Research ArticleCoupled Bending-Torsional Nonlinear Vibration andBifurcation Characteristics of Spiral Bevel Gear System

Jinli Xu Fancong Zeng and Xingyi Su

Wuhan University of Technology Wuhan 430070 China

Correspondence should be addressed to Fancong Zeng jyl zfc126com

Received 6 December 2016 Revised 13 February 2017 Accepted 15 February 2017 Published 5 March 2017

Academic Editor Lei Zuo

Copyright copy 2017 Jinli Xu 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

A spiral bevel gear system supported on thrust bearings considering the coupled bending-torsional nonlinear vibration is proposedand an eight degrees of freedom (8DOF) lumped parameter dynamic model of the spiral bevel gear system combined with time-varying stiffness static transmission error gear backlash and bearing clearances is investigated The spiral bevel gear system isanalyzed with the equations of motion and the dynamic response is solved using the Runge-Kutta method The effects of meshfrequency mesh damping coefficient load coefficient and gear backlash are revealed which describe the true mesh characteristicsof the spiral bevel gear systemThe bifurcation characteristics as jump discontinuities periodic windows and chaos are obtained bystudying time histories phase plane portraits Poincare maps Fourier spectra and global bifurcation diagrams of the gear systemThe results presented in this study provide some useful information for engineers in designing and controlling such gear systems

1 Introduction

As a typical gear transmission system the spiral bevel gearsystem has been used in main reducer of automobile drivingaxle particularly dynamic response is one of the most impor-tant factors affecting NVH performance its characteristicresearch has attracted the attention of domestic and foreignscholars Accordingly there are many related literaturesstudying the spiral bevel gear system in the past severaldecades

In earlier years many fundamental researches werefocused on the linear analysis of the spiral bevel gear system[1] A two degrees of freedom vibration model of a pair ofbevel gears was established by Kiyono et al [2] the modelwas applied to conduct a stability analysis in which the lineof action vector was simulated by a sine curve Kahramanand Singh [3] derived a single degree-of-freedommodel con-sidering the constant stiffness the dynamic equations withbacklash and transmission error were presented and solved inthe involute gear model Gosselin et al [4] analyzed the statictransmission error of spiral bevel gears and the effects ofthe shape and amplitude of the unloaded transmission errorcurve on the loaded dynamic behaviors were demonstrated

In order to get a basic understanding of dynamic behav-ior a lot of researches had been focusing on nonlinear vibra-tion analysis Litvin et al [5] presented tooth contact analysisand stress analysis in spiral bevel gears by means of finiteelement method Tang et al [6] studied the effect of statictransmission error on nonlinear dynamic response of the spi-ral bevel gear system combining with time-varying stiffnessgear backlash and observed various nonlinear phenomenaincluding periodic solutions bifurcations and chaos Yanget al [7] established a single degree-of-freedom hypoid gearpair dynamic equation which included the time-varyingstiffness transmission error and backlash and obtainedthe FFT responses Wang et al [8] described a generalizednonlinear time-varying (NLTV) dynamic model of a hypoidgear pair with backlash nonlinearity and time-dependentmesh point line of action mesh stiffness and kinematictransmission error Periodic motions were obtained by theincremental harmonic balance method (IHBM) Chang-Jian [9] performed dynamic analysis of bevel-geared rotorsystem supported on a thrust bearing and journal bearingsunder nonlinear suspension Theodossiades and Natsiavas[10] established a simplified dynamic model of motor-driven gear pair considering the gear backlash and bearing

HindawiShock and VibrationVolume 2017 Article ID 6835301 14 pageshttpsdoiorg10115520176835301

2 Shock and Vibration

clearance then numerical results are presented in the formof classical frequency response diagrams revealing the effectof the system parameters on its dynamics Cheng and Lim[11] pointed out that the gear kinematic transmission errorwas the primary source of the vibratory energy excita-tion a new analytical derivation of the hypoid gear-mesh-couplingmechanismbased on the simulation of tooth contactassuming idealized gear geometry was proposed Howeveronly in recent years did the related studies in findingnonlinear behaviors of spiral bevel gear system gain someattentionThe dynamic analysis of a spiral bevel-geared rotor-bearing system was studied by Li and Hu [12] The model-ing of coupled axial-lateral-torsional vibration of the rotorsystem geared by spiral bevel gears was discussed Differ-ent degrees of freedom (DOF) gear dynamic models wereimplemented byMohammed et al [13] their limitations wereevaluated by simulating different DOF for each gear discWang et al [14] established a nonlinear dynamic model ofthe spiral bevel gear with the dynamic relative transmissionerror backlash and time-varying stiffness The vibrationdisplacement and velocity in the torsional horizontal andvertical directions in the spiral bevel gear model underdifferent conditions were depicted and the dynamicalresponses of the geared system with harmonic internalexcitation and parameter excitation were obtained From theliteratures above the nonlinear characteristics of gear systemsuch as stability periodic solutions bifurcations and chaoshave become the most interesting research areas Differentnonlinear parameters will generate an obvious change onthe dynamic response However the bifurcation character-istics researches of nonlinear dynamic parameters as gearbacklash and bearing clearances seem a little deficient thedynamics analysis combined with thrust bearings clear-ances considering the coupled bending-torsional vibration isalso rarely seen In this paper a nonlinear dynamic modelof the spiral bevel gear system is formulated where the time-varying stiffness static transmission error gear backlashand bearing clearances are included The dimensionlessequations of the system are then solved using the Runge-Kutta numerical method The influence of the nonlinearparameters on the spiral bevel gear system is studied andthe nonlinear mesh characteristics are detected and analyzedby construction of the time histories phase plane por-traits Poincare maps Fourier spectra and global bifurcationdiagrams

2 Mathematical Modeling andEquations of Motion

Considering the supported stiffness of a spiral bevel gearsystem is large so the twist vibration can be neglected Thecomplex spiral bevel gear system is simplified by the lumpedmasses method Figure 1 shows a generalized dynamic modelfor eight degrees of freedom (8DOF) considering the coupledbending-torsional vibrationThe gear system ismodeled withrotational and translational displacements as their coordi-nates

z

y

o

b

k(t)

cm

x

e(t)

b11b21

b12

b32

b22

T1

T2

1

2

cx1

cz1cy1

cx2

cy2

cz2

kx2

ky2

kz2

kx1ky1

kz1

b31

Figure 1 Dynamic model of a spiral bevel gear system

The generalized coordinates vector of the nonlineardynamic model can be expressed as

119878 = 1199091 1199101 1199111 1205791 1199092 1199102 1199112 1205792119879 (1)

where 1199091 1199101 and 1199111 are the translations of the pinion alongthe axes 119909 119910 and 119911 1199092 1199102 and 1199112 are the translations of thegear along the axes 119909 119910 and 119911 1205791 and 1205792 are the torsionaldisplacements of the pinion and gear respectively

Here the term ldquopinionrdquo refers to the smaller gear which isa driver gear connected to the input shaft and the term ldquogearrdquorefers to the larger gear which is a driven gear connected tothe output shaft

Thepinion hasmass1198981 andmoment of inertia 1198681 the gearhas mass 1198982 and moment of inertia 1198682

Static transmission error 119890(119905) can be expressed in the form119890 (119905) = 1198900 + 119899sum

119897=1

119860119890119897 cos (119897Ωℎ119905 + Φ119890119897) (2)

where 1198900 is constant amplitude of static transmission error119860119890119897 is variable amplitude of static transmission error Ωℎ isexcitation frequency and Φ119890119897 is phase angle

Since the stiffness is periodically time-varying with themesh frequency its analytical formulation can be obtained bymeans of a Fourier expansion [15]

119896 (119905) = 119896119898 + 119873sum119897=1

119860119896119897 cos (119897Ωℎ119905 + Φ119896119897) (3)

Here 119896119898 is mean value of the mesh stiffness 119860119896119897 isstiffness fluctuation amplitude and Φ119896119897 is phase angle

The backlash function 119891(120575) and the bearing clearancesfunctions 119891(119909119895) 119891(119910119895) and 119891(119911119895) (119895 = 1 2) can be writtenas

119891 (120575) = 120575 minus 119887 120575 gt 1198870 |120575| le 119887120575 + 119887 120575 lt minus119887

Shock and Vibration 3

119891 (119909119895) = 119909119895 minus 1198871119895 119909119895 gt 11988711198950 1003816100381610038161003816100381611990911989510038161003816100381610038161003816 le 1198871119895119909119895 + 1198871119895 119909119895 lt minus1198871119895

119891 (119910119895) = 119910119895 minus 1198872119895 119910119895 gt 11988721198950 1003816100381610038161003816100381611991011989510038161003816100381610038161003816 le 1198872119895119910119895 + 1198872119895 119910119895 lt minus1198872119895

119891 (119911119895) = 119911119895 minus 1198873119895 119911119895 gt 11988731198950 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 le 1198873119895119911119895 + 1198873119895 119911119895 lt minus1198873119895

(4)

where 2119887 represents the total backlash 21198871119895 and 21198872119895 representthe bearing clearances in horizontal plane 21198873119895 represents thebearing clearances in vertical plane

The dynamic mesh force along the line of action 119865119899 canbe expressed as 119865119899 = 119896 (119905) 119891 (120575) + 119888119898 (5)

The dynamic mesh force along the coordinate directions119865119894 (119894 = 119909 119910 119911) can be expressed as119865119909 = minus1198861119865119899119865119910 = 1198862119865119899119865119911 = 1198863119865119899 (6)

where 1205721 = sin120572119899 cos 1205741 + cos120572119899 sin120573 sin 1205741 1205722 =sin120572119899 sin 1205741 minus cos120572119899 sin120573 cos 1205741 1205723 = cos120572119899 sin120573 119888119898 is meanvalue of the mesh damping 120572119899 is pressure angle 120573 is helixangle and 1205741 is cone angle of the pinion

119888119898 = 2120577radic 119896119898(11198981 + 11198982) (7)

Here 120577 is damping ratioFrom the proposed concept the equations of motion

of the coupled bending-torsional vibration model can bederived as11989811 + 11988811990911 + 1198961199091119891 (1199091) = minus119865119909 (8)11989811 + 11988811991011 + 1198961199101119891 (1199101) = minus119865119910 (9)11989811 + 11988811991111 + 1198961199111119891 (1199111) = minus119865119911 (10)11986811 = 1198791 minus 1198651198991199031 (11)11989822 + 11988811990922 + 1198961199092119891 (1199092) = 119865119909 (12)11989822 + 11988811991022 + 1198961199102119891 (1199102) = 119865119910 (13)11989822 + 11988811991122 + 1198961199112119891 (1199112) = 119865119911 (14)11986822 = minus1198792 + 119865119899119903119898 (15)

where 119888119894119895 (119894 = 119909 119910 119911 119895 = 1 2) are damping coefficientsof the supported structure for the thrust bearings 119896119894119895 (119894 =119909 119910 119911 119895 = 1 2) are stiffness coefficients of the supportedstructure for the thrust bearings 119879119895 (119895 = 1 2) are mean loadtorques on the pinion and gear respectively 119865119894 (119894 = 119909 119910 119911)are dynamic loads along the axes 119909 119910 and 119911 for the pinionand gear respectively 1199031 is base radius of the pinion and 119903119898is distance between the acting point of the normal force andthe center of the rotation of the gear

The motion equations of the spiral bevel gear system inmatrix form can be expressed as

119872 + 119862 + 119870 119878 = 119865 (16)

where 119872 is lumped mass matrix 119862 is damping matrix 119870 isstiffness matrix 119865 is external excitation force vector

Obviously this system is semidefinite [16] (11) and (15)can be merged to an equation by introducing a new variable120575 120575 = (1199091 minus 1199092) 1198864 minus (1199101 minus 1199102) 1198865minus (1199111 minus 1199112 + 11990311205791 minus 1199031198981205792) 1198866 minus 119890 (119905) (17)

where 1198864 = cos 1205741 sin120572119899 1198865 = cos 1205741 cos120572119899 sin120573 and 1198866 =cos120572119899 cos120573

And the equivalent mass and static load of gear transmis-sion are

119898119890 = 1198681119868211986811199031198982 + 119868211990312 119865119898 = 11987911199031 = 1198792119903119898 (18)

Next introducing the characteristic length 119887119888 and assum-ing the following set of dimensionless parameters

119883119894 = 119909119894119887119888 119884119894 = 119910119894119887119888 119885119894 = 119911119894119887119888 120575 = 120575119887119888 120591 = 120596119899119905

120596119899 = radic 119896119898119898119890 120596119894119895 = radic 119896119894119895119898119895

4 Shock and Vibration

120577119894119895 = 119888119894119895(2119898119894120596119899) 119896119894119895 = 12059611989411989521205962119899 120596ℎ = Ωℎ120596119899 120577119898119894119895 = 119888119898(2119898119895120596119899) 120577119898 = 119888119898(2119898119890120596119899)

119896119898119894119895 = 119896 (119905)(1198981198951205962119899) 119891119898 = 1198651198981198981198901198871198881205962119899 119891V = 119865V1198981198901198871198881205962119899 119891119890 = 119873sum119897=1

119860119890119897119887119888 (119897120596ℎ)2 cos (119897120596ℎ120591 + Φ119890119897) 119896 (119905) = 119896 (119905)119896119898 = 1 + 119873sum

119897=1

119860119896119897119896119898 cos (119897120596ℎ120591 + Φ119896119897) (119894 = 119909 119910 119911 119895 = 1 2)

119891 (120575) =

120575 minus 119887119887119888 120575 gt 1198871198871198880 1003816100381610038161003816100381612057510038161003816100381610038161003816 le 119887119887119888120575 + 119887119887119888 120575 lt minus 119887119887119888

119891 (119883119895) =

119883119895 minus 1198871119895119887119888 119883119895 gt 11988711198951198871198880 1003816100381610038161003816100381611988311989510038161003816100381610038161003816 le 1198871119895119887119888119883119895 + 1198871119895119887119888 119883119895 lt minus1198871119895119887119888

119891 (119884119895) =

119884119895 minus 1198872119895119887119888 119884119895 gt 11988721198951198871198880 1003816100381610038161003816100381611988411989510038161003816100381610038161003816 le 1198872119895119887119888119884119895 + 1198872119895119887119888 119884119895 lt minus1198872119895119887119888

119891 (119885119895) =

119885119895 minus 1198873119895119887119888 119885119895 gt 11988731198951198871198880 1003816100381610038161003816100381611988511989510038161003816100381610038161003816 le 1198873119895119887119888119885119895 + 1198873119895119887119888 119885119895 lt minus1198873119895119887119888 (19)

Input torque fluctuation is not considered in the currentstudy but only static torque load which means 119891V = 0 Inorder to reduce the complexity of the computation higherorders of the time-varying stiffness and static transmissionerror are truncated without losing generality So the dimen-sionless stiffness and the dimensionless static transmissionerror are calculated by the first-order component

119897 = 1Φ119890119897 = 0Φ119896119897 = 120587119896 (119905) = 1 minus 120576 cos (120596ℎ120591)

119891119890 = 120582120596ℎ2 sin (120596ℎ120591) (20)

where 120576 is stiffness coefficient 120582 is amplitude of statictransmission error

Equations (8)ndash(15) can be expressed in a matrix form

[[[[[[[[[[[[[[[[[

1 0 0 0 0 0 00 1 0 0 0 0 00 0 1 0 0 0 00 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 0minus1198864 1198865 1198866 1198864 minus1198865 minus1198866 1

]]]]]]]]]]]]]]]]]

(((((((((((((

111222

)))))))))))))

+ 2[[[[[[[[[[[[[[[[[

1205771199091 0 0 0 0 0 119886112057711989811990910 1205771199101 0 0 0 0 minus119886212057711989811991010 0 1205771199111 0 0 0 minus119886312057711989811991110 0 0 1205771199092 0 0 minus119886112057711989811990920 0 0 0 1205771199102 0 119886212057711989811991020 0 0 0 0 1205771199112 119886312057711989811991120 0 0 0 0 0 21205771198981198866

]]]]]]]]]]]]]]]]]

(((((((((((((

111222

)))))))))))))

Shock and Vibration 5

Table 1 The parameters of the spiral bevel gear system

Parameters Pinion GearTeeth numbers 8 41Module (mm) 4161Mass (kg) 1 5Inertia moment (kgmm2) 3000 11200Base circle (mm) 30 130Pressure angle (∘) 120572119899 = 2115Helix angle (∘) 120573 = 35Cone angle (∘) 1205741 = 1275 1205742 = 765Tooth width (mm) 28Mean mesh stiffness (Nmm) 119896119898 = 1191198646Bearing stiffness (Nmm) 119896119894119895 = 0351198646Half of bearing clearance (120583m) 1198871119895 = 1198872119895 = 1198873119895 = 10

+[[[[[[[[[[[[[[[[[[[

1198961199091 0 0 0 0 0 119886111989611989811990910 1198961199101 0 0 0 0 minus119886211989611989811991010 0 1198961199111 0 0 0 minus119886311989611989811991110 0 0 1198961199092 0 0 minus119886111989611989811990920 0 0 0 1198961199102 0 119886211989611989811991020 0 0 0 0 1198961199112 119886311989611989811991120 0 0 0 0 0 1198866119896 (119905)

]]]]]]]]]]]]]]]]]]]

((((((((((((

119891 (1198831)119891 (1198841)119891 (1198851)119891 (1198832)119891 (1198842)119891 (1198852)119891 (120575)

))))))))))))

=((((((((((

000000119891119898 + 119891V + 119891119890

))))))))))

(21)

3 Numerical Solutions

Due to the complexity of the spiral bevel gear system andalso the difficulty and limitation of the analytical methodsthe numerical method is commonly used to analyze thegear system In this paper the nonlinear dynamic equationsare solved using the fourth order Runge-Kutta methodwhich is generally applicable to strong nonlinearity [17] Thealgorithm is implemented in MATLAB that is a widely usedmatrix and numerical analysis program [18] Without losinggenerality the characteristic length is set to 119887119888 = 10 120583mand the same value can be found in Sunrsquos thesis [19] Theother parameters of the gear transmission system are givenin Table 1

31 Effect of Mesh Frequency A bifurcation diagram sum-marizes the essential dynamics of the gear system and istherefore a useful means of observing its nonlinear dynamicresponse [20] For our subsequent numerical study set halfof the backlash as 119887 = 40 120583m damping ratio as 120577 = 008amplitude of static transmission error as 120582 = 03 and stiffnesscoefficient as 120576 = 02 In this system the dimensionlessmesh frequency120596ℎ is commonly used as an important controlparameter Figure 2(a) shows the bifurcation diagram for thespiral bevel gear system displacement against the dimen-sionless mesh frequency 120596ℎ Figures 2(b) and 2(c) show theenhanced bifurcation diagrams with bifurcation parameters120596ℎ = 065sim12 and 120596ℎ = 14sim215 respectively The bifurcationresults show that the system behaves as 1T-periodic motionat low mesh frequency and the periodic motion persists until120596ℎ gt 0425 A jump phenomenon can be observed and2T-periodic motion appears at the region 120596ℎ = 0425sim08After a long 1T-periodic motion the system enters into atransient chaotic motion after 120596ℎ gt 115 then the dynamicbehavior behaves as nonperiodic motion until 120596ℎ = 15With the increase of mesh frequency a number of periodicwindows appear At the region 120596ℎ = 1sim2 1T-periodic motion2T-periodic motion and 119899T-periodic motion are shownin the chaotic area It can be observed that 119899T-periodicbifurcation enters into chaos and 7T-periodic bifurcationenters into 119899T-periodic bifurcation obviously in Figure 2(c)Finally 2T-periodic motion transits to 1T-periodic motionuntil 120596ℎ = 205

For a better clarity the other analytical methods forobserving nonlinear dynamic responses are necessary Fig-ures 3ndash5 illustrate the time histories phase plane portraitsPoincare maps and Fourier spectra for the spiral bevel gearmodel at various values of the dimensionless mesh frequency120596ℎ = 1120596ℎ = 16 and120596ℎ = 177 At120596ℎ = 1 the dynamic behaviorbehaves as 1T-periodicmotionAt120596ℎ = 16 chaotic behavior isclearly visible At120596ℎ = 177 the system enters into 6T-periodicmotion from chaotic motion

32 Effect of Damping Ratio In the following section whenthe damping ratio 120577 is changed and all the other parameters

6 Shock and Vibration

2

15

1

05

0

minus05

minus1

minus15

minus2

1816 2 2212 14080604 102

ℎ

(a)

08 09 1 1107

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)2

15

1

05

0

minus05

minus1

minus15

minus2

ℎ

212191817161514

(c)

Figure 2 Bifurcation diagrams using 120596ℎ as control parameter (a) general view 120596ℎ = 02sim23 (b) enhanced view 120596ℎ = 065sim12 (c) enhancedview 120596ℎ = 14sim215

047

048

049

05

051

300 400 500 600200

(a)

047 048 049 05 051

0

001

002

d

minus001

minus002

(b)

0 1 2

0

1

2

d

minus1

minus1minus2

minus2

(c)

0

20

40

60

80

100

102 03 04 05 06 07 08 09010

Frequency of spectrum

(d)

Figure 3 120596ℎ = 1 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 7

0

05

1

minus1

minus05

300 400 500 600200

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 4 120596ℎ = 16 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

0

05

1

15

minus1

minus05

300 400 500 600200

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 5 120596ℎ = 177 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

8 Shock and Vibration

2

15

1

05

0

minus05

minus1

minus15

minus21816 2 2212 14080604 102

ℎ

(a)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(c)

Figure 6 Bifurcation diagrams using 120577 as control parameter (a) 120577 = 001 (b) 120577 = 004 (c) 120577 = 012keep fixed the bifurcation diagrams are shown in Figure 6The chaotic motions are shown through the numericalsimulation for three different values of 120577 = 001 120577 = 004and 120577 = 012 (120577 = 008 see Figure 2(a)) According tothe comparison of the bifurcation diagrams with differentdamping ratio values it can be observed that chaotic regionbecomes narrower as 120577 is increased chaotic region getssmaller and smaller Therefore the increase in 120577 tends todecrease the nonlinearity dynamic responses

33 Effect of Load Coefficient Different vibration character-istics will be shown under different loads For heavy loadedcondition the value of the load coefficient parameter is fixedas 119891 = 119891119898120582 = 2 (119891119898 = 01 120582 = 005) For lightloaded condition the value of the load coefficient parameteris fixed as 119891 = 119891119898120582 = 13 (119891119898 = 01 120582 = 03) Thetime histories phase plane portraits Poincare maps andFourier spectra are shown in Figures 7 and 8 These figuresshowdynamic responses for heavy loaded condition and lightloaded condition at the dimensionless mesh frequency 120596ℎ= 14 respectively As illustrated in Figure 7 the system islinear system the steady-state response is harmonic responseThere is obviously no tooth impact and the system behavesas 1T-periodic motion It is illustrated that the dynamic

characteristics of the gear are not changedmuchunder loadedcondition when the dynamic behavior behaves as periodicmotion As illustrated in Figure 8 compared with the heavyloaded condition the magnitude of the spectrum becomeslarger The displacement along the line of action is greaterthan 0 and is less than 0 so the case of tooth impact can beobserved in the time history plot

Figures 9 and 10 show dynamic responses for heavyloaded condition and light loaded condition at the dimen-sionless mesh frequency 120596ℎ = 18 respectively Nonlinearvibration characteristics can be seen occurring in thesetwo situations However the intensities are not the sameAs illustrated in Figure 9 the motion state is 1T-periodicharmonic motion under heavy loaded condition It can beseen that the spiral bevel gear system is in chaotic motionunder light loaded condition and the response of the systemis irregular in the chaotic state from Figure 10 Furthermorechaotic phenomenon is more likely to occur in light loadedcondition compared with heavy loaded condition withoutchanging other conditions

34 Effect of Gear Backlash Gear backlash is another impor-tant parameter which affects the dynamic responses substan-tially Figure 11 presents the bifurcation diagrams for three

Shock and Vibration 9

200 300 400 500 6000

02

04

06

08

(a)

0 02 04 06 08

0

02

04

d

minus04

minus02

(b)

0 1 2

0

1

2

d

minus1

minus1minus2

minus2

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 7 120596ℎ = 18 and 119891 = 2 (a) time history (b) phase plane(c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus15minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 8 120596ℎ = 14 and 119891 = 13 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

10 Shock and Vibration

200 300 400 500 600

04

05

06

07

(a)

035 04 045 05 055 06 065 07

0

02

04

minus04

minus02

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 9 120596ℎ = 18 and 119891 = 2 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

15

minus1

minus05

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

Frequency of spectrum

01 02 03 04 05 06 07 08 09 10

(d)

Figure 10 120596ℎ = 18 and 119891 = 13 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 11

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(a)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(c)

Figure 11 Bifurcation diagrams using 119887 as control parameter (a) 119887 = 10 120583m (b) 119887 = 80 120583m (c) 119887 = 120 120583m

different values of half of the backlash 119887 = 10 120583m 119887 = 80 120583mand 119887 = 120120583m (119887 = 40 120583m see Figure 2(a)) A substantialregion of jump phenomenon subharmonic quasi-periodicand chaotic responses are contained in Figure 11 Withincreasing b the system becomes more and more unstableand consequently chaotic behavior expands

For the further study on the effect of gear backlash thetime histories phase plane portraits Poincare maps andFourier spectra of the dynamic responses at 120596ℎ = 223 fornumerous gear backlash values are compared in Figures12ndash15 According to these figures a transition from quasi-periodic motion to the chaotic dynamics is seen with theincrease of 119887 As observed in Poincare maps the variousforms of the system from the quasi-periodic motion areseen and with the increase of the bifurcation parameter theunstable attractive region is enlarged eventually leading tochaos

4 Conclusion

In the present paper a spiral bevel gear system supportedon thrust bearings considering the coupled bending-torsionalnonlinear vibration is proposed and an 8DOF lumpedparameter dynamic model of the spiral bevel gear system

combining with time-varying stiffness static transmissionerror gear backlash and bearing clearances is investigatedThe dimensionless equations of the system are solved usingthe Runge-Kutta numerical method The dynamics of thesystem are analyzed with reference to its bifurcation dia-grams time histories phase plane portraits Poincare mapsand Fourier spectra and jumpphenomena periodicmotionsand chaotic motions are found in this study

In this study mesh frequency is first expressed to studythe effects of their variations on dynamic response Withthe change of mesh frequency the repeated bifurcationphenomena and the complex chaotic motions are observedand the evolution process and rule of the system are revealedThe effect of damping ratio is secondly analyzed It is shownthat chaotic region becomes narrower as damping ratio isincreased Other important parameters like load coefficientand gear backlash are also considered in the current analysisLoad coefficient affects light loaded condition more thanheavy loaded condition The increase of gear backlash willexpand the effect of nonlinearity effect and increase thechaotic region of the gear system

The selection of the dynamic parameters plays an impor-tant role in designing and controlling such gear systems thesuitable values should be chosen so that chaotic behavior will

12 Shock and Vibration

200 300 400 500 600

0

02

04

06

minus02

minus04

(a)

0 02 04 06

0

05

1

15

minus05

minus04 minus02minus1

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 12 119887 = 10 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

minus05

(a)

0 05 1

0

1

2

minus2

minus1

minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 13 119887 = 40 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 13

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1minus15 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 14 119887 = 80 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

3

minus1

(a)

0 1 2 3

0

1

2

minus2

minus1

minus1minus3

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 15 119887 = 120 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

14 Shock and Vibration

be decreased A more precise model to better describe thespiral bevel gear system will be proposed and deep study willbe analyzed in the future

Competing Interests

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

Acknowledgments

This research was supported by the Natural Science Foun-dation of Hubei Province China (Grant no 2014 CFB827)The authors of the paper express their deep gratitude for thefinancial support of the research project

References

[1] HN Ozguven andD RHouser ldquoMathematicalmodels used ingear dynamicsmdasha reviewrdquo Journal of Sound and Vibration vol121 no 3 pp 383ndash411 1988

[2] S Kiyono Y Fujii and Y Suzuki ldquoAnalysis of vibration of bevelgearsrdquo Bulletin of the Japan Society of Mechanical Engineers vol24 no 188 pp 441ndash446 1981

[3] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[4] C Gosselin L Cloutier and Q D Nguyen ldquoA general formu-lation for the calculation of the load sharing and transmissionerror under load of spiral bevel and hypoid gearsrdquo Mechanismand Machine Theory vol 30 no 3 pp 433ndash450 1995

[5] F L Litvin A Fuentes Q Fan and R F Handschuh ldquoCom-puterized design simulation of meshing and contact and stressanalysis of face-milled formate generated spiral bevel gearsrdquoMechanism and Machine Theory vol 37 no 5 pp 441ndash4592002

[6] J-Y Tang Z-H Hu L-J Wu and S-Y Chen ldquoEffect of statictransmission error on dynamic responses of spiral bevel gearsrdquoJournal of Central South University vol 20 no 3 pp 640ndash6472013

[7] H B Yang J P Gao Z D Fang X Z Deng and Y W ZhouldquoNonlinear dynamics of hypoid gearsrdquoAutomotive Engineeringvol 22 no 1 pp 51ndash54 2002 (Chinese)

[8] J Wang T C Lim and M F Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibration vol308 no 1-2 pp 302ndash329 2007

[9] C-W Chang-Jian ldquoNonlinear dynamic analysis for bevel-gearsystem under nonlinear suspension-bifurcation and chaosrdquoApplied Mathematical Modelling vol 35 no 7 pp 3225ndash32372011

[10] S Theodossiades and S Natsiavas ldquoPeriodic and chaoticdynamics of motor-driven gear-pair systems with backlashrdquoChaos Solitons and Fractals vol 12 no 13 pp 2427ndash2440 2001

[11] Y Cheng and T C Lim ldquoVibration analysis of hypoid trans-missions applying an exact geometry-based gear mesh theoryrdquoJournal of Sound andVibration vol 240 no 3 pp 519ndash543 2001

[12] M Li and H Y Hu ldquoDynamic analysis of a spiral bevel-gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 259no 3 pp 605ndash624 2003

[13] O DMohammedM Rantatalo and J-O Aidanpaa ldquoDynamicmodelling of a one-stage spur gear system and vibration-basedtooth crack detection analysisrdquo Mechanical Systems and SignalProcessing vol 54-55 pp 293ndash305 2015

[14] S MWang YW Shen and H J Dong ldquoNon-linear dynamicalcharacteristics of a spiral bevel gear system with backlash andtime-varying stiffnessrdquo Chinese Journal of Mechanical Engineer-ing vol 39 no 2 pp 28ndash32 2003 (Chinese)

[15] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo NonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 75 no 4 pp 783ndash806 2014

[16] Y Shen S Yang and X Liu ldquoNonlinear dynamics of a spurgear pair with time-varying stiffness and backlash based onincremental harmonic balance methodrdquo International Journalof Mechanical Sciences vol 48 no 11 pp 1256ndash1263 2006

[17] R F Li and J J Wang Gear System Dynamics Science Publish-ing Beijing China 1997 (Chinese)

[18] H B Wilson L H Turcotte and D Halpern AdvancedMathematics and Mechanics Applications Using Matlab CRCPress Company Boca Raton Fla USA 3rd edition 2003

[19] T Sun and H Hu ldquoNonlinear dynamics of a planetary gearsystem with multiple clearancesrdquo Mechanism and MachineTheory vol 38 no 12 pp 1371ndash1390 2003

[20] Z Hu J Tang S Chen and D Lei ldquoEffect of mesh stiffness onthe dynamic response of face gear transmission systemrdquo Journalof Mechanical Design Transactions of the ASME vol 135 no 7Article ID 071005 7 pages 2013

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Shock and Vibration 3

119891 (119909119895) = 119909119895 minus 1198871119895 119909119895 gt 11988711198950 1003816100381610038161003816100381611990911989510038161003816100381610038161003816 le 1198871119895119909119895 + 1198871119895 119909119895 lt minus1198871119895

119891 (119910119895) = 119910119895 minus 1198872119895 119910119895 gt 11988721198950 1003816100381610038161003816100381611991011989510038161003816100381610038161003816 le 1198872119895119910119895 + 1198872119895 119910119895 lt minus1198872119895

119891 (119911119895) = 119911119895 minus 1198873119895 119911119895 gt 11988731198950 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 le 1198873119895119911119895 + 1198873119895 119911119895 lt minus1198873119895

(4)

where 2119887 represents the total backlash 21198871119895 and 21198872119895 representthe bearing clearances in horizontal plane 21198873119895 represents thebearing clearances in vertical plane

The dynamic mesh force along the line of action 119865119899 canbe expressed as 119865119899 = 119896 (119905) 119891 (120575) + 119888119898 (5)

The dynamic mesh force along the coordinate directions119865119894 (119894 = 119909 119910 119911) can be expressed as119865119909 = minus1198861119865119899119865119910 = 1198862119865119899119865119911 = 1198863119865119899 (6)

where 1205721 = sin120572119899 cos 1205741 + cos120572119899 sin120573 sin 1205741 1205722 =sin120572119899 sin 1205741 minus cos120572119899 sin120573 cos 1205741 1205723 = cos120572119899 sin120573 119888119898 is meanvalue of the mesh damping 120572119899 is pressure angle 120573 is helixangle and 1205741 is cone angle of the pinion

119888119898 = 2120577radic 119896119898(11198981 + 11198982) (7)

Here 120577 is damping ratioFrom the proposed concept the equations of motion

of the coupled bending-torsional vibration model can bederived as11989811 + 11988811990911 + 1198961199091119891 (1199091) = minus119865119909 (8)11989811 + 11988811991011 + 1198961199101119891 (1199101) = minus119865119910 (9)11989811 + 11988811991111 + 1198961199111119891 (1199111) = minus119865119911 (10)11986811 = 1198791 minus 1198651198991199031 (11)11989822 + 11988811990922 + 1198961199092119891 (1199092) = 119865119909 (12)11989822 + 11988811991022 + 1198961199102119891 (1199102) = 119865119910 (13)11989822 + 11988811991122 + 1198961199112119891 (1199112) = 119865119911 (14)11986822 = minus1198792 + 119865119899119903119898 (15)

where 119888119894119895 (119894 = 119909 119910 119911 119895 = 1 2) are damping coefficientsof the supported structure for the thrust bearings 119896119894119895 (119894 =119909 119910 119911 119895 = 1 2) are stiffness coefficients of the supportedstructure for the thrust bearings 119879119895 (119895 = 1 2) are mean loadtorques on the pinion and gear respectively 119865119894 (119894 = 119909 119910 119911)are dynamic loads along the axes 119909 119910 and 119911 for the pinionand gear respectively 1199031 is base radius of the pinion and 119903119898is distance between the acting point of the normal force andthe center of the rotation of the gear

The motion equations of the spiral bevel gear system inmatrix form can be expressed as

119872 + 119862 + 119870 119878 = 119865 (16)

where 119872 is lumped mass matrix 119862 is damping matrix 119870 isstiffness matrix 119865 is external excitation force vector

Obviously this system is semidefinite [16] (11) and (15)can be merged to an equation by introducing a new variable120575 120575 = (1199091 minus 1199092) 1198864 minus (1199101 minus 1199102) 1198865minus (1199111 minus 1199112 + 11990311205791 minus 1199031198981205792) 1198866 minus 119890 (119905) (17)

where 1198864 = cos 1205741 sin120572119899 1198865 = cos 1205741 cos120572119899 sin120573 and 1198866 =cos120572119899 cos120573

And the equivalent mass and static load of gear transmis-sion are

119898119890 = 1198681119868211986811199031198982 + 119868211990312 119865119898 = 11987911199031 = 1198792119903119898 (18)

Next introducing the characteristic length 119887119888 and assum-ing the following set of dimensionless parameters

119883119894 = 119909119894119887119888 119884119894 = 119910119894119887119888 119885119894 = 119911119894119887119888 120575 = 120575119887119888 120591 = 120596119899119905

120596119899 = radic 119896119898119898119890 120596119894119895 = radic 119896119894119895119898119895

4 Shock and Vibration

120577119894119895 = 119888119894119895(2119898119894120596119899) 119896119894119895 = 12059611989411989521205962119899 120596ℎ = Ωℎ120596119899 120577119898119894119895 = 119888119898(2119898119895120596119899) 120577119898 = 119888119898(2119898119890120596119899)

119896119898119894119895 = 119896 (119905)(1198981198951205962119899) 119891119898 = 1198651198981198981198901198871198881205962119899 119891V = 119865V1198981198901198871198881205962119899 119891119890 = 119873sum119897=1

119860119890119897119887119888 (119897120596ℎ)2 cos (119897120596ℎ120591 + Φ119890119897) 119896 (119905) = 119896 (119905)119896119898 = 1 + 119873sum

119897=1

119860119896119897119896119898 cos (119897120596ℎ120591 + Φ119896119897) (119894 = 119909 119910 119911 119895 = 1 2)

119891 (120575) =

120575 minus 119887119887119888 120575 gt 1198871198871198880 1003816100381610038161003816100381612057510038161003816100381610038161003816 le 119887119887119888120575 + 119887119887119888 120575 lt minus 119887119887119888

119891 (119883119895) =

119883119895 minus 1198871119895119887119888 119883119895 gt 11988711198951198871198880 1003816100381610038161003816100381611988311989510038161003816100381610038161003816 le 1198871119895119887119888119883119895 + 1198871119895119887119888 119883119895 lt minus1198871119895119887119888

119891 (119884119895) =

119884119895 minus 1198872119895119887119888 119884119895 gt 11988721198951198871198880 1003816100381610038161003816100381611988411989510038161003816100381610038161003816 le 1198872119895119887119888119884119895 + 1198872119895119887119888 119884119895 lt minus1198872119895119887119888

119891 (119885119895) =

119885119895 minus 1198873119895119887119888 119885119895 gt 11988731198951198871198880 1003816100381610038161003816100381611988511989510038161003816100381610038161003816 le 1198873119895119887119888119885119895 + 1198873119895119887119888 119885119895 lt minus1198873119895119887119888 (19)

Input torque fluctuation is not considered in the currentstudy but only static torque load which means 119891V = 0 Inorder to reduce the complexity of the computation higherorders of the time-varying stiffness and static transmissionerror are truncated without losing generality So the dimen-sionless stiffness and the dimensionless static transmissionerror are calculated by the first-order component

119897 = 1Φ119890119897 = 0Φ119896119897 = 120587119896 (119905) = 1 minus 120576 cos (120596ℎ120591)

119891119890 = 120582120596ℎ2 sin (120596ℎ120591) (20)

where 120576 is stiffness coefficient 120582 is amplitude of statictransmission error

Equations (8)ndash(15) can be expressed in a matrix form

[[[[[[[[[[[[[[[[[

1 0 0 0 0 0 00 1 0 0 0 0 00 0 1 0 0 0 00 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 0minus1198864 1198865 1198866 1198864 minus1198865 minus1198866 1

]]]]]]]]]]]]]]]]]

(((((((((((((

111222

)))))))))))))

+ 2[[[[[[[[[[[[[[[[[

1205771199091 0 0 0 0 0 119886112057711989811990910 1205771199101 0 0 0 0 minus119886212057711989811991010 0 1205771199111 0 0 0 minus119886312057711989811991110 0 0 1205771199092 0 0 minus119886112057711989811990920 0 0 0 1205771199102 0 119886212057711989811991020 0 0 0 0 1205771199112 119886312057711989811991120 0 0 0 0 0 21205771198981198866

]]]]]]]]]]]]]]]]]

(((((((((((((

111222

)))))))))))))

Shock and Vibration 5

Table 1 The parameters of the spiral bevel gear system

Parameters Pinion GearTeeth numbers 8 41Module (mm) 4161Mass (kg) 1 5Inertia moment (kgmm2) 3000 11200Base circle (mm) 30 130Pressure angle (∘) 120572119899 = 2115Helix angle (∘) 120573 = 35Cone angle (∘) 1205741 = 1275 1205742 = 765Tooth width (mm) 28Mean mesh stiffness (Nmm) 119896119898 = 1191198646Bearing stiffness (Nmm) 119896119894119895 = 0351198646Half of bearing clearance (120583m) 1198871119895 = 1198872119895 = 1198873119895 = 10

+[[[[[[[[[[[[[[[[[[[

1198961199091 0 0 0 0 0 119886111989611989811990910 1198961199101 0 0 0 0 minus119886211989611989811991010 0 1198961199111 0 0 0 minus119886311989611989811991110 0 0 1198961199092 0 0 minus119886111989611989811990920 0 0 0 1198961199102 0 119886211989611989811991020 0 0 0 0 1198961199112 119886311989611989811991120 0 0 0 0 0 1198866119896 (119905)

]]]]]]]]]]]]]]]]]]]

((((((((((((

119891 (1198831)119891 (1198841)119891 (1198851)119891 (1198832)119891 (1198842)119891 (1198852)119891 (120575)

))))))))))))

=((((((((((

000000119891119898 + 119891V + 119891119890

))))))))))

(21)

3 Numerical Solutions

Due to the complexity of the spiral bevel gear system andalso the difficulty and limitation of the analytical methodsthe numerical method is commonly used to analyze thegear system In this paper the nonlinear dynamic equationsare solved using the fourth order Runge-Kutta methodwhich is generally applicable to strong nonlinearity [17] Thealgorithm is implemented in MATLAB that is a widely usedmatrix and numerical analysis program [18] Without losinggenerality the characteristic length is set to 119887119888 = 10 120583mand the same value can be found in Sunrsquos thesis [19] Theother parameters of the gear transmission system are givenin Table 1

31 Effect of Mesh Frequency A bifurcation diagram sum-marizes the essential dynamics of the gear system and istherefore a useful means of observing its nonlinear dynamicresponse [20] For our subsequent numerical study set halfof the backlash as 119887 = 40 120583m damping ratio as 120577 = 008amplitude of static transmission error as 120582 = 03 and stiffnesscoefficient as 120576 = 02 In this system the dimensionlessmesh frequency120596ℎ is commonly used as an important controlparameter Figure 2(a) shows the bifurcation diagram for thespiral bevel gear system displacement against the dimen-sionless mesh frequency 120596ℎ Figures 2(b) and 2(c) show theenhanced bifurcation diagrams with bifurcation parameters120596ℎ = 065sim12 and 120596ℎ = 14sim215 respectively The bifurcationresults show that the system behaves as 1T-periodic motionat low mesh frequency and the periodic motion persists until120596ℎ gt 0425 A jump phenomenon can be observed and2T-periodic motion appears at the region 120596ℎ = 0425sim08After a long 1T-periodic motion the system enters into atransient chaotic motion after 120596ℎ gt 115 then the dynamicbehavior behaves as nonperiodic motion until 120596ℎ = 15With the increase of mesh frequency a number of periodicwindows appear At the region 120596ℎ = 1sim2 1T-periodic motion2T-periodic motion and 119899T-periodic motion are shownin the chaotic area It can be observed that 119899T-periodicbifurcation enters into chaos and 7T-periodic bifurcationenters into 119899T-periodic bifurcation obviously in Figure 2(c)Finally 2T-periodic motion transits to 1T-periodic motionuntil 120596ℎ = 205

For a better clarity the other analytical methods forobserving nonlinear dynamic responses are necessary Fig-ures 3ndash5 illustrate the time histories phase plane portraitsPoincare maps and Fourier spectra for the spiral bevel gearmodel at various values of the dimensionless mesh frequency120596ℎ = 1120596ℎ = 16 and120596ℎ = 177 At120596ℎ = 1 the dynamic behaviorbehaves as 1T-periodicmotionAt120596ℎ = 16 chaotic behavior isclearly visible At120596ℎ = 177 the system enters into 6T-periodicmotion from chaotic motion

32 Effect of Damping Ratio In the following section whenthe damping ratio 120577 is changed and all the other parameters

6 Shock and Vibration

2

15

1

05

0

minus05

minus1

minus15

minus2

1816 2 2212 14080604 102

ℎ

(a)

08 09 1 1107

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)2

15

1

05

0

minus05

minus1

minus15

minus2

ℎ

212191817161514

(c)

Figure 2 Bifurcation diagrams using 120596ℎ as control parameter (a) general view 120596ℎ = 02sim23 (b) enhanced view 120596ℎ = 065sim12 (c) enhancedview 120596ℎ = 14sim215

047

048

049

05

051

300 400 500 600200

(a)

047 048 049 05 051

0

001

002

d

minus001

minus002

(b)

0 1 2

0

1

2

d

minus1

minus1minus2

minus2

(c)

0

20

40

60

80

100

102 03 04 05 06 07 08 09010

Frequency of spectrum

(d)

Figure 3 120596ℎ = 1 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 7

0

05

1

minus1

minus05

300 400 500 600200

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 4 120596ℎ = 16 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

0

05

1

15

minus1

minus05

300 400 500 600200

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 5 120596ℎ = 177 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

8 Shock and Vibration

2

15

1

05

0

minus05

minus1

minus15

minus21816 2 2212 14080604 102

ℎ

(a)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(c)

Figure 6 Bifurcation diagrams using 120577 as control parameter (a) 120577 = 001 (b) 120577 = 004 (c) 120577 = 012keep fixed the bifurcation diagrams are shown in Figure 6The chaotic motions are shown through the numericalsimulation for three different values of 120577 = 001 120577 = 004and 120577 = 012 (120577 = 008 see Figure 2(a)) According tothe comparison of the bifurcation diagrams with differentdamping ratio values it can be observed that chaotic regionbecomes narrower as 120577 is increased chaotic region getssmaller and smaller Therefore the increase in 120577 tends todecrease the nonlinearity dynamic responses

33 Effect of Load Coefficient Different vibration character-istics will be shown under different loads For heavy loadedcondition the value of the load coefficient parameter is fixedas 119891 = 119891119898120582 = 2 (119891119898 = 01 120582 = 005) For lightloaded condition the value of the load coefficient parameteris fixed as 119891 = 119891119898120582 = 13 (119891119898 = 01 120582 = 03) Thetime histories phase plane portraits Poincare maps andFourier spectra are shown in Figures 7 and 8 These figuresshowdynamic responses for heavy loaded condition and lightloaded condition at the dimensionless mesh frequency 120596ℎ= 14 respectively As illustrated in Figure 7 the system islinear system the steady-state response is harmonic responseThere is obviously no tooth impact and the system behavesas 1T-periodic motion It is illustrated that the dynamic

characteristics of the gear are not changedmuchunder loadedcondition when the dynamic behavior behaves as periodicmotion As illustrated in Figure 8 compared with the heavyloaded condition the magnitude of the spectrum becomeslarger The displacement along the line of action is greaterthan 0 and is less than 0 so the case of tooth impact can beobserved in the time history plot

Figures 9 and 10 show dynamic responses for heavyloaded condition and light loaded condition at the dimen-sionless mesh frequency 120596ℎ = 18 respectively Nonlinearvibration characteristics can be seen occurring in thesetwo situations However the intensities are not the sameAs illustrated in Figure 9 the motion state is 1T-periodicharmonic motion under heavy loaded condition It can beseen that the spiral bevel gear system is in chaotic motionunder light loaded condition and the response of the systemis irregular in the chaotic state from Figure 10 Furthermorechaotic phenomenon is more likely to occur in light loadedcondition compared with heavy loaded condition withoutchanging other conditions

34 Effect of Gear Backlash Gear backlash is another impor-tant parameter which affects the dynamic responses substan-tially Figure 11 presents the bifurcation diagrams for three

Shock and Vibration 9

200 300 400 500 6000

02

04

06

08

(a)

0 02 04 06 08

0

02

04

d

minus04

minus02

(b)

0 1 2

0

1

2

d

minus1

minus1minus2

minus2

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 7 120596ℎ = 18 and 119891 = 2 (a) time history (b) phase plane(c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus15minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 8 120596ℎ = 14 and 119891 = 13 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

10 Shock and Vibration

200 300 400 500 600

04

05

06

07

(a)

035 04 045 05 055 06 065 07

0

02

04

minus04

minus02

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 9 120596ℎ = 18 and 119891 = 2 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

15

minus1

minus05

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

Frequency of spectrum

01 02 03 04 05 06 07 08 09 10

(d)

Figure 10 120596ℎ = 18 and 119891 = 13 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 11

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(a)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(c)

Figure 11 Bifurcation diagrams using 119887 as control parameter (a) 119887 = 10 120583m (b) 119887 = 80 120583m (c) 119887 = 120 120583m

different values of half of the backlash 119887 = 10 120583m 119887 = 80 120583mand 119887 = 120120583m (119887 = 40 120583m see Figure 2(a)) A substantialregion of jump phenomenon subharmonic quasi-periodicand chaotic responses are contained in Figure 11 Withincreasing b the system becomes more and more unstableand consequently chaotic behavior expands

For the further study on the effect of gear backlash thetime histories phase plane portraits Poincare maps andFourier spectra of the dynamic responses at 120596ℎ = 223 fornumerous gear backlash values are compared in Figures12ndash15 According to these figures a transition from quasi-periodic motion to the chaotic dynamics is seen with theincrease of 119887 As observed in Poincare maps the variousforms of the system from the quasi-periodic motion areseen and with the increase of the bifurcation parameter theunstable attractive region is enlarged eventually leading tochaos

4 Conclusion

In the present paper a spiral bevel gear system supportedon thrust bearings considering the coupled bending-torsionalnonlinear vibration is proposed and an 8DOF lumpedparameter dynamic model of the spiral bevel gear system

combining with time-varying stiffness static transmissionerror gear backlash and bearing clearances is investigatedThe dimensionless equations of the system are solved usingthe Runge-Kutta numerical method The dynamics of thesystem are analyzed with reference to its bifurcation dia-grams time histories phase plane portraits Poincare mapsand Fourier spectra and jumpphenomena periodicmotionsand chaotic motions are found in this study

In this study mesh frequency is first expressed to studythe effects of their variations on dynamic response Withthe change of mesh frequency the repeated bifurcationphenomena and the complex chaotic motions are observedand the evolution process and rule of the system are revealedThe effect of damping ratio is secondly analyzed It is shownthat chaotic region becomes narrower as damping ratio isincreased Other important parameters like load coefficientand gear backlash are also considered in the current analysisLoad coefficient affects light loaded condition more thanheavy loaded condition The increase of gear backlash willexpand the effect of nonlinearity effect and increase thechaotic region of the gear system

The selection of the dynamic parameters plays an impor-tant role in designing and controlling such gear systems thesuitable values should be chosen so that chaotic behavior will

12 Shock and Vibration

200 300 400 500 600

0

02

04

06

minus02

minus04

(a)

0 02 04 06

0

05

1

15

minus05

minus04 minus02minus1

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 12 119887 = 10 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

minus05

(a)

0 05 1

0

1

2

minus2

minus1

minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 13 119887 = 40 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 13

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1minus15 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 14 119887 = 80 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

3

minus1

(a)

0 1 2 3

0

1

2

minus2

minus1

minus1minus3

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 15 119887 = 120 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

14 Shock and Vibration

be decreased A more precise model to better describe thespiral bevel gear system will be proposed and deep study willbe analyzed in the future

Competing Interests

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

Acknowledgments

This research was supported by the Natural Science Foun-dation of Hubei Province China (Grant no 2014 CFB827)The authors of the paper express their deep gratitude for thefinancial support of the research project

References

[1] HN Ozguven andD RHouser ldquoMathematicalmodels used ingear dynamicsmdasha reviewrdquo Journal of Sound and Vibration vol121 no 3 pp 383ndash411 1988

[2] S Kiyono Y Fujii and Y Suzuki ldquoAnalysis of vibration of bevelgearsrdquo Bulletin of the Japan Society of Mechanical Engineers vol24 no 188 pp 441ndash446 1981

[3] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[4] C Gosselin L Cloutier and Q D Nguyen ldquoA general formu-lation for the calculation of the load sharing and transmissionerror under load of spiral bevel and hypoid gearsrdquo Mechanismand Machine Theory vol 30 no 3 pp 433ndash450 1995

[5] F L Litvin A Fuentes Q Fan and R F Handschuh ldquoCom-puterized design simulation of meshing and contact and stressanalysis of face-milled formate generated spiral bevel gearsrdquoMechanism and Machine Theory vol 37 no 5 pp 441ndash4592002

[6] J-Y Tang Z-H Hu L-J Wu and S-Y Chen ldquoEffect of statictransmission error on dynamic responses of spiral bevel gearsrdquoJournal of Central South University vol 20 no 3 pp 640ndash6472013

[7] H B Yang J P Gao Z D Fang X Z Deng and Y W ZhouldquoNonlinear dynamics of hypoid gearsrdquoAutomotive Engineeringvol 22 no 1 pp 51ndash54 2002 (Chinese)

[8] J Wang T C Lim and M F Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibration vol308 no 1-2 pp 302ndash329 2007

[9] C-W Chang-Jian ldquoNonlinear dynamic analysis for bevel-gearsystem under nonlinear suspension-bifurcation and chaosrdquoApplied Mathematical Modelling vol 35 no 7 pp 3225ndash32372011

[10] S Theodossiades and S Natsiavas ldquoPeriodic and chaoticdynamics of motor-driven gear-pair systems with backlashrdquoChaos Solitons and Fractals vol 12 no 13 pp 2427ndash2440 2001

[11] Y Cheng and T C Lim ldquoVibration analysis of hypoid trans-missions applying an exact geometry-based gear mesh theoryrdquoJournal of Sound andVibration vol 240 no 3 pp 519ndash543 2001

[12] M Li and H Y Hu ldquoDynamic analysis of a spiral bevel-gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 259no 3 pp 605ndash624 2003

[13] O DMohammedM Rantatalo and J-O Aidanpaa ldquoDynamicmodelling of a one-stage spur gear system and vibration-basedtooth crack detection analysisrdquo Mechanical Systems and SignalProcessing vol 54-55 pp 293ndash305 2015

[14] S MWang YW Shen and H J Dong ldquoNon-linear dynamicalcharacteristics of a spiral bevel gear system with backlash andtime-varying stiffnessrdquo Chinese Journal of Mechanical Engineer-ing vol 39 no 2 pp 28ndash32 2003 (Chinese)

[15] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo NonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 75 no 4 pp 783ndash806 2014

[16] Y Shen S Yang and X Liu ldquoNonlinear dynamics of a spurgear pair with time-varying stiffness and backlash based onincremental harmonic balance methodrdquo International Journalof Mechanical Sciences vol 48 no 11 pp 1256ndash1263 2006

[17] R F Li and J J Wang Gear System Dynamics Science Publish-ing Beijing China 1997 (Chinese)

[18] H B Wilson L H Turcotte and D Halpern AdvancedMathematics and Mechanics Applications Using Matlab CRCPress Company Boca Raton Fla USA 3rd edition 2003

[19] T Sun and H Hu ldquoNonlinear dynamics of a planetary gearsystem with multiple clearancesrdquo Mechanism and MachineTheory vol 38 no 12 pp 1371ndash1390 2003

[20] Z Hu J Tang S Chen and D Lei ldquoEffect of mesh stiffness onthe dynamic response of face gear transmission systemrdquo Journalof Mechanical Design Transactions of the ASME vol 135 no 7Article ID 071005 7 pages 2013

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DistributedSensor Networks

International Journal of

4 Shock and Vibration

120577119894119895 = 119888119894119895(2119898119894120596119899) 119896119894119895 = 12059611989411989521205962119899 120596ℎ = Ωℎ120596119899 120577119898119894119895 = 119888119898(2119898119895120596119899) 120577119898 = 119888119898(2119898119890120596119899)

119896119898119894119895 = 119896 (119905)(1198981198951205962119899) 119891119898 = 1198651198981198981198901198871198881205962119899 119891V = 119865V1198981198901198871198881205962119899 119891119890 = 119873sum119897=1

119860119890119897119887119888 (119897120596ℎ)2 cos (119897120596ℎ120591 + Φ119890119897) 119896 (119905) = 119896 (119905)119896119898 = 1 + 119873sum

119897=1

119860119896119897119896119898 cos (119897120596ℎ120591 + Φ119896119897) (119894 = 119909 119910 119911 119895 = 1 2)

119891 (120575) =

120575 minus 119887119887119888 120575 gt 1198871198871198880 1003816100381610038161003816100381612057510038161003816100381610038161003816 le 119887119887119888120575 + 119887119887119888 120575 lt minus 119887119887119888

119891 (119883119895) =

119883119895 minus 1198871119895119887119888 119883119895 gt 11988711198951198871198880 1003816100381610038161003816100381611988311989510038161003816100381610038161003816 le 1198871119895119887119888119883119895 + 1198871119895119887119888 119883119895 lt minus1198871119895119887119888

119891 (119884119895) =

119884119895 minus 1198872119895119887119888 119884119895 gt 11988721198951198871198880 1003816100381610038161003816100381611988411989510038161003816100381610038161003816 le 1198872119895119887119888119884119895 + 1198872119895119887119888 119884119895 lt minus1198872119895119887119888

119891 (119885119895) =

119885119895 minus 1198873119895119887119888 119885119895 gt 11988731198951198871198880 1003816100381610038161003816100381611988511989510038161003816100381610038161003816 le 1198873119895119887119888119885119895 + 1198873119895119887119888 119885119895 lt minus1198873119895119887119888 (19)

Input torque fluctuation is not considered in the currentstudy but only static torque load which means 119891V = 0 Inorder to reduce the complexity of the computation higherorders of the time-varying stiffness and static transmissionerror are truncated without losing generality So the dimen-sionless stiffness and the dimensionless static transmissionerror are calculated by the first-order component

119897 = 1Φ119890119897 = 0Φ119896119897 = 120587119896 (119905) = 1 minus 120576 cos (120596ℎ120591)

119891119890 = 120582120596ℎ2 sin (120596ℎ120591) (20)

where 120576 is stiffness coefficient 120582 is amplitude of statictransmission error

Equations (8)ndash(15) can be expressed in a matrix form

[[[[[[[[[[[[[[[[[

1 0 0 0 0 0 00 1 0 0 0 0 00 0 1 0 0 0 00 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 0minus1198864 1198865 1198866 1198864 minus1198865 minus1198866 1

]]]]]]]]]]]]]]]]]

(((((((((((((

111222

)))))))))))))

+ 2[[[[[[[[[[[[[[[[[

1205771199091 0 0 0 0 0 119886112057711989811990910 1205771199101 0 0 0 0 minus119886212057711989811991010 0 1205771199111 0 0 0 minus119886312057711989811991110 0 0 1205771199092 0 0 minus119886112057711989811990920 0 0 0 1205771199102 0 119886212057711989811991020 0 0 0 0 1205771199112 119886312057711989811991120 0 0 0 0 0 21205771198981198866

]]]]]]]]]]]]]]]]]

(((((((((((((

111222

)))))))))))))

Shock and Vibration 5

Table 1 The parameters of the spiral bevel gear system

Parameters Pinion GearTeeth numbers 8 41Module (mm) 4161Mass (kg) 1 5Inertia moment (kgmm2) 3000 11200Base circle (mm) 30 130Pressure angle (∘) 120572119899 = 2115Helix angle (∘) 120573 = 35Cone angle (∘) 1205741 = 1275 1205742 = 765Tooth width (mm) 28Mean mesh stiffness (Nmm) 119896119898 = 1191198646Bearing stiffness (Nmm) 119896119894119895 = 0351198646Half of bearing clearance (120583m) 1198871119895 = 1198872119895 = 1198873119895 = 10

+[[[[[[[[[[[[[[[[[[[

1198961199091 0 0 0 0 0 119886111989611989811990910 1198961199101 0 0 0 0 minus119886211989611989811991010 0 1198961199111 0 0 0 minus119886311989611989811991110 0 0 1198961199092 0 0 minus119886111989611989811990920 0 0 0 1198961199102 0 119886211989611989811991020 0 0 0 0 1198961199112 119886311989611989811991120 0 0 0 0 0 1198866119896 (119905)

]]]]]]]]]]]]]]]]]]]

((((((((((((

119891 (1198831)119891 (1198841)119891 (1198851)119891 (1198832)119891 (1198842)119891 (1198852)119891 (120575)

))))))))))))

=((((((((((

000000119891119898 + 119891V + 119891119890

))))))))))

(21)

3 Numerical Solutions

Due to the complexity of the spiral bevel gear system andalso the difficulty and limitation of the analytical methodsthe numerical method is commonly used to analyze thegear system In this paper the nonlinear dynamic equationsare solved using the fourth order Runge-Kutta methodwhich is generally applicable to strong nonlinearity [17] Thealgorithm is implemented in MATLAB that is a widely usedmatrix and numerical analysis program [18] Without losinggenerality the characteristic length is set to 119887119888 = 10 120583mand the same value can be found in Sunrsquos thesis [19] Theother parameters of the gear transmission system are givenin Table 1

31 Effect of Mesh Frequency A bifurcation diagram sum-marizes the essential dynamics of the gear system and istherefore a useful means of observing its nonlinear dynamicresponse [20] For our subsequent numerical study set halfof the backlash as 119887 = 40 120583m damping ratio as 120577 = 008amplitude of static transmission error as 120582 = 03 and stiffnesscoefficient as 120576 = 02 In this system the dimensionlessmesh frequency120596ℎ is commonly used as an important controlparameter Figure 2(a) shows the bifurcation diagram for thespiral bevel gear system displacement against the dimen-sionless mesh frequency 120596ℎ Figures 2(b) and 2(c) show theenhanced bifurcation diagrams with bifurcation parameters120596ℎ = 065sim12 and 120596ℎ = 14sim215 respectively The bifurcationresults show that the system behaves as 1T-periodic motionat low mesh frequency and the periodic motion persists until120596ℎ gt 0425 A jump phenomenon can be observed and2T-periodic motion appears at the region 120596ℎ = 0425sim08After a long 1T-periodic motion the system enters into atransient chaotic motion after 120596ℎ gt 115 then the dynamicbehavior behaves as nonperiodic motion until 120596ℎ = 15With the increase of mesh frequency a number of periodicwindows appear At the region 120596ℎ = 1sim2 1T-periodic motion2T-periodic motion and 119899T-periodic motion are shownin the chaotic area It can be observed that 119899T-periodicbifurcation enters into chaos and 7T-periodic bifurcationenters into 119899T-periodic bifurcation obviously in Figure 2(c)Finally 2T-periodic motion transits to 1T-periodic motionuntil 120596ℎ = 205

For a better clarity the other analytical methods forobserving nonlinear dynamic responses are necessary Fig-ures 3ndash5 illustrate the time histories phase plane portraitsPoincare maps and Fourier spectra for the spiral bevel gearmodel at various values of the dimensionless mesh frequency120596ℎ = 1120596ℎ = 16 and120596ℎ = 177 At120596ℎ = 1 the dynamic behaviorbehaves as 1T-periodicmotionAt120596ℎ = 16 chaotic behavior isclearly visible At120596ℎ = 177 the system enters into 6T-periodicmotion from chaotic motion

32 Effect of Damping Ratio In the following section whenthe damping ratio 120577 is changed and all the other parameters

6 Shock and Vibration

2

15

1

05

0

minus05

minus1

minus15

minus2

1816 2 2212 14080604 102

ℎ

(a)

08 09 1 1107

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)2

15

1

05

0

minus05

minus1

minus15

minus2

ℎ

212191817161514

(c)

Figure 2 Bifurcation diagrams using 120596ℎ as control parameter (a) general view 120596ℎ = 02sim23 (b) enhanced view 120596ℎ = 065sim12 (c) enhancedview 120596ℎ = 14sim215

047

048

049

05

051

300 400 500 600200

(a)

047 048 049 05 051

0

001

002

d

minus001

minus002

(b)

0 1 2

0

1

2

d

minus1

minus1minus2

minus2

(c)

0

20

40

60

80

100

102 03 04 05 06 07 08 09010

Frequency of spectrum

(d)

Figure 3 120596ℎ = 1 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 7

0

05

1

minus1

minus05

300 400 500 600200

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 4 120596ℎ = 16 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

0

05

1

15

minus1

minus05

300 400 500 600200

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 5 120596ℎ = 177 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

8 Shock and Vibration

2

15

1

05

0

minus05

minus1

minus15

minus21816 2 2212 14080604 102

ℎ

(a)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(c)

Figure 6 Bifurcation diagrams using 120577 as control parameter (a) 120577 = 001 (b) 120577 = 004 (c) 120577 = 012keep fixed the bifurcation diagrams are shown in Figure 6The chaotic motions are shown through the numericalsimulation for three different values of 120577 = 001 120577 = 004and 120577 = 012 (120577 = 008 see Figure 2(a)) According tothe comparison of the bifurcation diagrams with differentdamping ratio values it can be observed that chaotic regionbecomes narrower as 120577 is increased chaotic region getssmaller and smaller Therefore the increase in 120577 tends todecrease the nonlinearity dynamic responses

33 Effect of Load Coefficient Different vibration character-istics will be shown under different loads For heavy loadedcondition the value of the load coefficient parameter is fixedas 119891 = 119891119898120582 = 2 (119891119898 = 01 120582 = 005) For lightloaded condition the value of the load coefficient parameteris fixed as 119891 = 119891119898120582 = 13 (119891119898 = 01 120582 = 03) Thetime histories phase plane portraits Poincare maps andFourier spectra are shown in Figures 7 and 8 These figuresshowdynamic responses for heavy loaded condition and lightloaded condition at the dimensionless mesh frequency 120596ℎ= 14 respectively As illustrated in Figure 7 the system islinear system the steady-state response is harmonic responseThere is obviously no tooth impact and the system behavesas 1T-periodic motion It is illustrated that the dynamic

characteristics of the gear are not changedmuchunder loadedcondition when the dynamic behavior behaves as periodicmotion As illustrated in Figure 8 compared with the heavyloaded condition the magnitude of the spectrum becomeslarger The displacement along the line of action is greaterthan 0 and is less than 0 so the case of tooth impact can beobserved in the time history plot

Figures 9 and 10 show dynamic responses for heavyloaded condition and light loaded condition at the dimen-sionless mesh frequency 120596ℎ = 18 respectively Nonlinearvibration characteristics can be seen occurring in thesetwo situations However the intensities are not the sameAs illustrated in Figure 9 the motion state is 1T-periodicharmonic motion under heavy loaded condition It can beseen that the spiral bevel gear system is in chaotic motionunder light loaded condition and the response of the systemis irregular in the chaotic state from Figure 10 Furthermorechaotic phenomenon is more likely to occur in light loadedcondition compared with heavy loaded condition withoutchanging other conditions

34 Effect of Gear Backlash Gear backlash is another impor-tant parameter which affects the dynamic responses substan-tially Figure 11 presents the bifurcation diagrams for three

Shock and Vibration 9

200 300 400 500 6000

02

04

06

08

(a)

0 02 04 06 08

0

02

04

d

minus04

minus02

(b)

0 1 2

0

1

2

d

minus1

minus1minus2

minus2

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 7 120596ℎ = 18 and 119891 = 2 (a) time history (b) phase plane(c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus15minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 8 120596ℎ = 14 and 119891 = 13 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

10 Shock and Vibration

200 300 400 500 600

04

05

06

07

(a)

035 04 045 05 055 06 065 07

0

02

04

minus04

minus02

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 9 120596ℎ = 18 and 119891 = 2 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

15

minus1

minus05

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

Frequency of spectrum

01 02 03 04 05 06 07 08 09 10

(d)

Figure 10 120596ℎ = 18 and 119891 = 13 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 11

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(a)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(c)

Figure 11 Bifurcation diagrams using 119887 as control parameter (a) 119887 = 10 120583m (b) 119887 = 80 120583m (c) 119887 = 120 120583m

different values of half of the backlash 119887 = 10 120583m 119887 = 80 120583mand 119887 = 120120583m (119887 = 40 120583m see Figure 2(a)) A substantialregion of jump phenomenon subharmonic quasi-periodicand chaotic responses are contained in Figure 11 Withincreasing b the system becomes more and more unstableand consequently chaotic behavior expands

For the further study on the effect of gear backlash thetime histories phase plane portraits Poincare maps andFourier spectra of the dynamic responses at 120596ℎ = 223 fornumerous gear backlash values are compared in Figures12ndash15 According to these figures a transition from quasi-periodic motion to the chaotic dynamics is seen with theincrease of 119887 As observed in Poincare maps the variousforms of the system from the quasi-periodic motion areseen and with the increase of the bifurcation parameter theunstable attractive region is enlarged eventually leading tochaos

4 Conclusion

In the present paper a spiral bevel gear system supportedon thrust bearings considering the coupled bending-torsionalnonlinear vibration is proposed and an 8DOF lumpedparameter dynamic model of the spiral bevel gear system

combining with time-varying stiffness static transmissionerror gear backlash and bearing clearances is investigatedThe dimensionless equations of the system are solved usingthe Runge-Kutta numerical method The dynamics of thesystem are analyzed with reference to its bifurcation dia-grams time histories phase plane portraits Poincare mapsand Fourier spectra and jumpphenomena periodicmotionsand chaotic motions are found in this study

In this study mesh frequency is first expressed to studythe effects of their variations on dynamic response Withthe change of mesh frequency the repeated bifurcationphenomena and the complex chaotic motions are observedand the evolution process and rule of the system are revealedThe effect of damping ratio is secondly analyzed It is shownthat chaotic region becomes narrower as damping ratio isincreased Other important parameters like load coefficientand gear backlash are also considered in the current analysisLoad coefficient affects light loaded condition more thanheavy loaded condition The increase of gear backlash willexpand the effect of nonlinearity effect and increase thechaotic region of the gear system

The selection of the dynamic parameters plays an impor-tant role in designing and controlling such gear systems thesuitable values should be chosen so that chaotic behavior will

12 Shock and Vibration

200 300 400 500 600

0

02

04

06

minus02

minus04

(a)

0 02 04 06

0

05

1

15

minus05

minus04 minus02minus1

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 12 119887 = 10 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

minus05

(a)

0 05 1

0

1

2

minus2

minus1

minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 13 119887 = 40 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 13

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1minus15 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 14 119887 = 80 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

3

minus1

(a)

0 1 2 3

0

1

2

minus2

minus1

minus1minus3

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 15 119887 = 120 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

14 Shock and Vibration

be decreased A more precise model to better describe thespiral bevel gear system will be proposed and deep study willbe analyzed in the future

Competing Interests

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

Acknowledgments

This research was supported by the Natural Science Foun-dation of Hubei Province China (Grant no 2014 CFB827)The authors of the paper express their deep gratitude for thefinancial support of the research project

References

[1] HN Ozguven andD RHouser ldquoMathematicalmodels used ingear dynamicsmdasha reviewrdquo Journal of Sound and Vibration vol121 no 3 pp 383ndash411 1988

[2] S Kiyono Y Fujii and Y Suzuki ldquoAnalysis of vibration of bevelgearsrdquo Bulletin of the Japan Society of Mechanical Engineers vol24 no 188 pp 441ndash446 1981

[3] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[4] C Gosselin L Cloutier and Q D Nguyen ldquoA general formu-lation for the calculation of the load sharing and transmissionerror under load of spiral bevel and hypoid gearsrdquo Mechanismand Machine Theory vol 30 no 3 pp 433ndash450 1995

[5] F L Litvin A Fuentes Q Fan and R F Handschuh ldquoCom-puterized design simulation of meshing and contact and stressanalysis of face-milled formate generated spiral bevel gearsrdquoMechanism and Machine Theory vol 37 no 5 pp 441ndash4592002

[6] J-Y Tang Z-H Hu L-J Wu and S-Y Chen ldquoEffect of statictransmission error on dynamic responses of spiral bevel gearsrdquoJournal of Central South University vol 20 no 3 pp 640ndash6472013

[7] H B Yang J P Gao Z D Fang X Z Deng and Y W ZhouldquoNonlinear dynamics of hypoid gearsrdquoAutomotive Engineeringvol 22 no 1 pp 51ndash54 2002 (Chinese)

[8] J Wang T C Lim and M F Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibration vol308 no 1-2 pp 302ndash329 2007

[9] C-W Chang-Jian ldquoNonlinear dynamic analysis for bevel-gearsystem under nonlinear suspension-bifurcation and chaosrdquoApplied Mathematical Modelling vol 35 no 7 pp 3225ndash32372011

[10] S Theodossiades and S Natsiavas ldquoPeriodic and chaoticdynamics of motor-driven gear-pair systems with backlashrdquoChaos Solitons and Fractals vol 12 no 13 pp 2427ndash2440 2001

[11] Y Cheng and T C Lim ldquoVibration analysis of hypoid trans-missions applying an exact geometry-based gear mesh theoryrdquoJournal of Sound andVibration vol 240 no 3 pp 519ndash543 2001

[12] M Li and H Y Hu ldquoDynamic analysis of a spiral bevel-gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 259no 3 pp 605ndash624 2003

[13] O DMohammedM Rantatalo and J-O Aidanpaa ldquoDynamicmodelling of a one-stage spur gear system and vibration-basedtooth crack detection analysisrdquo Mechanical Systems and SignalProcessing vol 54-55 pp 293ndash305 2015

[14] S MWang YW Shen and H J Dong ldquoNon-linear dynamicalcharacteristics of a spiral bevel gear system with backlash andtime-varying stiffnessrdquo Chinese Journal of Mechanical Engineer-ing vol 39 no 2 pp 28ndash32 2003 (Chinese)

[15] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo NonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 75 no 4 pp 783ndash806 2014

[16] Y Shen S Yang and X Liu ldquoNonlinear dynamics of a spurgear pair with time-varying stiffness and backlash based onincremental harmonic balance methodrdquo International Journalof Mechanical Sciences vol 48 no 11 pp 1256ndash1263 2006

[17] R F Li and J J Wang Gear System Dynamics Science Publish-ing Beijing China 1997 (Chinese)

[18] H B Wilson L H Turcotte and D Halpern AdvancedMathematics and Mechanics Applications Using Matlab CRCPress Company Boca Raton Fla USA 3rd edition 2003

[19] T Sun and H Hu ldquoNonlinear dynamics of a planetary gearsystem with multiple clearancesrdquo Mechanism and MachineTheory vol 38 no 12 pp 1371ndash1390 2003

[20] Z Hu J Tang S Chen and D Lei ldquoEffect of mesh stiffness onthe dynamic response of face gear transmission systemrdquo Journalof Mechanical Design Transactions of the ASME vol 135 no 7Article ID 071005 7 pages 2013

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Shock and Vibration 5

Table 1 The parameters of the spiral bevel gear system

Parameters Pinion GearTeeth numbers 8 41Module (mm) 4161Mass (kg) 1 5Inertia moment (kgmm2) 3000 11200Base circle (mm) 30 130Pressure angle (∘) 120572119899 = 2115Helix angle (∘) 120573 = 35Cone angle (∘) 1205741 = 1275 1205742 = 765Tooth width (mm) 28Mean mesh stiffness (Nmm) 119896119898 = 1191198646Bearing stiffness (Nmm) 119896119894119895 = 0351198646Half of bearing clearance (120583m) 1198871119895 = 1198872119895 = 1198873119895 = 10

+[[[[[[[[[[[[[[[[[[[

1198961199091 0 0 0 0 0 119886111989611989811990910 1198961199101 0 0 0 0 minus119886211989611989811991010 0 1198961199111 0 0 0 minus119886311989611989811991110 0 0 1198961199092 0 0 minus119886111989611989811990920 0 0 0 1198961199102 0 119886211989611989811991020 0 0 0 0 1198961199112 119886311989611989811991120 0 0 0 0 0 1198866119896 (119905)

]]]]]]]]]]]]]]]]]]]

((((((((((((

119891 (1198831)119891 (1198841)119891 (1198851)119891 (1198832)119891 (1198842)119891 (1198852)119891 (120575)

))))))))))))

=((((((((((

000000119891119898 + 119891V + 119891119890

))))))))))

(21)

3 Numerical Solutions

Due to the complexity of the spiral bevel gear system andalso the difficulty and limitation of the analytical methodsthe numerical method is commonly used to analyze thegear system In this paper the nonlinear dynamic equationsare solved using the fourth order Runge-Kutta methodwhich is generally applicable to strong nonlinearity [17] Thealgorithm is implemented in MATLAB that is a widely usedmatrix and numerical analysis program [18] Without losinggenerality the characteristic length is set to 119887119888 = 10 120583mand the same value can be found in Sunrsquos thesis [19] Theother parameters of the gear transmission system are givenin Table 1

31 Effect of Mesh Frequency A bifurcation diagram sum-marizes the essential dynamics of the gear system and istherefore a useful means of observing its nonlinear dynamicresponse [20] For our subsequent numerical study set halfof the backlash as 119887 = 40 120583m damping ratio as 120577 = 008amplitude of static transmission error as 120582 = 03 and stiffnesscoefficient as 120576 = 02 In this system the dimensionlessmesh frequency120596ℎ is commonly used as an important controlparameter Figure 2(a) shows the bifurcation diagram for thespiral bevel gear system displacement against the dimen-sionless mesh frequency 120596ℎ Figures 2(b) and 2(c) show theenhanced bifurcation diagrams with bifurcation parameters120596ℎ = 065sim12 and 120596ℎ = 14sim215 respectively The bifurcationresults show that the system behaves as 1T-periodic motionat low mesh frequency and the periodic motion persists until120596ℎ gt 0425 A jump phenomenon can be observed and2T-periodic motion appears at the region 120596ℎ = 0425sim08After a long 1T-periodic motion the system enters into atransient chaotic motion after 120596ℎ gt 115 then the dynamicbehavior behaves as nonperiodic motion until 120596ℎ = 15With the increase of mesh frequency a number of periodicwindows appear At the region 120596ℎ = 1sim2 1T-periodic motion2T-periodic motion and 119899T-periodic motion are shownin the chaotic area It can be observed that 119899T-periodicbifurcation enters into chaos and 7T-periodic bifurcationenters into 119899T-periodic bifurcation obviously in Figure 2(c)Finally 2T-periodic motion transits to 1T-periodic motionuntil 120596ℎ = 205

For a better clarity the other analytical methods forobserving nonlinear dynamic responses are necessary Fig-ures 3ndash5 illustrate the time histories phase plane portraitsPoincare maps and Fourier spectra for the spiral bevel gearmodel at various values of the dimensionless mesh frequency120596ℎ = 1120596ℎ = 16 and120596ℎ = 177 At120596ℎ = 1 the dynamic behaviorbehaves as 1T-periodicmotionAt120596ℎ = 16 chaotic behavior isclearly visible At120596ℎ = 177 the system enters into 6T-periodicmotion from chaotic motion

32 Effect of Damping Ratio In the following section whenthe damping ratio 120577 is changed and all the other parameters

6 Shock and Vibration

2

15

1

05

0

minus05

minus1

minus15

minus2

1816 2 2212 14080604 102

ℎ

(a)

08 09 1 1107

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)2

15

1

05

0

minus05

minus1

minus15

minus2

ℎ

212191817161514

(c)

Figure 2 Bifurcation diagrams using 120596ℎ as control parameter (a) general view 120596ℎ = 02sim23 (b) enhanced view 120596ℎ = 065sim12 (c) enhancedview 120596ℎ = 14sim215

047

048

049

05

051

300 400 500 600200

(a)

047 048 049 05 051

0

001

002

d

minus001

minus002

(b)

0 1 2

0

1

2

d

minus1

minus1minus2

minus2

(c)

0

20

40

60

80

100

102 03 04 05 06 07 08 09010

Frequency of spectrum

(d)

Figure 3 120596ℎ = 1 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 7

0

05

1

minus1

minus05

300 400 500 600200

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 4 120596ℎ = 16 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

0

05

1

15

minus1

minus05

300 400 500 600200

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 5 120596ℎ = 177 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

8 Shock and Vibration

2

15

1

05

0

minus05

minus1

minus15

minus21816 2 2212 14080604 102

ℎ

(a)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(c)

Figure 6 Bifurcation diagrams using 120577 as control parameter (a) 120577 = 001 (b) 120577 = 004 (c) 120577 = 012keep fixed the bifurcation diagrams are shown in Figure 6The chaotic motions are shown through the numericalsimulation for three different values of 120577 = 001 120577 = 004and 120577 = 012 (120577 = 008 see Figure 2(a)) According tothe comparison of the bifurcation diagrams with differentdamping ratio values it can be observed that chaotic regionbecomes narrower as 120577 is increased chaotic region getssmaller and smaller Therefore the increase in 120577 tends todecrease the nonlinearity dynamic responses

33 Effect of Load Coefficient Different vibration character-istics will be shown under different loads For heavy loadedcondition the value of the load coefficient parameter is fixedas 119891 = 119891119898120582 = 2 (119891119898 = 01 120582 = 005) For lightloaded condition the value of the load coefficient parameteris fixed as 119891 = 119891119898120582 = 13 (119891119898 = 01 120582 = 03) Thetime histories phase plane portraits Poincare maps andFourier spectra are shown in Figures 7 and 8 These figuresshowdynamic responses for heavy loaded condition and lightloaded condition at the dimensionless mesh frequency 120596ℎ= 14 respectively As illustrated in Figure 7 the system islinear system the steady-state response is harmonic responseThere is obviously no tooth impact and the system behavesas 1T-periodic motion It is illustrated that the dynamic

characteristics of the gear are not changedmuchunder loadedcondition when the dynamic behavior behaves as periodicmotion As illustrated in Figure 8 compared with the heavyloaded condition the magnitude of the spectrum becomeslarger The displacement along the line of action is greaterthan 0 and is less than 0 so the case of tooth impact can beobserved in the time history plot

Figures 9 and 10 show dynamic responses for heavyloaded condition and light loaded condition at the dimen-sionless mesh frequency 120596ℎ = 18 respectively Nonlinearvibration characteristics can be seen occurring in thesetwo situations However the intensities are not the sameAs illustrated in Figure 9 the motion state is 1T-periodicharmonic motion under heavy loaded condition It can beseen that the spiral bevel gear system is in chaotic motionunder light loaded condition and the response of the systemis irregular in the chaotic state from Figure 10 Furthermorechaotic phenomenon is more likely to occur in light loadedcondition compared with heavy loaded condition withoutchanging other conditions

34 Effect of Gear Backlash Gear backlash is another impor-tant parameter which affects the dynamic responses substan-tially Figure 11 presents the bifurcation diagrams for three

Shock and Vibration 9

200 300 400 500 6000

02

04

06

08

(a)

0 02 04 06 08

0

02

04

d

minus04

minus02

(b)

0 1 2

0

1

2

d

minus1

minus1minus2

minus2

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 7 120596ℎ = 18 and 119891 = 2 (a) time history (b) phase plane(c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus15minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 8 120596ℎ = 14 and 119891 = 13 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

10 Shock and Vibration

200 300 400 500 600

04

05

06

07

(a)

035 04 045 05 055 06 065 07

0

02

04

minus04

minus02

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 9 120596ℎ = 18 and 119891 = 2 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

15

minus1

minus05

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

Frequency of spectrum

01 02 03 04 05 06 07 08 09 10

(d)

Figure 10 120596ℎ = 18 and 119891 = 13 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 11

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(a)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(c)

Figure 11 Bifurcation diagrams using 119887 as control parameter (a) 119887 = 10 120583m (b) 119887 = 80 120583m (c) 119887 = 120 120583m

different values of half of the backlash 119887 = 10 120583m 119887 = 80 120583mand 119887 = 120120583m (119887 = 40 120583m see Figure 2(a)) A substantialregion of jump phenomenon subharmonic quasi-periodicand chaotic responses are contained in Figure 11 Withincreasing b the system becomes more and more unstableand consequently chaotic behavior expands

For the further study on the effect of gear backlash thetime histories phase plane portraits Poincare maps andFourier spectra of the dynamic responses at 120596ℎ = 223 fornumerous gear backlash values are compared in Figures12ndash15 According to these figures a transition from quasi-periodic motion to the chaotic dynamics is seen with theincrease of 119887 As observed in Poincare maps the variousforms of the system from the quasi-periodic motion areseen and with the increase of the bifurcation parameter theunstable attractive region is enlarged eventually leading tochaos

4 Conclusion

In the present paper a spiral bevel gear system supportedon thrust bearings considering the coupled bending-torsionalnonlinear vibration is proposed and an 8DOF lumpedparameter dynamic model of the spiral bevel gear system

combining with time-varying stiffness static transmissionerror gear backlash and bearing clearances is investigatedThe dimensionless equations of the system are solved usingthe Runge-Kutta numerical method The dynamics of thesystem are analyzed with reference to its bifurcation dia-grams time histories phase plane portraits Poincare mapsand Fourier spectra and jumpphenomena periodicmotionsand chaotic motions are found in this study

In this study mesh frequency is first expressed to studythe effects of their variations on dynamic response Withthe change of mesh frequency the repeated bifurcationphenomena and the complex chaotic motions are observedand the evolution process and rule of the system are revealedThe effect of damping ratio is secondly analyzed It is shownthat chaotic region becomes narrower as damping ratio isincreased Other important parameters like load coefficientand gear backlash are also considered in the current analysisLoad coefficient affects light loaded condition more thanheavy loaded condition The increase of gear backlash willexpand the effect of nonlinearity effect and increase thechaotic region of the gear system

The selection of the dynamic parameters plays an impor-tant role in designing and controlling such gear systems thesuitable values should be chosen so that chaotic behavior will

12 Shock and Vibration

200 300 400 500 600

0

02

04

06

minus02

minus04

(a)

0 02 04 06

0

05

1

15

minus05

minus04 minus02minus1

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 12 119887 = 10 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

minus05

(a)

0 05 1

0

1

2

minus2

minus1

minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 13 119887 = 40 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 13

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1minus15 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 14 119887 = 80 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

3

minus1

(a)

0 1 2 3

0

1

2

minus2

minus1

minus1minus3

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 15 119887 = 120 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

14 Shock and Vibration

be decreased A more precise model to better describe thespiral bevel gear system will be proposed and deep study willbe analyzed in the future

Competing Interests

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

Acknowledgments

This research was supported by the Natural Science Foun-dation of Hubei Province China (Grant no 2014 CFB827)The authors of the paper express their deep gratitude for thefinancial support of the research project

References

[1] HN Ozguven andD RHouser ldquoMathematicalmodels used ingear dynamicsmdasha reviewrdquo Journal of Sound and Vibration vol121 no 3 pp 383ndash411 1988

[2] S Kiyono Y Fujii and Y Suzuki ldquoAnalysis of vibration of bevelgearsrdquo Bulletin of the Japan Society of Mechanical Engineers vol24 no 188 pp 441ndash446 1981

[3] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[4] C Gosselin L Cloutier and Q D Nguyen ldquoA general formu-lation for the calculation of the load sharing and transmissionerror under load of spiral bevel and hypoid gearsrdquo Mechanismand Machine Theory vol 30 no 3 pp 433ndash450 1995

[5] F L Litvin A Fuentes Q Fan and R F Handschuh ldquoCom-puterized design simulation of meshing and contact and stressanalysis of face-milled formate generated spiral bevel gearsrdquoMechanism and Machine Theory vol 37 no 5 pp 441ndash4592002

[6] J-Y Tang Z-H Hu L-J Wu and S-Y Chen ldquoEffect of statictransmission error on dynamic responses of spiral bevel gearsrdquoJournal of Central South University vol 20 no 3 pp 640ndash6472013

[7] H B Yang J P Gao Z D Fang X Z Deng and Y W ZhouldquoNonlinear dynamics of hypoid gearsrdquoAutomotive Engineeringvol 22 no 1 pp 51ndash54 2002 (Chinese)

[8] J Wang T C Lim and M F Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibration vol308 no 1-2 pp 302ndash329 2007

[9] C-W Chang-Jian ldquoNonlinear dynamic analysis for bevel-gearsystem under nonlinear suspension-bifurcation and chaosrdquoApplied Mathematical Modelling vol 35 no 7 pp 3225ndash32372011

[10] S Theodossiades and S Natsiavas ldquoPeriodic and chaoticdynamics of motor-driven gear-pair systems with backlashrdquoChaos Solitons and Fractals vol 12 no 13 pp 2427ndash2440 2001

[11] Y Cheng and T C Lim ldquoVibration analysis of hypoid trans-missions applying an exact geometry-based gear mesh theoryrdquoJournal of Sound andVibration vol 240 no 3 pp 519ndash543 2001

[12] M Li and H Y Hu ldquoDynamic analysis of a spiral bevel-gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 259no 3 pp 605ndash624 2003

[13] O DMohammedM Rantatalo and J-O Aidanpaa ldquoDynamicmodelling of a one-stage spur gear system and vibration-basedtooth crack detection analysisrdquo Mechanical Systems and SignalProcessing vol 54-55 pp 293ndash305 2015

[14] S MWang YW Shen and H J Dong ldquoNon-linear dynamicalcharacteristics of a spiral bevel gear system with backlash andtime-varying stiffnessrdquo Chinese Journal of Mechanical Engineer-ing vol 39 no 2 pp 28ndash32 2003 (Chinese)

[15] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo NonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 75 no 4 pp 783ndash806 2014

[16] Y Shen S Yang and X Liu ldquoNonlinear dynamics of a spurgear pair with time-varying stiffness and backlash based onincremental harmonic balance methodrdquo International Journalof Mechanical Sciences vol 48 no 11 pp 1256ndash1263 2006

[17] R F Li and J J Wang Gear System Dynamics Science Publish-ing Beijing China 1997 (Chinese)

[18] H B Wilson L H Turcotte and D Halpern AdvancedMathematics and Mechanics Applications Using Matlab CRCPress Company Boca Raton Fla USA 3rd edition 2003

[19] T Sun and H Hu ldquoNonlinear dynamics of a planetary gearsystem with multiple clearancesrdquo Mechanism and MachineTheory vol 38 no 12 pp 1371ndash1390 2003

[20] Z Hu J Tang S Chen and D Lei ldquoEffect of mesh stiffness onthe dynamic response of face gear transmission systemrdquo Journalof Mechanical Design Transactions of the ASME vol 135 no 7Article ID 071005 7 pages 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Shock and Vibration

2

15

1

05

0

minus05

minus1

minus15

minus2

1816 2 2212 14080604 102

ℎ

(a)

08 09 1 1107

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)2

15

1

05

0

minus05

minus1

minus15

minus2

ℎ

212191817161514

(c)

Figure 2 Bifurcation diagrams using 120596ℎ as control parameter (a) general view 120596ℎ = 02sim23 (b) enhanced view 120596ℎ = 065sim12 (c) enhancedview 120596ℎ = 14sim215

047

048

049

05

051

300 400 500 600200

(a)

047 048 049 05 051

0

001

002

d

minus001

minus002

(b)

0 1 2

0

1

2

d

minus1

minus1minus2

minus2

(c)

0

20

40

60

80

100

102 03 04 05 06 07 08 09010

Frequency of spectrum

(d)

Figure 3 120596ℎ = 1 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 7

0

05

1

minus1

minus05

300 400 500 600200

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 4 120596ℎ = 16 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

0

05

1

15

minus1

minus05

300 400 500 600200

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 5 120596ℎ = 177 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

8 Shock and Vibration

2

15

1

05

0

minus05

minus1

minus15

minus21816 2 2212 14080604 102

ℎ

(a)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(c)

Figure 6 Bifurcation diagrams using 120577 as control parameter (a) 120577 = 001 (b) 120577 = 004 (c) 120577 = 012keep fixed the bifurcation diagrams are shown in Figure 6The chaotic motions are shown through the numericalsimulation for three different values of 120577 = 001 120577 = 004and 120577 = 012 (120577 = 008 see Figure 2(a)) According tothe comparison of the bifurcation diagrams with differentdamping ratio values it can be observed that chaotic regionbecomes narrower as 120577 is increased chaotic region getssmaller and smaller Therefore the increase in 120577 tends todecrease the nonlinearity dynamic responses

33 Effect of Load Coefficient Different vibration character-istics will be shown under different loads For heavy loadedcondition the value of the load coefficient parameter is fixedas 119891 = 119891119898120582 = 2 (119891119898 = 01 120582 = 005) For lightloaded condition the value of the load coefficient parameteris fixed as 119891 = 119891119898120582 = 13 (119891119898 = 01 120582 = 03) Thetime histories phase plane portraits Poincare maps andFourier spectra are shown in Figures 7 and 8 These figuresshowdynamic responses for heavy loaded condition and lightloaded condition at the dimensionless mesh frequency 120596ℎ= 14 respectively As illustrated in Figure 7 the system islinear system the steady-state response is harmonic responseThere is obviously no tooth impact and the system behavesas 1T-periodic motion It is illustrated that the dynamic

characteristics of the gear are not changedmuchunder loadedcondition when the dynamic behavior behaves as periodicmotion As illustrated in Figure 8 compared with the heavyloaded condition the magnitude of the spectrum becomeslarger The displacement along the line of action is greaterthan 0 and is less than 0 so the case of tooth impact can beobserved in the time history plot

Figures 9 and 10 show dynamic responses for heavyloaded condition and light loaded condition at the dimen-sionless mesh frequency 120596ℎ = 18 respectively Nonlinearvibration characteristics can be seen occurring in thesetwo situations However the intensities are not the sameAs illustrated in Figure 9 the motion state is 1T-periodicharmonic motion under heavy loaded condition It can beseen that the spiral bevel gear system is in chaotic motionunder light loaded condition and the response of the systemis irregular in the chaotic state from Figure 10 Furthermorechaotic phenomenon is more likely to occur in light loadedcondition compared with heavy loaded condition withoutchanging other conditions

34 Effect of Gear Backlash Gear backlash is another impor-tant parameter which affects the dynamic responses substan-tially Figure 11 presents the bifurcation diagrams for three

Shock and Vibration 9

200 300 400 500 6000

02

04

06

08

(a)

0 02 04 06 08

0

02

04

d

minus04

minus02

(b)

0 1 2

0

1

2

d

minus1

minus1minus2

minus2

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 7 120596ℎ = 18 and 119891 = 2 (a) time history (b) phase plane(c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus15minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 8 120596ℎ = 14 and 119891 = 13 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

10 Shock and Vibration

200 300 400 500 600

04

05

06

07

(a)

035 04 045 05 055 06 065 07

0

02

04

minus04

minus02

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 9 120596ℎ = 18 and 119891 = 2 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

15

minus1

minus05

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

Frequency of spectrum

01 02 03 04 05 06 07 08 09 10

(d)

Figure 10 120596ℎ = 18 and 119891 = 13 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 11

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(a)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(c)

Figure 11 Bifurcation diagrams using 119887 as control parameter (a) 119887 = 10 120583m (b) 119887 = 80 120583m (c) 119887 = 120 120583m

different values of half of the backlash 119887 = 10 120583m 119887 = 80 120583mand 119887 = 120120583m (119887 = 40 120583m see Figure 2(a)) A substantialregion of jump phenomenon subharmonic quasi-periodicand chaotic responses are contained in Figure 11 Withincreasing b the system becomes more and more unstableand consequently chaotic behavior expands

For the further study on the effect of gear backlash thetime histories phase plane portraits Poincare maps andFourier spectra of the dynamic responses at 120596ℎ = 223 fornumerous gear backlash values are compared in Figures12ndash15 According to these figures a transition from quasi-periodic motion to the chaotic dynamics is seen with theincrease of 119887 As observed in Poincare maps the variousforms of the system from the quasi-periodic motion areseen and with the increase of the bifurcation parameter theunstable attractive region is enlarged eventually leading tochaos

4 Conclusion

In the present paper a spiral bevel gear system supportedon thrust bearings considering the coupled bending-torsionalnonlinear vibration is proposed and an 8DOF lumpedparameter dynamic model of the spiral bevel gear system

combining with time-varying stiffness static transmissionerror gear backlash and bearing clearances is investigatedThe dimensionless equations of the system are solved usingthe Runge-Kutta numerical method The dynamics of thesystem are analyzed with reference to its bifurcation dia-grams time histories phase plane portraits Poincare mapsand Fourier spectra and jumpphenomena periodicmotionsand chaotic motions are found in this study

In this study mesh frequency is first expressed to studythe effects of their variations on dynamic response Withthe change of mesh frequency the repeated bifurcationphenomena and the complex chaotic motions are observedand the evolution process and rule of the system are revealedThe effect of damping ratio is secondly analyzed It is shownthat chaotic region becomes narrower as damping ratio isincreased Other important parameters like load coefficientand gear backlash are also considered in the current analysisLoad coefficient affects light loaded condition more thanheavy loaded condition The increase of gear backlash willexpand the effect of nonlinearity effect and increase thechaotic region of the gear system

The selection of the dynamic parameters plays an impor-tant role in designing and controlling such gear systems thesuitable values should be chosen so that chaotic behavior will

12 Shock and Vibration

200 300 400 500 600

0

02

04

06

minus02

minus04

(a)

0 02 04 06

0

05

1

15

minus05

minus04 minus02minus1

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 12 119887 = 10 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

minus05

(a)

0 05 1

0

1

2

minus2

minus1

minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 13 119887 = 40 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 13

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1minus15 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 14 119887 = 80 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

3

minus1

(a)

0 1 2 3

0

1

2

minus2

minus1

minus1minus3

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 15 119887 = 120 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

14 Shock and Vibration

be decreased A more precise model to better describe thespiral bevel gear system will be proposed and deep study willbe analyzed in the future

Competing Interests

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

Acknowledgments

This research was supported by the Natural Science Foun-dation of Hubei Province China (Grant no 2014 CFB827)The authors of the paper express their deep gratitude for thefinancial support of the research project

References

[1] HN Ozguven andD RHouser ldquoMathematicalmodels used ingear dynamicsmdasha reviewrdquo Journal of Sound and Vibration vol121 no 3 pp 383ndash411 1988

[2] S Kiyono Y Fujii and Y Suzuki ldquoAnalysis of vibration of bevelgearsrdquo Bulletin of the Japan Society of Mechanical Engineers vol24 no 188 pp 441ndash446 1981

[3] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[4] C Gosselin L Cloutier and Q D Nguyen ldquoA general formu-lation for the calculation of the load sharing and transmissionerror under load of spiral bevel and hypoid gearsrdquo Mechanismand Machine Theory vol 30 no 3 pp 433ndash450 1995

[5] F L Litvin A Fuentes Q Fan and R F Handschuh ldquoCom-puterized design simulation of meshing and contact and stressanalysis of face-milled formate generated spiral bevel gearsrdquoMechanism and Machine Theory vol 37 no 5 pp 441ndash4592002

[6] J-Y Tang Z-H Hu L-J Wu and S-Y Chen ldquoEffect of statictransmission error on dynamic responses of spiral bevel gearsrdquoJournal of Central South University vol 20 no 3 pp 640ndash6472013

[7] H B Yang J P Gao Z D Fang X Z Deng and Y W ZhouldquoNonlinear dynamics of hypoid gearsrdquoAutomotive Engineeringvol 22 no 1 pp 51ndash54 2002 (Chinese)

[8] J Wang T C Lim and M F Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibration vol308 no 1-2 pp 302ndash329 2007

[9] C-W Chang-Jian ldquoNonlinear dynamic analysis for bevel-gearsystem under nonlinear suspension-bifurcation and chaosrdquoApplied Mathematical Modelling vol 35 no 7 pp 3225ndash32372011

[10] S Theodossiades and S Natsiavas ldquoPeriodic and chaoticdynamics of motor-driven gear-pair systems with backlashrdquoChaos Solitons and Fractals vol 12 no 13 pp 2427ndash2440 2001

[11] Y Cheng and T C Lim ldquoVibration analysis of hypoid trans-missions applying an exact geometry-based gear mesh theoryrdquoJournal of Sound andVibration vol 240 no 3 pp 519ndash543 2001

[12] M Li and H Y Hu ldquoDynamic analysis of a spiral bevel-gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 259no 3 pp 605ndash624 2003

[13] O DMohammedM Rantatalo and J-O Aidanpaa ldquoDynamicmodelling of a one-stage spur gear system and vibration-basedtooth crack detection analysisrdquo Mechanical Systems and SignalProcessing vol 54-55 pp 293ndash305 2015

[14] S MWang YW Shen and H J Dong ldquoNon-linear dynamicalcharacteristics of a spiral bevel gear system with backlash andtime-varying stiffnessrdquo Chinese Journal of Mechanical Engineer-ing vol 39 no 2 pp 28ndash32 2003 (Chinese)

[15] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo NonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 75 no 4 pp 783ndash806 2014

[16] Y Shen S Yang and X Liu ldquoNonlinear dynamics of a spurgear pair with time-varying stiffness and backlash based onincremental harmonic balance methodrdquo International Journalof Mechanical Sciences vol 48 no 11 pp 1256ndash1263 2006

[17] R F Li and J J Wang Gear System Dynamics Science Publish-ing Beijing China 1997 (Chinese)

[18] H B Wilson L H Turcotte and D Halpern AdvancedMathematics and Mechanics Applications Using Matlab CRCPress Company Boca Raton Fla USA 3rd edition 2003

[19] T Sun and H Hu ldquoNonlinear dynamics of a planetary gearsystem with multiple clearancesrdquo Mechanism and MachineTheory vol 38 no 12 pp 1371ndash1390 2003

[20] Z Hu J Tang S Chen and D Lei ldquoEffect of mesh stiffness onthe dynamic response of face gear transmission systemrdquo Journalof Mechanical Design Transactions of the ASME vol 135 no 7Article ID 071005 7 pages 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

0

05

1

minus1

minus05

300 400 500 600200

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 4 120596ℎ = 16 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

0

05

1

15

minus1

minus05

300 400 500 600200

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 5 120596ℎ = 177 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

8 Shock and Vibration

2

15

1

05

0

minus05

minus1

minus15

minus21816 2 2212 14080604 102

ℎ

(a)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(c)

Figure 6 Bifurcation diagrams using 120577 as control parameter (a) 120577 = 001 (b) 120577 = 004 (c) 120577 = 012keep fixed the bifurcation diagrams are shown in Figure 6The chaotic motions are shown through the numericalsimulation for three different values of 120577 = 001 120577 = 004and 120577 = 012 (120577 = 008 see Figure 2(a)) According tothe comparison of the bifurcation diagrams with differentdamping ratio values it can be observed that chaotic regionbecomes narrower as 120577 is increased chaotic region getssmaller and smaller Therefore the increase in 120577 tends todecrease the nonlinearity dynamic responses

33 Effect of Load Coefficient Different vibration character-istics will be shown under different loads For heavy loadedcondition the value of the load coefficient parameter is fixedas 119891 = 119891119898120582 = 2 (119891119898 = 01 120582 = 005) For lightloaded condition the value of the load coefficient parameteris fixed as 119891 = 119891119898120582 = 13 (119891119898 = 01 120582 = 03) Thetime histories phase plane portraits Poincare maps andFourier spectra are shown in Figures 7 and 8 These figuresshowdynamic responses for heavy loaded condition and lightloaded condition at the dimensionless mesh frequency 120596ℎ= 14 respectively As illustrated in Figure 7 the system islinear system the steady-state response is harmonic responseThere is obviously no tooth impact and the system behavesas 1T-periodic motion It is illustrated that the dynamic

characteristics of the gear are not changedmuchunder loadedcondition when the dynamic behavior behaves as periodicmotion As illustrated in Figure 8 compared with the heavyloaded condition the magnitude of the spectrum becomeslarger The displacement along the line of action is greaterthan 0 and is less than 0 so the case of tooth impact can beobserved in the time history plot

Figures 9 and 10 show dynamic responses for heavyloaded condition and light loaded condition at the dimen-sionless mesh frequency 120596ℎ = 18 respectively Nonlinearvibration characteristics can be seen occurring in thesetwo situations However the intensities are not the sameAs illustrated in Figure 9 the motion state is 1T-periodicharmonic motion under heavy loaded condition It can beseen that the spiral bevel gear system is in chaotic motionunder light loaded condition and the response of the systemis irregular in the chaotic state from Figure 10 Furthermorechaotic phenomenon is more likely to occur in light loadedcondition compared with heavy loaded condition withoutchanging other conditions

34 Effect of Gear Backlash Gear backlash is another impor-tant parameter which affects the dynamic responses substan-tially Figure 11 presents the bifurcation diagrams for three

Shock and Vibration 9

200 300 400 500 6000

02

04

06

08

(a)

0 02 04 06 08

0

02

04

d

minus04

minus02

(b)

0 1 2

0

1

2

d

minus1

minus1minus2

minus2

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 7 120596ℎ = 18 and 119891 = 2 (a) time history (b) phase plane(c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus15minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 8 120596ℎ = 14 and 119891 = 13 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

10 Shock and Vibration

200 300 400 500 600

04

05

06

07

(a)

035 04 045 05 055 06 065 07

0

02

04

minus04

minus02

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 9 120596ℎ = 18 and 119891 = 2 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

15

minus1

minus05

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

Frequency of spectrum

01 02 03 04 05 06 07 08 09 10

(d)

Figure 10 120596ℎ = 18 and 119891 = 13 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 11

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(a)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(c)

Figure 11 Bifurcation diagrams using 119887 as control parameter (a) 119887 = 10 120583m (b) 119887 = 80 120583m (c) 119887 = 120 120583m

different values of half of the backlash 119887 = 10 120583m 119887 = 80 120583mand 119887 = 120120583m (119887 = 40 120583m see Figure 2(a)) A substantialregion of jump phenomenon subharmonic quasi-periodicand chaotic responses are contained in Figure 11 Withincreasing b the system becomes more and more unstableand consequently chaotic behavior expands

For the further study on the effect of gear backlash thetime histories phase plane portraits Poincare maps andFourier spectra of the dynamic responses at 120596ℎ = 223 fornumerous gear backlash values are compared in Figures12ndash15 According to these figures a transition from quasi-periodic motion to the chaotic dynamics is seen with theincrease of 119887 As observed in Poincare maps the variousforms of the system from the quasi-periodic motion areseen and with the increase of the bifurcation parameter theunstable attractive region is enlarged eventually leading tochaos

4 Conclusion

In the present paper a spiral bevel gear system supportedon thrust bearings considering the coupled bending-torsionalnonlinear vibration is proposed and an 8DOF lumpedparameter dynamic model of the spiral bevel gear system

combining with time-varying stiffness static transmissionerror gear backlash and bearing clearances is investigatedThe dimensionless equations of the system are solved usingthe Runge-Kutta numerical method The dynamics of thesystem are analyzed with reference to its bifurcation dia-grams time histories phase plane portraits Poincare mapsand Fourier spectra and jumpphenomena periodicmotionsand chaotic motions are found in this study

In this study mesh frequency is first expressed to studythe effects of their variations on dynamic response Withthe change of mesh frequency the repeated bifurcationphenomena and the complex chaotic motions are observedand the evolution process and rule of the system are revealedThe effect of damping ratio is secondly analyzed It is shownthat chaotic region becomes narrower as damping ratio isincreased Other important parameters like load coefficientand gear backlash are also considered in the current analysisLoad coefficient affects light loaded condition more thanheavy loaded condition The increase of gear backlash willexpand the effect of nonlinearity effect and increase thechaotic region of the gear system

The selection of the dynamic parameters plays an impor-tant role in designing and controlling such gear systems thesuitable values should be chosen so that chaotic behavior will

12 Shock and Vibration

200 300 400 500 600

0

02

04

06

minus02

minus04

(a)

0 02 04 06

0

05

1

15

minus05

minus04 minus02minus1

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 12 119887 = 10 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

minus05

(a)

0 05 1

0

1

2

minus2

minus1

minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 13 119887 = 40 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 13

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1minus15 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 14 119887 = 80 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

3

minus1

(a)

0 1 2 3

0

1

2

minus2

minus1

minus1minus3

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 15 119887 = 120 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

14 Shock and Vibration

be decreased A more precise model to better describe thespiral bevel gear system will be proposed and deep study willbe analyzed in the future

Competing Interests

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

Acknowledgments

This research was supported by the Natural Science Foun-dation of Hubei Province China (Grant no 2014 CFB827)The authors of the paper express their deep gratitude for thefinancial support of the research project

References

[1] HN Ozguven andD RHouser ldquoMathematicalmodels used ingear dynamicsmdasha reviewrdquo Journal of Sound and Vibration vol121 no 3 pp 383ndash411 1988

[2] S Kiyono Y Fujii and Y Suzuki ldquoAnalysis of vibration of bevelgearsrdquo Bulletin of the Japan Society of Mechanical Engineers vol24 no 188 pp 441ndash446 1981

[3] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[4] C Gosselin L Cloutier and Q D Nguyen ldquoA general formu-lation for the calculation of the load sharing and transmissionerror under load of spiral bevel and hypoid gearsrdquo Mechanismand Machine Theory vol 30 no 3 pp 433ndash450 1995

[5] F L Litvin A Fuentes Q Fan and R F Handschuh ldquoCom-puterized design simulation of meshing and contact and stressanalysis of face-milled formate generated spiral bevel gearsrdquoMechanism and Machine Theory vol 37 no 5 pp 441ndash4592002

[6] J-Y Tang Z-H Hu L-J Wu and S-Y Chen ldquoEffect of statictransmission error on dynamic responses of spiral bevel gearsrdquoJournal of Central South University vol 20 no 3 pp 640ndash6472013

[7] H B Yang J P Gao Z D Fang X Z Deng and Y W ZhouldquoNonlinear dynamics of hypoid gearsrdquoAutomotive Engineeringvol 22 no 1 pp 51ndash54 2002 (Chinese)

[8] J Wang T C Lim and M F Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibration vol308 no 1-2 pp 302ndash329 2007

[9] C-W Chang-Jian ldquoNonlinear dynamic analysis for bevel-gearsystem under nonlinear suspension-bifurcation and chaosrdquoApplied Mathematical Modelling vol 35 no 7 pp 3225ndash32372011

[10] S Theodossiades and S Natsiavas ldquoPeriodic and chaoticdynamics of motor-driven gear-pair systems with backlashrdquoChaos Solitons and Fractals vol 12 no 13 pp 2427ndash2440 2001

[11] Y Cheng and T C Lim ldquoVibration analysis of hypoid trans-missions applying an exact geometry-based gear mesh theoryrdquoJournal of Sound andVibration vol 240 no 3 pp 519ndash543 2001

[12] M Li and H Y Hu ldquoDynamic analysis of a spiral bevel-gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 259no 3 pp 605ndash624 2003

[13] O DMohammedM Rantatalo and J-O Aidanpaa ldquoDynamicmodelling of a one-stage spur gear system and vibration-basedtooth crack detection analysisrdquo Mechanical Systems and SignalProcessing vol 54-55 pp 293ndash305 2015

[14] S MWang YW Shen and H J Dong ldquoNon-linear dynamicalcharacteristics of a spiral bevel gear system with backlash andtime-varying stiffnessrdquo Chinese Journal of Mechanical Engineer-ing vol 39 no 2 pp 28ndash32 2003 (Chinese)

[15] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo NonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 75 no 4 pp 783ndash806 2014

[16] Y Shen S Yang and X Liu ldquoNonlinear dynamics of a spurgear pair with time-varying stiffness and backlash based onincremental harmonic balance methodrdquo International Journalof Mechanical Sciences vol 48 no 11 pp 1256ndash1263 2006

[17] R F Li and J J Wang Gear System Dynamics Science Publish-ing Beijing China 1997 (Chinese)

[18] H B Wilson L H Turcotte and D Halpern AdvancedMathematics and Mechanics Applications Using Matlab CRCPress Company Boca Raton Fla USA 3rd edition 2003

[19] T Sun and H Hu ldquoNonlinear dynamics of a planetary gearsystem with multiple clearancesrdquo Mechanism and MachineTheory vol 38 no 12 pp 1371ndash1390 2003

[20] Z Hu J Tang S Chen and D Lei ldquoEffect of mesh stiffness onthe dynamic response of face gear transmission systemrdquo Journalof Mechanical Design Transactions of the ASME vol 135 no 7Article ID 071005 7 pages 2013

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Shock and Vibration

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Navigation and Observation

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DistributedSensor Networks

International Journal of

8 Shock and Vibration

2

15

1

05

0

minus05

minus1

minus15

minus21816 2 2212 14080604 102

ℎ

(a)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(c)

Figure 6 Bifurcation diagrams using 120577 as control parameter (a) 120577 = 001 (b) 120577 = 004 (c) 120577 = 012keep fixed the bifurcation diagrams are shown in Figure 6The chaotic motions are shown through the numericalsimulation for three different values of 120577 = 001 120577 = 004and 120577 = 012 (120577 = 008 see Figure 2(a)) According tothe comparison of the bifurcation diagrams with differentdamping ratio values it can be observed that chaotic regionbecomes narrower as 120577 is increased chaotic region getssmaller and smaller Therefore the increase in 120577 tends todecrease the nonlinearity dynamic responses

33 Effect of Load Coefficient Different vibration character-istics will be shown under different loads For heavy loadedcondition the value of the load coefficient parameter is fixedas 119891 = 119891119898120582 = 2 (119891119898 = 01 120582 = 005) For lightloaded condition the value of the load coefficient parameteris fixed as 119891 = 119891119898120582 = 13 (119891119898 = 01 120582 = 03) Thetime histories phase plane portraits Poincare maps andFourier spectra are shown in Figures 7 and 8 These figuresshowdynamic responses for heavy loaded condition and lightloaded condition at the dimensionless mesh frequency 120596ℎ= 14 respectively As illustrated in Figure 7 the system islinear system the steady-state response is harmonic responseThere is obviously no tooth impact and the system behavesas 1T-periodic motion It is illustrated that the dynamic

characteristics of the gear are not changedmuchunder loadedcondition when the dynamic behavior behaves as periodicmotion As illustrated in Figure 8 compared with the heavyloaded condition the magnitude of the spectrum becomeslarger The displacement along the line of action is greaterthan 0 and is less than 0 so the case of tooth impact can beobserved in the time history plot

Figures 9 and 10 show dynamic responses for heavyloaded condition and light loaded condition at the dimen-sionless mesh frequency 120596ℎ = 18 respectively Nonlinearvibration characteristics can be seen occurring in thesetwo situations However the intensities are not the sameAs illustrated in Figure 9 the motion state is 1T-periodicharmonic motion under heavy loaded condition It can beseen that the spiral bevel gear system is in chaotic motionunder light loaded condition and the response of the systemis irregular in the chaotic state from Figure 10 Furthermorechaotic phenomenon is more likely to occur in light loadedcondition compared with heavy loaded condition withoutchanging other conditions

34 Effect of Gear Backlash Gear backlash is another impor-tant parameter which affects the dynamic responses substan-tially Figure 11 presents the bifurcation diagrams for three

Shock and Vibration 9

200 300 400 500 6000

02

04

06

08

(a)

0 02 04 06 08

0

02

04

d

minus04

minus02

(b)

0 1 2

0

1

2

d

minus1

minus1minus2

minus2

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 7 120596ℎ = 18 and 119891 = 2 (a) time history (b) phase plane(c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus15minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 8 120596ℎ = 14 and 119891 = 13 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

10 Shock and Vibration

200 300 400 500 600

04

05

06

07

(a)

035 04 045 05 055 06 065 07

0

02

04

minus04

minus02

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 9 120596ℎ = 18 and 119891 = 2 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

15

minus1

minus05

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

Frequency of spectrum

01 02 03 04 05 06 07 08 09 10

(d)

Figure 10 120596ℎ = 18 and 119891 = 13 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 11

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(a)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(c)

Figure 11 Bifurcation diagrams using 119887 as control parameter (a) 119887 = 10 120583m (b) 119887 = 80 120583m (c) 119887 = 120 120583m

different values of half of the backlash 119887 = 10 120583m 119887 = 80 120583mand 119887 = 120120583m (119887 = 40 120583m see Figure 2(a)) A substantialregion of jump phenomenon subharmonic quasi-periodicand chaotic responses are contained in Figure 11 Withincreasing b the system becomes more and more unstableand consequently chaotic behavior expands

For the further study on the effect of gear backlash thetime histories phase plane portraits Poincare maps andFourier spectra of the dynamic responses at 120596ℎ = 223 fornumerous gear backlash values are compared in Figures12ndash15 According to these figures a transition from quasi-periodic motion to the chaotic dynamics is seen with theincrease of 119887 As observed in Poincare maps the variousforms of the system from the quasi-periodic motion areseen and with the increase of the bifurcation parameter theunstable attractive region is enlarged eventually leading tochaos

4 Conclusion

In the present paper a spiral bevel gear system supportedon thrust bearings considering the coupled bending-torsionalnonlinear vibration is proposed and an 8DOF lumpedparameter dynamic model of the spiral bevel gear system

combining with time-varying stiffness static transmissionerror gear backlash and bearing clearances is investigatedThe dimensionless equations of the system are solved usingthe Runge-Kutta numerical method The dynamics of thesystem are analyzed with reference to its bifurcation dia-grams time histories phase plane portraits Poincare mapsand Fourier spectra and jumpphenomena periodicmotionsand chaotic motions are found in this study

In this study mesh frequency is first expressed to studythe effects of their variations on dynamic response Withthe change of mesh frequency the repeated bifurcationphenomena and the complex chaotic motions are observedand the evolution process and rule of the system are revealedThe effect of damping ratio is secondly analyzed It is shownthat chaotic region becomes narrower as damping ratio isincreased Other important parameters like load coefficientand gear backlash are also considered in the current analysisLoad coefficient affects light loaded condition more thanheavy loaded condition The increase of gear backlash willexpand the effect of nonlinearity effect and increase thechaotic region of the gear system

The selection of the dynamic parameters plays an impor-tant role in designing and controlling such gear systems thesuitable values should be chosen so that chaotic behavior will

12 Shock and Vibration

200 300 400 500 600

0

02

04

06

minus02

minus04

(a)

0 02 04 06

0

05

1

15

minus05

minus04 minus02minus1

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 12 119887 = 10 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

minus05

(a)

0 05 1

0

1

2

minus2

minus1

minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 13 119887 = 40 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 13

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1minus15 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 14 119887 = 80 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

3

minus1

(a)

0 1 2 3

0

1

2

minus2

minus1

minus1minus3

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 15 119887 = 120 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

14 Shock and Vibration

be decreased A more precise model to better describe thespiral bevel gear system will be proposed and deep study willbe analyzed in the future

Competing Interests

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

Acknowledgments

This research was supported by the Natural Science Foun-dation of Hubei Province China (Grant no 2014 CFB827)The authors of the paper express their deep gratitude for thefinancial support of the research project

References

[1] HN Ozguven andD RHouser ldquoMathematicalmodels used ingear dynamicsmdasha reviewrdquo Journal of Sound and Vibration vol121 no 3 pp 383ndash411 1988

[2] S Kiyono Y Fujii and Y Suzuki ldquoAnalysis of vibration of bevelgearsrdquo Bulletin of the Japan Society of Mechanical Engineers vol24 no 188 pp 441ndash446 1981

[3] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[4] C Gosselin L Cloutier and Q D Nguyen ldquoA general formu-lation for the calculation of the load sharing and transmissionerror under load of spiral bevel and hypoid gearsrdquo Mechanismand Machine Theory vol 30 no 3 pp 433ndash450 1995

[5] F L Litvin A Fuentes Q Fan and R F Handschuh ldquoCom-puterized design simulation of meshing and contact and stressanalysis of face-milled formate generated spiral bevel gearsrdquoMechanism and Machine Theory vol 37 no 5 pp 441ndash4592002

[6] J-Y Tang Z-H Hu L-J Wu and S-Y Chen ldquoEffect of statictransmission error on dynamic responses of spiral bevel gearsrdquoJournal of Central South University vol 20 no 3 pp 640ndash6472013

[7] H B Yang J P Gao Z D Fang X Z Deng and Y W ZhouldquoNonlinear dynamics of hypoid gearsrdquoAutomotive Engineeringvol 22 no 1 pp 51ndash54 2002 (Chinese)

[8] J Wang T C Lim and M F Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibration vol308 no 1-2 pp 302ndash329 2007

[9] C-W Chang-Jian ldquoNonlinear dynamic analysis for bevel-gearsystem under nonlinear suspension-bifurcation and chaosrdquoApplied Mathematical Modelling vol 35 no 7 pp 3225ndash32372011

[10] S Theodossiades and S Natsiavas ldquoPeriodic and chaoticdynamics of motor-driven gear-pair systems with backlashrdquoChaos Solitons and Fractals vol 12 no 13 pp 2427ndash2440 2001

[11] Y Cheng and T C Lim ldquoVibration analysis of hypoid trans-missions applying an exact geometry-based gear mesh theoryrdquoJournal of Sound andVibration vol 240 no 3 pp 519ndash543 2001

[12] M Li and H Y Hu ldquoDynamic analysis of a spiral bevel-gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 259no 3 pp 605ndash624 2003

[13] O DMohammedM Rantatalo and J-O Aidanpaa ldquoDynamicmodelling of a one-stage spur gear system and vibration-basedtooth crack detection analysisrdquo Mechanical Systems and SignalProcessing vol 54-55 pp 293ndash305 2015

[14] S MWang YW Shen and H J Dong ldquoNon-linear dynamicalcharacteristics of a spiral bevel gear system with backlash andtime-varying stiffnessrdquo Chinese Journal of Mechanical Engineer-ing vol 39 no 2 pp 28ndash32 2003 (Chinese)

[15] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo NonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 75 no 4 pp 783ndash806 2014

[16] Y Shen S Yang and X Liu ldquoNonlinear dynamics of a spurgear pair with time-varying stiffness and backlash based onincremental harmonic balance methodrdquo International Journalof Mechanical Sciences vol 48 no 11 pp 1256ndash1263 2006

[17] R F Li and J J Wang Gear System Dynamics Science Publish-ing Beijing China 1997 (Chinese)

[18] H B Wilson L H Turcotte and D Halpern AdvancedMathematics and Mechanics Applications Using Matlab CRCPress Company Boca Raton Fla USA 3rd edition 2003

[19] T Sun and H Hu ldquoNonlinear dynamics of a planetary gearsystem with multiple clearancesrdquo Mechanism and MachineTheory vol 38 no 12 pp 1371ndash1390 2003

[20] Z Hu J Tang S Chen and D Lei ldquoEffect of mesh stiffness onthe dynamic response of face gear transmission systemrdquo Journalof Mechanical Design Transactions of the ASME vol 135 no 7Article ID 071005 7 pages 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

200 300 400 500 6000

02

04

06

08

(a)

0 02 04 06 08

0

02

04

d

minus04

minus02

(b)

0 1 2

0

1

2

d

minus1

minus1minus2

minus2

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 7 120596ℎ = 18 and 119891 = 2 (a) time history (b) phase plane(c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus1

minus1 minus05minus15minus2

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 8 120596ℎ = 14 and 119891 = 13 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

10 Shock and Vibration

200 300 400 500 600

04

05

06

07

(a)

035 04 045 05 055 06 065 07

0

02

04

minus04

minus02

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 9 120596ℎ = 18 and 119891 = 2 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

15

minus1

minus05

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

Frequency of spectrum

01 02 03 04 05 06 07 08 09 10

(d)

Figure 10 120596ℎ = 18 and 119891 = 13 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 11

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(a)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(c)

Figure 11 Bifurcation diagrams using 119887 as control parameter (a) 119887 = 10 120583m (b) 119887 = 80 120583m (c) 119887 = 120 120583m

different values of half of the backlash 119887 = 10 120583m 119887 = 80 120583mand 119887 = 120120583m (119887 = 40 120583m see Figure 2(a)) A substantialregion of jump phenomenon subharmonic quasi-periodicand chaotic responses are contained in Figure 11 Withincreasing b the system becomes more and more unstableand consequently chaotic behavior expands

For the further study on the effect of gear backlash thetime histories phase plane portraits Poincare maps andFourier spectra of the dynamic responses at 120596ℎ = 223 fornumerous gear backlash values are compared in Figures12ndash15 According to these figures a transition from quasi-periodic motion to the chaotic dynamics is seen with theincrease of 119887 As observed in Poincare maps the variousforms of the system from the quasi-periodic motion areseen and with the increase of the bifurcation parameter theunstable attractive region is enlarged eventually leading tochaos

4 Conclusion

In the present paper a spiral bevel gear system supportedon thrust bearings considering the coupled bending-torsionalnonlinear vibration is proposed and an 8DOF lumpedparameter dynamic model of the spiral bevel gear system

combining with time-varying stiffness static transmissionerror gear backlash and bearing clearances is investigatedThe dimensionless equations of the system are solved usingthe Runge-Kutta numerical method The dynamics of thesystem are analyzed with reference to its bifurcation dia-grams time histories phase plane portraits Poincare mapsand Fourier spectra and jumpphenomena periodicmotionsand chaotic motions are found in this study

In this study mesh frequency is first expressed to studythe effects of their variations on dynamic response Withthe change of mesh frequency the repeated bifurcationphenomena and the complex chaotic motions are observedand the evolution process and rule of the system are revealedThe effect of damping ratio is secondly analyzed It is shownthat chaotic region becomes narrower as damping ratio isincreased Other important parameters like load coefficientand gear backlash are also considered in the current analysisLoad coefficient affects light loaded condition more thanheavy loaded condition The increase of gear backlash willexpand the effect of nonlinearity effect and increase thechaotic region of the gear system

The selection of the dynamic parameters plays an impor-tant role in designing and controlling such gear systems thesuitable values should be chosen so that chaotic behavior will

12 Shock and Vibration

200 300 400 500 600

0

02

04

06

minus02

minus04

(a)

0 02 04 06

0

05

1

15

minus05

minus04 minus02minus1

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 12 119887 = 10 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

minus05

(a)

0 05 1

0

1

2

minus2

minus1

minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 13 119887 = 40 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 13

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1minus15 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 14 119887 = 80 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

3

minus1

(a)

0 1 2 3

0

1

2

minus2

minus1

minus1minus3

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 15 119887 = 120 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

14 Shock and Vibration

be decreased A more precise model to better describe thespiral bevel gear system will be proposed and deep study willbe analyzed in the future

Competing Interests

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

Acknowledgments

This research was supported by the Natural Science Foun-dation of Hubei Province China (Grant no 2014 CFB827)The authors of the paper express their deep gratitude for thefinancial support of the research project

References

[1] HN Ozguven andD RHouser ldquoMathematicalmodels used ingear dynamicsmdasha reviewrdquo Journal of Sound and Vibration vol121 no 3 pp 383ndash411 1988

[2] S Kiyono Y Fujii and Y Suzuki ldquoAnalysis of vibration of bevelgearsrdquo Bulletin of the Japan Society of Mechanical Engineers vol24 no 188 pp 441ndash446 1981

[3] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[4] C Gosselin L Cloutier and Q D Nguyen ldquoA general formu-lation for the calculation of the load sharing and transmissionerror under load of spiral bevel and hypoid gearsrdquo Mechanismand Machine Theory vol 30 no 3 pp 433ndash450 1995

[5] F L Litvin A Fuentes Q Fan and R F Handschuh ldquoCom-puterized design simulation of meshing and contact and stressanalysis of face-milled formate generated spiral bevel gearsrdquoMechanism and Machine Theory vol 37 no 5 pp 441ndash4592002

[6] J-Y Tang Z-H Hu L-J Wu and S-Y Chen ldquoEffect of statictransmission error on dynamic responses of spiral bevel gearsrdquoJournal of Central South University vol 20 no 3 pp 640ndash6472013

[7] H B Yang J P Gao Z D Fang X Z Deng and Y W ZhouldquoNonlinear dynamics of hypoid gearsrdquoAutomotive Engineeringvol 22 no 1 pp 51ndash54 2002 (Chinese)

[8] J Wang T C Lim and M F Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibration vol308 no 1-2 pp 302ndash329 2007

[9] C-W Chang-Jian ldquoNonlinear dynamic analysis for bevel-gearsystem under nonlinear suspension-bifurcation and chaosrdquoApplied Mathematical Modelling vol 35 no 7 pp 3225ndash32372011

[10] S Theodossiades and S Natsiavas ldquoPeriodic and chaoticdynamics of motor-driven gear-pair systems with backlashrdquoChaos Solitons and Fractals vol 12 no 13 pp 2427ndash2440 2001

[11] Y Cheng and T C Lim ldquoVibration analysis of hypoid trans-missions applying an exact geometry-based gear mesh theoryrdquoJournal of Sound andVibration vol 240 no 3 pp 519ndash543 2001

[12] M Li and H Y Hu ldquoDynamic analysis of a spiral bevel-gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 259no 3 pp 605ndash624 2003

[13] O DMohammedM Rantatalo and J-O Aidanpaa ldquoDynamicmodelling of a one-stage spur gear system and vibration-basedtooth crack detection analysisrdquo Mechanical Systems and SignalProcessing vol 54-55 pp 293ndash305 2015

[14] S MWang YW Shen and H J Dong ldquoNon-linear dynamicalcharacteristics of a spiral bevel gear system with backlash andtime-varying stiffnessrdquo Chinese Journal of Mechanical Engineer-ing vol 39 no 2 pp 28ndash32 2003 (Chinese)

[15] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo NonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 75 no 4 pp 783ndash806 2014

[16] Y Shen S Yang and X Liu ldquoNonlinear dynamics of a spurgear pair with time-varying stiffness and backlash based onincremental harmonic balance methodrdquo International Journalof Mechanical Sciences vol 48 no 11 pp 1256ndash1263 2006

[17] R F Li and J J Wang Gear System Dynamics Science Publish-ing Beijing China 1997 (Chinese)

[18] H B Wilson L H Turcotte and D Halpern AdvancedMathematics and Mechanics Applications Using Matlab CRCPress Company Boca Raton Fla USA 3rd edition 2003

[19] T Sun and H Hu ldquoNonlinear dynamics of a planetary gearsystem with multiple clearancesrdquo Mechanism and MachineTheory vol 38 no 12 pp 1371ndash1390 2003

[20] Z Hu J Tang S Chen and D Lei ldquoEffect of mesh stiffness onthe dynamic response of face gear transmission systemrdquo Journalof Mechanical Design Transactions of the ASME vol 135 no 7Article ID 071005 7 pages 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

200 300 400 500 600

04

05

06

07

(a)

035 04 045 05 055 06 065 07

0

02

04

minus04

minus02

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 9 120596ℎ = 18 and 119891 = 2 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

15

minus1

minus05

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

Frequency of spectrum

01 02 03 04 05 06 07 08 09 10

(d)

Figure 10 120596ℎ = 18 and 119891 = 13 (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 11

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(a)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(c)

Figure 11 Bifurcation diagrams using 119887 as control parameter (a) 119887 = 10 120583m (b) 119887 = 80 120583m (c) 119887 = 120 120583m

different values of half of the backlash 119887 = 10 120583m 119887 = 80 120583mand 119887 = 120120583m (119887 = 40 120583m see Figure 2(a)) A substantialregion of jump phenomenon subharmonic quasi-periodicand chaotic responses are contained in Figure 11 Withincreasing b the system becomes more and more unstableand consequently chaotic behavior expands

For the further study on the effect of gear backlash thetime histories phase plane portraits Poincare maps andFourier spectra of the dynamic responses at 120596ℎ = 223 fornumerous gear backlash values are compared in Figures12ndash15 According to these figures a transition from quasi-periodic motion to the chaotic dynamics is seen with theincrease of 119887 As observed in Poincare maps the variousforms of the system from the quasi-periodic motion areseen and with the increase of the bifurcation parameter theunstable attractive region is enlarged eventually leading tochaos

4 Conclusion

In the present paper a spiral bevel gear system supportedon thrust bearings considering the coupled bending-torsionalnonlinear vibration is proposed and an 8DOF lumpedparameter dynamic model of the spiral bevel gear system

combining with time-varying stiffness static transmissionerror gear backlash and bearing clearances is investigatedThe dimensionless equations of the system are solved usingthe Runge-Kutta numerical method The dynamics of thesystem are analyzed with reference to its bifurcation dia-grams time histories phase plane portraits Poincare mapsand Fourier spectra and jumpphenomena periodicmotionsand chaotic motions are found in this study

In this study mesh frequency is first expressed to studythe effects of their variations on dynamic response Withthe change of mesh frequency the repeated bifurcationphenomena and the complex chaotic motions are observedand the evolution process and rule of the system are revealedThe effect of damping ratio is secondly analyzed It is shownthat chaotic region becomes narrower as damping ratio isincreased Other important parameters like load coefficientand gear backlash are also considered in the current analysisLoad coefficient affects light loaded condition more thanheavy loaded condition The increase of gear backlash willexpand the effect of nonlinearity effect and increase thechaotic region of the gear system

The selection of the dynamic parameters plays an impor-tant role in designing and controlling such gear systems thesuitable values should be chosen so that chaotic behavior will

12 Shock and Vibration

200 300 400 500 600

0

02

04

06

minus02

minus04

(a)

0 02 04 06

0

05

1

15

minus05

minus04 minus02minus1

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 12 119887 = 10 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

minus05

(a)

0 05 1

0

1

2

minus2

minus1

minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 13 119887 = 40 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 13

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1minus15 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 14 119887 = 80 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

3

minus1

(a)

0 1 2 3

0

1

2

minus2

minus1

minus1minus3

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 15 119887 = 120 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

14 Shock and Vibration

be decreased A more precise model to better describe thespiral bevel gear system will be proposed and deep study willbe analyzed in the future

Competing Interests

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

Acknowledgments

This research was supported by the Natural Science Foun-dation of Hubei Province China (Grant no 2014 CFB827)The authors of the paper express their deep gratitude for thefinancial support of the research project

References

[1] HN Ozguven andD RHouser ldquoMathematicalmodels used ingear dynamicsmdasha reviewrdquo Journal of Sound and Vibration vol121 no 3 pp 383ndash411 1988

[2] S Kiyono Y Fujii and Y Suzuki ldquoAnalysis of vibration of bevelgearsrdquo Bulletin of the Japan Society of Mechanical Engineers vol24 no 188 pp 441ndash446 1981

[3] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[4] C Gosselin L Cloutier and Q D Nguyen ldquoA general formu-lation for the calculation of the load sharing and transmissionerror under load of spiral bevel and hypoid gearsrdquo Mechanismand Machine Theory vol 30 no 3 pp 433ndash450 1995

[5] F L Litvin A Fuentes Q Fan and R F Handschuh ldquoCom-puterized design simulation of meshing and contact and stressanalysis of face-milled formate generated spiral bevel gearsrdquoMechanism and Machine Theory vol 37 no 5 pp 441ndash4592002

[6] J-Y Tang Z-H Hu L-J Wu and S-Y Chen ldquoEffect of statictransmission error on dynamic responses of spiral bevel gearsrdquoJournal of Central South University vol 20 no 3 pp 640ndash6472013

[7] H B Yang J P Gao Z D Fang X Z Deng and Y W ZhouldquoNonlinear dynamics of hypoid gearsrdquoAutomotive Engineeringvol 22 no 1 pp 51ndash54 2002 (Chinese)

[8] J Wang T C Lim and M F Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibration vol308 no 1-2 pp 302ndash329 2007

[9] C-W Chang-Jian ldquoNonlinear dynamic analysis for bevel-gearsystem under nonlinear suspension-bifurcation and chaosrdquoApplied Mathematical Modelling vol 35 no 7 pp 3225ndash32372011

[10] S Theodossiades and S Natsiavas ldquoPeriodic and chaoticdynamics of motor-driven gear-pair systems with backlashrdquoChaos Solitons and Fractals vol 12 no 13 pp 2427ndash2440 2001

[11] Y Cheng and T C Lim ldquoVibration analysis of hypoid trans-missions applying an exact geometry-based gear mesh theoryrdquoJournal of Sound andVibration vol 240 no 3 pp 519ndash543 2001

[12] M Li and H Y Hu ldquoDynamic analysis of a spiral bevel-gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 259no 3 pp 605ndash624 2003

[13] O DMohammedM Rantatalo and J-O Aidanpaa ldquoDynamicmodelling of a one-stage spur gear system and vibration-basedtooth crack detection analysisrdquo Mechanical Systems and SignalProcessing vol 54-55 pp 293ndash305 2015

[14] S MWang YW Shen and H J Dong ldquoNon-linear dynamicalcharacteristics of a spiral bevel gear system with backlash andtime-varying stiffnessrdquo Chinese Journal of Mechanical Engineer-ing vol 39 no 2 pp 28ndash32 2003 (Chinese)

[15] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo NonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 75 no 4 pp 783ndash806 2014

[16] Y Shen S Yang and X Liu ldquoNonlinear dynamics of a spurgear pair with time-varying stiffness and backlash based onincremental harmonic balance methodrdquo International Journalof Mechanical Sciences vol 48 no 11 pp 1256ndash1263 2006

[17] R F Li and J J Wang Gear System Dynamics Science Publish-ing Beijing China 1997 (Chinese)

[18] H B Wilson L H Turcotte and D Halpern AdvancedMathematics and Mechanics Applications Using Matlab CRCPress Company Boca Raton Fla USA 3rd edition 2003

[19] T Sun and H Hu ldquoNonlinear dynamics of a planetary gearsystem with multiple clearancesrdquo Mechanism and MachineTheory vol 38 no 12 pp 1371ndash1390 2003

[20] Z Hu J Tang S Chen and D Lei ldquoEffect of mesh stiffness onthe dynamic response of face gear transmission systemrdquo Journalof Mechanical Design Transactions of the ASME vol 135 no 7Article ID 071005 7 pages 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(a)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(b)

1816 2 2212 14080604 102

ℎ

2

15

1

05

0

minus05

minus1

minus15

minus2

(c)

Figure 11 Bifurcation diagrams using 119887 as control parameter (a) 119887 = 10 120583m (b) 119887 = 80 120583m (c) 119887 = 120 120583m

different values of half of the backlash 119887 = 10 120583m 119887 = 80 120583mand 119887 = 120120583m (119887 = 40 120583m see Figure 2(a)) A substantialregion of jump phenomenon subharmonic quasi-periodicand chaotic responses are contained in Figure 11 Withincreasing b the system becomes more and more unstableand consequently chaotic behavior expands

For the further study on the effect of gear backlash thetime histories phase plane portraits Poincare maps andFourier spectra of the dynamic responses at 120596ℎ = 223 fornumerous gear backlash values are compared in Figures12ndash15 According to these figures a transition from quasi-periodic motion to the chaotic dynamics is seen with theincrease of 119887 As observed in Poincare maps the variousforms of the system from the quasi-periodic motion areseen and with the increase of the bifurcation parameter theunstable attractive region is enlarged eventually leading tochaos

4 Conclusion

In the present paper a spiral bevel gear system supportedon thrust bearings considering the coupled bending-torsionalnonlinear vibration is proposed and an 8DOF lumpedparameter dynamic model of the spiral bevel gear system

combining with time-varying stiffness static transmissionerror gear backlash and bearing clearances is investigatedThe dimensionless equations of the system are solved usingthe Runge-Kutta numerical method The dynamics of thesystem are analyzed with reference to its bifurcation dia-grams time histories phase plane portraits Poincare mapsand Fourier spectra and jumpphenomena periodicmotionsand chaotic motions are found in this study

In this study mesh frequency is first expressed to studythe effects of their variations on dynamic response Withthe change of mesh frequency the repeated bifurcationphenomena and the complex chaotic motions are observedand the evolution process and rule of the system are revealedThe effect of damping ratio is secondly analyzed It is shownthat chaotic region becomes narrower as damping ratio isincreased Other important parameters like load coefficientand gear backlash are also considered in the current analysisLoad coefficient affects light loaded condition more thanheavy loaded condition The increase of gear backlash willexpand the effect of nonlinearity effect and increase thechaotic region of the gear system

The selection of the dynamic parameters plays an impor-tant role in designing and controlling such gear systems thesuitable values should be chosen so that chaotic behavior will

12 Shock and Vibration

200 300 400 500 600

0

02

04

06

minus02

minus04

(a)

0 02 04 06

0

05

1

15

minus05

minus04 minus02minus1

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 12 119887 = 10 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

minus05

(a)

0 05 1

0

1

2

minus2

minus1

minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 13 119887 = 40 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 13

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1minus15 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 14 119887 = 80 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

3

minus1

(a)

0 1 2 3

0

1

2

minus2

minus1

minus1minus3

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 15 119887 = 120 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

14 Shock and Vibration

be decreased A more precise model to better describe thespiral bevel gear system will be proposed and deep study willbe analyzed in the future

Competing Interests

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

Acknowledgments

This research was supported by the Natural Science Foun-dation of Hubei Province China (Grant no 2014 CFB827)The authors of the paper express their deep gratitude for thefinancial support of the research project

References

[1] HN Ozguven andD RHouser ldquoMathematicalmodels used ingear dynamicsmdasha reviewrdquo Journal of Sound and Vibration vol121 no 3 pp 383ndash411 1988

[2] S Kiyono Y Fujii and Y Suzuki ldquoAnalysis of vibration of bevelgearsrdquo Bulletin of the Japan Society of Mechanical Engineers vol24 no 188 pp 441ndash446 1981

[3] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[4] C Gosselin L Cloutier and Q D Nguyen ldquoA general formu-lation for the calculation of the load sharing and transmissionerror under load of spiral bevel and hypoid gearsrdquo Mechanismand Machine Theory vol 30 no 3 pp 433ndash450 1995

[5] F L Litvin A Fuentes Q Fan and R F Handschuh ldquoCom-puterized design simulation of meshing and contact and stressanalysis of face-milled formate generated spiral bevel gearsrdquoMechanism and Machine Theory vol 37 no 5 pp 441ndash4592002

[6] J-Y Tang Z-H Hu L-J Wu and S-Y Chen ldquoEffect of statictransmission error on dynamic responses of spiral bevel gearsrdquoJournal of Central South University vol 20 no 3 pp 640ndash6472013

[7] H B Yang J P Gao Z D Fang X Z Deng and Y W ZhouldquoNonlinear dynamics of hypoid gearsrdquoAutomotive Engineeringvol 22 no 1 pp 51ndash54 2002 (Chinese)

[8] J Wang T C Lim and M F Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibration vol308 no 1-2 pp 302ndash329 2007

[9] C-W Chang-Jian ldquoNonlinear dynamic analysis for bevel-gearsystem under nonlinear suspension-bifurcation and chaosrdquoApplied Mathematical Modelling vol 35 no 7 pp 3225ndash32372011

[10] S Theodossiades and S Natsiavas ldquoPeriodic and chaoticdynamics of motor-driven gear-pair systems with backlashrdquoChaos Solitons and Fractals vol 12 no 13 pp 2427ndash2440 2001

[11] Y Cheng and T C Lim ldquoVibration analysis of hypoid trans-missions applying an exact geometry-based gear mesh theoryrdquoJournal of Sound andVibration vol 240 no 3 pp 519ndash543 2001

[12] M Li and H Y Hu ldquoDynamic analysis of a spiral bevel-gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 259no 3 pp 605ndash624 2003

[13] O DMohammedM Rantatalo and J-O Aidanpaa ldquoDynamicmodelling of a one-stage spur gear system and vibration-basedtooth crack detection analysisrdquo Mechanical Systems and SignalProcessing vol 54-55 pp 293ndash305 2015

[14] S MWang YW Shen and H J Dong ldquoNon-linear dynamicalcharacteristics of a spiral bevel gear system with backlash andtime-varying stiffnessrdquo Chinese Journal of Mechanical Engineer-ing vol 39 no 2 pp 28ndash32 2003 (Chinese)

[15] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo NonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 75 no 4 pp 783ndash806 2014

[16] Y Shen S Yang and X Liu ldquoNonlinear dynamics of a spurgear pair with time-varying stiffness and backlash based onincremental harmonic balance methodrdquo International Journalof Mechanical Sciences vol 48 no 11 pp 1256ndash1263 2006

[17] R F Li and J J Wang Gear System Dynamics Science Publish-ing Beijing China 1997 (Chinese)

[18] H B Wilson L H Turcotte and D Halpern AdvancedMathematics and Mechanics Applications Using Matlab CRCPress Company Boca Raton Fla USA 3rd edition 2003

[19] T Sun and H Hu ldquoNonlinear dynamics of a planetary gearsystem with multiple clearancesrdquo Mechanism and MachineTheory vol 38 no 12 pp 1371ndash1390 2003

[20] Z Hu J Tang S Chen and D Lei ldquoEffect of mesh stiffness onthe dynamic response of face gear transmission systemrdquo Journalof Mechanical Design Transactions of the ASME vol 135 no 7Article ID 071005 7 pages 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

200 300 400 500 600

0

02

04

06

minus02

minus04

(a)

0 02 04 06

0

05

1

15

minus05

minus04 minus02minus1

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 12 119887 = 10 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

05

1

minus05

(a)

0 05 1

0

1

2

minus2

minus1

minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 13 119887 = 40 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

Shock and Vibration 13

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1minus15 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 14 119887 = 80 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

3

minus1

(a)

0 1 2 3

0

1

2

minus2

minus1

minus1minus3

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 15 119887 = 120 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

14 Shock and Vibration

be decreased A more precise model to better describe thespiral bevel gear system will be proposed and deep study willbe analyzed in the future

Competing Interests

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

Acknowledgments

This research was supported by the Natural Science Foun-dation of Hubei Province China (Grant no 2014 CFB827)The authors of the paper express their deep gratitude for thefinancial support of the research project

References

[1] HN Ozguven andD RHouser ldquoMathematicalmodels used ingear dynamicsmdasha reviewrdquo Journal of Sound and Vibration vol121 no 3 pp 383ndash411 1988

[2] S Kiyono Y Fujii and Y Suzuki ldquoAnalysis of vibration of bevelgearsrdquo Bulletin of the Japan Society of Mechanical Engineers vol24 no 188 pp 441ndash446 1981

[3] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[4] C Gosselin L Cloutier and Q D Nguyen ldquoA general formu-lation for the calculation of the load sharing and transmissionerror under load of spiral bevel and hypoid gearsrdquo Mechanismand Machine Theory vol 30 no 3 pp 433ndash450 1995

[5] F L Litvin A Fuentes Q Fan and R F Handschuh ldquoCom-puterized design simulation of meshing and contact and stressanalysis of face-milled formate generated spiral bevel gearsrdquoMechanism and Machine Theory vol 37 no 5 pp 441ndash4592002

[6] J-Y Tang Z-H Hu L-J Wu and S-Y Chen ldquoEffect of statictransmission error on dynamic responses of spiral bevel gearsrdquoJournal of Central South University vol 20 no 3 pp 640ndash6472013

[7] H B Yang J P Gao Z D Fang X Z Deng and Y W ZhouldquoNonlinear dynamics of hypoid gearsrdquoAutomotive Engineeringvol 22 no 1 pp 51ndash54 2002 (Chinese)

[8] J Wang T C Lim and M F Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibration vol308 no 1-2 pp 302ndash329 2007

[9] C-W Chang-Jian ldquoNonlinear dynamic analysis for bevel-gearsystem under nonlinear suspension-bifurcation and chaosrdquoApplied Mathematical Modelling vol 35 no 7 pp 3225ndash32372011

[10] S Theodossiades and S Natsiavas ldquoPeriodic and chaoticdynamics of motor-driven gear-pair systems with backlashrdquoChaos Solitons and Fractals vol 12 no 13 pp 2427ndash2440 2001

[11] Y Cheng and T C Lim ldquoVibration analysis of hypoid trans-missions applying an exact geometry-based gear mesh theoryrdquoJournal of Sound andVibration vol 240 no 3 pp 519ndash543 2001

[12] M Li and H Y Hu ldquoDynamic analysis of a spiral bevel-gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 259no 3 pp 605ndash624 2003

[13] O DMohammedM Rantatalo and J-O Aidanpaa ldquoDynamicmodelling of a one-stage spur gear system and vibration-basedtooth crack detection analysisrdquo Mechanical Systems and SignalProcessing vol 54-55 pp 293ndash305 2015

[14] S MWang YW Shen and H J Dong ldquoNon-linear dynamicalcharacteristics of a spiral bevel gear system with backlash andtime-varying stiffnessrdquo Chinese Journal of Mechanical Engineer-ing vol 39 no 2 pp 28ndash32 2003 (Chinese)

[15] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo NonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 75 no 4 pp 783ndash806 2014

[16] Y Shen S Yang and X Liu ldquoNonlinear dynamics of a spurgear pair with time-varying stiffness and backlash based onincremental harmonic balance methodrdquo International Journalof Mechanical Sciences vol 48 no 11 pp 1256ndash1263 2006

[17] R F Li and J J Wang Gear System Dynamics Science Publish-ing Beijing China 1997 (Chinese)

[18] H B Wilson L H Turcotte and D Halpern AdvancedMathematics and Mechanics Applications Using Matlab CRCPress Company Boca Raton Fla USA 3rd edition 2003

[19] T Sun and H Hu ldquoNonlinear dynamics of a planetary gearsystem with multiple clearancesrdquo Mechanism and MachineTheory vol 38 no 12 pp 1371ndash1390 2003

[20] Z Hu J Tang S Chen and D Lei ldquoEffect of mesh stiffness onthe dynamic response of face gear transmission systemrdquo Journalof Mechanical Design Transactions of the ASME vol 135 no 7Article ID 071005 7 pages 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 13

200 300 400 500 600

0

1

2

minus1

minus2

(a)

0 05 1 15

0

1

2

minus2

minus1

minus1minus15 minus05

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 14 119887 = 80 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

200 300 400 500 600

0

1

2

3

minus1

(a)

0 1 2 3

0

1

2

minus2

minus1

minus1minus3

d

(b)

0 1 2

0

1

2

minus1

minus1minus2

minus2

d

(c)

0

20

40

60

80

100

01 02 03 04 05 06 07 08 09 10

Frequency of spectrum

(d)

Figure 15 119887 = 120 120583m (a) time history (b) phase plane (c) Poincare map (d) Fourier spectrum

14 Shock and Vibration

be decreased A more precise model to better describe thespiral bevel gear system will be proposed and deep study willbe analyzed in the future

Competing Interests

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

Acknowledgments

This research was supported by the Natural Science Foun-dation of Hubei Province China (Grant no 2014 CFB827)The authors of the paper express their deep gratitude for thefinancial support of the research project

References

[1] HN Ozguven andD RHouser ldquoMathematicalmodels used ingear dynamicsmdasha reviewrdquo Journal of Sound and Vibration vol121 no 3 pp 383ndash411 1988

[2] S Kiyono Y Fujii and Y Suzuki ldquoAnalysis of vibration of bevelgearsrdquo Bulletin of the Japan Society of Mechanical Engineers vol24 no 188 pp 441ndash446 1981

[3] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[4] C Gosselin L Cloutier and Q D Nguyen ldquoA general formu-lation for the calculation of the load sharing and transmissionerror under load of spiral bevel and hypoid gearsrdquo Mechanismand Machine Theory vol 30 no 3 pp 433ndash450 1995

[5] F L Litvin A Fuentes Q Fan and R F Handschuh ldquoCom-puterized design simulation of meshing and contact and stressanalysis of face-milled formate generated spiral bevel gearsrdquoMechanism and Machine Theory vol 37 no 5 pp 441ndash4592002

[6] J-Y Tang Z-H Hu L-J Wu and S-Y Chen ldquoEffect of statictransmission error on dynamic responses of spiral bevel gearsrdquoJournal of Central South University vol 20 no 3 pp 640ndash6472013

[7] H B Yang J P Gao Z D Fang X Z Deng and Y W ZhouldquoNonlinear dynamics of hypoid gearsrdquoAutomotive Engineeringvol 22 no 1 pp 51ndash54 2002 (Chinese)

[8] J Wang T C Lim and M F Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibration vol308 no 1-2 pp 302ndash329 2007

[9] C-W Chang-Jian ldquoNonlinear dynamic analysis for bevel-gearsystem under nonlinear suspension-bifurcation and chaosrdquoApplied Mathematical Modelling vol 35 no 7 pp 3225ndash32372011

[10] S Theodossiades and S Natsiavas ldquoPeriodic and chaoticdynamics of motor-driven gear-pair systems with backlashrdquoChaos Solitons and Fractals vol 12 no 13 pp 2427ndash2440 2001

[11] Y Cheng and T C Lim ldquoVibration analysis of hypoid trans-missions applying an exact geometry-based gear mesh theoryrdquoJournal of Sound andVibration vol 240 no 3 pp 519ndash543 2001

[12] M Li and H Y Hu ldquoDynamic analysis of a spiral bevel-gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 259no 3 pp 605ndash624 2003

[13] O DMohammedM Rantatalo and J-O Aidanpaa ldquoDynamicmodelling of a one-stage spur gear system and vibration-basedtooth crack detection analysisrdquo Mechanical Systems and SignalProcessing vol 54-55 pp 293ndash305 2015

[14] S MWang YW Shen and H J Dong ldquoNon-linear dynamicalcharacteristics of a spiral bevel gear system with backlash andtime-varying stiffnessrdquo Chinese Journal of Mechanical Engineer-ing vol 39 no 2 pp 28ndash32 2003 (Chinese)

[15] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo NonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 75 no 4 pp 783ndash806 2014

[16] Y Shen S Yang and X Liu ldquoNonlinear dynamics of a spurgear pair with time-varying stiffness and backlash based onincremental harmonic balance methodrdquo International Journalof Mechanical Sciences vol 48 no 11 pp 1256ndash1263 2006

[17] R F Li and J J Wang Gear System Dynamics Science Publish-ing Beijing China 1997 (Chinese)

[18] H B Wilson L H Turcotte and D Halpern AdvancedMathematics and Mechanics Applications Using Matlab CRCPress Company Boca Raton Fla USA 3rd edition 2003

[19] T Sun and H Hu ldquoNonlinear dynamics of a planetary gearsystem with multiple clearancesrdquo Mechanism and MachineTheory vol 38 no 12 pp 1371ndash1390 2003

[20] Z Hu J Tang S Chen and D Lei ldquoEffect of mesh stiffness onthe dynamic response of face gear transmission systemrdquo Journalof Mechanical Design Transactions of the ASME vol 135 no 7Article ID 071005 7 pages 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Shock and Vibration

be decreased A more precise model to better describe thespiral bevel gear system will be proposed and deep study willbe analyzed in the future

Competing Interests

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

Acknowledgments

This research was supported by the Natural Science Foun-dation of Hubei Province China (Grant no 2014 CFB827)The authors of the paper express their deep gratitude for thefinancial support of the research project

References

[1] HN Ozguven andD RHouser ldquoMathematicalmodels used ingear dynamicsmdasha reviewrdquo Journal of Sound and Vibration vol121 no 3 pp 383ndash411 1988

[2] S Kiyono Y Fujii and Y Suzuki ldquoAnalysis of vibration of bevelgearsrdquo Bulletin of the Japan Society of Mechanical Engineers vol24 no 188 pp 441ndash446 1981

[3] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[4] C Gosselin L Cloutier and Q D Nguyen ldquoA general formu-lation for the calculation of the load sharing and transmissionerror under load of spiral bevel and hypoid gearsrdquo Mechanismand Machine Theory vol 30 no 3 pp 433ndash450 1995

[5] F L Litvin A Fuentes Q Fan and R F Handschuh ldquoCom-puterized design simulation of meshing and contact and stressanalysis of face-milled formate generated spiral bevel gearsrdquoMechanism and Machine Theory vol 37 no 5 pp 441ndash4592002

[6] J-Y Tang Z-H Hu L-J Wu and S-Y Chen ldquoEffect of statictransmission error on dynamic responses of spiral bevel gearsrdquoJournal of Central South University vol 20 no 3 pp 640ndash6472013

[7] H B Yang J P Gao Z D Fang X Z Deng and Y W ZhouldquoNonlinear dynamics of hypoid gearsrdquoAutomotive Engineeringvol 22 no 1 pp 51ndash54 2002 (Chinese)

[8] J Wang T C Lim and M F Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibration vol308 no 1-2 pp 302ndash329 2007

[9] C-W Chang-Jian ldquoNonlinear dynamic analysis for bevel-gearsystem under nonlinear suspension-bifurcation and chaosrdquoApplied Mathematical Modelling vol 35 no 7 pp 3225ndash32372011

[10] S Theodossiades and S Natsiavas ldquoPeriodic and chaoticdynamics of motor-driven gear-pair systems with backlashrdquoChaos Solitons and Fractals vol 12 no 13 pp 2427ndash2440 2001

[11] Y Cheng and T C Lim ldquoVibration analysis of hypoid trans-missions applying an exact geometry-based gear mesh theoryrdquoJournal of Sound andVibration vol 240 no 3 pp 519ndash543 2001

[12] M Li and H Y Hu ldquoDynamic analysis of a spiral bevel-gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 259no 3 pp 605ndash624 2003

[13] O DMohammedM Rantatalo and J-O Aidanpaa ldquoDynamicmodelling of a one-stage spur gear system and vibration-basedtooth crack detection analysisrdquo Mechanical Systems and SignalProcessing vol 54-55 pp 293ndash305 2015

[14] S MWang YW Shen and H J Dong ldquoNon-linear dynamicalcharacteristics of a spiral bevel gear system with backlash andtime-varying stiffnessrdquo Chinese Journal of Mechanical Engineer-ing vol 39 no 2 pp 28ndash32 2003 (Chinese)

[15] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo NonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 75 no 4 pp 783ndash806 2014

[16] Y Shen S Yang and X Liu ldquoNonlinear dynamics of a spurgear pair with time-varying stiffness and backlash based onincremental harmonic balance methodrdquo International Journalof Mechanical Sciences vol 48 no 11 pp 1256ndash1263 2006

[17] R F Li and J J Wang Gear System Dynamics Science Publish-ing Beijing China 1997 (Chinese)

[18] H B Wilson L H Turcotte and D Halpern AdvancedMathematics and Mechanics Applications Using Matlab CRCPress Company Boca Raton Fla USA 3rd edition 2003

[19] T Sun and H Hu ldquoNonlinear dynamics of a planetary gearsystem with multiple clearancesrdquo Mechanism and MachineTheory vol 38 no 12 pp 1371ndash1390 2003

[20] Z Hu J Tang S Chen and D Lei ldquoEffect of mesh stiffness onthe dynamic response of face gear transmission systemrdquo Journalof Mechanical Design Transactions of the ASME vol 135 no 7Article ID 071005 7 pages 2013

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of