Research ArticleModeling andAnalysis of Amplitude-Frequency Characteristics ofTorsional Vibration for Automotive Powertrain

Jinli Xu Jiwei Zhu and Feifan Xia

School of Mechanical and Electronic Engineering Wuhan University of Technology 122 Luoshi Road Hongshan DistrictWuhan 430070 Hubei China

Correspondence should be addressed to Jiwei Zhu edwinzjw163com

Received 3 December 2019 Revised 21 July 2020 Accepted 31 July 2020 Published 29 August 2020

Academic Editor F Viadero

Copyright copy 2020 Jinli Xu et alis 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

In the present paper the amplitude-frequency characteristics of torsional vibration are discussed theoretically and experimentallyfor automotive powertrain A bending-torsional-lateral-rocking coupled dynamic model with time-dependent mesh stiffnessbacklash transmission error etc is proposed by the lumped-mass method to analysis the amplitude-frequency characteristic oftorsional vibration for practical purposes and equations of motive are derived e RungendashKutta method is employed to conducta sweep frequency response analysis numerically Furthermore a torsional experiment is performed and validates the feasibility ofthe theoretical model As a result some torsional characteristics of automotive powertrain are obtained e first three-ordernature torsional frequencies are predicted Torsional behaviors only affect the vibration characteristics of a complete vehicle atlow-speed condition and will be reinforced expectedly while increasing torque fluctuation Gear mesh excitations have little effectson torsional responses for such components located before mesh point but a lot for ones behind it In particular it is noted that thetorsional system has a stiffness-softening characteristic with respect to torque fluctuation

1 Introduction

e front-engine rear-wheel drive layout (FR) vehicle has acomplicated powertrain consisting of engine clutchtransmission drive shaft rear axle and tire as shown inFigure 1 which indicates that more dynamic behaviorsoccur In a typical dynamic operating condition the fluc-tuation of engine output torque U-joint dynamic charac-teristics will induce torsional vibration of the wholepowertrain Internal excitations originating from inherentcharacteristics of hypoid gear set also are responsible for thetransverse-torsional-rocking coupled vibration of rear axleese vibrations are eventually transmitted to the vehiclebody through intermediate support and suspension to act asthe main aspects of excitations causing vibration and noisefor the complete vehicle In order to improve the dynamicperformances for vehicle a number of research studies hadbeen performed

Vesali et al [1] obtained the relationship between inputand output angular speed for different U-joint with

variations in structure Lu et al [2] proposed a dynamicmodel with clearance by the lumped-mass method to discussdynamics of a cross shaft type universal joint and analyze theeffects of clearance on output speed and dynamic torqueresponse Wu et al [3] optimized the phase differencesbetween adjacent U-joints which reduced the torsionalvibration of transmission shaft A classical model wasproposed by Porter [4] in which the transmission systemcontaining a U-joint was modeled as a torsional system withone degree-of-freedom Asokanthan and Wang [5] intro-duced a torsional model with two degree-of-freedom toinvestigate stability and bifurcation performances by usingthe maximum Lyapunov exponent method based on Porterrsquosmodel Farshidianfar et al [6] developed a lumped-massmodel for a driveline forced by torsional excitation to analyzethe formation mechanism of noise Friction was consideredin Abd Elmaksoudrsquos paper [7] whose investigation indicatedthat the friction moment is a main aspect of excitation-inducing torsional vibration for the automobile drivelineJuang et al [8] discussed effects of the sliding-tube-type

HindawiShock and VibrationVolume 2020 Article ID 6403413 17 pageshttpsdoiorg10115520206403413

driveshaft on a complete vehicle by experimental and finiteelement method ey reached a conclusion that fluctuationof nonlinear contact force in sliding mechanism accounts forthe complete vehicle vibration Coutinho and Tamagna [9]analyzed the bending natural frequency of half-shaft toprevent from resonance under power excitation e effectsof intermediate support on complete vehicle vibration arediscussed in Yiting Kangrsquos work [10] Xu et al [11] andZhang et al [12] analyzed dynamics of transmission shaft-rear driving axle based on ADAMS and experimentaldemonstration and the coupling effects between trans-mission shaft and rear axle were discussed Xu et al [13] andXu et al [14] developed several different dynamic modelsconsidering coupling interaction between automobiletransmission shaft and main reducer In their studies effectsof bearing stiffness backlash intermediate support etc onthe system were discussed numerically It is noted that thekey of analytical solution for the main reducer is the de-velopment of the dynamic model of hypoid gear with time-varying meshing stiffness damping clearance bear andother line or nonlinear factors which inherits the researchstudies on spur and helical gear pair conduct by Kahramanand Singh [15ndash17] Cai and Hayashi [18] and Velex et al[19ndash21] and remarkable achievements can be found withindocuments of Limrsquos team [22ndash26] and other researchers

As introduced in preceding section a few investigationstreating engine clutch transmission transmission shaft andrear axle as a whole system were conducted mathematicallyAlthough many models have been introduced in the liter-ature very few considered the influence of transverse androcking vibration on torsional performances theoreticallyfor discussing torsional characteristics of powertrain In thepresent paper the amplitude-frequency characteristics oftorsional vibration for automotive powertrain are analyzedtheoretically and experimentally considering the interac-tions between powertrain and the main reducer gear system

As for the layout of the present paper it mainly containsfive sections besides introduction In the second section alumped-parameter dynamic model with 29 degree-of-free-dom is proposed considering the transverse torsional androcking vibration and equations of motion are derived Inthe third section the introduced model is adopted to discussthe amplitude-frequency characteristics of torsional vibra-tion for the coupling system by sweep frequency responseanalysis numerically In the fourth part an experiment isperformed to validate the feasibility of the considered modeland analyze the effect of torque fluctuation on the ampli-tude-frequency characteristics of torsional vibration As thefinal part a conclusion is offered

2 Dynamic Modeling of Powertrain

As shown in Figure 1 the powertrain discussed in this paperconsists of engine transmission drive shaft system and rearaxle Considering the complexity of the system a series ofsimplifying processes are adopted to establish the dynamicmodel of the system by using lumped-parameter methodereby a generalized model with twenty-nine degree-of-freedom (29DOF) is proposed as shown in Figure 2 in whichU-joint dynamic actions intermediate support stiffnesstime-varying mesh stiffness gear backlash static trans-mission error and other factors are considered

e generalized coordinate vector of the dynamic modelis given by

S θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 θ9 θ10 θ11 θ12 θ13 θ14 θ15 θ16 θ17

θpy θpz θgx θgz ym zm xp yp zp xg yg zg

⎧⎨

⎩

⎫⎬

⎭

Τ

(1)

where θi(i 1 minus 5) represents the torsional displacement ofthe equivalent element as their coordinates ym and zm arethe translations of the intermediate support along the axes y

and z xpx xpy and xpz are the translations of the pinionalong the axes x y and z xgx xgy andxgz are the trans-lations of the gear along the axes x y and z θpx is thetorsional displacement of the pinion around the x axisθpy and θpz are the rocking displacements of the pinionaround the y and z axes and θgx and θgz are the rockingdisplacements of the gear around the x and z axesrespectively

e equivalent mass and moment of inertia are denotedby mi and Ji

21 Engine In order to satisfy limitations of experimentalbench parameters of engine are replaced by ones of inputmotor which is processed as two equivalent moments ofinertia corresponding to the equivalent moments of inertiaof the motor flywheel and clutch

Output torque of engine as a main external excitation ofthe system is written as in Fourier series form

TransmissionIntermediate

drive sha

Half-sha

Main drive sha

Intermediate support

Main reducer anddifferential

U-joint

U-jointU-jointEngine

and clutch

Figure 1 Schematic of the automotive powertrain system dis-cussed in this paper

2 Shock and Vibration

TD Tm + 1113944N

i1Tdi cos iωft + φi1113872 1113873 (2)

where Tm is mean output torque Tdi is the amplitude of ithharmonic term ωf is output speed of engine and φi is phaseangel of ith harmonic term

Based on previous research [14] only mean componentand the second-order harmonic term are considered here

TD Tm + Td2 cos 2ωft + φ21113872 1113873 (3)

22 Transmission e transmission is simplified as achained system with six equivalent moments of inertia andi1 is introduced to describe the drive ratio of transmissionFurthermore gears are assumed as rigid and gear meshcharacteristics frictions etc are neglected Without loss ofgenerality the case of i1 1350 is employed as an example inthe present work

In order to simplify a multimesh gear train into achained system a shaft should be chosen as the main bodyfirstly as illustrated in Figure 3 in which Ji (i 1ndash4) andJiprime(i 3 minus 4) are equivalent moments of inertia and Ki(i 1ndash4) and Ki

prime(i 3 minus 4) denote equivalent stiffness of gearshaft

Kinetic energy of output shaft for the original system andchained system can be calculated as follows

E J3ω2

2 + J4ω22

2

Eprime J3primeω2

1 + J4primeω21

2

(4)

where E is the kinetic energy of output shaft for the originalsystem J3 and J4 are the equivalent moments of inertia forthe original system which are derived by dynamic balancetheory ω1 and ω2 are the angular velocities Eprime is the kineticenergy of output shaft for the original system and J3prime and J4primeare equivalent moments of inertia for the chained system

From the conservation law of energy the followingrelationships are obtained

E Eprime

J3prime J3

i2

J4prime J4

i2

(5)

where i is drive ratio

(a)

(b) (c)

Figure 2 e dynamic model of the powertrain shown in Figure 1 (a) dynamic model of powertrain without rear axle (b) torsional modelof rear axle (c) dynamic model of the main reducer in rear axle

Shock and Vibration 3

From the conservation of strain energy an equation isderived

12K3primeφ

21

12K3φ

22 (6)

where K3prime andK3 are torsional stiffness of output shaft for thechained system and original system respectively andφ1 andφ2 are torsional angels of driving and driven shaft

Owing to the assumption concerning a gear as rigid thegear meshing stiffness K2prime and K2 is treated as infinitequantities e following equations are derived

K3prime K3

i2 i

φ1φ2

T2prime T2

i1113896 (7)

where T2prime is the input torque of output shaft and the T2 is theoutput torque of input shaft

According to preceding derivation the equivalent mo-ments of inertia for the chained system are calculated by

J3 Ia1 +Ia2

2

J4 Ia2

2+ Ig1

J5 Ib1 + Ig2 + Ib221113872 1113873

i22 1

J6 Ib22( 1113857 + Ig3 + Ib3 + Ig4 + Ig11i211 41113872 1113873 + Ib4 + Ig5 + middot middot middot + Ig12i212 51113872 1113873 + Ib5 + Id + Ib6 + Ig8 + Ig9 + Ig15i215 91113872 1113873i29 81113872 1113873 + Ib71113960 1113961

i22 1

J7 Ic1 + Id + Ig10 + Ic2 + Ic3 + Ic421113872 1113873 + Id1113872 1113873

i210 3 middot i22 1( 1113857

J8 Ic42 + Ig13 + Ig6i26 13 + Ic5 + Ig14 + Ig7i27 14 + Ic6 + Ic7 + Id1113960 1113961

i210 3 middot i22 1( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

J1J2

J1J2

J3

J3primeJ4prime

J4

K1

K2

K3

K1 K2prime K3prime

First gearshaft

Secondgearshaft

Gear 1Gear 2

Gear 3Gear 4

First gearshaft

Secondgearshaft

Gear 1

Gear 2Gear 3

Gear 4

Figure 3 e schematic diagram of equivalent principle for simplifying a multimesh gear train into a chained system

4 Shock and Vibration

where Iaj(j 1 2) is the equivalent moment of inertia of thejth node of input shaft Ibj(j 1 2 7) is the equivalentmoment of inertia of the jth node of middle shaft Icj(j

1 2 7) is the equivalent moment of inertia of the jthnode of output shaft Igj(j 1 2 15) is the equivalentmoment of inertia of jth gear Id is the equivalent moment ofinertia of the synchronizer ring and im n is the drive ratio ofthe mating gear pair

23 Drive Shaft System Drive shaft system comprises in-termediate drive shaft main drive shaft three universaljoints and intermediate support In this paper the rotationalinertia of shaft is equivalent to the universal joint by thelumped-mass method averagely e lubrication clearancefriction manufacturing and assembly errors and flexibilityof universal joint are not considered

In order to describe the dynamics of a U-joint a set ofparameters are introduced as follows

Ai 1 + cos2αi

2 cos αi

Bi 1 minus cos2αi

2 cos αi

i 1 2 3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

where αi(i 1 2 3) is the intersection angel between shaftslinked by a universal joint (U-joint)

According to kinematics analysis the following equa-tions are derived

θ9 arctantan θ8cos α1

θ9prime θ8prime

A1 minus B1 cos 2θ8

θ9Prime θ8Prime

A1 minus B1 cos 2θ8minus

2B1 sin 2θ8A1 minus B1 cos 2θ8( 1113857

2 θ8prime( 11138572

T2 A1 minus B1 cos 2θ8( 1113857T1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

θ11 arctantan θ10cos α2

θ11prime θ10prime

A2 minus B2 cos 2θ10

θ11Prime θ10Prime

A2 minus B2 cos 2θ10minus

2B2 sin 2θ10A2 minus B2 cos 2θ10( 1113857

2 θ10prime( 11138572

T4 A2 minus B2 cos 2θ10( 1113857T3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

θ13 arctantan θ12cos α3

θ13prime θ12prime

A3 minus B3 cos 2θ12

θ13Prime θ12Prime

A3 minus B3 cos 2θ12minus

2B3 sin 2θ12A3 minus B3 cos 2θ12( 1113857

2 θ12prime( 11138572

T6 A3 minus B3 cos 2θ12( 1113857T5

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

Shock and Vibration 5

where α1 is the intersection angel between output shaft oftransmission and intermediate drive shaft α2 is the inter-section angel between main drive shaft and intermediatedrive shaft α3 is the intersection angel between main driveshaft and input shaft of rear axle

e equivalent moments of inertia for intermediate driveshaft and main drive shaft are calculated by

J9 J10 12i21

Jids

J11 J12 12i21

Jmds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(11)

where i1 is the drive ratio of transmission Jids is the momentof inertia of intermediate drive shaft and Jmds is the momentof inertia of main shaft

Equivalent torsional stiffness k and damping coefficient c

of shaft are given by

k M

ϕ

Gapπ D4 minus d4( 1113857

32L

c 2ζ

kJaJb

Ja + Jb

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(12)

where ζ is the damping ratio and Ja and Jb are the equivalentmoment of inertia of shaft end

24 Rear Axle In this paper the main reducer system issimplified as a geared rotor-bearing system with bending-torsional-lateral-rocking coupled vibration Pinion shaft issimplified as two lumpedmasses linked by a spring-dampingpair with infinite stiffness and the differential assembly ismodeled as a rigid part with a concentratedmass Half-shaftsare described by twomass-spring-damping pairs Otherwisepinion and gear are treated as rigid bodies within torsionalvibration analysis

e equivalent moments of inertia for rear axle arecalculated by

J13 15Jps

i21

J14 45Jps

i21

J15 Jd + 05Jhs

i21i22

J16 J17 05Jhs + Jhs

i21i22

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)

where i1 (i1 1350) is the drive ratio of transmission i2(i2 41) is the drive ratio of the main reducer Jps is themoment of inertia of pinion shaft Jd is the moment of inertia

of differential assembly and Jhs is the moment of inertia ofhalf-shaft

Equivalent stiffness of bearing is expressed as [27]

Kr 7253l08

1 z091 cos20α1 middot F01

a0cos01α1

Ka 29011l081 cos19α1 middot F01

a0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(14)

where Kr is the equivalent radial stiffness of bearing Ka isthe equivalent axial stiffness of bearing l1 is the nominalcontact length z1 is the number of rolling elementα1 is thepressure angle and Fa0 is the pretightening force

In addition a new model for pinion shaft is proposedwhich is different from previous works Owing to negligibleelastic deformation pinion shaft is modeled as rigid bodyBearing is modeled as a spring-damping pair Without lossof generality the derivation procedure on displacements ofthe mounting point in the x-o-y plane is used as an example

In Figure 4 Xpx and Xpy are introduced to representtranslational displacements for centroid of pinion shaft θpz

denotes rocking displacement around the z axisFrom the geometry deformation relationship and pro-

posed concepts the displacement for mounting point iscalculated by

xpy1 xpy minus lp1 tan θpz

xpy2 xpy minus lp2 tan θpz

⎧⎨

⎩ (15)

ereby the following expressions are derived

xpy1prime xpyprime minus lp1

θpzprime

cos2θpz

xpy2prime xpyprime minus lp2

θpzprime

cos2θpz

1113896

(16)

Static transmission error e(t) [22ndash26 28] is simulated inFourier series form by

e(t) em + 1113944N

i1eAi cos iωmt + φe( 1113857 (17)

where em is the average value of transmission error eAi arethe amplitude of the ith order harmonic φe are the initialphase angle of the ith order harmonic and ωm is the meshingfrequency

e dynamic meshing stiffness is the periodically time-varying parameter with respect to the mesh frequency asshown in Figure 5 which can be simulated by means of aFourier expansion [13 14 22ndash26]

km(t) km + 1113944N

i1kAi cos iωmt + φk( 1113857 (18)

where km is the mean value of meshing stiffness kAi is thefluctuation amplitude of ith order meshing stiffness ωm isthe mesh frequency and φk is the initial phase of meshingstiffness

Defining me denotes mean mass of pinion and gear

6 Shock and Vibration

me 1

1mp1113872 1113873 + 1mg1113872 1113873

mpmg

mp + mg

(19)

e following equation can be derived

ωn

km

me

1113971

(20)

where ωn is the equivalent natural frequency of pinion andgear

Equivalent mesh damping coefficient is calculated whichis different from Wang and Limrsquos analysis [25]

cm ξ lowast 2meωn 2ξ

km

1mp1113872 1113873 + 1mg1113872 1113873

1113971

(21)

where ζ is the damping ratio generally set as a range of 003to 017 [25]mp andmg are equivalent mass of pinion and gearrespectively

e relative displacement along the normal direction atthe meshing point xn is given as follows

xn xpx + Eyθpz1113872 1113873 minus xgx + Ezθgy + lg3θgz1113872 11138731113960 1113961

middot cos αn sin βm cos δ1 + sin αn sin δ1( 1113857

+ xpy + lp3θpz1113872 1113873 minus xgy + Ezθgx minus Exθgz1113872 11138731113960 1113961

middot minuscos αn sin βm sin δ1 + sin αn cos δ1( 1113857

+ xpz minus Rpθpx + lp3θpy1113872 1113873 minus xgz minus lg3θgx minus Rgθgy1113872 11138731113960 1113961

middot cos αn cos βm( 1113857 minus e(t)

ε1 xpx minus xgx minus Ezθgy + Eyθpz minus lg3θgz1113872 1113873

+ ε2 xpy minus xgy minus Ezθgx + lp3θpz + Exθgz1113872 1113873

+ ε3 xpz minus xgz minus Rpθpx + lg3θgx + lp3θpy + Rgθgy1113872 1113873

minus e(t)

(22)

e backlash function is expressed as [23]

f xn( 1113857

xn minus b xn ge b

0 minusblexn le b

xn + b xn le minus b

⎧⎪⎪⎨

⎪⎪⎩(23)

where b is half of gear backlashe dynamic meshing force along the line of action Fn

can be calculated byFn km(t)f xn( 1113857 + cmxn

prime (24)

wherecm represents thevalueof equivalentmeshdampingcoefficiente components of dynamic mesh force along the co-

ordinate directions are expressed asFx Fn cos αn sin βm cos δ1 + Fn sin αn sin δ1

Fy minusFn cos αn sin βm sin δ1 + Fn sin αn cos δ1

Fz Fn cos αn cos βm

⎧⎪⎪⎨

⎪⎪⎩(25)

where δ1 is cone angle of the pinion αn is the normal pressureangle and βm is the helix angle at the midpoint of the pinion

In additional the following equations are derived byintroducing the parameter εi

ε1 cos αn sin βm cos δ1 + sin αn sin δ1

ε2 minuscos αn sin βm sin δ1 + sin αn cos δ1

ε3 cos αn cos βm

⎧⎪⎪⎨

⎪⎪⎩(26)

Substituting equations (26) into (25) the expressions ofdynamicmesh force applied on pinion and gear can bewritten as

Fpx minusFgx ε1Fn

Fpy minusFgy ε2Fn

Fpz minusFgz ε3Fn

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(27)

where the subscript p is the label for pinion and the subscriptg is the label for gear

Mesh impact due to teeth separation is neglected in this paper

25 Equations of Motion From the preceding derivationsthe equations of motion considering bending-torsional-lateral-rocking vibration are expressed as

Fpz

kpz2 cpz2

kpx

cpx

θpy

Xpx

kpz1 cpz1

FpxFpy

Op1

cpy2

θpz

mp

kpy2cpy1

Lp1

Lp2 Lp3

Op2 Op0

kpy1y

x

z

Xpy1 Xpy2

Xpy

Op

Op3 Ey

θpz

θpx

Figure 4 e dynamic model of pinion shaft

Shock and Vibration 7

J1θ1Prime + c1 θ1prime minus θ2prime( 1113857 + k1 θ1 minus θ2( 1113857 TD

J2θ2Prime + c1 θ2prime minus θ1prime( 1113857 + k1 θ2 minus θ1( 1113857 + c2 θ2prime minus θ3prime( 1113857 + k2 θ2 minus θ3( 1113857 0

J3θ3Prime + c2 θ3prime minus θ2prime( 1113857 + k2 θ3 minus θ2( 1113857 + c3 θ3prime minus θ4prime( 1113857 + k3 θ3 minus θ4( 1113857 0

J4θ4Prime + c3 θ4prime minus θ3prime( 1113857 + k3 θ4 minus θ3( 1113857 + c4 θ4prime minus θ5prime( 1113857 + k4 θ4 minus θ5( 1113857 0

J5θ5Prime + c4 θ5prime minus θ4prime( 1113857 + k4 θ5 minus θ4( 1113857 + c5 θ5prime minus θ6prime( 1113857 + k5 θ5 minus θ6( 1113857 0

J6θ6Prime + c5 θ6prime minus θ5prime( 1113857 + k5 θ6 minus θ5( 1113857 + c6 θ6prime minus θ7prime( 1113857 + k6 θ6 minus θ7( 1113857 0

J7θ7Prime + c6 θ7prime minus θ6prime( 1113857 + k6 θ7 minus θ6( 1113857 + c7 θ7prime minus θ8prime( 1113857 + k7 θ7 minus θ8( 1113857 0

J8θ8Prime + c7 θ8prime minus θ7prime( 1113857 + k7 θ8 minus θ7( 1113857 minusT1

i1

J9θ9Prime + c8 θ9prime minus θ10prime( 1113857 + k8 θ9 minus θ10( 1113857 T2

i1

J10θ10Prime + c8 θ10prime minus θ9prime( 1113857 + k8 θ10 minus θ9( 1113857 minusT3

i1

J11θ11Prime + c9 θ11prime minus θ12prime( 1113857 + k9 θ11 minus θ12( 1113857 T4

i1

J12θ12Prime + c9 θ12prime minus θ11prime( 1113857 + k9 θ12 minus θ11( 1113857 minusT5

i1

J13θ13Prime + c10 θ13prime minus θ14prime( 1113857 + k10 θ13 minus θ14( 1113857 T6

i1

J14θ14Prime + c10 θ14prime minus θ13prime( 1113857 + k10 θ14 minus θ131113857( 1113857 minusFpzEy

i1

J15θ15Prime + c11 θ15prime minus θ16prime( 1113857 + k11 θ15 minus θ16( 1113857 + c12 θ15prime minus θ17prime( 1113857 + k12 θ15 minus θ17( 1113857 FgzEx + FgxEz

i1i2

J16θ16Prime + c11 θ16prime minus θ15prime( 1113857 + k11 θ16 minus θ15( 1113857 minusTRL

i1i2

J17θ17Prime + c12 θ17prime minus θ15prime( 1113857 + k12 θ17 minus θ15( 1113857 minusTRR

i1i2

mmymPrime + cmyym

prime + kmyym Fmy

mmzmPrime + cmzzm

prime + kmzzm Fmz

mpxpxPrime + kpxxpx + cpxxpx

prime minusFpx

mpxpyPrime + kpy1 xpy minus lp1 tan θpz1113872 1113873 + cpy1 xpy

prime minus lp1θpzprime

cos2θpz

1113888 1113889 + kpy2 xpy minus lp2 tan θpz1113872 1113873 + cpy2 xpyprime minus lp2

θpzprime

cos2θpz

1113888 1113889 Fpy

mpxpzPrime + kpz1 xpz minus lp1 tan θpy1113872 1113873 + cpz1 xpz

prime minus lp1θpyprime

cos2θpy

1113888 1113889 + kpz2 xpz minus lp2 tan θpy1113872 1113873 + cpz2 xpzprime minus lp2

θpyprime

cos2θpy

1113888 1113889 Fpz

JpyθpyPrime minus lp1kpz1 xpz minus lp1 tan θpy1113872 1113873 minus lp1cpz1 xpz

prime minus lp1θpyprime

cos2θpy

1113888 1113889 minus lp2kpz2 xpz minus lp2 tan θpy1113872 1113873 minus lp2cpz2 xpzprime minus lp2

θpyprime

cos2θpy

1113888 1113889 Fpzlp3

JpzθpzPrime minus lp1kpz1 xpy minus lp1 tan θpy1113872 1113873 minus lp1cp1 xpy

prime minus lp1θpzprime

cos2θpz

1113888 1113889 minus lp2kpy2 xpy minus lp2 tan θpz1113872 1113873 minus lp2cpy2 xpyprime minus lp2

θpzprime

cos2θpz

1113888 1113889 minusFpxEy + Fpylp3

mgxgxPrime + kgx1 xgx minus lg1 tan θgz1113872 1113873 + cgx1 xgx

Prime minus lg1θgzprime

cos2θgz

1113888 1113889 + kgx2 xgx + lg2 tan θgz1113872 1113873 + cgx2 xgxprime + lg2

θgzprime

cos2θgz

1113888 1113889 Fgx

mgxgyPrime + kgyxgy + cgyxgy

Prime minusFgy

mgxgzPrime + kgz1 xgz + lg1 tan θgx1113872 1113873 + cgz1 xgz

prime + lg1θgxprime

cos2θgx

1113888 1113889 + kgz2 xgz minus lg2 tan θgx1113872 1113873 + cgz2 xgzprime minus lg2

θgxprime

cos2θgx

1113888 1113889 minusFgz

JgxθgxPrime + lg1kgz1 xgz + lg1 tan θgx1113872 1113873 + lg1cgz1 xgz

prime + lg1θgxprime

cos2θgx

1113888 1113889 minus lg2kgz2 xgz minus lg2 tan θgx1113872 1113873 + lg2cgz2 xgzprime minus lg2

θgxprime

cos2θgx

1113888 1113889 Fgzlg3 minus FgyEz

JgzθgzPrime + lg1kgx1 xgx + lg1 tan θgz1113872 1113873 + lg1cgx1 xgx

prime + lg1θgzprime

cos2θgz

1113888 1113889 minus lg2kgx2 xgx minus lg2 tan θgz1113872 1113873 + lg2cgx2 xgxprime minus lg2

θgzprime

cos2θgz

1113888 1113889 Fgxlg3 minus FgyEx

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

8 Shock and Vibration

where TD is the input torque of motor T1 is the outputtorque of transmission T2 andT3 are the input and outputtorque for intermediate drive shaft T4 andT5 are the inputand output torque for main drive shaft T6 is the inputtorque of rear axle ki(i 1 2 7) are the equivalenttorsional stiffness ci(i 1 2 7) are the equivalent tor-sional damping ki(i 8 9) are the equivalent torsionalstiffness ci(i 8 9) are the equivalent torsional dampingmm is the mass of transmission shaft ki(i 10 11 12) arethe equivalent torsional stiffness for pinion shaft and dif-ferential axle ci(i 10 11 12) are the equivalent torsionaldamping for pinion shaft and differential axle cmy and cmz

are the equivalent damping of intermediate support alongaxes y and z kmy and kmz are the equivalent stiffness ofintermediate support along axes y and z Fmy and Fmz arecomponents of additional force Fi along axes y and z forintermediate support Ei(i x y z) are coordinates ofmeshing point along axes x y and z and TRL andTRR areload torque for wheels

26 Numerical Solution Procedure e fourth-order adap-tive step RungendashKutta method which is generally applicableto strong nonlinearity is adopted in the present work to solve

the equations In order to program a new state space isneeded

In consideration of equations (12)mdash(14) it can be de-rived that kinetic equations of 3 universal joints reduce thenumber of degree-of-freedom from 29 to 26 for the wholesystem e following state space coordinate vector alsodescribes the dynamic behaviors of the system

Ss θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 θ10 θ12 θ14 θ15 θ16 θ17θpy θpz θgx θgz ym zm xp yp zp xg yg zg

⎧⎨

⎩

⎫⎬

⎭

Τ

(29)

In order to eliminate the difference in magnitude be-tween parameters which will reduce the solvability and evenlead to no convergence for solving equations of motionmentioned above b is defined as the characteristic lengthand a new time parameter τ is introduced as follows

τ tωn (30)

where t denotes timeFor the sake of nominalizing equations a generalized

state space coordinate vector with 26 parameters is proposedto characterize dimensionless equations as

st sr1 sr2 sr3 sr4 sr5 sr6 sr7 sr8 sr10 sr12 sr16 sr17 srpx srpy srpz srgx srgy srgz smy smz spx spy spz sgx sgy sgz1113966 1113967T

(31)

e following equations are derived as

sri θi timesRi

b i 1 sim 8 10 12 16 17

srpx θ14 timesRpx

b

srgy θ15 timesRgy

b

srij θij timesRij

b i p g j x y z

six xi

b i p g

siy yi

b i m p g

siz zi

b i m p g

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

where b is the half of backlash and Ri is the equivalent gyrationradius which can be calculated by the following equation

Ri

Ji

mi

1113971

i 1 sim 8 10 12 16 17

Rij

Jij

mi

1113971

i p g j x y z

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(33)

where Ji and Jij are the equivalent moment of inertia and mi

is equivalent massA set of dimensionless parameters are defined as

xnτ xn

b

f xnτ( 1113857 f xt

n( 1113857

b

c(ijs) ci

mjRjRsωn

k(ijs) ki

mjRjRsω2n

cmi cm

miωn

kmi km

miω2n

T(xj) Tx

mjRjω2nb

lijτ lij

b

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)

As equations (30) (32) and (34) are substituted intoequation (28) the dimensionless equations of motion can bederived

Shock and Vibration 9

e equivalent inertia parameters are obtained in UGand stiffness parameters of powertrain are calculated byusing the relationships given in modeling section as listed inTable 1 Subsequently equivalent damping coefficients canbe obtained e parameters of the main reducer can becalculated by substituting system parameters listed in Table 2into equations (17)ndash(31) e solution procedures for di-mensionless parameter are carried out by using precedingcalculated parameters and the relationships proposed inprevious parts

e exciting and configuration parameters are set asθ1 1500 rpm Td2 100Nmiddotm km 9697e8Nm i1 135i2 41 2b 015mm em 0 α1 1695 degree α2 13044degree and α3 1554 degree respectively Without signif-icant loss of accuracy the fundamental harmonic form isadopted for time-varying mesh stiffness and static trans-mission error function whose initial phases are set at 0respectively

3 Numerical Results

31 Analysis of Coupled Torsional Vibration In order toanalysis coupled interaction of automotive powertrainFourier spectrums of torsional vibration response are cal-culated at the initial parameters as shown in Figure 6 Twopeaks and their sidebands are observed e torsional vi-brational responses are governed by the component of 50Hzwhich equals with exciting frequency of input torque asshown in equation (3) (ωf 150060 lowast 2 50Hz) isimplies that engine input excitation is responsible for tor-sional vibration and a valuable approach to decreasingtorsional vibration of powertrain is inferred In addition thecomponent with a frequency of around 185Hz is observedobviously for each considered part As one knows themeshing frequency undergoing this condition is presumably185185Hz (f 150060135185185Hz) which is quite inagreement with the frequency of the second peak Coupledvibration occurs between the torsional system and hypoidgear set and inner excitation caused by gear pair meshingprocess which is affected by clearance and time-varyingmeshing stiffness [25] accounts a lot for torsional vibrationat 185Hz of powertrain and affects torsional responsesslightly After simply calculation bandwidths reaching thesame value of around 37Hz of sidebands around two peaksare got U-joint is a transmission mechanism with non-constant angular speed which will excite a doubled fre-quency torsional vibration of input as shown in equations(12)ndash(14) Under this condition the doubled frequencyexcited by U-joint is calculated

fuminusjoint n

i1 lowast 60lowast 2

1500135lowast 60

lowast 2 3704 (35)

It can be conjectured that modulation phenomenonappears when automotive powertrain works between exci-tations attributed to input U-joint and gear pair As one cansee in Figure 6 compared torsional response of gear withthat located front of meshing point a lower value of tor-sional angular is observed and it is inferred that gear meshexcitations reduce torsional vibration for components

located behind meshing point It is noted that taking gearmesh behaviors into account is quite necessary to when atorsional characteristic analysis is conducted

32 Amplitude-Frequency Response of Torsional Vibration forAutomotive Powertrain e previous works [29] analyzedtorsional characteristics of the coupled system consisted ofdrive shaft and rear axle which denoted that resonance ofdrive shaft at the first two-order nature frequency accountsfor dramatic torsional vibration responses undergoing high-rotation-speed condition However it cannot reveal thetorsional vibration mechanism of automotive under low-and middle-speed condition

In order to obtain a more realistic understanding foramplitude-frequency response of automotive powertrainundergoing middle-speed condition theoretically the pre-ceding equations of motion are adopted A sweep frequencyanalysis procedure is carried out in a range from 1000 rpm to3500 rpm which respectively corresponds to frequencyinternal of 166ndash583Hz for input speed 3333ndash1167Hz forinput excitation and 12346ndash43209Hz for meshing exci-tation e response at input end of rear axle is taken as anexample in this work

Figure 7 shows the amplitude-frequency response inwhich abscissa denotes torsion speed of input and ordinate isRMS value of angular displacement As one can see threeobvious peaks which correspond to input speed at about1000 rpm 1600 rpm and 2800 rpm respectively are ob-served Owing to the assumption that the dramatic responsesare attributed to resonance of the system the first three-order nature frequency respect to the second-order term ofinput torque fluctuation is proposed at 333Hz 533Hz and933Hz successively It is noted that on the basis of previousanalysis these three peaks also correspond to frequencies of247Hz 395Hz and 691Hz for the U-joint excitation and1234Hz 1975Hz and 3456Hz for the meshing excitationPreliminary works [29] showed that resonance is attributedto input excitation of drive shaft and the meshing excitationof hypoid gear pair with higher exciting frequency thaninput will not excite prominent resonance behaviors of driveshaft as illustrated in Figure 8 ie meshing excitation hasslight influence on the torsional system which reaches thesame conclusion with this work According to analysis inSection 31 the sidebands respect to each exciting frequencyare (86ndash58) Hz at 333Hz (138ndash928) Hz at 533Hz and(242ndash1624) Hz at 933Hz for input torque and (987ndash1481)Hz at 1234Hz (158ndash237) Hz at 1975Hz and (2765ndash4147)Hz at 3456Hz for meshing behavior of the gear pair re-spectively As listed in Table 3 the first peak can be attributedto the first two-order resonances of drive shaft excited bysideband around 1234Hz Sidebands at 533Hz and1975Hz result in the second-order resonance which gen-erates the second peak Moreover the second-order reso-nances excited by the sideband at 933Hz and third-orderone caused by sideband at 3456Hz explain the appearanceof the third peak A typical phenomenon is observed that thetorsional vibration reduces a lot while one compares theamplitude of the third peak with another two which

10 Shock and Vibration

Table 1 e equivalent parameters of powertrain

Equivalent mass (kg) Equivalent moment of inertia (kgmiddotm2) Equivalent stiffness (Nm)m1 2 J1 0091125 k1 1002375m2 10 J2 03801735 k2 164025m3 06150 J3 000008103 k3 27500m4 13798 J4 0001001 k4 10e11m5 23581 J5 0001323 k5 479000m6 114323 J6 000249 k6 10e11m7 45323 J7 0001865 k7 21400m8 75567 J8 0002098 mdashm9 1284 J9 0001695 k8 43921799m10 1284 J10 0001695 mdashm11 13125 J11 000173 k9 41933635m12 13125 J12 000173 mdashm13 04254 J13 000002772 k10 10e11m14 17016 J14 00001109 km 96965e8m15 1375 J15 0034177 mdashm16 845 J16 99 k11 12032114m17 845 J17 99 k12 12032114

Table 2 e system parameters of the main reducer

Parameter Pinion GearNumber of teeth 10 41Module (mm) 400Mean pressure angle (deg) 20 20Helix angle at midpoint of the pinion (deg) 50 2713Cone angle (deg) 2312 6514Tooth width (mm) 4066 3452Tooth addendum (mm) 359 644Pinion outset (mm) minus38Mounting distance (mm) 112 64Pitch radius (mm) 40 108Backlash (mm) 01536Distance from pinion reference point to the x-o-z plane for pinion (mm) 40Distance from pinion reference point to the y-o-z plane for gear (mm) 101Distance between projection point and centroid of pinion shaft (mm) lp3 10 lg3 5

Distance between bearing reference point and centroid of pinion shaft (mm) lp1 100 lg1 90lp2 175 lg2 93

Equivalent mass (kg) mg 2127 mp 1375Jpx J14 Jgx 0050315

Equivalent moment of inertia (kgmiddotm2) Jpy 0002982 Jgx J15Jpz 0002982 Jpz 0037551

Meshing stiffnessof (i + 1)th tooth pair

Meshing stiffnessof ith tooth pair

t

k

Meshing stiffnessof (i ndash 1)th tooth pair

Total meshingstiffness km(t)

Figure 5 Schematic diagram of time-varying meshing stiffness on hypoid gear

Shock and Vibration 11

indicates that torsional behaviors of automotive powertrainaffect the vibration characteristics of the complete vehicleunder a low- or middle-input-speed condition In order tovalidate the preceding result an experimental study isnecessary

4 Experimental Analysis

A test bench which can be used to test the vibration behavior ofa transmission system used for four-wheel drive vehicle be-longing to the China Automotive Technology and ResearchCenter is adopted to perform this experiment ere are fivemotors in the test bench and three of them are used in thisexperiment One of them drives automotive powertrain whichcan generate a high frequency input torque with fluctuationvaried in accordance with necessary and others are used asloads In this test two photoelectric sensors and two magne-toelectric sensors are used to detect torsional vibration signalsand other five accelerometers are mounted to monitor trans-verse vibration Installation site for each sensor can be found inFigure 9 Signals are sampled and processed by LMS TestLab

In order to validate the feasibility of the theoreticalmodel the input torque simulated by input motor and loadsare consistent with parameters employed in numericalanalysis (mean value and the second-order term are con-tained only and fluctuation amplitude is set at 100NmiddotM forinput torque) Within the experiment the output speed ofmotor varies from 1000 rpm to 3500 rpm to conduct a sweepfrequency response analysis experimentally

As illustrated in Figure 10 the amplitude-frequencyresponses of torsional vibration have the similar feature for

different parts of powertrain Peaks can be observed forcarves at a similar input speed It is inferred that a presumeresonance occurs at each peak As a result the first three-order nature torsional frequencies can be predictedMeanwhile the same results are reached As shown inFigure 10 each curve has a higher value nearby the first andsecond peak With the further increase in frequency itreaches a lower plateau value Torsional behaviors have aneffect on the vibration characteristics under a low-input-speed condition Furthermore the part located behind themesh point such as output end of half-shaft has a lowerRMS value than transmission and drive shaft which can beattributed to reduction effect of mesh excitation on torsionalvibration

e experimental result of input end of rear axle iscompared with the numerical one As shown in Figure 11 aquite consistency is observed and the first three-order fre-quencies are inferred at 35Hz (1050 rpm) 557Hz(1670 rpm) and 883Hz (2650 rpm) for experimental resultby the same way mentioned in preceding section Comparedwith numerical predictions an acceptable difference ofabout 5 (the difference is 51 at 333Hz 45 at 533Hzand 536 at 933Hz) can be obtained ese indicate thatthe theoretical model is feasible for predicting nature tor-sional frequencies Moreover numerical torsional dis-placement is a little lower than the experimental one at thesame input speed which is below about 2450 rpm and a littlehigher with the further increase is can be presumablyattributed to assumptions made in this present work withrespect to the complexity of the system and the limitations ofthe lumped-mass method

03503

02502

01501

0050

0

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(a)

13Hz

50Hz

87Hz 150Hz185Hz222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(b)

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(c)

13Hz

50Hz

87Hz150Hz

185Hz222Hz

004

003

002

001

00 50 100 150 200 250

Frequency (Hz)To

rsio

nal d

ispla

cem

ent (

deg)300 350 400 450 500 550 600

(d)

Figure 6 e Fourier spectrum of torsional responses (a) output shaft of transmission (b) output end of the drive shaft system (c)torsional displacement of the pinion along the axis x (d) torsional displacement of the gear along the axis y

12 Shock and Vibration

e previous section indicates numerically that excita-tion of input dominates torsion vibration of the system andgear set affects responses slightly In order to validate thecorrectness of numerical analysis and feasibility decreasing

torsional vibration by reducing input excitation effect ofinput torque fluctuation on torsional response is studiedexperimentally e mean value of input torque is set at150NmiddotM and fluctuation of the second-order term is set at

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

2800rpm1600rpm

Figure 7 Amplitude-frequency response at input end of rear axle

8

7

6

5

4

2

3

1

00 1000 2000 3000

Rotation speed (RPM)

Vibr

atio

n am

plitu

de (micro

m)

4000 5000 6000

x directiony directionz direction

Figure 8 Influence of input rotation speed of drive shaft on vibration of the main reducer [29]

Table 3 e first six-order nature frequency of drive shaft [29]

Order 1 2 3 4 5 6Frequency (Hz) 1309 1402 4152 4422 6362 9066

Shock and Vibration 13

25NmiddotM 50NmiddotM 75NmiddotM 100NmiddotM 125NmiddotM 150NmiddotM175NmiddotM and 200NmiddotM in order

Together Figures 12 and 13 expectably show that thetorsional vibration is reinforced while increasing torquefluctuation Meanwhile an obvious increase in amplitudeoccurs at the second peak when fluctuation is increased from175NmiddotM to 200NmiddotM which indicates that a limitation should

be considered e accurate value can be determined by afurther study In summary reducing torque fluctuation is abeneficial approach for vibration damping Furthermore it isnoted that such a nonlinear phenomenon as natural fre-quency shift occurs while changing torque fluctuation Takingthe second-order torsional frequency as an example thefrequency value decreases with elevating fluctuation of input

Transmission

Photoelectricsensor

Photoelectricsensor

Magnetoelectricsensor

Magnetoelectricsensor

Rear axle

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Drive sha system

Figure 9 e experiment bench and arrangement of sensors

042

040

038

035

033

030

028

025

023

020

018

015

013

010

008

005

003

Amplitude-frequency curve ofle wheel

Amplitude-frequency curve oftransmission output sha

Amplitude-frequency curve ofrear axle output end

Amplitude-frequency curve ofright wheel

000

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (K

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Curve 167000

041

035

003

003

5567 8817

167000 264500

rpm

deg

deg

deg

deg

264500

012

005

003

764endash3

Figure 10 Amplitude-frequency responses of torsional vibration for powertrain at a torque fluctuation of 100NmiddotM

14 Shock and Vibration

036

034

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

Numerical result

Experimental result

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

Figure 11 Comparison of numerical and experimental results for input end of rear axle

160 5567 8817

264500

150

140

130

120

110

100

090

080

070

060

050

040

030

020

010

000 167000

25Nmiddotm

50Nmiddotm

75Nmiddotm

100Nmiddotm

125Nmiddotm150Nmiddotm

175Nmiddotm

200Nmiddotm

Curve 167000147078057045035025017013

rpm--------

264500009008007006005004004003

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (R

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Figure 12 Amplitude-frequency responses of torsional vibration for input end of rear axle at different torque fluctuations

Shock and Vibration 15

torque which shows that the torsional system has a stiffness-softening characteristic with respect to torque fluctuationis result can be accounted for a number of nonlinearitiessuch as friction clearance and sliding and the existence ofwhich sophisticates system nonlinear characteristics

5 Conclusion

In the present paper a lumped-parameter dynamic modelwith 29 degree-of-freedom is proposed considering thetransverse torsional and rocking coupled vibration for abetter understanding of the automotive powertrain systemdynamics and equations of motion are derived e nu-merical result shows that gear mesh excitations have littleeffects on torsional response for such components locatedbefore the mesh point as transmission transmission shaftand pinion but significantly reduce the torsional responsesfor ones behind the mesh point e amplitude-frequencycharacteristics of torsional vibration graphically indicate thattorsional behaviors of automotive powertrain only affect thevibration characteristics of complete vehicle under low-speed condition and predict the first three-order naturetorsional frequencies of the whole system An experimentalexamination is performed and validates the feasibility of theconsidered model e torsional vibration is reinforcedwhile increasing torque fluctuation In addition a stiffnesssoftening characteristic is observed for torsional vibrationexperimentally In order to reach a better understanding inamplitude-frequency response at low- and middle-input-speed condition a procedure to analyze mode shapes ofautomotive powertrain will be conducted in future works

Data Availability

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

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported and funded by the ApplicationTechnology Research and Development Plan of LiuzhouCity (grant no 2017AA10102) and the Key Projects ofUniversities in Inner Mongolia Autonomous Region (grantno NJZZ16369)e authors of this paper express their deepgratitude for the financial support of the research project

References

[1] F Vesali M A Rezvani and M Kashfi ldquoDynamics of uni-versal joints its failures and some propositions for practicallyimproving its performance and life expectancyrdquo Journal ofMechanical Science and Technology vol 26 no 8 pp 2439ndash2449 2012

[2] J-W Lu G-C Wang H Chen A F Vakakis andL A Bergman ldquoDynamic analysis of cross shaft type universaljoint with clearancerdquo Journal of Mechanical Science andTechnology vol 27 no 11 pp 3201ndash3205 2013

[3] G Wu W Shi and Z Chen ldquoe effect of multi-universalcoupling phase on torsional vibration of drive shaft and vi-bration of vehiclerdquo in Proceedings of the SAE 2013 WorldCongress amp Exhibition Detroit MI USA April 2013

0113333 4000 4500 5000 5500

5567

6000 6500 7000Frequency (Hz)

7500 8000 85008777

263300

9000 9500 10000 10500 11000 11667100

Am

plitu

de

Am

plitu

de (R

MS)

010

009

008

007

006

005

004

003

002

001

000100000 120000 140000 160000 180000

167000

25Nmiddotm

50Nmiddotm

75Nmiddotm100Nmiddotm

125Nmiddotm

150Nmiddotm

175Nmiddotm200Nmiddotm

Curve 167000010006004003003002001

982endash3

263300006005004003003002001

690endash3

rpm--------

200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000000

Figure 13 Amplitude-frequency responses of torsional vibration for output end of rear axle at different torque fluctuations

16 Shock and Vibration

[4] B A Porter ldquoeoretical analysis of the torsional oscillationof a system incorporating a Hookersquos jointrdquo Journal of Me-chanical Engineering Science vol 3 no 4 pp 324ndash329 1961

[5] S F Asokanthan and X-H Wang ldquoCharacterization oftorsional instabilities in a Hookersquos joint driven system viamaximal Lyapunov exponentsrdquo Journal of Sound and Vi-bration vol 194 no 1 pp 83ndash91 1996

[6] A Farshidianfar M Ebrahimi and H Bartlett ldquoHybridmodelling and simulation of the torsional vibration of vehicledriveline systemsrdquo Proceedings of the Institution of Me-chanical Engineers Part D Journal of Automobile Engineeringvol 215 no 2 pp 217ndash229 2001

[7] E O A Abd Elmaksoud E M Rabeih N A Abdel-halimand S M El Demerdash ldquoInvestigation of self excited tor-sional vibrations of different configurations of automatictransmission systems during engagementrdquo Engineeringvol 3 no 12 pp 1171ndash1181 2011

[8] T B Juang M Cheng and L Na ldquoExperimental and finiteelement analyses of a sliding-tube-type driveshaft-inducedvehicle vibrationrdquo Proceedings of the Institution of MechanicalEngineers Part K Journal of Multi-Body Dynamics vol 221no 3 pp 375ndash385 2007

[9] L F Coutinho and A Tamagna ldquoFlexural vibration analysisof constant velocity half-shaftsrdquo in Proceedings of the Inter-national Congress amp Exposition Chicago IL ISA 1996

[10] Y T Kang Y H Shen W Zhang and J Yang ldquoStabilityregion of floating intermediate support in a shaft system withmultiple universal jointsrdquo Journal of Mechanical Science andTechnology vol 28 no 6 pp 2733ndash2742 2014

[11] J L Xu T Yan B Peng and B Wei ldquoEffects of the trans-mission shaft on the main reducer vibration based onADAMS and experimental demonstrationrdquo AustralianJournal of Mechanical Engineering vol 15 no 1 pp 2ndash102015

[12] C X Zhang H B Li and D Q Jin ldquoSimulation on dynamicsof transmission shaft-rear driving axle of minibus withADAMSrdquo Applied Mechanics and Materials vol 651ndash653pp 862ndash865 2014

[13] J L Xu X Y Su and P Bo ldquoNumerical analysis anddemonstration transmission shaft influence on meshing vi-bration in driving and driven gearsrdquo Shock and Vibrationvol 2015 Article ID 365084 10 pages 2015

[14] J L Xu W Lei andW X Luo ldquoInfluence of bearing stiffnesson the nonlinear dynamics of a shaft-final drive systemrdquoShock and Vibration vol 2016 Article ID 3524609 14 pages2016

[15] A Kahraman and R Singh ldquoNon-linear dynamics of a spurgear pairrdquo Journal of Sound and Vibration vol 142 no 1pp 49ndash75 1990

[16] A Kahraman and R Singh ldquoInteractions between time-varying mesh stiffness and clearance non-linearities in ageared systemrdquo Journal of Sound and Vibration vol 146no 1 pp 135ndash156 1991

[17] A Kahraman ldquoEffect of axial vibrations on the dynamics of ahelical gear pairrdquo Journal of Vibration and Acoustics vol 115no 1 pp 33ndash39 1993

[18] Y Cai and T Hayashi ldquoe linear approximated equation ofvibration of a pair of spur gears (theory and experiment)rdquoJournal of Mechanical Design vol 116 no 2 pp 558ndash5641994

[19] P Velex andMMaatar ldquoAmathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound and Vibration vol 191no 5 pp 629ndash660 1996

[20] P Velex and V Cahouet ldquoExperimental and numerical in-vestigations on the influence of tooth friction in spur andhelical gear dynamicsrdquo Journal of Mechanical Design vol 122no 4 pp 515ndash522 2000

[21] P Velex and P Sainsot ldquoAn analytical study of tooth frictionexcitations in errorless spur and helical gearsrdquo Mechanismand Machine Ieory vol 37 no 7 pp 641ndash658 2002

[22] T C Lim and Y Cheng ldquoA theoretical study of the effect ofpinion offset on the dynamics of hypoid geared rotor systemrdquoJournal of Mechanical Design vol 121 no 4 pp 594ndash6011999

[23] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[24] Y Cheng and T C Lim ldquoDynamics of hypoid gear trans-mission with nonlinear time-varying mesh characteristicsrdquoJournal of Mechanical Design vol 125 no 2 pp 373ndash3822003

[25] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 2017 pp 302ndash329 2007

[26] J Wang and T C Lim ldquoEffect of tooth mesh stiffnessasymmetric nonlinearity for drive and coast sides on hypoidgear dynamicsrdquo Journal of Sound and Vibration vol 319no 3-5 pp 885ndash903 2009

[27] W M Li and H T Wang ldquoRigidity calculation of axialposition preload bearingsrdquo Journal of Hebei University ofTechnology vol 30 no 2 pp 15ndash19 2001

[28] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo Non-linear Dynamics vol 75 no 4 pp 783ndash806 2014

[29] X Liu Z Wu J Lu and J Xu ldquoInvestigation of the effect ofrotation speed on vibration responses of transmission sys-temrdquo Matec Web of Conferences vol 256 no 3 Article ID02019 2019

Shock and Vibration 17

TD Tm + 1113944N

i1Tdi cos iωft + φi1113872 1113873 (2)

where Tm is mean output torque Tdi is the amplitude of ithharmonic term ωf is output speed of engine and φi is phaseangel of ith harmonic term

Based on previous research [14] only mean componentand the second-order harmonic term are considered here

TD Tm + Td2 cos 2ωft + φ21113872 1113873 (3)

22 Transmission e transmission is simplified as achained system with six equivalent moments of inertia andi1 is introduced to describe the drive ratio of transmissionFurthermore gears are assumed as rigid and gear meshcharacteristics frictions etc are neglected Without loss ofgenerality the case of i1 1350 is employed as an example inthe present work

In order to simplify a multimesh gear train into achained system a shaft should be chosen as the main bodyfirstly as illustrated in Figure 3 in which Ji (i 1ndash4) andJiprime(i 3 minus 4) are equivalent moments of inertia and Ki(i 1ndash4) and Ki

prime(i 3 minus 4) denote equivalent stiffness of gearshaft

Kinetic energy of output shaft for the original system andchained system can be calculated as follows

E J3ω2

2 + J4ω22

2

Eprime J3primeω2

1 + J4primeω21

2

(4)

where E is the kinetic energy of output shaft for the originalsystem J3 and J4 are the equivalent moments of inertia forthe original system which are derived by dynamic balancetheory ω1 and ω2 are the angular velocities Eprime is the kineticenergy of output shaft for the original system and J3prime and J4primeare equivalent moments of inertia for the chained system

From the conservation law of energy the followingrelationships are obtained

E Eprime

J3prime J3

i2

J4prime J4

i2

(5)

where i is drive ratio

(a)

(b) (c)

Figure 2 e dynamic model of the powertrain shown in Figure 1 (a) dynamic model of powertrain without rear axle (b) torsional modelof rear axle (c) dynamic model of the main reducer in rear axle

Shock and Vibration 3

From the conservation of strain energy an equation isderived

12K3primeφ

21

12K3φ

22 (6)

where K3prime andK3 are torsional stiffness of output shaft for thechained system and original system respectively andφ1 andφ2 are torsional angels of driving and driven shaft

Owing to the assumption concerning a gear as rigid thegear meshing stiffness K2prime and K2 is treated as infinitequantities e following equations are derived

K3prime K3

i2 i

φ1φ2

T2prime T2

i1113896 (7)

where T2prime is the input torque of output shaft and the T2 is theoutput torque of input shaft

According to preceding derivation the equivalent mo-ments of inertia for the chained system are calculated by

J3 Ia1 +Ia2

2

J4 Ia2

2+ Ig1

J5 Ib1 + Ig2 + Ib221113872 1113873

i22 1

J6 Ib22( 1113857 + Ig3 + Ib3 + Ig4 + Ig11i211 41113872 1113873 + Ib4 + Ig5 + middot middot middot + Ig12i212 51113872 1113873 + Ib5 + Id + Ib6 + Ig8 + Ig9 + Ig15i215 91113872 1113873i29 81113872 1113873 + Ib71113960 1113961

i22 1

J7 Ic1 + Id + Ig10 + Ic2 + Ic3 + Ic421113872 1113873 + Id1113872 1113873

i210 3 middot i22 1( 1113857

J8 Ic42 + Ig13 + Ig6i26 13 + Ic5 + Ig14 + Ig7i27 14 + Ic6 + Ic7 + Id1113960 1113961

i210 3 middot i22 1( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

J1J2

J1J2

J3

J3primeJ4prime

J4

K1

K2

K3

K1 K2prime K3prime

First gearshaft

Secondgearshaft

Gear 1Gear 2

Gear 3Gear 4

First gearshaft

Secondgearshaft

Gear 1

Gear 2Gear 3

Gear 4

Figure 3 e schematic diagram of equivalent principle for simplifying a multimesh gear train into a chained system

4 Shock and Vibration

where Iaj(j 1 2) is the equivalent moment of inertia of thejth node of input shaft Ibj(j 1 2 7) is the equivalentmoment of inertia of the jth node of middle shaft Icj(j

1 2 7) is the equivalent moment of inertia of the jthnode of output shaft Igj(j 1 2 15) is the equivalentmoment of inertia of jth gear Id is the equivalent moment ofinertia of the synchronizer ring and im n is the drive ratio ofthe mating gear pair

23 Drive Shaft System Drive shaft system comprises in-termediate drive shaft main drive shaft three universaljoints and intermediate support In this paper the rotationalinertia of shaft is equivalent to the universal joint by thelumped-mass method averagely e lubrication clearancefriction manufacturing and assembly errors and flexibilityof universal joint are not considered

In order to describe the dynamics of a U-joint a set ofparameters are introduced as follows

Ai 1 + cos2αi

2 cos αi

Bi 1 minus cos2αi

2 cos αi

i 1 2 3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

where αi(i 1 2 3) is the intersection angel between shaftslinked by a universal joint (U-joint)

According to kinematics analysis the following equa-tions are derived

θ9 arctantan θ8cos α1

θ9prime θ8prime

A1 minus B1 cos 2θ8

θ9Prime θ8Prime

A1 minus B1 cos 2θ8minus

2B1 sin 2θ8A1 minus B1 cos 2θ8( 1113857

2 θ8prime( 11138572

T2 A1 minus B1 cos 2θ8( 1113857T1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

θ11 arctantan θ10cos α2

θ11prime θ10prime

A2 minus B2 cos 2θ10

θ11Prime θ10Prime

A2 minus B2 cos 2θ10minus

2B2 sin 2θ10A2 minus B2 cos 2θ10( 1113857

2 θ10prime( 11138572

T4 A2 minus B2 cos 2θ10( 1113857T3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

θ13 arctantan θ12cos α3

θ13prime θ12prime

A3 minus B3 cos 2θ12

θ13Prime θ12Prime

A3 minus B3 cos 2θ12minus

2B3 sin 2θ12A3 minus B3 cos 2θ12( 1113857

2 θ12prime( 11138572

T6 A3 minus B3 cos 2θ12( 1113857T5

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

Shock and Vibration 5

where α1 is the intersection angel between output shaft oftransmission and intermediate drive shaft α2 is the inter-section angel between main drive shaft and intermediatedrive shaft α3 is the intersection angel between main driveshaft and input shaft of rear axle

e equivalent moments of inertia for intermediate driveshaft and main drive shaft are calculated by

J9 J10 12i21

Jids

J11 J12 12i21

Jmds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(11)

where i1 is the drive ratio of transmission Jids is the momentof inertia of intermediate drive shaft and Jmds is the momentof inertia of main shaft

Equivalent torsional stiffness k and damping coefficient c

of shaft are given by

k M

ϕ

Gapπ D4 minus d4( 1113857

32L

c 2ζ

kJaJb

Ja + Jb

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(12)

where ζ is the damping ratio and Ja and Jb are the equivalentmoment of inertia of shaft end

24 Rear Axle In this paper the main reducer system issimplified as a geared rotor-bearing system with bending-torsional-lateral-rocking coupled vibration Pinion shaft issimplified as two lumpedmasses linked by a spring-dampingpair with infinite stiffness and the differential assembly ismodeled as a rigid part with a concentratedmass Half-shaftsare described by twomass-spring-damping pairs Otherwisepinion and gear are treated as rigid bodies within torsionalvibration analysis

e equivalent moments of inertia for rear axle arecalculated by

J13 15Jps

i21

J14 45Jps

i21

J15 Jd + 05Jhs

i21i22

J16 J17 05Jhs + Jhs

i21i22

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)

where i1 (i1 1350) is the drive ratio of transmission i2(i2 41) is the drive ratio of the main reducer Jps is themoment of inertia of pinion shaft Jd is the moment of inertia

of differential assembly and Jhs is the moment of inertia ofhalf-shaft

Equivalent stiffness of bearing is expressed as [27]

Kr 7253l08

1 z091 cos20α1 middot F01

a0cos01α1

Ka 29011l081 cos19α1 middot F01

a0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(14)

where Kr is the equivalent radial stiffness of bearing Ka isthe equivalent axial stiffness of bearing l1 is the nominalcontact length z1 is the number of rolling elementα1 is thepressure angle and Fa0 is the pretightening force

In addition a new model for pinion shaft is proposedwhich is different from previous works Owing to negligibleelastic deformation pinion shaft is modeled as rigid bodyBearing is modeled as a spring-damping pair Without lossof generality the derivation procedure on displacements ofthe mounting point in the x-o-y plane is used as an example

In Figure 4 Xpx and Xpy are introduced to representtranslational displacements for centroid of pinion shaft θpz

denotes rocking displacement around the z axisFrom the geometry deformation relationship and pro-

posed concepts the displacement for mounting point iscalculated by

xpy1 xpy minus lp1 tan θpz

xpy2 xpy minus lp2 tan θpz

⎧⎨

⎩ (15)

ereby the following expressions are derived

xpy1prime xpyprime minus lp1

θpzprime

cos2θpz

xpy2prime xpyprime minus lp2

θpzprime

cos2θpz

1113896

(16)

Static transmission error e(t) [22ndash26 28] is simulated inFourier series form by

e(t) em + 1113944N

i1eAi cos iωmt + φe( 1113857 (17)

where em is the average value of transmission error eAi arethe amplitude of the ith order harmonic φe are the initialphase angle of the ith order harmonic and ωm is the meshingfrequency

e dynamic meshing stiffness is the periodically time-varying parameter with respect to the mesh frequency asshown in Figure 5 which can be simulated by means of aFourier expansion [13 14 22ndash26]

km(t) km + 1113944N

i1kAi cos iωmt + φk( 1113857 (18)

where km is the mean value of meshing stiffness kAi is thefluctuation amplitude of ith order meshing stiffness ωm isthe mesh frequency and φk is the initial phase of meshingstiffness

Defining me denotes mean mass of pinion and gear

6 Shock and Vibration

me 1

1mp1113872 1113873 + 1mg1113872 1113873

mpmg

mp + mg

(19)

e following equation can be derived

ωn

km

me

1113971

(20)

where ωn is the equivalent natural frequency of pinion andgear

Equivalent mesh damping coefficient is calculated whichis different from Wang and Limrsquos analysis [25]

cm ξ lowast 2meωn 2ξ

km

1mp1113872 1113873 + 1mg1113872 1113873

1113971

(21)

where ζ is the damping ratio generally set as a range of 003to 017 [25]mp andmg are equivalent mass of pinion and gearrespectively

e relative displacement along the normal direction atthe meshing point xn is given as follows

xn xpx + Eyθpz1113872 1113873 minus xgx + Ezθgy + lg3θgz1113872 11138731113960 1113961

middot cos αn sin βm cos δ1 + sin αn sin δ1( 1113857

+ xpy + lp3θpz1113872 1113873 minus xgy + Ezθgx minus Exθgz1113872 11138731113960 1113961

middot minuscos αn sin βm sin δ1 + sin αn cos δ1( 1113857

+ xpz minus Rpθpx + lp3θpy1113872 1113873 minus xgz minus lg3θgx minus Rgθgy1113872 11138731113960 1113961

middot cos αn cos βm( 1113857 minus e(t)

ε1 xpx minus xgx minus Ezθgy + Eyθpz minus lg3θgz1113872 1113873

+ ε2 xpy minus xgy minus Ezθgx + lp3θpz + Exθgz1113872 1113873

+ ε3 xpz minus xgz minus Rpθpx + lg3θgx + lp3θpy + Rgθgy1113872 1113873

minus e(t)

(22)

e backlash function is expressed as [23]

f xn( 1113857

xn minus b xn ge b

0 minusblexn le b

xn + b xn le minus b

⎧⎪⎪⎨

⎪⎪⎩(23)

where b is half of gear backlashe dynamic meshing force along the line of action Fn

can be calculated byFn km(t)f xn( 1113857 + cmxn

prime (24)

wherecm represents thevalueof equivalentmeshdampingcoefficiente components of dynamic mesh force along the co-

ordinate directions are expressed asFx Fn cos αn sin βm cos δ1 + Fn sin αn sin δ1

Fy minusFn cos αn sin βm sin δ1 + Fn sin αn cos δ1

Fz Fn cos αn cos βm

⎧⎪⎪⎨

⎪⎪⎩(25)

where δ1 is cone angle of the pinion αn is the normal pressureangle and βm is the helix angle at the midpoint of the pinion

In additional the following equations are derived byintroducing the parameter εi

ε1 cos αn sin βm cos δ1 + sin αn sin δ1

ε2 minuscos αn sin βm sin δ1 + sin αn cos δ1

ε3 cos αn cos βm

⎧⎪⎪⎨

⎪⎪⎩(26)

Substituting equations (26) into (25) the expressions ofdynamicmesh force applied on pinion and gear can bewritten as

Fpx minusFgx ε1Fn

Fpy minusFgy ε2Fn

Fpz minusFgz ε3Fn

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(27)

where the subscript p is the label for pinion and the subscriptg is the label for gear

Mesh impact due to teeth separation is neglected in this paper

25 Equations of Motion From the preceding derivationsthe equations of motion considering bending-torsional-lateral-rocking vibration are expressed as

Fpz

kpz2 cpz2

kpx

cpx

θpy

Xpx

kpz1 cpz1

FpxFpy

Op1

cpy2

θpz

mp

kpy2cpy1

Lp1

Lp2 Lp3

Op2 Op0

kpy1y

x

z

Xpy1 Xpy2

Xpy

Op

Op3 Ey

θpz

θpx

Figure 4 e dynamic model of pinion shaft

Shock and Vibration 7

J1θ1Prime + c1 θ1prime minus θ2prime( 1113857 + k1 θ1 minus θ2( 1113857 TD

J2θ2Prime + c1 θ2prime minus θ1prime( 1113857 + k1 θ2 minus θ1( 1113857 + c2 θ2prime minus θ3prime( 1113857 + k2 θ2 minus θ3( 1113857 0

J3θ3Prime + c2 θ3prime minus θ2prime( 1113857 + k2 θ3 minus θ2( 1113857 + c3 θ3prime minus θ4prime( 1113857 + k3 θ3 minus θ4( 1113857 0

J4θ4Prime + c3 θ4prime minus θ3prime( 1113857 + k3 θ4 minus θ3( 1113857 + c4 θ4prime minus θ5prime( 1113857 + k4 θ4 minus θ5( 1113857 0

J5θ5Prime + c4 θ5prime minus θ4prime( 1113857 + k4 θ5 minus θ4( 1113857 + c5 θ5prime minus θ6prime( 1113857 + k5 θ5 minus θ6( 1113857 0

J6θ6Prime + c5 θ6prime minus θ5prime( 1113857 + k5 θ6 minus θ5( 1113857 + c6 θ6prime minus θ7prime( 1113857 + k6 θ6 minus θ7( 1113857 0

J7θ7Prime + c6 θ7prime minus θ6prime( 1113857 + k6 θ7 minus θ6( 1113857 + c7 θ7prime minus θ8prime( 1113857 + k7 θ7 minus θ8( 1113857 0

J8θ8Prime + c7 θ8prime minus θ7prime( 1113857 + k7 θ8 minus θ7( 1113857 minusT1

i1

J9θ9Prime + c8 θ9prime minus θ10prime( 1113857 + k8 θ9 minus θ10( 1113857 T2

i1

J10θ10Prime + c8 θ10prime minus θ9prime( 1113857 + k8 θ10 minus θ9( 1113857 minusT3

i1

J11θ11Prime + c9 θ11prime minus θ12prime( 1113857 + k9 θ11 minus θ12( 1113857 T4

i1

J12θ12Prime + c9 θ12prime minus θ11prime( 1113857 + k9 θ12 minus θ11( 1113857 minusT5

i1

J13θ13Prime + c10 θ13prime minus θ14prime( 1113857 + k10 θ13 minus θ14( 1113857 T6

i1

J14θ14Prime + c10 θ14prime minus θ13prime( 1113857 + k10 θ14 minus θ131113857( 1113857 minusFpzEy

i1

J15θ15Prime + c11 θ15prime minus θ16prime( 1113857 + k11 θ15 minus θ16( 1113857 + c12 θ15prime minus θ17prime( 1113857 + k12 θ15 minus θ17( 1113857 FgzEx + FgxEz

i1i2

J16θ16Prime + c11 θ16prime minus θ15prime( 1113857 + k11 θ16 minus θ15( 1113857 minusTRL

i1i2

J17θ17Prime + c12 θ17prime minus θ15prime( 1113857 + k12 θ17 minus θ15( 1113857 minusTRR

i1i2

mmymPrime + cmyym

prime + kmyym Fmy

mmzmPrime + cmzzm

prime + kmzzm Fmz

mpxpxPrime + kpxxpx + cpxxpx

prime minusFpx

mpxpyPrime + kpy1 xpy minus lp1 tan θpz1113872 1113873 + cpy1 xpy

prime minus lp1θpzprime

cos2θpz

1113888 1113889 + kpy2 xpy minus lp2 tan θpz1113872 1113873 + cpy2 xpyprime minus lp2

θpzprime

cos2θpz

1113888 1113889 Fpy

mpxpzPrime + kpz1 xpz minus lp1 tan θpy1113872 1113873 + cpz1 xpz

prime minus lp1θpyprime

cos2θpy

1113888 1113889 + kpz2 xpz minus lp2 tan θpy1113872 1113873 + cpz2 xpzprime minus lp2

θpyprime

cos2θpy

1113888 1113889 Fpz

JpyθpyPrime minus lp1kpz1 xpz minus lp1 tan θpy1113872 1113873 minus lp1cpz1 xpz

prime minus lp1θpyprime

cos2θpy

1113888 1113889 minus lp2kpz2 xpz minus lp2 tan θpy1113872 1113873 minus lp2cpz2 xpzprime minus lp2

θpyprime

cos2θpy

1113888 1113889 Fpzlp3

JpzθpzPrime minus lp1kpz1 xpy minus lp1 tan θpy1113872 1113873 minus lp1cp1 xpy

prime minus lp1θpzprime

cos2θpz

1113888 1113889 minus lp2kpy2 xpy minus lp2 tan θpz1113872 1113873 minus lp2cpy2 xpyprime minus lp2

θpzprime

cos2θpz

1113888 1113889 minusFpxEy + Fpylp3

mgxgxPrime + kgx1 xgx minus lg1 tan θgz1113872 1113873 + cgx1 xgx

Prime minus lg1θgzprime

cos2θgz

1113888 1113889 + kgx2 xgx + lg2 tan θgz1113872 1113873 + cgx2 xgxprime + lg2

θgzprime

cos2θgz

1113888 1113889 Fgx

mgxgyPrime + kgyxgy + cgyxgy

Prime minusFgy

mgxgzPrime + kgz1 xgz + lg1 tan θgx1113872 1113873 + cgz1 xgz

prime + lg1θgxprime

cos2θgx

1113888 1113889 + kgz2 xgz minus lg2 tan θgx1113872 1113873 + cgz2 xgzprime minus lg2

θgxprime

cos2θgx

1113888 1113889 minusFgz

JgxθgxPrime + lg1kgz1 xgz + lg1 tan θgx1113872 1113873 + lg1cgz1 xgz

prime + lg1θgxprime

cos2θgx

1113888 1113889 minus lg2kgz2 xgz minus lg2 tan θgx1113872 1113873 + lg2cgz2 xgzprime minus lg2

θgxprime

cos2θgx

1113888 1113889 Fgzlg3 minus FgyEz

JgzθgzPrime + lg1kgx1 xgx + lg1 tan θgz1113872 1113873 + lg1cgx1 xgx

prime + lg1θgzprime

cos2θgz

1113888 1113889 minus lg2kgx2 xgx minus lg2 tan θgz1113872 1113873 + lg2cgx2 xgxprime minus lg2

θgzprime

cos2θgz

1113888 1113889 Fgxlg3 minus FgyEx

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

8 Shock and Vibration

where TD is the input torque of motor T1 is the outputtorque of transmission T2 andT3 are the input and outputtorque for intermediate drive shaft T4 andT5 are the inputand output torque for main drive shaft T6 is the inputtorque of rear axle ki(i 1 2 7) are the equivalenttorsional stiffness ci(i 1 2 7) are the equivalent tor-sional damping ki(i 8 9) are the equivalent torsionalstiffness ci(i 8 9) are the equivalent torsional dampingmm is the mass of transmission shaft ki(i 10 11 12) arethe equivalent torsional stiffness for pinion shaft and dif-ferential axle ci(i 10 11 12) are the equivalent torsionaldamping for pinion shaft and differential axle cmy and cmz

are the equivalent damping of intermediate support alongaxes y and z kmy and kmz are the equivalent stiffness ofintermediate support along axes y and z Fmy and Fmz arecomponents of additional force Fi along axes y and z forintermediate support Ei(i x y z) are coordinates ofmeshing point along axes x y and z and TRL andTRR areload torque for wheels

26 Numerical Solution Procedure e fourth-order adap-tive step RungendashKutta method which is generally applicableto strong nonlinearity is adopted in the present work to solve

the equations In order to program a new state space isneeded

In consideration of equations (12)mdash(14) it can be de-rived that kinetic equations of 3 universal joints reduce thenumber of degree-of-freedom from 29 to 26 for the wholesystem e following state space coordinate vector alsodescribes the dynamic behaviors of the system

Ss θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 θ10 θ12 θ14 θ15 θ16 θ17θpy θpz θgx θgz ym zm xp yp zp xg yg zg

⎧⎨

⎩

⎫⎬

⎭

Τ

(29)

In order to eliminate the difference in magnitude be-tween parameters which will reduce the solvability and evenlead to no convergence for solving equations of motionmentioned above b is defined as the characteristic lengthand a new time parameter τ is introduced as follows

τ tωn (30)

where t denotes timeFor the sake of nominalizing equations a generalized

state space coordinate vector with 26 parameters is proposedto characterize dimensionless equations as

st sr1 sr2 sr3 sr4 sr5 sr6 sr7 sr8 sr10 sr12 sr16 sr17 srpx srpy srpz srgx srgy srgz smy smz spx spy spz sgx sgy sgz1113966 1113967T

(31)

e following equations are derived as

sri θi timesRi

b i 1 sim 8 10 12 16 17

srpx θ14 timesRpx

b

srgy θ15 timesRgy

b

srij θij timesRij

b i p g j x y z

six xi

b i p g

siy yi

b i m p g

siz zi

b i m p g

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

where b is the half of backlash and Ri is the equivalent gyrationradius which can be calculated by the following equation

Ri

Ji

mi

1113971

i 1 sim 8 10 12 16 17

Rij

Jij

mi

1113971

i p g j x y z

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(33)

where Ji and Jij are the equivalent moment of inertia and mi

is equivalent massA set of dimensionless parameters are defined as

xnτ xn

b

f xnτ( 1113857 f xt

n( 1113857

b

c(ijs) ci

mjRjRsωn

k(ijs) ki

mjRjRsω2n

cmi cm

miωn

kmi km

miω2n

T(xj) Tx

mjRjω2nb

lijτ lij

b

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)

As equations (30) (32) and (34) are substituted intoequation (28) the dimensionless equations of motion can bederived

Shock and Vibration 9

e equivalent inertia parameters are obtained in UGand stiffness parameters of powertrain are calculated byusing the relationships given in modeling section as listed inTable 1 Subsequently equivalent damping coefficients canbe obtained e parameters of the main reducer can becalculated by substituting system parameters listed in Table 2into equations (17)ndash(31) e solution procedures for di-mensionless parameter are carried out by using precedingcalculated parameters and the relationships proposed inprevious parts

e exciting and configuration parameters are set asθ1 1500 rpm Td2 100Nmiddotm km 9697e8Nm i1 135i2 41 2b 015mm em 0 α1 1695 degree α2 13044degree and α3 1554 degree respectively Without signif-icant loss of accuracy the fundamental harmonic form isadopted for time-varying mesh stiffness and static trans-mission error function whose initial phases are set at 0respectively

3 Numerical Results

31 Analysis of Coupled Torsional Vibration In order toanalysis coupled interaction of automotive powertrainFourier spectrums of torsional vibration response are cal-culated at the initial parameters as shown in Figure 6 Twopeaks and their sidebands are observed e torsional vi-brational responses are governed by the component of 50Hzwhich equals with exciting frequency of input torque asshown in equation (3) (ωf 150060 lowast 2 50Hz) isimplies that engine input excitation is responsible for tor-sional vibration and a valuable approach to decreasingtorsional vibration of powertrain is inferred In addition thecomponent with a frequency of around 185Hz is observedobviously for each considered part As one knows themeshing frequency undergoing this condition is presumably185185Hz (f 150060135185185Hz) which is quite inagreement with the frequency of the second peak Coupledvibration occurs between the torsional system and hypoidgear set and inner excitation caused by gear pair meshingprocess which is affected by clearance and time-varyingmeshing stiffness [25] accounts a lot for torsional vibrationat 185Hz of powertrain and affects torsional responsesslightly After simply calculation bandwidths reaching thesame value of around 37Hz of sidebands around two peaksare got U-joint is a transmission mechanism with non-constant angular speed which will excite a doubled fre-quency torsional vibration of input as shown in equations(12)ndash(14) Under this condition the doubled frequencyexcited by U-joint is calculated

fuminusjoint n

i1 lowast 60lowast 2

1500135lowast 60

lowast 2 3704 (35)

It can be conjectured that modulation phenomenonappears when automotive powertrain works between exci-tations attributed to input U-joint and gear pair As one cansee in Figure 6 compared torsional response of gear withthat located front of meshing point a lower value of tor-sional angular is observed and it is inferred that gear meshexcitations reduce torsional vibration for components

located behind meshing point It is noted that taking gearmesh behaviors into account is quite necessary to when atorsional characteristic analysis is conducted

32 Amplitude-Frequency Response of Torsional Vibration forAutomotive Powertrain e previous works [29] analyzedtorsional characteristics of the coupled system consisted ofdrive shaft and rear axle which denoted that resonance ofdrive shaft at the first two-order nature frequency accountsfor dramatic torsional vibration responses undergoing high-rotation-speed condition However it cannot reveal thetorsional vibration mechanism of automotive under low-and middle-speed condition

In order to obtain a more realistic understanding foramplitude-frequency response of automotive powertrainundergoing middle-speed condition theoretically the pre-ceding equations of motion are adopted A sweep frequencyanalysis procedure is carried out in a range from 1000 rpm to3500 rpm which respectively corresponds to frequencyinternal of 166ndash583Hz for input speed 3333ndash1167Hz forinput excitation and 12346ndash43209Hz for meshing exci-tation e response at input end of rear axle is taken as anexample in this work

Figure 7 shows the amplitude-frequency response inwhich abscissa denotes torsion speed of input and ordinate isRMS value of angular displacement As one can see threeobvious peaks which correspond to input speed at about1000 rpm 1600 rpm and 2800 rpm respectively are ob-served Owing to the assumption that the dramatic responsesare attributed to resonance of the system the first three-order nature frequency respect to the second-order term ofinput torque fluctuation is proposed at 333Hz 533Hz and933Hz successively It is noted that on the basis of previousanalysis these three peaks also correspond to frequencies of247Hz 395Hz and 691Hz for the U-joint excitation and1234Hz 1975Hz and 3456Hz for the meshing excitationPreliminary works [29] showed that resonance is attributedto input excitation of drive shaft and the meshing excitationof hypoid gear pair with higher exciting frequency thaninput will not excite prominent resonance behaviors of driveshaft as illustrated in Figure 8 ie meshing excitation hasslight influence on the torsional system which reaches thesame conclusion with this work According to analysis inSection 31 the sidebands respect to each exciting frequencyare (86ndash58) Hz at 333Hz (138ndash928) Hz at 533Hz and(242ndash1624) Hz at 933Hz for input torque and (987ndash1481)Hz at 1234Hz (158ndash237) Hz at 1975Hz and (2765ndash4147)Hz at 3456Hz for meshing behavior of the gear pair re-spectively As listed in Table 3 the first peak can be attributedto the first two-order resonances of drive shaft excited bysideband around 1234Hz Sidebands at 533Hz and1975Hz result in the second-order resonance which gen-erates the second peak Moreover the second-order reso-nances excited by the sideband at 933Hz and third-orderone caused by sideband at 3456Hz explain the appearanceof the third peak A typical phenomenon is observed that thetorsional vibration reduces a lot while one compares theamplitude of the third peak with another two which

10 Shock and Vibration

Table 1 e equivalent parameters of powertrain

Equivalent mass (kg) Equivalent moment of inertia (kgmiddotm2) Equivalent stiffness (Nm)m1 2 J1 0091125 k1 1002375m2 10 J2 03801735 k2 164025m3 06150 J3 000008103 k3 27500m4 13798 J4 0001001 k4 10e11m5 23581 J5 0001323 k5 479000m6 114323 J6 000249 k6 10e11m7 45323 J7 0001865 k7 21400m8 75567 J8 0002098 mdashm9 1284 J9 0001695 k8 43921799m10 1284 J10 0001695 mdashm11 13125 J11 000173 k9 41933635m12 13125 J12 000173 mdashm13 04254 J13 000002772 k10 10e11m14 17016 J14 00001109 km 96965e8m15 1375 J15 0034177 mdashm16 845 J16 99 k11 12032114m17 845 J17 99 k12 12032114

Table 2 e system parameters of the main reducer

Parameter Pinion GearNumber of teeth 10 41Module (mm) 400Mean pressure angle (deg) 20 20Helix angle at midpoint of the pinion (deg) 50 2713Cone angle (deg) 2312 6514Tooth width (mm) 4066 3452Tooth addendum (mm) 359 644Pinion outset (mm) minus38Mounting distance (mm) 112 64Pitch radius (mm) 40 108Backlash (mm) 01536Distance from pinion reference point to the x-o-z plane for pinion (mm) 40Distance from pinion reference point to the y-o-z plane for gear (mm) 101Distance between projection point and centroid of pinion shaft (mm) lp3 10 lg3 5

Distance between bearing reference point and centroid of pinion shaft (mm) lp1 100 lg1 90lp2 175 lg2 93

Equivalent mass (kg) mg 2127 mp 1375Jpx J14 Jgx 0050315

Equivalent moment of inertia (kgmiddotm2) Jpy 0002982 Jgx J15Jpz 0002982 Jpz 0037551

Meshing stiffnessof (i + 1)th tooth pair

Meshing stiffnessof ith tooth pair

t

k

Meshing stiffnessof (i ndash 1)th tooth pair

Total meshingstiffness km(t)

Figure 5 Schematic diagram of time-varying meshing stiffness on hypoid gear

Shock and Vibration 11

indicates that torsional behaviors of automotive powertrainaffect the vibration characteristics of the complete vehicleunder a low- or middle-input-speed condition In order tovalidate the preceding result an experimental study isnecessary

4 Experimental Analysis

A test bench which can be used to test the vibration behavior ofa transmission system used for four-wheel drive vehicle be-longing to the China Automotive Technology and ResearchCenter is adopted to perform this experiment ere are fivemotors in the test bench and three of them are used in thisexperiment One of them drives automotive powertrain whichcan generate a high frequency input torque with fluctuationvaried in accordance with necessary and others are used asloads In this test two photoelectric sensors and two magne-toelectric sensors are used to detect torsional vibration signalsand other five accelerometers are mounted to monitor trans-verse vibration Installation site for each sensor can be found inFigure 9 Signals are sampled and processed by LMS TestLab

In order to validate the feasibility of the theoreticalmodel the input torque simulated by input motor and loadsare consistent with parameters employed in numericalanalysis (mean value and the second-order term are con-tained only and fluctuation amplitude is set at 100NmiddotM forinput torque) Within the experiment the output speed ofmotor varies from 1000 rpm to 3500 rpm to conduct a sweepfrequency response analysis experimentally

As illustrated in Figure 10 the amplitude-frequencyresponses of torsional vibration have the similar feature for

different parts of powertrain Peaks can be observed forcarves at a similar input speed It is inferred that a presumeresonance occurs at each peak As a result the first three-order nature torsional frequencies can be predictedMeanwhile the same results are reached As shown inFigure 10 each curve has a higher value nearby the first andsecond peak With the further increase in frequency itreaches a lower plateau value Torsional behaviors have aneffect on the vibration characteristics under a low-input-speed condition Furthermore the part located behind themesh point such as output end of half-shaft has a lowerRMS value than transmission and drive shaft which can beattributed to reduction effect of mesh excitation on torsionalvibration

e experimental result of input end of rear axle iscompared with the numerical one As shown in Figure 11 aquite consistency is observed and the first three-order fre-quencies are inferred at 35Hz (1050 rpm) 557Hz(1670 rpm) and 883Hz (2650 rpm) for experimental resultby the same way mentioned in preceding section Comparedwith numerical predictions an acceptable difference ofabout 5 (the difference is 51 at 333Hz 45 at 533Hzand 536 at 933Hz) can be obtained ese indicate thatthe theoretical model is feasible for predicting nature tor-sional frequencies Moreover numerical torsional dis-placement is a little lower than the experimental one at thesame input speed which is below about 2450 rpm and a littlehigher with the further increase is can be presumablyattributed to assumptions made in this present work withrespect to the complexity of the system and the limitations ofthe lumped-mass method

03503

02502

01501

0050

0

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(a)

13Hz

50Hz

87Hz 150Hz185Hz222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(b)

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(c)

13Hz

50Hz

87Hz150Hz

185Hz222Hz

004

003

002

001

00 50 100 150 200 250

Frequency (Hz)To

rsio

nal d

ispla

cem

ent (

deg)300 350 400 450 500 550 600

(d)

Figure 6 e Fourier spectrum of torsional responses (a) output shaft of transmission (b) output end of the drive shaft system (c)torsional displacement of the pinion along the axis x (d) torsional displacement of the gear along the axis y

12 Shock and Vibration

e previous section indicates numerically that excita-tion of input dominates torsion vibration of the system andgear set affects responses slightly In order to validate thecorrectness of numerical analysis and feasibility decreasing

torsional vibration by reducing input excitation effect ofinput torque fluctuation on torsional response is studiedexperimentally e mean value of input torque is set at150NmiddotM and fluctuation of the second-order term is set at

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

2800rpm1600rpm

Figure 7 Amplitude-frequency response at input end of rear axle

8

7

6

5

4

2

3

1

00 1000 2000 3000

Rotation speed (RPM)

Vibr

atio

n am

plitu

de (micro

m)

4000 5000 6000

x directiony directionz direction

Figure 8 Influence of input rotation speed of drive shaft on vibration of the main reducer [29]

Table 3 e first six-order nature frequency of drive shaft [29]

Order 1 2 3 4 5 6Frequency (Hz) 1309 1402 4152 4422 6362 9066

Shock and Vibration 13

25NmiddotM 50NmiddotM 75NmiddotM 100NmiddotM 125NmiddotM 150NmiddotM175NmiddotM and 200NmiddotM in order

Together Figures 12 and 13 expectably show that thetorsional vibration is reinforced while increasing torquefluctuation Meanwhile an obvious increase in amplitudeoccurs at the second peak when fluctuation is increased from175NmiddotM to 200NmiddotM which indicates that a limitation should

be considered e accurate value can be determined by afurther study In summary reducing torque fluctuation is abeneficial approach for vibration damping Furthermore it isnoted that such a nonlinear phenomenon as natural fre-quency shift occurs while changing torque fluctuation Takingthe second-order torsional frequency as an example thefrequency value decreases with elevating fluctuation of input

Transmission

Photoelectricsensor

Photoelectricsensor

Magnetoelectricsensor

Magnetoelectricsensor

Rear axle

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Drive sha system

Figure 9 e experiment bench and arrangement of sensors

042

040

038

035

033

030

028

025

023

020

018

015

013

010

008

005

003

Amplitude-frequency curve ofle wheel

Amplitude-frequency curve oftransmission output sha

Amplitude-frequency curve ofrear axle output end

Amplitude-frequency curve ofright wheel

000

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (K

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Curve 167000

041

035

003

003

5567 8817

167000 264500

rpm

deg

deg

deg

deg

264500

012

005

003

764endash3

Figure 10 Amplitude-frequency responses of torsional vibration for powertrain at a torque fluctuation of 100NmiddotM

14 Shock and Vibration

036

034

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

Numerical result

Experimental result

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

Figure 11 Comparison of numerical and experimental results for input end of rear axle

160 5567 8817

264500

150

140

130

120

110

100

090

080

070

060

050

040

030

020

010

000 167000

25Nmiddotm

50Nmiddotm

75Nmiddotm

100Nmiddotm

125Nmiddotm150Nmiddotm

175Nmiddotm

200Nmiddotm

Curve 167000147078057045035025017013

rpm--------

264500009008007006005004004003

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (R

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Figure 12 Amplitude-frequency responses of torsional vibration for input end of rear axle at different torque fluctuations

Shock and Vibration 15

torque which shows that the torsional system has a stiffness-softening characteristic with respect to torque fluctuationis result can be accounted for a number of nonlinearitiessuch as friction clearance and sliding and the existence ofwhich sophisticates system nonlinear characteristics

5 Conclusion

In the present paper a lumped-parameter dynamic modelwith 29 degree-of-freedom is proposed considering thetransverse torsional and rocking coupled vibration for abetter understanding of the automotive powertrain systemdynamics and equations of motion are derived e nu-merical result shows that gear mesh excitations have littleeffects on torsional response for such components locatedbefore the mesh point as transmission transmission shaftand pinion but significantly reduce the torsional responsesfor ones behind the mesh point e amplitude-frequencycharacteristics of torsional vibration graphically indicate thattorsional behaviors of automotive powertrain only affect thevibration characteristics of complete vehicle under low-speed condition and predict the first three-order naturetorsional frequencies of the whole system An experimentalexamination is performed and validates the feasibility of theconsidered model e torsional vibration is reinforcedwhile increasing torque fluctuation In addition a stiffnesssoftening characteristic is observed for torsional vibrationexperimentally In order to reach a better understanding inamplitude-frequency response at low- and middle-input-speed condition a procedure to analyze mode shapes ofautomotive powertrain will be conducted in future works

Data Availability

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

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported and funded by the ApplicationTechnology Research and Development Plan of LiuzhouCity (grant no 2017AA10102) and the Key Projects ofUniversities in Inner Mongolia Autonomous Region (grantno NJZZ16369)e authors of this paper express their deepgratitude for the financial support of the research project

References

[1] F Vesali M A Rezvani and M Kashfi ldquoDynamics of uni-versal joints its failures and some propositions for practicallyimproving its performance and life expectancyrdquo Journal ofMechanical Science and Technology vol 26 no 8 pp 2439ndash2449 2012

[2] J-W Lu G-C Wang H Chen A F Vakakis andL A Bergman ldquoDynamic analysis of cross shaft type universaljoint with clearancerdquo Journal of Mechanical Science andTechnology vol 27 no 11 pp 3201ndash3205 2013

[3] G Wu W Shi and Z Chen ldquoe effect of multi-universalcoupling phase on torsional vibration of drive shaft and vi-bration of vehiclerdquo in Proceedings of the SAE 2013 WorldCongress amp Exhibition Detroit MI USA April 2013

0113333 4000 4500 5000 5500

5567

6000 6500 7000Frequency (Hz)

7500 8000 85008777

263300

9000 9500 10000 10500 11000 11667100

Am

plitu

de

Am

plitu

de (R

MS)

010

009

008

007

006

005

004

003

002

001

000100000 120000 140000 160000 180000

167000

25Nmiddotm

50Nmiddotm

75Nmiddotm100Nmiddotm

125Nmiddotm

150Nmiddotm

175Nmiddotm200Nmiddotm

Curve 167000010006004003003002001

982endash3

263300006005004003003002001

690endash3

rpm--------

200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000000

Figure 13 Amplitude-frequency responses of torsional vibration for output end of rear axle at different torque fluctuations

16 Shock and Vibration

[4] B A Porter ldquoeoretical analysis of the torsional oscillationof a system incorporating a Hookersquos jointrdquo Journal of Me-chanical Engineering Science vol 3 no 4 pp 324ndash329 1961

[5] S F Asokanthan and X-H Wang ldquoCharacterization oftorsional instabilities in a Hookersquos joint driven system viamaximal Lyapunov exponentsrdquo Journal of Sound and Vi-bration vol 194 no 1 pp 83ndash91 1996

[6] A Farshidianfar M Ebrahimi and H Bartlett ldquoHybridmodelling and simulation of the torsional vibration of vehicledriveline systemsrdquo Proceedings of the Institution of Me-chanical Engineers Part D Journal of Automobile Engineeringvol 215 no 2 pp 217ndash229 2001

[7] E O A Abd Elmaksoud E M Rabeih N A Abdel-halimand S M El Demerdash ldquoInvestigation of self excited tor-sional vibrations of different configurations of automatictransmission systems during engagementrdquo Engineeringvol 3 no 12 pp 1171ndash1181 2011

[8] T B Juang M Cheng and L Na ldquoExperimental and finiteelement analyses of a sliding-tube-type driveshaft-inducedvehicle vibrationrdquo Proceedings of the Institution of MechanicalEngineers Part K Journal of Multi-Body Dynamics vol 221no 3 pp 375ndash385 2007

[9] L F Coutinho and A Tamagna ldquoFlexural vibration analysisof constant velocity half-shaftsrdquo in Proceedings of the Inter-national Congress amp Exposition Chicago IL ISA 1996

[10] Y T Kang Y H Shen W Zhang and J Yang ldquoStabilityregion of floating intermediate support in a shaft system withmultiple universal jointsrdquo Journal of Mechanical Science andTechnology vol 28 no 6 pp 2733ndash2742 2014

[11] J L Xu T Yan B Peng and B Wei ldquoEffects of the trans-mission shaft on the main reducer vibration based onADAMS and experimental demonstrationrdquo AustralianJournal of Mechanical Engineering vol 15 no 1 pp 2ndash102015

[12] C X Zhang H B Li and D Q Jin ldquoSimulation on dynamicsof transmission shaft-rear driving axle of minibus withADAMSrdquo Applied Mechanics and Materials vol 651ndash653pp 862ndash865 2014

[13] J L Xu X Y Su and P Bo ldquoNumerical analysis anddemonstration transmission shaft influence on meshing vi-bration in driving and driven gearsrdquo Shock and Vibrationvol 2015 Article ID 365084 10 pages 2015

[14] J L Xu W Lei andW X Luo ldquoInfluence of bearing stiffnesson the nonlinear dynamics of a shaft-final drive systemrdquoShock and Vibration vol 2016 Article ID 3524609 14 pages2016

[15] A Kahraman and R Singh ldquoNon-linear dynamics of a spurgear pairrdquo Journal of Sound and Vibration vol 142 no 1pp 49ndash75 1990

[16] A Kahraman and R Singh ldquoInteractions between time-varying mesh stiffness and clearance non-linearities in ageared systemrdquo Journal of Sound and Vibration vol 146no 1 pp 135ndash156 1991

[17] A Kahraman ldquoEffect of axial vibrations on the dynamics of ahelical gear pairrdquo Journal of Vibration and Acoustics vol 115no 1 pp 33ndash39 1993

[18] Y Cai and T Hayashi ldquoe linear approximated equation ofvibration of a pair of spur gears (theory and experiment)rdquoJournal of Mechanical Design vol 116 no 2 pp 558ndash5641994

[19] P Velex andMMaatar ldquoAmathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound and Vibration vol 191no 5 pp 629ndash660 1996

[20] P Velex and V Cahouet ldquoExperimental and numerical in-vestigations on the influence of tooth friction in spur andhelical gear dynamicsrdquo Journal of Mechanical Design vol 122no 4 pp 515ndash522 2000

[21] P Velex and P Sainsot ldquoAn analytical study of tooth frictionexcitations in errorless spur and helical gearsrdquo Mechanismand Machine Ieory vol 37 no 7 pp 641ndash658 2002

[22] T C Lim and Y Cheng ldquoA theoretical study of the effect ofpinion offset on the dynamics of hypoid geared rotor systemrdquoJournal of Mechanical Design vol 121 no 4 pp 594ndash6011999

[23] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[24] Y Cheng and T C Lim ldquoDynamics of hypoid gear trans-mission with nonlinear time-varying mesh characteristicsrdquoJournal of Mechanical Design vol 125 no 2 pp 373ndash3822003

[25] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 2017 pp 302ndash329 2007

[26] J Wang and T C Lim ldquoEffect of tooth mesh stiffnessasymmetric nonlinearity for drive and coast sides on hypoidgear dynamicsrdquo Journal of Sound and Vibration vol 319no 3-5 pp 885ndash903 2009

[27] W M Li and H T Wang ldquoRigidity calculation of axialposition preload bearingsrdquo Journal of Hebei University ofTechnology vol 30 no 2 pp 15ndash19 2001

[28] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo Non-linear Dynamics vol 75 no 4 pp 783ndash806 2014

[29] X Liu Z Wu J Lu and J Xu ldquoInvestigation of the effect ofrotation speed on vibration responses of transmission sys-temrdquo Matec Web of Conferences vol 256 no 3 Article ID02019 2019

Shock and Vibration 17

From the conservation of strain energy an equation isderived

12K3primeφ

21

12K3φ

22 (6)

where K3prime andK3 are torsional stiffness of output shaft for thechained system and original system respectively andφ1 andφ2 are torsional angels of driving and driven shaft

Owing to the assumption concerning a gear as rigid thegear meshing stiffness K2prime and K2 is treated as infinitequantities e following equations are derived

K3prime K3

i2 i

φ1φ2

T2prime T2

i1113896 (7)

where T2prime is the input torque of output shaft and the T2 is theoutput torque of input shaft

According to preceding derivation the equivalent mo-ments of inertia for the chained system are calculated by

J3 Ia1 +Ia2

2

J4 Ia2

2+ Ig1

J5 Ib1 + Ig2 + Ib221113872 1113873

i22 1

J6 Ib22( 1113857 + Ig3 + Ib3 + Ig4 + Ig11i211 41113872 1113873 + Ib4 + Ig5 + middot middot middot + Ig12i212 51113872 1113873 + Ib5 + Id + Ib6 + Ig8 + Ig9 + Ig15i215 91113872 1113873i29 81113872 1113873 + Ib71113960 1113961

i22 1

J7 Ic1 + Id + Ig10 + Ic2 + Ic3 + Ic421113872 1113873 + Id1113872 1113873

i210 3 middot i22 1( 1113857

J8 Ic42 + Ig13 + Ig6i26 13 + Ic5 + Ig14 + Ig7i27 14 + Ic6 + Ic7 + Id1113960 1113961

i210 3 middot i22 1( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

J1J2

J1J2

J3

J3primeJ4prime

J4

K1

K2

K3

K1 K2prime K3prime

First gearshaft

Secondgearshaft

Gear 1Gear 2

Gear 3Gear 4

First gearshaft

Secondgearshaft

Gear 1

Gear 2Gear 3

Gear 4

Figure 3 e schematic diagram of equivalent principle for simplifying a multimesh gear train into a chained system

4 Shock and Vibration

where Iaj(j 1 2) is the equivalent moment of inertia of thejth node of input shaft Ibj(j 1 2 7) is the equivalentmoment of inertia of the jth node of middle shaft Icj(j

1 2 7) is the equivalent moment of inertia of the jthnode of output shaft Igj(j 1 2 15) is the equivalentmoment of inertia of jth gear Id is the equivalent moment ofinertia of the synchronizer ring and im n is the drive ratio ofthe mating gear pair

23 Drive Shaft System Drive shaft system comprises in-termediate drive shaft main drive shaft three universaljoints and intermediate support In this paper the rotationalinertia of shaft is equivalent to the universal joint by thelumped-mass method averagely e lubrication clearancefriction manufacturing and assembly errors and flexibilityof universal joint are not considered

In order to describe the dynamics of a U-joint a set ofparameters are introduced as follows

Ai 1 + cos2αi

2 cos αi

Bi 1 minus cos2αi

2 cos αi

i 1 2 3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

where αi(i 1 2 3) is the intersection angel between shaftslinked by a universal joint (U-joint)

According to kinematics analysis the following equa-tions are derived

θ9 arctantan θ8cos α1

θ9prime θ8prime

A1 minus B1 cos 2θ8

θ9Prime θ8Prime

A1 minus B1 cos 2θ8minus

2B1 sin 2θ8A1 minus B1 cos 2θ8( 1113857

2 θ8prime( 11138572

T2 A1 minus B1 cos 2θ8( 1113857T1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

θ11 arctantan θ10cos α2

θ11prime θ10prime

A2 minus B2 cos 2θ10

θ11Prime θ10Prime

A2 minus B2 cos 2θ10minus

2B2 sin 2θ10A2 minus B2 cos 2θ10( 1113857

2 θ10prime( 11138572

T4 A2 minus B2 cos 2θ10( 1113857T3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

θ13 arctantan θ12cos α3

θ13prime θ12prime

A3 minus B3 cos 2θ12

θ13Prime θ12Prime

A3 minus B3 cos 2θ12minus

2B3 sin 2θ12A3 minus B3 cos 2θ12( 1113857

2 θ12prime( 11138572

T6 A3 minus B3 cos 2θ12( 1113857T5

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

Shock and Vibration 5

where α1 is the intersection angel between output shaft oftransmission and intermediate drive shaft α2 is the inter-section angel between main drive shaft and intermediatedrive shaft α3 is the intersection angel between main driveshaft and input shaft of rear axle

e equivalent moments of inertia for intermediate driveshaft and main drive shaft are calculated by

J9 J10 12i21

Jids

J11 J12 12i21

Jmds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(11)

where i1 is the drive ratio of transmission Jids is the momentof inertia of intermediate drive shaft and Jmds is the momentof inertia of main shaft

Equivalent torsional stiffness k and damping coefficient c

of shaft are given by

k M

ϕ

Gapπ D4 minus d4( 1113857

32L

c 2ζ

kJaJb

Ja + Jb

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(12)

where ζ is the damping ratio and Ja and Jb are the equivalentmoment of inertia of shaft end

24 Rear Axle In this paper the main reducer system issimplified as a geared rotor-bearing system with bending-torsional-lateral-rocking coupled vibration Pinion shaft issimplified as two lumpedmasses linked by a spring-dampingpair with infinite stiffness and the differential assembly ismodeled as a rigid part with a concentratedmass Half-shaftsare described by twomass-spring-damping pairs Otherwisepinion and gear are treated as rigid bodies within torsionalvibration analysis

e equivalent moments of inertia for rear axle arecalculated by

J13 15Jps

i21

J14 45Jps

i21

J15 Jd + 05Jhs

i21i22

J16 J17 05Jhs + Jhs

i21i22

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)

where i1 (i1 1350) is the drive ratio of transmission i2(i2 41) is the drive ratio of the main reducer Jps is themoment of inertia of pinion shaft Jd is the moment of inertia

of differential assembly and Jhs is the moment of inertia ofhalf-shaft

Equivalent stiffness of bearing is expressed as [27]

Kr 7253l08

1 z091 cos20α1 middot F01

a0cos01α1

Ka 29011l081 cos19α1 middot F01

a0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(14)

where Kr is the equivalent radial stiffness of bearing Ka isthe equivalent axial stiffness of bearing l1 is the nominalcontact length z1 is the number of rolling elementα1 is thepressure angle and Fa0 is the pretightening force

In addition a new model for pinion shaft is proposedwhich is different from previous works Owing to negligibleelastic deformation pinion shaft is modeled as rigid bodyBearing is modeled as a spring-damping pair Without lossof generality the derivation procedure on displacements ofthe mounting point in the x-o-y plane is used as an example

In Figure 4 Xpx and Xpy are introduced to representtranslational displacements for centroid of pinion shaft θpz

denotes rocking displacement around the z axisFrom the geometry deformation relationship and pro-

posed concepts the displacement for mounting point iscalculated by

xpy1 xpy minus lp1 tan θpz

xpy2 xpy minus lp2 tan θpz

⎧⎨

⎩ (15)

ereby the following expressions are derived

xpy1prime xpyprime minus lp1

θpzprime

cos2θpz

xpy2prime xpyprime minus lp2

θpzprime

cos2θpz

1113896

(16)

Static transmission error e(t) [22ndash26 28] is simulated inFourier series form by

e(t) em + 1113944N

i1eAi cos iωmt + φe( 1113857 (17)

where em is the average value of transmission error eAi arethe amplitude of the ith order harmonic φe are the initialphase angle of the ith order harmonic and ωm is the meshingfrequency

e dynamic meshing stiffness is the periodically time-varying parameter with respect to the mesh frequency asshown in Figure 5 which can be simulated by means of aFourier expansion [13 14 22ndash26]

km(t) km + 1113944N

i1kAi cos iωmt + φk( 1113857 (18)

where km is the mean value of meshing stiffness kAi is thefluctuation amplitude of ith order meshing stiffness ωm isthe mesh frequency and φk is the initial phase of meshingstiffness

Defining me denotes mean mass of pinion and gear

6 Shock and Vibration

me 1

1mp1113872 1113873 + 1mg1113872 1113873

mpmg

mp + mg

(19)

e following equation can be derived

ωn

km

me

1113971

(20)

where ωn is the equivalent natural frequency of pinion andgear

Equivalent mesh damping coefficient is calculated whichis different from Wang and Limrsquos analysis [25]

cm ξ lowast 2meωn 2ξ

km

1mp1113872 1113873 + 1mg1113872 1113873

1113971

(21)

where ζ is the damping ratio generally set as a range of 003to 017 [25]mp andmg are equivalent mass of pinion and gearrespectively

e relative displacement along the normal direction atthe meshing point xn is given as follows

xn xpx + Eyθpz1113872 1113873 minus xgx + Ezθgy + lg3θgz1113872 11138731113960 1113961

middot cos αn sin βm cos δ1 + sin αn sin δ1( 1113857

+ xpy + lp3θpz1113872 1113873 minus xgy + Ezθgx minus Exθgz1113872 11138731113960 1113961

middot minuscos αn sin βm sin δ1 + sin αn cos δ1( 1113857

+ xpz minus Rpθpx + lp3θpy1113872 1113873 minus xgz minus lg3θgx minus Rgθgy1113872 11138731113960 1113961

middot cos αn cos βm( 1113857 minus e(t)

ε1 xpx minus xgx minus Ezθgy + Eyθpz minus lg3θgz1113872 1113873

+ ε2 xpy minus xgy minus Ezθgx + lp3θpz + Exθgz1113872 1113873

+ ε3 xpz minus xgz minus Rpθpx + lg3θgx + lp3θpy + Rgθgy1113872 1113873

minus e(t)

(22)

e backlash function is expressed as [23]

f xn( 1113857

xn minus b xn ge b

0 minusblexn le b

xn + b xn le minus b

⎧⎪⎪⎨

⎪⎪⎩(23)

where b is half of gear backlashe dynamic meshing force along the line of action Fn

can be calculated byFn km(t)f xn( 1113857 + cmxn

prime (24)

wherecm represents thevalueof equivalentmeshdampingcoefficiente components of dynamic mesh force along the co-

ordinate directions are expressed asFx Fn cos αn sin βm cos δ1 + Fn sin αn sin δ1

Fy minusFn cos αn sin βm sin δ1 + Fn sin αn cos δ1

Fz Fn cos αn cos βm

⎧⎪⎪⎨

⎪⎪⎩(25)

where δ1 is cone angle of the pinion αn is the normal pressureangle and βm is the helix angle at the midpoint of the pinion

In additional the following equations are derived byintroducing the parameter εi

ε1 cos αn sin βm cos δ1 + sin αn sin δ1

ε2 minuscos αn sin βm sin δ1 + sin αn cos δ1

ε3 cos αn cos βm

⎧⎪⎪⎨

⎪⎪⎩(26)

Substituting equations (26) into (25) the expressions ofdynamicmesh force applied on pinion and gear can bewritten as

Fpx minusFgx ε1Fn

Fpy minusFgy ε2Fn

Fpz minusFgz ε3Fn

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(27)

where the subscript p is the label for pinion and the subscriptg is the label for gear

Mesh impact due to teeth separation is neglected in this paper

25 Equations of Motion From the preceding derivationsthe equations of motion considering bending-torsional-lateral-rocking vibration are expressed as

Fpz

kpz2 cpz2

kpx

cpx

θpy

Xpx

kpz1 cpz1

FpxFpy

Op1

cpy2

θpz

mp

kpy2cpy1

Lp1

Lp2 Lp3

Op2 Op0

kpy1y

x

z

Xpy1 Xpy2

Xpy

Op

Op3 Ey

θpz

θpx

Figure 4 e dynamic model of pinion shaft

Shock and Vibration 7

J1θ1Prime + c1 θ1prime minus θ2prime( 1113857 + k1 θ1 minus θ2( 1113857 TD

J2θ2Prime + c1 θ2prime minus θ1prime( 1113857 + k1 θ2 minus θ1( 1113857 + c2 θ2prime minus θ3prime( 1113857 + k2 θ2 minus θ3( 1113857 0

J3θ3Prime + c2 θ3prime minus θ2prime( 1113857 + k2 θ3 minus θ2( 1113857 + c3 θ3prime minus θ4prime( 1113857 + k3 θ3 minus θ4( 1113857 0

J4θ4Prime + c3 θ4prime minus θ3prime( 1113857 + k3 θ4 minus θ3( 1113857 + c4 θ4prime minus θ5prime( 1113857 + k4 θ4 minus θ5( 1113857 0

J5θ5Prime + c4 θ5prime minus θ4prime( 1113857 + k4 θ5 minus θ4( 1113857 + c5 θ5prime minus θ6prime( 1113857 + k5 θ5 minus θ6( 1113857 0

J6θ6Prime + c5 θ6prime minus θ5prime( 1113857 + k5 θ6 minus θ5( 1113857 + c6 θ6prime minus θ7prime( 1113857 + k6 θ6 minus θ7( 1113857 0

J7θ7Prime + c6 θ7prime minus θ6prime( 1113857 + k6 θ7 minus θ6( 1113857 + c7 θ7prime minus θ8prime( 1113857 + k7 θ7 minus θ8( 1113857 0

J8θ8Prime + c7 θ8prime minus θ7prime( 1113857 + k7 θ8 minus θ7( 1113857 minusT1

i1

J9θ9Prime + c8 θ9prime minus θ10prime( 1113857 + k8 θ9 minus θ10( 1113857 T2

i1

J10θ10Prime + c8 θ10prime minus θ9prime( 1113857 + k8 θ10 minus θ9( 1113857 minusT3

i1

J11θ11Prime + c9 θ11prime minus θ12prime( 1113857 + k9 θ11 minus θ12( 1113857 T4

i1

J12θ12Prime + c9 θ12prime minus θ11prime( 1113857 + k9 θ12 minus θ11( 1113857 minusT5

i1

J13θ13Prime + c10 θ13prime minus θ14prime( 1113857 + k10 θ13 minus θ14( 1113857 T6

i1

J14θ14Prime + c10 θ14prime minus θ13prime( 1113857 + k10 θ14 minus θ131113857( 1113857 minusFpzEy

i1

J15θ15Prime + c11 θ15prime minus θ16prime( 1113857 + k11 θ15 minus θ16( 1113857 + c12 θ15prime minus θ17prime( 1113857 + k12 θ15 minus θ17( 1113857 FgzEx + FgxEz

i1i2

J16θ16Prime + c11 θ16prime minus θ15prime( 1113857 + k11 θ16 minus θ15( 1113857 minusTRL

i1i2

J17θ17Prime + c12 θ17prime minus θ15prime( 1113857 + k12 θ17 minus θ15( 1113857 minusTRR

i1i2

mmymPrime + cmyym

prime + kmyym Fmy

mmzmPrime + cmzzm

prime + kmzzm Fmz

mpxpxPrime + kpxxpx + cpxxpx

prime minusFpx

mpxpyPrime + kpy1 xpy minus lp1 tan θpz1113872 1113873 + cpy1 xpy

prime minus lp1θpzprime

cos2θpz

1113888 1113889 + kpy2 xpy minus lp2 tan θpz1113872 1113873 + cpy2 xpyprime minus lp2

θpzprime

cos2θpz

1113888 1113889 Fpy

mpxpzPrime + kpz1 xpz minus lp1 tan θpy1113872 1113873 + cpz1 xpz

prime minus lp1θpyprime

cos2θpy

1113888 1113889 + kpz2 xpz minus lp2 tan θpy1113872 1113873 + cpz2 xpzprime minus lp2

θpyprime

cos2θpy

1113888 1113889 Fpz

JpyθpyPrime minus lp1kpz1 xpz minus lp1 tan θpy1113872 1113873 minus lp1cpz1 xpz

prime minus lp1θpyprime

cos2θpy

1113888 1113889 minus lp2kpz2 xpz minus lp2 tan θpy1113872 1113873 minus lp2cpz2 xpzprime minus lp2

θpyprime

cos2θpy

1113888 1113889 Fpzlp3

JpzθpzPrime minus lp1kpz1 xpy minus lp1 tan θpy1113872 1113873 minus lp1cp1 xpy

prime minus lp1θpzprime

cos2θpz

1113888 1113889 minus lp2kpy2 xpy minus lp2 tan θpz1113872 1113873 minus lp2cpy2 xpyprime minus lp2

θpzprime

cos2θpz

1113888 1113889 minusFpxEy + Fpylp3

mgxgxPrime + kgx1 xgx minus lg1 tan θgz1113872 1113873 + cgx1 xgx

Prime minus lg1θgzprime

cos2θgz

1113888 1113889 + kgx2 xgx + lg2 tan θgz1113872 1113873 + cgx2 xgxprime + lg2

θgzprime

cos2θgz

1113888 1113889 Fgx

mgxgyPrime + kgyxgy + cgyxgy

Prime minusFgy

mgxgzPrime + kgz1 xgz + lg1 tan θgx1113872 1113873 + cgz1 xgz

prime + lg1θgxprime

cos2θgx

1113888 1113889 + kgz2 xgz minus lg2 tan θgx1113872 1113873 + cgz2 xgzprime minus lg2

θgxprime

cos2θgx

1113888 1113889 minusFgz

JgxθgxPrime + lg1kgz1 xgz + lg1 tan θgx1113872 1113873 + lg1cgz1 xgz

prime + lg1θgxprime

cos2θgx

1113888 1113889 minus lg2kgz2 xgz minus lg2 tan θgx1113872 1113873 + lg2cgz2 xgzprime minus lg2

θgxprime

cos2θgx

1113888 1113889 Fgzlg3 minus FgyEz

JgzθgzPrime + lg1kgx1 xgx + lg1 tan θgz1113872 1113873 + lg1cgx1 xgx

prime + lg1θgzprime

cos2θgz

1113888 1113889 minus lg2kgx2 xgx minus lg2 tan θgz1113872 1113873 + lg2cgx2 xgxprime minus lg2

θgzprime

cos2θgz

1113888 1113889 Fgxlg3 minus FgyEx

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

8 Shock and Vibration

where TD is the input torque of motor T1 is the outputtorque of transmission T2 andT3 are the input and outputtorque for intermediate drive shaft T4 andT5 are the inputand output torque for main drive shaft T6 is the inputtorque of rear axle ki(i 1 2 7) are the equivalenttorsional stiffness ci(i 1 2 7) are the equivalent tor-sional damping ki(i 8 9) are the equivalent torsionalstiffness ci(i 8 9) are the equivalent torsional dampingmm is the mass of transmission shaft ki(i 10 11 12) arethe equivalent torsional stiffness for pinion shaft and dif-ferential axle ci(i 10 11 12) are the equivalent torsionaldamping for pinion shaft and differential axle cmy and cmz

are the equivalent damping of intermediate support alongaxes y and z kmy and kmz are the equivalent stiffness ofintermediate support along axes y and z Fmy and Fmz arecomponents of additional force Fi along axes y and z forintermediate support Ei(i x y z) are coordinates ofmeshing point along axes x y and z and TRL andTRR areload torque for wheels

26 Numerical Solution Procedure e fourth-order adap-tive step RungendashKutta method which is generally applicableto strong nonlinearity is adopted in the present work to solve

the equations In order to program a new state space isneeded

In consideration of equations (12)mdash(14) it can be de-rived that kinetic equations of 3 universal joints reduce thenumber of degree-of-freedom from 29 to 26 for the wholesystem e following state space coordinate vector alsodescribes the dynamic behaviors of the system

Ss θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 θ10 θ12 θ14 θ15 θ16 θ17θpy θpz θgx θgz ym zm xp yp zp xg yg zg

⎧⎨

⎩

⎫⎬

⎭

Τ

(29)

In order to eliminate the difference in magnitude be-tween parameters which will reduce the solvability and evenlead to no convergence for solving equations of motionmentioned above b is defined as the characteristic lengthand a new time parameter τ is introduced as follows

τ tωn (30)

where t denotes timeFor the sake of nominalizing equations a generalized

state space coordinate vector with 26 parameters is proposedto characterize dimensionless equations as

st sr1 sr2 sr3 sr4 sr5 sr6 sr7 sr8 sr10 sr12 sr16 sr17 srpx srpy srpz srgx srgy srgz smy smz spx spy spz sgx sgy sgz1113966 1113967T

(31)

e following equations are derived as

sri θi timesRi

b i 1 sim 8 10 12 16 17

srpx θ14 timesRpx

b

srgy θ15 timesRgy

b

srij θij timesRij

b i p g j x y z

six xi

b i p g

siy yi

b i m p g

siz zi

b i m p g

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

where b is the half of backlash and Ri is the equivalent gyrationradius which can be calculated by the following equation

Ri

Ji

mi

1113971

i 1 sim 8 10 12 16 17

Rij

Jij

mi

1113971

i p g j x y z

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(33)

where Ji and Jij are the equivalent moment of inertia and mi

is equivalent massA set of dimensionless parameters are defined as

xnτ xn

b

f xnτ( 1113857 f xt

n( 1113857

b

c(ijs) ci

mjRjRsωn

k(ijs) ki

mjRjRsω2n

cmi cm

miωn

kmi km

miω2n

T(xj) Tx

mjRjω2nb

lijτ lij

b

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)

As equations (30) (32) and (34) are substituted intoequation (28) the dimensionless equations of motion can bederived

Shock and Vibration 9

e equivalent inertia parameters are obtained in UGand stiffness parameters of powertrain are calculated byusing the relationships given in modeling section as listed inTable 1 Subsequently equivalent damping coefficients canbe obtained e parameters of the main reducer can becalculated by substituting system parameters listed in Table 2into equations (17)ndash(31) e solution procedures for di-mensionless parameter are carried out by using precedingcalculated parameters and the relationships proposed inprevious parts

e exciting and configuration parameters are set asθ1 1500 rpm Td2 100Nmiddotm km 9697e8Nm i1 135i2 41 2b 015mm em 0 α1 1695 degree α2 13044degree and α3 1554 degree respectively Without signif-icant loss of accuracy the fundamental harmonic form isadopted for time-varying mesh stiffness and static trans-mission error function whose initial phases are set at 0respectively

3 Numerical Results

31 Analysis of Coupled Torsional Vibration In order toanalysis coupled interaction of automotive powertrainFourier spectrums of torsional vibration response are cal-culated at the initial parameters as shown in Figure 6 Twopeaks and their sidebands are observed e torsional vi-brational responses are governed by the component of 50Hzwhich equals with exciting frequency of input torque asshown in equation (3) (ωf 150060 lowast 2 50Hz) isimplies that engine input excitation is responsible for tor-sional vibration and a valuable approach to decreasingtorsional vibration of powertrain is inferred In addition thecomponent with a frequency of around 185Hz is observedobviously for each considered part As one knows themeshing frequency undergoing this condition is presumably185185Hz (f 150060135185185Hz) which is quite inagreement with the frequency of the second peak Coupledvibration occurs between the torsional system and hypoidgear set and inner excitation caused by gear pair meshingprocess which is affected by clearance and time-varyingmeshing stiffness [25] accounts a lot for torsional vibrationat 185Hz of powertrain and affects torsional responsesslightly After simply calculation bandwidths reaching thesame value of around 37Hz of sidebands around two peaksare got U-joint is a transmission mechanism with non-constant angular speed which will excite a doubled fre-quency torsional vibration of input as shown in equations(12)ndash(14) Under this condition the doubled frequencyexcited by U-joint is calculated

fuminusjoint n

i1 lowast 60lowast 2

1500135lowast 60

lowast 2 3704 (35)

It can be conjectured that modulation phenomenonappears when automotive powertrain works between exci-tations attributed to input U-joint and gear pair As one cansee in Figure 6 compared torsional response of gear withthat located front of meshing point a lower value of tor-sional angular is observed and it is inferred that gear meshexcitations reduce torsional vibration for components

located behind meshing point It is noted that taking gearmesh behaviors into account is quite necessary to when atorsional characteristic analysis is conducted

32 Amplitude-Frequency Response of Torsional Vibration forAutomotive Powertrain e previous works [29] analyzedtorsional characteristics of the coupled system consisted ofdrive shaft and rear axle which denoted that resonance ofdrive shaft at the first two-order nature frequency accountsfor dramatic torsional vibration responses undergoing high-rotation-speed condition However it cannot reveal thetorsional vibration mechanism of automotive under low-and middle-speed condition

In order to obtain a more realistic understanding foramplitude-frequency response of automotive powertrainundergoing middle-speed condition theoretically the pre-ceding equations of motion are adopted A sweep frequencyanalysis procedure is carried out in a range from 1000 rpm to3500 rpm which respectively corresponds to frequencyinternal of 166ndash583Hz for input speed 3333ndash1167Hz forinput excitation and 12346ndash43209Hz for meshing exci-tation e response at input end of rear axle is taken as anexample in this work

Figure 7 shows the amplitude-frequency response inwhich abscissa denotes torsion speed of input and ordinate isRMS value of angular displacement As one can see threeobvious peaks which correspond to input speed at about1000 rpm 1600 rpm and 2800 rpm respectively are ob-served Owing to the assumption that the dramatic responsesare attributed to resonance of the system the first three-order nature frequency respect to the second-order term ofinput torque fluctuation is proposed at 333Hz 533Hz and933Hz successively It is noted that on the basis of previousanalysis these three peaks also correspond to frequencies of247Hz 395Hz and 691Hz for the U-joint excitation and1234Hz 1975Hz and 3456Hz for the meshing excitationPreliminary works [29] showed that resonance is attributedto input excitation of drive shaft and the meshing excitationof hypoid gear pair with higher exciting frequency thaninput will not excite prominent resonance behaviors of driveshaft as illustrated in Figure 8 ie meshing excitation hasslight influence on the torsional system which reaches thesame conclusion with this work According to analysis inSection 31 the sidebands respect to each exciting frequencyare (86ndash58) Hz at 333Hz (138ndash928) Hz at 533Hz and(242ndash1624) Hz at 933Hz for input torque and (987ndash1481)Hz at 1234Hz (158ndash237) Hz at 1975Hz and (2765ndash4147)Hz at 3456Hz for meshing behavior of the gear pair re-spectively As listed in Table 3 the first peak can be attributedto the first two-order resonances of drive shaft excited bysideband around 1234Hz Sidebands at 533Hz and1975Hz result in the second-order resonance which gen-erates the second peak Moreover the second-order reso-nances excited by the sideband at 933Hz and third-orderone caused by sideband at 3456Hz explain the appearanceof the third peak A typical phenomenon is observed that thetorsional vibration reduces a lot while one compares theamplitude of the third peak with another two which

10 Shock and Vibration

Table 1 e equivalent parameters of powertrain

Equivalent mass (kg) Equivalent moment of inertia (kgmiddotm2) Equivalent stiffness (Nm)m1 2 J1 0091125 k1 1002375m2 10 J2 03801735 k2 164025m3 06150 J3 000008103 k3 27500m4 13798 J4 0001001 k4 10e11m5 23581 J5 0001323 k5 479000m6 114323 J6 000249 k6 10e11m7 45323 J7 0001865 k7 21400m8 75567 J8 0002098 mdashm9 1284 J9 0001695 k8 43921799m10 1284 J10 0001695 mdashm11 13125 J11 000173 k9 41933635m12 13125 J12 000173 mdashm13 04254 J13 000002772 k10 10e11m14 17016 J14 00001109 km 96965e8m15 1375 J15 0034177 mdashm16 845 J16 99 k11 12032114m17 845 J17 99 k12 12032114

Table 2 e system parameters of the main reducer

Parameter Pinion GearNumber of teeth 10 41Module (mm) 400Mean pressure angle (deg) 20 20Helix angle at midpoint of the pinion (deg) 50 2713Cone angle (deg) 2312 6514Tooth width (mm) 4066 3452Tooth addendum (mm) 359 644Pinion outset (mm) minus38Mounting distance (mm) 112 64Pitch radius (mm) 40 108Backlash (mm) 01536Distance from pinion reference point to the x-o-z plane for pinion (mm) 40Distance from pinion reference point to the y-o-z plane for gear (mm) 101Distance between projection point and centroid of pinion shaft (mm) lp3 10 lg3 5

Distance between bearing reference point and centroid of pinion shaft (mm) lp1 100 lg1 90lp2 175 lg2 93

Equivalent mass (kg) mg 2127 mp 1375Jpx J14 Jgx 0050315

Equivalent moment of inertia (kgmiddotm2) Jpy 0002982 Jgx J15Jpz 0002982 Jpz 0037551

Meshing stiffnessof (i + 1)th tooth pair

Meshing stiffnessof ith tooth pair

t

k

Meshing stiffnessof (i ndash 1)th tooth pair

Total meshingstiffness km(t)

Figure 5 Schematic diagram of time-varying meshing stiffness on hypoid gear

Shock and Vibration 11

indicates that torsional behaviors of automotive powertrainaffect the vibration characteristics of the complete vehicleunder a low- or middle-input-speed condition In order tovalidate the preceding result an experimental study isnecessary

4 Experimental Analysis

A test bench which can be used to test the vibration behavior ofa transmission system used for four-wheel drive vehicle be-longing to the China Automotive Technology and ResearchCenter is adopted to perform this experiment ere are fivemotors in the test bench and three of them are used in thisexperiment One of them drives automotive powertrain whichcan generate a high frequency input torque with fluctuationvaried in accordance with necessary and others are used asloads In this test two photoelectric sensors and two magne-toelectric sensors are used to detect torsional vibration signalsand other five accelerometers are mounted to monitor trans-verse vibration Installation site for each sensor can be found inFigure 9 Signals are sampled and processed by LMS TestLab

In order to validate the feasibility of the theoreticalmodel the input torque simulated by input motor and loadsare consistent with parameters employed in numericalanalysis (mean value and the second-order term are con-tained only and fluctuation amplitude is set at 100NmiddotM forinput torque) Within the experiment the output speed ofmotor varies from 1000 rpm to 3500 rpm to conduct a sweepfrequency response analysis experimentally

As illustrated in Figure 10 the amplitude-frequencyresponses of torsional vibration have the similar feature for

different parts of powertrain Peaks can be observed forcarves at a similar input speed It is inferred that a presumeresonance occurs at each peak As a result the first three-order nature torsional frequencies can be predictedMeanwhile the same results are reached As shown inFigure 10 each curve has a higher value nearby the first andsecond peak With the further increase in frequency itreaches a lower plateau value Torsional behaviors have aneffect on the vibration characteristics under a low-input-speed condition Furthermore the part located behind themesh point such as output end of half-shaft has a lowerRMS value than transmission and drive shaft which can beattributed to reduction effect of mesh excitation on torsionalvibration

e experimental result of input end of rear axle iscompared with the numerical one As shown in Figure 11 aquite consistency is observed and the first three-order fre-quencies are inferred at 35Hz (1050 rpm) 557Hz(1670 rpm) and 883Hz (2650 rpm) for experimental resultby the same way mentioned in preceding section Comparedwith numerical predictions an acceptable difference ofabout 5 (the difference is 51 at 333Hz 45 at 533Hzand 536 at 933Hz) can be obtained ese indicate thatthe theoretical model is feasible for predicting nature tor-sional frequencies Moreover numerical torsional dis-placement is a little lower than the experimental one at thesame input speed which is below about 2450 rpm and a littlehigher with the further increase is can be presumablyattributed to assumptions made in this present work withrespect to the complexity of the system and the limitations ofthe lumped-mass method

03503

02502

01501

0050

0

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(a)

13Hz

50Hz

87Hz 150Hz185Hz222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(b)

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(c)

13Hz

50Hz

87Hz150Hz

185Hz222Hz

004

003

002

001

00 50 100 150 200 250

Frequency (Hz)To

rsio

nal d

ispla

cem

ent (

deg)300 350 400 450 500 550 600

(d)

Figure 6 e Fourier spectrum of torsional responses (a) output shaft of transmission (b) output end of the drive shaft system (c)torsional displacement of the pinion along the axis x (d) torsional displacement of the gear along the axis y

12 Shock and Vibration

e previous section indicates numerically that excita-tion of input dominates torsion vibration of the system andgear set affects responses slightly In order to validate thecorrectness of numerical analysis and feasibility decreasing

torsional vibration by reducing input excitation effect ofinput torque fluctuation on torsional response is studiedexperimentally e mean value of input torque is set at150NmiddotM and fluctuation of the second-order term is set at

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

2800rpm1600rpm

Figure 7 Amplitude-frequency response at input end of rear axle

8

7

6

5

4

2

3

1

00 1000 2000 3000

Rotation speed (RPM)

Vibr

atio

n am

plitu

de (micro

m)

4000 5000 6000

x directiony directionz direction

Figure 8 Influence of input rotation speed of drive shaft on vibration of the main reducer [29]

Table 3 e first six-order nature frequency of drive shaft [29]

Order 1 2 3 4 5 6Frequency (Hz) 1309 1402 4152 4422 6362 9066

Shock and Vibration 13

25NmiddotM 50NmiddotM 75NmiddotM 100NmiddotM 125NmiddotM 150NmiddotM175NmiddotM and 200NmiddotM in order

Together Figures 12 and 13 expectably show that thetorsional vibration is reinforced while increasing torquefluctuation Meanwhile an obvious increase in amplitudeoccurs at the second peak when fluctuation is increased from175NmiddotM to 200NmiddotM which indicates that a limitation should

be considered e accurate value can be determined by afurther study In summary reducing torque fluctuation is abeneficial approach for vibration damping Furthermore it isnoted that such a nonlinear phenomenon as natural fre-quency shift occurs while changing torque fluctuation Takingthe second-order torsional frequency as an example thefrequency value decreases with elevating fluctuation of input

Transmission

Photoelectricsensor

Photoelectricsensor

Magnetoelectricsensor

Magnetoelectricsensor

Rear axle

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Drive sha system

Figure 9 e experiment bench and arrangement of sensors

042

040

038

035

033

030

028

025

023

020

018

015

013

010

008

005

003

Amplitude-frequency curve ofle wheel

Amplitude-frequency curve oftransmission output sha

Amplitude-frequency curve ofrear axle output end

Amplitude-frequency curve ofright wheel

000

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (K

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Curve 167000

041

035

003

003

5567 8817

167000 264500

rpm

deg

deg

deg

deg

264500

012

005

003

764endash3

Figure 10 Amplitude-frequency responses of torsional vibration for powertrain at a torque fluctuation of 100NmiddotM

14 Shock and Vibration

036

034

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

Numerical result

Experimental result

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

Figure 11 Comparison of numerical and experimental results for input end of rear axle

160 5567 8817

264500

150

140

130

120

110

100

090

080

070

060

050

040

030

020

010

000 167000

25Nmiddotm

50Nmiddotm

75Nmiddotm

100Nmiddotm

125Nmiddotm150Nmiddotm

175Nmiddotm

200Nmiddotm

Curve 167000147078057045035025017013

rpm--------

264500009008007006005004004003

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (R

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Figure 12 Amplitude-frequency responses of torsional vibration for input end of rear axle at different torque fluctuations

Shock and Vibration 15

torque which shows that the torsional system has a stiffness-softening characteristic with respect to torque fluctuationis result can be accounted for a number of nonlinearitiessuch as friction clearance and sliding and the existence ofwhich sophisticates system nonlinear characteristics

5 Conclusion

In the present paper a lumped-parameter dynamic modelwith 29 degree-of-freedom is proposed considering thetransverse torsional and rocking coupled vibration for abetter understanding of the automotive powertrain systemdynamics and equations of motion are derived e nu-merical result shows that gear mesh excitations have littleeffects on torsional response for such components locatedbefore the mesh point as transmission transmission shaftand pinion but significantly reduce the torsional responsesfor ones behind the mesh point e amplitude-frequencycharacteristics of torsional vibration graphically indicate thattorsional behaviors of automotive powertrain only affect thevibration characteristics of complete vehicle under low-speed condition and predict the first three-order naturetorsional frequencies of the whole system An experimentalexamination is performed and validates the feasibility of theconsidered model e torsional vibration is reinforcedwhile increasing torque fluctuation In addition a stiffnesssoftening characteristic is observed for torsional vibrationexperimentally In order to reach a better understanding inamplitude-frequency response at low- and middle-input-speed condition a procedure to analyze mode shapes ofautomotive powertrain will be conducted in future works

Data Availability

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

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported and funded by the ApplicationTechnology Research and Development Plan of LiuzhouCity (grant no 2017AA10102) and the Key Projects ofUniversities in Inner Mongolia Autonomous Region (grantno NJZZ16369)e authors of this paper express their deepgratitude for the financial support of the research project

References

[1] F Vesali M A Rezvani and M Kashfi ldquoDynamics of uni-versal joints its failures and some propositions for practicallyimproving its performance and life expectancyrdquo Journal ofMechanical Science and Technology vol 26 no 8 pp 2439ndash2449 2012

[2] J-W Lu G-C Wang H Chen A F Vakakis andL A Bergman ldquoDynamic analysis of cross shaft type universaljoint with clearancerdquo Journal of Mechanical Science andTechnology vol 27 no 11 pp 3201ndash3205 2013

[3] G Wu W Shi and Z Chen ldquoe effect of multi-universalcoupling phase on torsional vibration of drive shaft and vi-bration of vehiclerdquo in Proceedings of the SAE 2013 WorldCongress amp Exhibition Detroit MI USA April 2013

0113333 4000 4500 5000 5500

5567

6000 6500 7000Frequency (Hz)

7500 8000 85008777

263300

9000 9500 10000 10500 11000 11667100

Am

plitu

de

Am

plitu

de (R

MS)

010

009

008

007

006

005

004

003

002

001

000100000 120000 140000 160000 180000

167000

25Nmiddotm

50Nmiddotm

75Nmiddotm100Nmiddotm

125Nmiddotm

150Nmiddotm

175Nmiddotm200Nmiddotm

Curve 167000010006004003003002001

982endash3

263300006005004003003002001

690endash3

rpm--------

200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000000

Figure 13 Amplitude-frequency responses of torsional vibration for output end of rear axle at different torque fluctuations

16 Shock and Vibration

[4] B A Porter ldquoeoretical analysis of the torsional oscillationof a system incorporating a Hookersquos jointrdquo Journal of Me-chanical Engineering Science vol 3 no 4 pp 324ndash329 1961

[5] S F Asokanthan and X-H Wang ldquoCharacterization oftorsional instabilities in a Hookersquos joint driven system viamaximal Lyapunov exponentsrdquo Journal of Sound and Vi-bration vol 194 no 1 pp 83ndash91 1996

[6] A Farshidianfar M Ebrahimi and H Bartlett ldquoHybridmodelling and simulation of the torsional vibration of vehicledriveline systemsrdquo Proceedings of the Institution of Me-chanical Engineers Part D Journal of Automobile Engineeringvol 215 no 2 pp 217ndash229 2001

[7] E O A Abd Elmaksoud E M Rabeih N A Abdel-halimand S M El Demerdash ldquoInvestigation of self excited tor-sional vibrations of different configurations of automatictransmission systems during engagementrdquo Engineeringvol 3 no 12 pp 1171ndash1181 2011

[8] T B Juang M Cheng and L Na ldquoExperimental and finiteelement analyses of a sliding-tube-type driveshaft-inducedvehicle vibrationrdquo Proceedings of the Institution of MechanicalEngineers Part K Journal of Multi-Body Dynamics vol 221no 3 pp 375ndash385 2007

[9] L F Coutinho and A Tamagna ldquoFlexural vibration analysisof constant velocity half-shaftsrdquo in Proceedings of the Inter-national Congress amp Exposition Chicago IL ISA 1996

[10] Y T Kang Y H Shen W Zhang and J Yang ldquoStabilityregion of floating intermediate support in a shaft system withmultiple universal jointsrdquo Journal of Mechanical Science andTechnology vol 28 no 6 pp 2733ndash2742 2014

[11] J L Xu T Yan B Peng and B Wei ldquoEffects of the trans-mission shaft on the main reducer vibration based onADAMS and experimental demonstrationrdquo AustralianJournal of Mechanical Engineering vol 15 no 1 pp 2ndash102015

[12] C X Zhang H B Li and D Q Jin ldquoSimulation on dynamicsof transmission shaft-rear driving axle of minibus withADAMSrdquo Applied Mechanics and Materials vol 651ndash653pp 862ndash865 2014

[13] J L Xu X Y Su and P Bo ldquoNumerical analysis anddemonstration transmission shaft influence on meshing vi-bration in driving and driven gearsrdquo Shock and Vibrationvol 2015 Article ID 365084 10 pages 2015

[14] J L Xu W Lei andW X Luo ldquoInfluence of bearing stiffnesson the nonlinear dynamics of a shaft-final drive systemrdquoShock and Vibration vol 2016 Article ID 3524609 14 pages2016

[15] A Kahraman and R Singh ldquoNon-linear dynamics of a spurgear pairrdquo Journal of Sound and Vibration vol 142 no 1pp 49ndash75 1990

[16] A Kahraman and R Singh ldquoInteractions between time-varying mesh stiffness and clearance non-linearities in ageared systemrdquo Journal of Sound and Vibration vol 146no 1 pp 135ndash156 1991

[17] A Kahraman ldquoEffect of axial vibrations on the dynamics of ahelical gear pairrdquo Journal of Vibration and Acoustics vol 115no 1 pp 33ndash39 1993

[18] Y Cai and T Hayashi ldquoe linear approximated equation ofvibration of a pair of spur gears (theory and experiment)rdquoJournal of Mechanical Design vol 116 no 2 pp 558ndash5641994

[19] P Velex andMMaatar ldquoAmathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound and Vibration vol 191no 5 pp 629ndash660 1996

[20] P Velex and V Cahouet ldquoExperimental and numerical in-vestigations on the influence of tooth friction in spur andhelical gear dynamicsrdquo Journal of Mechanical Design vol 122no 4 pp 515ndash522 2000

[21] P Velex and P Sainsot ldquoAn analytical study of tooth frictionexcitations in errorless spur and helical gearsrdquo Mechanismand Machine Ieory vol 37 no 7 pp 641ndash658 2002

[22] T C Lim and Y Cheng ldquoA theoretical study of the effect ofpinion offset on the dynamics of hypoid geared rotor systemrdquoJournal of Mechanical Design vol 121 no 4 pp 594ndash6011999

[23] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[24] Y Cheng and T C Lim ldquoDynamics of hypoid gear trans-mission with nonlinear time-varying mesh characteristicsrdquoJournal of Mechanical Design vol 125 no 2 pp 373ndash3822003

[25] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 2017 pp 302ndash329 2007

[26] J Wang and T C Lim ldquoEffect of tooth mesh stiffnessasymmetric nonlinearity for drive and coast sides on hypoidgear dynamicsrdquo Journal of Sound and Vibration vol 319no 3-5 pp 885ndash903 2009

[27] W M Li and H T Wang ldquoRigidity calculation of axialposition preload bearingsrdquo Journal of Hebei University ofTechnology vol 30 no 2 pp 15ndash19 2001

[28] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo Non-linear Dynamics vol 75 no 4 pp 783ndash806 2014

[29] X Liu Z Wu J Lu and J Xu ldquoInvestigation of the effect ofrotation speed on vibration responses of transmission sys-temrdquo Matec Web of Conferences vol 256 no 3 Article ID02019 2019

Shock and Vibration 17

where Iaj(j 1 2) is the equivalent moment of inertia of thejth node of input shaft Ibj(j 1 2 7) is the equivalentmoment of inertia of the jth node of middle shaft Icj(j

1 2 7) is the equivalent moment of inertia of the jthnode of output shaft Igj(j 1 2 15) is the equivalentmoment of inertia of jth gear Id is the equivalent moment ofinertia of the synchronizer ring and im n is the drive ratio ofthe mating gear pair

23 Drive Shaft System Drive shaft system comprises in-termediate drive shaft main drive shaft three universaljoints and intermediate support In this paper the rotationalinertia of shaft is equivalent to the universal joint by thelumped-mass method averagely e lubrication clearancefriction manufacturing and assembly errors and flexibilityof universal joint are not considered

In order to describe the dynamics of a U-joint a set ofparameters are introduced as follows

Ai 1 + cos2αi

2 cos αi

Bi 1 minus cos2αi

2 cos αi

i 1 2 3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

where αi(i 1 2 3) is the intersection angel between shaftslinked by a universal joint (U-joint)

According to kinematics analysis the following equa-tions are derived

θ9 arctantan θ8cos α1

θ9prime θ8prime

A1 minus B1 cos 2θ8

θ9Prime θ8Prime

A1 minus B1 cos 2θ8minus

2B1 sin 2θ8A1 minus B1 cos 2θ8( 1113857

2 θ8prime( 11138572

T2 A1 minus B1 cos 2θ8( 1113857T1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

θ11 arctantan θ10cos α2

θ11prime θ10prime

A2 minus B2 cos 2θ10

θ11Prime θ10Prime

A2 minus B2 cos 2θ10minus

2B2 sin 2θ10A2 minus B2 cos 2θ10( 1113857

2 θ10prime( 11138572

T4 A2 minus B2 cos 2θ10( 1113857T3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

θ13 arctantan θ12cos α3

θ13prime θ12prime

A3 minus B3 cos 2θ12

θ13Prime θ12Prime

A3 minus B3 cos 2θ12minus

2B3 sin 2θ12A3 minus B3 cos 2θ12( 1113857

2 θ12prime( 11138572

T6 A3 minus B3 cos 2θ12( 1113857T5

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

Shock and Vibration 5

where α1 is the intersection angel between output shaft oftransmission and intermediate drive shaft α2 is the inter-section angel between main drive shaft and intermediatedrive shaft α3 is the intersection angel between main driveshaft and input shaft of rear axle

e equivalent moments of inertia for intermediate driveshaft and main drive shaft are calculated by

J9 J10 12i21

Jids

J11 J12 12i21

Jmds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(11)

where i1 is the drive ratio of transmission Jids is the momentof inertia of intermediate drive shaft and Jmds is the momentof inertia of main shaft

Equivalent torsional stiffness k and damping coefficient c

of shaft are given by

k M

ϕ

Gapπ D4 minus d4( 1113857

32L

c 2ζ

kJaJb

Ja + Jb

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(12)

where ζ is the damping ratio and Ja and Jb are the equivalentmoment of inertia of shaft end

24 Rear Axle In this paper the main reducer system issimplified as a geared rotor-bearing system with bending-torsional-lateral-rocking coupled vibration Pinion shaft issimplified as two lumpedmasses linked by a spring-dampingpair with infinite stiffness and the differential assembly ismodeled as a rigid part with a concentratedmass Half-shaftsare described by twomass-spring-damping pairs Otherwisepinion and gear are treated as rigid bodies within torsionalvibration analysis

e equivalent moments of inertia for rear axle arecalculated by

J13 15Jps

i21

J14 45Jps

i21

J15 Jd + 05Jhs

i21i22

J16 J17 05Jhs + Jhs

i21i22

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)

where i1 (i1 1350) is the drive ratio of transmission i2(i2 41) is the drive ratio of the main reducer Jps is themoment of inertia of pinion shaft Jd is the moment of inertia

of differential assembly and Jhs is the moment of inertia ofhalf-shaft

Equivalent stiffness of bearing is expressed as [27]

Kr 7253l08

1 z091 cos20α1 middot F01

a0cos01α1

Ka 29011l081 cos19α1 middot F01

a0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(14)

where Kr is the equivalent radial stiffness of bearing Ka isthe equivalent axial stiffness of bearing l1 is the nominalcontact length z1 is the number of rolling elementα1 is thepressure angle and Fa0 is the pretightening force

In addition a new model for pinion shaft is proposedwhich is different from previous works Owing to negligibleelastic deformation pinion shaft is modeled as rigid bodyBearing is modeled as a spring-damping pair Without lossof generality the derivation procedure on displacements ofthe mounting point in the x-o-y plane is used as an example

In Figure 4 Xpx and Xpy are introduced to representtranslational displacements for centroid of pinion shaft θpz

denotes rocking displacement around the z axisFrom the geometry deformation relationship and pro-

posed concepts the displacement for mounting point iscalculated by

xpy1 xpy minus lp1 tan θpz

xpy2 xpy minus lp2 tan θpz

⎧⎨

⎩ (15)

ereby the following expressions are derived

xpy1prime xpyprime minus lp1

θpzprime

cos2θpz

xpy2prime xpyprime minus lp2

θpzprime

cos2θpz

1113896

(16)

Static transmission error e(t) [22ndash26 28] is simulated inFourier series form by

e(t) em + 1113944N

i1eAi cos iωmt + φe( 1113857 (17)

where em is the average value of transmission error eAi arethe amplitude of the ith order harmonic φe are the initialphase angle of the ith order harmonic and ωm is the meshingfrequency

e dynamic meshing stiffness is the periodically time-varying parameter with respect to the mesh frequency asshown in Figure 5 which can be simulated by means of aFourier expansion [13 14 22ndash26]

km(t) km + 1113944N

i1kAi cos iωmt + φk( 1113857 (18)

where km is the mean value of meshing stiffness kAi is thefluctuation amplitude of ith order meshing stiffness ωm isthe mesh frequency and φk is the initial phase of meshingstiffness

Defining me denotes mean mass of pinion and gear

6 Shock and Vibration

me 1

1mp1113872 1113873 + 1mg1113872 1113873

mpmg

mp + mg

(19)

e following equation can be derived

ωn

km

me

1113971

(20)

where ωn is the equivalent natural frequency of pinion andgear

Equivalent mesh damping coefficient is calculated whichis different from Wang and Limrsquos analysis [25]

cm ξ lowast 2meωn 2ξ

km

1mp1113872 1113873 + 1mg1113872 1113873

1113971

(21)

where ζ is the damping ratio generally set as a range of 003to 017 [25]mp andmg are equivalent mass of pinion and gearrespectively

e relative displacement along the normal direction atthe meshing point xn is given as follows

xn xpx + Eyθpz1113872 1113873 minus xgx + Ezθgy + lg3θgz1113872 11138731113960 1113961

middot cos αn sin βm cos δ1 + sin αn sin δ1( 1113857

+ xpy + lp3θpz1113872 1113873 minus xgy + Ezθgx minus Exθgz1113872 11138731113960 1113961

middot minuscos αn sin βm sin δ1 + sin αn cos δ1( 1113857

+ xpz minus Rpθpx + lp3θpy1113872 1113873 minus xgz minus lg3θgx minus Rgθgy1113872 11138731113960 1113961

middot cos αn cos βm( 1113857 minus e(t)

ε1 xpx minus xgx minus Ezθgy + Eyθpz minus lg3θgz1113872 1113873

+ ε2 xpy minus xgy minus Ezθgx + lp3θpz + Exθgz1113872 1113873

+ ε3 xpz minus xgz minus Rpθpx + lg3θgx + lp3θpy + Rgθgy1113872 1113873

minus e(t)

(22)

e backlash function is expressed as [23]

f xn( 1113857

xn minus b xn ge b

0 minusblexn le b

xn + b xn le minus b

⎧⎪⎪⎨

⎪⎪⎩(23)

where b is half of gear backlashe dynamic meshing force along the line of action Fn

can be calculated byFn km(t)f xn( 1113857 + cmxn

prime (24)

wherecm represents thevalueof equivalentmeshdampingcoefficiente components of dynamic mesh force along the co-

ordinate directions are expressed asFx Fn cos αn sin βm cos δ1 + Fn sin αn sin δ1

Fy minusFn cos αn sin βm sin δ1 + Fn sin αn cos δ1

Fz Fn cos αn cos βm

⎧⎪⎪⎨

⎪⎪⎩(25)

where δ1 is cone angle of the pinion αn is the normal pressureangle and βm is the helix angle at the midpoint of the pinion

In additional the following equations are derived byintroducing the parameter εi

ε1 cos αn sin βm cos δ1 + sin αn sin δ1

ε2 minuscos αn sin βm sin δ1 + sin αn cos δ1

ε3 cos αn cos βm

⎧⎪⎪⎨

⎪⎪⎩(26)

Substituting equations (26) into (25) the expressions ofdynamicmesh force applied on pinion and gear can bewritten as

Fpx minusFgx ε1Fn

Fpy minusFgy ε2Fn

Fpz minusFgz ε3Fn

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(27)

where the subscript p is the label for pinion and the subscriptg is the label for gear

Mesh impact due to teeth separation is neglected in this paper

25 Equations of Motion From the preceding derivationsthe equations of motion considering bending-torsional-lateral-rocking vibration are expressed as

Fpz

kpz2 cpz2

kpx

cpx

θpy

Xpx

kpz1 cpz1

FpxFpy

Op1

cpy2

θpz

mp

kpy2cpy1

Lp1

Lp2 Lp3

Op2 Op0

kpy1y

x

z

Xpy1 Xpy2

Xpy

Op

Op3 Ey

θpz

θpx

Figure 4 e dynamic model of pinion shaft

Shock and Vibration 7

J1θ1Prime + c1 θ1prime minus θ2prime( 1113857 + k1 θ1 minus θ2( 1113857 TD

J2θ2Prime + c1 θ2prime minus θ1prime( 1113857 + k1 θ2 minus θ1( 1113857 + c2 θ2prime minus θ3prime( 1113857 + k2 θ2 minus θ3( 1113857 0

J3θ3Prime + c2 θ3prime minus θ2prime( 1113857 + k2 θ3 minus θ2( 1113857 + c3 θ3prime minus θ4prime( 1113857 + k3 θ3 minus θ4( 1113857 0

J4θ4Prime + c3 θ4prime minus θ3prime( 1113857 + k3 θ4 minus θ3( 1113857 + c4 θ4prime minus θ5prime( 1113857 + k4 θ4 minus θ5( 1113857 0

J5θ5Prime + c4 θ5prime minus θ4prime( 1113857 + k4 θ5 minus θ4( 1113857 + c5 θ5prime minus θ6prime( 1113857 + k5 θ5 minus θ6( 1113857 0

J6θ6Prime + c5 θ6prime minus θ5prime( 1113857 + k5 θ6 minus θ5( 1113857 + c6 θ6prime minus θ7prime( 1113857 + k6 θ6 minus θ7( 1113857 0

J7θ7Prime + c6 θ7prime minus θ6prime( 1113857 + k6 θ7 minus θ6( 1113857 + c7 θ7prime minus θ8prime( 1113857 + k7 θ7 minus θ8( 1113857 0

J8θ8Prime + c7 θ8prime minus θ7prime( 1113857 + k7 θ8 minus θ7( 1113857 minusT1

i1

J9θ9Prime + c8 θ9prime minus θ10prime( 1113857 + k8 θ9 minus θ10( 1113857 T2

i1

J10θ10Prime + c8 θ10prime minus θ9prime( 1113857 + k8 θ10 minus θ9( 1113857 minusT3

i1

J11θ11Prime + c9 θ11prime minus θ12prime( 1113857 + k9 θ11 minus θ12( 1113857 T4

i1

J12θ12Prime + c9 θ12prime minus θ11prime( 1113857 + k9 θ12 minus θ11( 1113857 minusT5

i1

J13θ13Prime + c10 θ13prime minus θ14prime( 1113857 + k10 θ13 minus θ14( 1113857 T6

i1

J14θ14Prime + c10 θ14prime minus θ13prime( 1113857 + k10 θ14 minus θ131113857( 1113857 minusFpzEy

i1

J15θ15Prime + c11 θ15prime minus θ16prime( 1113857 + k11 θ15 minus θ16( 1113857 + c12 θ15prime minus θ17prime( 1113857 + k12 θ15 minus θ17( 1113857 FgzEx + FgxEz

i1i2

J16θ16Prime + c11 θ16prime minus θ15prime( 1113857 + k11 θ16 minus θ15( 1113857 minusTRL

i1i2

J17θ17Prime + c12 θ17prime minus θ15prime( 1113857 + k12 θ17 minus θ15( 1113857 minusTRR

i1i2

mmymPrime + cmyym

prime + kmyym Fmy

mmzmPrime + cmzzm

prime + kmzzm Fmz

mpxpxPrime + kpxxpx + cpxxpx

prime minusFpx

mpxpyPrime + kpy1 xpy minus lp1 tan θpz1113872 1113873 + cpy1 xpy

prime minus lp1θpzprime

cos2θpz

1113888 1113889 + kpy2 xpy minus lp2 tan θpz1113872 1113873 + cpy2 xpyprime minus lp2

θpzprime

cos2θpz

1113888 1113889 Fpy

mpxpzPrime + kpz1 xpz minus lp1 tan θpy1113872 1113873 + cpz1 xpz

prime minus lp1θpyprime

cos2θpy

1113888 1113889 + kpz2 xpz minus lp2 tan θpy1113872 1113873 + cpz2 xpzprime minus lp2

θpyprime

cos2θpy

1113888 1113889 Fpz

JpyθpyPrime minus lp1kpz1 xpz minus lp1 tan θpy1113872 1113873 minus lp1cpz1 xpz

prime minus lp1θpyprime

cos2θpy

1113888 1113889 minus lp2kpz2 xpz minus lp2 tan θpy1113872 1113873 minus lp2cpz2 xpzprime minus lp2

θpyprime

cos2θpy

1113888 1113889 Fpzlp3

JpzθpzPrime minus lp1kpz1 xpy minus lp1 tan θpy1113872 1113873 minus lp1cp1 xpy

prime minus lp1θpzprime

cos2θpz

1113888 1113889 minus lp2kpy2 xpy minus lp2 tan θpz1113872 1113873 minus lp2cpy2 xpyprime minus lp2

θpzprime

cos2θpz

1113888 1113889 minusFpxEy + Fpylp3

mgxgxPrime + kgx1 xgx minus lg1 tan θgz1113872 1113873 + cgx1 xgx

Prime minus lg1θgzprime

cos2θgz

1113888 1113889 + kgx2 xgx + lg2 tan θgz1113872 1113873 + cgx2 xgxprime + lg2

θgzprime

cos2θgz

1113888 1113889 Fgx

mgxgyPrime + kgyxgy + cgyxgy

Prime minusFgy

mgxgzPrime + kgz1 xgz + lg1 tan θgx1113872 1113873 + cgz1 xgz

prime + lg1θgxprime

cos2θgx

1113888 1113889 + kgz2 xgz minus lg2 tan θgx1113872 1113873 + cgz2 xgzprime minus lg2

θgxprime

cos2θgx

1113888 1113889 minusFgz

JgxθgxPrime + lg1kgz1 xgz + lg1 tan θgx1113872 1113873 + lg1cgz1 xgz

prime + lg1θgxprime

cos2θgx

1113888 1113889 minus lg2kgz2 xgz minus lg2 tan θgx1113872 1113873 + lg2cgz2 xgzprime minus lg2

θgxprime

cos2θgx

1113888 1113889 Fgzlg3 minus FgyEz

JgzθgzPrime + lg1kgx1 xgx + lg1 tan θgz1113872 1113873 + lg1cgx1 xgx

prime + lg1θgzprime

cos2θgz

1113888 1113889 minus lg2kgx2 xgx minus lg2 tan θgz1113872 1113873 + lg2cgx2 xgxprime minus lg2

θgzprime

cos2θgz

1113888 1113889 Fgxlg3 minus FgyEx

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

8 Shock and Vibration

where TD is the input torque of motor T1 is the outputtorque of transmission T2 andT3 are the input and outputtorque for intermediate drive shaft T4 andT5 are the inputand output torque for main drive shaft T6 is the inputtorque of rear axle ki(i 1 2 7) are the equivalenttorsional stiffness ci(i 1 2 7) are the equivalent tor-sional damping ki(i 8 9) are the equivalent torsionalstiffness ci(i 8 9) are the equivalent torsional dampingmm is the mass of transmission shaft ki(i 10 11 12) arethe equivalent torsional stiffness for pinion shaft and dif-ferential axle ci(i 10 11 12) are the equivalent torsionaldamping for pinion shaft and differential axle cmy and cmz

are the equivalent damping of intermediate support alongaxes y and z kmy and kmz are the equivalent stiffness ofintermediate support along axes y and z Fmy and Fmz arecomponents of additional force Fi along axes y and z forintermediate support Ei(i x y z) are coordinates ofmeshing point along axes x y and z and TRL andTRR areload torque for wheels

26 Numerical Solution Procedure e fourth-order adap-tive step RungendashKutta method which is generally applicableto strong nonlinearity is adopted in the present work to solve

the equations In order to program a new state space isneeded

In consideration of equations (12)mdash(14) it can be de-rived that kinetic equations of 3 universal joints reduce thenumber of degree-of-freedom from 29 to 26 for the wholesystem e following state space coordinate vector alsodescribes the dynamic behaviors of the system

Ss θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 θ10 θ12 θ14 θ15 θ16 θ17θpy θpz θgx θgz ym zm xp yp zp xg yg zg

⎧⎨

⎩

⎫⎬

⎭

Τ

(29)

In order to eliminate the difference in magnitude be-tween parameters which will reduce the solvability and evenlead to no convergence for solving equations of motionmentioned above b is defined as the characteristic lengthand a new time parameter τ is introduced as follows

τ tωn (30)

where t denotes timeFor the sake of nominalizing equations a generalized

state space coordinate vector with 26 parameters is proposedto characterize dimensionless equations as

st sr1 sr2 sr3 sr4 sr5 sr6 sr7 sr8 sr10 sr12 sr16 sr17 srpx srpy srpz srgx srgy srgz smy smz spx spy spz sgx sgy sgz1113966 1113967T

(31)

e following equations are derived as

sri θi timesRi

b i 1 sim 8 10 12 16 17

srpx θ14 timesRpx

b

srgy θ15 timesRgy

b

srij θij timesRij

b i p g j x y z

six xi

b i p g

siy yi

b i m p g

siz zi

b i m p g

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

where b is the half of backlash and Ri is the equivalent gyrationradius which can be calculated by the following equation

Ri

Ji

mi

1113971

i 1 sim 8 10 12 16 17

Rij

Jij

mi

1113971

i p g j x y z

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(33)

where Ji and Jij are the equivalent moment of inertia and mi

is equivalent massA set of dimensionless parameters are defined as

xnτ xn

b

f xnτ( 1113857 f xt

n( 1113857

b

c(ijs) ci

mjRjRsωn

k(ijs) ki

mjRjRsω2n

cmi cm

miωn

kmi km

miω2n

T(xj) Tx

mjRjω2nb

lijτ lij

b

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)

As equations (30) (32) and (34) are substituted intoequation (28) the dimensionless equations of motion can bederived

Shock and Vibration 9

e equivalent inertia parameters are obtained in UGand stiffness parameters of powertrain are calculated byusing the relationships given in modeling section as listed inTable 1 Subsequently equivalent damping coefficients canbe obtained e parameters of the main reducer can becalculated by substituting system parameters listed in Table 2into equations (17)ndash(31) e solution procedures for di-mensionless parameter are carried out by using precedingcalculated parameters and the relationships proposed inprevious parts

e exciting and configuration parameters are set asθ1 1500 rpm Td2 100Nmiddotm km 9697e8Nm i1 135i2 41 2b 015mm em 0 α1 1695 degree α2 13044degree and α3 1554 degree respectively Without signif-icant loss of accuracy the fundamental harmonic form isadopted for time-varying mesh stiffness and static trans-mission error function whose initial phases are set at 0respectively

3 Numerical Results

31 Analysis of Coupled Torsional Vibration In order toanalysis coupled interaction of automotive powertrainFourier spectrums of torsional vibration response are cal-culated at the initial parameters as shown in Figure 6 Twopeaks and their sidebands are observed e torsional vi-brational responses are governed by the component of 50Hzwhich equals with exciting frequency of input torque asshown in equation (3) (ωf 150060 lowast 2 50Hz) isimplies that engine input excitation is responsible for tor-sional vibration and a valuable approach to decreasingtorsional vibration of powertrain is inferred In addition thecomponent with a frequency of around 185Hz is observedobviously for each considered part As one knows themeshing frequency undergoing this condition is presumably185185Hz (f 150060135185185Hz) which is quite inagreement with the frequency of the second peak Coupledvibration occurs between the torsional system and hypoidgear set and inner excitation caused by gear pair meshingprocess which is affected by clearance and time-varyingmeshing stiffness [25] accounts a lot for torsional vibrationat 185Hz of powertrain and affects torsional responsesslightly After simply calculation bandwidths reaching thesame value of around 37Hz of sidebands around two peaksare got U-joint is a transmission mechanism with non-constant angular speed which will excite a doubled fre-quency torsional vibration of input as shown in equations(12)ndash(14) Under this condition the doubled frequencyexcited by U-joint is calculated

fuminusjoint n

i1 lowast 60lowast 2

1500135lowast 60

lowast 2 3704 (35)

It can be conjectured that modulation phenomenonappears when automotive powertrain works between exci-tations attributed to input U-joint and gear pair As one cansee in Figure 6 compared torsional response of gear withthat located front of meshing point a lower value of tor-sional angular is observed and it is inferred that gear meshexcitations reduce torsional vibration for components

located behind meshing point It is noted that taking gearmesh behaviors into account is quite necessary to when atorsional characteristic analysis is conducted

32 Amplitude-Frequency Response of Torsional Vibration forAutomotive Powertrain e previous works [29] analyzedtorsional characteristics of the coupled system consisted ofdrive shaft and rear axle which denoted that resonance ofdrive shaft at the first two-order nature frequency accountsfor dramatic torsional vibration responses undergoing high-rotation-speed condition However it cannot reveal thetorsional vibration mechanism of automotive under low-and middle-speed condition

In order to obtain a more realistic understanding foramplitude-frequency response of automotive powertrainundergoing middle-speed condition theoretically the pre-ceding equations of motion are adopted A sweep frequencyanalysis procedure is carried out in a range from 1000 rpm to3500 rpm which respectively corresponds to frequencyinternal of 166ndash583Hz for input speed 3333ndash1167Hz forinput excitation and 12346ndash43209Hz for meshing exci-tation e response at input end of rear axle is taken as anexample in this work

Figure 7 shows the amplitude-frequency response inwhich abscissa denotes torsion speed of input and ordinate isRMS value of angular displacement As one can see threeobvious peaks which correspond to input speed at about1000 rpm 1600 rpm and 2800 rpm respectively are ob-served Owing to the assumption that the dramatic responsesare attributed to resonance of the system the first three-order nature frequency respect to the second-order term ofinput torque fluctuation is proposed at 333Hz 533Hz and933Hz successively It is noted that on the basis of previousanalysis these three peaks also correspond to frequencies of247Hz 395Hz and 691Hz for the U-joint excitation and1234Hz 1975Hz and 3456Hz for the meshing excitationPreliminary works [29] showed that resonance is attributedto input excitation of drive shaft and the meshing excitationof hypoid gear pair with higher exciting frequency thaninput will not excite prominent resonance behaviors of driveshaft as illustrated in Figure 8 ie meshing excitation hasslight influence on the torsional system which reaches thesame conclusion with this work According to analysis inSection 31 the sidebands respect to each exciting frequencyare (86ndash58) Hz at 333Hz (138ndash928) Hz at 533Hz and(242ndash1624) Hz at 933Hz for input torque and (987ndash1481)Hz at 1234Hz (158ndash237) Hz at 1975Hz and (2765ndash4147)Hz at 3456Hz for meshing behavior of the gear pair re-spectively As listed in Table 3 the first peak can be attributedto the first two-order resonances of drive shaft excited bysideband around 1234Hz Sidebands at 533Hz and1975Hz result in the second-order resonance which gen-erates the second peak Moreover the second-order reso-nances excited by the sideband at 933Hz and third-orderone caused by sideband at 3456Hz explain the appearanceof the third peak A typical phenomenon is observed that thetorsional vibration reduces a lot while one compares theamplitude of the third peak with another two which

10 Shock and Vibration

Table 1 e equivalent parameters of powertrain

Equivalent mass (kg) Equivalent moment of inertia (kgmiddotm2) Equivalent stiffness (Nm)m1 2 J1 0091125 k1 1002375m2 10 J2 03801735 k2 164025m3 06150 J3 000008103 k3 27500m4 13798 J4 0001001 k4 10e11m5 23581 J5 0001323 k5 479000m6 114323 J6 000249 k6 10e11m7 45323 J7 0001865 k7 21400m8 75567 J8 0002098 mdashm9 1284 J9 0001695 k8 43921799m10 1284 J10 0001695 mdashm11 13125 J11 000173 k9 41933635m12 13125 J12 000173 mdashm13 04254 J13 000002772 k10 10e11m14 17016 J14 00001109 km 96965e8m15 1375 J15 0034177 mdashm16 845 J16 99 k11 12032114m17 845 J17 99 k12 12032114

Table 2 e system parameters of the main reducer

Parameter Pinion GearNumber of teeth 10 41Module (mm) 400Mean pressure angle (deg) 20 20Helix angle at midpoint of the pinion (deg) 50 2713Cone angle (deg) 2312 6514Tooth width (mm) 4066 3452Tooth addendum (mm) 359 644Pinion outset (mm) minus38Mounting distance (mm) 112 64Pitch radius (mm) 40 108Backlash (mm) 01536Distance from pinion reference point to the x-o-z plane for pinion (mm) 40Distance from pinion reference point to the y-o-z plane for gear (mm) 101Distance between projection point and centroid of pinion shaft (mm) lp3 10 lg3 5

Distance between bearing reference point and centroid of pinion shaft (mm) lp1 100 lg1 90lp2 175 lg2 93

Equivalent mass (kg) mg 2127 mp 1375Jpx J14 Jgx 0050315

Equivalent moment of inertia (kgmiddotm2) Jpy 0002982 Jgx J15Jpz 0002982 Jpz 0037551

Meshing stiffnessof (i + 1)th tooth pair

Meshing stiffnessof ith tooth pair

t

k

Meshing stiffnessof (i ndash 1)th tooth pair

Total meshingstiffness km(t)

Figure 5 Schematic diagram of time-varying meshing stiffness on hypoid gear

Shock and Vibration 11

indicates that torsional behaviors of automotive powertrainaffect the vibration characteristics of the complete vehicleunder a low- or middle-input-speed condition In order tovalidate the preceding result an experimental study isnecessary

4 Experimental Analysis

A test bench which can be used to test the vibration behavior ofa transmission system used for four-wheel drive vehicle be-longing to the China Automotive Technology and ResearchCenter is adopted to perform this experiment ere are fivemotors in the test bench and three of them are used in thisexperiment One of them drives automotive powertrain whichcan generate a high frequency input torque with fluctuationvaried in accordance with necessary and others are used asloads In this test two photoelectric sensors and two magne-toelectric sensors are used to detect torsional vibration signalsand other five accelerometers are mounted to monitor trans-verse vibration Installation site for each sensor can be found inFigure 9 Signals are sampled and processed by LMS TestLab

In order to validate the feasibility of the theoreticalmodel the input torque simulated by input motor and loadsare consistent with parameters employed in numericalanalysis (mean value and the second-order term are con-tained only and fluctuation amplitude is set at 100NmiddotM forinput torque) Within the experiment the output speed ofmotor varies from 1000 rpm to 3500 rpm to conduct a sweepfrequency response analysis experimentally

As illustrated in Figure 10 the amplitude-frequencyresponses of torsional vibration have the similar feature for

different parts of powertrain Peaks can be observed forcarves at a similar input speed It is inferred that a presumeresonance occurs at each peak As a result the first three-order nature torsional frequencies can be predictedMeanwhile the same results are reached As shown inFigure 10 each curve has a higher value nearby the first andsecond peak With the further increase in frequency itreaches a lower plateau value Torsional behaviors have aneffect on the vibration characteristics under a low-input-speed condition Furthermore the part located behind themesh point such as output end of half-shaft has a lowerRMS value than transmission and drive shaft which can beattributed to reduction effect of mesh excitation on torsionalvibration

e experimental result of input end of rear axle iscompared with the numerical one As shown in Figure 11 aquite consistency is observed and the first three-order fre-quencies are inferred at 35Hz (1050 rpm) 557Hz(1670 rpm) and 883Hz (2650 rpm) for experimental resultby the same way mentioned in preceding section Comparedwith numerical predictions an acceptable difference ofabout 5 (the difference is 51 at 333Hz 45 at 533Hzand 536 at 933Hz) can be obtained ese indicate thatthe theoretical model is feasible for predicting nature tor-sional frequencies Moreover numerical torsional dis-placement is a little lower than the experimental one at thesame input speed which is below about 2450 rpm and a littlehigher with the further increase is can be presumablyattributed to assumptions made in this present work withrespect to the complexity of the system and the limitations ofthe lumped-mass method

03503

02502

01501

0050

0

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(a)

13Hz

50Hz

87Hz 150Hz185Hz222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(b)

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(c)

13Hz

50Hz

87Hz150Hz

185Hz222Hz

004

003

002

001

00 50 100 150 200 250

Frequency (Hz)To

rsio

nal d

ispla

cem

ent (

deg)300 350 400 450 500 550 600

(d)

Figure 6 e Fourier spectrum of torsional responses (a) output shaft of transmission (b) output end of the drive shaft system (c)torsional displacement of the pinion along the axis x (d) torsional displacement of the gear along the axis y

12 Shock and Vibration

e previous section indicates numerically that excita-tion of input dominates torsion vibration of the system andgear set affects responses slightly In order to validate thecorrectness of numerical analysis and feasibility decreasing

torsional vibration by reducing input excitation effect ofinput torque fluctuation on torsional response is studiedexperimentally e mean value of input torque is set at150NmiddotM and fluctuation of the second-order term is set at

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

2800rpm1600rpm

Figure 7 Amplitude-frequency response at input end of rear axle

8

7

6

5

4

2

3

1

00 1000 2000 3000

Rotation speed (RPM)

Vibr

atio

n am

plitu

de (micro

m)

4000 5000 6000

x directiony directionz direction

Figure 8 Influence of input rotation speed of drive shaft on vibration of the main reducer [29]

Table 3 e first six-order nature frequency of drive shaft [29]

Order 1 2 3 4 5 6Frequency (Hz) 1309 1402 4152 4422 6362 9066

Shock and Vibration 13

25NmiddotM 50NmiddotM 75NmiddotM 100NmiddotM 125NmiddotM 150NmiddotM175NmiddotM and 200NmiddotM in order

Together Figures 12 and 13 expectably show that thetorsional vibration is reinforced while increasing torquefluctuation Meanwhile an obvious increase in amplitudeoccurs at the second peak when fluctuation is increased from175NmiddotM to 200NmiddotM which indicates that a limitation should

be considered e accurate value can be determined by afurther study In summary reducing torque fluctuation is abeneficial approach for vibration damping Furthermore it isnoted that such a nonlinear phenomenon as natural fre-quency shift occurs while changing torque fluctuation Takingthe second-order torsional frequency as an example thefrequency value decreases with elevating fluctuation of input

Transmission

Photoelectricsensor

Photoelectricsensor

Magnetoelectricsensor

Magnetoelectricsensor

Rear axle

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Drive sha system

Figure 9 e experiment bench and arrangement of sensors

042

040

038

035

033

030

028

025

023

020

018

015

013

010

008

005

003

Amplitude-frequency curve ofle wheel

Amplitude-frequency curve oftransmission output sha

Amplitude-frequency curve ofrear axle output end

Amplitude-frequency curve ofright wheel

000

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (K

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Curve 167000

041

035

003

003

5567 8817

167000 264500

rpm

deg

deg

deg

deg

264500

012

005

003

764endash3

Figure 10 Amplitude-frequency responses of torsional vibration for powertrain at a torque fluctuation of 100NmiddotM

14 Shock and Vibration

036

034

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

Numerical result

Experimental result

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

Figure 11 Comparison of numerical and experimental results for input end of rear axle

160 5567 8817

264500

150

140

130

120

110

100

090

080

070

060

050

040

030

020

010

000 167000

25Nmiddotm

50Nmiddotm

75Nmiddotm

100Nmiddotm

125Nmiddotm150Nmiddotm

175Nmiddotm

200Nmiddotm

Curve 167000147078057045035025017013

rpm--------

264500009008007006005004004003

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (R

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Figure 12 Amplitude-frequency responses of torsional vibration for input end of rear axle at different torque fluctuations

Shock and Vibration 15

torque which shows that the torsional system has a stiffness-softening characteristic with respect to torque fluctuationis result can be accounted for a number of nonlinearitiessuch as friction clearance and sliding and the existence ofwhich sophisticates system nonlinear characteristics

5 Conclusion

In the present paper a lumped-parameter dynamic modelwith 29 degree-of-freedom is proposed considering thetransverse torsional and rocking coupled vibration for abetter understanding of the automotive powertrain systemdynamics and equations of motion are derived e nu-merical result shows that gear mesh excitations have littleeffects on torsional response for such components locatedbefore the mesh point as transmission transmission shaftand pinion but significantly reduce the torsional responsesfor ones behind the mesh point e amplitude-frequencycharacteristics of torsional vibration graphically indicate thattorsional behaviors of automotive powertrain only affect thevibration characteristics of complete vehicle under low-speed condition and predict the first three-order naturetorsional frequencies of the whole system An experimentalexamination is performed and validates the feasibility of theconsidered model e torsional vibration is reinforcedwhile increasing torque fluctuation In addition a stiffnesssoftening characteristic is observed for torsional vibrationexperimentally In order to reach a better understanding inamplitude-frequency response at low- and middle-input-speed condition a procedure to analyze mode shapes ofautomotive powertrain will be conducted in future works

Data Availability

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

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported and funded by the ApplicationTechnology Research and Development Plan of LiuzhouCity (grant no 2017AA10102) and the Key Projects ofUniversities in Inner Mongolia Autonomous Region (grantno NJZZ16369)e authors of this paper express their deepgratitude for the financial support of the research project

References

[1] F Vesali M A Rezvani and M Kashfi ldquoDynamics of uni-versal joints its failures and some propositions for practicallyimproving its performance and life expectancyrdquo Journal ofMechanical Science and Technology vol 26 no 8 pp 2439ndash2449 2012

[2] J-W Lu G-C Wang H Chen A F Vakakis andL A Bergman ldquoDynamic analysis of cross shaft type universaljoint with clearancerdquo Journal of Mechanical Science andTechnology vol 27 no 11 pp 3201ndash3205 2013

[3] G Wu W Shi and Z Chen ldquoe effect of multi-universalcoupling phase on torsional vibration of drive shaft and vi-bration of vehiclerdquo in Proceedings of the SAE 2013 WorldCongress amp Exhibition Detroit MI USA April 2013

0113333 4000 4500 5000 5500

5567

6000 6500 7000Frequency (Hz)

7500 8000 85008777

263300

9000 9500 10000 10500 11000 11667100

Am

plitu

de

Am

plitu

de (R

MS)

010

009

008

007

006

005

004

003

002

001

000100000 120000 140000 160000 180000

167000

25Nmiddotm

50Nmiddotm

75Nmiddotm100Nmiddotm

125Nmiddotm

150Nmiddotm

175Nmiddotm200Nmiddotm

Curve 167000010006004003003002001

982endash3

263300006005004003003002001

690endash3

rpm--------

200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000000

Figure 13 Amplitude-frequency responses of torsional vibration for output end of rear axle at different torque fluctuations

16 Shock and Vibration

[4] B A Porter ldquoeoretical analysis of the torsional oscillationof a system incorporating a Hookersquos jointrdquo Journal of Me-chanical Engineering Science vol 3 no 4 pp 324ndash329 1961

[5] S F Asokanthan and X-H Wang ldquoCharacterization oftorsional instabilities in a Hookersquos joint driven system viamaximal Lyapunov exponentsrdquo Journal of Sound and Vi-bration vol 194 no 1 pp 83ndash91 1996

[6] A Farshidianfar M Ebrahimi and H Bartlett ldquoHybridmodelling and simulation of the torsional vibration of vehicledriveline systemsrdquo Proceedings of the Institution of Me-chanical Engineers Part D Journal of Automobile Engineeringvol 215 no 2 pp 217ndash229 2001

[7] E O A Abd Elmaksoud E M Rabeih N A Abdel-halimand S M El Demerdash ldquoInvestigation of self excited tor-sional vibrations of different configurations of automatictransmission systems during engagementrdquo Engineeringvol 3 no 12 pp 1171ndash1181 2011

[8] T B Juang M Cheng and L Na ldquoExperimental and finiteelement analyses of a sliding-tube-type driveshaft-inducedvehicle vibrationrdquo Proceedings of the Institution of MechanicalEngineers Part K Journal of Multi-Body Dynamics vol 221no 3 pp 375ndash385 2007

[9] L F Coutinho and A Tamagna ldquoFlexural vibration analysisof constant velocity half-shaftsrdquo in Proceedings of the Inter-national Congress amp Exposition Chicago IL ISA 1996

[10] Y T Kang Y H Shen W Zhang and J Yang ldquoStabilityregion of floating intermediate support in a shaft system withmultiple universal jointsrdquo Journal of Mechanical Science andTechnology vol 28 no 6 pp 2733ndash2742 2014

[11] J L Xu T Yan B Peng and B Wei ldquoEffects of the trans-mission shaft on the main reducer vibration based onADAMS and experimental demonstrationrdquo AustralianJournal of Mechanical Engineering vol 15 no 1 pp 2ndash102015

[12] C X Zhang H B Li and D Q Jin ldquoSimulation on dynamicsof transmission shaft-rear driving axle of minibus withADAMSrdquo Applied Mechanics and Materials vol 651ndash653pp 862ndash865 2014

[13] J L Xu X Y Su and P Bo ldquoNumerical analysis anddemonstration transmission shaft influence on meshing vi-bration in driving and driven gearsrdquo Shock and Vibrationvol 2015 Article ID 365084 10 pages 2015

[14] J L Xu W Lei andW X Luo ldquoInfluence of bearing stiffnesson the nonlinear dynamics of a shaft-final drive systemrdquoShock and Vibration vol 2016 Article ID 3524609 14 pages2016

[15] A Kahraman and R Singh ldquoNon-linear dynamics of a spurgear pairrdquo Journal of Sound and Vibration vol 142 no 1pp 49ndash75 1990

[16] A Kahraman and R Singh ldquoInteractions between time-varying mesh stiffness and clearance non-linearities in ageared systemrdquo Journal of Sound and Vibration vol 146no 1 pp 135ndash156 1991

[17] A Kahraman ldquoEffect of axial vibrations on the dynamics of ahelical gear pairrdquo Journal of Vibration and Acoustics vol 115no 1 pp 33ndash39 1993

[18] Y Cai and T Hayashi ldquoe linear approximated equation ofvibration of a pair of spur gears (theory and experiment)rdquoJournal of Mechanical Design vol 116 no 2 pp 558ndash5641994

[19] P Velex andMMaatar ldquoAmathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound and Vibration vol 191no 5 pp 629ndash660 1996

[20] P Velex and V Cahouet ldquoExperimental and numerical in-vestigations on the influence of tooth friction in spur andhelical gear dynamicsrdquo Journal of Mechanical Design vol 122no 4 pp 515ndash522 2000

[21] P Velex and P Sainsot ldquoAn analytical study of tooth frictionexcitations in errorless spur and helical gearsrdquo Mechanismand Machine Ieory vol 37 no 7 pp 641ndash658 2002

[22] T C Lim and Y Cheng ldquoA theoretical study of the effect ofpinion offset on the dynamics of hypoid geared rotor systemrdquoJournal of Mechanical Design vol 121 no 4 pp 594ndash6011999

[23] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[24] Y Cheng and T C Lim ldquoDynamics of hypoid gear trans-mission with nonlinear time-varying mesh characteristicsrdquoJournal of Mechanical Design vol 125 no 2 pp 373ndash3822003

[25] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 2017 pp 302ndash329 2007

[26] J Wang and T C Lim ldquoEffect of tooth mesh stiffnessasymmetric nonlinearity for drive and coast sides on hypoidgear dynamicsrdquo Journal of Sound and Vibration vol 319no 3-5 pp 885ndash903 2009

[27] W M Li and H T Wang ldquoRigidity calculation of axialposition preload bearingsrdquo Journal of Hebei University ofTechnology vol 30 no 2 pp 15ndash19 2001

[28] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo Non-linear Dynamics vol 75 no 4 pp 783ndash806 2014

[29] X Liu Z Wu J Lu and J Xu ldquoInvestigation of the effect ofrotation speed on vibration responses of transmission sys-temrdquo Matec Web of Conferences vol 256 no 3 Article ID02019 2019

Shock and Vibration 17

where α1 is the intersection angel between output shaft oftransmission and intermediate drive shaft α2 is the inter-section angel between main drive shaft and intermediatedrive shaft α3 is the intersection angel between main driveshaft and input shaft of rear axle

e equivalent moments of inertia for intermediate driveshaft and main drive shaft are calculated by

J9 J10 12i21

Jids

J11 J12 12i21

Jmds

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(11)

where i1 is the drive ratio of transmission Jids is the momentof inertia of intermediate drive shaft and Jmds is the momentof inertia of main shaft

Equivalent torsional stiffness k and damping coefficient c

of shaft are given by

k M

ϕ

Gapπ D4 minus d4( 1113857

32L

c 2ζ

kJaJb

Ja + Jb

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(12)

where ζ is the damping ratio and Ja and Jb are the equivalentmoment of inertia of shaft end

24 Rear Axle In this paper the main reducer system issimplified as a geared rotor-bearing system with bending-torsional-lateral-rocking coupled vibration Pinion shaft issimplified as two lumpedmasses linked by a spring-dampingpair with infinite stiffness and the differential assembly ismodeled as a rigid part with a concentratedmass Half-shaftsare described by twomass-spring-damping pairs Otherwisepinion and gear are treated as rigid bodies within torsionalvibration analysis

e equivalent moments of inertia for rear axle arecalculated by

J13 15Jps

i21

J14 45Jps

i21

J15 Jd + 05Jhs

i21i22

J16 J17 05Jhs + Jhs

i21i22

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)

where i1 (i1 1350) is the drive ratio of transmission i2(i2 41) is the drive ratio of the main reducer Jps is themoment of inertia of pinion shaft Jd is the moment of inertia

of differential assembly and Jhs is the moment of inertia ofhalf-shaft

Equivalent stiffness of bearing is expressed as [27]

Kr 7253l08

1 z091 cos20α1 middot F01

a0cos01α1

Ka 29011l081 cos19α1 middot F01

a0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(14)

where Kr is the equivalent radial stiffness of bearing Ka isthe equivalent axial stiffness of bearing l1 is the nominalcontact length z1 is the number of rolling elementα1 is thepressure angle and Fa0 is the pretightening force

In addition a new model for pinion shaft is proposedwhich is different from previous works Owing to negligibleelastic deformation pinion shaft is modeled as rigid bodyBearing is modeled as a spring-damping pair Without lossof generality the derivation procedure on displacements ofthe mounting point in the x-o-y plane is used as an example

In Figure 4 Xpx and Xpy are introduced to representtranslational displacements for centroid of pinion shaft θpz

denotes rocking displacement around the z axisFrom the geometry deformation relationship and pro-

posed concepts the displacement for mounting point iscalculated by

xpy1 xpy minus lp1 tan θpz

xpy2 xpy minus lp2 tan θpz

⎧⎨

⎩ (15)

ereby the following expressions are derived

xpy1prime xpyprime minus lp1

θpzprime

cos2θpz

xpy2prime xpyprime minus lp2

θpzprime

cos2θpz

1113896

(16)

Static transmission error e(t) [22ndash26 28] is simulated inFourier series form by

e(t) em + 1113944N

i1eAi cos iωmt + φe( 1113857 (17)

where em is the average value of transmission error eAi arethe amplitude of the ith order harmonic φe are the initialphase angle of the ith order harmonic and ωm is the meshingfrequency

e dynamic meshing stiffness is the periodically time-varying parameter with respect to the mesh frequency asshown in Figure 5 which can be simulated by means of aFourier expansion [13 14 22ndash26]

km(t) km + 1113944N

i1kAi cos iωmt + φk( 1113857 (18)

where km is the mean value of meshing stiffness kAi is thefluctuation amplitude of ith order meshing stiffness ωm isthe mesh frequency and φk is the initial phase of meshingstiffness

Defining me denotes mean mass of pinion and gear

6 Shock and Vibration

me 1

1mp1113872 1113873 + 1mg1113872 1113873

mpmg

mp + mg

(19)

e following equation can be derived

ωn

km

me

1113971

(20)

where ωn is the equivalent natural frequency of pinion andgear

Equivalent mesh damping coefficient is calculated whichis different from Wang and Limrsquos analysis [25]

cm ξ lowast 2meωn 2ξ

km

1mp1113872 1113873 + 1mg1113872 1113873

1113971

(21)

where ζ is the damping ratio generally set as a range of 003to 017 [25]mp andmg are equivalent mass of pinion and gearrespectively

e relative displacement along the normal direction atthe meshing point xn is given as follows

xn xpx + Eyθpz1113872 1113873 minus xgx + Ezθgy + lg3θgz1113872 11138731113960 1113961

middot cos αn sin βm cos δ1 + sin αn sin δ1( 1113857

+ xpy + lp3θpz1113872 1113873 minus xgy + Ezθgx minus Exθgz1113872 11138731113960 1113961

middot minuscos αn sin βm sin δ1 + sin αn cos δ1( 1113857

+ xpz minus Rpθpx + lp3θpy1113872 1113873 minus xgz minus lg3θgx minus Rgθgy1113872 11138731113960 1113961

middot cos αn cos βm( 1113857 minus e(t)

ε1 xpx minus xgx minus Ezθgy + Eyθpz minus lg3θgz1113872 1113873

+ ε2 xpy minus xgy minus Ezθgx + lp3θpz + Exθgz1113872 1113873

+ ε3 xpz minus xgz minus Rpθpx + lg3θgx + lp3θpy + Rgθgy1113872 1113873

minus e(t)

(22)

e backlash function is expressed as [23]

f xn( 1113857

xn minus b xn ge b

0 minusblexn le b

xn + b xn le minus b

⎧⎪⎪⎨

⎪⎪⎩(23)

where b is half of gear backlashe dynamic meshing force along the line of action Fn

can be calculated byFn km(t)f xn( 1113857 + cmxn

prime (24)

wherecm represents thevalueof equivalentmeshdampingcoefficiente components of dynamic mesh force along the co-

ordinate directions are expressed asFx Fn cos αn sin βm cos δ1 + Fn sin αn sin δ1

Fy minusFn cos αn sin βm sin δ1 + Fn sin αn cos δ1

Fz Fn cos αn cos βm

⎧⎪⎪⎨

⎪⎪⎩(25)

where δ1 is cone angle of the pinion αn is the normal pressureangle and βm is the helix angle at the midpoint of the pinion

In additional the following equations are derived byintroducing the parameter εi

ε1 cos αn sin βm cos δ1 + sin αn sin δ1

ε2 minuscos αn sin βm sin δ1 + sin αn cos δ1

ε3 cos αn cos βm

⎧⎪⎪⎨

⎪⎪⎩(26)

Substituting equations (26) into (25) the expressions ofdynamicmesh force applied on pinion and gear can bewritten as

Fpx minusFgx ε1Fn

Fpy minusFgy ε2Fn

Fpz minusFgz ε3Fn

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(27)

where the subscript p is the label for pinion and the subscriptg is the label for gear

Mesh impact due to teeth separation is neglected in this paper

25 Equations of Motion From the preceding derivationsthe equations of motion considering bending-torsional-lateral-rocking vibration are expressed as

Fpz

kpz2 cpz2

kpx

cpx

θpy

Xpx

kpz1 cpz1

FpxFpy

Op1

cpy2

θpz

mp

kpy2cpy1

Lp1

Lp2 Lp3

Op2 Op0

kpy1y

x

z

Xpy1 Xpy2

Xpy

Op

Op3 Ey

θpz

θpx

Figure 4 e dynamic model of pinion shaft

Shock and Vibration 7

J1θ1Prime + c1 θ1prime minus θ2prime( 1113857 + k1 θ1 minus θ2( 1113857 TD

J2θ2Prime + c1 θ2prime minus θ1prime( 1113857 + k1 θ2 minus θ1( 1113857 + c2 θ2prime minus θ3prime( 1113857 + k2 θ2 minus θ3( 1113857 0

J3θ3Prime + c2 θ3prime minus θ2prime( 1113857 + k2 θ3 minus θ2( 1113857 + c3 θ3prime minus θ4prime( 1113857 + k3 θ3 minus θ4( 1113857 0

J4θ4Prime + c3 θ4prime minus θ3prime( 1113857 + k3 θ4 minus θ3( 1113857 + c4 θ4prime minus θ5prime( 1113857 + k4 θ4 minus θ5( 1113857 0

J5θ5Prime + c4 θ5prime minus θ4prime( 1113857 + k4 θ5 minus θ4( 1113857 + c5 θ5prime minus θ6prime( 1113857 + k5 θ5 minus θ6( 1113857 0

J6θ6Prime + c5 θ6prime minus θ5prime( 1113857 + k5 θ6 minus θ5( 1113857 + c6 θ6prime minus θ7prime( 1113857 + k6 θ6 minus θ7( 1113857 0

J7θ7Prime + c6 θ7prime minus θ6prime( 1113857 + k6 θ7 minus θ6( 1113857 + c7 θ7prime minus θ8prime( 1113857 + k7 θ7 minus θ8( 1113857 0

J8θ8Prime + c7 θ8prime minus θ7prime( 1113857 + k7 θ8 minus θ7( 1113857 minusT1

i1

J9θ9Prime + c8 θ9prime minus θ10prime( 1113857 + k8 θ9 minus θ10( 1113857 T2

i1

J10θ10Prime + c8 θ10prime minus θ9prime( 1113857 + k8 θ10 minus θ9( 1113857 minusT3

i1

J11θ11Prime + c9 θ11prime minus θ12prime( 1113857 + k9 θ11 minus θ12( 1113857 T4

i1

J12θ12Prime + c9 θ12prime minus θ11prime( 1113857 + k9 θ12 minus θ11( 1113857 minusT5

i1

J13θ13Prime + c10 θ13prime minus θ14prime( 1113857 + k10 θ13 minus θ14( 1113857 T6

i1

J14θ14Prime + c10 θ14prime minus θ13prime( 1113857 + k10 θ14 minus θ131113857( 1113857 minusFpzEy

i1

J15θ15Prime + c11 θ15prime minus θ16prime( 1113857 + k11 θ15 minus θ16( 1113857 + c12 θ15prime minus θ17prime( 1113857 + k12 θ15 minus θ17( 1113857 FgzEx + FgxEz

i1i2

J16θ16Prime + c11 θ16prime minus θ15prime( 1113857 + k11 θ16 minus θ15( 1113857 minusTRL

i1i2

J17θ17Prime + c12 θ17prime minus θ15prime( 1113857 + k12 θ17 minus θ15( 1113857 minusTRR

i1i2

mmymPrime + cmyym

prime + kmyym Fmy

mmzmPrime + cmzzm

prime + kmzzm Fmz

mpxpxPrime + kpxxpx + cpxxpx

prime minusFpx

mpxpyPrime + kpy1 xpy minus lp1 tan θpz1113872 1113873 + cpy1 xpy

prime minus lp1θpzprime

cos2θpz

1113888 1113889 + kpy2 xpy minus lp2 tan θpz1113872 1113873 + cpy2 xpyprime minus lp2

θpzprime

cos2θpz

1113888 1113889 Fpy

mpxpzPrime + kpz1 xpz minus lp1 tan θpy1113872 1113873 + cpz1 xpz

prime minus lp1θpyprime

cos2θpy

1113888 1113889 + kpz2 xpz minus lp2 tan θpy1113872 1113873 + cpz2 xpzprime minus lp2

θpyprime

cos2θpy

1113888 1113889 Fpz

JpyθpyPrime minus lp1kpz1 xpz minus lp1 tan θpy1113872 1113873 minus lp1cpz1 xpz

prime minus lp1θpyprime

cos2θpy

1113888 1113889 minus lp2kpz2 xpz minus lp2 tan θpy1113872 1113873 minus lp2cpz2 xpzprime minus lp2

θpyprime

cos2θpy

1113888 1113889 Fpzlp3

JpzθpzPrime minus lp1kpz1 xpy minus lp1 tan θpy1113872 1113873 minus lp1cp1 xpy

prime minus lp1θpzprime

cos2θpz

1113888 1113889 minus lp2kpy2 xpy minus lp2 tan θpz1113872 1113873 minus lp2cpy2 xpyprime minus lp2

θpzprime

cos2θpz

1113888 1113889 minusFpxEy + Fpylp3

mgxgxPrime + kgx1 xgx minus lg1 tan θgz1113872 1113873 + cgx1 xgx

Prime minus lg1θgzprime

cos2θgz

1113888 1113889 + kgx2 xgx + lg2 tan θgz1113872 1113873 + cgx2 xgxprime + lg2

θgzprime

cos2θgz

1113888 1113889 Fgx

mgxgyPrime + kgyxgy + cgyxgy

Prime minusFgy

mgxgzPrime + kgz1 xgz + lg1 tan θgx1113872 1113873 + cgz1 xgz

prime + lg1θgxprime

cos2θgx

1113888 1113889 + kgz2 xgz minus lg2 tan θgx1113872 1113873 + cgz2 xgzprime minus lg2

θgxprime

cos2θgx

1113888 1113889 minusFgz

JgxθgxPrime + lg1kgz1 xgz + lg1 tan θgx1113872 1113873 + lg1cgz1 xgz

prime + lg1θgxprime

cos2θgx

1113888 1113889 minus lg2kgz2 xgz minus lg2 tan θgx1113872 1113873 + lg2cgz2 xgzprime minus lg2

θgxprime

cos2θgx

1113888 1113889 Fgzlg3 minus FgyEz

JgzθgzPrime + lg1kgx1 xgx + lg1 tan θgz1113872 1113873 + lg1cgx1 xgx

prime + lg1θgzprime

cos2θgz

1113888 1113889 minus lg2kgx2 xgx minus lg2 tan θgz1113872 1113873 + lg2cgx2 xgxprime minus lg2

θgzprime

cos2θgz

1113888 1113889 Fgxlg3 minus FgyEx

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

8 Shock and Vibration

where TD is the input torque of motor T1 is the outputtorque of transmission T2 andT3 are the input and outputtorque for intermediate drive shaft T4 andT5 are the inputand output torque for main drive shaft T6 is the inputtorque of rear axle ki(i 1 2 7) are the equivalenttorsional stiffness ci(i 1 2 7) are the equivalent tor-sional damping ki(i 8 9) are the equivalent torsionalstiffness ci(i 8 9) are the equivalent torsional dampingmm is the mass of transmission shaft ki(i 10 11 12) arethe equivalent torsional stiffness for pinion shaft and dif-ferential axle ci(i 10 11 12) are the equivalent torsionaldamping for pinion shaft and differential axle cmy and cmz

are the equivalent damping of intermediate support alongaxes y and z kmy and kmz are the equivalent stiffness ofintermediate support along axes y and z Fmy and Fmz arecomponents of additional force Fi along axes y and z forintermediate support Ei(i x y z) are coordinates ofmeshing point along axes x y and z and TRL andTRR areload torque for wheels

26 Numerical Solution Procedure e fourth-order adap-tive step RungendashKutta method which is generally applicableto strong nonlinearity is adopted in the present work to solve

the equations In order to program a new state space isneeded

In consideration of equations (12)mdash(14) it can be de-rived that kinetic equations of 3 universal joints reduce thenumber of degree-of-freedom from 29 to 26 for the wholesystem e following state space coordinate vector alsodescribes the dynamic behaviors of the system

Ss θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 θ10 θ12 θ14 θ15 θ16 θ17θpy θpz θgx θgz ym zm xp yp zp xg yg zg

⎧⎨

⎩

⎫⎬

⎭

Τ

(29)

In order to eliminate the difference in magnitude be-tween parameters which will reduce the solvability and evenlead to no convergence for solving equations of motionmentioned above b is defined as the characteristic lengthand a new time parameter τ is introduced as follows

τ tωn (30)

where t denotes timeFor the sake of nominalizing equations a generalized

state space coordinate vector with 26 parameters is proposedto characterize dimensionless equations as

st sr1 sr2 sr3 sr4 sr5 sr6 sr7 sr8 sr10 sr12 sr16 sr17 srpx srpy srpz srgx srgy srgz smy smz spx spy spz sgx sgy sgz1113966 1113967T

(31)

e following equations are derived as

sri θi timesRi

b i 1 sim 8 10 12 16 17

srpx θ14 timesRpx

b

srgy θ15 timesRgy

b

srij θij timesRij

b i p g j x y z

six xi

b i p g

siy yi

b i m p g

siz zi

b i m p g

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

where b is the half of backlash and Ri is the equivalent gyrationradius which can be calculated by the following equation

Ri

Ji

mi

1113971

i 1 sim 8 10 12 16 17

Rij

Jij

mi

1113971

i p g j x y z

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(33)

where Ji and Jij are the equivalent moment of inertia and mi

is equivalent massA set of dimensionless parameters are defined as

xnτ xn

b

f xnτ( 1113857 f xt

n( 1113857

b

c(ijs) ci

mjRjRsωn

k(ijs) ki

mjRjRsω2n

cmi cm

miωn

kmi km

miω2n

T(xj) Tx

mjRjω2nb

lijτ lij

b

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)

As equations (30) (32) and (34) are substituted intoequation (28) the dimensionless equations of motion can bederived

Shock and Vibration 9

e equivalent inertia parameters are obtained in UGand stiffness parameters of powertrain are calculated byusing the relationships given in modeling section as listed inTable 1 Subsequently equivalent damping coefficients canbe obtained e parameters of the main reducer can becalculated by substituting system parameters listed in Table 2into equations (17)ndash(31) e solution procedures for di-mensionless parameter are carried out by using precedingcalculated parameters and the relationships proposed inprevious parts

e exciting and configuration parameters are set asθ1 1500 rpm Td2 100Nmiddotm km 9697e8Nm i1 135i2 41 2b 015mm em 0 α1 1695 degree α2 13044degree and α3 1554 degree respectively Without signif-icant loss of accuracy the fundamental harmonic form isadopted for time-varying mesh stiffness and static trans-mission error function whose initial phases are set at 0respectively

3 Numerical Results

31 Analysis of Coupled Torsional Vibration In order toanalysis coupled interaction of automotive powertrainFourier spectrums of torsional vibration response are cal-culated at the initial parameters as shown in Figure 6 Twopeaks and their sidebands are observed e torsional vi-brational responses are governed by the component of 50Hzwhich equals with exciting frequency of input torque asshown in equation (3) (ωf 150060 lowast 2 50Hz) isimplies that engine input excitation is responsible for tor-sional vibration and a valuable approach to decreasingtorsional vibration of powertrain is inferred In addition thecomponent with a frequency of around 185Hz is observedobviously for each considered part As one knows themeshing frequency undergoing this condition is presumably185185Hz (f 150060135185185Hz) which is quite inagreement with the frequency of the second peak Coupledvibration occurs between the torsional system and hypoidgear set and inner excitation caused by gear pair meshingprocess which is affected by clearance and time-varyingmeshing stiffness [25] accounts a lot for torsional vibrationat 185Hz of powertrain and affects torsional responsesslightly After simply calculation bandwidths reaching thesame value of around 37Hz of sidebands around two peaksare got U-joint is a transmission mechanism with non-constant angular speed which will excite a doubled fre-quency torsional vibration of input as shown in equations(12)ndash(14) Under this condition the doubled frequencyexcited by U-joint is calculated

fuminusjoint n

i1 lowast 60lowast 2

1500135lowast 60

lowast 2 3704 (35)

It can be conjectured that modulation phenomenonappears when automotive powertrain works between exci-tations attributed to input U-joint and gear pair As one cansee in Figure 6 compared torsional response of gear withthat located front of meshing point a lower value of tor-sional angular is observed and it is inferred that gear meshexcitations reduce torsional vibration for components

located behind meshing point It is noted that taking gearmesh behaviors into account is quite necessary to when atorsional characteristic analysis is conducted

32 Amplitude-Frequency Response of Torsional Vibration forAutomotive Powertrain e previous works [29] analyzedtorsional characteristics of the coupled system consisted ofdrive shaft and rear axle which denoted that resonance ofdrive shaft at the first two-order nature frequency accountsfor dramatic torsional vibration responses undergoing high-rotation-speed condition However it cannot reveal thetorsional vibration mechanism of automotive under low-and middle-speed condition

In order to obtain a more realistic understanding foramplitude-frequency response of automotive powertrainundergoing middle-speed condition theoretically the pre-ceding equations of motion are adopted A sweep frequencyanalysis procedure is carried out in a range from 1000 rpm to3500 rpm which respectively corresponds to frequencyinternal of 166ndash583Hz for input speed 3333ndash1167Hz forinput excitation and 12346ndash43209Hz for meshing exci-tation e response at input end of rear axle is taken as anexample in this work

Figure 7 shows the amplitude-frequency response inwhich abscissa denotes torsion speed of input and ordinate isRMS value of angular displacement As one can see threeobvious peaks which correspond to input speed at about1000 rpm 1600 rpm and 2800 rpm respectively are ob-served Owing to the assumption that the dramatic responsesare attributed to resonance of the system the first three-order nature frequency respect to the second-order term ofinput torque fluctuation is proposed at 333Hz 533Hz and933Hz successively It is noted that on the basis of previousanalysis these three peaks also correspond to frequencies of247Hz 395Hz and 691Hz for the U-joint excitation and1234Hz 1975Hz and 3456Hz for the meshing excitationPreliminary works [29] showed that resonance is attributedto input excitation of drive shaft and the meshing excitationof hypoid gear pair with higher exciting frequency thaninput will not excite prominent resonance behaviors of driveshaft as illustrated in Figure 8 ie meshing excitation hasslight influence on the torsional system which reaches thesame conclusion with this work According to analysis inSection 31 the sidebands respect to each exciting frequencyare (86ndash58) Hz at 333Hz (138ndash928) Hz at 533Hz and(242ndash1624) Hz at 933Hz for input torque and (987ndash1481)Hz at 1234Hz (158ndash237) Hz at 1975Hz and (2765ndash4147)Hz at 3456Hz for meshing behavior of the gear pair re-spectively As listed in Table 3 the first peak can be attributedto the first two-order resonances of drive shaft excited bysideband around 1234Hz Sidebands at 533Hz and1975Hz result in the second-order resonance which gen-erates the second peak Moreover the second-order reso-nances excited by the sideband at 933Hz and third-orderone caused by sideband at 3456Hz explain the appearanceof the third peak A typical phenomenon is observed that thetorsional vibration reduces a lot while one compares theamplitude of the third peak with another two which

10 Shock and Vibration

Table 1 e equivalent parameters of powertrain

Equivalent mass (kg) Equivalent moment of inertia (kgmiddotm2) Equivalent stiffness (Nm)m1 2 J1 0091125 k1 1002375m2 10 J2 03801735 k2 164025m3 06150 J3 000008103 k3 27500m4 13798 J4 0001001 k4 10e11m5 23581 J5 0001323 k5 479000m6 114323 J6 000249 k6 10e11m7 45323 J7 0001865 k7 21400m8 75567 J8 0002098 mdashm9 1284 J9 0001695 k8 43921799m10 1284 J10 0001695 mdashm11 13125 J11 000173 k9 41933635m12 13125 J12 000173 mdashm13 04254 J13 000002772 k10 10e11m14 17016 J14 00001109 km 96965e8m15 1375 J15 0034177 mdashm16 845 J16 99 k11 12032114m17 845 J17 99 k12 12032114

Table 2 e system parameters of the main reducer

Parameter Pinion GearNumber of teeth 10 41Module (mm) 400Mean pressure angle (deg) 20 20Helix angle at midpoint of the pinion (deg) 50 2713Cone angle (deg) 2312 6514Tooth width (mm) 4066 3452Tooth addendum (mm) 359 644Pinion outset (mm) minus38Mounting distance (mm) 112 64Pitch radius (mm) 40 108Backlash (mm) 01536Distance from pinion reference point to the x-o-z plane for pinion (mm) 40Distance from pinion reference point to the y-o-z plane for gear (mm) 101Distance between projection point and centroid of pinion shaft (mm) lp3 10 lg3 5

Distance between bearing reference point and centroid of pinion shaft (mm) lp1 100 lg1 90lp2 175 lg2 93

Equivalent mass (kg) mg 2127 mp 1375Jpx J14 Jgx 0050315

Equivalent moment of inertia (kgmiddotm2) Jpy 0002982 Jgx J15Jpz 0002982 Jpz 0037551

Meshing stiffnessof (i + 1)th tooth pair

Meshing stiffnessof ith tooth pair

t

k

Meshing stiffnessof (i ndash 1)th tooth pair

Total meshingstiffness km(t)

Figure 5 Schematic diagram of time-varying meshing stiffness on hypoid gear

Shock and Vibration 11

indicates that torsional behaviors of automotive powertrainaffect the vibration characteristics of the complete vehicleunder a low- or middle-input-speed condition In order tovalidate the preceding result an experimental study isnecessary

4 Experimental Analysis

A test bench which can be used to test the vibration behavior ofa transmission system used for four-wheel drive vehicle be-longing to the China Automotive Technology and ResearchCenter is adopted to perform this experiment ere are fivemotors in the test bench and three of them are used in thisexperiment One of them drives automotive powertrain whichcan generate a high frequency input torque with fluctuationvaried in accordance with necessary and others are used asloads In this test two photoelectric sensors and two magne-toelectric sensors are used to detect torsional vibration signalsand other five accelerometers are mounted to monitor trans-verse vibration Installation site for each sensor can be found inFigure 9 Signals are sampled and processed by LMS TestLab

In order to validate the feasibility of the theoreticalmodel the input torque simulated by input motor and loadsare consistent with parameters employed in numericalanalysis (mean value and the second-order term are con-tained only and fluctuation amplitude is set at 100NmiddotM forinput torque) Within the experiment the output speed ofmotor varies from 1000 rpm to 3500 rpm to conduct a sweepfrequency response analysis experimentally

As illustrated in Figure 10 the amplitude-frequencyresponses of torsional vibration have the similar feature for

different parts of powertrain Peaks can be observed forcarves at a similar input speed It is inferred that a presumeresonance occurs at each peak As a result the first three-order nature torsional frequencies can be predictedMeanwhile the same results are reached As shown inFigure 10 each curve has a higher value nearby the first andsecond peak With the further increase in frequency itreaches a lower plateau value Torsional behaviors have aneffect on the vibration characteristics under a low-input-speed condition Furthermore the part located behind themesh point such as output end of half-shaft has a lowerRMS value than transmission and drive shaft which can beattributed to reduction effect of mesh excitation on torsionalvibration

e experimental result of input end of rear axle iscompared with the numerical one As shown in Figure 11 aquite consistency is observed and the first three-order fre-quencies are inferred at 35Hz (1050 rpm) 557Hz(1670 rpm) and 883Hz (2650 rpm) for experimental resultby the same way mentioned in preceding section Comparedwith numerical predictions an acceptable difference ofabout 5 (the difference is 51 at 333Hz 45 at 533Hzand 536 at 933Hz) can be obtained ese indicate thatthe theoretical model is feasible for predicting nature tor-sional frequencies Moreover numerical torsional dis-placement is a little lower than the experimental one at thesame input speed which is below about 2450 rpm and a littlehigher with the further increase is can be presumablyattributed to assumptions made in this present work withrespect to the complexity of the system and the limitations ofthe lumped-mass method

03503

02502

01501

0050

0

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(a)

13Hz

50Hz

87Hz 150Hz185Hz222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(b)

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(c)

13Hz

50Hz

87Hz150Hz

185Hz222Hz

004

003

002

001

00 50 100 150 200 250

Frequency (Hz)To

rsio

nal d

ispla

cem

ent (

deg)300 350 400 450 500 550 600

(d)

Figure 6 e Fourier spectrum of torsional responses (a) output shaft of transmission (b) output end of the drive shaft system (c)torsional displacement of the pinion along the axis x (d) torsional displacement of the gear along the axis y

12 Shock and Vibration

e previous section indicates numerically that excita-tion of input dominates torsion vibration of the system andgear set affects responses slightly In order to validate thecorrectness of numerical analysis and feasibility decreasing

torsional vibration by reducing input excitation effect ofinput torque fluctuation on torsional response is studiedexperimentally e mean value of input torque is set at150NmiddotM and fluctuation of the second-order term is set at

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

2800rpm1600rpm

Figure 7 Amplitude-frequency response at input end of rear axle

8

7

6

5

4

2

3

1

00 1000 2000 3000

Rotation speed (RPM)

Vibr

atio

n am

plitu

de (micro

m)

4000 5000 6000

x directiony directionz direction

Figure 8 Influence of input rotation speed of drive shaft on vibration of the main reducer [29]

Table 3 e first six-order nature frequency of drive shaft [29]

Order 1 2 3 4 5 6Frequency (Hz) 1309 1402 4152 4422 6362 9066

Shock and Vibration 13

25NmiddotM 50NmiddotM 75NmiddotM 100NmiddotM 125NmiddotM 150NmiddotM175NmiddotM and 200NmiddotM in order

Together Figures 12 and 13 expectably show that thetorsional vibration is reinforced while increasing torquefluctuation Meanwhile an obvious increase in amplitudeoccurs at the second peak when fluctuation is increased from175NmiddotM to 200NmiddotM which indicates that a limitation should

be considered e accurate value can be determined by afurther study In summary reducing torque fluctuation is abeneficial approach for vibration damping Furthermore it isnoted that such a nonlinear phenomenon as natural fre-quency shift occurs while changing torque fluctuation Takingthe second-order torsional frequency as an example thefrequency value decreases with elevating fluctuation of input

Transmission

Photoelectricsensor

Photoelectricsensor

Magnetoelectricsensor

Magnetoelectricsensor

Rear axle

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Drive sha system

Figure 9 e experiment bench and arrangement of sensors

042

040

038

035

033

030

028

025

023

020

018

015

013

010

008

005

003

Amplitude-frequency curve ofle wheel

Amplitude-frequency curve oftransmission output sha

Amplitude-frequency curve ofrear axle output end

Amplitude-frequency curve ofright wheel

000

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (K

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Curve 167000

041

035

003

003

5567 8817

167000 264500

rpm

deg

deg

deg

deg

264500

012

005

003

764endash3

Figure 10 Amplitude-frequency responses of torsional vibration for powertrain at a torque fluctuation of 100NmiddotM

14 Shock and Vibration

036

034

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

Numerical result

Experimental result

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

Figure 11 Comparison of numerical and experimental results for input end of rear axle

160 5567 8817

264500

150

140

130

120

110

100

090

080

070

060

050

040

030

020

010

000 167000

25Nmiddotm

50Nmiddotm

75Nmiddotm

100Nmiddotm

125Nmiddotm150Nmiddotm

175Nmiddotm

200Nmiddotm

Curve 167000147078057045035025017013

rpm--------

264500009008007006005004004003

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (R

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Figure 12 Amplitude-frequency responses of torsional vibration for input end of rear axle at different torque fluctuations

Shock and Vibration 15

torque which shows that the torsional system has a stiffness-softening characteristic with respect to torque fluctuationis result can be accounted for a number of nonlinearitiessuch as friction clearance and sliding and the existence ofwhich sophisticates system nonlinear characteristics

5 Conclusion

In the present paper a lumped-parameter dynamic modelwith 29 degree-of-freedom is proposed considering thetransverse torsional and rocking coupled vibration for abetter understanding of the automotive powertrain systemdynamics and equations of motion are derived e nu-merical result shows that gear mesh excitations have littleeffects on torsional response for such components locatedbefore the mesh point as transmission transmission shaftand pinion but significantly reduce the torsional responsesfor ones behind the mesh point e amplitude-frequencycharacteristics of torsional vibration graphically indicate thattorsional behaviors of automotive powertrain only affect thevibration characteristics of complete vehicle under low-speed condition and predict the first three-order naturetorsional frequencies of the whole system An experimentalexamination is performed and validates the feasibility of theconsidered model e torsional vibration is reinforcedwhile increasing torque fluctuation In addition a stiffnesssoftening characteristic is observed for torsional vibrationexperimentally In order to reach a better understanding inamplitude-frequency response at low- and middle-input-speed condition a procedure to analyze mode shapes ofautomotive powertrain will be conducted in future works

Data Availability

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

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported and funded by the ApplicationTechnology Research and Development Plan of LiuzhouCity (grant no 2017AA10102) and the Key Projects ofUniversities in Inner Mongolia Autonomous Region (grantno NJZZ16369)e authors of this paper express their deepgratitude for the financial support of the research project

References

[1] F Vesali M A Rezvani and M Kashfi ldquoDynamics of uni-versal joints its failures and some propositions for practicallyimproving its performance and life expectancyrdquo Journal ofMechanical Science and Technology vol 26 no 8 pp 2439ndash2449 2012

[2] J-W Lu G-C Wang H Chen A F Vakakis andL A Bergman ldquoDynamic analysis of cross shaft type universaljoint with clearancerdquo Journal of Mechanical Science andTechnology vol 27 no 11 pp 3201ndash3205 2013

[3] G Wu W Shi and Z Chen ldquoe effect of multi-universalcoupling phase on torsional vibration of drive shaft and vi-bration of vehiclerdquo in Proceedings of the SAE 2013 WorldCongress amp Exhibition Detroit MI USA April 2013

0113333 4000 4500 5000 5500

5567

6000 6500 7000Frequency (Hz)

7500 8000 85008777

263300

9000 9500 10000 10500 11000 11667100

Am

plitu

de

Am

plitu

de (R

MS)

010

009

008

007

006

005

004

003

002

001

000100000 120000 140000 160000 180000

167000

25Nmiddotm

50Nmiddotm

75Nmiddotm100Nmiddotm

125Nmiddotm

150Nmiddotm

175Nmiddotm200Nmiddotm

Curve 167000010006004003003002001

982endash3

263300006005004003003002001

690endash3

rpm--------

200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000000

Figure 13 Amplitude-frequency responses of torsional vibration for output end of rear axle at different torque fluctuations

16 Shock and Vibration

[4] B A Porter ldquoeoretical analysis of the torsional oscillationof a system incorporating a Hookersquos jointrdquo Journal of Me-chanical Engineering Science vol 3 no 4 pp 324ndash329 1961

[5] S F Asokanthan and X-H Wang ldquoCharacterization oftorsional instabilities in a Hookersquos joint driven system viamaximal Lyapunov exponentsrdquo Journal of Sound and Vi-bration vol 194 no 1 pp 83ndash91 1996

[6] A Farshidianfar M Ebrahimi and H Bartlett ldquoHybridmodelling and simulation of the torsional vibration of vehicledriveline systemsrdquo Proceedings of the Institution of Me-chanical Engineers Part D Journal of Automobile Engineeringvol 215 no 2 pp 217ndash229 2001

[7] E O A Abd Elmaksoud E M Rabeih N A Abdel-halimand S M El Demerdash ldquoInvestigation of self excited tor-sional vibrations of different configurations of automatictransmission systems during engagementrdquo Engineeringvol 3 no 12 pp 1171ndash1181 2011

[8] T B Juang M Cheng and L Na ldquoExperimental and finiteelement analyses of a sliding-tube-type driveshaft-inducedvehicle vibrationrdquo Proceedings of the Institution of MechanicalEngineers Part K Journal of Multi-Body Dynamics vol 221no 3 pp 375ndash385 2007

[9] L F Coutinho and A Tamagna ldquoFlexural vibration analysisof constant velocity half-shaftsrdquo in Proceedings of the Inter-national Congress amp Exposition Chicago IL ISA 1996

[10] Y T Kang Y H Shen W Zhang and J Yang ldquoStabilityregion of floating intermediate support in a shaft system withmultiple universal jointsrdquo Journal of Mechanical Science andTechnology vol 28 no 6 pp 2733ndash2742 2014

[11] J L Xu T Yan B Peng and B Wei ldquoEffects of the trans-mission shaft on the main reducer vibration based onADAMS and experimental demonstrationrdquo AustralianJournal of Mechanical Engineering vol 15 no 1 pp 2ndash102015

[12] C X Zhang H B Li and D Q Jin ldquoSimulation on dynamicsof transmission shaft-rear driving axle of minibus withADAMSrdquo Applied Mechanics and Materials vol 651ndash653pp 862ndash865 2014

[13] J L Xu X Y Su and P Bo ldquoNumerical analysis anddemonstration transmission shaft influence on meshing vi-bration in driving and driven gearsrdquo Shock and Vibrationvol 2015 Article ID 365084 10 pages 2015

[14] J L Xu W Lei andW X Luo ldquoInfluence of bearing stiffnesson the nonlinear dynamics of a shaft-final drive systemrdquoShock and Vibration vol 2016 Article ID 3524609 14 pages2016

[15] A Kahraman and R Singh ldquoNon-linear dynamics of a spurgear pairrdquo Journal of Sound and Vibration vol 142 no 1pp 49ndash75 1990

[16] A Kahraman and R Singh ldquoInteractions between time-varying mesh stiffness and clearance non-linearities in ageared systemrdquo Journal of Sound and Vibration vol 146no 1 pp 135ndash156 1991

[17] A Kahraman ldquoEffect of axial vibrations on the dynamics of ahelical gear pairrdquo Journal of Vibration and Acoustics vol 115no 1 pp 33ndash39 1993

[18] Y Cai and T Hayashi ldquoe linear approximated equation ofvibration of a pair of spur gears (theory and experiment)rdquoJournal of Mechanical Design vol 116 no 2 pp 558ndash5641994

[19] P Velex andMMaatar ldquoAmathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound and Vibration vol 191no 5 pp 629ndash660 1996

[20] P Velex and V Cahouet ldquoExperimental and numerical in-vestigations on the influence of tooth friction in spur andhelical gear dynamicsrdquo Journal of Mechanical Design vol 122no 4 pp 515ndash522 2000

[21] P Velex and P Sainsot ldquoAn analytical study of tooth frictionexcitations in errorless spur and helical gearsrdquo Mechanismand Machine Ieory vol 37 no 7 pp 641ndash658 2002

[22] T C Lim and Y Cheng ldquoA theoretical study of the effect ofpinion offset on the dynamics of hypoid geared rotor systemrdquoJournal of Mechanical Design vol 121 no 4 pp 594ndash6011999

[23] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[24] Y Cheng and T C Lim ldquoDynamics of hypoid gear trans-mission with nonlinear time-varying mesh characteristicsrdquoJournal of Mechanical Design vol 125 no 2 pp 373ndash3822003

[25] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 2017 pp 302ndash329 2007

[26] J Wang and T C Lim ldquoEffect of tooth mesh stiffnessasymmetric nonlinearity for drive and coast sides on hypoidgear dynamicsrdquo Journal of Sound and Vibration vol 319no 3-5 pp 885ndash903 2009

[27] W M Li and H T Wang ldquoRigidity calculation of axialposition preload bearingsrdquo Journal of Hebei University ofTechnology vol 30 no 2 pp 15ndash19 2001

[28] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo Non-linear Dynamics vol 75 no 4 pp 783ndash806 2014

[29] X Liu Z Wu J Lu and J Xu ldquoInvestigation of the effect ofrotation speed on vibration responses of transmission sys-temrdquo Matec Web of Conferences vol 256 no 3 Article ID02019 2019

Shock and Vibration 17

me 1

1mp1113872 1113873 + 1mg1113872 1113873

mpmg

mp + mg

(19)

e following equation can be derived

ωn

km

me

1113971

(20)

where ωn is the equivalent natural frequency of pinion andgear

Equivalent mesh damping coefficient is calculated whichis different from Wang and Limrsquos analysis [25]

cm ξ lowast 2meωn 2ξ

km

1mp1113872 1113873 + 1mg1113872 1113873

1113971

(21)

where ζ is the damping ratio generally set as a range of 003to 017 [25]mp andmg are equivalent mass of pinion and gearrespectively

e relative displacement along the normal direction atthe meshing point xn is given as follows

xn xpx + Eyθpz1113872 1113873 minus xgx + Ezθgy + lg3θgz1113872 11138731113960 1113961

middot cos αn sin βm cos δ1 + sin αn sin δ1( 1113857

+ xpy + lp3θpz1113872 1113873 minus xgy + Ezθgx minus Exθgz1113872 11138731113960 1113961

middot minuscos αn sin βm sin δ1 + sin αn cos δ1( 1113857

+ xpz minus Rpθpx + lp3θpy1113872 1113873 minus xgz minus lg3θgx minus Rgθgy1113872 11138731113960 1113961

middot cos αn cos βm( 1113857 minus e(t)

ε1 xpx minus xgx minus Ezθgy + Eyθpz minus lg3θgz1113872 1113873

+ ε2 xpy minus xgy minus Ezθgx + lp3θpz + Exθgz1113872 1113873

+ ε3 xpz minus xgz minus Rpθpx + lg3θgx + lp3θpy + Rgθgy1113872 1113873

minus e(t)

(22)

e backlash function is expressed as [23]

f xn( 1113857

xn minus b xn ge b

0 minusblexn le b

xn + b xn le minus b

⎧⎪⎪⎨

⎪⎪⎩(23)

where b is half of gear backlashe dynamic meshing force along the line of action Fn

can be calculated byFn km(t)f xn( 1113857 + cmxn

prime (24)

wherecm represents thevalueof equivalentmeshdampingcoefficiente components of dynamic mesh force along the co-

ordinate directions are expressed asFx Fn cos αn sin βm cos δ1 + Fn sin αn sin δ1

Fy minusFn cos αn sin βm sin δ1 + Fn sin αn cos δ1

Fz Fn cos αn cos βm

⎧⎪⎪⎨

⎪⎪⎩(25)

where δ1 is cone angle of the pinion αn is the normal pressureangle and βm is the helix angle at the midpoint of the pinion

In additional the following equations are derived byintroducing the parameter εi

ε1 cos αn sin βm cos δ1 + sin αn sin δ1

ε2 minuscos αn sin βm sin δ1 + sin αn cos δ1

ε3 cos αn cos βm

⎧⎪⎪⎨

⎪⎪⎩(26)

Substituting equations (26) into (25) the expressions ofdynamicmesh force applied on pinion and gear can bewritten as

Fpx minusFgx ε1Fn

Fpy minusFgy ε2Fn

Fpz minusFgz ε3Fn

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(27)

where the subscript p is the label for pinion and the subscriptg is the label for gear

Mesh impact due to teeth separation is neglected in this paper

25 Equations of Motion From the preceding derivationsthe equations of motion considering bending-torsional-lateral-rocking vibration are expressed as

Fpz

kpz2 cpz2

kpx

cpx

θpy

Xpx

kpz1 cpz1

FpxFpy

Op1

cpy2

θpz

mp

kpy2cpy1

Lp1

Lp2 Lp3

Op2 Op0

kpy1y

x

z

Xpy1 Xpy2

Xpy

Op

Op3 Ey

θpz

θpx

Figure 4 e dynamic model of pinion shaft

Shock and Vibration 7

J1θ1Prime + c1 θ1prime minus θ2prime( 1113857 + k1 θ1 minus θ2( 1113857 TD

J2θ2Prime + c1 θ2prime minus θ1prime( 1113857 + k1 θ2 minus θ1( 1113857 + c2 θ2prime minus θ3prime( 1113857 + k2 θ2 minus θ3( 1113857 0

J3θ3Prime + c2 θ3prime minus θ2prime( 1113857 + k2 θ3 minus θ2( 1113857 + c3 θ3prime minus θ4prime( 1113857 + k3 θ3 minus θ4( 1113857 0

J4θ4Prime + c3 θ4prime minus θ3prime( 1113857 + k3 θ4 minus θ3( 1113857 + c4 θ4prime minus θ5prime( 1113857 + k4 θ4 minus θ5( 1113857 0

J5θ5Prime + c4 θ5prime minus θ4prime( 1113857 + k4 θ5 minus θ4( 1113857 + c5 θ5prime minus θ6prime( 1113857 + k5 θ5 minus θ6( 1113857 0

J6θ6Prime + c5 θ6prime minus θ5prime( 1113857 + k5 θ6 minus θ5( 1113857 + c6 θ6prime minus θ7prime( 1113857 + k6 θ6 minus θ7( 1113857 0

J7θ7Prime + c6 θ7prime minus θ6prime( 1113857 + k6 θ7 minus θ6( 1113857 + c7 θ7prime minus θ8prime( 1113857 + k7 θ7 minus θ8( 1113857 0

J8θ8Prime + c7 θ8prime minus θ7prime( 1113857 + k7 θ8 minus θ7( 1113857 minusT1

i1

J9θ9Prime + c8 θ9prime minus θ10prime( 1113857 + k8 θ9 minus θ10( 1113857 T2

i1

J10θ10Prime + c8 θ10prime minus θ9prime( 1113857 + k8 θ10 minus θ9( 1113857 minusT3

i1

J11θ11Prime + c9 θ11prime minus θ12prime( 1113857 + k9 θ11 minus θ12( 1113857 T4

i1

J12θ12Prime + c9 θ12prime minus θ11prime( 1113857 + k9 θ12 minus θ11( 1113857 minusT5

i1

J13θ13Prime + c10 θ13prime minus θ14prime( 1113857 + k10 θ13 minus θ14( 1113857 T6

i1

J14θ14Prime + c10 θ14prime minus θ13prime( 1113857 + k10 θ14 minus θ131113857( 1113857 minusFpzEy

i1

J15θ15Prime + c11 θ15prime minus θ16prime( 1113857 + k11 θ15 minus θ16( 1113857 + c12 θ15prime minus θ17prime( 1113857 + k12 θ15 minus θ17( 1113857 FgzEx + FgxEz

i1i2

J16θ16Prime + c11 θ16prime minus θ15prime( 1113857 + k11 θ16 minus θ15( 1113857 minusTRL

i1i2

J17θ17Prime + c12 θ17prime minus θ15prime( 1113857 + k12 θ17 minus θ15( 1113857 minusTRR

i1i2

mmymPrime + cmyym

prime + kmyym Fmy

mmzmPrime + cmzzm

prime + kmzzm Fmz

mpxpxPrime + kpxxpx + cpxxpx

prime minusFpx

mpxpyPrime + kpy1 xpy minus lp1 tan θpz1113872 1113873 + cpy1 xpy

prime minus lp1θpzprime

cos2θpz

1113888 1113889 + kpy2 xpy minus lp2 tan θpz1113872 1113873 + cpy2 xpyprime minus lp2

θpzprime

cos2θpz

1113888 1113889 Fpy

mpxpzPrime + kpz1 xpz minus lp1 tan θpy1113872 1113873 + cpz1 xpz

prime minus lp1θpyprime

cos2θpy

1113888 1113889 + kpz2 xpz minus lp2 tan θpy1113872 1113873 + cpz2 xpzprime minus lp2

θpyprime

cos2θpy

1113888 1113889 Fpz

JpyθpyPrime minus lp1kpz1 xpz minus lp1 tan θpy1113872 1113873 minus lp1cpz1 xpz

prime minus lp1θpyprime

cos2θpy

1113888 1113889 minus lp2kpz2 xpz minus lp2 tan θpy1113872 1113873 minus lp2cpz2 xpzprime minus lp2

θpyprime

cos2θpy

1113888 1113889 Fpzlp3

JpzθpzPrime minus lp1kpz1 xpy minus lp1 tan θpy1113872 1113873 minus lp1cp1 xpy

prime minus lp1θpzprime

cos2θpz

1113888 1113889 minus lp2kpy2 xpy minus lp2 tan θpz1113872 1113873 minus lp2cpy2 xpyprime minus lp2

θpzprime

cos2θpz

1113888 1113889 minusFpxEy + Fpylp3

mgxgxPrime + kgx1 xgx minus lg1 tan θgz1113872 1113873 + cgx1 xgx

Prime minus lg1θgzprime

cos2θgz

1113888 1113889 + kgx2 xgx + lg2 tan θgz1113872 1113873 + cgx2 xgxprime + lg2

θgzprime

cos2θgz

1113888 1113889 Fgx

mgxgyPrime + kgyxgy + cgyxgy

Prime minusFgy

mgxgzPrime + kgz1 xgz + lg1 tan θgx1113872 1113873 + cgz1 xgz

prime + lg1θgxprime

cos2θgx

1113888 1113889 + kgz2 xgz minus lg2 tan θgx1113872 1113873 + cgz2 xgzprime minus lg2

θgxprime

cos2θgx

1113888 1113889 minusFgz

JgxθgxPrime + lg1kgz1 xgz + lg1 tan θgx1113872 1113873 + lg1cgz1 xgz

prime + lg1θgxprime

cos2θgx

1113888 1113889 minus lg2kgz2 xgz minus lg2 tan θgx1113872 1113873 + lg2cgz2 xgzprime minus lg2

θgxprime

cos2θgx

1113888 1113889 Fgzlg3 minus FgyEz

JgzθgzPrime + lg1kgx1 xgx + lg1 tan θgz1113872 1113873 + lg1cgx1 xgx

prime + lg1θgzprime

cos2θgz

1113888 1113889 minus lg2kgx2 xgx minus lg2 tan θgz1113872 1113873 + lg2cgx2 xgxprime minus lg2

θgzprime

cos2θgz

1113888 1113889 Fgxlg3 minus FgyEx

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

8 Shock and Vibration

where TD is the input torque of motor T1 is the outputtorque of transmission T2 andT3 are the input and outputtorque for intermediate drive shaft T4 andT5 are the inputand output torque for main drive shaft T6 is the inputtorque of rear axle ki(i 1 2 7) are the equivalenttorsional stiffness ci(i 1 2 7) are the equivalent tor-sional damping ki(i 8 9) are the equivalent torsionalstiffness ci(i 8 9) are the equivalent torsional dampingmm is the mass of transmission shaft ki(i 10 11 12) arethe equivalent torsional stiffness for pinion shaft and dif-ferential axle ci(i 10 11 12) are the equivalent torsionaldamping for pinion shaft and differential axle cmy and cmz

are the equivalent damping of intermediate support alongaxes y and z kmy and kmz are the equivalent stiffness ofintermediate support along axes y and z Fmy and Fmz arecomponents of additional force Fi along axes y and z forintermediate support Ei(i x y z) are coordinates ofmeshing point along axes x y and z and TRL andTRR areload torque for wheels

26 Numerical Solution Procedure e fourth-order adap-tive step RungendashKutta method which is generally applicableto strong nonlinearity is adopted in the present work to solve

the equations In order to program a new state space isneeded

In consideration of equations (12)mdash(14) it can be de-rived that kinetic equations of 3 universal joints reduce thenumber of degree-of-freedom from 29 to 26 for the wholesystem e following state space coordinate vector alsodescribes the dynamic behaviors of the system

Ss θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 θ10 θ12 θ14 θ15 θ16 θ17θpy θpz θgx θgz ym zm xp yp zp xg yg zg

⎧⎨

⎩

⎫⎬

⎭

Τ

(29)

In order to eliminate the difference in magnitude be-tween parameters which will reduce the solvability and evenlead to no convergence for solving equations of motionmentioned above b is defined as the characteristic lengthand a new time parameter τ is introduced as follows

τ tωn (30)

where t denotes timeFor the sake of nominalizing equations a generalized

state space coordinate vector with 26 parameters is proposedto characterize dimensionless equations as

st sr1 sr2 sr3 sr4 sr5 sr6 sr7 sr8 sr10 sr12 sr16 sr17 srpx srpy srpz srgx srgy srgz smy smz spx spy spz sgx sgy sgz1113966 1113967T

(31)

e following equations are derived as

sri θi timesRi

b i 1 sim 8 10 12 16 17

srpx θ14 timesRpx

b

srgy θ15 timesRgy

b

srij θij timesRij

b i p g j x y z

six xi

b i p g

siy yi

b i m p g

siz zi

b i m p g

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

where b is the half of backlash and Ri is the equivalent gyrationradius which can be calculated by the following equation

Ri

Ji

mi

1113971

i 1 sim 8 10 12 16 17

Rij

Jij

mi

1113971

i p g j x y z

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(33)

where Ji and Jij are the equivalent moment of inertia and mi

is equivalent massA set of dimensionless parameters are defined as

xnτ xn

b

f xnτ( 1113857 f xt

n( 1113857

b

c(ijs) ci

mjRjRsωn

k(ijs) ki

mjRjRsω2n

cmi cm

miωn

kmi km

miω2n

T(xj) Tx

mjRjω2nb

lijτ lij

b

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)

As equations (30) (32) and (34) are substituted intoequation (28) the dimensionless equations of motion can bederived

Shock and Vibration 9

e equivalent inertia parameters are obtained in UGand stiffness parameters of powertrain are calculated byusing the relationships given in modeling section as listed inTable 1 Subsequently equivalent damping coefficients canbe obtained e parameters of the main reducer can becalculated by substituting system parameters listed in Table 2into equations (17)ndash(31) e solution procedures for di-mensionless parameter are carried out by using precedingcalculated parameters and the relationships proposed inprevious parts

e exciting and configuration parameters are set asθ1 1500 rpm Td2 100Nmiddotm km 9697e8Nm i1 135i2 41 2b 015mm em 0 α1 1695 degree α2 13044degree and α3 1554 degree respectively Without signif-icant loss of accuracy the fundamental harmonic form isadopted for time-varying mesh stiffness and static trans-mission error function whose initial phases are set at 0respectively

3 Numerical Results

31 Analysis of Coupled Torsional Vibration In order toanalysis coupled interaction of automotive powertrainFourier spectrums of torsional vibration response are cal-culated at the initial parameters as shown in Figure 6 Twopeaks and their sidebands are observed e torsional vi-brational responses are governed by the component of 50Hzwhich equals with exciting frequency of input torque asshown in equation (3) (ωf 150060 lowast 2 50Hz) isimplies that engine input excitation is responsible for tor-sional vibration and a valuable approach to decreasingtorsional vibration of powertrain is inferred In addition thecomponent with a frequency of around 185Hz is observedobviously for each considered part As one knows themeshing frequency undergoing this condition is presumably185185Hz (f 150060135185185Hz) which is quite inagreement with the frequency of the second peak Coupledvibration occurs between the torsional system and hypoidgear set and inner excitation caused by gear pair meshingprocess which is affected by clearance and time-varyingmeshing stiffness [25] accounts a lot for torsional vibrationat 185Hz of powertrain and affects torsional responsesslightly After simply calculation bandwidths reaching thesame value of around 37Hz of sidebands around two peaksare got U-joint is a transmission mechanism with non-constant angular speed which will excite a doubled fre-quency torsional vibration of input as shown in equations(12)ndash(14) Under this condition the doubled frequencyexcited by U-joint is calculated

fuminusjoint n

i1 lowast 60lowast 2

1500135lowast 60

lowast 2 3704 (35)

It can be conjectured that modulation phenomenonappears when automotive powertrain works between exci-tations attributed to input U-joint and gear pair As one cansee in Figure 6 compared torsional response of gear withthat located front of meshing point a lower value of tor-sional angular is observed and it is inferred that gear meshexcitations reduce torsional vibration for components

located behind meshing point It is noted that taking gearmesh behaviors into account is quite necessary to when atorsional characteristic analysis is conducted

32 Amplitude-Frequency Response of Torsional Vibration forAutomotive Powertrain e previous works [29] analyzedtorsional characteristics of the coupled system consisted ofdrive shaft and rear axle which denoted that resonance ofdrive shaft at the first two-order nature frequency accountsfor dramatic torsional vibration responses undergoing high-rotation-speed condition However it cannot reveal thetorsional vibration mechanism of automotive under low-and middle-speed condition

In order to obtain a more realistic understanding foramplitude-frequency response of automotive powertrainundergoing middle-speed condition theoretically the pre-ceding equations of motion are adopted A sweep frequencyanalysis procedure is carried out in a range from 1000 rpm to3500 rpm which respectively corresponds to frequencyinternal of 166ndash583Hz for input speed 3333ndash1167Hz forinput excitation and 12346ndash43209Hz for meshing exci-tation e response at input end of rear axle is taken as anexample in this work

Figure 7 shows the amplitude-frequency response inwhich abscissa denotes torsion speed of input and ordinate isRMS value of angular displacement As one can see threeobvious peaks which correspond to input speed at about1000 rpm 1600 rpm and 2800 rpm respectively are ob-served Owing to the assumption that the dramatic responsesare attributed to resonance of the system the first three-order nature frequency respect to the second-order term ofinput torque fluctuation is proposed at 333Hz 533Hz and933Hz successively It is noted that on the basis of previousanalysis these three peaks also correspond to frequencies of247Hz 395Hz and 691Hz for the U-joint excitation and1234Hz 1975Hz and 3456Hz for the meshing excitationPreliminary works [29] showed that resonance is attributedto input excitation of drive shaft and the meshing excitationof hypoid gear pair with higher exciting frequency thaninput will not excite prominent resonance behaviors of driveshaft as illustrated in Figure 8 ie meshing excitation hasslight influence on the torsional system which reaches thesame conclusion with this work According to analysis inSection 31 the sidebands respect to each exciting frequencyare (86ndash58) Hz at 333Hz (138ndash928) Hz at 533Hz and(242ndash1624) Hz at 933Hz for input torque and (987ndash1481)Hz at 1234Hz (158ndash237) Hz at 1975Hz and (2765ndash4147)Hz at 3456Hz for meshing behavior of the gear pair re-spectively As listed in Table 3 the first peak can be attributedto the first two-order resonances of drive shaft excited bysideband around 1234Hz Sidebands at 533Hz and1975Hz result in the second-order resonance which gen-erates the second peak Moreover the second-order reso-nances excited by the sideband at 933Hz and third-orderone caused by sideband at 3456Hz explain the appearanceof the third peak A typical phenomenon is observed that thetorsional vibration reduces a lot while one compares theamplitude of the third peak with another two which

10 Shock and Vibration

Table 1 e equivalent parameters of powertrain

Equivalent mass (kg) Equivalent moment of inertia (kgmiddotm2) Equivalent stiffness (Nm)m1 2 J1 0091125 k1 1002375m2 10 J2 03801735 k2 164025m3 06150 J3 000008103 k3 27500m4 13798 J4 0001001 k4 10e11m5 23581 J5 0001323 k5 479000m6 114323 J6 000249 k6 10e11m7 45323 J7 0001865 k7 21400m8 75567 J8 0002098 mdashm9 1284 J9 0001695 k8 43921799m10 1284 J10 0001695 mdashm11 13125 J11 000173 k9 41933635m12 13125 J12 000173 mdashm13 04254 J13 000002772 k10 10e11m14 17016 J14 00001109 km 96965e8m15 1375 J15 0034177 mdashm16 845 J16 99 k11 12032114m17 845 J17 99 k12 12032114

Table 2 e system parameters of the main reducer

Parameter Pinion GearNumber of teeth 10 41Module (mm) 400Mean pressure angle (deg) 20 20Helix angle at midpoint of the pinion (deg) 50 2713Cone angle (deg) 2312 6514Tooth width (mm) 4066 3452Tooth addendum (mm) 359 644Pinion outset (mm) minus38Mounting distance (mm) 112 64Pitch radius (mm) 40 108Backlash (mm) 01536Distance from pinion reference point to the x-o-z plane for pinion (mm) 40Distance from pinion reference point to the y-o-z plane for gear (mm) 101Distance between projection point and centroid of pinion shaft (mm) lp3 10 lg3 5

Distance between bearing reference point and centroid of pinion shaft (mm) lp1 100 lg1 90lp2 175 lg2 93

Equivalent mass (kg) mg 2127 mp 1375Jpx J14 Jgx 0050315

Equivalent moment of inertia (kgmiddotm2) Jpy 0002982 Jgx J15Jpz 0002982 Jpz 0037551

Meshing stiffnessof (i + 1)th tooth pair

Meshing stiffnessof ith tooth pair

t

k

Meshing stiffnessof (i ndash 1)th tooth pair

Total meshingstiffness km(t)

Figure 5 Schematic diagram of time-varying meshing stiffness on hypoid gear

Shock and Vibration 11

indicates that torsional behaviors of automotive powertrainaffect the vibration characteristics of the complete vehicleunder a low- or middle-input-speed condition In order tovalidate the preceding result an experimental study isnecessary

4 Experimental Analysis

A test bench which can be used to test the vibration behavior ofa transmission system used for four-wheel drive vehicle be-longing to the China Automotive Technology and ResearchCenter is adopted to perform this experiment ere are fivemotors in the test bench and three of them are used in thisexperiment One of them drives automotive powertrain whichcan generate a high frequency input torque with fluctuationvaried in accordance with necessary and others are used asloads In this test two photoelectric sensors and two magne-toelectric sensors are used to detect torsional vibration signalsand other five accelerometers are mounted to monitor trans-verse vibration Installation site for each sensor can be found inFigure 9 Signals are sampled and processed by LMS TestLab

In order to validate the feasibility of the theoreticalmodel the input torque simulated by input motor and loadsare consistent with parameters employed in numericalanalysis (mean value and the second-order term are con-tained only and fluctuation amplitude is set at 100NmiddotM forinput torque) Within the experiment the output speed ofmotor varies from 1000 rpm to 3500 rpm to conduct a sweepfrequency response analysis experimentally

As illustrated in Figure 10 the amplitude-frequencyresponses of torsional vibration have the similar feature for

different parts of powertrain Peaks can be observed forcarves at a similar input speed It is inferred that a presumeresonance occurs at each peak As a result the first three-order nature torsional frequencies can be predictedMeanwhile the same results are reached As shown inFigure 10 each curve has a higher value nearby the first andsecond peak With the further increase in frequency itreaches a lower plateau value Torsional behaviors have aneffect on the vibration characteristics under a low-input-speed condition Furthermore the part located behind themesh point such as output end of half-shaft has a lowerRMS value than transmission and drive shaft which can beattributed to reduction effect of mesh excitation on torsionalvibration

e experimental result of input end of rear axle iscompared with the numerical one As shown in Figure 11 aquite consistency is observed and the first three-order fre-quencies are inferred at 35Hz (1050 rpm) 557Hz(1670 rpm) and 883Hz (2650 rpm) for experimental resultby the same way mentioned in preceding section Comparedwith numerical predictions an acceptable difference ofabout 5 (the difference is 51 at 333Hz 45 at 533Hzand 536 at 933Hz) can be obtained ese indicate thatthe theoretical model is feasible for predicting nature tor-sional frequencies Moreover numerical torsional dis-placement is a little lower than the experimental one at thesame input speed which is below about 2450 rpm and a littlehigher with the further increase is can be presumablyattributed to assumptions made in this present work withrespect to the complexity of the system and the limitations ofthe lumped-mass method

03503

02502

01501

0050

0

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(a)

13Hz

50Hz

87Hz 150Hz185Hz222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(b)

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(c)

13Hz

50Hz

87Hz150Hz

185Hz222Hz

004

003

002

001

00 50 100 150 200 250

Frequency (Hz)To

rsio

nal d

ispla

cem

ent (

deg)300 350 400 450 500 550 600

(d)

Figure 6 e Fourier spectrum of torsional responses (a) output shaft of transmission (b) output end of the drive shaft system (c)torsional displacement of the pinion along the axis x (d) torsional displacement of the gear along the axis y

12 Shock and Vibration

e previous section indicates numerically that excita-tion of input dominates torsion vibration of the system andgear set affects responses slightly In order to validate thecorrectness of numerical analysis and feasibility decreasing

torsional vibration by reducing input excitation effect ofinput torque fluctuation on torsional response is studiedexperimentally e mean value of input torque is set at150NmiddotM and fluctuation of the second-order term is set at

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

2800rpm1600rpm

Figure 7 Amplitude-frequency response at input end of rear axle

8

7

6

5

4

2

3

1

00 1000 2000 3000

Rotation speed (RPM)

Vibr

atio

n am

plitu

de (micro

m)

4000 5000 6000

x directiony directionz direction

Figure 8 Influence of input rotation speed of drive shaft on vibration of the main reducer [29]

Table 3 e first six-order nature frequency of drive shaft [29]

Order 1 2 3 4 5 6Frequency (Hz) 1309 1402 4152 4422 6362 9066

Shock and Vibration 13

25NmiddotM 50NmiddotM 75NmiddotM 100NmiddotM 125NmiddotM 150NmiddotM175NmiddotM and 200NmiddotM in order

Together Figures 12 and 13 expectably show that thetorsional vibration is reinforced while increasing torquefluctuation Meanwhile an obvious increase in amplitudeoccurs at the second peak when fluctuation is increased from175NmiddotM to 200NmiddotM which indicates that a limitation should

be considered e accurate value can be determined by afurther study In summary reducing torque fluctuation is abeneficial approach for vibration damping Furthermore it isnoted that such a nonlinear phenomenon as natural fre-quency shift occurs while changing torque fluctuation Takingthe second-order torsional frequency as an example thefrequency value decreases with elevating fluctuation of input

Transmission

Photoelectricsensor

Photoelectricsensor

Magnetoelectricsensor

Magnetoelectricsensor

Rear axle

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Drive sha system

Figure 9 e experiment bench and arrangement of sensors

042

040

038

035

033

030

028

025

023

020

018

015

013

010

008

005

003

Amplitude-frequency curve ofle wheel

Amplitude-frequency curve oftransmission output sha

Amplitude-frequency curve ofrear axle output end

Amplitude-frequency curve ofright wheel

000

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (K

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Curve 167000

041

035

003

003

5567 8817

167000 264500

rpm

deg

deg

deg

deg

264500

012

005

003

764endash3

Figure 10 Amplitude-frequency responses of torsional vibration for powertrain at a torque fluctuation of 100NmiddotM

14 Shock and Vibration

036

034

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

Numerical result

Experimental result

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

Figure 11 Comparison of numerical and experimental results for input end of rear axle

160 5567 8817

264500

150

140

130

120

110

100

090

080

070

060

050

040

030

020

010

000 167000

25Nmiddotm

50Nmiddotm

75Nmiddotm

100Nmiddotm

125Nmiddotm150Nmiddotm

175Nmiddotm

200Nmiddotm

Curve 167000147078057045035025017013

rpm--------

264500009008007006005004004003

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (R

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Figure 12 Amplitude-frequency responses of torsional vibration for input end of rear axle at different torque fluctuations

Shock and Vibration 15

torque which shows that the torsional system has a stiffness-softening characteristic with respect to torque fluctuationis result can be accounted for a number of nonlinearitiessuch as friction clearance and sliding and the existence ofwhich sophisticates system nonlinear characteristics

5 Conclusion

In the present paper a lumped-parameter dynamic modelwith 29 degree-of-freedom is proposed considering thetransverse torsional and rocking coupled vibration for abetter understanding of the automotive powertrain systemdynamics and equations of motion are derived e nu-merical result shows that gear mesh excitations have littleeffects on torsional response for such components locatedbefore the mesh point as transmission transmission shaftand pinion but significantly reduce the torsional responsesfor ones behind the mesh point e amplitude-frequencycharacteristics of torsional vibration graphically indicate thattorsional behaviors of automotive powertrain only affect thevibration characteristics of complete vehicle under low-speed condition and predict the first three-order naturetorsional frequencies of the whole system An experimentalexamination is performed and validates the feasibility of theconsidered model e torsional vibration is reinforcedwhile increasing torque fluctuation In addition a stiffnesssoftening characteristic is observed for torsional vibrationexperimentally In order to reach a better understanding inamplitude-frequency response at low- and middle-input-speed condition a procedure to analyze mode shapes ofautomotive powertrain will be conducted in future works

Data Availability

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

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported and funded by the ApplicationTechnology Research and Development Plan of LiuzhouCity (grant no 2017AA10102) and the Key Projects ofUniversities in Inner Mongolia Autonomous Region (grantno NJZZ16369)e authors of this paper express their deepgratitude for the financial support of the research project

References

[1] F Vesali M A Rezvani and M Kashfi ldquoDynamics of uni-versal joints its failures and some propositions for practicallyimproving its performance and life expectancyrdquo Journal ofMechanical Science and Technology vol 26 no 8 pp 2439ndash2449 2012

[2] J-W Lu G-C Wang H Chen A F Vakakis andL A Bergman ldquoDynamic analysis of cross shaft type universaljoint with clearancerdquo Journal of Mechanical Science andTechnology vol 27 no 11 pp 3201ndash3205 2013

[3] G Wu W Shi and Z Chen ldquoe effect of multi-universalcoupling phase on torsional vibration of drive shaft and vi-bration of vehiclerdquo in Proceedings of the SAE 2013 WorldCongress amp Exhibition Detroit MI USA April 2013

0113333 4000 4500 5000 5500

5567

6000 6500 7000Frequency (Hz)

7500 8000 85008777

263300

9000 9500 10000 10500 11000 11667100

Am

plitu

de

Am

plitu

de (R

MS)

010

009

008

007

006

005

004

003

002

001

000100000 120000 140000 160000 180000

167000

25Nmiddotm

50Nmiddotm

75Nmiddotm100Nmiddotm

125Nmiddotm

150Nmiddotm

175Nmiddotm200Nmiddotm

Curve 167000010006004003003002001

982endash3

263300006005004003003002001

690endash3

rpm--------

200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000000

Figure 13 Amplitude-frequency responses of torsional vibration for output end of rear axle at different torque fluctuations

16 Shock and Vibration

[4] B A Porter ldquoeoretical analysis of the torsional oscillationof a system incorporating a Hookersquos jointrdquo Journal of Me-chanical Engineering Science vol 3 no 4 pp 324ndash329 1961

[5] S F Asokanthan and X-H Wang ldquoCharacterization oftorsional instabilities in a Hookersquos joint driven system viamaximal Lyapunov exponentsrdquo Journal of Sound and Vi-bration vol 194 no 1 pp 83ndash91 1996

[6] A Farshidianfar M Ebrahimi and H Bartlett ldquoHybridmodelling and simulation of the torsional vibration of vehicledriveline systemsrdquo Proceedings of the Institution of Me-chanical Engineers Part D Journal of Automobile Engineeringvol 215 no 2 pp 217ndash229 2001

[7] E O A Abd Elmaksoud E M Rabeih N A Abdel-halimand S M El Demerdash ldquoInvestigation of self excited tor-sional vibrations of different configurations of automatictransmission systems during engagementrdquo Engineeringvol 3 no 12 pp 1171ndash1181 2011

[8] T B Juang M Cheng and L Na ldquoExperimental and finiteelement analyses of a sliding-tube-type driveshaft-inducedvehicle vibrationrdquo Proceedings of the Institution of MechanicalEngineers Part K Journal of Multi-Body Dynamics vol 221no 3 pp 375ndash385 2007

[9] L F Coutinho and A Tamagna ldquoFlexural vibration analysisof constant velocity half-shaftsrdquo in Proceedings of the Inter-national Congress amp Exposition Chicago IL ISA 1996

[10] Y T Kang Y H Shen W Zhang and J Yang ldquoStabilityregion of floating intermediate support in a shaft system withmultiple universal jointsrdquo Journal of Mechanical Science andTechnology vol 28 no 6 pp 2733ndash2742 2014

[11] J L Xu T Yan B Peng and B Wei ldquoEffects of the trans-mission shaft on the main reducer vibration based onADAMS and experimental demonstrationrdquo AustralianJournal of Mechanical Engineering vol 15 no 1 pp 2ndash102015

[12] C X Zhang H B Li and D Q Jin ldquoSimulation on dynamicsof transmission shaft-rear driving axle of minibus withADAMSrdquo Applied Mechanics and Materials vol 651ndash653pp 862ndash865 2014

[13] J L Xu X Y Su and P Bo ldquoNumerical analysis anddemonstration transmission shaft influence on meshing vi-bration in driving and driven gearsrdquo Shock and Vibrationvol 2015 Article ID 365084 10 pages 2015

[14] J L Xu W Lei andW X Luo ldquoInfluence of bearing stiffnesson the nonlinear dynamics of a shaft-final drive systemrdquoShock and Vibration vol 2016 Article ID 3524609 14 pages2016

[15] A Kahraman and R Singh ldquoNon-linear dynamics of a spurgear pairrdquo Journal of Sound and Vibration vol 142 no 1pp 49ndash75 1990

[16] A Kahraman and R Singh ldquoInteractions between time-varying mesh stiffness and clearance non-linearities in ageared systemrdquo Journal of Sound and Vibration vol 146no 1 pp 135ndash156 1991

[17] A Kahraman ldquoEffect of axial vibrations on the dynamics of ahelical gear pairrdquo Journal of Vibration and Acoustics vol 115no 1 pp 33ndash39 1993

[18] Y Cai and T Hayashi ldquoe linear approximated equation ofvibration of a pair of spur gears (theory and experiment)rdquoJournal of Mechanical Design vol 116 no 2 pp 558ndash5641994

[19] P Velex andMMaatar ldquoAmathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound and Vibration vol 191no 5 pp 629ndash660 1996

[20] P Velex and V Cahouet ldquoExperimental and numerical in-vestigations on the influence of tooth friction in spur andhelical gear dynamicsrdquo Journal of Mechanical Design vol 122no 4 pp 515ndash522 2000

[21] P Velex and P Sainsot ldquoAn analytical study of tooth frictionexcitations in errorless spur and helical gearsrdquo Mechanismand Machine Ieory vol 37 no 7 pp 641ndash658 2002

[22] T C Lim and Y Cheng ldquoA theoretical study of the effect ofpinion offset on the dynamics of hypoid geared rotor systemrdquoJournal of Mechanical Design vol 121 no 4 pp 594ndash6011999

[23] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[24] Y Cheng and T C Lim ldquoDynamics of hypoid gear trans-mission with nonlinear time-varying mesh characteristicsrdquoJournal of Mechanical Design vol 125 no 2 pp 373ndash3822003

[25] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 2017 pp 302ndash329 2007

[26] J Wang and T C Lim ldquoEffect of tooth mesh stiffnessasymmetric nonlinearity for drive and coast sides on hypoidgear dynamicsrdquo Journal of Sound and Vibration vol 319no 3-5 pp 885ndash903 2009

[27] W M Li and H T Wang ldquoRigidity calculation of axialposition preload bearingsrdquo Journal of Hebei University ofTechnology vol 30 no 2 pp 15ndash19 2001

[28] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo Non-linear Dynamics vol 75 no 4 pp 783ndash806 2014

[29] X Liu Z Wu J Lu and J Xu ldquoInvestigation of the effect ofrotation speed on vibration responses of transmission sys-temrdquo Matec Web of Conferences vol 256 no 3 Article ID02019 2019

Shock and Vibration 17

J1θ1Prime + c1 θ1prime minus θ2prime( 1113857 + k1 θ1 minus θ2( 1113857 TD

J2θ2Prime + c1 θ2prime minus θ1prime( 1113857 + k1 θ2 minus θ1( 1113857 + c2 θ2prime minus θ3prime( 1113857 + k2 θ2 minus θ3( 1113857 0

J3θ3Prime + c2 θ3prime minus θ2prime( 1113857 + k2 θ3 minus θ2( 1113857 + c3 θ3prime minus θ4prime( 1113857 + k3 θ3 minus θ4( 1113857 0

J4θ4Prime + c3 θ4prime minus θ3prime( 1113857 + k3 θ4 minus θ3( 1113857 + c4 θ4prime minus θ5prime( 1113857 + k4 θ4 minus θ5( 1113857 0

J5θ5Prime + c4 θ5prime minus θ4prime( 1113857 + k4 θ5 minus θ4( 1113857 + c5 θ5prime minus θ6prime( 1113857 + k5 θ5 minus θ6( 1113857 0

J6θ6Prime + c5 θ6prime minus θ5prime( 1113857 + k5 θ6 minus θ5( 1113857 + c6 θ6prime minus θ7prime( 1113857 + k6 θ6 minus θ7( 1113857 0

J7θ7Prime + c6 θ7prime minus θ6prime( 1113857 + k6 θ7 minus θ6( 1113857 + c7 θ7prime minus θ8prime( 1113857 + k7 θ7 minus θ8( 1113857 0

J8θ8Prime + c7 θ8prime minus θ7prime( 1113857 + k7 θ8 minus θ7( 1113857 minusT1

i1

J9θ9Prime + c8 θ9prime minus θ10prime( 1113857 + k8 θ9 minus θ10( 1113857 T2

i1

J10θ10Prime + c8 θ10prime minus θ9prime( 1113857 + k8 θ10 minus θ9( 1113857 minusT3

i1

J11θ11Prime + c9 θ11prime minus θ12prime( 1113857 + k9 θ11 minus θ12( 1113857 T4

i1

J12θ12Prime + c9 θ12prime minus θ11prime( 1113857 + k9 θ12 minus θ11( 1113857 minusT5

i1

J13θ13Prime + c10 θ13prime minus θ14prime( 1113857 + k10 θ13 minus θ14( 1113857 T6

i1

J14θ14Prime + c10 θ14prime minus θ13prime( 1113857 + k10 θ14 minus θ131113857( 1113857 minusFpzEy

i1

J15θ15Prime + c11 θ15prime minus θ16prime( 1113857 + k11 θ15 minus θ16( 1113857 + c12 θ15prime minus θ17prime( 1113857 + k12 θ15 minus θ17( 1113857 FgzEx + FgxEz

i1i2

J16θ16Prime + c11 θ16prime minus θ15prime( 1113857 + k11 θ16 minus θ15( 1113857 minusTRL

i1i2

J17θ17Prime + c12 θ17prime minus θ15prime( 1113857 + k12 θ17 minus θ15( 1113857 minusTRR

i1i2

mmymPrime + cmyym

prime + kmyym Fmy

mmzmPrime + cmzzm

prime + kmzzm Fmz

mpxpxPrime + kpxxpx + cpxxpx

prime minusFpx

mpxpyPrime + kpy1 xpy minus lp1 tan θpz1113872 1113873 + cpy1 xpy

prime minus lp1θpzprime

cos2θpz

1113888 1113889 + kpy2 xpy minus lp2 tan θpz1113872 1113873 + cpy2 xpyprime minus lp2

θpzprime

cos2θpz

1113888 1113889 Fpy

mpxpzPrime + kpz1 xpz minus lp1 tan θpy1113872 1113873 + cpz1 xpz

prime minus lp1θpyprime

cos2θpy

1113888 1113889 + kpz2 xpz minus lp2 tan θpy1113872 1113873 + cpz2 xpzprime minus lp2

θpyprime

cos2θpy

1113888 1113889 Fpz

JpyθpyPrime minus lp1kpz1 xpz minus lp1 tan θpy1113872 1113873 minus lp1cpz1 xpz

prime minus lp1θpyprime

cos2θpy

1113888 1113889 minus lp2kpz2 xpz minus lp2 tan θpy1113872 1113873 minus lp2cpz2 xpzprime minus lp2

θpyprime

cos2θpy

1113888 1113889 Fpzlp3

JpzθpzPrime minus lp1kpz1 xpy minus lp1 tan θpy1113872 1113873 minus lp1cp1 xpy

prime minus lp1θpzprime

cos2θpz

1113888 1113889 minus lp2kpy2 xpy minus lp2 tan θpz1113872 1113873 minus lp2cpy2 xpyprime minus lp2

θpzprime

cos2θpz

1113888 1113889 minusFpxEy + Fpylp3

mgxgxPrime + kgx1 xgx minus lg1 tan θgz1113872 1113873 + cgx1 xgx

Prime minus lg1θgzprime

cos2θgz

1113888 1113889 + kgx2 xgx + lg2 tan θgz1113872 1113873 + cgx2 xgxprime + lg2

θgzprime

cos2θgz

1113888 1113889 Fgx

mgxgyPrime + kgyxgy + cgyxgy

Prime minusFgy

mgxgzPrime + kgz1 xgz + lg1 tan θgx1113872 1113873 + cgz1 xgz

prime + lg1θgxprime

cos2θgx

1113888 1113889 + kgz2 xgz minus lg2 tan θgx1113872 1113873 + cgz2 xgzprime minus lg2

θgxprime

cos2θgx

1113888 1113889 minusFgz

JgxθgxPrime + lg1kgz1 xgz + lg1 tan θgx1113872 1113873 + lg1cgz1 xgz

prime + lg1θgxprime

cos2θgx

1113888 1113889 minus lg2kgz2 xgz minus lg2 tan θgx1113872 1113873 + lg2cgz2 xgzprime minus lg2

θgxprime

cos2θgx

1113888 1113889 Fgzlg3 minus FgyEz

JgzθgzPrime + lg1kgx1 xgx + lg1 tan θgz1113872 1113873 + lg1cgx1 xgx

prime + lg1θgzprime

cos2θgz

1113888 1113889 minus lg2kgx2 xgx minus lg2 tan θgz1113872 1113873 + lg2cgx2 xgxprime minus lg2

θgzprime

cos2θgz

1113888 1113889 Fgxlg3 minus FgyEx

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

8 Shock and Vibration

where TD is the input torque of motor T1 is the outputtorque of transmission T2 andT3 are the input and outputtorque for intermediate drive shaft T4 andT5 are the inputand output torque for main drive shaft T6 is the inputtorque of rear axle ki(i 1 2 7) are the equivalenttorsional stiffness ci(i 1 2 7) are the equivalent tor-sional damping ki(i 8 9) are the equivalent torsionalstiffness ci(i 8 9) are the equivalent torsional dampingmm is the mass of transmission shaft ki(i 10 11 12) arethe equivalent torsional stiffness for pinion shaft and dif-ferential axle ci(i 10 11 12) are the equivalent torsionaldamping for pinion shaft and differential axle cmy and cmz

are the equivalent damping of intermediate support alongaxes y and z kmy and kmz are the equivalent stiffness ofintermediate support along axes y and z Fmy and Fmz arecomponents of additional force Fi along axes y and z forintermediate support Ei(i x y z) are coordinates ofmeshing point along axes x y and z and TRL andTRR areload torque for wheels

26 Numerical Solution Procedure e fourth-order adap-tive step RungendashKutta method which is generally applicableto strong nonlinearity is adopted in the present work to solve

the equations In order to program a new state space isneeded

In consideration of equations (12)mdash(14) it can be de-rived that kinetic equations of 3 universal joints reduce thenumber of degree-of-freedom from 29 to 26 for the wholesystem e following state space coordinate vector alsodescribes the dynamic behaviors of the system

Ss θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 θ10 θ12 θ14 θ15 θ16 θ17θpy θpz θgx θgz ym zm xp yp zp xg yg zg

⎧⎨

⎩

⎫⎬

⎭

Τ

(29)

In order to eliminate the difference in magnitude be-tween parameters which will reduce the solvability and evenlead to no convergence for solving equations of motionmentioned above b is defined as the characteristic lengthand a new time parameter τ is introduced as follows

τ tωn (30)

where t denotes timeFor the sake of nominalizing equations a generalized

state space coordinate vector with 26 parameters is proposedto characterize dimensionless equations as

st sr1 sr2 sr3 sr4 sr5 sr6 sr7 sr8 sr10 sr12 sr16 sr17 srpx srpy srpz srgx srgy srgz smy smz spx spy spz sgx sgy sgz1113966 1113967T

(31)

e following equations are derived as

sri θi timesRi

b i 1 sim 8 10 12 16 17

srpx θ14 timesRpx

b

srgy θ15 timesRgy

b

srij θij timesRij

b i p g j x y z

six xi

b i p g

siy yi

b i m p g

siz zi

b i m p g

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

where b is the half of backlash and Ri is the equivalent gyrationradius which can be calculated by the following equation

Ri

Ji

mi

1113971

i 1 sim 8 10 12 16 17

Rij

Jij

mi

1113971

i p g j x y z

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(33)

where Ji and Jij are the equivalent moment of inertia and mi

is equivalent massA set of dimensionless parameters are defined as

xnτ xn

b

f xnτ( 1113857 f xt

n( 1113857

b

c(ijs) ci

mjRjRsωn

k(ijs) ki

mjRjRsω2n

cmi cm

miωn

kmi km

miω2n

T(xj) Tx

mjRjω2nb

lijτ lij

b

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)

As equations (30) (32) and (34) are substituted intoequation (28) the dimensionless equations of motion can bederived

Shock and Vibration 9

e equivalent inertia parameters are obtained in UGand stiffness parameters of powertrain are calculated byusing the relationships given in modeling section as listed inTable 1 Subsequently equivalent damping coefficients canbe obtained e parameters of the main reducer can becalculated by substituting system parameters listed in Table 2into equations (17)ndash(31) e solution procedures for di-mensionless parameter are carried out by using precedingcalculated parameters and the relationships proposed inprevious parts

e exciting and configuration parameters are set asθ1 1500 rpm Td2 100Nmiddotm km 9697e8Nm i1 135i2 41 2b 015mm em 0 α1 1695 degree α2 13044degree and α3 1554 degree respectively Without signif-icant loss of accuracy the fundamental harmonic form isadopted for time-varying mesh stiffness and static trans-mission error function whose initial phases are set at 0respectively

3 Numerical Results

31 Analysis of Coupled Torsional Vibration In order toanalysis coupled interaction of automotive powertrainFourier spectrums of torsional vibration response are cal-culated at the initial parameters as shown in Figure 6 Twopeaks and their sidebands are observed e torsional vi-brational responses are governed by the component of 50Hzwhich equals with exciting frequency of input torque asshown in equation (3) (ωf 150060 lowast 2 50Hz) isimplies that engine input excitation is responsible for tor-sional vibration and a valuable approach to decreasingtorsional vibration of powertrain is inferred In addition thecomponent with a frequency of around 185Hz is observedobviously for each considered part As one knows themeshing frequency undergoing this condition is presumably185185Hz (f 150060135185185Hz) which is quite inagreement with the frequency of the second peak Coupledvibration occurs between the torsional system and hypoidgear set and inner excitation caused by gear pair meshingprocess which is affected by clearance and time-varyingmeshing stiffness [25] accounts a lot for torsional vibrationat 185Hz of powertrain and affects torsional responsesslightly After simply calculation bandwidths reaching thesame value of around 37Hz of sidebands around two peaksare got U-joint is a transmission mechanism with non-constant angular speed which will excite a doubled fre-quency torsional vibration of input as shown in equations(12)ndash(14) Under this condition the doubled frequencyexcited by U-joint is calculated

fuminusjoint n

i1 lowast 60lowast 2

1500135lowast 60

lowast 2 3704 (35)

It can be conjectured that modulation phenomenonappears when automotive powertrain works between exci-tations attributed to input U-joint and gear pair As one cansee in Figure 6 compared torsional response of gear withthat located front of meshing point a lower value of tor-sional angular is observed and it is inferred that gear meshexcitations reduce torsional vibration for components

located behind meshing point It is noted that taking gearmesh behaviors into account is quite necessary to when atorsional characteristic analysis is conducted

32 Amplitude-Frequency Response of Torsional Vibration forAutomotive Powertrain e previous works [29] analyzedtorsional characteristics of the coupled system consisted ofdrive shaft and rear axle which denoted that resonance ofdrive shaft at the first two-order nature frequency accountsfor dramatic torsional vibration responses undergoing high-rotation-speed condition However it cannot reveal thetorsional vibration mechanism of automotive under low-and middle-speed condition

In order to obtain a more realistic understanding foramplitude-frequency response of automotive powertrainundergoing middle-speed condition theoretically the pre-ceding equations of motion are adopted A sweep frequencyanalysis procedure is carried out in a range from 1000 rpm to3500 rpm which respectively corresponds to frequencyinternal of 166ndash583Hz for input speed 3333ndash1167Hz forinput excitation and 12346ndash43209Hz for meshing exci-tation e response at input end of rear axle is taken as anexample in this work

Figure 7 shows the amplitude-frequency response inwhich abscissa denotes torsion speed of input and ordinate isRMS value of angular displacement As one can see threeobvious peaks which correspond to input speed at about1000 rpm 1600 rpm and 2800 rpm respectively are ob-served Owing to the assumption that the dramatic responsesare attributed to resonance of the system the first three-order nature frequency respect to the second-order term ofinput torque fluctuation is proposed at 333Hz 533Hz and933Hz successively It is noted that on the basis of previousanalysis these three peaks also correspond to frequencies of247Hz 395Hz and 691Hz for the U-joint excitation and1234Hz 1975Hz and 3456Hz for the meshing excitationPreliminary works [29] showed that resonance is attributedto input excitation of drive shaft and the meshing excitationof hypoid gear pair with higher exciting frequency thaninput will not excite prominent resonance behaviors of driveshaft as illustrated in Figure 8 ie meshing excitation hasslight influence on the torsional system which reaches thesame conclusion with this work According to analysis inSection 31 the sidebands respect to each exciting frequencyare (86ndash58) Hz at 333Hz (138ndash928) Hz at 533Hz and(242ndash1624) Hz at 933Hz for input torque and (987ndash1481)Hz at 1234Hz (158ndash237) Hz at 1975Hz and (2765ndash4147)Hz at 3456Hz for meshing behavior of the gear pair re-spectively As listed in Table 3 the first peak can be attributedto the first two-order resonances of drive shaft excited bysideband around 1234Hz Sidebands at 533Hz and1975Hz result in the second-order resonance which gen-erates the second peak Moreover the second-order reso-nances excited by the sideband at 933Hz and third-orderone caused by sideband at 3456Hz explain the appearanceof the third peak A typical phenomenon is observed that thetorsional vibration reduces a lot while one compares theamplitude of the third peak with another two which

10 Shock and Vibration

Table 1 e equivalent parameters of powertrain

Equivalent mass (kg) Equivalent moment of inertia (kgmiddotm2) Equivalent stiffness (Nm)m1 2 J1 0091125 k1 1002375m2 10 J2 03801735 k2 164025m3 06150 J3 000008103 k3 27500m4 13798 J4 0001001 k4 10e11m5 23581 J5 0001323 k5 479000m6 114323 J6 000249 k6 10e11m7 45323 J7 0001865 k7 21400m8 75567 J8 0002098 mdashm9 1284 J9 0001695 k8 43921799m10 1284 J10 0001695 mdashm11 13125 J11 000173 k9 41933635m12 13125 J12 000173 mdashm13 04254 J13 000002772 k10 10e11m14 17016 J14 00001109 km 96965e8m15 1375 J15 0034177 mdashm16 845 J16 99 k11 12032114m17 845 J17 99 k12 12032114

Table 2 e system parameters of the main reducer

Parameter Pinion GearNumber of teeth 10 41Module (mm) 400Mean pressure angle (deg) 20 20Helix angle at midpoint of the pinion (deg) 50 2713Cone angle (deg) 2312 6514Tooth width (mm) 4066 3452Tooth addendum (mm) 359 644Pinion outset (mm) minus38Mounting distance (mm) 112 64Pitch radius (mm) 40 108Backlash (mm) 01536Distance from pinion reference point to the x-o-z plane for pinion (mm) 40Distance from pinion reference point to the y-o-z plane for gear (mm) 101Distance between projection point and centroid of pinion shaft (mm) lp3 10 lg3 5

Distance between bearing reference point and centroid of pinion shaft (mm) lp1 100 lg1 90lp2 175 lg2 93

Equivalent mass (kg) mg 2127 mp 1375Jpx J14 Jgx 0050315

Equivalent moment of inertia (kgmiddotm2) Jpy 0002982 Jgx J15Jpz 0002982 Jpz 0037551

Meshing stiffnessof (i + 1)th tooth pair

Meshing stiffnessof ith tooth pair

t

k

Meshing stiffnessof (i ndash 1)th tooth pair

Total meshingstiffness km(t)

Figure 5 Schematic diagram of time-varying meshing stiffness on hypoid gear

Shock and Vibration 11

indicates that torsional behaviors of automotive powertrainaffect the vibration characteristics of the complete vehicleunder a low- or middle-input-speed condition In order tovalidate the preceding result an experimental study isnecessary

4 Experimental Analysis

A test bench which can be used to test the vibration behavior ofa transmission system used for four-wheel drive vehicle be-longing to the China Automotive Technology and ResearchCenter is adopted to perform this experiment ere are fivemotors in the test bench and three of them are used in thisexperiment One of them drives automotive powertrain whichcan generate a high frequency input torque with fluctuationvaried in accordance with necessary and others are used asloads In this test two photoelectric sensors and two magne-toelectric sensors are used to detect torsional vibration signalsand other five accelerometers are mounted to monitor trans-verse vibration Installation site for each sensor can be found inFigure 9 Signals are sampled and processed by LMS TestLab

In order to validate the feasibility of the theoreticalmodel the input torque simulated by input motor and loadsare consistent with parameters employed in numericalanalysis (mean value and the second-order term are con-tained only and fluctuation amplitude is set at 100NmiddotM forinput torque) Within the experiment the output speed ofmotor varies from 1000 rpm to 3500 rpm to conduct a sweepfrequency response analysis experimentally

As illustrated in Figure 10 the amplitude-frequencyresponses of torsional vibration have the similar feature for

different parts of powertrain Peaks can be observed forcarves at a similar input speed It is inferred that a presumeresonance occurs at each peak As a result the first three-order nature torsional frequencies can be predictedMeanwhile the same results are reached As shown inFigure 10 each curve has a higher value nearby the first andsecond peak With the further increase in frequency itreaches a lower plateau value Torsional behaviors have aneffect on the vibration characteristics under a low-input-speed condition Furthermore the part located behind themesh point such as output end of half-shaft has a lowerRMS value than transmission and drive shaft which can beattributed to reduction effect of mesh excitation on torsionalvibration

e experimental result of input end of rear axle iscompared with the numerical one As shown in Figure 11 aquite consistency is observed and the first three-order fre-quencies are inferred at 35Hz (1050 rpm) 557Hz(1670 rpm) and 883Hz (2650 rpm) for experimental resultby the same way mentioned in preceding section Comparedwith numerical predictions an acceptable difference ofabout 5 (the difference is 51 at 333Hz 45 at 533Hzand 536 at 933Hz) can be obtained ese indicate thatthe theoretical model is feasible for predicting nature tor-sional frequencies Moreover numerical torsional dis-placement is a little lower than the experimental one at thesame input speed which is below about 2450 rpm and a littlehigher with the further increase is can be presumablyattributed to assumptions made in this present work withrespect to the complexity of the system and the limitations ofthe lumped-mass method

03503

02502

01501

0050

0

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(a)

13Hz

50Hz

87Hz 150Hz185Hz222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(b)

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(c)

13Hz

50Hz

87Hz150Hz

185Hz222Hz

004

003

002

001

00 50 100 150 200 250

Frequency (Hz)To

rsio

nal d

ispla

cem

ent (

deg)300 350 400 450 500 550 600

(d)

Figure 6 e Fourier spectrum of torsional responses (a) output shaft of transmission (b) output end of the drive shaft system (c)torsional displacement of the pinion along the axis x (d) torsional displacement of the gear along the axis y

12 Shock and Vibration

e previous section indicates numerically that excita-tion of input dominates torsion vibration of the system andgear set affects responses slightly In order to validate thecorrectness of numerical analysis and feasibility decreasing

torsional vibration by reducing input excitation effect ofinput torque fluctuation on torsional response is studiedexperimentally e mean value of input torque is set at150NmiddotM and fluctuation of the second-order term is set at

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

2800rpm1600rpm

Figure 7 Amplitude-frequency response at input end of rear axle

8

7

6

5

4

2

3

1

00 1000 2000 3000

Rotation speed (RPM)

Vibr

atio

n am

plitu

de (micro

m)

4000 5000 6000

x directiony directionz direction

Figure 8 Influence of input rotation speed of drive shaft on vibration of the main reducer [29]

Table 3 e first six-order nature frequency of drive shaft [29]

Order 1 2 3 4 5 6Frequency (Hz) 1309 1402 4152 4422 6362 9066

Shock and Vibration 13

25NmiddotM 50NmiddotM 75NmiddotM 100NmiddotM 125NmiddotM 150NmiddotM175NmiddotM and 200NmiddotM in order

Together Figures 12 and 13 expectably show that thetorsional vibration is reinforced while increasing torquefluctuation Meanwhile an obvious increase in amplitudeoccurs at the second peak when fluctuation is increased from175NmiddotM to 200NmiddotM which indicates that a limitation should

be considered e accurate value can be determined by afurther study In summary reducing torque fluctuation is abeneficial approach for vibration damping Furthermore it isnoted that such a nonlinear phenomenon as natural fre-quency shift occurs while changing torque fluctuation Takingthe second-order torsional frequency as an example thefrequency value decreases with elevating fluctuation of input

Transmission

Photoelectricsensor

Photoelectricsensor

Magnetoelectricsensor

Magnetoelectricsensor

Rear axle

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Drive sha system

Figure 9 e experiment bench and arrangement of sensors

042

040

038

035

033

030

028

025

023

020

018

015

013

010

008

005

003

Amplitude-frequency curve ofle wheel

Amplitude-frequency curve oftransmission output sha

Amplitude-frequency curve ofrear axle output end

Amplitude-frequency curve ofright wheel

000

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (K

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Curve 167000

041

035

003

003

5567 8817

167000 264500

rpm

deg

deg

deg

deg

264500

012

005

003

764endash3

Figure 10 Amplitude-frequency responses of torsional vibration for powertrain at a torque fluctuation of 100NmiddotM

14 Shock and Vibration

036

034

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

Numerical result

Experimental result

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

Figure 11 Comparison of numerical and experimental results for input end of rear axle

160 5567 8817

264500

150

140

130

120

110

100

090

080

070

060

050

040

030

020

010

000 167000

25Nmiddotm

50Nmiddotm

75Nmiddotm

100Nmiddotm

125Nmiddotm150Nmiddotm

175Nmiddotm

200Nmiddotm

Curve 167000147078057045035025017013

rpm--------

264500009008007006005004004003

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (R

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Figure 12 Amplitude-frequency responses of torsional vibration for input end of rear axle at different torque fluctuations

Shock and Vibration 15

torque which shows that the torsional system has a stiffness-softening characteristic with respect to torque fluctuationis result can be accounted for a number of nonlinearitiessuch as friction clearance and sliding and the existence ofwhich sophisticates system nonlinear characteristics

5 Conclusion

In the present paper a lumped-parameter dynamic modelwith 29 degree-of-freedom is proposed considering thetransverse torsional and rocking coupled vibration for abetter understanding of the automotive powertrain systemdynamics and equations of motion are derived e nu-merical result shows that gear mesh excitations have littleeffects on torsional response for such components locatedbefore the mesh point as transmission transmission shaftand pinion but significantly reduce the torsional responsesfor ones behind the mesh point e amplitude-frequencycharacteristics of torsional vibration graphically indicate thattorsional behaviors of automotive powertrain only affect thevibration characteristics of complete vehicle under low-speed condition and predict the first three-order naturetorsional frequencies of the whole system An experimentalexamination is performed and validates the feasibility of theconsidered model e torsional vibration is reinforcedwhile increasing torque fluctuation In addition a stiffnesssoftening characteristic is observed for torsional vibrationexperimentally In order to reach a better understanding inamplitude-frequency response at low- and middle-input-speed condition a procedure to analyze mode shapes ofautomotive powertrain will be conducted in future works

Data Availability

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

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported and funded by the ApplicationTechnology Research and Development Plan of LiuzhouCity (grant no 2017AA10102) and the Key Projects ofUniversities in Inner Mongolia Autonomous Region (grantno NJZZ16369)e authors of this paper express their deepgratitude for the financial support of the research project

References

[1] F Vesali M A Rezvani and M Kashfi ldquoDynamics of uni-versal joints its failures and some propositions for practicallyimproving its performance and life expectancyrdquo Journal ofMechanical Science and Technology vol 26 no 8 pp 2439ndash2449 2012

[2] J-W Lu G-C Wang H Chen A F Vakakis andL A Bergman ldquoDynamic analysis of cross shaft type universaljoint with clearancerdquo Journal of Mechanical Science andTechnology vol 27 no 11 pp 3201ndash3205 2013

[3] G Wu W Shi and Z Chen ldquoe effect of multi-universalcoupling phase on torsional vibration of drive shaft and vi-bration of vehiclerdquo in Proceedings of the SAE 2013 WorldCongress amp Exhibition Detroit MI USA April 2013

0113333 4000 4500 5000 5500

5567

6000 6500 7000Frequency (Hz)

7500 8000 85008777

263300

9000 9500 10000 10500 11000 11667100

Am

plitu

de

Am

plitu

de (R

MS)

010

009

008

007

006

005

004

003

002

001

000100000 120000 140000 160000 180000

167000

25Nmiddotm

50Nmiddotm

75Nmiddotm100Nmiddotm

125Nmiddotm

150Nmiddotm

175Nmiddotm200Nmiddotm

Curve 167000010006004003003002001

982endash3

263300006005004003003002001

690endash3

rpm--------

200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000000

Figure 13 Amplitude-frequency responses of torsional vibration for output end of rear axle at different torque fluctuations

16 Shock and Vibration

[4] B A Porter ldquoeoretical analysis of the torsional oscillationof a system incorporating a Hookersquos jointrdquo Journal of Me-chanical Engineering Science vol 3 no 4 pp 324ndash329 1961

[5] S F Asokanthan and X-H Wang ldquoCharacterization oftorsional instabilities in a Hookersquos joint driven system viamaximal Lyapunov exponentsrdquo Journal of Sound and Vi-bration vol 194 no 1 pp 83ndash91 1996

[6] A Farshidianfar M Ebrahimi and H Bartlett ldquoHybridmodelling and simulation of the torsional vibration of vehicledriveline systemsrdquo Proceedings of the Institution of Me-chanical Engineers Part D Journal of Automobile Engineeringvol 215 no 2 pp 217ndash229 2001

[7] E O A Abd Elmaksoud E M Rabeih N A Abdel-halimand S M El Demerdash ldquoInvestigation of self excited tor-sional vibrations of different configurations of automatictransmission systems during engagementrdquo Engineeringvol 3 no 12 pp 1171ndash1181 2011

[8] T B Juang M Cheng and L Na ldquoExperimental and finiteelement analyses of a sliding-tube-type driveshaft-inducedvehicle vibrationrdquo Proceedings of the Institution of MechanicalEngineers Part K Journal of Multi-Body Dynamics vol 221no 3 pp 375ndash385 2007

[9] L F Coutinho and A Tamagna ldquoFlexural vibration analysisof constant velocity half-shaftsrdquo in Proceedings of the Inter-national Congress amp Exposition Chicago IL ISA 1996

[10] Y T Kang Y H Shen W Zhang and J Yang ldquoStabilityregion of floating intermediate support in a shaft system withmultiple universal jointsrdquo Journal of Mechanical Science andTechnology vol 28 no 6 pp 2733ndash2742 2014

[11] J L Xu T Yan B Peng and B Wei ldquoEffects of the trans-mission shaft on the main reducer vibration based onADAMS and experimental demonstrationrdquo AustralianJournal of Mechanical Engineering vol 15 no 1 pp 2ndash102015

[12] C X Zhang H B Li and D Q Jin ldquoSimulation on dynamicsof transmission shaft-rear driving axle of minibus withADAMSrdquo Applied Mechanics and Materials vol 651ndash653pp 862ndash865 2014

[13] J L Xu X Y Su and P Bo ldquoNumerical analysis anddemonstration transmission shaft influence on meshing vi-bration in driving and driven gearsrdquo Shock and Vibrationvol 2015 Article ID 365084 10 pages 2015

[14] J L Xu W Lei andW X Luo ldquoInfluence of bearing stiffnesson the nonlinear dynamics of a shaft-final drive systemrdquoShock and Vibration vol 2016 Article ID 3524609 14 pages2016

[15] A Kahraman and R Singh ldquoNon-linear dynamics of a spurgear pairrdquo Journal of Sound and Vibration vol 142 no 1pp 49ndash75 1990

[16] A Kahraman and R Singh ldquoInteractions between time-varying mesh stiffness and clearance non-linearities in ageared systemrdquo Journal of Sound and Vibration vol 146no 1 pp 135ndash156 1991

[17] A Kahraman ldquoEffect of axial vibrations on the dynamics of ahelical gear pairrdquo Journal of Vibration and Acoustics vol 115no 1 pp 33ndash39 1993

[18] Y Cai and T Hayashi ldquoe linear approximated equation ofvibration of a pair of spur gears (theory and experiment)rdquoJournal of Mechanical Design vol 116 no 2 pp 558ndash5641994

[19] P Velex andMMaatar ldquoAmathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound and Vibration vol 191no 5 pp 629ndash660 1996

[20] P Velex and V Cahouet ldquoExperimental and numerical in-vestigations on the influence of tooth friction in spur andhelical gear dynamicsrdquo Journal of Mechanical Design vol 122no 4 pp 515ndash522 2000

[21] P Velex and P Sainsot ldquoAn analytical study of tooth frictionexcitations in errorless spur and helical gearsrdquo Mechanismand Machine Ieory vol 37 no 7 pp 641ndash658 2002

[22] T C Lim and Y Cheng ldquoA theoretical study of the effect ofpinion offset on the dynamics of hypoid geared rotor systemrdquoJournal of Mechanical Design vol 121 no 4 pp 594ndash6011999

[23] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[24] Y Cheng and T C Lim ldquoDynamics of hypoid gear trans-mission with nonlinear time-varying mesh characteristicsrdquoJournal of Mechanical Design vol 125 no 2 pp 373ndash3822003

[25] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 2017 pp 302ndash329 2007

[26] J Wang and T C Lim ldquoEffect of tooth mesh stiffnessasymmetric nonlinearity for drive and coast sides on hypoidgear dynamicsrdquo Journal of Sound and Vibration vol 319no 3-5 pp 885ndash903 2009

[27] W M Li and H T Wang ldquoRigidity calculation of axialposition preload bearingsrdquo Journal of Hebei University ofTechnology vol 30 no 2 pp 15ndash19 2001

[28] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo Non-linear Dynamics vol 75 no 4 pp 783ndash806 2014

[29] X Liu Z Wu J Lu and J Xu ldquoInvestigation of the effect ofrotation speed on vibration responses of transmission sys-temrdquo Matec Web of Conferences vol 256 no 3 Article ID02019 2019

Shock and Vibration 17

where TD is the input torque of motor T1 is the outputtorque of transmission T2 andT3 are the input and outputtorque for intermediate drive shaft T4 andT5 are the inputand output torque for main drive shaft T6 is the inputtorque of rear axle ki(i 1 2 7) are the equivalenttorsional stiffness ci(i 1 2 7) are the equivalent tor-sional damping ki(i 8 9) are the equivalent torsionalstiffness ci(i 8 9) are the equivalent torsional dampingmm is the mass of transmission shaft ki(i 10 11 12) arethe equivalent torsional stiffness for pinion shaft and dif-ferential axle ci(i 10 11 12) are the equivalent torsionaldamping for pinion shaft and differential axle cmy and cmz

are the equivalent damping of intermediate support alongaxes y and z kmy and kmz are the equivalent stiffness ofintermediate support along axes y and z Fmy and Fmz arecomponents of additional force Fi along axes y and z forintermediate support Ei(i x y z) are coordinates ofmeshing point along axes x y and z and TRL andTRR areload torque for wheels

26 Numerical Solution Procedure e fourth-order adap-tive step RungendashKutta method which is generally applicableto strong nonlinearity is adopted in the present work to solve

the equations In order to program a new state space isneeded

In consideration of equations (12)mdash(14) it can be de-rived that kinetic equations of 3 universal joints reduce thenumber of degree-of-freedom from 29 to 26 for the wholesystem e following state space coordinate vector alsodescribes the dynamic behaviors of the system

Ss θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 θ10 θ12 θ14 θ15 θ16 θ17θpy θpz θgx θgz ym zm xp yp zp xg yg zg

⎧⎨

⎩

⎫⎬

⎭

Τ

(29)

In order to eliminate the difference in magnitude be-tween parameters which will reduce the solvability and evenlead to no convergence for solving equations of motionmentioned above b is defined as the characteristic lengthand a new time parameter τ is introduced as follows

τ tωn (30)

where t denotes timeFor the sake of nominalizing equations a generalized

state space coordinate vector with 26 parameters is proposedto characterize dimensionless equations as

st sr1 sr2 sr3 sr4 sr5 sr6 sr7 sr8 sr10 sr12 sr16 sr17 srpx srpy srpz srgx srgy srgz smy smz spx spy spz sgx sgy sgz1113966 1113967T

(31)

e following equations are derived as

sri θi timesRi

b i 1 sim 8 10 12 16 17

srpx θ14 timesRpx

b

srgy θ15 timesRgy

b

srij θij timesRij

b i p g j x y z

six xi

b i p g

siy yi

b i m p g

siz zi

b i m p g

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

where b is the half of backlash and Ri is the equivalent gyrationradius which can be calculated by the following equation

Ri

Ji

mi

1113971

i 1 sim 8 10 12 16 17

Rij

Jij

mi

1113971

i p g j x y z

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(33)

where Ji and Jij are the equivalent moment of inertia and mi

is equivalent massA set of dimensionless parameters are defined as

xnτ xn

b

f xnτ( 1113857 f xt

n( 1113857

b

c(ijs) ci

mjRjRsωn

k(ijs) ki

mjRjRsω2n

cmi cm

miωn

kmi km

miω2n

T(xj) Tx

mjRjω2nb

lijτ lij

b

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)

As equations (30) (32) and (34) are substituted intoequation (28) the dimensionless equations of motion can bederived

Shock and Vibration 9

e equivalent inertia parameters are obtained in UGand stiffness parameters of powertrain are calculated byusing the relationships given in modeling section as listed inTable 1 Subsequently equivalent damping coefficients canbe obtained e parameters of the main reducer can becalculated by substituting system parameters listed in Table 2into equations (17)ndash(31) e solution procedures for di-mensionless parameter are carried out by using precedingcalculated parameters and the relationships proposed inprevious parts

e exciting and configuration parameters are set asθ1 1500 rpm Td2 100Nmiddotm km 9697e8Nm i1 135i2 41 2b 015mm em 0 α1 1695 degree α2 13044degree and α3 1554 degree respectively Without signif-icant loss of accuracy the fundamental harmonic form isadopted for time-varying mesh stiffness and static trans-mission error function whose initial phases are set at 0respectively

3 Numerical Results

31 Analysis of Coupled Torsional Vibration In order toanalysis coupled interaction of automotive powertrainFourier spectrums of torsional vibration response are cal-culated at the initial parameters as shown in Figure 6 Twopeaks and their sidebands are observed e torsional vi-brational responses are governed by the component of 50Hzwhich equals with exciting frequency of input torque asshown in equation (3) (ωf 150060 lowast 2 50Hz) isimplies that engine input excitation is responsible for tor-sional vibration and a valuable approach to decreasingtorsional vibration of powertrain is inferred In addition thecomponent with a frequency of around 185Hz is observedobviously for each considered part As one knows themeshing frequency undergoing this condition is presumably185185Hz (f 150060135185185Hz) which is quite inagreement with the frequency of the second peak Coupledvibration occurs between the torsional system and hypoidgear set and inner excitation caused by gear pair meshingprocess which is affected by clearance and time-varyingmeshing stiffness [25] accounts a lot for torsional vibrationat 185Hz of powertrain and affects torsional responsesslightly After simply calculation bandwidths reaching thesame value of around 37Hz of sidebands around two peaksare got U-joint is a transmission mechanism with non-constant angular speed which will excite a doubled fre-quency torsional vibration of input as shown in equations(12)ndash(14) Under this condition the doubled frequencyexcited by U-joint is calculated

fuminusjoint n

i1 lowast 60lowast 2

1500135lowast 60

lowast 2 3704 (35)

It can be conjectured that modulation phenomenonappears when automotive powertrain works between exci-tations attributed to input U-joint and gear pair As one cansee in Figure 6 compared torsional response of gear withthat located front of meshing point a lower value of tor-sional angular is observed and it is inferred that gear meshexcitations reduce torsional vibration for components

located behind meshing point It is noted that taking gearmesh behaviors into account is quite necessary to when atorsional characteristic analysis is conducted

32 Amplitude-Frequency Response of Torsional Vibration forAutomotive Powertrain e previous works [29] analyzedtorsional characteristics of the coupled system consisted ofdrive shaft and rear axle which denoted that resonance ofdrive shaft at the first two-order nature frequency accountsfor dramatic torsional vibration responses undergoing high-rotation-speed condition However it cannot reveal thetorsional vibration mechanism of automotive under low-and middle-speed condition

In order to obtain a more realistic understanding foramplitude-frequency response of automotive powertrainundergoing middle-speed condition theoretically the pre-ceding equations of motion are adopted A sweep frequencyanalysis procedure is carried out in a range from 1000 rpm to3500 rpm which respectively corresponds to frequencyinternal of 166ndash583Hz for input speed 3333ndash1167Hz forinput excitation and 12346ndash43209Hz for meshing exci-tation e response at input end of rear axle is taken as anexample in this work

Figure 7 shows the amplitude-frequency response inwhich abscissa denotes torsion speed of input and ordinate isRMS value of angular displacement As one can see threeobvious peaks which correspond to input speed at about1000 rpm 1600 rpm and 2800 rpm respectively are ob-served Owing to the assumption that the dramatic responsesare attributed to resonance of the system the first three-order nature frequency respect to the second-order term ofinput torque fluctuation is proposed at 333Hz 533Hz and933Hz successively It is noted that on the basis of previousanalysis these three peaks also correspond to frequencies of247Hz 395Hz and 691Hz for the U-joint excitation and1234Hz 1975Hz and 3456Hz for the meshing excitationPreliminary works [29] showed that resonance is attributedto input excitation of drive shaft and the meshing excitationof hypoid gear pair with higher exciting frequency thaninput will not excite prominent resonance behaviors of driveshaft as illustrated in Figure 8 ie meshing excitation hasslight influence on the torsional system which reaches thesame conclusion with this work According to analysis inSection 31 the sidebands respect to each exciting frequencyare (86ndash58) Hz at 333Hz (138ndash928) Hz at 533Hz and(242ndash1624) Hz at 933Hz for input torque and (987ndash1481)Hz at 1234Hz (158ndash237) Hz at 1975Hz and (2765ndash4147)Hz at 3456Hz for meshing behavior of the gear pair re-spectively As listed in Table 3 the first peak can be attributedto the first two-order resonances of drive shaft excited bysideband around 1234Hz Sidebands at 533Hz and1975Hz result in the second-order resonance which gen-erates the second peak Moreover the second-order reso-nances excited by the sideband at 933Hz and third-orderone caused by sideband at 3456Hz explain the appearanceof the third peak A typical phenomenon is observed that thetorsional vibration reduces a lot while one compares theamplitude of the third peak with another two which

10 Shock and Vibration

Table 1 e equivalent parameters of powertrain

Equivalent mass (kg) Equivalent moment of inertia (kgmiddotm2) Equivalent stiffness (Nm)m1 2 J1 0091125 k1 1002375m2 10 J2 03801735 k2 164025m3 06150 J3 000008103 k3 27500m4 13798 J4 0001001 k4 10e11m5 23581 J5 0001323 k5 479000m6 114323 J6 000249 k6 10e11m7 45323 J7 0001865 k7 21400m8 75567 J8 0002098 mdashm9 1284 J9 0001695 k8 43921799m10 1284 J10 0001695 mdashm11 13125 J11 000173 k9 41933635m12 13125 J12 000173 mdashm13 04254 J13 000002772 k10 10e11m14 17016 J14 00001109 km 96965e8m15 1375 J15 0034177 mdashm16 845 J16 99 k11 12032114m17 845 J17 99 k12 12032114

Table 2 e system parameters of the main reducer

Parameter Pinion GearNumber of teeth 10 41Module (mm) 400Mean pressure angle (deg) 20 20Helix angle at midpoint of the pinion (deg) 50 2713Cone angle (deg) 2312 6514Tooth width (mm) 4066 3452Tooth addendum (mm) 359 644Pinion outset (mm) minus38Mounting distance (mm) 112 64Pitch radius (mm) 40 108Backlash (mm) 01536Distance from pinion reference point to the x-o-z plane for pinion (mm) 40Distance from pinion reference point to the y-o-z plane for gear (mm) 101Distance between projection point and centroid of pinion shaft (mm) lp3 10 lg3 5

Distance between bearing reference point and centroid of pinion shaft (mm) lp1 100 lg1 90lp2 175 lg2 93

Equivalent mass (kg) mg 2127 mp 1375Jpx J14 Jgx 0050315

Equivalent moment of inertia (kgmiddotm2) Jpy 0002982 Jgx J15Jpz 0002982 Jpz 0037551

Meshing stiffnessof (i + 1)th tooth pair

Meshing stiffnessof ith tooth pair

t

k

Meshing stiffnessof (i ndash 1)th tooth pair

Total meshingstiffness km(t)

Figure 5 Schematic diagram of time-varying meshing stiffness on hypoid gear

Shock and Vibration 11

indicates that torsional behaviors of automotive powertrainaffect the vibration characteristics of the complete vehicleunder a low- or middle-input-speed condition In order tovalidate the preceding result an experimental study isnecessary

4 Experimental Analysis

A test bench which can be used to test the vibration behavior ofa transmission system used for four-wheel drive vehicle be-longing to the China Automotive Technology and ResearchCenter is adopted to perform this experiment ere are fivemotors in the test bench and three of them are used in thisexperiment One of them drives automotive powertrain whichcan generate a high frequency input torque with fluctuationvaried in accordance with necessary and others are used asloads In this test two photoelectric sensors and two magne-toelectric sensors are used to detect torsional vibration signalsand other five accelerometers are mounted to monitor trans-verse vibration Installation site for each sensor can be found inFigure 9 Signals are sampled and processed by LMS TestLab

In order to validate the feasibility of the theoreticalmodel the input torque simulated by input motor and loadsare consistent with parameters employed in numericalanalysis (mean value and the second-order term are con-tained only and fluctuation amplitude is set at 100NmiddotM forinput torque) Within the experiment the output speed ofmotor varies from 1000 rpm to 3500 rpm to conduct a sweepfrequency response analysis experimentally

As illustrated in Figure 10 the amplitude-frequencyresponses of torsional vibration have the similar feature for

different parts of powertrain Peaks can be observed forcarves at a similar input speed It is inferred that a presumeresonance occurs at each peak As a result the first three-order nature torsional frequencies can be predictedMeanwhile the same results are reached As shown inFigure 10 each curve has a higher value nearby the first andsecond peak With the further increase in frequency itreaches a lower plateau value Torsional behaviors have aneffect on the vibration characteristics under a low-input-speed condition Furthermore the part located behind themesh point such as output end of half-shaft has a lowerRMS value than transmission and drive shaft which can beattributed to reduction effect of mesh excitation on torsionalvibration

e experimental result of input end of rear axle iscompared with the numerical one As shown in Figure 11 aquite consistency is observed and the first three-order fre-quencies are inferred at 35Hz (1050 rpm) 557Hz(1670 rpm) and 883Hz (2650 rpm) for experimental resultby the same way mentioned in preceding section Comparedwith numerical predictions an acceptable difference ofabout 5 (the difference is 51 at 333Hz 45 at 533Hzand 536 at 933Hz) can be obtained ese indicate thatthe theoretical model is feasible for predicting nature tor-sional frequencies Moreover numerical torsional dis-placement is a little lower than the experimental one at thesame input speed which is below about 2450 rpm and a littlehigher with the further increase is can be presumablyattributed to assumptions made in this present work withrespect to the complexity of the system and the limitations ofthe lumped-mass method

03503

02502

01501

0050

0

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(a)

13Hz

50Hz

87Hz 150Hz185Hz222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(b)

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(c)

13Hz

50Hz

87Hz150Hz

185Hz222Hz

004

003

002

001

00 50 100 150 200 250

Frequency (Hz)To

rsio

nal d

ispla

cem

ent (

deg)300 350 400 450 500 550 600

(d)

Figure 6 e Fourier spectrum of torsional responses (a) output shaft of transmission (b) output end of the drive shaft system (c)torsional displacement of the pinion along the axis x (d) torsional displacement of the gear along the axis y

12 Shock and Vibration

e previous section indicates numerically that excita-tion of input dominates torsion vibration of the system andgear set affects responses slightly In order to validate thecorrectness of numerical analysis and feasibility decreasing

torsional vibration by reducing input excitation effect ofinput torque fluctuation on torsional response is studiedexperimentally e mean value of input torque is set at150NmiddotM and fluctuation of the second-order term is set at

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

2800rpm1600rpm

Figure 7 Amplitude-frequency response at input end of rear axle

8

7

6

5

4

2

3

1

00 1000 2000 3000

Rotation speed (RPM)

Vibr

atio

n am

plitu

de (micro

m)

4000 5000 6000

x directiony directionz direction

Figure 8 Influence of input rotation speed of drive shaft on vibration of the main reducer [29]

Table 3 e first six-order nature frequency of drive shaft [29]

Order 1 2 3 4 5 6Frequency (Hz) 1309 1402 4152 4422 6362 9066

Shock and Vibration 13

25NmiddotM 50NmiddotM 75NmiddotM 100NmiddotM 125NmiddotM 150NmiddotM175NmiddotM and 200NmiddotM in order

Together Figures 12 and 13 expectably show that thetorsional vibration is reinforced while increasing torquefluctuation Meanwhile an obvious increase in amplitudeoccurs at the second peak when fluctuation is increased from175NmiddotM to 200NmiddotM which indicates that a limitation should

be considered e accurate value can be determined by afurther study In summary reducing torque fluctuation is abeneficial approach for vibration damping Furthermore it isnoted that such a nonlinear phenomenon as natural fre-quency shift occurs while changing torque fluctuation Takingthe second-order torsional frequency as an example thefrequency value decreases with elevating fluctuation of input

Transmission

Photoelectricsensor

Photoelectricsensor

Magnetoelectricsensor

Magnetoelectricsensor

Rear axle

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Drive sha system

Figure 9 e experiment bench and arrangement of sensors

042

040

038

035

033

030

028

025

023

020

018

015

013

010

008

005

003

Amplitude-frequency curve ofle wheel

Amplitude-frequency curve oftransmission output sha

Amplitude-frequency curve ofrear axle output end

Amplitude-frequency curve ofright wheel

000

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (K

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Curve 167000

041

035

003

003

5567 8817

167000 264500

rpm

deg

deg

deg

deg

264500

012

005

003

764endash3

Figure 10 Amplitude-frequency responses of torsional vibration for powertrain at a torque fluctuation of 100NmiddotM

14 Shock and Vibration

036

034

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

Numerical result

Experimental result

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

Figure 11 Comparison of numerical and experimental results for input end of rear axle

160 5567 8817

264500

150

140

130

120

110

100

090

080

070

060

050

040

030

020

010

000 167000

25Nmiddotm

50Nmiddotm

75Nmiddotm

100Nmiddotm

125Nmiddotm150Nmiddotm

175Nmiddotm

200Nmiddotm

Curve 167000147078057045035025017013

rpm--------

264500009008007006005004004003

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (R

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Figure 12 Amplitude-frequency responses of torsional vibration for input end of rear axle at different torque fluctuations

Shock and Vibration 15

torque which shows that the torsional system has a stiffness-softening characteristic with respect to torque fluctuationis result can be accounted for a number of nonlinearitiessuch as friction clearance and sliding and the existence ofwhich sophisticates system nonlinear characteristics

5 Conclusion

In the present paper a lumped-parameter dynamic modelwith 29 degree-of-freedom is proposed considering thetransverse torsional and rocking coupled vibration for abetter understanding of the automotive powertrain systemdynamics and equations of motion are derived e nu-merical result shows that gear mesh excitations have littleeffects on torsional response for such components locatedbefore the mesh point as transmission transmission shaftand pinion but significantly reduce the torsional responsesfor ones behind the mesh point e amplitude-frequencycharacteristics of torsional vibration graphically indicate thattorsional behaviors of automotive powertrain only affect thevibration characteristics of complete vehicle under low-speed condition and predict the first three-order naturetorsional frequencies of the whole system An experimentalexamination is performed and validates the feasibility of theconsidered model e torsional vibration is reinforcedwhile increasing torque fluctuation In addition a stiffnesssoftening characteristic is observed for torsional vibrationexperimentally In order to reach a better understanding inamplitude-frequency response at low- and middle-input-speed condition a procedure to analyze mode shapes ofautomotive powertrain will be conducted in future works

Data Availability

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

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported and funded by the ApplicationTechnology Research and Development Plan of LiuzhouCity (grant no 2017AA10102) and the Key Projects ofUniversities in Inner Mongolia Autonomous Region (grantno NJZZ16369)e authors of this paper express their deepgratitude for the financial support of the research project

References

[1] F Vesali M A Rezvani and M Kashfi ldquoDynamics of uni-versal joints its failures and some propositions for practicallyimproving its performance and life expectancyrdquo Journal ofMechanical Science and Technology vol 26 no 8 pp 2439ndash2449 2012

[2] J-W Lu G-C Wang H Chen A F Vakakis andL A Bergman ldquoDynamic analysis of cross shaft type universaljoint with clearancerdquo Journal of Mechanical Science andTechnology vol 27 no 11 pp 3201ndash3205 2013

[3] G Wu W Shi and Z Chen ldquoe effect of multi-universalcoupling phase on torsional vibration of drive shaft and vi-bration of vehiclerdquo in Proceedings of the SAE 2013 WorldCongress amp Exhibition Detroit MI USA April 2013

0113333 4000 4500 5000 5500

5567

6000 6500 7000Frequency (Hz)

7500 8000 85008777

263300

9000 9500 10000 10500 11000 11667100

Am

plitu

de

Am

plitu

de (R

MS)

010

009

008

007

006

005

004

003

002

001

000100000 120000 140000 160000 180000

167000

25Nmiddotm

50Nmiddotm

75Nmiddotm100Nmiddotm

125Nmiddotm

150Nmiddotm

175Nmiddotm200Nmiddotm

Curve 167000010006004003003002001

982endash3

263300006005004003003002001

690endash3

rpm--------

200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000000

Figure 13 Amplitude-frequency responses of torsional vibration for output end of rear axle at different torque fluctuations

16 Shock and Vibration

[4] B A Porter ldquoeoretical analysis of the torsional oscillationof a system incorporating a Hookersquos jointrdquo Journal of Me-chanical Engineering Science vol 3 no 4 pp 324ndash329 1961

[5] S F Asokanthan and X-H Wang ldquoCharacterization oftorsional instabilities in a Hookersquos joint driven system viamaximal Lyapunov exponentsrdquo Journal of Sound and Vi-bration vol 194 no 1 pp 83ndash91 1996

[6] A Farshidianfar M Ebrahimi and H Bartlett ldquoHybridmodelling and simulation of the torsional vibration of vehicledriveline systemsrdquo Proceedings of the Institution of Me-chanical Engineers Part D Journal of Automobile Engineeringvol 215 no 2 pp 217ndash229 2001

[7] E O A Abd Elmaksoud E M Rabeih N A Abdel-halimand S M El Demerdash ldquoInvestigation of self excited tor-sional vibrations of different configurations of automatictransmission systems during engagementrdquo Engineeringvol 3 no 12 pp 1171ndash1181 2011

[8] T B Juang M Cheng and L Na ldquoExperimental and finiteelement analyses of a sliding-tube-type driveshaft-inducedvehicle vibrationrdquo Proceedings of the Institution of MechanicalEngineers Part K Journal of Multi-Body Dynamics vol 221no 3 pp 375ndash385 2007

[9] L F Coutinho and A Tamagna ldquoFlexural vibration analysisof constant velocity half-shaftsrdquo in Proceedings of the Inter-national Congress amp Exposition Chicago IL ISA 1996

[10] Y T Kang Y H Shen W Zhang and J Yang ldquoStabilityregion of floating intermediate support in a shaft system withmultiple universal jointsrdquo Journal of Mechanical Science andTechnology vol 28 no 6 pp 2733ndash2742 2014

[11] J L Xu T Yan B Peng and B Wei ldquoEffects of the trans-mission shaft on the main reducer vibration based onADAMS and experimental demonstrationrdquo AustralianJournal of Mechanical Engineering vol 15 no 1 pp 2ndash102015

[12] C X Zhang H B Li and D Q Jin ldquoSimulation on dynamicsof transmission shaft-rear driving axle of minibus withADAMSrdquo Applied Mechanics and Materials vol 651ndash653pp 862ndash865 2014

[13] J L Xu X Y Su and P Bo ldquoNumerical analysis anddemonstration transmission shaft influence on meshing vi-bration in driving and driven gearsrdquo Shock and Vibrationvol 2015 Article ID 365084 10 pages 2015

[14] J L Xu W Lei andW X Luo ldquoInfluence of bearing stiffnesson the nonlinear dynamics of a shaft-final drive systemrdquoShock and Vibration vol 2016 Article ID 3524609 14 pages2016

[15] A Kahraman and R Singh ldquoNon-linear dynamics of a spurgear pairrdquo Journal of Sound and Vibration vol 142 no 1pp 49ndash75 1990

[16] A Kahraman and R Singh ldquoInteractions between time-varying mesh stiffness and clearance non-linearities in ageared systemrdquo Journal of Sound and Vibration vol 146no 1 pp 135ndash156 1991

[17] A Kahraman ldquoEffect of axial vibrations on the dynamics of ahelical gear pairrdquo Journal of Vibration and Acoustics vol 115no 1 pp 33ndash39 1993

[18] Y Cai and T Hayashi ldquoe linear approximated equation ofvibration of a pair of spur gears (theory and experiment)rdquoJournal of Mechanical Design vol 116 no 2 pp 558ndash5641994

[19] P Velex andMMaatar ldquoAmathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound and Vibration vol 191no 5 pp 629ndash660 1996

[20] P Velex and V Cahouet ldquoExperimental and numerical in-vestigations on the influence of tooth friction in spur andhelical gear dynamicsrdquo Journal of Mechanical Design vol 122no 4 pp 515ndash522 2000

[21] P Velex and P Sainsot ldquoAn analytical study of tooth frictionexcitations in errorless spur and helical gearsrdquo Mechanismand Machine Ieory vol 37 no 7 pp 641ndash658 2002

[22] T C Lim and Y Cheng ldquoA theoretical study of the effect ofpinion offset on the dynamics of hypoid geared rotor systemrdquoJournal of Mechanical Design vol 121 no 4 pp 594ndash6011999

[23] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[24] Y Cheng and T C Lim ldquoDynamics of hypoid gear trans-mission with nonlinear time-varying mesh characteristicsrdquoJournal of Mechanical Design vol 125 no 2 pp 373ndash3822003

[25] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 2017 pp 302ndash329 2007

[26] J Wang and T C Lim ldquoEffect of tooth mesh stiffnessasymmetric nonlinearity for drive and coast sides on hypoidgear dynamicsrdquo Journal of Sound and Vibration vol 319no 3-5 pp 885ndash903 2009

[27] W M Li and H T Wang ldquoRigidity calculation of axialposition preload bearingsrdquo Journal of Hebei University ofTechnology vol 30 no 2 pp 15ndash19 2001

[28] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo Non-linear Dynamics vol 75 no 4 pp 783ndash806 2014

[29] X Liu Z Wu J Lu and J Xu ldquoInvestigation of the effect ofrotation speed on vibration responses of transmission sys-temrdquo Matec Web of Conferences vol 256 no 3 Article ID02019 2019

Shock and Vibration 17

e equivalent inertia parameters are obtained in UGand stiffness parameters of powertrain are calculated byusing the relationships given in modeling section as listed inTable 1 Subsequently equivalent damping coefficients canbe obtained e parameters of the main reducer can becalculated by substituting system parameters listed in Table 2into equations (17)ndash(31) e solution procedures for di-mensionless parameter are carried out by using precedingcalculated parameters and the relationships proposed inprevious parts

e exciting and configuration parameters are set asθ1 1500 rpm Td2 100Nmiddotm km 9697e8Nm i1 135i2 41 2b 015mm em 0 α1 1695 degree α2 13044degree and α3 1554 degree respectively Without signif-icant loss of accuracy the fundamental harmonic form isadopted for time-varying mesh stiffness and static trans-mission error function whose initial phases are set at 0respectively

3 Numerical Results

31 Analysis of Coupled Torsional Vibration In order toanalysis coupled interaction of automotive powertrainFourier spectrums of torsional vibration response are cal-culated at the initial parameters as shown in Figure 6 Twopeaks and their sidebands are observed e torsional vi-brational responses are governed by the component of 50Hzwhich equals with exciting frequency of input torque asshown in equation (3) (ωf 150060 lowast 2 50Hz) isimplies that engine input excitation is responsible for tor-sional vibration and a valuable approach to decreasingtorsional vibration of powertrain is inferred In addition thecomponent with a frequency of around 185Hz is observedobviously for each considered part As one knows themeshing frequency undergoing this condition is presumably185185Hz (f 150060135185185Hz) which is quite inagreement with the frequency of the second peak Coupledvibration occurs between the torsional system and hypoidgear set and inner excitation caused by gear pair meshingprocess which is affected by clearance and time-varyingmeshing stiffness [25] accounts a lot for torsional vibrationat 185Hz of powertrain and affects torsional responsesslightly After simply calculation bandwidths reaching thesame value of around 37Hz of sidebands around two peaksare got U-joint is a transmission mechanism with non-constant angular speed which will excite a doubled fre-quency torsional vibration of input as shown in equations(12)ndash(14) Under this condition the doubled frequencyexcited by U-joint is calculated

fuminusjoint n

i1 lowast 60lowast 2

1500135lowast 60

lowast 2 3704 (35)

It can be conjectured that modulation phenomenonappears when automotive powertrain works between exci-tations attributed to input U-joint and gear pair As one cansee in Figure 6 compared torsional response of gear withthat located front of meshing point a lower value of tor-sional angular is observed and it is inferred that gear meshexcitations reduce torsional vibration for components

located behind meshing point It is noted that taking gearmesh behaviors into account is quite necessary to when atorsional characteristic analysis is conducted

32 Amplitude-Frequency Response of Torsional Vibration forAutomotive Powertrain e previous works [29] analyzedtorsional characteristics of the coupled system consisted ofdrive shaft and rear axle which denoted that resonance ofdrive shaft at the first two-order nature frequency accountsfor dramatic torsional vibration responses undergoing high-rotation-speed condition However it cannot reveal thetorsional vibration mechanism of automotive under low-and middle-speed condition

In order to obtain a more realistic understanding foramplitude-frequency response of automotive powertrainundergoing middle-speed condition theoretically the pre-ceding equations of motion are adopted A sweep frequencyanalysis procedure is carried out in a range from 1000 rpm to3500 rpm which respectively corresponds to frequencyinternal of 166ndash583Hz for input speed 3333ndash1167Hz forinput excitation and 12346ndash43209Hz for meshing exci-tation e response at input end of rear axle is taken as anexample in this work

Figure 7 shows the amplitude-frequency response inwhich abscissa denotes torsion speed of input and ordinate isRMS value of angular displacement As one can see threeobvious peaks which correspond to input speed at about1000 rpm 1600 rpm and 2800 rpm respectively are ob-served Owing to the assumption that the dramatic responsesare attributed to resonance of the system the first three-order nature frequency respect to the second-order term ofinput torque fluctuation is proposed at 333Hz 533Hz and933Hz successively It is noted that on the basis of previousanalysis these three peaks also correspond to frequencies of247Hz 395Hz and 691Hz for the U-joint excitation and1234Hz 1975Hz and 3456Hz for the meshing excitationPreliminary works [29] showed that resonance is attributedto input excitation of drive shaft and the meshing excitationof hypoid gear pair with higher exciting frequency thaninput will not excite prominent resonance behaviors of driveshaft as illustrated in Figure 8 ie meshing excitation hasslight influence on the torsional system which reaches thesame conclusion with this work According to analysis inSection 31 the sidebands respect to each exciting frequencyare (86ndash58) Hz at 333Hz (138ndash928) Hz at 533Hz and(242ndash1624) Hz at 933Hz for input torque and (987ndash1481)Hz at 1234Hz (158ndash237) Hz at 1975Hz and (2765ndash4147)Hz at 3456Hz for meshing behavior of the gear pair re-spectively As listed in Table 3 the first peak can be attributedto the first two-order resonances of drive shaft excited bysideband around 1234Hz Sidebands at 533Hz and1975Hz result in the second-order resonance which gen-erates the second peak Moreover the second-order reso-nances excited by the sideband at 933Hz and third-orderone caused by sideband at 3456Hz explain the appearanceof the third peak A typical phenomenon is observed that thetorsional vibration reduces a lot while one compares theamplitude of the third peak with another two which

10 Shock and Vibration

Table 1 e equivalent parameters of powertrain

Equivalent mass (kg) Equivalent moment of inertia (kgmiddotm2) Equivalent stiffness (Nm)m1 2 J1 0091125 k1 1002375m2 10 J2 03801735 k2 164025m3 06150 J3 000008103 k3 27500m4 13798 J4 0001001 k4 10e11m5 23581 J5 0001323 k5 479000m6 114323 J6 000249 k6 10e11m7 45323 J7 0001865 k7 21400m8 75567 J8 0002098 mdashm9 1284 J9 0001695 k8 43921799m10 1284 J10 0001695 mdashm11 13125 J11 000173 k9 41933635m12 13125 J12 000173 mdashm13 04254 J13 000002772 k10 10e11m14 17016 J14 00001109 km 96965e8m15 1375 J15 0034177 mdashm16 845 J16 99 k11 12032114m17 845 J17 99 k12 12032114

Table 2 e system parameters of the main reducer

Parameter Pinion GearNumber of teeth 10 41Module (mm) 400Mean pressure angle (deg) 20 20Helix angle at midpoint of the pinion (deg) 50 2713Cone angle (deg) 2312 6514Tooth width (mm) 4066 3452Tooth addendum (mm) 359 644Pinion outset (mm) minus38Mounting distance (mm) 112 64Pitch radius (mm) 40 108Backlash (mm) 01536Distance from pinion reference point to the x-o-z plane for pinion (mm) 40Distance from pinion reference point to the y-o-z plane for gear (mm) 101Distance between projection point and centroid of pinion shaft (mm) lp3 10 lg3 5

Distance between bearing reference point and centroid of pinion shaft (mm) lp1 100 lg1 90lp2 175 lg2 93

Equivalent mass (kg) mg 2127 mp 1375Jpx J14 Jgx 0050315

Equivalent moment of inertia (kgmiddotm2) Jpy 0002982 Jgx J15Jpz 0002982 Jpz 0037551

Meshing stiffnessof (i + 1)th tooth pair

Meshing stiffnessof ith tooth pair

t

k

Meshing stiffnessof (i ndash 1)th tooth pair

Total meshingstiffness km(t)

Figure 5 Schematic diagram of time-varying meshing stiffness on hypoid gear

Shock and Vibration 11

indicates that torsional behaviors of automotive powertrainaffect the vibration characteristics of the complete vehicleunder a low- or middle-input-speed condition In order tovalidate the preceding result an experimental study isnecessary

4 Experimental Analysis

A test bench which can be used to test the vibration behavior ofa transmission system used for four-wheel drive vehicle be-longing to the China Automotive Technology and ResearchCenter is adopted to perform this experiment ere are fivemotors in the test bench and three of them are used in thisexperiment One of them drives automotive powertrain whichcan generate a high frequency input torque with fluctuationvaried in accordance with necessary and others are used asloads In this test two photoelectric sensors and two magne-toelectric sensors are used to detect torsional vibration signalsand other five accelerometers are mounted to monitor trans-verse vibration Installation site for each sensor can be found inFigure 9 Signals are sampled and processed by LMS TestLab

In order to validate the feasibility of the theoreticalmodel the input torque simulated by input motor and loadsare consistent with parameters employed in numericalanalysis (mean value and the second-order term are con-tained only and fluctuation amplitude is set at 100NmiddotM forinput torque) Within the experiment the output speed ofmotor varies from 1000 rpm to 3500 rpm to conduct a sweepfrequency response analysis experimentally

As illustrated in Figure 10 the amplitude-frequencyresponses of torsional vibration have the similar feature for

different parts of powertrain Peaks can be observed forcarves at a similar input speed It is inferred that a presumeresonance occurs at each peak As a result the first three-order nature torsional frequencies can be predictedMeanwhile the same results are reached As shown inFigure 10 each curve has a higher value nearby the first andsecond peak With the further increase in frequency itreaches a lower plateau value Torsional behaviors have aneffect on the vibration characteristics under a low-input-speed condition Furthermore the part located behind themesh point such as output end of half-shaft has a lowerRMS value than transmission and drive shaft which can beattributed to reduction effect of mesh excitation on torsionalvibration

e experimental result of input end of rear axle iscompared with the numerical one As shown in Figure 11 aquite consistency is observed and the first three-order fre-quencies are inferred at 35Hz (1050 rpm) 557Hz(1670 rpm) and 883Hz (2650 rpm) for experimental resultby the same way mentioned in preceding section Comparedwith numerical predictions an acceptable difference ofabout 5 (the difference is 51 at 333Hz 45 at 533Hzand 536 at 933Hz) can be obtained ese indicate thatthe theoretical model is feasible for predicting nature tor-sional frequencies Moreover numerical torsional dis-placement is a little lower than the experimental one at thesame input speed which is below about 2450 rpm and a littlehigher with the further increase is can be presumablyattributed to assumptions made in this present work withrespect to the complexity of the system and the limitations ofthe lumped-mass method

03503

02502

01501

0050

0

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(a)

13Hz

50Hz

87Hz 150Hz185Hz222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(b)

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(c)

13Hz

50Hz

87Hz150Hz

185Hz222Hz

004

003

002

001

00 50 100 150 200 250

Frequency (Hz)To

rsio

nal d

ispla

cem

ent (

deg)300 350 400 450 500 550 600

(d)

Figure 6 e Fourier spectrum of torsional responses (a) output shaft of transmission (b) output end of the drive shaft system (c)torsional displacement of the pinion along the axis x (d) torsional displacement of the gear along the axis y

12 Shock and Vibration

e previous section indicates numerically that excita-tion of input dominates torsion vibration of the system andgear set affects responses slightly In order to validate thecorrectness of numerical analysis and feasibility decreasing

torsional vibration by reducing input excitation effect ofinput torque fluctuation on torsional response is studiedexperimentally e mean value of input torque is set at150NmiddotM and fluctuation of the second-order term is set at

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

2800rpm1600rpm

Figure 7 Amplitude-frequency response at input end of rear axle

8

7

6

5

4

2

3

1

00 1000 2000 3000

Rotation speed (RPM)

Vibr

atio

n am

plitu

de (micro

m)

4000 5000 6000

x directiony directionz direction

Figure 8 Influence of input rotation speed of drive shaft on vibration of the main reducer [29]

Table 3 e first six-order nature frequency of drive shaft [29]

Order 1 2 3 4 5 6Frequency (Hz) 1309 1402 4152 4422 6362 9066

Shock and Vibration 13

25NmiddotM 50NmiddotM 75NmiddotM 100NmiddotM 125NmiddotM 150NmiddotM175NmiddotM and 200NmiddotM in order

Together Figures 12 and 13 expectably show that thetorsional vibration is reinforced while increasing torquefluctuation Meanwhile an obvious increase in amplitudeoccurs at the second peak when fluctuation is increased from175NmiddotM to 200NmiddotM which indicates that a limitation should

be considered e accurate value can be determined by afurther study In summary reducing torque fluctuation is abeneficial approach for vibration damping Furthermore it isnoted that such a nonlinear phenomenon as natural fre-quency shift occurs while changing torque fluctuation Takingthe second-order torsional frequency as an example thefrequency value decreases with elevating fluctuation of input

Transmission

Photoelectricsensor

Photoelectricsensor

Magnetoelectricsensor

Magnetoelectricsensor

Rear axle

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Drive sha system

Figure 9 e experiment bench and arrangement of sensors

042

040

038

035

033

030

028

025

023

020

018

015

013

010

008

005

003

Amplitude-frequency curve ofle wheel

Amplitude-frequency curve oftransmission output sha

Amplitude-frequency curve ofrear axle output end

Amplitude-frequency curve ofright wheel

000

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (K

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Curve 167000

041

035

003

003

5567 8817

167000 264500

rpm

deg

deg

deg

deg

264500

012

005

003

764endash3

Figure 10 Amplitude-frequency responses of torsional vibration for powertrain at a torque fluctuation of 100NmiddotM

14 Shock and Vibration

036

034

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

Numerical result

Experimental result

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

Figure 11 Comparison of numerical and experimental results for input end of rear axle

160 5567 8817

264500

150

140

130

120

110

100

090

080

070

060

050

040

030

020

010

000 167000

25Nmiddotm

50Nmiddotm

75Nmiddotm

100Nmiddotm

125Nmiddotm150Nmiddotm

175Nmiddotm

200Nmiddotm

Curve 167000147078057045035025017013

rpm--------

264500009008007006005004004003

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (R

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Figure 12 Amplitude-frequency responses of torsional vibration for input end of rear axle at different torque fluctuations

Shock and Vibration 15

torque which shows that the torsional system has a stiffness-softening characteristic with respect to torque fluctuationis result can be accounted for a number of nonlinearitiessuch as friction clearance and sliding and the existence ofwhich sophisticates system nonlinear characteristics

5 Conclusion

In the present paper a lumped-parameter dynamic modelwith 29 degree-of-freedom is proposed considering thetransverse torsional and rocking coupled vibration for abetter understanding of the automotive powertrain systemdynamics and equations of motion are derived e nu-merical result shows that gear mesh excitations have littleeffects on torsional response for such components locatedbefore the mesh point as transmission transmission shaftand pinion but significantly reduce the torsional responsesfor ones behind the mesh point e amplitude-frequencycharacteristics of torsional vibration graphically indicate thattorsional behaviors of automotive powertrain only affect thevibration characteristics of complete vehicle under low-speed condition and predict the first three-order naturetorsional frequencies of the whole system An experimentalexamination is performed and validates the feasibility of theconsidered model e torsional vibration is reinforcedwhile increasing torque fluctuation In addition a stiffnesssoftening characteristic is observed for torsional vibrationexperimentally In order to reach a better understanding inamplitude-frequency response at low- and middle-input-speed condition a procedure to analyze mode shapes ofautomotive powertrain will be conducted in future works

Data Availability

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

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported and funded by the ApplicationTechnology Research and Development Plan of LiuzhouCity (grant no 2017AA10102) and the Key Projects ofUniversities in Inner Mongolia Autonomous Region (grantno NJZZ16369)e authors of this paper express their deepgratitude for the financial support of the research project

References

[1] F Vesali M A Rezvani and M Kashfi ldquoDynamics of uni-versal joints its failures and some propositions for practicallyimproving its performance and life expectancyrdquo Journal ofMechanical Science and Technology vol 26 no 8 pp 2439ndash2449 2012

[2] J-W Lu G-C Wang H Chen A F Vakakis andL A Bergman ldquoDynamic analysis of cross shaft type universaljoint with clearancerdquo Journal of Mechanical Science andTechnology vol 27 no 11 pp 3201ndash3205 2013

[3] G Wu W Shi and Z Chen ldquoe effect of multi-universalcoupling phase on torsional vibration of drive shaft and vi-bration of vehiclerdquo in Proceedings of the SAE 2013 WorldCongress amp Exhibition Detroit MI USA April 2013

0113333 4000 4500 5000 5500

5567

6000 6500 7000Frequency (Hz)

7500 8000 85008777

263300

9000 9500 10000 10500 11000 11667100

Am

plitu

de

Am

plitu

de (R

MS)

010

009

008

007

006

005

004

003

002

001

000100000 120000 140000 160000 180000

167000

25Nmiddotm

50Nmiddotm

75Nmiddotm100Nmiddotm

125Nmiddotm

150Nmiddotm

175Nmiddotm200Nmiddotm

Curve 167000010006004003003002001

982endash3

263300006005004003003002001

690endash3

rpm--------

200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000000

Figure 13 Amplitude-frequency responses of torsional vibration for output end of rear axle at different torque fluctuations

16 Shock and Vibration

[4] B A Porter ldquoeoretical analysis of the torsional oscillationof a system incorporating a Hookersquos jointrdquo Journal of Me-chanical Engineering Science vol 3 no 4 pp 324ndash329 1961

[5] S F Asokanthan and X-H Wang ldquoCharacterization oftorsional instabilities in a Hookersquos joint driven system viamaximal Lyapunov exponentsrdquo Journal of Sound and Vi-bration vol 194 no 1 pp 83ndash91 1996

[6] A Farshidianfar M Ebrahimi and H Bartlett ldquoHybridmodelling and simulation of the torsional vibration of vehicledriveline systemsrdquo Proceedings of the Institution of Me-chanical Engineers Part D Journal of Automobile Engineeringvol 215 no 2 pp 217ndash229 2001

[7] E O A Abd Elmaksoud E M Rabeih N A Abdel-halimand S M El Demerdash ldquoInvestigation of self excited tor-sional vibrations of different configurations of automatictransmission systems during engagementrdquo Engineeringvol 3 no 12 pp 1171ndash1181 2011

[8] T B Juang M Cheng and L Na ldquoExperimental and finiteelement analyses of a sliding-tube-type driveshaft-inducedvehicle vibrationrdquo Proceedings of the Institution of MechanicalEngineers Part K Journal of Multi-Body Dynamics vol 221no 3 pp 375ndash385 2007

[9] L F Coutinho and A Tamagna ldquoFlexural vibration analysisof constant velocity half-shaftsrdquo in Proceedings of the Inter-national Congress amp Exposition Chicago IL ISA 1996

[10] Y T Kang Y H Shen W Zhang and J Yang ldquoStabilityregion of floating intermediate support in a shaft system withmultiple universal jointsrdquo Journal of Mechanical Science andTechnology vol 28 no 6 pp 2733ndash2742 2014

[11] J L Xu T Yan B Peng and B Wei ldquoEffects of the trans-mission shaft on the main reducer vibration based onADAMS and experimental demonstrationrdquo AustralianJournal of Mechanical Engineering vol 15 no 1 pp 2ndash102015

[12] C X Zhang H B Li and D Q Jin ldquoSimulation on dynamicsof transmission shaft-rear driving axle of minibus withADAMSrdquo Applied Mechanics and Materials vol 651ndash653pp 862ndash865 2014

[13] J L Xu X Y Su and P Bo ldquoNumerical analysis anddemonstration transmission shaft influence on meshing vi-bration in driving and driven gearsrdquo Shock and Vibrationvol 2015 Article ID 365084 10 pages 2015

[14] J L Xu W Lei andW X Luo ldquoInfluence of bearing stiffnesson the nonlinear dynamics of a shaft-final drive systemrdquoShock and Vibration vol 2016 Article ID 3524609 14 pages2016

[15] A Kahraman and R Singh ldquoNon-linear dynamics of a spurgear pairrdquo Journal of Sound and Vibration vol 142 no 1pp 49ndash75 1990

[16] A Kahraman and R Singh ldquoInteractions between time-varying mesh stiffness and clearance non-linearities in ageared systemrdquo Journal of Sound and Vibration vol 146no 1 pp 135ndash156 1991

[17] A Kahraman ldquoEffect of axial vibrations on the dynamics of ahelical gear pairrdquo Journal of Vibration and Acoustics vol 115no 1 pp 33ndash39 1993

[18] Y Cai and T Hayashi ldquoe linear approximated equation ofvibration of a pair of spur gears (theory and experiment)rdquoJournal of Mechanical Design vol 116 no 2 pp 558ndash5641994

[19] P Velex andMMaatar ldquoAmathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound and Vibration vol 191no 5 pp 629ndash660 1996

[20] P Velex and V Cahouet ldquoExperimental and numerical in-vestigations on the influence of tooth friction in spur andhelical gear dynamicsrdquo Journal of Mechanical Design vol 122no 4 pp 515ndash522 2000

[21] P Velex and P Sainsot ldquoAn analytical study of tooth frictionexcitations in errorless spur and helical gearsrdquo Mechanismand Machine Ieory vol 37 no 7 pp 641ndash658 2002

[22] T C Lim and Y Cheng ldquoA theoretical study of the effect ofpinion offset on the dynamics of hypoid geared rotor systemrdquoJournal of Mechanical Design vol 121 no 4 pp 594ndash6011999

[23] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[24] Y Cheng and T C Lim ldquoDynamics of hypoid gear trans-mission with nonlinear time-varying mesh characteristicsrdquoJournal of Mechanical Design vol 125 no 2 pp 373ndash3822003

[25] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 2017 pp 302ndash329 2007

[26] J Wang and T C Lim ldquoEffect of tooth mesh stiffnessasymmetric nonlinearity for drive and coast sides on hypoidgear dynamicsrdquo Journal of Sound and Vibration vol 319no 3-5 pp 885ndash903 2009

[27] W M Li and H T Wang ldquoRigidity calculation of axialposition preload bearingsrdquo Journal of Hebei University ofTechnology vol 30 no 2 pp 15ndash19 2001

[28] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo Non-linear Dynamics vol 75 no 4 pp 783ndash806 2014

[29] X Liu Z Wu J Lu and J Xu ldquoInvestigation of the effect ofrotation speed on vibration responses of transmission sys-temrdquo Matec Web of Conferences vol 256 no 3 Article ID02019 2019

Shock and Vibration 17

Table 1 e equivalent parameters of powertrain

Equivalent mass (kg) Equivalent moment of inertia (kgmiddotm2) Equivalent stiffness (Nm)m1 2 J1 0091125 k1 1002375m2 10 J2 03801735 k2 164025m3 06150 J3 000008103 k3 27500m4 13798 J4 0001001 k4 10e11m5 23581 J5 0001323 k5 479000m6 114323 J6 000249 k6 10e11m7 45323 J7 0001865 k7 21400m8 75567 J8 0002098 mdashm9 1284 J9 0001695 k8 43921799m10 1284 J10 0001695 mdashm11 13125 J11 000173 k9 41933635m12 13125 J12 000173 mdashm13 04254 J13 000002772 k10 10e11m14 17016 J14 00001109 km 96965e8m15 1375 J15 0034177 mdashm16 845 J16 99 k11 12032114m17 845 J17 99 k12 12032114

Table 2 e system parameters of the main reducer

Parameter Pinion GearNumber of teeth 10 41Module (mm) 400Mean pressure angle (deg) 20 20Helix angle at midpoint of the pinion (deg) 50 2713Cone angle (deg) 2312 6514Tooth width (mm) 4066 3452Tooth addendum (mm) 359 644Pinion outset (mm) minus38Mounting distance (mm) 112 64Pitch radius (mm) 40 108Backlash (mm) 01536Distance from pinion reference point to the x-o-z plane for pinion (mm) 40Distance from pinion reference point to the y-o-z plane for gear (mm) 101Distance between projection point and centroid of pinion shaft (mm) lp3 10 lg3 5

Distance between bearing reference point and centroid of pinion shaft (mm) lp1 100 lg1 90lp2 175 lg2 93

Equivalent mass (kg) mg 2127 mp 1375Jpx J14 Jgx 0050315

Equivalent moment of inertia (kgmiddotm2) Jpy 0002982 Jgx J15Jpz 0002982 Jpz 0037551

Meshing stiffnessof (i + 1)th tooth pair

Meshing stiffnessof ith tooth pair

t

k

Meshing stiffnessof (i ndash 1)th tooth pair

Total meshingstiffness km(t)

Figure 5 Schematic diagram of time-varying meshing stiffness on hypoid gear

Shock and Vibration 11

indicates that torsional behaviors of automotive powertrainaffect the vibration characteristics of the complete vehicleunder a low- or middle-input-speed condition In order tovalidate the preceding result an experimental study isnecessary

4 Experimental Analysis

A test bench which can be used to test the vibration behavior ofa transmission system used for four-wheel drive vehicle be-longing to the China Automotive Technology and ResearchCenter is adopted to perform this experiment ere are fivemotors in the test bench and three of them are used in thisexperiment One of them drives automotive powertrain whichcan generate a high frequency input torque with fluctuationvaried in accordance with necessary and others are used asloads In this test two photoelectric sensors and two magne-toelectric sensors are used to detect torsional vibration signalsand other five accelerometers are mounted to monitor trans-verse vibration Installation site for each sensor can be found inFigure 9 Signals are sampled and processed by LMS TestLab

In order to validate the feasibility of the theoreticalmodel the input torque simulated by input motor and loadsare consistent with parameters employed in numericalanalysis (mean value and the second-order term are con-tained only and fluctuation amplitude is set at 100NmiddotM forinput torque) Within the experiment the output speed ofmotor varies from 1000 rpm to 3500 rpm to conduct a sweepfrequency response analysis experimentally

As illustrated in Figure 10 the amplitude-frequencyresponses of torsional vibration have the similar feature for

different parts of powertrain Peaks can be observed forcarves at a similar input speed It is inferred that a presumeresonance occurs at each peak As a result the first three-order nature torsional frequencies can be predictedMeanwhile the same results are reached As shown inFigure 10 each curve has a higher value nearby the first andsecond peak With the further increase in frequency itreaches a lower plateau value Torsional behaviors have aneffect on the vibration characteristics under a low-input-speed condition Furthermore the part located behind themesh point such as output end of half-shaft has a lowerRMS value than transmission and drive shaft which can beattributed to reduction effect of mesh excitation on torsionalvibration

e experimental result of input end of rear axle iscompared with the numerical one As shown in Figure 11 aquite consistency is observed and the first three-order fre-quencies are inferred at 35Hz (1050 rpm) 557Hz(1670 rpm) and 883Hz (2650 rpm) for experimental resultby the same way mentioned in preceding section Comparedwith numerical predictions an acceptable difference ofabout 5 (the difference is 51 at 333Hz 45 at 533Hzand 536 at 933Hz) can be obtained ese indicate thatthe theoretical model is feasible for predicting nature tor-sional frequencies Moreover numerical torsional dis-placement is a little lower than the experimental one at thesame input speed which is below about 2450 rpm and a littlehigher with the further increase is can be presumablyattributed to assumptions made in this present work withrespect to the complexity of the system and the limitations ofthe lumped-mass method

03503

02502

01501

0050

0

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(a)

13Hz

50Hz

87Hz 150Hz185Hz222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(b)

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(c)

13Hz

50Hz

87Hz150Hz

185Hz222Hz

004

003

002

001

00 50 100 150 200 250

Frequency (Hz)To

rsio

nal d

ispla

cem

ent (

deg)300 350 400 450 500 550 600

(d)

Figure 6 e Fourier spectrum of torsional responses (a) output shaft of transmission (b) output end of the drive shaft system (c)torsional displacement of the pinion along the axis x (d) torsional displacement of the gear along the axis y

12 Shock and Vibration

e previous section indicates numerically that excita-tion of input dominates torsion vibration of the system andgear set affects responses slightly In order to validate thecorrectness of numerical analysis and feasibility decreasing

torsional vibration by reducing input excitation effect ofinput torque fluctuation on torsional response is studiedexperimentally e mean value of input torque is set at150NmiddotM and fluctuation of the second-order term is set at

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

2800rpm1600rpm

Figure 7 Amplitude-frequency response at input end of rear axle

8

7

6

5

4

2

3

1

00 1000 2000 3000

Rotation speed (RPM)

Vibr

atio

n am

plitu

de (micro

m)

4000 5000 6000

x directiony directionz direction

Figure 8 Influence of input rotation speed of drive shaft on vibration of the main reducer [29]

Table 3 e first six-order nature frequency of drive shaft [29]

Order 1 2 3 4 5 6Frequency (Hz) 1309 1402 4152 4422 6362 9066

Shock and Vibration 13

25NmiddotM 50NmiddotM 75NmiddotM 100NmiddotM 125NmiddotM 150NmiddotM175NmiddotM and 200NmiddotM in order

Together Figures 12 and 13 expectably show that thetorsional vibration is reinforced while increasing torquefluctuation Meanwhile an obvious increase in amplitudeoccurs at the second peak when fluctuation is increased from175NmiddotM to 200NmiddotM which indicates that a limitation should

be considered e accurate value can be determined by afurther study In summary reducing torque fluctuation is abeneficial approach for vibration damping Furthermore it isnoted that such a nonlinear phenomenon as natural fre-quency shift occurs while changing torque fluctuation Takingthe second-order torsional frequency as an example thefrequency value decreases with elevating fluctuation of input

Transmission

Photoelectricsensor

Photoelectricsensor

Magnetoelectricsensor

Magnetoelectricsensor

Rear axle

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Drive sha system

Figure 9 e experiment bench and arrangement of sensors

042

040

038

035

033

030

028

025

023

020

018

015

013

010

008

005

003

Amplitude-frequency curve ofle wheel

Amplitude-frequency curve oftransmission output sha

Amplitude-frequency curve ofrear axle output end

Amplitude-frequency curve ofright wheel

000

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (K

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Curve 167000

041

035

003

003

5567 8817

167000 264500

rpm

deg

deg

deg

deg

264500

012

005

003

764endash3

Figure 10 Amplitude-frequency responses of torsional vibration for powertrain at a torque fluctuation of 100NmiddotM

14 Shock and Vibration

036

034

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

Numerical result

Experimental result

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

Figure 11 Comparison of numerical and experimental results for input end of rear axle

160 5567 8817

264500

150

140

130

120

110

100

090

080

070

060

050

040

030

020

010

000 167000

25Nmiddotm

50Nmiddotm

75Nmiddotm

100Nmiddotm

125Nmiddotm150Nmiddotm

175Nmiddotm

200Nmiddotm

Curve 167000147078057045035025017013

rpm--------

264500009008007006005004004003

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (R

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Figure 12 Amplitude-frequency responses of torsional vibration for input end of rear axle at different torque fluctuations

Shock and Vibration 15

torque which shows that the torsional system has a stiffness-softening characteristic with respect to torque fluctuationis result can be accounted for a number of nonlinearitiessuch as friction clearance and sliding and the existence ofwhich sophisticates system nonlinear characteristics

5 Conclusion

In the present paper a lumped-parameter dynamic modelwith 29 degree-of-freedom is proposed considering thetransverse torsional and rocking coupled vibration for abetter understanding of the automotive powertrain systemdynamics and equations of motion are derived e nu-merical result shows that gear mesh excitations have littleeffects on torsional response for such components locatedbefore the mesh point as transmission transmission shaftand pinion but significantly reduce the torsional responsesfor ones behind the mesh point e amplitude-frequencycharacteristics of torsional vibration graphically indicate thattorsional behaviors of automotive powertrain only affect thevibration characteristics of complete vehicle under low-speed condition and predict the first three-order naturetorsional frequencies of the whole system An experimentalexamination is performed and validates the feasibility of theconsidered model e torsional vibration is reinforcedwhile increasing torque fluctuation In addition a stiffnesssoftening characteristic is observed for torsional vibrationexperimentally In order to reach a better understanding inamplitude-frequency response at low- and middle-input-speed condition a procedure to analyze mode shapes ofautomotive powertrain will be conducted in future works

Data Availability

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

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported and funded by the ApplicationTechnology Research and Development Plan of LiuzhouCity (grant no 2017AA10102) and the Key Projects ofUniversities in Inner Mongolia Autonomous Region (grantno NJZZ16369)e authors of this paper express their deepgratitude for the financial support of the research project

References

[1] F Vesali M A Rezvani and M Kashfi ldquoDynamics of uni-versal joints its failures and some propositions for practicallyimproving its performance and life expectancyrdquo Journal ofMechanical Science and Technology vol 26 no 8 pp 2439ndash2449 2012

[2] J-W Lu G-C Wang H Chen A F Vakakis andL A Bergman ldquoDynamic analysis of cross shaft type universaljoint with clearancerdquo Journal of Mechanical Science andTechnology vol 27 no 11 pp 3201ndash3205 2013

[3] G Wu W Shi and Z Chen ldquoe effect of multi-universalcoupling phase on torsional vibration of drive shaft and vi-bration of vehiclerdquo in Proceedings of the SAE 2013 WorldCongress amp Exhibition Detroit MI USA April 2013

0113333 4000 4500 5000 5500

5567

6000 6500 7000Frequency (Hz)

7500 8000 85008777

263300

9000 9500 10000 10500 11000 11667100

Am

plitu

de

Am

plitu

de (R

MS)

010

009

008

007

006

005

004

003

002

001

000100000 120000 140000 160000 180000

167000

25Nmiddotm

50Nmiddotm

75Nmiddotm100Nmiddotm

125Nmiddotm

150Nmiddotm

175Nmiddotm200Nmiddotm

Curve 167000010006004003003002001

982endash3

263300006005004003003002001

690endash3

rpm--------

200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000000

Figure 13 Amplitude-frequency responses of torsional vibration for output end of rear axle at different torque fluctuations

16 Shock and Vibration

[4] B A Porter ldquoeoretical analysis of the torsional oscillationof a system incorporating a Hookersquos jointrdquo Journal of Me-chanical Engineering Science vol 3 no 4 pp 324ndash329 1961

[5] S F Asokanthan and X-H Wang ldquoCharacterization oftorsional instabilities in a Hookersquos joint driven system viamaximal Lyapunov exponentsrdquo Journal of Sound and Vi-bration vol 194 no 1 pp 83ndash91 1996

[6] A Farshidianfar M Ebrahimi and H Bartlett ldquoHybridmodelling and simulation of the torsional vibration of vehicledriveline systemsrdquo Proceedings of the Institution of Me-chanical Engineers Part D Journal of Automobile Engineeringvol 215 no 2 pp 217ndash229 2001

[7] E O A Abd Elmaksoud E M Rabeih N A Abdel-halimand S M El Demerdash ldquoInvestigation of self excited tor-sional vibrations of different configurations of automatictransmission systems during engagementrdquo Engineeringvol 3 no 12 pp 1171ndash1181 2011

[8] T B Juang M Cheng and L Na ldquoExperimental and finiteelement analyses of a sliding-tube-type driveshaft-inducedvehicle vibrationrdquo Proceedings of the Institution of MechanicalEngineers Part K Journal of Multi-Body Dynamics vol 221no 3 pp 375ndash385 2007

[9] L F Coutinho and A Tamagna ldquoFlexural vibration analysisof constant velocity half-shaftsrdquo in Proceedings of the Inter-national Congress amp Exposition Chicago IL ISA 1996

[10] Y T Kang Y H Shen W Zhang and J Yang ldquoStabilityregion of floating intermediate support in a shaft system withmultiple universal jointsrdquo Journal of Mechanical Science andTechnology vol 28 no 6 pp 2733ndash2742 2014

[11] J L Xu T Yan B Peng and B Wei ldquoEffects of the trans-mission shaft on the main reducer vibration based onADAMS and experimental demonstrationrdquo AustralianJournal of Mechanical Engineering vol 15 no 1 pp 2ndash102015

[12] C X Zhang H B Li and D Q Jin ldquoSimulation on dynamicsof transmission shaft-rear driving axle of minibus withADAMSrdquo Applied Mechanics and Materials vol 651ndash653pp 862ndash865 2014

[13] J L Xu X Y Su and P Bo ldquoNumerical analysis anddemonstration transmission shaft influence on meshing vi-bration in driving and driven gearsrdquo Shock and Vibrationvol 2015 Article ID 365084 10 pages 2015

[14] J L Xu W Lei andW X Luo ldquoInfluence of bearing stiffnesson the nonlinear dynamics of a shaft-final drive systemrdquoShock and Vibration vol 2016 Article ID 3524609 14 pages2016

[15] A Kahraman and R Singh ldquoNon-linear dynamics of a spurgear pairrdquo Journal of Sound and Vibration vol 142 no 1pp 49ndash75 1990

[16] A Kahraman and R Singh ldquoInteractions between time-varying mesh stiffness and clearance non-linearities in ageared systemrdquo Journal of Sound and Vibration vol 146no 1 pp 135ndash156 1991

[17] A Kahraman ldquoEffect of axial vibrations on the dynamics of ahelical gear pairrdquo Journal of Vibration and Acoustics vol 115no 1 pp 33ndash39 1993

[18] Y Cai and T Hayashi ldquoe linear approximated equation ofvibration of a pair of spur gears (theory and experiment)rdquoJournal of Mechanical Design vol 116 no 2 pp 558ndash5641994

[19] P Velex andMMaatar ldquoAmathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound and Vibration vol 191no 5 pp 629ndash660 1996

[20] P Velex and V Cahouet ldquoExperimental and numerical in-vestigations on the influence of tooth friction in spur andhelical gear dynamicsrdquo Journal of Mechanical Design vol 122no 4 pp 515ndash522 2000

[21] P Velex and P Sainsot ldquoAn analytical study of tooth frictionexcitations in errorless spur and helical gearsrdquo Mechanismand Machine Ieory vol 37 no 7 pp 641ndash658 2002

[22] T C Lim and Y Cheng ldquoA theoretical study of the effect ofpinion offset on the dynamics of hypoid geared rotor systemrdquoJournal of Mechanical Design vol 121 no 4 pp 594ndash6011999

[23] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[24] Y Cheng and T C Lim ldquoDynamics of hypoid gear trans-mission with nonlinear time-varying mesh characteristicsrdquoJournal of Mechanical Design vol 125 no 2 pp 373ndash3822003

[25] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 2017 pp 302ndash329 2007

[26] J Wang and T C Lim ldquoEffect of tooth mesh stiffnessasymmetric nonlinearity for drive and coast sides on hypoidgear dynamicsrdquo Journal of Sound and Vibration vol 319no 3-5 pp 885ndash903 2009

[27] W M Li and H T Wang ldquoRigidity calculation of axialposition preload bearingsrdquo Journal of Hebei University ofTechnology vol 30 no 2 pp 15ndash19 2001

[28] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo Non-linear Dynamics vol 75 no 4 pp 783ndash806 2014

[29] X Liu Z Wu J Lu and J Xu ldquoInvestigation of the effect ofrotation speed on vibration responses of transmission sys-temrdquo Matec Web of Conferences vol 256 no 3 Article ID02019 2019

Shock and Vibration 17

indicates that torsional behaviors of automotive powertrainaffect the vibration characteristics of the complete vehicleunder a low- or middle-input-speed condition In order tovalidate the preceding result an experimental study isnecessary

4 Experimental Analysis

A test bench which can be used to test the vibration behavior ofa transmission system used for four-wheel drive vehicle be-longing to the China Automotive Technology and ResearchCenter is adopted to perform this experiment ere are fivemotors in the test bench and three of them are used in thisexperiment One of them drives automotive powertrain whichcan generate a high frequency input torque with fluctuationvaried in accordance with necessary and others are used asloads In this test two photoelectric sensors and two magne-toelectric sensors are used to detect torsional vibration signalsand other five accelerometers are mounted to monitor trans-verse vibration Installation site for each sensor can be found inFigure 9 Signals are sampled and processed by LMS TestLab

In order to validate the feasibility of the theoreticalmodel the input torque simulated by input motor and loadsare consistent with parameters employed in numericalanalysis (mean value and the second-order term are con-tained only and fluctuation amplitude is set at 100NmiddotM forinput torque) Within the experiment the output speed ofmotor varies from 1000 rpm to 3500 rpm to conduct a sweepfrequency response analysis experimentally

As illustrated in Figure 10 the amplitude-frequencyresponses of torsional vibration have the similar feature for

different parts of powertrain Peaks can be observed forcarves at a similar input speed It is inferred that a presumeresonance occurs at each peak As a result the first three-order nature torsional frequencies can be predictedMeanwhile the same results are reached As shown inFigure 10 each curve has a higher value nearby the first andsecond peak With the further increase in frequency itreaches a lower plateau value Torsional behaviors have aneffect on the vibration characteristics under a low-input-speed condition Furthermore the part located behind themesh point such as output end of half-shaft has a lowerRMS value than transmission and drive shaft which can beattributed to reduction effect of mesh excitation on torsionalvibration

e experimental result of input end of rear axle iscompared with the numerical one As shown in Figure 11 aquite consistency is observed and the first three-order fre-quencies are inferred at 35Hz (1050 rpm) 557Hz(1670 rpm) and 883Hz (2650 rpm) for experimental resultby the same way mentioned in preceding section Comparedwith numerical predictions an acceptable difference ofabout 5 (the difference is 51 at 333Hz 45 at 533Hzand 536 at 933Hz) can be obtained ese indicate thatthe theoretical model is feasible for predicting nature tor-sional frequencies Moreover numerical torsional dis-placement is a little lower than the experimental one at thesame input speed which is below about 2450 rpm and a littlehigher with the further increase is can be presumablyattributed to assumptions made in this present work withrespect to the complexity of the system and the limitations ofthe lumped-mass method

03503

02502

01501

0050

0

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(a)

13Hz

50Hz

87Hz 150Hz185Hz222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(b)

13Hz

50Hz

87Hz 150Hz185Hz

222Hz

03503

02502

01501

0050

0 50 100 150 200 250Frequency (Hz)

Tors

iona

l disp

lace

men

t (deg)

300 350 400 450 500 550 600

(c)

13Hz

50Hz

87Hz150Hz

185Hz222Hz

004

003

002

001

00 50 100 150 200 250

Frequency (Hz)To

rsio

nal d

ispla

cem

ent (

deg)300 350 400 450 500 550 600

(d)

Figure 6 e Fourier spectrum of torsional responses (a) output shaft of transmission (b) output end of the drive shaft system (c)torsional displacement of the pinion along the axis x (d) torsional displacement of the gear along the axis y

12 Shock and Vibration

e previous section indicates numerically that excita-tion of input dominates torsion vibration of the system andgear set affects responses slightly In order to validate thecorrectness of numerical analysis and feasibility decreasing

torsional vibration by reducing input excitation effect ofinput torque fluctuation on torsional response is studiedexperimentally e mean value of input torque is set at150NmiddotM and fluctuation of the second-order term is set at

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

2800rpm1600rpm

Figure 7 Amplitude-frequency response at input end of rear axle

8

7

6

5

4

2

3

1

00 1000 2000 3000

Rotation speed (RPM)

Vibr

atio

n am

plitu

de (micro

m)

4000 5000 6000

x directiony directionz direction

Figure 8 Influence of input rotation speed of drive shaft on vibration of the main reducer [29]

Table 3 e first six-order nature frequency of drive shaft [29]

Order 1 2 3 4 5 6Frequency (Hz) 1309 1402 4152 4422 6362 9066

Shock and Vibration 13

25NmiddotM 50NmiddotM 75NmiddotM 100NmiddotM 125NmiddotM 150NmiddotM175NmiddotM and 200NmiddotM in order

Together Figures 12 and 13 expectably show that thetorsional vibration is reinforced while increasing torquefluctuation Meanwhile an obvious increase in amplitudeoccurs at the second peak when fluctuation is increased from175NmiddotM to 200NmiddotM which indicates that a limitation should

be considered e accurate value can be determined by afurther study In summary reducing torque fluctuation is abeneficial approach for vibration damping Furthermore it isnoted that such a nonlinear phenomenon as natural fre-quency shift occurs while changing torque fluctuation Takingthe second-order torsional frequency as an example thefrequency value decreases with elevating fluctuation of input

Transmission

Photoelectricsensor

Photoelectricsensor

Magnetoelectricsensor

Magnetoelectricsensor

Rear axle

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Drive sha system

Figure 9 e experiment bench and arrangement of sensors

042

040

038

035

033

030

028

025

023

020

018

015

013

010

008

005

003

Amplitude-frequency curve ofle wheel

Amplitude-frequency curve oftransmission output sha

Amplitude-frequency curve ofrear axle output end

Amplitude-frequency curve ofright wheel

000

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (K

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Curve 167000

041

035

003

003

5567 8817

167000 264500

rpm

deg

deg

deg

deg

264500

012

005

003

764endash3

Figure 10 Amplitude-frequency responses of torsional vibration for powertrain at a torque fluctuation of 100NmiddotM

14 Shock and Vibration

036

034

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

Numerical result

Experimental result

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

Figure 11 Comparison of numerical and experimental results for input end of rear axle

160 5567 8817

264500

150

140

130

120

110

100

090

080

070

060

050

040

030

020

010

000 167000

25Nmiddotm

50Nmiddotm

75Nmiddotm

100Nmiddotm

125Nmiddotm150Nmiddotm

175Nmiddotm

200Nmiddotm

Curve 167000147078057045035025017013

rpm--------

264500009008007006005004004003

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (R

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Figure 12 Amplitude-frequency responses of torsional vibration for input end of rear axle at different torque fluctuations

Shock and Vibration 15

torque which shows that the torsional system has a stiffness-softening characteristic with respect to torque fluctuationis result can be accounted for a number of nonlinearitiessuch as friction clearance and sliding and the existence ofwhich sophisticates system nonlinear characteristics

5 Conclusion

In the present paper a lumped-parameter dynamic modelwith 29 degree-of-freedom is proposed considering thetransverse torsional and rocking coupled vibration for abetter understanding of the automotive powertrain systemdynamics and equations of motion are derived e nu-merical result shows that gear mesh excitations have littleeffects on torsional response for such components locatedbefore the mesh point as transmission transmission shaftand pinion but significantly reduce the torsional responsesfor ones behind the mesh point e amplitude-frequencycharacteristics of torsional vibration graphically indicate thattorsional behaviors of automotive powertrain only affect thevibration characteristics of complete vehicle under low-speed condition and predict the first three-order naturetorsional frequencies of the whole system An experimentalexamination is performed and validates the feasibility of theconsidered model e torsional vibration is reinforcedwhile increasing torque fluctuation In addition a stiffnesssoftening characteristic is observed for torsional vibrationexperimentally In order to reach a better understanding inamplitude-frequency response at low- and middle-input-speed condition a procedure to analyze mode shapes ofautomotive powertrain will be conducted in future works

Data Availability

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

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported and funded by the ApplicationTechnology Research and Development Plan of LiuzhouCity (grant no 2017AA10102) and the Key Projects ofUniversities in Inner Mongolia Autonomous Region (grantno NJZZ16369)e authors of this paper express their deepgratitude for the financial support of the research project

References

[1] F Vesali M A Rezvani and M Kashfi ldquoDynamics of uni-versal joints its failures and some propositions for practicallyimproving its performance and life expectancyrdquo Journal ofMechanical Science and Technology vol 26 no 8 pp 2439ndash2449 2012

[2] J-W Lu G-C Wang H Chen A F Vakakis andL A Bergman ldquoDynamic analysis of cross shaft type universaljoint with clearancerdquo Journal of Mechanical Science andTechnology vol 27 no 11 pp 3201ndash3205 2013

[3] G Wu W Shi and Z Chen ldquoe effect of multi-universalcoupling phase on torsional vibration of drive shaft and vi-bration of vehiclerdquo in Proceedings of the SAE 2013 WorldCongress amp Exhibition Detroit MI USA April 2013

0113333 4000 4500 5000 5500

5567

6000 6500 7000Frequency (Hz)

7500 8000 85008777

263300

9000 9500 10000 10500 11000 11667100

Am

plitu

de

Am

plitu

de (R

MS)

010

009

008

007

006

005

004

003

002

001

000100000 120000 140000 160000 180000

167000

25Nmiddotm

50Nmiddotm

75Nmiddotm100Nmiddotm

125Nmiddotm

150Nmiddotm

175Nmiddotm200Nmiddotm

Curve 167000010006004003003002001

982endash3

263300006005004003003002001

690endash3

rpm--------

200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000000

Figure 13 Amplitude-frequency responses of torsional vibration for output end of rear axle at different torque fluctuations

16 Shock and Vibration

[4] B A Porter ldquoeoretical analysis of the torsional oscillationof a system incorporating a Hookersquos jointrdquo Journal of Me-chanical Engineering Science vol 3 no 4 pp 324ndash329 1961

[5] S F Asokanthan and X-H Wang ldquoCharacterization oftorsional instabilities in a Hookersquos joint driven system viamaximal Lyapunov exponentsrdquo Journal of Sound and Vi-bration vol 194 no 1 pp 83ndash91 1996

[6] A Farshidianfar M Ebrahimi and H Bartlett ldquoHybridmodelling and simulation of the torsional vibration of vehicledriveline systemsrdquo Proceedings of the Institution of Me-chanical Engineers Part D Journal of Automobile Engineeringvol 215 no 2 pp 217ndash229 2001

[7] E O A Abd Elmaksoud E M Rabeih N A Abdel-halimand S M El Demerdash ldquoInvestigation of self excited tor-sional vibrations of different configurations of automatictransmission systems during engagementrdquo Engineeringvol 3 no 12 pp 1171ndash1181 2011

[8] T B Juang M Cheng and L Na ldquoExperimental and finiteelement analyses of a sliding-tube-type driveshaft-inducedvehicle vibrationrdquo Proceedings of the Institution of MechanicalEngineers Part K Journal of Multi-Body Dynamics vol 221no 3 pp 375ndash385 2007

[9] L F Coutinho and A Tamagna ldquoFlexural vibration analysisof constant velocity half-shaftsrdquo in Proceedings of the Inter-national Congress amp Exposition Chicago IL ISA 1996

[10] Y T Kang Y H Shen W Zhang and J Yang ldquoStabilityregion of floating intermediate support in a shaft system withmultiple universal jointsrdquo Journal of Mechanical Science andTechnology vol 28 no 6 pp 2733ndash2742 2014

[11] J L Xu T Yan B Peng and B Wei ldquoEffects of the trans-mission shaft on the main reducer vibration based onADAMS and experimental demonstrationrdquo AustralianJournal of Mechanical Engineering vol 15 no 1 pp 2ndash102015

[12] C X Zhang H B Li and D Q Jin ldquoSimulation on dynamicsof transmission shaft-rear driving axle of minibus withADAMSrdquo Applied Mechanics and Materials vol 651ndash653pp 862ndash865 2014

[13] J L Xu X Y Su and P Bo ldquoNumerical analysis anddemonstration transmission shaft influence on meshing vi-bration in driving and driven gearsrdquo Shock and Vibrationvol 2015 Article ID 365084 10 pages 2015

[14] J L Xu W Lei andW X Luo ldquoInfluence of bearing stiffnesson the nonlinear dynamics of a shaft-final drive systemrdquoShock and Vibration vol 2016 Article ID 3524609 14 pages2016

[15] A Kahraman and R Singh ldquoNon-linear dynamics of a spurgear pairrdquo Journal of Sound and Vibration vol 142 no 1pp 49ndash75 1990

[16] A Kahraman and R Singh ldquoInteractions between time-varying mesh stiffness and clearance non-linearities in ageared systemrdquo Journal of Sound and Vibration vol 146no 1 pp 135ndash156 1991

[17] A Kahraman ldquoEffect of axial vibrations on the dynamics of ahelical gear pairrdquo Journal of Vibration and Acoustics vol 115no 1 pp 33ndash39 1993

[18] Y Cai and T Hayashi ldquoe linear approximated equation ofvibration of a pair of spur gears (theory and experiment)rdquoJournal of Mechanical Design vol 116 no 2 pp 558ndash5641994

[19] P Velex andMMaatar ldquoAmathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound and Vibration vol 191no 5 pp 629ndash660 1996

[20] P Velex and V Cahouet ldquoExperimental and numerical in-vestigations on the influence of tooth friction in spur andhelical gear dynamicsrdquo Journal of Mechanical Design vol 122no 4 pp 515ndash522 2000

[21] P Velex and P Sainsot ldquoAn analytical study of tooth frictionexcitations in errorless spur and helical gearsrdquo Mechanismand Machine Ieory vol 37 no 7 pp 641ndash658 2002

[22] T C Lim and Y Cheng ldquoA theoretical study of the effect ofpinion offset on the dynamics of hypoid geared rotor systemrdquoJournal of Mechanical Design vol 121 no 4 pp 594ndash6011999

[23] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[24] Y Cheng and T C Lim ldquoDynamics of hypoid gear trans-mission with nonlinear time-varying mesh characteristicsrdquoJournal of Mechanical Design vol 125 no 2 pp 373ndash3822003

[25] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 2017 pp 302ndash329 2007

[26] J Wang and T C Lim ldquoEffect of tooth mesh stiffnessasymmetric nonlinearity for drive and coast sides on hypoidgear dynamicsrdquo Journal of Sound and Vibration vol 319no 3-5 pp 885ndash903 2009

[27] W M Li and H T Wang ldquoRigidity calculation of axialposition preload bearingsrdquo Journal of Hebei University ofTechnology vol 30 no 2 pp 15ndash19 2001

[28] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo Non-linear Dynamics vol 75 no 4 pp 783ndash806 2014

[29] X Liu Z Wu J Lu and J Xu ldquoInvestigation of the effect ofrotation speed on vibration responses of transmission sys-temrdquo Matec Web of Conferences vol 256 no 3 Article ID02019 2019

Shock and Vibration 17

e previous section indicates numerically that excita-tion of input dominates torsion vibration of the system andgear set affects responses slightly In order to validate thecorrectness of numerical analysis and feasibility decreasing

torsional vibration by reducing input excitation effect ofinput torque fluctuation on torsional response is studiedexperimentally e mean value of input torque is set at150NmiddotM and fluctuation of the second-order term is set at

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

2800rpm1600rpm

Figure 7 Amplitude-frequency response at input end of rear axle

8

7

6

5

4

2

3

1

00 1000 2000 3000

Rotation speed (RPM)

Vibr

atio

n am

plitu

de (micro

m)

4000 5000 6000

x directiony directionz direction

Figure 8 Influence of input rotation speed of drive shaft on vibration of the main reducer [29]

Table 3 e first six-order nature frequency of drive shaft [29]

Order 1 2 3 4 5 6Frequency (Hz) 1309 1402 4152 4422 6362 9066

Shock and Vibration 13

25NmiddotM 50NmiddotM 75NmiddotM 100NmiddotM 125NmiddotM 150NmiddotM175NmiddotM and 200NmiddotM in order

Together Figures 12 and 13 expectably show that thetorsional vibration is reinforced while increasing torquefluctuation Meanwhile an obvious increase in amplitudeoccurs at the second peak when fluctuation is increased from175NmiddotM to 200NmiddotM which indicates that a limitation should

be considered e accurate value can be determined by afurther study In summary reducing torque fluctuation is abeneficial approach for vibration damping Furthermore it isnoted that such a nonlinear phenomenon as natural fre-quency shift occurs while changing torque fluctuation Takingthe second-order torsional frequency as an example thefrequency value decreases with elevating fluctuation of input

Transmission

Photoelectricsensor

Photoelectricsensor

Magnetoelectricsensor

Magnetoelectricsensor

Rear axle

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Drive sha system

Figure 9 e experiment bench and arrangement of sensors

042

040

038

035

033

030

028

025

023

020

018

015

013

010

008

005

003

Amplitude-frequency curve ofle wheel

Amplitude-frequency curve oftransmission output sha

Amplitude-frequency curve ofrear axle output end

Amplitude-frequency curve ofright wheel

000

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (K

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Curve 167000

041

035

003

003

5567 8817

167000 264500

rpm

deg

deg

deg

deg

264500

012

005

003

764endash3

Figure 10 Amplitude-frequency responses of torsional vibration for powertrain at a torque fluctuation of 100NmiddotM

14 Shock and Vibration

036

034

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

Numerical result

Experimental result

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

Figure 11 Comparison of numerical and experimental results for input end of rear axle

160 5567 8817

264500

150

140

130

120

110

100

090

080

070

060

050

040

030

020

010

000 167000

25Nmiddotm

50Nmiddotm

75Nmiddotm

100Nmiddotm

125Nmiddotm150Nmiddotm

175Nmiddotm

200Nmiddotm

Curve 167000147078057045035025017013

rpm--------

264500009008007006005004004003

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (R

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Figure 12 Amplitude-frequency responses of torsional vibration for input end of rear axle at different torque fluctuations

Shock and Vibration 15

torque which shows that the torsional system has a stiffness-softening characteristic with respect to torque fluctuationis result can be accounted for a number of nonlinearitiessuch as friction clearance and sliding and the existence ofwhich sophisticates system nonlinear characteristics

5 Conclusion

In the present paper a lumped-parameter dynamic modelwith 29 degree-of-freedom is proposed considering thetransverse torsional and rocking coupled vibration for abetter understanding of the automotive powertrain systemdynamics and equations of motion are derived e nu-merical result shows that gear mesh excitations have littleeffects on torsional response for such components locatedbefore the mesh point as transmission transmission shaftand pinion but significantly reduce the torsional responsesfor ones behind the mesh point e amplitude-frequencycharacteristics of torsional vibration graphically indicate thattorsional behaviors of automotive powertrain only affect thevibration characteristics of complete vehicle under low-speed condition and predict the first three-order naturetorsional frequencies of the whole system An experimentalexamination is performed and validates the feasibility of theconsidered model e torsional vibration is reinforcedwhile increasing torque fluctuation In addition a stiffnesssoftening characteristic is observed for torsional vibrationexperimentally In order to reach a better understanding inamplitude-frequency response at low- and middle-input-speed condition a procedure to analyze mode shapes ofautomotive powertrain will be conducted in future works

Data Availability

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

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported and funded by the ApplicationTechnology Research and Development Plan of LiuzhouCity (grant no 2017AA10102) and the Key Projects ofUniversities in Inner Mongolia Autonomous Region (grantno NJZZ16369)e authors of this paper express their deepgratitude for the financial support of the research project

References

[1] F Vesali M A Rezvani and M Kashfi ldquoDynamics of uni-versal joints its failures and some propositions for practicallyimproving its performance and life expectancyrdquo Journal ofMechanical Science and Technology vol 26 no 8 pp 2439ndash2449 2012

[2] J-W Lu G-C Wang H Chen A F Vakakis andL A Bergman ldquoDynamic analysis of cross shaft type universaljoint with clearancerdquo Journal of Mechanical Science andTechnology vol 27 no 11 pp 3201ndash3205 2013

[3] G Wu W Shi and Z Chen ldquoe effect of multi-universalcoupling phase on torsional vibration of drive shaft and vi-bration of vehiclerdquo in Proceedings of the SAE 2013 WorldCongress amp Exhibition Detroit MI USA April 2013

0113333 4000 4500 5000 5500

5567

6000 6500 7000Frequency (Hz)

7500 8000 85008777

263300

9000 9500 10000 10500 11000 11667100

Am

plitu

de

Am

plitu

de (R

MS)

010

009

008

007

006

005

004

003

002

001

000100000 120000 140000 160000 180000

167000

25Nmiddotm

50Nmiddotm

75Nmiddotm100Nmiddotm

125Nmiddotm

150Nmiddotm

175Nmiddotm200Nmiddotm

Curve 167000010006004003003002001

982endash3

263300006005004003003002001

690endash3

rpm--------

200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000000

Figure 13 Amplitude-frequency responses of torsional vibration for output end of rear axle at different torque fluctuations

16 Shock and Vibration

[4] B A Porter ldquoeoretical analysis of the torsional oscillationof a system incorporating a Hookersquos jointrdquo Journal of Me-chanical Engineering Science vol 3 no 4 pp 324ndash329 1961

[5] S F Asokanthan and X-H Wang ldquoCharacterization oftorsional instabilities in a Hookersquos joint driven system viamaximal Lyapunov exponentsrdquo Journal of Sound and Vi-bration vol 194 no 1 pp 83ndash91 1996

[6] A Farshidianfar M Ebrahimi and H Bartlett ldquoHybridmodelling and simulation of the torsional vibration of vehicledriveline systemsrdquo Proceedings of the Institution of Me-chanical Engineers Part D Journal of Automobile Engineeringvol 215 no 2 pp 217ndash229 2001

[7] E O A Abd Elmaksoud E M Rabeih N A Abdel-halimand S M El Demerdash ldquoInvestigation of self excited tor-sional vibrations of different configurations of automatictransmission systems during engagementrdquo Engineeringvol 3 no 12 pp 1171ndash1181 2011

[8] T B Juang M Cheng and L Na ldquoExperimental and finiteelement analyses of a sliding-tube-type driveshaft-inducedvehicle vibrationrdquo Proceedings of the Institution of MechanicalEngineers Part K Journal of Multi-Body Dynamics vol 221no 3 pp 375ndash385 2007

[9] L F Coutinho and A Tamagna ldquoFlexural vibration analysisof constant velocity half-shaftsrdquo in Proceedings of the Inter-national Congress amp Exposition Chicago IL ISA 1996

[10] Y T Kang Y H Shen W Zhang and J Yang ldquoStabilityregion of floating intermediate support in a shaft system withmultiple universal jointsrdquo Journal of Mechanical Science andTechnology vol 28 no 6 pp 2733ndash2742 2014

[11] J L Xu T Yan B Peng and B Wei ldquoEffects of the trans-mission shaft on the main reducer vibration based onADAMS and experimental demonstrationrdquo AustralianJournal of Mechanical Engineering vol 15 no 1 pp 2ndash102015

[12] C X Zhang H B Li and D Q Jin ldquoSimulation on dynamicsof transmission shaft-rear driving axle of minibus withADAMSrdquo Applied Mechanics and Materials vol 651ndash653pp 862ndash865 2014

[13] J L Xu X Y Su and P Bo ldquoNumerical analysis anddemonstration transmission shaft influence on meshing vi-bration in driving and driven gearsrdquo Shock and Vibrationvol 2015 Article ID 365084 10 pages 2015

[14] J L Xu W Lei andW X Luo ldquoInfluence of bearing stiffnesson the nonlinear dynamics of a shaft-final drive systemrdquoShock and Vibration vol 2016 Article ID 3524609 14 pages2016

[15] A Kahraman and R Singh ldquoNon-linear dynamics of a spurgear pairrdquo Journal of Sound and Vibration vol 142 no 1pp 49ndash75 1990

[16] A Kahraman and R Singh ldquoInteractions between time-varying mesh stiffness and clearance non-linearities in ageared systemrdquo Journal of Sound and Vibration vol 146no 1 pp 135ndash156 1991

[17] A Kahraman ldquoEffect of axial vibrations on the dynamics of ahelical gear pairrdquo Journal of Vibration and Acoustics vol 115no 1 pp 33ndash39 1993

[18] Y Cai and T Hayashi ldquoe linear approximated equation ofvibration of a pair of spur gears (theory and experiment)rdquoJournal of Mechanical Design vol 116 no 2 pp 558ndash5641994

[19] P Velex andMMaatar ldquoAmathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound and Vibration vol 191no 5 pp 629ndash660 1996

[20] P Velex and V Cahouet ldquoExperimental and numerical in-vestigations on the influence of tooth friction in spur andhelical gear dynamicsrdquo Journal of Mechanical Design vol 122no 4 pp 515ndash522 2000

[21] P Velex and P Sainsot ldquoAn analytical study of tooth frictionexcitations in errorless spur and helical gearsrdquo Mechanismand Machine Ieory vol 37 no 7 pp 641ndash658 2002

[22] T C Lim and Y Cheng ldquoA theoretical study of the effect ofpinion offset on the dynamics of hypoid geared rotor systemrdquoJournal of Mechanical Design vol 121 no 4 pp 594ndash6011999

[23] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[24] Y Cheng and T C Lim ldquoDynamics of hypoid gear trans-mission with nonlinear time-varying mesh characteristicsrdquoJournal of Mechanical Design vol 125 no 2 pp 373ndash3822003

[25] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 2017 pp 302ndash329 2007

[26] J Wang and T C Lim ldquoEffect of tooth mesh stiffnessasymmetric nonlinearity for drive and coast sides on hypoidgear dynamicsrdquo Journal of Sound and Vibration vol 319no 3-5 pp 885ndash903 2009

[27] W M Li and H T Wang ldquoRigidity calculation of axialposition preload bearingsrdquo Journal of Hebei University ofTechnology vol 30 no 2 pp 15ndash19 2001

[28] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo Non-linear Dynamics vol 75 no 4 pp 783ndash806 2014

[29] X Liu Z Wu J Lu and J Xu ldquoInvestigation of the effect ofrotation speed on vibration responses of transmission sys-temrdquo Matec Web of Conferences vol 256 no 3 Article ID02019 2019

Shock and Vibration 17

25NmiddotM 50NmiddotM 75NmiddotM 100NmiddotM 125NmiddotM 150NmiddotM175NmiddotM and 200NmiddotM in order

Together Figures 12 and 13 expectably show that thetorsional vibration is reinforced while increasing torquefluctuation Meanwhile an obvious increase in amplitudeoccurs at the second peak when fluctuation is increased from175NmiddotM to 200NmiddotM which indicates that a limitation should

be considered e accurate value can be determined by afurther study In summary reducing torque fluctuation is abeneficial approach for vibration damping Furthermore it isnoted that such a nonlinear phenomenon as natural fre-quency shift occurs while changing torque fluctuation Takingthe second-order torsional frequency as an example thefrequency value decreases with elevating fluctuation of input

Transmission

Photoelectricsensor

Photoelectricsensor

Magnetoelectricsensor

Magnetoelectricsensor

Rear axle

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Accelerationsensor

Drive sha system

Figure 9 e experiment bench and arrangement of sensors

042

040

038

035

033

030

028

025

023

020

018

015

013

010

008

005

003

Amplitude-frequency curve ofle wheel

Amplitude-frequency curve oftransmission output sha

Amplitude-frequency curve ofrear axle output end

Amplitude-frequency curve ofright wheel

000

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (K

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Curve 167000

041

035

003

003

5567 8817

167000 264500

rpm

deg

deg

deg

deg

264500

012

005

003

764endash3

Figure 10 Amplitude-frequency responses of torsional vibration for powertrain at a torque fluctuation of 100NmiddotM

14 Shock and Vibration

036

034

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

Numerical result

Experimental result

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

Figure 11 Comparison of numerical and experimental results for input end of rear axle

160 5567 8817

264500

150

140

130

120

110

100

090

080

070

060

050

040

030

020

010

000 167000

25Nmiddotm

50Nmiddotm

75Nmiddotm

100Nmiddotm

125Nmiddotm150Nmiddotm

175Nmiddotm

200Nmiddotm

Curve 167000147078057045035025017013

rpm--------

264500009008007006005004004003

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (R

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Figure 12 Amplitude-frequency responses of torsional vibration for input end of rear axle at different torque fluctuations

Shock and Vibration 15

torque which shows that the torsional system has a stiffness-softening characteristic with respect to torque fluctuationis result can be accounted for a number of nonlinearitiessuch as friction clearance and sliding and the existence ofwhich sophisticates system nonlinear characteristics

5 Conclusion

In the present paper a lumped-parameter dynamic modelwith 29 degree-of-freedom is proposed considering thetransverse torsional and rocking coupled vibration for abetter understanding of the automotive powertrain systemdynamics and equations of motion are derived e nu-merical result shows that gear mesh excitations have littleeffects on torsional response for such components locatedbefore the mesh point as transmission transmission shaftand pinion but significantly reduce the torsional responsesfor ones behind the mesh point e amplitude-frequencycharacteristics of torsional vibration graphically indicate thattorsional behaviors of automotive powertrain only affect thevibration characteristics of complete vehicle under low-speed condition and predict the first three-order naturetorsional frequencies of the whole system An experimentalexamination is performed and validates the feasibility of theconsidered model e torsional vibration is reinforcedwhile increasing torque fluctuation In addition a stiffnesssoftening characteristic is observed for torsional vibrationexperimentally In order to reach a better understanding inamplitude-frequency response at low- and middle-input-speed condition a procedure to analyze mode shapes ofautomotive powertrain will be conducted in future works

Data Availability

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

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported and funded by the ApplicationTechnology Research and Development Plan of LiuzhouCity (grant no 2017AA10102) and the Key Projects ofUniversities in Inner Mongolia Autonomous Region (grantno NJZZ16369)e authors of this paper express their deepgratitude for the financial support of the research project

References

[1] F Vesali M A Rezvani and M Kashfi ldquoDynamics of uni-versal joints its failures and some propositions for practicallyimproving its performance and life expectancyrdquo Journal ofMechanical Science and Technology vol 26 no 8 pp 2439ndash2449 2012

[2] J-W Lu G-C Wang H Chen A F Vakakis andL A Bergman ldquoDynamic analysis of cross shaft type universaljoint with clearancerdquo Journal of Mechanical Science andTechnology vol 27 no 11 pp 3201ndash3205 2013

[3] G Wu W Shi and Z Chen ldquoe effect of multi-universalcoupling phase on torsional vibration of drive shaft and vi-bration of vehiclerdquo in Proceedings of the SAE 2013 WorldCongress amp Exhibition Detroit MI USA April 2013

0113333 4000 4500 5000 5500

5567

6000 6500 7000Frequency (Hz)

7500 8000 85008777

263300

9000 9500 10000 10500 11000 11667100

Am

plitu

de

Am

plitu

de (R

MS)

010

009

008

007

006

005

004

003

002

001

000100000 120000 140000 160000 180000

167000

25Nmiddotm

50Nmiddotm

75Nmiddotm100Nmiddotm

125Nmiddotm

150Nmiddotm

175Nmiddotm200Nmiddotm

Curve 167000010006004003003002001

982endash3

263300006005004003003002001

690endash3

rpm--------

200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000000

Figure 13 Amplitude-frequency responses of torsional vibration for output end of rear axle at different torque fluctuations

16 Shock and Vibration

[4] B A Porter ldquoeoretical analysis of the torsional oscillationof a system incorporating a Hookersquos jointrdquo Journal of Me-chanical Engineering Science vol 3 no 4 pp 324ndash329 1961

[5] S F Asokanthan and X-H Wang ldquoCharacterization oftorsional instabilities in a Hookersquos joint driven system viamaximal Lyapunov exponentsrdquo Journal of Sound and Vi-bration vol 194 no 1 pp 83ndash91 1996

[6] A Farshidianfar M Ebrahimi and H Bartlett ldquoHybridmodelling and simulation of the torsional vibration of vehicledriveline systemsrdquo Proceedings of the Institution of Me-chanical Engineers Part D Journal of Automobile Engineeringvol 215 no 2 pp 217ndash229 2001

[7] E O A Abd Elmaksoud E M Rabeih N A Abdel-halimand S M El Demerdash ldquoInvestigation of self excited tor-sional vibrations of different configurations of automatictransmission systems during engagementrdquo Engineeringvol 3 no 12 pp 1171ndash1181 2011

[8] T B Juang M Cheng and L Na ldquoExperimental and finiteelement analyses of a sliding-tube-type driveshaft-inducedvehicle vibrationrdquo Proceedings of the Institution of MechanicalEngineers Part K Journal of Multi-Body Dynamics vol 221no 3 pp 375ndash385 2007

[9] L F Coutinho and A Tamagna ldquoFlexural vibration analysisof constant velocity half-shaftsrdquo in Proceedings of the Inter-national Congress amp Exposition Chicago IL ISA 1996

[10] Y T Kang Y H Shen W Zhang and J Yang ldquoStabilityregion of floating intermediate support in a shaft system withmultiple universal jointsrdquo Journal of Mechanical Science andTechnology vol 28 no 6 pp 2733ndash2742 2014

[11] J L Xu T Yan B Peng and B Wei ldquoEffects of the trans-mission shaft on the main reducer vibration based onADAMS and experimental demonstrationrdquo AustralianJournal of Mechanical Engineering vol 15 no 1 pp 2ndash102015

[12] C X Zhang H B Li and D Q Jin ldquoSimulation on dynamicsof transmission shaft-rear driving axle of minibus withADAMSrdquo Applied Mechanics and Materials vol 651ndash653pp 862ndash865 2014

[13] J L Xu X Y Su and P Bo ldquoNumerical analysis anddemonstration transmission shaft influence on meshing vi-bration in driving and driven gearsrdquo Shock and Vibrationvol 2015 Article ID 365084 10 pages 2015

[14] J L Xu W Lei andW X Luo ldquoInfluence of bearing stiffnesson the nonlinear dynamics of a shaft-final drive systemrdquoShock and Vibration vol 2016 Article ID 3524609 14 pages2016

[15] A Kahraman and R Singh ldquoNon-linear dynamics of a spurgear pairrdquo Journal of Sound and Vibration vol 142 no 1pp 49ndash75 1990

[16] A Kahraman and R Singh ldquoInteractions between time-varying mesh stiffness and clearance non-linearities in ageared systemrdquo Journal of Sound and Vibration vol 146no 1 pp 135ndash156 1991

[17] A Kahraman ldquoEffect of axial vibrations on the dynamics of ahelical gear pairrdquo Journal of Vibration and Acoustics vol 115no 1 pp 33ndash39 1993

[18] Y Cai and T Hayashi ldquoe linear approximated equation ofvibration of a pair of spur gears (theory and experiment)rdquoJournal of Mechanical Design vol 116 no 2 pp 558ndash5641994

[19] P Velex andMMaatar ldquoAmathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound and Vibration vol 191no 5 pp 629ndash660 1996

[20] P Velex and V Cahouet ldquoExperimental and numerical in-vestigations on the influence of tooth friction in spur andhelical gear dynamicsrdquo Journal of Mechanical Design vol 122no 4 pp 515ndash522 2000

[21] P Velex and P Sainsot ldquoAn analytical study of tooth frictionexcitations in errorless spur and helical gearsrdquo Mechanismand Machine Ieory vol 37 no 7 pp 641ndash658 2002

[22] T C Lim and Y Cheng ldquoA theoretical study of the effect ofpinion offset on the dynamics of hypoid geared rotor systemrdquoJournal of Mechanical Design vol 121 no 4 pp 594ndash6011999

[23] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[24] Y Cheng and T C Lim ldquoDynamics of hypoid gear trans-mission with nonlinear time-varying mesh characteristicsrdquoJournal of Mechanical Design vol 125 no 2 pp 373ndash3822003

[25] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 2017 pp 302ndash329 2007

[26] J Wang and T C Lim ldquoEffect of tooth mesh stiffnessasymmetric nonlinearity for drive and coast sides on hypoidgear dynamicsrdquo Journal of Sound and Vibration vol 319no 3-5 pp 885ndash903 2009

[27] W M Li and H T Wang ldquoRigidity calculation of axialposition preload bearingsrdquo Journal of Hebei University ofTechnology vol 30 no 2 pp 15ndash19 2001

[28] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo Non-linear Dynamics vol 75 no 4 pp 783ndash806 2014

[29] X Liu Z Wu J Lu and J Xu ldquoInvestigation of the effect ofrotation speed on vibration responses of transmission sys-temrdquo Matec Web of Conferences vol 256 no 3 Article ID02019 2019

Shock and Vibration 17

036

034

032

03

028

026

024

022

018

016

014

012

01

008

006

004

0021000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Speed (rpm)

Tors

iona

l disp

lace

men

t (de

gree

)

Numerical result

Experimental result

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

02

Figure 11 Comparison of numerical and experimental results for input end of rear axle

160 5567 8817

264500

150

140

130

120

110

100

090

080

070

060

050

040

030

020

010

000 167000

25Nmiddotm

50Nmiddotm

75Nmiddotm

100Nmiddotm

125Nmiddotm150Nmiddotm

175Nmiddotm

200Nmiddotm

Curve 167000147078057045035025017013

rpm--------

264500009008007006005004004003

3333 4000 4500 5000 5500 6000 6500 7000Frequency (Hz)

7500 8000 8500 9000 9500 10000 10500 11000 11667

Am

plitu

de (R

MS)

100000 120000 140000 160000 180000 200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000

100

Am

plitu

de

000

Figure 12 Amplitude-frequency responses of torsional vibration for input end of rear axle at different torque fluctuations

Shock and Vibration 15

torque which shows that the torsional system has a stiffness-softening characteristic with respect to torque fluctuationis result can be accounted for a number of nonlinearitiessuch as friction clearance and sliding and the existence ofwhich sophisticates system nonlinear characteristics

5 Conclusion

In the present paper a lumped-parameter dynamic modelwith 29 degree-of-freedom is proposed considering thetransverse torsional and rocking coupled vibration for abetter understanding of the automotive powertrain systemdynamics and equations of motion are derived e nu-merical result shows that gear mesh excitations have littleeffects on torsional response for such components locatedbefore the mesh point as transmission transmission shaftand pinion but significantly reduce the torsional responsesfor ones behind the mesh point e amplitude-frequencycharacteristics of torsional vibration graphically indicate thattorsional behaviors of automotive powertrain only affect thevibration characteristics of complete vehicle under low-speed condition and predict the first three-order naturetorsional frequencies of the whole system An experimentalexamination is performed and validates the feasibility of theconsidered model e torsional vibration is reinforcedwhile increasing torque fluctuation In addition a stiffnesssoftening characteristic is observed for torsional vibrationexperimentally In order to reach a better understanding inamplitude-frequency response at low- and middle-input-speed condition a procedure to analyze mode shapes ofautomotive powertrain will be conducted in future works

Data Availability

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

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported and funded by the ApplicationTechnology Research and Development Plan of LiuzhouCity (grant no 2017AA10102) and the Key Projects ofUniversities in Inner Mongolia Autonomous Region (grantno NJZZ16369)e authors of this paper express their deepgratitude for the financial support of the research project

References

[1] F Vesali M A Rezvani and M Kashfi ldquoDynamics of uni-versal joints its failures and some propositions for practicallyimproving its performance and life expectancyrdquo Journal ofMechanical Science and Technology vol 26 no 8 pp 2439ndash2449 2012

[2] J-W Lu G-C Wang H Chen A F Vakakis andL A Bergman ldquoDynamic analysis of cross shaft type universaljoint with clearancerdquo Journal of Mechanical Science andTechnology vol 27 no 11 pp 3201ndash3205 2013

[3] G Wu W Shi and Z Chen ldquoe effect of multi-universalcoupling phase on torsional vibration of drive shaft and vi-bration of vehiclerdquo in Proceedings of the SAE 2013 WorldCongress amp Exhibition Detroit MI USA April 2013

0113333 4000 4500 5000 5500

5567

6000 6500 7000Frequency (Hz)

7500 8000 85008777

263300

9000 9500 10000 10500 11000 11667100

Am

plitu

de

Am

plitu

de (R

MS)

010

009

008

007

006

005

004

003

002

001

000100000 120000 140000 160000 180000

167000

25Nmiddotm

50Nmiddotm

75Nmiddotm100Nmiddotm

125Nmiddotm

150Nmiddotm

175Nmiddotm200Nmiddotm

Curve 167000010006004003003002001

982endash3

263300006005004003003002001

690endash3

rpm--------

200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000000

Figure 13 Amplitude-frequency responses of torsional vibration for output end of rear axle at different torque fluctuations

16 Shock and Vibration

[4] B A Porter ldquoeoretical analysis of the torsional oscillationof a system incorporating a Hookersquos jointrdquo Journal of Me-chanical Engineering Science vol 3 no 4 pp 324ndash329 1961

[5] S F Asokanthan and X-H Wang ldquoCharacterization oftorsional instabilities in a Hookersquos joint driven system viamaximal Lyapunov exponentsrdquo Journal of Sound and Vi-bration vol 194 no 1 pp 83ndash91 1996

[6] A Farshidianfar M Ebrahimi and H Bartlett ldquoHybridmodelling and simulation of the torsional vibration of vehicledriveline systemsrdquo Proceedings of the Institution of Me-chanical Engineers Part D Journal of Automobile Engineeringvol 215 no 2 pp 217ndash229 2001

[7] E O A Abd Elmaksoud E M Rabeih N A Abdel-halimand S M El Demerdash ldquoInvestigation of self excited tor-sional vibrations of different configurations of automatictransmission systems during engagementrdquo Engineeringvol 3 no 12 pp 1171ndash1181 2011

[8] T B Juang M Cheng and L Na ldquoExperimental and finiteelement analyses of a sliding-tube-type driveshaft-inducedvehicle vibrationrdquo Proceedings of the Institution of MechanicalEngineers Part K Journal of Multi-Body Dynamics vol 221no 3 pp 375ndash385 2007

[9] L F Coutinho and A Tamagna ldquoFlexural vibration analysisof constant velocity half-shaftsrdquo in Proceedings of the Inter-national Congress amp Exposition Chicago IL ISA 1996

[10] Y T Kang Y H Shen W Zhang and J Yang ldquoStabilityregion of floating intermediate support in a shaft system withmultiple universal jointsrdquo Journal of Mechanical Science andTechnology vol 28 no 6 pp 2733ndash2742 2014

[11] J L Xu T Yan B Peng and B Wei ldquoEffects of the trans-mission shaft on the main reducer vibration based onADAMS and experimental demonstrationrdquo AustralianJournal of Mechanical Engineering vol 15 no 1 pp 2ndash102015

[12] C X Zhang H B Li and D Q Jin ldquoSimulation on dynamicsof transmission shaft-rear driving axle of minibus withADAMSrdquo Applied Mechanics and Materials vol 651ndash653pp 862ndash865 2014

[13] J L Xu X Y Su and P Bo ldquoNumerical analysis anddemonstration transmission shaft influence on meshing vi-bration in driving and driven gearsrdquo Shock and Vibrationvol 2015 Article ID 365084 10 pages 2015

[14] J L Xu W Lei andW X Luo ldquoInfluence of bearing stiffnesson the nonlinear dynamics of a shaft-final drive systemrdquoShock and Vibration vol 2016 Article ID 3524609 14 pages2016

[15] A Kahraman and R Singh ldquoNon-linear dynamics of a spurgear pairrdquo Journal of Sound and Vibration vol 142 no 1pp 49ndash75 1990

[16] A Kahraman and R Singh ldquoInteractions between time-varying mesh stiffness and clearance non-linearities in ageared systemrdquo Journal of Sound and Vibration vol 146no 1 pp 135ndash156 1991

[17] A Kahraman ldquoEffect of axial vibrations on the dynamics of ahelical gear pairrdquo Journal of Vibration and Acoustics vol 115no 1 pp 33ndash39 1993

[18] Y Cai and T Hayashi ldquoe linear approximated equation ofvibration of a pair of spur gears (theory and experiment)rdquoJournal of Mechanical Design vol 116 no 2 pp 558ndash5641994

[19] P Velex andMMaatar ldquoAmathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound and Vibration vol 191no 5 pp 629ndash660 1996

[20] P Velex and V Cahouet ldquoExperimental and numerical in-vestigations on the influence of tooth friction in spur andhelical gear dynamicsrdquo Journal of Mechanical Design vol 122no 4 pp 515ndash522 2000

[21] P Velex and P Sainsot ldquoAn analytical study of tooth frictionexcitations in errorless spur and helical gearsrdquo Mechanismand Machine Ieory vol 37 no 7 pp 641ndash658 2002

[22] T C Lim and Y Cheng ldquoA theoretical study of the effect ofpinion offset on the dynamics of hypoid geared rotor systemrdquoJournal of Mechanical Design vol 121 no 4 pp 594ndash6011999

[23] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[24] Y Cheng and T C Lim ldquoDynamics of hypoid gear trans-mission with nonlinear time-varying mesh characteristicsrdquoJournal of Mechanical Design vol 125 no 2 pp 373ndash3822003

[25] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 2017 pp 302ndash329 2007

[26] J Wang and T C Lim ldquoEffect of tooth mesh stiffnessasymmetric nonlinearity for drive and coast sides on hypoidgear dynamicsrdquo Journal of Sound and Vibration vol 319no 3-5 pp 885ndash903 2009

[27] W M Li and H T Wang ldquoRigidity calculation of axialposition preload bearingsrdquo Journal of Hebei University ofTechnology vol 30 no 2 pp 15ndash19 2001

[28] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo Non-linear Dynamics vol 75 no 4 pp 783ndash806 2014

[29] X Liu Z Wu J Lu and J Xu ldquoInvestigation of the effect ofrotation speed on vibration responses of transmission sys-temrdquo Matec Web of Conferences vol 256 no 3 Article ID02019 2019

Shock and Vibration 17

torque which shows that the torsional system has a stiffness-softening characteristic with respect to torque fluctuationis result can be accounted for a number of nonlinearitiessuch as friction clearance and sliding and the existence ofwhich sophisticates system nonlinear characteristics

5 Conclusion

In the present paper a lumped-parameter dynamic modelwith 29 degree-of-freedom is proposed considering thetransverse torsional and rocking coupled vibration for abetter understanding of the automotive powertrain systemdynamics and equations of motion are derived e nu-merical result shows that gear mesh excitations have littleeffects on torsional response for such components locatedbefore the mesh point as transmission transmission shaftand pinion but significantly reduce the torsional responsesfor ones behind the mesh point e amplitude-frequencycharacteristics of torsional vibration graphically indicate thattorsional behaviors of automotive powertrain only affect thevibration characteristics of complete vehicle under low-speed condition and predict the first three-order naturetorsional frequencies of the whole system An experimentalexamination is performed and validates the feasibility of theconsidered model e torsional vibration is reinforcedwhile increasing torque fluctuation In addition a stiffnesssoftening characteristic is observed for torsional vibrationexperimentally In order to reach a better understanding inamplitude-frequency response at low- and middle-input-speed condition a procedure to analyze mode shapes ofautomotive powertrain will be conducted in future works

Data Availability

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

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported and funded by the ApplicationTechnology Research and Development Plan of LiuzhouCity (grant no 2017AA10102) and the Key Projects ofUniversities in Inner Mongolia Autonomous Region (grantno NJZZ16369)e authors of this paper express their deepgratitude for the financial support of the research project

References

[1] F Vesali M A Rezvani and M Kashfi ldquoDynamics of uni-versal joints its failures and some propositions for practicallyimproving its performance and life expectancyrdquo Journal ofMechanical Science and Technology vol 26 no 8 pp 2439ndash2449 2012

[2] J-W Lu G-C Wang H Chen A F Vakakis andL A Bergman ldquoDynamic analysis of cross shaft type universaljoint with clearancerdquo Journal of Mechanical Science andTechnology vol 27 no 11 pp 3201ndash3205 2013

[3] G Wu W Shi and Z Chen ldquoe effect of multi-universalcoupling phase on torsional vibration of drive shaft and vi-bration of vehiclerdquo in Proceedings of the SAE 2013 WorldCongress amp Exhibition Detroit MI USA April 2013

0113333 4000 4500 5000 5500

5567

6000 6500 7000Frequency (Hz)

7500 8000 85008777

263300

9000 9500 10000 10500 11000 11667100

Am

plitu

de

Am

plitu

de (R

MS)

010

009

008

007

006

005

004

003

002

001

000100000 120000 140000 160000 180000

167000

25Nmiddotm

50Nmiddotm

75Nmiddotm100Nmiddotm

125Nmiddotm

150Nmiddotm

175Nmiddotm200Nmiddotm

Curve 167000010006004003003002001

982endash3

263300006005004003003002001

690endash3

rpm--------

200000 220000Speed (rpm)

240000 260000 280000 300000 320000 350000000

Figure 13 Amplitude-frequency responses of torsional vibration for output end of rear axle at different torque fluctuations

16 Shock and Vibration

[4] B A Porter ldquoeoretical analysis of the torsional oscillationof a system incorporating a Hookersquos jointrdquo Journal of Me-chanical Engineering Science vol 3 no 4 pp 324ndash329 1961

[5] S F Asokanthan and X-H Wang ldquoCharacterization oftorsional instabilities in a Hookersquos joint driven system viamaximal Lyapunov exponentsrdquo Journal of Sound and Vi-bration vol 194 no 1 pp 83ndash91 1996

[6] A Farshidianfar M Ebrahimi and H Bartlett ldquoHybridmodelling and simulation of the torsional vibration of vehicledriveline systemsrdquo Proceedings of the Institution of Me-chanical Engineers Part D Journal of Automobile Engineeringvol 215 no 2 pp 217ndash229 2001

[7] E O A Abd Elmaksoud E M Rabeih N A Abdel-halimand S M El Demerdash ldquoInvestigation of self excited tor-sional vibrations of different configurations of automatictransmission systems during engagementrdquo Engineeringvol 3 no 12 pp 1171ndash1181 2011

[8] T B Juang M Cheng and L Na ldquoExperimental and finiteelement analyses of a sliding-tube-type driveshaft-inducedvehicle vibrationrdquo Proceedings of the Institution of MechanicalEngineers Part K Journal of Multi-Body Dynamics vol 221no 3 pp 375ndash385 2007

[9] L F Coutinho and A Tamagna ldquoFlexural vibration analysisof constant velocity half-shaftsrdquo in Proceedings of the Inter-national Congress amp Exposition Chicago IL ISA 1996

[10] Y T Kang Y H Shen W Zhang and J Yang ldquoStabilityregion of floating intermediate support in a shaft system withmultiple universal jointsrdquo Journal of Mechanical Science andTechnology vol 28 no 6 pp 2733ndash2742 2014

[11] J L Xu T Yan B Peng and B Wei ldquoEffects of the trans-mission shaft on the main reducer vibration based onADAMS and experimental demonstrationrdquo AustralianJournal of Mechanical Engineering vol 15 no 1 pp 2ndash102015

[12] C X Zhang H B Li and D Q Jin ldquoSimulation on dynamicsof transmission shaft-rear driving axle of minibus withADAMSrdquo Applied Mechanics and Materials vol 651ndash653pp 862ndash865 2014

[13] J L Xu X Y Su and P Bo ldquoNumerical analysis anddemonstration transmission shaft influence on meshing vi-bration in driving and driven gearsrdquo Shock and Vibrationvol 2015 Article ID 365084 10 pages 2015

[14] J L Xu W Lei andW X Luo ldquoInfluence of bearing stiffnesson the nonlinear dynamics of a shaft-final drive systemrdquoShock and Vibration vol 2016 Article ID 3524609 14 pages2016

[15] A Kahraman and R Singh ldquoNon-linear dynamics of a spurgear pairrdquo Journal of Sound and Vibration vol 142 no 1pp 49ndash75 1990

[16] A Kahraman and R Singh ldquoInteractions between time-varying mesh stiffness and clearance non-linearities in ageared systemrdquo Journal of Sound and Vibration vol 146no 1 pp 135ndash156 1991

[17] A Kahraman ldquoEffect of axial vibrations on the dynamics of ahelical gear pairrdquo Journal of Vibration and Acoustics vol 115no 1 pp 33ndash39 1993

[18] Y Cai and T Hayashi ldquoe linear approximated equation ofvibration of a pair of spur gears (theory and experiment)rdquoJournal of Mechanical Design vol 116 no 2 pp 558ndash5641994

[19] P Velex andMMaatar ldquoAmathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound and Vibration vol 191no 5 pp 629ndash660 1996

[20] P Velex and V Cahouet ldquoExperimental and numerical in-vestigations on the influence of tooth friction in spur andhelical gear dynamicsrdquo Journal of Mechanical Design vol 122no 4 pp 515ndash522 2000

[21] P Velex and P Sainsot ldquoAn analytical study of tooth frictionexcitations in errorless spur and helical gearsrdquo Mechanismand Machine Ieory vol 37 no 7 pp 641ndash658 2002

[22] T C Lim and Y Cheng ldquoA theoretical study of the effect ofpinion offset on the dynamics of hypoid geared rotor systemrdquoJournal of Mechanical Design vol 121 no 4 pp 594ndash6011999

[23] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[24] Y Cheng and T C Lim ldquoDynamics of hypoid gear trans-mission with nonlinear time-varying mesh characteristicsrdquoJournal of Mechanical Design vol 125 no 2 pp 373ndash3822003

[25] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 2017 pp 302ndash329 2007

[26] J Wang and T C Lim ldquoEffect of tooth mesh stiffnessasymmetric nonlinearity for drive and coast sides on hypoidgear dynamicsrdquo Journal of Sound and Vibration vol 319no 3-5 pp 885ndash903 2009

[27] W M Li and H T Wang ldquoRigidity calculation of axialposition preload bearingsrdquo Journal of Hebei University ofTechnology vol 30 no 2 pp 15ndash19 2001

[28] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo Non-linear Dynamics vol 75 no 4 pp 783ndash806 2014

[29] X Liu Z Wu J Lu and J Xu ldquoInvestigation of the effect ofrotation speed on vibration responses of transmission sys-temrdquo Matec Web of Conferences vol 256 no 3 Article ID02019 2019

Shock and Vibration 17

[4] B A Porter ldquoeoretical analysis of the torsional oscillationof a system incorporating a Hookersquos jointrdquo Journal of Me-chanical Engineering Science vol 3 no 4 pp 324ndash329 1961

[5] S F Asokanthan and X-H Wang ldquoCharacterization oftorsional instabilities in a Hookersquos joint driven system viamaximal Lyapunov exponentsrdquo Journal of Sound and Vi-bration vol 194 no 1 pp 83ndash91 1996

[6] A Farshidianfar M Ebrahimi and H Bartlett ldquoHybridmodelling and simulation of the torsional vibration of vehicledriveline systemsrdquo Proceedings of the Institution of Me-chanical Engineers Part D Journal of Automobile Engineeringvol 215 no 2 pp 217ndash229 2001

[7] E O A Abd Elmaksoud E M Rabeih N A Abdel-halimand S M El Demerdash ldquoInvestigation of self excited tor-sional vibrations of different configurations of automatictransmission systems during engagementrdquo Engineeringvol 3 no 12 pp 1171ndash1181 2011

[8] T B Juang M Cheng and L Na ldquoExperimental and finiteelement analyses of a sliding-tube-type driveshaft-inducedvehicle vibrationrdquo Proceedings of the Institution of MechanicalEngineers Part K Journal of Multi-Body Dynamics vol 221no 3 pp 375ndash385 2007

[9] L F Coutinho and A Tamagna ldquoFlexural vibration analysisof constant velocity half-shaftsrdquo in Proceedings of the Inter-national Congress amp Exposition Chicago IL ISA 1996

[10] Y T Kang Y H Shen W Zhang and J Yang ldquoStabilityregion of floating intermediate support in a shaft system withmultiple universal jointsrdquo Journal of Mechanical Science andTechnology vol 28 no 6 pp 2733ndash2742 2014

[11] J L Xu T Yan B Peng and B Wei ldquoEffects of the trans-mission shaft on the main reducer vibration based onADAMS and experimental demonstrationrdquo AustralianJournal of Mechanical Engineering vol 15 no 1 pp 2ndash102015

[12] C X Zhang H B Li and D Q Jin ldquoSimulation on dynamicsof transmission shaft-rear driving axle of minibus withADAMSrdquo Applied Mechanics and Materials vol 651ndash653pp 862ndash865 2014

[13] J L Xu X Y Su and P Bo ldquoNumerical analysis anddemonstration transmission shaft influence on meshing vi-bration in driving and driven gearsrdquo Shock and Vibrationvol 2015 Article ID 365084 10 pages 2015

[14] J L Xu W Lei andW X Luo ldquoInfluence of bearing stiffnesson the nonlinear dynamics of a shaft-final drive systemrdquoShock and Vibration vol 2016 Article ID 3524609 14 pages2016

[15] A Kahraman and R Singh ldquoNon-linear dynamics of a spurgear pairrdquo Journal of Sound and Vibration vol 142 no 1pp 49ndash75 1990

[16] A Kahraman and R Singh ldquoInteractions between time-varying mesh stiffness and clearance non-linearities in ageared systemrdquo Journal of Sound and Vibration vol 146no 1 pp 135ndash156 1991

[17] A Kahraman ldquoEffect of axial vibrations on the dynamics of ahelical gear pairrdquo Journal of Vibration and Acoustics vol 115no 1 pp 33ndash39 1993

[18] Y Cai and T Hayashi ldquoe linear approximated equation ofvibration of a pair of spur gears (theory and experiment)rdquoJournal of Mechanical Design vol 116 no 2 pp 558ndash5641994

[19] P Velex andMMaatar ldquoAmathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound and Vibration vol 191no 5 pp 629ndash660 1996

[20] P Velex and V Cahouet ldquoExperimental and numerical in-vestigations on the influence of tooth friction in spur andhelical gear dynamicsrdquo Journal of Mechanical Design vol 122no 4 pp 515ndash522 2000

[21] P Velex and P Sainsot ldquoAn analytical study of tooth frictionexcitations in errorless spur and helical gearsrdquo Mechanismand Machine Ieory vol 37 no 7 pp 641ndash658 2002

[22] T C Lim and Y Cheng ldquoA theoretical study of the effect ofpinion offset on the dynamics of hypoid geared rotor systemrdquoJournal of Mechanical Design vol 121 no 4 pp 594ndash6011999

[23] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[24] Y Cheng and T C Lim ldquoDynamics of hypoid gear trans-mission with nonlinear time-varying mesh characteristicsrdquoJournal of Mechanical Design vol 125 no 2 pp 373ndash3822003

[25] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 2017 pp 302ndash329 2007

[26] J Wang and T C Lim ldquoEffect of tooth mesh stiffnessasymmetric nonlinearity for drive and coast sides on hypoidgear dynamicsrdquo Journal of Sound and Vibration vol 319no 3-5 pp 885ndash903 2009

[27] W M Li and H T Wang ldquoRigidity calculation of axialposition preload bearingsrdquo Journal of Hebei University ofTechnology vol 30 no 2 pp 15ndash19 2001

[28] A Farshidianfar and A Saghafi ldquoGlobal bifurcation and chaosanalysis in nonlinear vibration of spur gear systemsrdquo Non-linear Dynamics vol 75 no 4 pp 783ndash806 2014

[29] X Liu Z Wu J Lu and J Xu ldquoInvestigation of the effect ofrotation speed on vibration responses of transmission sys-temrdquo Matec Web of Conferences vol 256 no 3 Article ID02019 2019

Shock and Vibration 17