear fault wereobtained

11 Torsion Dynamics Model of a Multistage Gear Trans-mission System e system studied in this paper is a testrig with a multistage gear transmission system thatcontains a two-stage fixed-axis gear and a one-stageplanetary gear where spur gears 1 and 2 compose the1st stage fixed-axis gear for the input spur gears 3 and 4compose the 2nd stage fixed-axis gear and the planetcarrier is for the output e torsional dynamic model isestablished using the lumped mass method (Figure 1) emodel does not consider the transverse vibration dis-placement of the gears e gear parameters are simulatedwith a spring and a damper

In Figure 1 θs θc θpn θ1 θ2 θ3 and θ4 represent theangular displacements of the sun gear planet carrier plan-etary gear n (n 1 2 3 4) and spur gears 1 2 3 and 4respectively roughout this paper the subscripts s c pn rand 1 2 3 and 4 denote the sun gear planet carrier planetarygear ring gear and spur gears 1 2 3 and 4 respectively equantities rs rc rpn r1 r2 r3 and r4 are the base circle radiusvalues of the gears e quantities Kspn(t) Krpn(t) K1(t) andK2(t) denote the meshing stiffness of the sun gear withplanetary gear n the ring gear with planetary gear n the 1ststage fixed gear and the 2nd stage fixed gear respectively equantities Cspn Crpn C1 and C2 denote the damping of thesun gear with planetary gear n the ring gear with planetarygear n the 1st stage fixed gear and the 2nd stage fixed gearrespectively Tin is the input and Tout is the output

111 Motion Differential Equations of the System By usingthe Lagrangian equation the clearance time-varyingmeshingstiffness and composite error are considered to establish themotion differential equations of the system After that themotion differential equation is processed in a dimensionlessway is process has been deduced and described in detail inpaper [18] Only the final system dimensionless motiondifferential equations are listed here

eurox1 +C1

me1wh

_x1 +K1(τ)

me1w2h

f x1( 1113857minusC2

me2wh

_x2 minusK2(τ)

me2w2h

f x2( 1113857

Tinr1

J1w2hb1

+ea1

b1Ω21 sin Ω1τ + ϕ1( 1113857

eurox2 minusC1

me1wh

_x1 minusK1(τ)

me2w2h

f x1( 1113857 +C2

me3wh

_x2 +K2(τ)

me3w2h

f x2( 1113857

minus 11139444

n1

Cspn

m4swh

_xspn minus 11139444

n1

Kspn(τ)

m4sw2h

f xspn1113872 1113873

ea2

b1Ω22 sin Ω2τ + ϕ2( 1113857

euroxspn +1

m4swh1113944

4

n1Cspn

_xspn +1

mcwh1113944

4

n1Cspn

_xspn

+1

mpnwhCspn

_xspn +1

m4sw2h

1113944

4

n1Kspn(τ)f xspn1113872 1113873

+1

mcw2h

1113944

4

n1Kspn(τ)f xspn1113872 1113873 +

1mpnw2

hKspn(τ)f xspn1113872 1113873

minus1

mpnwhCrpn

_xrpn +1

mcwh1113944

4

n1Crpn

_xrpn minus1

mpnw2hKrpn(τ)f xrpn1113872 1113873

+1

mcw2h

1113944

4

n1Krpn(τ)f xrpn1113872 1113873minus

r4C2

m4srswh

_x2

minusr4K2(τ)

m4srsw2h

f x2( 1113857 rcTout

Jcew2hb1

+easpn

b1Ω2spn sin Ωspnτ + ϕspn1113872 1113873

euroxrpn minus1

mpnwhCspn

_xspn +1

mcwh1113944

4

n1Cspn

_xspn

minus1

mpnw2hKspn(τ) xspn1113872 1113873 +

1mcw

2h

1113944

4

n1Kspn(τ)f xspn1113872 1113873

+1

mpnwhCrpn

_xrpn minus1

mcwh1113944

4

n1Crpn

_xrpn

+1

mpnw2hKrpn(τ)f xrpn1113872 1113873minus

1mcw

2h

1113944

4

n1Krpn(τ)f xrpn1113872 1113873

rcTout

Jcew2hb1minus

earpn

b1Ω2rpn sin Ωrpnτ + ϕrpn1113872 1113873

(1)

112 Wear Fault e gear tooth surface will wear out withan increase in the running time e clearance between theteeth will change when wear occurs When gear teeth areevenly worn the tooth clearance will be larger than normalTo model wear failure Flodin and Andersson [19] simulated

2 Shock and Vibration

mild wear in spur gears Park et al [20] presented an ap-proximate method to predict the surface wear of hypoidgears using surface interpolation In this paper the clearancewas controlled by changing the size of the dimensionlesscomposite error amplitude eai [21] e bifurcation diagramof the multistage gear transmission system with the changein excitation frequency was calculated in the normal stateAccording to the clearance calculation formula given by theChina national standard GB 2363-90 the dimensionlessclearance interval of the meshing gear in the normal state is[01 1]

12 Nonlinear Dynamic Behavior Analysis of the System withIncreased Gear Wear

121 System Bifurcation Diagrams with Increased GearWear By increasing the wear degree of fixed-axis spur gear1 the influence of fixed-axis gear wear on the excitationfrequency range of the multistage gear transmission systemwas investigated e minimum clearance of the multistagegear transmission system was selected as the initial state thatis eai 01 At this point the system is in a periodic motionstate and it is easy to observe the changes caused by thefixed-axis gear weare variable-step RungendashKutta methodwas used to solve the nonlinear differential equations nu-merically (1) e bifurcation diagrams were investigatedwhen the planetary gear clearance was constant(easpn earpn 01) and the fixed-axis clearance increasedgradually (ea1 ea2 01sim1) on the meshing points of the 1ststage fixed-axis gear and the planetary gear with the ringgear as shown in Figure 2 e gear parameters of themultistage gear transmission system are shown in Table 1e pressure angle α 20deg Tin 65Nmiddotm Tout 85Nmiddotmbi 5 μm and the damping ratio of the meshing pairs

ξi 007 ese parameters were derived from the multistagegear transmission system test rig We set the sampling in-terval as Ω1 001 001 5 Each Ω1 calculates 80 rotatingperiodse number of samples for each rotating period was256 points e initial velocity and displacement were set to0 e calculation time of the bifurcation diagram was2820 s

Figure 2(a) shows that the system is in a single-periodicmotion state from the beginning and then changes fromsingle-periodic motion to multiperiodic motion at Ω1 1When Ω1 35 a quasi-periodic motion of the large oscil-lation occurs When ea1 ea2 02 (Figure 2(c)) chaos oc-curs at Ω1 15 However the chaos soon disappears andbecomes periodic motion again e large oscillation ad-vances to Ω1 32 e wear continues to increase(Figure 2(e)) and the chaos range increases e interval ofΩ1isin[05 16] is all chaotic When ea1 ea2 08 (Figure 2(g))the interval of Ω1isin[05 085] degenerates into periodicmotion However the vibration amplitude of the chaoticmotion range Ω1isin[085 16] increases dramatically Whenea1 ea2 1 (Figure 2(i)) the chaotic interval is [09 16] andthe chaotic amplitude increases further e large oscillationrange is in advance of the raising process of ea1 which isalready Ω1 23

On the meshing point of the planetary gear the systemis always in a state of periodic motion because of its smallinitial setting When the fixed-axis gear clearance beginsto increase the motion state of the planetary gear is notaffected (Figure 2(d)) e fixed-axis gear clearancecontinues to increase the chaotic motion of the fixed-axisgear is transmitted to the planetary gear and the plane-tary gear also presents a small chaotic characteristic(Figure 2(f )) When the fixed-axis gear clearance continuesto increase the amplitude of the fixed-axis gear chaoticmotion increases significantly leading to the synchronous

Tin

Tout θc

θ1

θ2 θ3

θ4

θp4

θs

θp3 θp2

θp1

R

K1(t)

K2(t) K rp4(t)

Krp1 (t)

Ksp4 (t)

C rp4

Crp1

C rp2

Crp3

K rp2(t)

Ksp2 (t)

Krp3 (t)

Ksp1(t)

Ksp3(t)

Csp3 Csp2

Csp1

Csp4

C1

C2

Figure 1 Torsional dynamic model of the multistage gear transmission system [18] (reproduced from Wang [18] (under the creativecommons attribution licensepublic domain))

Shock and Vibration 3

0 1 2 3 4 509

095

1

105

11

115

12

125

13

135

14xndash 1

Ω1

(a)

Ω1

0 1 2 3 4 50

02

04

06

08

1

12

14

xndash rpn

(b)

0 1 2 3 4 502

04

06

08

1

12

14

16

Ω1

xndash 1

(c)

Ω1

0 1 2 3 4 502

04

06

08

1

12

14

16

xndash rpn

(d)

0 1 2 3 4 5ndash15

ndash1

ndash05

0

05

1

15

2

xndash 1

Ω1

(e)

Ω1

0 1 2 3 4 5ndash02

0

02

04

06

08

1

12

14

xndash rpn

(f )

Figure 2 Continued


chaotic motion of the planetary gear in the interval ofΩ1isin[085 16] (Figure 2(h)) In Figure 2(j) the planetarygear and the fixed-axis gear show synchronization in thechaotic motion and periodic oscillation interval At thistime the planetary gear is mainly affected by the fixed-axis

gear vibration and it reflects the vibration characteristics ofthe fixed-axis gears

Figure 2 shows that the increased gear clearance of thefixed axis mainly affects the excitation frequency range of[05 16] showing that the vibration of the system in this

Table 1 Gear parameters

Gear Number of teeth Rri (mm) Rbi (mm) Mass (mig) Ji (gmiddotm2) Face width (mm)1 29 192 204 125 005 302 100 689 705 12245 6 303 36 239 253 224 014 304 90 615 634 1111 4 20s 28 123 13 41 0007 20pn 36 16 17 346 001 20c 30 8487 076 20R 100 456 47 20

0 1 2 3 4 5ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Ω1

xndash 1

(g)

Ω1

0 1 2 3 4 5ndash3

ndash2

ndash1

0

1

2

3

xndash rpn

(h)

0 1 2 3 4 5ndash4

ndash3

ndash2

ndash1

0

1

2

3

Ω1

xndash 1

(i)

0 1 2 3 4 5ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Ω1

xndash rpn

(j)

Figure 2 Bifurcation diagrams of the system with fixed-axis wear fault (a) and (b) ea1 ea2 01 (c) and (d) ea1 ea2 02 (e) and (f)ea1 ea2 04 (g) and (h) ea1 ea2 08 (i) and (j) ea1 ea2 1


interval is mainly affected by the characteristics of the fixed-axis gear e clearance of the planetary gear mainly in-fluences the interval where the excitation frequency isgreater than 3 and this result has been described in anotherpaper by the author in [22]

122 Frequency Characteristics of the Fixed-Axis Gear underDifferent Excitation Frequencies Section 121 shows thatwith the increase in the clearance the system changes fromsingle-periodic motion to multiperiodic chaos and quasi-periodic motions To understand the fault frequency char-acteristics of the system in each motion state more detailedanalyses of the time domain the frequency domain thephase diagram and the Poincare section of eachmotion stateare required e large clearance state is (Figure 2(g)) se-lected as the study object At this time the system includes alltypes of motion states e wear failure is relatively seriousand the characteristics are more obvious e dimensionlessgear characteristic frequencies in the multistage geartransmission system are shown in Table 2

In Figure 2(g) when ea1 ea2 08 the time-domaindiagrams frequency-domain diagrams phase diagramsand Poincare section diagrams of the 1st stage fixed-axis gearare calculated on each excitation frequency to obtainFigures 3ndash14

As shown in Figure 3 when the excitation frequency isvery low the system is in single-periodic motion eamplitude of vibration is very small e Poincare sectionis a point ere is only one frequency component in thespectrum that is the 1st stage meshing frequency f1 Asthe excitation frequency increases (Figures 4 and 5) thesystem enters multiperiodic motion e time-domainvibration amplitude increases e Poincare sectionchanges from a point to double ellipses Since the periodicmotion is maintained the frequency component is still 1ststage meshing frequency f1 and the frequency amplitudeincreases significantly As the excitation frequency con-tinues to increase (Figures 6ndash9(c)) the system movesfrom multiperiodic motion to chaos due to increasedcollision motion In the spectrum shown in Figures 6ndash9(d) the 2nd stage meshing frequency f2 appears enthe chaotic motion enters the limit cycle through Hopfbifurcation as shown in Figures 10ndash12(c) As the exci-tation frequency continues to increase the systemmaintains a quasi-periodic motion state (limit cycle state)and presents torus doubling (Figures 13 and 14(c)) eamplitude of the 1st stage meshing frequency f1 is heldconstant in the spectrum and the amplitude of the 2nd

stage meshing frequency f2 gradually decreases (Fig-ures 13 and 14(d))

When the clearance of the fixed-axis gear increases thechaotic motion of the system is caused by the collision andthe fault feature of the 2nd stage meshing frequency appearsin the spectrum e clearance size determines the speed ofthe system entering chaos e larger the clearance is thefaster the systemmoves into the chaotic motion state and thefaster the system will degenerate from chaos to quasi-periodic motion e wear degree of the system can be

monitored through the motion state and frequency char-acteristics of the system to predict the system wear fault inadvance

13 Test Signal Identification andDiagnosis Figure 15 showsa multistage gear transmission test rig consisting of a two-stage fixed-axis gear and a one-stage planetary gear

According to the actual rotation speed of the motor theoperating state occurs when the excitation frequency Ω1 1(corresponding to a motor frequency of 30Hz) is selectedSignal testing and analysis were performed on the test rige parameters are shown in Table 1 To facilitate com-parison with simulation results the measured spectrum isdimensionless Figure 16 shows the dimensionless spectrumof the axial measuring point on the drive side of the fixed-axis gearbox when Ω1 1 e dimensionless characteristicfrequencies of the gears in the multistage gear transmissionsystem are shown in Table 2

Comparing Figure 16 with Figure 6(d) reveals that themain peaks of the two spectra are similar all are f1 f2 and2f2 In Figure 16 the planetary gear meshing frequency f3 islarger indicating that the planetary gear clearance is morethan 01 (easpn earpngt01) at this time Due to the obviousamplitude of f2 the motion state of the fixed-axis gear isinitially judged to be chaotic motion and the clearancecondition is ea1 ea2 08 However according toFigures 2(e) 2(g) and 2(i) when the excitation frequency is1 the interval [04 1] is chaotic thus it is necessary toperform more detailed spectrum analysis (Figure 17) todetermine the degree of wear

Figure 17 shows that the main peaks are f1 and (12)f1when ea1 ea2 04-05 For ea1 ea2 06ndash1 the mainpeaks are f1 f2 and 2f2 e amplitudes of f2 and 2f2 arerelatively stable but the amplitude of f1 increases as theclearance increases According to the proportional re-lationship between the amplitudes of f1 and f2 in Figure 16the degree of wear failure is predicted to be approximately08-09

An open-case inspection of the test rig revealed thatthere was severe wear on the 1st stage fixed-axis gears(Figure 18) and the gear clearance was ea1asymp4 μm isconclusion is the same conclusion that was reached in thetheoretical study

2 Conclusions

ewear degree of the fixed-axis gear in a test rig was judgedbased on the frequency spectrum characteristics of the wearfault By changing the fixed-axis gear clearance the wearfailure of the fixed-axis gear was simulated and the bi-furcation diagrams of the fixed-axis gear with differentclearances were analyzed e changes of the motion state ofthe system were investigated e vibration mechanism ofthe system with wear failure was summarized e studyfound that the wear failure of the fixed-axis gear mainlyaffects the excitation frequency range of [05 16] With theincrease in wear the periodic motion of the system becomeschaotic e more serious the wear is the more the chaotic


Table 2 Dimensionless characteristic frequencies of the multistage gear transmission system (Hz)

Characteristic frequency Dimensional frequencyMeshing frequency of the 1st stage fixed-axis gear f1 1Meshing frequency of the 2nd stage fixed-axis gear f2 03599Meshing frequency of the planetary gear f3 00877

0 200 400 6000

05

1

15

2

τ

x 1

(a)

08 1 12 14 16ndash04

ndash02

0

02

04

x1

x 1

(b)

0 05 1 15 2ndash05

0

05

x1

x 1

(c)

0 05 1

2

4

6times104

Ω1

Amplitu

de f1

(d)

Figure 3 Vibration characteristics when Ω1 03 (a) time domain (b) phase diagram (c) Poincare section and (d) frequency domain

0 500 1000ndash2

0

2

4

τ

x 1

(a)

ndash2 0 2 4ndash1

0

1

2

x1

x 1

(b)

ndash2 ndash15 ndash1 ndash05ndash06

ndash04

ndash02

0

02

x1

x 1

(c)

0 05 1 150

05

1

15

2times105

Ω1

Amplitu

de

f1

(d)


ndash5

0

5

τ0 200 400 600 800

x 1

(a)

ndash4

ndash2

0

2

4

ndash5 0 5x1

x 1

(b)


ndash14

ndash12

ndash1

ndash08

x1

x 1

(c)

0 05 1 150

2

4

6

Ω1

Amplitu

de

f1

times105

(d)


ndash4

ndash2

0

2

4

0 200 400 600τ

x 1

(a)

ndash4 ndash2 0 2 4ndash4

ndash2

0

2

4

x1

x 1

(b)

ndash2 0 2 4ndash3

ndash2

ndash1

0

1

x1

x 1

(c)

0 05 1 150

05

1

15

2 times105

Ω1

Am

plitu

de

2f2

f1

f2f1 ndash f2

(d)



motion advances By studying the time domain frequencydomain phase diagram and Poincare section of each mo-tion state when there is a large clearance the transition

processes and frequency characteristics of each motion statewere analyzed e fault frequency characteristics of thewear faults were obtained

0 100 200 300 400ndash2

0

2

4

τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)

ndash1 0 1 2 3ndash3

ndash2

ndash1

0

x1

x 1

(c)

0 05 1 150

5

10

15 times104

Ω1

Amplitu

de

f1

f2 12f1

(d)


0 100 200 300 400ndash2

0

2

4

τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)

ndash2 0 2 4ndash3

ndash2

ndash1

0

x1

x 1

(c)

times104

0 05 1 150

5

10

15

Ω1

Amplitu

de

f1f2

(d)


τ

x 1

0 100 200 300 400ndash2

0

2

4

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

ndash2 0 2 4ndash3

ndash2

ndash1

0

(c)

0 05 1 150

5

10

15 times104

Ω1

Amplitu

de

f1f2

(d)


x 1

0 100 200 300ndash4

ndash2

0

2

4

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

ndash2 0 2 4ndash3

ndash2

ndash1

0

(c)

times104

0 05 1 150

5

10

15

Ω1

Amplitu

de f1

f2

(d)



Using a test signal from the test rig at a certain excitationfrequency the motion state of the system was determinedbased on the motion state characteristics in the theoretical

results and the wear degree of the gear was roughly inferreden the frequency spectra of different wear degrees werecompared at the excitation frequency Finally the degree of

x 1

0 100 200 300ndash2

0

2

4

τ

(a)

x1

x 1

ndash2 0 2 4ndash4

ndash2

0

2

4

(b)

x1

x 1

0 05 1 15 2ndash25

ndash2

ndash15

ndash1

ndash05

(c)

0 05 1 150

05

1

15

2 times105

Ω1

Amplitu

de

f1f2

(d)


x 1

0 100 200 300ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 105 1 15 2

ndash2

ndash18

ndash16

ndash14

ndash12

(c)

times105

0 05 1 150

05

1

15

2

Ω1

Amplitu

de

f1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

1 12 14 16 18ndash22

ndash2

ndash18

ndash16

ndash14

(c)

0 05 1 150

1

2

3 times105

Ω1

Amplitu

def1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

τ

(a)

x1

x 1

ndash1 0 1 2ndash4

ndash2

0

2

4

(b)

x1

x 1

13 14 15 16ndash23

ndash22

ndash21

ndash2

ndash19

(c)

times105

0 05 1 150

1

2

3

Ω1

Amplitu

de

f1f2

(d)



Figure 15emultistage gear transmission system test rig 1 motor 2 torque sensor and encoder 3 two-stage fixed-axis gearbox 4 radialbearing load 5 one-stage planetary gearbox 6 brake [18] (reproduced from Wang [18] (under the creative commons attribution licensepublic domain))

0 05 1 150

2

4

6

8

Amplitu

de

Ω1

f1

f2

f3

2f2

Figure 16 Dimensionless test signal frequency spectrum

Amplitu

de

10

times104

05 1Ω1

15

5

0

12f1

f1

(a)

05 1Ω1

15

Amplitu

de 10

15 times104

5

00

12f1

f1

(b)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2 f1

f1 ndash f22f2

(c)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2

f1

f1 ndash f2 2f2

(d)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(e)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(f )

times105

Ω1

f2

f1

2f2Amplitu

de

0 05 1 150

1

2

3

(g)

Figure 17 Frequency spectra for each clearance when Ω1 1 (a) ea1 ea2 04 (b) ea1 ea2 05 (c) ea1 ea2 06 (d) ea1 ea2 07 (e)ea1 ea2 08 (f ) ea1 ea2 09 and (g) ea1 ea2 1


wear of the test rig gears was determined e experimentalresults were the same as the theoretical results

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e author declares no conflicts of interest

Acknowledgments

e author gratefully acknowledges the financial supportfrom Baoji Science and Technology Plan Project (2017JH2-10)

References

[1] Z M Fan and H X Yang ldquoResearch on friction and wear ofgear transmission with meshed medium beltrdquo AdvancedMaterials Research vol 753ndash755 pp 1719ndash1722 2013

[2] V Onishchenko ldquoInvestigation of tooth wears from scuffingof heavy duty machine spur gearsrdquo Mechanism and Machineeory vol 83 pp 38ndash55 2015

[3] J A Brandatildeo P Cerqueira J H O Seabra andM J D Castro ldquoMeasurement of mean wear coefficientduring gear tests under various operating conditionsrdquo Tri-bology International vol 102 pp 61ndash69 2016

[4] M S Tunalioglu and B Tuccedil ldquoeoretical and experimentalinvestigation of wear in internal gearsrdquoWear vol 309 no 1-2 pp 208ndash215 2014

[5] MMasjedi andMM Khonsari ldquoOn the prediction of steady-state wear rate in spur gearsrdquo Wear vol 342-343 pp 234ndash243 2015

[6] A C Petare and N K Jain ldquoOn simultaneous improvementof wear characteristics surface finish and microgeometry ofstraight bevel gears by abrasive flow finishing processrdquoWearvol 404-405 pp 38ndash49 2018

[7] J Wojnarowski and V Onishchenko ldquoTooth wear effects onspur gear dynamicsrdquoMechanism andMachineeory vol 38no 2 pp 161ndash178 2003

[8] Z Liu Z Liu J Zhao and G Zhang ldquoStudy on interactionsbetween tooth backlash and journal bearing clearance non-linearity in spur gear pair systemrdquo Mechanism and Machineeory vol 107 pp 229ndash245 2017

[9] H Ding and A Kahraman ldquoInteractions between nonlinearspur gear dynamics and surface wearrdquo Journal of Sound andVibration vol 307 no 3ndash5 pp 662ndash679 2007

[10] T Osman and Ph Velex ldquoStatic and dynamic simulations ofmild abrasive wear in wide-faced solid spur and helical gearsrdquoMechanism and Machine eory vol 45 no 6 pp 911ndash9242010

[11] Y J Shen S P Yang H J Xing and X Y Wang ldquoNonlineardynamics of a spur gear pair with faultrdquo Key EngineeringMaterials vol 353ndash358 pp 1177ndash1180 2007

[12] R Zhang F Gu H Mansaf T Wang and A D Ball ldquoGearwear monitoring by modulation signal bispectrum based onmotor current signal analysisrdquoMechanical Systems and SignalProcessing vol 94 pp 202ndash213 2017

[13] K F Brethee D Zhen F Gu and A D Ball ldquoHelical gearwear monitoring modelling and experimental validationrdquoMechanism and Machine eory vol 117 pp 210ndash229 2017

[14] H Tian X Y Kang Y J Li and J N Zhang ldquoFault diagnosisof gear wearing based on order cepstrum analysisrdquo AppliedMechanics and Materials vol 543ndash547 pp 922ndash925 2014

[15] M A Plint and A F Alliston-Greiner ldquoe energy pulse anew wear criterion and its relevance to wear in gear teeth andautomotive engine valve trainsrdquo Lubrication Science vol 8no 3 pp 233ndash251 1996

[16] Q B He ldquoTimendashfrequency manifold for nonlinear featureextraction in machinery fault diagnosisrdquo Mechanical Systemsand Signal Processing vol 35 no 1-2 pp 200ndash218 2013

[17] X Liu Y Yang and J Zhang ldquoInvestigation on couplingeffects between surface wear and dynamics in a spur gearsystemrdquo Tribology International vol 101 pp 383ndash394 2016

[18] X Wang ldquoStability research of multistage gear transmissionsystem with crack faultrdquo Journal of Sound and Vibrationvol 434 pp 63ndash77 2018

[19] A Flodin and S Andersson ldquoSimulation of mild wear in spurgearsrdquo Wear vol 207 no 1-2 pp 16ndash23 1997

[20] D Park M Kolivand and A Kahraman ldquoAn approximatemethod to predict surface wear of hypoid gears using surfaceinterpolationrdquo Mechanism and Machine eory vol 71pp 64ndash78 2014

[21] Y GWang H Q Zheng T Q Yang Z Z Guan and J YangldquoNonlinear dynamics behavior of gear system with faultparametersrdquo Journal of Vibration Measurement amp Diagnosisvol 31 no 5 pp 570ndash573 2011

[22] X Wang ldquoA study on coupling faultsrsquo characteristics of fixed-axis gear crack and planetary gear wearrdquo Shock and Vibrationvol 2018 Article ID 4692796 13 pages 2018

Figure 18 Fixed-axis wear failure gear


International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018


Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation


Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

[15] proposed a new wear criterion the energy pulse He [16]proposed a novel nonlinear time-frequency feature based ona time-frequency manifold (TFM) technique Liu et al [17]proposed a dynamic wear prediction methodology to in-vestigate the coupling effects between surface wear and thedynamics of spur gear systems is paper will also use thenonlinear wear fault time-frequency characteristics topresent a diagnostic method for wear faults

In this paper a nonlinear dynamic model of a multistagegear transmission system consisting of a two-stage fixed-axisgear and a one-stage planetary gear was established esimulation method was used to analyze the bifurcationcharacteristics of the system under different clearances todetermine the wear excitation frequency range and thechanges in the system motion state to understand the fre-quency characteristics of the system e transition processand fault frequency characteristics of each motion state underdifferent excitation frequencies are discussed e vibrationmechanism and a diagnostic method of wear fault wereobtained

11 Torsion Dynamics Model of a Multistage Gear Trans-mission System e system studied in this paper is a testrig with a multistage gear transmission system thatcontains a two-stage fixed-axis gear and a one-stageplanetary gear where spur gears 1 and 2 compose the1st stage fixed-axis gear for the input spur gears 3 and 4compose the 2nd stage fixed-axis gear and the planetcarrier is for the output e torsional dynamic model isestablished using the lumped mass method (Figure 1) emodel does not consider the transverse vibration dis-placement of the gears e gear parameters are simulatedwith a spring and a damper

In Figure 1 θs θc θpn θ1 θ2 θ3 and θ4 represent theangular displacements of the sun gear planet carrier plan-etary gear n (n 1 2 3 4) and spur gears 1 2 3 and 4respectively roughout this paper the subscripts s c pn rand 1 2 3 and 4 denote the sun gear planet carrier planetarygear ring gear and spur gears 1 2 3 and 4 respectively equantities rs rc rpn r1 r2 r3 and r4 are the base circle radiusvalues of the gears e quantities Kspn(t) Krpn(t) K1(t) andK2(t) denote the meshing stiffness of the sun gear withplanetary gear n the ring gear with planetary gear n the 1ststage fixed gear and the 2nd stage fixed gear respectively equantities Cspn Crpn C1 and C2 denote the damping of thesun gear with planetary gear n the ring gear with planetarygear n the 1st stage fixed gear and the 2nd stage fixed gearrespectively Tin is the input and Tout is the output

111 Motion Differential Equations of the System By usingthe Lagrangian equation the clearance time-varyingmeshingstiffness and composite error are considered to establish themotion differential equations of the system After that themotion differential equation is processed in a dimensionlessway is process has been deduced and described in detail inpaper [18] Only the final system dimensionless motiondifferential equations are listed here

eurox1 +C1

me1wh

_x1 +K1(τ)

me1w2h

f x1( 1113857minusC2

me2wh

_x2 minusK2(τ)

me2w2h

f x2( 1113857

Tinr1

J1w2hb1

+ea1

b1Ω21 sin Ω1τ + ϕ1( 1113857

eurox2 minusC1

me1wh

_x1 minusK1(τ)

me2w2h

f x1( 1113857 +C2

me3wh

_x2 +K2(τ)

me3w2h

f x2( 1113857

minus 11139444

n1

Cspn

m4swh

_xspn minus 11139444

n1

Kspn(τ)

m4sw2h

f xspn1113872 1113873

ea2

b1Ω22 sin Ω2τ + ϕ2( 1113857

euroxspn +1

m4swh1113944

4

n1Cspn

_xspn +1

mcwh1113944

4

n1Cspn

_xspn

+1

mpnwhCspn

_xspn +1

m4sw2h

1113944

4

n1Kspn(τ)f xspn1113872 1113873

+1

mcw2h

1113944

4

n1Kspn(τ)f xspn1113872 1113873 +

1mpnw2

hKspn(τ)f xspn1113872 1113873

minus1

mpnwhCrpn

_xrpn +1

mcwh1113944

4

n1Crpn

_xrpn minus1

mpnw2hKrpn(τ)f xrpn1113872 1113873

+1

mcw2h

1113944

4

n1Krpn(τ)f xrpn1113872 1113873minus

r4C2

m4srswh

_x2

minusr4K2(τ)

m4srsw2h

f x2( 1113857 rcTout

Jcew2hb1

+easpn

b1Ω2spn sin Ωspnτ + ϕspn1113872 1113873

euroxrpn minus1

mpnwhCspn

_xspn +1

mcwh1113944

4

n1Cspn

_xspn

minus1

mpnw2hKspn(τ) xspn1113872 1113873 +

1mcw

2h

1113944

4

n1Kspn(τ)f xspn1113872 1113873

+1

mpnwhCrpn

_xrpn minus1

mcwh1113944

4

n1Crpn

_xrpn

+1

mpnw2hKrpn(τ)f xrpn1113872 1113873minus

1mcw

2h

1113944

4

n1Krpn(τ)f xrpn1113872 1113873

rcTout

Jcew2hb1minus

earpn

b1Ω2rpn sin Ωrpnτ + ϕrpn1113872 1113873

(1)

112 Wear Fault e gear tooth surface will wear out withan increase in the running time e clearance between theteeth will change when wear occurs When gear teeth areevenly worn the tooth clearance will be larger than normalTo model wear failure Flodin and Andersson [19] simulated








Tin

Tout θc

θ1

θ2 θ3

θ4

θp4

θs

θp3 θp2

θp1

R

K1(t)

K2(t) K rp4(t)

Krp1 (t)

Ksp4 (t)

C rp4

Crp1

C rp2

Crp3

K rp2(t)

Ksp2 (t)

Krp3 (t)

Ksp1(t)

Ksp3(t)

Csp3 Csp2

Csp1

Csp4

C1

C2



0 1 2 3 4 509

095

1

105

11

115

12

125

13

135

14xndash 1

Ω1

(a)

Ω1

0 1 2 3 4 50

02

04

06

08

1

12

14

xndash rpn

(b)

0 1 2 3 4 502

04

06

08

1

12

14

16

Ω1

xndash 1

(c)

Ω1

0 1 2 3 4 502

04

06

08

1

12

14

16

xndash rpn

(d)

0 1 2 3 4 5ndash15

ndash1

ndash05

0

05

1

15

2

xndash 1

Ω1

(e)

Ω1

0 1 2 3 4 5ndash02

0

02

04

06

08

1

12

14

xndash rpn

(f )

Figure 2 Continued







0 1 2 3 4 5ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Ω1

xndash 1

(g)

Ω1

0 1 2 3 4 5ndash3

ndash2

ndash1

0

1

2

3

xndash rpn

(h)

0 1 2 3 4 5ndash4

ndash3

ndash2

ndash1

0

1

2

3

Ω1

xndash 1

(i)

0 1 2 3 4 5ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Ω1

xndash rpn

(j)















2 Conclusions





0 200 400 6000

05

1

15

2

τ

x 1

(a)

08 1 12 14 16ndash04

ndash02

0

02

04

x1

x 1

(b)

0 05 1 15 2ndash05

0

05

x1

x 1

(c)

0 05 1

2

4

6times104

Ω1

Amplitu

de f1

(d)


0 500 1000ndash2

0

2

4

τ

x 1

(a)

ndash2 0 2 4ndash1

0

1

2

x1

x 1

(b)


ndash04

ndash02

0

02

x1

x 1

(c)

0 05 1 150

05

1

15

2times105

Ω1

Amplitu

de

f1

(d)


ndash5

0

5

τ0 200 400 600 800

x 1

(a)

ndash4

ndash2

0

2

4

ndash5 0 5x1

x 1

(b)


ndash14

ndash12

ndash1

ndash08

x1

x 1

(c)

0 05 1 150

2

4

6

Ω1

Amplitu

de

f1

times105

(d)


ndash4

ndash2

0

2

4

0 200 400 600τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)

ndash2 0 2 4ndash3

ndash2

ndash1

0

1

x1

x 1

(c)

0 05 1 150

05

1

15

2 times105

Ω1

Am

plitu

de

2f2

f1

f2f1 ndash f2

(d)





0 100 200 300 400ndash2

0

2

4

τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)


ndash2

ndash1

0

x1

x 1

(c)

0 05 1 150

5

10

15 times104

Ω1

Amplitu

de

f1

f2 12f1

(d)


0 100 200 300 400ndash2

0

2

4

τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)

ndash2 0 2 4ndash3

ndash2

ndash1

0

x1

x 1

(c)

times104

0 05 1 150

5

10

15

Ω1

Amplitu

de

f1f2

(d)


τ

x 1

0 100 200 300 400ndash2

0

2

4

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

ndash2 0 2 4ndash3

ndash2

ndash1

0

(c)

0 05 1 150

5

10

15 times104

Ω1

Amplitu

de

f1f2

(d)


x 1

0 100 200 300ndash4

ndash2

0

2

4

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

ndash2 0 2 4ndash3

ndash2

ndash1

0

(c)

times104

0 05 1 150

5

10

15

Ω1

Amplitu

de f1

f2

(d)





x 1

0 100 200 300ndash2

0

2

4

τ

(a)

x1

x 1

ndash2 0 2 4ndash4

ndash2

0

2

4

(b)

x1

x 1

0 05 1 15 2ndash25

ndash2

ndash15

ndash1

ndash05

(c)

0 05 1 150

05

1

15

2 times105

Ω1

Amplitu

de

f1f2

(d)


x 1

0 100 200 300ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 105 1 15 2

ndash2

ndash18

ndash16

ndash14

ndash12

(c)

times105

0 05 1 150

05

1

15

2

Ω1

Amplitu

de

f1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

1 12 14 16 18ndash22

ndash2

ndash18

ndash16

ndash14

(c)

0 05 1 150

1

2

3 times105

Ω1

Amplitu

def1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

τ

(a)

x1

x 1

ndash1 0 1 2ndash4

ndash2

0

2

4

(b)

x1

x 1

13 14 15 16ndash23

ndash22

ndash21

ndash2

ndash19

(c)

times105

0 05 1 150

1

2

3

Ω1

Amplitu

de

f1f2

(d)




0 05 1 150

2

4

6

8

Amplitu

de

Ω1

f1

f2

f3

2f2


Amplitu

de

10

times104

05 1Ω1

15

5

0

12f1

f1

(a)

05 1Ω1

15

Amplitu

de 10

15 times104

5

00

12f1

f1

(b)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2 f1

f1 ndash f22f2

(c)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2

f1

f1 ndash f2 2f2

(d)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(e)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(f )

times105

Ω1

f2

f1

2f2Amplitu

de

0 05 1 150

1

2

3

(g)




Data Availability




Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








Tin

Tout θc

θ1

θ2 θ3

θ4

θp4

θs

θp3 θp2

θp1

R

K1(t)

K2(t) K rp4(t)

Krp1 (t)

Ksp4 (t)

C rp4

Crp1

C rp2

Crp3

K rp2(t)

Ksp2 (t)

Krp3 (t)

Ksp1(t)

Ksp3(t)

Csp3 Csp2

Csp1

Csp4

C1

C2



0 1 2 3 4 509

095

1

105

11

115

12

125

13

135

14xndash 1

Ω1

(a)

Ω1

0 1 2 3 4 50

02

04

06

08

1

12

14

xndash rpn

(b)

0 1 2 3 4 502

04

06

08

1

12

14

16

Ω1

xndash 1

(c)

Ω1

0 1 2 3 4 502

04

06

08

1

12

14

16

xndash rpn

(d)

0 1 2 3 4 5ndash15

ndash1

ndash05

0

05

1

15

2

xndash 1

Ω1

(e)

Ω1

0 1 2 3 4 5ndash02

0

02

04

06

08

1

12

14

xndash rpn

(f )

Figure 2 Continued







0 1 2 3 4 5ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Ω1

xndash 1

(g)

Ω1

0 1 2 3 4 5ndash3

ndash2

ndash1

0

1

2

3

xndash rpn

(h)

0 1 2 3 4 5ndash4

ndash3

ndash2

ndash1

0

1

2

3

Ω1

xndash 1

(i)

0 1 2 3 4 5ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Ω1

xndash rpn

(j)















2 Conclusions





0 200 400 6000

05

1

15

2

τ

x 1

(a)

08 1 12 14 16ndash04

ndash02

0

02

04

x1

x 1

(b)

0 05 1 15 2ndash05

0

05

x1

x 1

(c)

0 05 1

2

4

6times104

Ω1

Amplitu

de f1

(d)


0 500 1000ndash2

0

2

4

τ

x 1

(a)

ndash2 0 2 4ndash1

0

1

2

x1

x 1

(b)


ndash04

ndash02

0

02

x1

x 1

(c)

0 05 1 150

05

1

15

2times105

Ω1

Amplitu

de

f1

(d)


ndash5

0

5

τ0 200 400 600 800

x 1

(a)

ndash4

ndash2

0

2

4

ndash5 0 5x1

x 1

(b)


ndash14

ndash12

ndash1

ndash08

x1

x 1

(c)

0 05 1 150

2

4

6

Ω1

Amplitu

de

f1

times105

(d)


ndash4

ndash2

0

2

4

0 200 400 600τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)

ndash2 0 2 4ndash3

ndash2

ndash1

0

1

x1

x 1

(c)

0 05 1 150

05

1

15

2 times105

Ω1

Am

plitu

de

2f2

f1

f2f1 ndash f2

(d)





0 100 200 300 400ndash2

0

2

4

τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)


ndash2

ndash1

0

x1

x 1

(c)

0 05 1 150

5

10

15 times104

Ω1

Amplitu

de

f1

f2 12f1

(d)


0 100 200 300 400ndash2

0

2

4

τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)

ndash2 0 2 4ndash3

ndash2

ndash1

0

x1

x 1

(c)

times104

0 05 1 150

5

10

15

Ω1

Amplitu

de

f1f2

(d)


τ

x 1

0 100 200 300 400ndash2

0

2

4

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

ndash2 0 2 4ndash3

ndash2

ndash1

0

(c)

0 05 1 150

5

10

15 times104

Ω1

Amplitu

de

f1f2

(d)


x 1

0 100 200 300ndash4

ndash2

0

2

4

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

ndash2 0 2 4ndash3

ndash2

ndash1

0

(c)

times104

0 05 1 150

5

10

15

Ω1

Amplitu

de f1

f2

(d)





x 1

0 100 200 300ndash2

0

2

4

τ

(a)

x1

x 1

ndash2 0 2 4ndash4

ndash2

0

2

4

(b)

x1

x 1

0 05 1 15 2ndash25

ndash2

ndash15

ndash1

ndash05

(c)

0 05 1 150

05

1

15

2 times105

Ω1

Amplitu

de

f1f2

(d)


x 1

0 100 200 300ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 105 1 15 2

ndash2

ndash18

ndash16

ndash14

ndash12

(c)

times105

0 05 1 150

05

1

15

2

Ω1

Amplitu

de

f1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

1 12 14 16 18ndash22

ndash2

ndash18

ndash16

ndash14

(c)

0 05 1 150

1

2

3 times105

Ω1

Amplitu

def1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

τ

(a)

x1

x 1

ndash1 0 1 2ndash4

ndash2

0

2

4

(b)

x1

x 1

13 14 15 16ndash23

ndash22

ndash21

ndash2

ndash19

(c)

times105

0 05 1 150

1

2

3

Ω1

Amplitu

de

f1f2

(d)




0 05 1 150

2

4

6

8

Amplitu

de

Ω1

f1

f2

f3

2f2


Amplitu

de

10

times104

05 1Ω1

15

5

0

12f1

f1

(a)

05 1Ω1

15

Amplitu

de 10

15 times104

5

00

12f1

f1

(b)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2 f1

f1 ndash f22f2

(c)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2

f1

f1 ndash f2 2f2

(d)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(e)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(f )

times105

Ω1

f2

f1

2f2Amplitu

de

0 05 1 150

1

2

3

(g)




Data Availability




Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


0 1 2 3 4 509

095

1

105

11

115

12

125

13

135

14xndash 1

Ω1

(a)

Ω1

0 1 2 3 4 50

02

04

06

08

1

12

14

xndash rpn

(b)

0 1 2 3 4 502

04

06

08

1

12

14

16

Ω1

xndash 1

(c)

Ω1

0 1 2 3 4 502

04

06

08

1

12

14

16

xndash rpn

(d)

0 1 2 3 4 5ndash15

ndash1

ndash05

0

05

1

15

2

xndash 1

Ω1

(e)

Ω1

0 1 2 3 4 5ndash02

0

02

04

06

08

1

12

14

xndash rpn

(f )

Figure 2 Continued







0 1 2 3 4 5ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Ω1

xndash 1

(g)

Ω1

0 1 2 3 4 5ndash3

ndash2

ndash1

0

1

2

3

xndash rpn

(h)

0 1 2 3 4 5ndash4

ndash3

ndash2

ndash1

0

1

2

3

Ω1

xndash 1

(i)

0 1 2 3 4 5ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Ω1

xndash rpn

(j)















2 Conclusions





0 200 400 6000

05

1

15

2

τ

x 1

(a)

08 1 12 14 16ndash04

ndash02

0

02

04

x1

x 1

(b)

0 05 1 15 2ndash05

0

05

x1

x 1

(c)

0 05 1

2

4

6times104

Ω1

Amplitu

de f1

(d)


0 500 1000ndash2

0

2

4

τ

x 1

(a)

ndash2 0 2 4ndash1

0

1

2

x1

x 1

(b)


ndash04

ndash02

0

02

x1

x 1

(c)

0 05 1 150

05

1

15

2times105

Ω1

Amplitu

de

f1

(d)


ndash5

0

5

τ0 200 400 600 800

x 1

(a)

ndash4

ndash2

0

2

4

ndash5 0 5x1

x 1

(b)


ndash14

ndash12

ndash1

ndash08

x1

x 1

(c)

0 05 1 150

2

4

6

Ω1

Amplitu

de

f1

times105

(d)


ndash4

ndash2

0

2

4

0 200 400 600τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)

ndash2 0 2 4ndash3

ndash2

ndash1

0

1

x1

x 1

(c)

0 05 1 150

05

1

15

2 times105

Ω1

Am

plitu

de

2f2

f1

f2f1 ndash f2

(d)





0 100 200 300 400ndash2

0

2

4

τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)


ndash2

ndash1

0

x1

x 1

(c)

0 05 1 150

5

10

15 times104

Ω1

Amplitu

de

f1

f2 12f1

(d)


0 100 200 300 400ndash2

0

2

4

τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)

ndash2 0 2 4ndash3

ndash2

ndash1

0

x1

x 1

(c)

times104

0 05 1 150

5

10

15

Ω1

Amplitu

de

f1f2

(d)


τ

x 1

0 100 200 300 400ndash2

0

2

4

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

ndash2 0 2 4ndash3

ndash2

ndash1

0

(c)

0 05 1 150

5

10

15 times104

Ω1

Amplitu

de

f1f2

(d)


x 1

0 100 200 300ndash4

ndash2

0

2

4

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

ndash2 0 2 4ndash3

ndash2

ndash1

0

(c)

times104

0 05 1 150

5

10

15

Ω1

Amplitu

de f1

f2

(d)





x 1

0 100 200 300ndash2

0

2

4

τ

(a)

x1

x 1

ndash2 0 2 4ndash4

ndash2

0

2

4

(b)

x1

x 1

0 05 1 15 2ndash25

ndash2

ndash15

ndash1

ndash05

(c)

0 05 1 150

05

1

15

2 times105

Ω1

Amplitu

de

f1f2

(d)


x 1

0 100 200 300ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 105 1 15 2

ndash2

ndash18

ndash16

ndash14

ndash12

(c)

times105

0 05 1 150

05

1

15

2

Ω1

Amplitu

de

f1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

1 12 14 16 18ndash22

ndash2

ndash18

ndash16

ndash14

(c)

0 05 1 150

1

2

3 times105

Ω1

Amplitu

def1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

τ

(a)

x1

x 1

ndash1 0 1 2ndash4

ndash2

0

2

4

(b)

x1

x 1

13 14 15 16ndash23

ndash22

ndash21

ndash2

ndash19

(c)

times105

0 05 1 150

1

2

3

Ω1

Amplitu

de

f1f2

(d)




0 05 1 150

2

4

6

8

Amplitu

de

Ω1

f1

f2

f3

2f2


Amplitu

de

10

times104

05 1Ω1

15

5

0

12f1

f1

(a)

05 1Ω1

15

Amplitu

de 10

15 times104

5

00

12f1

f1

(b)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2 f1

f1 ndash f22f2

(c)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2

f1

f1 ndash f2 2f2

(d)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(e)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(f )

times105

Ω1

f2

f1

2f2Amplitu

de

0 05 1 150

1

2

3

(g)




Data Availability




Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







0 1 2 3 4 5ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Ω1

xndash 1

(g)

Ω1

0 1 2 3 4 5ndash3

ndash2

ndash1

0

1

2

3

xndash rpn

(h)

0 1 2 3 4 5ndash4

ndash3

ndash2

ndash1

0

1

2

3

Ω1

xndash 1

(i)

0 1 2 3 4 5ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Ω1

xndash rpn

(j)















2 Conclusions





0 200 400 6000

05

1

15

2

τ

x 1

(a)

08 1 12 14 16ndash04

ndash02

0

02

04

x1

x 1

(b)

0 05 1 15 2ndash05

0

05

x1

x 1

(c)

0 05 1

2

4

6times104

Ω1

Amplitu

de f1

(d)


0 500 1000ndash2

0

2

4

τ

x 1

(a)

ndash2 0 2 4ndash1

0

1

2

x1

x 1

(b)


ndash04

ndash02

0

02

x1

x 1

(c)

0 05 1 150

05

1

15

2times105

Ω1

Amplitu

de

f1

(d)


ndash5

0

5

τ0 200 400 600 800

x 1

(a)

ndash4

ndash2

0

2

4

ndash5 0 5x1

x 1

(b)


ndash14

ndash12

ndash1

ndash08

x1

x 1

(c)

0 05 1 150

2

4

6

Ω1

Amplitu

de

f1

times105

(d)


ndash4

ndash2

0

2

4

0 200 400 600τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)

ndash2 0 2 4ndash3

ndash2

ndash1

0

1

x1

x 1

(c)

0 05 1 150

05

1

15

2 times105

Ω1

Am

plitu

de

2f2

f1

f2f1 ndash f2

(d)





0 100 200 300 400ndash2

0

2

4

τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)


ndash2

ndash1

0

x1

x 1

(c)

0 05 1 150

5

10

15 times104

Ω1

Amplitu

de

f1

f2 12f1

(d)


0 100 200 300 400ndash2

0

2

4

τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)

ndash2 0 2 4ndash3

ndash2

ndash1

0

x1

x 1

(c)

times104

0 05 1 150

5

10

15

Ω1

Amplitu

de

f1f2

(d)


τ

x 1

0 100 200 300 400ndash2

0

2

4

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

ndash2 0 2 4ndash3

ndash2

ndash1

0

(c)

0 05 1 150

5

10

15 times104

Ω1

Amplitu

de

f1f2

(d)


x 1

0 100 200 300ndash4

ndash2

0

2

4

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

ndash2 0 2 4ndash3

ndash2

ndash1

0

(c)

times104

0 05 1 150

5

10

15

Ω1

Amplitu

de f1

f2

(d)





x 1

0 100 200 300ndash2

0

2

4

τ

(a)

x1

x 1

ndash2 0 2 4ndash4

ndash2

0

2

4

(b)

x1

x 1

0 05 1 15 2ndash25

ndash2

ndash15

ndash1

ndash05

(c)

0 05 1 150

05

1

15

2 times105

Ω1

Amplitu

de

f1f2

(d)


x 1

0 100 200 300ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 105 1 15 2

ndash2

ndash18

ndash16

ndash14

ndash12

(c)

times105

0 05 1 150

05

1

15

2

Ω1

Amplitu

de

f1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

1 12 14 16 18ndash22

ndash2

ndash18

ndash16

ndash14

(c)

0 05 1 150

1

2

3 times105

Ω1

Amplitu

def1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

τ

(a)

x1

x 1

ndash1 0 1 2ndash4

ndash2

0

2

4

(b)

x1

x 1

13 14 15 16ndash23

ndash22

ndash21

ndash2

ndash19

(c)

times105

0 05 1 150

1

2

3

Ω1

Amplitu

de

f1f2

(d)




0 05 1 150

2

4

6

8

Amplitu

de

Ω1

f1

f2

f3

2f2


Amplitu

de

10

times104

05 1Ω1

15

5

0

12f1

f1

(a)

05 1Ω1

15

Amplitu

de 10

15 times104

5

00

12f1

f1

(b)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2 f1

f1 ndash f22f2

(c)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2

f1

f1 ndash f2 2f2

(d)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(e)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(f )

times105

Ω1

f2

f1

2f2Amplitu

de

0 05 1 150

1

2

3

(g)




Data Availability




Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia














2 Conclusions





0 200 400 6000

05

1

15

2

τ

x 1

(a)

08 1 12 14 16ndash04

ndash02

0

02

04

x1

x 1

(b)

0 05 1 15 2ndash05

0

05

x1

x 1

(c)

0 05 1

2

4

6times104

Ω1

Amplitu

de f1

(d)


0 500 1000ndash2

0

2

4

τ

x 1

(a)

ndash2 0 2 4ndash1

0

1

2

x1

x 1

(b)


ndash04

ndash02

0

02

x1

x 1

(c)

0 05 1 150

05

1

15

2times105

Ω1

Amplitu

de

f1

(d)


ndash5

0

5

τ0 200 400 600 800

x 1

(a)

ndash4

ndash2

0

2

4

ndash5 0 5x1

x 1

(b)


ndash14

ndash12

ndash1

ndash08

x1

x 1

(c)

0 05 1 150

2

4

6

Ω1

Amplitu

de

f1

times105

(d)


ndash4

ndash2

0

2

4

0 200 400 600τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)

ndash2 0 2 4ndash3

ndash2

ndash1

0

1

x1

x 1

(c)

0 05 1 150

05

1

15

2 times105

Ω1

Am

plitu

de

2f2

f1

f2f1 ndash f2

(d)





0 100 200 300 400ndash2

0

2

4

τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)


ndash2

ndash1

0

x1

x 1

(c)

0 05 1 150

5

10

15 times104

Ω1

Amplitu

de

f1

f2 12f1

(d)


0 100 200 300 400ndash2

0

2

4

τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)

ndash2 0 2 4ndash3

ndash2

ndash1

0

x1

x 1

(c)

times104

0 05 1 150

5

10

15

Ω1

Amplitu

de

f1f2

(d)


τ

x 1

0 100 200 300 400ndash2

0

2

4

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

ndash2 0 2 4ndash3

ndash2

ndash1

0

(c)

0 05 1 150

5

10

15 times104

Ω1

Amplitu

de

f1f2

(d)


x 1

0 100 200 300ndash4

ndash2

0

2

4

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

ndash2 0 2 4ndash3

ndash2

ndash1

0

(c)

times104

0 05 1 150

5

10

15

Ω1

Amplitu

de f1

f2

(d)





x 1

0 100 200 300ndash2

0

2

4

τ

(a)

x1

x 1

ndash2 0 2 4ndash4

ndash2

0

2

4

(b)

x1

x 1

0 05 1 15 2ndash25

ndash2

ndash15

ndash1

ndash05

(c)

0 05 1 150

05

1

15

2 times105

Ω1

Amplitu

de

f1f2

(d)


x 1

0 100 200 300ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 105 1 15 2

ndash2

ndash18

ndash16

ndash14

ndash12

(c)

times105

0 05 1 150

05

1

15

2

Ω1

Amplitu

de

f1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

1 12 14 16 18ndash22

ndash2

ndash18

ndash16

ndash14

(c)

0 05 1 150

1

2

3 times105

Ω1

Amplitu

def1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

τ

(a)

x1

x 1

ndash1 0 1 2ndash4

ndash2

0

2

4

(b)

x1

x 1

13 14 15 16ndash23

ndash22

ndash21

ndash2

ndash19

(c)

times105

0 05 1 150

1

2

3

Ω1

Amplitu

de

f1f2

(d)




0 05 1 150

2

4

6

8

Amplitu

de

Ω1

f1

f2

f3

2f2


Amplitu

de

10

times104

05 1Ω1

15

5

0

12f1

f1

(a)

05 1Ω1

15

Amplitu

de 10

15 times104

5

00

12f1

f1

(b)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2 f1

f1 ndash f22f2

(c)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2

f1

f1 ndash f2 2f2

(d)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(e)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(f )

times105

Ω1

f2

f1

2f2Amplitu

de

0 05 1 150

1

2

3

(g)




Data Availability




Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




0 200 400 6000

05

1

15

2

τ

x 1

(a)

08 1 12 14 16ndash04

ndash02

0

02

04

x1

x 1

(b)

0 05 1 15 2ndash05

0

05

x1

x 1

(c)

0 05 1

2

4

6times104

Ω1

Amplitu

de f1

(d)


0 500 1000ndash2

0

2

4

τ

x 1

(a)

ndash2 0 2 4ndash1

0

1

2

x1

x 1

(b)


ndash04

ndash02

0

02

x1

x 1

(c)

0 05 1 150

05

1

15

2times105

Ω1

Amplitu

de

f1

(d)


ndash5

0

5

τ0 200 400 600 800

x 1

(a)

ndash4

ndash2

0

2

4

ndash5 0 5x1

x 1

(b)


ndash14

ndash12

ndash1

ndash08

x1

x 1

(c)

0 05 1 150

2

4

6

Ω1

Amplitu

de

f1

times105

(d)


ndash4

ndash2

0

2

4

0 200 400 600τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)

ndash2 0 2 4ndash3

ndash2

ndash1

0

1

x1

x 1

(c)

0 05 1 150

05

1

15

2 times105

Ω1

Am

plitu

de

2f2

f1

f2f1 ndash f2

(d)





0 100 200 300 400ndash2

0

2

4

τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)


ndash2

ndash1

0

x1

x 1

(c)

0 05 1 150

5

10

15 times104

Ω1

Amplitu

de

f1

f2 12f1

(d)


0 100 200 300 400ndash2

0

2

4

τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)

ndash2 0 2 4ndash3

ndash2

ndash1

0

x1

x 1

(c)

times104

0 05 1 150

5

10

15

Ω1

Amplitu

de

f1f2

(d)


τ

x 1

0 100 200 300 400ndash2

0

2

4

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

ndash2 0 2 4ndash3

ndash2

ndash1

0

(c)

0 05 1 150

5

10

15 times104

Ω1

Amplitu

de

f1f2

(d)


x 1

0 100 200 300ndash4

ndash2

0

2

4

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

ndash2 0 2 4ndash3

ndash2

ndash1

0

(c)

times104

0 05 1 150

5

10

15

Ω1

Amplitu

de f1

f2

(d)





x 1

0 100 200 300ndash2

0

2

4

τ

(a)

x1

x 1

ndash2 0 2 4ndash4

ndash2

0

2

4

(b)

x1

x 1

0 05 1 15 2ndash25

ndash2

ndash15

ndash1

ndash05

(c)

0 05 1 150

05

1

15

2 times105

Ω1

Amplitu

de

f1f2

(d)


x 1

0 100 200 300ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 105 1 15 2

ndash2

ndash18

ndash16

ndash14

ndash12

(c)

times105

0 05 1 150

05

1

15

2

Ω1

Amplitu

de

f1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

1 12 14 16 18ndash22

ndash2

ndash18

ndash16

ndash14

(c)

0 05 1 150

1

2

3 times105

Ω1

Amplitu

def1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

τ

(a)

x1

x 1

ndash1 0 1 2ndash4

ndash2

0

2

4

(b)

x1

x 1

13 14 15 16ndash23

ndash22

ndash21

ndash2

ndash19

(c)

times105

0 05 1 150

1

2

3

Ω1

Amplitu

de

f1f2

(d)




0 05 1 150

2

4

6

8

Amplitu

de

Ω1

f1

f2

f3

2f2


Amplitu

de

10

times104

05 1Ω1

15

5

0

12f1

f1

(a)

05 1Ω1

15

Amplitu

de 10

15 times104

5

00

12f1

f1

(b)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2 f1

f1 ndash f22f2

(c)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2

f1

f1 ndash f2 2f2

(d)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(e)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(f )

times105

Ω1

f2

f1

2f2Amplitu

de

0 05 1 150

1

2

3

(g)




Data Availability




Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




0 100 200 300 400ndash2

0

2

4

τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)


ndash2

ndash1

0

x1

x 1

(c)

0 05 1 150

5

10

15 times104

Ω1

Amplitu

de

f1

f2 12f1

(d)


0 100 200 300 400ndash2

0

2

4

τ

x 1

(a)


ndash2

0

2

4

x1

x 1

(b)

ndash2 0 2 4ndash3

ndash2

ndash1

0

x1

x 1

(c)

times104

0 05 1 150

5

10

15

Ω1

Amplitu

de

f1f2

(d)


τ

x 1

0 100 200 300 400ndash2

0

2

4

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

ndash2 0 2 4ndash3

ndash2

ndash1

0

(c)

0 05 1 150

5

10

15 times104

Ω1

Amplitu

de

f1f2

(d)


x 1

0 100 200 300ndash4

ndash2

0

2

4

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

ndash2 0 2 4ndash3

ndash2

ndash1

0

(c)

times104

0 05 1 150

5

10

15

Ω1

Amplitu

de f1

f2

(d)





x 1

0 100 200 300ndash2

0

2

4

τ

(a)

x1

x 1

ndash2 0 2 4ndash4

ndash2

0

2

4

(b)

x1

x 1

0 05 1 15 2ndash25

ndash2

ndash15

ndash1

ndash05

(c)

0 05 1 150

05

1

15

2 times105

Ω1

Amplitu

de

f1f2

(d)


x 1

0 100 200 300ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 105 1 15 2

ndash2

ndash18

ndash16

ndash14

ndash12

(c)

times105

0 05 1 150

05

1

15

2

Ω1

Amplitu

de

f1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

1 12 14 16 18ndash22

ndash2

ndash18

ndash16

ndash14

(c)

0 05 1 150

1

2

3 times105

Ω1

Amplitu

def1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

τ

(a)

x1

x 1

ndash1 0 1 2ndash4

ndash2

0

2

4

(b)

x1

x 1

13 14 15 16ndash23

ndash22

ndash21

ndash2

ndash19

(c)

times105

0 05 1 150

1

2

3

Ω1

Amplitu

de

f1f2

(d)




0 05 1 150

2

4

6

8

Amplitu

de

Ω1

f1

f2

f3

2f2


Amplitu

de

10

times104

05 1Ω1

15

5

0

12f1

f1

(a)

05 1Ω1

15

Amplitu

de 10

15 times104

5

00

12f1

f1

(b)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2 f1

f1 ndash f22f2

(c)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2

f1

f1 ndash f2 2f2

(d)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(e)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(f )

times105

Ω1

f2

f1

2f2Amplitu

de

0 05 1 150

1

2

3

(g)




Data Availability




Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




x 1

0 100 200 300ndash2

0

2

4

τ

(a)

x1

x 1

ndash2 0 2 4ndash4

ndash2

0

2

4

(b)

x1

x 1

0 05 1 15 2ndash25

ndash2

ndash15

ndash1

ndash05

(c)

0 05 1 150

05

1

15

2 times105

Ω1

Amplitu

de

f1f2

(d)


x 1

0 100 200 300ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 105 1 15 2

ndash2

ndash18

ndash16

ndash14

ndash12

(c)

times105

0 05 1 150

05

1

15

2

Ω1

Amplitu

de

f1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

3

τ

(a)

x1

x 1


ndash2

0

2

4

(b)

x1

x 1

1 12 14 16 18ndash22

ndash2

ndash18

ndash16

ndash14

(c)

0 05 1 150

1

2

3 times105

Ω1

Amplitu

def1

f2

(d)


x 1

0 50 100 150 200ndash1

0

1

2

τ

(a)

x1

x 1

ndash1 0 1 2ndash4

ndash2

0

2

4

(b)

x1

x 1

13 14 15 16ndash23

ndash22

ndash21

ndash2

ndash19

(c)

times105

0 05 1 150

1

2

3

Ω1

Amplitu

de

f1f2

(d)




0 05 1 150

2

4

6

8

Amplitu

de

Ω1

f1

f2

f3

2f2


Amplitu

de

10

times104

05 1Ω1

15

5

0

12f1

f1

(a)

05 1Ω1

15

Amplitu

de 10

15 times104

5

00

12f1

f1

(b)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2 f1

f1 ndash f22f2

(c)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2

f1

f1 ndash f2 2f2

(d)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(e)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(f )

times105

Ω1

f2

f1

2f2Amplitu

de

0 05 1 150

1

2

3

(g)




Data Availability




Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0 05 1 150

2

4

6

8

Amplitu

de

Ω1

f1

f2

f3

2f2


Amplitu

de

10

times104

05 1Ω1

15

5

0

12f1

f1

(a)

05 1Ω1

15

Amplitu

de 10

15 times104

5

00

12f1

f1

(b)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2 f1

f1 ndash f22f2

(c)

times104

05 1Ω1

150

Amplitu

de 10

15

5

0

f2

f1

f1 ndash f2 2f2

(d)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(e)

times105

Ω1

f2

f1

f1 ndash f22f2

0 05 1 150

05

1

15

2

Amplitu

de

(f )

times105

Ω1

f2

f1

2f2Amplitu

de

0 05 1 150

1

2

3

(g)




Data Availability




Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Data Availability




Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Documents

Nonlinear Dynamic Characteristics of Fixed-Axis Gear Wear in Multistage … · 2019. 7. 30. · p4 o s o p3 p2 p1 “ – 1(t) – 2(t) – 4 (t) – 1 t) – sp4 (t) ff 4 ff ff