ReviewArticledownloads.hindawi.com/journals/sv/2020/9834939.pdf · 2020-08-06 · ReviewArticle...

Review ArticleClassifying Predicting and Reducing Strategies of the MeshExcitations of Gear Whine Noise A Survey

Menglei Sun1 Chihua Lu123 Zhien Liu 12 Yi Sun1 Hao Chen1 and Cunrui Shen1

1Hubei Key Laboratory of Advanced Technology for Automotive Components Wuhan University of TechnologyWuhan 430070 China2Hubei Collaborative Innovation Center for Automotive Components Technology Wuhan University of TechnologyWuhan 430070 China3Hubei Research Center for New Energy amp Intelligent Connected Vehicle Wuhan University of TechnologyWuhan 430070 China

Correspondence should be addressed to Zhien Liu lzenwhuteducn

Received 22 October 2019 Accepted 20 July 2020 Published 6 August 2020

Academic Editor F Viadero

Copyright copy 2020 Menglei Sun et al +is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Gear whine noise has attracted increasing attention from researchers in both the academe and the industry over the past twodecades+e wide range of research topics demonstrates that there is a huge technical challenge in understanding the source-path-receiver mechanisms deeply and predicting the gear whine noise precisely +oroughly understanding the sources of gear whinenoise is the first step to solving this issue In this paper the authors summarize a certain number of published articles regarding thesources of gear whine noise +e excitations of gear whine noise are classified into three groups transmission error along the lineof action direction frictional excitations along the off-line of action direction and shuttling excitation along the axial direction+e mechanisms characteristics predicting approaches measuring methods and decreasing strategies for these excitations aresummarized Current research characteristics and future research recommendations are presented at the end

1 Introduction

With the rapid development of technology in the automotiveindustry electric vehicles fiercely shock the automobile marketand are expected to occupy the market in a few years +eincreasingly intense competition in the vehicle market and therapid emergence of electric vehicles pose a series of challengesin improving interior acoustic comfort Internal combustionengine noise exhaust noise and gearbox noise are major in-terior noise sources in traditional vehicles However for electricvehicles gearbox noise can be perceived more easily given theabsence of the masking effect from the internal combustionengine and exhaust noise +erefore the key to reduce interiornoise and improve interior acoustic comfort for electric ve-hicles is to reduce the noise from the gearbox

According to the literature reviewed there are two kindsof noise emitted from the automotive gearbox namely gearrattle noise and gear whine noise Gear rattle noise is

generally caused by fluctuations in the engine torque andspeed [1ndash4] which usually lead to contact loss and impactsbetween lightly loaded mating gears Gear rattle noise alsohas a close relationship with lubricant conditions [5ndash7]Details of gear rattle noise for traditional and hybrid vehiclescan be found in publications by Singh et al [8] and Zhanget al [9] respectively Mesh excitations of gear rattle noiseare beyond the scope of this survey However gear whinenoise with its pure tonal characteristic and high frequencycan be much more annoying in electric vehicles where themask effect of the engine noise is weak +erefore gearwhine noise represents the main concern for acousticcomfort in an electric vehicle +is article focuses mainly onpublications related to gear whine noise limited space doesnot allow for publications related to gear rattle noise to beincluded in this paper +e objectives are limited to tradi-tional cylinder involute gears +e objective gears discussedin this survey are restricted to those in parallel gearboxes

HindawiShock and VibrationVolume 2020 Article ID 9834939 20 pageshttpsdoiorg10115520209834939

In geared systems if the loads transmitted by the gearswere constant the geometry of the gears were perfect andthe motions of the gears were smooth there would not beany vibration However in real conditions the profiles ofgears are imperfect because of manufacturing errors andintentional modifications In addition the teeth will deflectsignificantly when they are subjected to transmitted torqueMoreover supporting structures of gears are also deform-able when subjected to operation loads which induces in-evitable misalignments Deviations of the real gear profilefrom the ideal involute one deflections of teeth underoperating conditions and inevitable misalignments allcontribute to periodic displacement which induces varyingmeshing forces along the line of action (LOA) +e frictionforces between gear surfaces which act along the off-line ofaction (OLOA) change directions before and after the teethpasses through the pitch point For helical gears and spurgears with severe misalignments the centroid of the meshingforces shifts back and forth axially along the tooth facewidth +e displacement along the LOA and the frictionforces along the OLOA together with the axially back-and-forth shifts of the centroid of the meshing forces change theamplitudes action positions and directions of the meshingforces between mating gears +e oscillating meshing forcesare transmitted to supporting bearings by shafts resulting invarying bearing forces +e varying bearing forces vibrategearbox-housing plates which finally radiate undesired gearwhine noise

+e complexity of gear whine noise has inspired a largenumber of researchers to study this issue Dating back to1958 Harris [10] noticed the gear whine noise phenomenonand investigated the effect of the transmission error (TE) ongear whine noise Because of computer technology break-throughs in 1980s a large number of studies on topics re-lated to gear whine noise were conducted Coming into the21st century the number of the publications in this areashows an exponential growth trend as shown in Figure 1Many outstanding researchers published review papers re-lated to gear vibration and noise such as gear system dy-namic models [11] nonlinear dynamics of gear-drivensystems [12] condition monitoring and fault diagnosis[13 14] and planetary dynamics and vibration [15] re-spectively Conversely so far no one has made a compre-hensive summary on the sources of gear whine noise+erefore there is an urgent need for a systematic literaturereview on the sources of gear whine noise

During the generation of gear whine noise excitations ofgear meshes act as sources shafts and bearings as vibrationtransfer paths and the gearbox-housing plates as the re-ceiver +e reduction of gear whine noise can be reachedonly via the reduction of the amplitudes of the housing platevibrations Both the strength of the sources and the prop-agation property of the transfer paths influence the mag-nitudes of the vibrations of the housing plates In this reviewthe authors focus on the sources of gear whine noise

+is paper aims to summarize the excitations of gearwhine noise from the literature classify these excitationsaccording to the three action directions of the meshingforces present methods for predicting these excitations

conclude strategies to reduce these excitations and proposenew possibilities for gear whine noise reduction at its source+e remaining part of this paper is organized as follows InSection 2 the excitation along the LOA direction namelyTE is summarized the definition of TE harmonic contri-butions from TE strategies to calculate and measure TE andapproaches to reduce TE are also described In Section 3excitations along the OLOA are reviewed+e approaches toevaluate the frictional excitations and strategies to reducethem are presented In Section 4 shuttling excitations in theaxial direction and the properties of shuttling excitation aresummarized +e potential of vibration reduction forlightweight gears is discussed in Section 5 Section 6 gives asummary +e authors also point out essential existingproblems in reported research work and describe prospectsof future research directions regarding excitations of gearwhine noise

2 Transmission Error

If the profiles of gears were geometrically perfect the teethon gears were perfectly rigid and correctly spaced and thesupporting structures were rigid and accurately installedthere would be no variance inmeshing forces whenmeshingwhich results in no vibration being generated In reality thisideal scenario does not occur for a variety of reasons such asmanufacturing imperfections intentional modificationsand inevitable teeth deflections All of these imperfectionscontribute to a periodic displacement namely TE whichwas first defined by Harris [10] As stated by Munro [16] theperiodic variation of TE induces periodic advancement andretardation of the driven gear while the gears are rotating Inaddition if the rotation speed coincides with the frequencyof one component of TE then resonance will occur +eperiodic motions of the driven gear and the potential res-onance may give rise to large dynamic meshing loads andhigh noise levels Transmission error is therefore a primarysource of gear noise and vibration Since these two studies

100

50

18

35

71

101

0ndash1980 1981ndash1999

Year2000ndash2009 2010ndash2019

Publ

icat

ion

num

ber

Figure 1 Histogram of research papers on gear vibration andnoise

2 Shock and Vibration

gear designers have gradually appreciated the importance ofTE and tried to explore the relationship between TE and gearwhine noise Few researchers [17ndash20] showed that direc-tionally reducing the transmission error should reduce noiseas well However the direct relationship between TE and thelevel of gear whine noise remains unrevealed

In this section a detailed description of TE is presentedFor researchers who have just started their studies in thisarea this section will help them to quickly understand therelationship between TE and gear whine noise and the basicdefinition components prediction methods and reductionstrategies of TE

21 e Definition of Static Transmission Error Harris [10]proved that the periodic variations in the velocity ratio andthe variance and nonlinearity of mesh stiffness which allcontribute to static measured relative displacement were themain internal sources of vibration for spur gears +estatically measured relative displacement in that paper wasthe well-known TE that was subsequently defined [21] forany instantaneous angular position of one gear as the an-gular displacement (given by equation (1)) of the matinggear from the position it would occupy if the teeth wereperfect To the authorsrsquo knowledge Harris was the first whodefined TE and plotted the TE curves under different loadsfor rotating gears +ese curves were known as the Harrismap and were of great importance in understanding gearmotions Transmission error can also be described as lineardisplacement as in equation (2) along the LOA direction Itis much more convenient to describe TE as a linear dis-placement than as an angular difference because the lineardisplacement of gear pairs along LOA closely relates to andtriggers angular vibrations

TE(θ) θgminus

Rbp

Rbg

θp (1)

TE(d) Rbgθg

minus Rbpθp (2)

in which θi is the rotating angular of gear i as in Figure 2 andRbi is the base radius of gear i i p g in which p and g

represent the pinion and the gear respectivelyTransmission error is widely recognized as the dominant

excitation of gear whine noise [10 11 21ndash28] A thoroughresearch in TE is beneficial for deep understanding of gearwhine noise From the publications of Mark [29ndash32] TE canbe expressed accurately using equation (3a) and (3b) asfollows

TE(l) W

KM(l)+

1113936jη(p)

Kj (l) + η(g)

Kj (l)

KM(l) (3a)

KM(l) KminusM times 1 +

δKM(l)

KminusM

1113888 1113889 asymp KminusM times 1 minus

δKM(l)

KminusM

1113888 1113889

minus 1

(3b)

in which l is the rotating position KM(l) is the total time-varying mesh stiffness Kminus

M is the mean component of mesh

stiffness δKM(l) is the varying component of mesh stiffnessW is the transmitted torque j is the tooth number i p gand η(i)

Kj(l) is the deviation of the tooth surface for gear i+e first term in the right-hand side of equation (3a) is the

elastic deformation which also contains the mesh stiffnessvariation +e second term consists of deviations from perfectinvolute surfaces these deviations include manufacturing er-rors intentionally designed microgeometry modifications andsupporting misalignments +is analytical equation is suitablefor both helical and spur-gear pairs In a word there are fourprimary sources of TE for a gear pair gear teeth elastic de-formation [33 34] and stiffness variance [35ndash47]manufacturing errors [30 48ndash50] misalignments [51ndash58] dueto supporting structures and intentional profile modifications[33 53 59ndash66] In addition gear surface roughness [67ndash70]also contributes to TE It is worth mentioning that gearmicrogeometry correction is a significant approach for mini-mizing TE excitation the details of the approach are describedin the subsequent sections

22 Harmonic Contributions for Transmission ErrorTransmission error is generally analysed in the frequencydomain +ere are four sets of harmonics generated by the TEexcitation of a meshing gear pair tooth mesh harmonics twosets of rotational harmonics and the fundamental harmonicsof the meshing gear pair As described by Mark [31 32] themean deviations of the tooth working surfaces from equallyspaced perfect involute surfaces include the average elasticdeflections mesh stiffness mean component of manufacturingerrors [71] and intentional modifications all of which con-tribute to gear mesh harmonics +e tooth-to-tooth variationssuch as individual tooth manufacturing errors [72] contributeto rotational harmonics In addition shaft misalignment [73]gear plastic deformation [74 75] and gear damages [76ndash79]also provide contributions to TE rotational harmonics Vari-ations associated with tooth surface contact points contributeto the fundamental harmonics of the gear pair

Among the four sets mentioned above the toothmeshing harmonics are the strongest excitation the

Op

Og

Rbp

Rbg

θp

θg

Line of action

Pitch line

Pitch circleBase circle

α

Figure 2 Gear motions along the LOA

Shock and Vibration 3

rotational ones are the second strongest and the one as-sociated with fundamental harmonics of the gear pair is theweakest Because of frequency modulation there are side-bands around each harmonic of the mesh frequency asshown in Figure 3 For the analysis of gear whine noise it isuseful to examine how various TE sources are exhibiting inthe frequency domain

23 Transmission Error Calculation Methods As we knowTE is the primary source of gear whine noise and serves asthe forcing term in gear system dynamic analysis [80 81]Moreover at the design stage TE is the guideline to de-termine optimal tooth modifications [82] TE analysis is alsoan important tool for gear fault detection [74ndash79] at the earlystage +us the ability to accurately predict the TE of thetarget gears is essential in evaluating and minimizing gearwhine noise As described in equations (3a) and (3b) thereare two parts of TE tooth elastic deflections and surfacedeviations respectively Surface deviations which consist ofmanufacturing errors determined by gear accuracy andintentional modifications designed by engineers [83] can bemeasured accurately +erefore TE calculation mainly fo-cuses on the prediction of gear teeth elastic deflection andthe varying mesh stiffness

231 Analytical Method When spur-gear teeth are con-sidered as nonuniform cantilever beams bending shearingand contact deformations all contribute to the teeth de-flections Mesh stiffness is the transmitted force divided bythe elastic deformations In 1963 Gregory et al [21] de-veloped a simple approximate TE formula which was basedon Weberrsquos deflection equation [34] and varied as a sinu-soidal function Based on the publications by Cornell [84]and Lin et al [85 86] there are three components in geartooth deflection which are bending and shearing contactdeformation and deflections related to foundational effectsSubsequent TE calculations [87ndash90] considered cornercontact and proposed an additional approximate equationfor TE outside the normal path of contact Another methodfor calculating the deflections of gear teeth [91 92] was basedon conformal mapping of complex variables in planeelasticity

+e teeth of helical gear pairs are considered as infinitecantilever plates when their deformations are calculated[93 94] For helical gears the calculation of elastic defor-mation is more difficult than for spur gears because thegeometry and load distribution are more complicated Basedon the cantilever plate theory and the mathematic pro-gramming approach Conry and Seireg [95 96] developed aload distribution program to determine the load distributionand elastic deformation of the gear tooth for both helical andspur gears Many recent research studies [59 97ndash100] relatedto TE prediction apply this computer program

232 Numerical Method Because of the vast assumptionsand empirical parameters the analytical equations for TEwere not accurate enough to perform exact calculations

With recent computer hardware and software improve-ments numerical simulation becomes increasingly popularin calculating TE As long as the detailed 3D model of theteeth is established and the tooth surface is described pre-cisely by high-quality elements given the material propertyand boundary conditions the finite element (FE) model cancalculate the TE for a pair of gears accurately

Finite element method is widely applied in calculatingtooth deformation and mesh stiffness for spur [101 102] andhelical gears [103 104] +e challenge is the modelling ofcontact deformation with its nonlinearity For the im-provement of computational efficiency a properly simplified2D FEmodel [105 106] is popular for calculating TE for spurgears When misalignment and nonuniform load distribu-tion along contact lines are considered a 3D FE model[107ndash111] of the spur gear should be applied However forhelical gears [112 113] a 3D FE model is essential for thecomplex geometry Generally the FE model of a spur orhelical gear is specific for one gear pair so parameter analysisbased on FEmodels seems impractical However parametricFE modelling approaches [114ndash119] overcome this difficultyand offer an opportunity to study the effects of micro-geometry parameters on TE

For both spur and helical gears the contact between twosurfaces is localized in a very small region so there is anurgent need for refined mesh [120] in the contact areaAccordingly preparing gear element models with highquality is a time-consuming burden requiring high levels ofskill +e microgeometry complexities of the tooth profileand tooth leadmake it more difficult to model every detail onthe tooth surface In addition too many details in FE modelsslow down the numerical simulation

233 Analytical-Numerical Method Due to too many as-sumptions the analytical models are efficient but not ac-curate Although the results of FE simulation are veryaccurate its efficiency is not satisfactory +erefore re-searchers need to find a way to balance the efficiency andaccuracy of the methodologies for calculating TE A com-bination of analytical and FE methods is a compromisemethod offering quick simulation and high accuracysimultaneously

Rotational harmonics Fundamental meshfrequency

Second harmonicof mesh frequency

ird harmonicof mesh frequency

Sideband

Figure 3 Harmonic contribution for the transmission error


For the gear teeth the bending and shearing deflectionsare easy to calculate using the FE method while the cal-culation of Hertzian contact deformations is more com-plicated due to the nonlinearity of the deformations+erefore Umeyama et al [130] and Rincon et al [40]estimated the deformations of a helical gear tooth by ana-lytical Hertzian formulae for the contact deformation and bythe FE method for the bending deflections Simon [122 123]calculated the tooth deflection of helical gears using ana-lytical equations which were obtained via regression anal-ysis and interpolation of the results of the 3D FE model Raoand Yoon [131] subsequently adopted the formulae pro-posed by Simon to calculate the deformations of helicalgears Another combined method is the integral equationmethod [62 124ndash126 129] consisting of the load distribu-tion coefficient and mesh stiffness +e integral equationscan calculate TE faster than the normal FE method Nor-mally the mesh stiffness in the integral equations is obtainedvia the FE method while the load distribution coefficient isdetermined analytically [127 128] Li [52 53 121] combinedthe mathematical approach with the FE method to calculatetooth load distribution and TE Based on the surface re-sponse approach [132] the peak-to-peak TE derived fromthe FE model is represented and optimized via a mathe-matical formula Many papers related to TE reduction viagear microgeometry modification adopt combined analyti-cal-FE models A summary of the analytical equationsadopted in the reviewed publications is presented in Table 1

24 Transmission Error Measurement As described inequations (3a) and (3b) tooth surface deviations are theprimary source of TE Transmission error predictive ap-proaches cannot accurately calculate the deviations due tomanufacturing imperfections which have a significant in-fluence on the gear whine noise +erefore the prediction ofTE mainly focuses on the calculations of deflections anddeformations of gear teeth +e measurement strategies forTE make these calculations possible and can obtain toothsurface deviations and deflections simultaneously

Transmission error can be measured statically or dy-namically (under low or high speed) as described byAkerblom [133] Gregory et al [134] proposed the first staticTE measurement equipment but it is only suitable for 1 1ratio gear pairs Houser and Blankenship [135] summarizedfour methods for measuring static and dynamic TETransmission error measurements in the early stage wereusually limited to isolated gears tested under extremely light-load and low-speed condition using the single-flank test rigTraditionally the measurements from single-flank tests weremainly for verifying gear manufacturing accuracy Morerecently measurements are conducted under quasistaticloaded conditions [136ndash141] +e measurements of loadedgears are the interactions of microgeometry deviations andtooth deflections +ese experiments are generally con-ducted on a noncirculating power test rig [141] or on apower-circulating test rig [142] Optical encoders withproper resolution are generally installed at the shaft ends torecord the rotating displacements of the gears+e challenge

with TEmeasurement under quasistatic loaded conditions isthat the magnitude of TE is extremely small and often in themicron order +us selection of encoders with properresolution is an important step in measuring TE

Nowadays the need for dynamic TEmeasurement growsfaster because dynamic TE has a closer relationship with gearwhine noise +ere are three methods suitable for measuringdynamic TE one based on optical encoders [143 144]another based on vibrometers [145] and the last based onaccelerations [142 146ndash148]

Methods based on encoders are conducted on a power-circulating test rig under high-speed conditions +e chal-lenge with this method is that it is difficult to eliminate thedynamic effects of the slave gearbox In addition the testedgears are generally mounted on two isolated shafts or in aspecially designed gearbox +e boundary conditions of thegears are extremely different from the ones that gears aresubject to in a real gearbox +us there is an urgent need fortechnology that can successfully measure the static anddynamic TE in a real gearbox Methods based on vibr-ometers provide a chance to measure the dynamic TE in areal gearbox [145] However the outputs of this method arenot as good as those produced by encoders and require anintegration to obtain the dynamic TE In addition thereshould be several holes on the housing plates for laser access+us the lubricant of the gearbox may be a great challengefor accurate measurement Methods based on accelerationsare conducted on a power-circulating test rig and the ac-celerometers are installed either on the walls of the gearbox[142] or on the flanks of the gears [148] +e outputs of thismethod are the second derivative of the TE which meansthat there would be a large error when calculating TE Whenthe accelerations are measured precise calibration tightmisalignment and adequate mounting space are necessaryconditions

25 Methods to Reduce Transmission Error As mentionedbefore the meshing variations during the meshing processare the primary excitations of gear whine noise A reductionin the level of excitations directly results in the decrease inthe gear whine noise One-micrometre reduction of TEresults in 5 dB reduction of the gear whine noise [149] Geartooth modification is such an approach for gear whine noisereduction by decreasing the meshing excitations A smoothload transfer is crucial for reducing the gear whine noise andcan be achieved by intentional tooth surface modificationsMicrogeometries should be optimized carefully to obtain aquiet gearbox design

For uniform transfer of motions tip and root relieves arefrequently implemented on the gear tooth profile In the1930s Walker [33] first proposed the gear tip relief theoryHarris [10] observed that the amount of tip relief could beappropriately chosen so that TE can be constant at the designload Since then many researchers devote their efforts inclarifying the effects of linear [21 59 150 151] parabolic[106 152ndash156] cubic [88] and other complicated types[157 158] of relief on TE reduction For helical gears withbias modification [159] the TE is less sensitive to the


transmitted torque +e common function of lead andprofile crowning is the reduction of contact and root stress[160 161] To understand the influence of the combinedprofile and lead crown modifications on TE and contactstress for helical gears Zhang et al [98] conducted a nu-merical investigation from which one could learn that eachpair of modification parameters gives an optimum peak-to-peak TE at different torques

+ese investigations mentioned above are limited to casestudies and numerical analysis for specific gear applicationsAlthough the outputs of the analysis are rather accurate theglobal trend of TE when changing modification parametersis difficult to illustrate+e optimal modificationsmentionedabove are valid solely at the design load while gears usuallywork under multiple load conditions +ere is an urgentneed for a robust modification optimization that canguarantee a relatively small excitation level under the wholeoperating range Analytical methods seem to have moreadvantages in finding robust modification designs [82 162]and allow for parameter analysis [129 163ndash165] MinimizingTE is the primary objective for gear-form modificationsMeanwhile optimal TE achieves good performance onlyunder static conditions Reduction of meshing forces andbearing forces for gear systems is of great importance in thereduction of gear whine noise +erefore researchers shouldconsider dynamic criteria [166ndash169] when the setup entailsgear profile modifications

3 Frictional Excitations

During the process of meshing the teeth undergo purerolling at the pitch point +e motions before and after thepitch point are known as approaching and recessing re-spectively +e directions of the friction forces are oppositein these two periods +e sudden direction reversal of thefriction forces near the pitch point has a significant influenceon dynamic meshing forces and induces varying bearingforces In addition the arms of friction moments vary as thegears rotate Time-varying sliding friction forces and frictionmoments between meshing teeth are significant excitations

of gear whine noise Friction forces and moments are usuallyreferred as the secondary excitation when compared to TE

31 Frictional Excitations To the knowledge of the authorsIida et al [170] were the first researchers who investigatedthe vibrational characteristics of gears due to friction forcesBorner and Houser [171] quantitatively evaluated the in-fluences of friction forces and reached the conclusion thatfriction forces should be considered as an excitation of gearwhine noise when TE is low Hochmann [172] proved viaexperimentation that friction force is a potential excitationVedmar andHenriksson [101] highlighted the importance ofthe motions along the OLOA direction on dynamic meshingforces Velex and Cahouet [173] revealed that friction forceshave a significant contribution on gear vibration and noise atlow and medium speeds under high torque levels Houseret al [174] conducted a series of experiments to identifyfrictional excitation and to study the influences of tribologyparameters From the experimental results one can learnthat friction force is a dominant excitation for gear whinenoise in higher harmonics of mesh frequencies Vaishya andSingh [175 176] proposed an analytical model to evaluatethe contribution of friction torque as an external excitationon TE Results show that frictional excitations have limitedinfluence on lower harmonics of mesh frequency but havesignificant influence on higher harmonics Velex and Sainsot[177] concluded that translational responses were sensitiveto frictional excitations and that torsional responses wereless sensitive to them According to Gunda and Singhrsquos [178]observation frictional excitations could not only change theshape of the TE curves but also increase the amplitude of thesecond harmonic of the mesh frequency In Lundvall et alrsquosinvestigation [179] the friction forces increased the peak-to-peak TE In He et alrsquos [180] study friction forces inducedlarge oscillations in bearing forces along the OLOA direc-tion+e authors also emphasized the importance of frictionforces when TEwas small He et al [181] then proposed a 12-degree-of-freedom (DOF) helical gear model which coupledrotation motions translation motions and axial motionsSimulation showed that friction forces are a potential

Table 1 Formulae for transmission error calculation

Formula Referencex ε cos(NΩt) [21]x xp + xg

xj xbj + xfj + xcj j p g[84ndash86]

xopc asymp (c22)((1r1) + (1r2)) [89 90]x cε minus 1113936

7i1(ci(ai + ε)) [91 92]

x wk + εk + P

wk 1113936Ni1 akiFi

aki wkbi + wkTi + wkpi + wkHi

[94ndash96 121]

x (151537FnEm)nf1f2f3nzminus 10622n(α020)minus 03879n(1 + (β010))008219n(1 + xp)minus 02165

n(hfm)05563n(hkm)06971n(rfilm)000043n(bfm)minus 0604 [122 123]

x 1113938 Kb(xL ζL)P(ζL)dζL + 1113938 Kc(xL ζL)P(ζL)dζL

F 1113938 P(ζL)cos βdζL

[124ndash128]

x(τ) δmn((1 minus 1113938 1113954K(M)n(e(M)δm)dM)cos βbn1113954k(τ xs)) [129]

x is the TE and the definition and value of other coefficients in the formula can be found in the corresponding reference


excitation in the LOA direction significant excitation in theOLOA direction and insignificant excitation in the axialdirection According to another publication of He and Singh[182] motions along the LOA are insensitive to frictionforces Singh et al [183] assessed the contributions of TE andfriction forces on gear vibration and noise+e bearing forcecurves indicated that besides the excitation role along theOLOA friction forces might influence the forces along theLOA significantly He and Singh [184] stated that the co-efficient of the friction forces affected only the first twoharmonics of TE Kahraman et al [185] concluded that themotions along the OLOA were sensitive to friction forceswhile the motions along the LOA were insensitive to thesefriction forces He et al [186] observed that the gear whinenoise induced by friction forces was comparable to thatinduced by optimized TE Liu and Parker [187] reported thattooth bending induced by friction moments and forcesaffected system dynamics significantly Chen et al [188]suggested that friction force might reduce the dynamicresponses of gear systems in high-frequency and low-speedconditions Wang et al [189] stated that friction forces couldchange the motion stability of a gear system Brethee et al[190] discovered that the motions along the OLOA weresensitive to the variance of friction forces

All the research studies mentioned above reach anagreement that friction forces and moments are indeedsignificant excitations of gear vibration and noise along theOLOA direction Some investigators [171 180 186] em-phasize the importance of frictional excitations when TE islow However there is no consistent conclusion about therole of frictional excitations in the LOA direction A smallnumber of scholars [175ndash177 182 185] believe that in theLOA direction the influence of friction forces is ignorableFrom other studies the effects of frictional excitations on theamplitudes [179 184] and curve shape [178] of TE dynamicmeshing forces [183 188 189] and gear bending [187]cannot be easily concluded To the authorsrsquo knowledge Heet al [181] were the only researchers who studied the in-fluence of frictional excitations on motions along the axialdirection indicating that this area needs further researchFrictional excitations have a significant influence on geardynamic response a fact that is recognized by manyscholars Meanwhile Jiang et al [191] reported that fric-tional excitations and gear vibration were interactive andcoupled Vibration affected the sliding velocities and di-rections of the frictional excitations and hence magnifiedvibrations along the OLOA direction

32 Friction Force Prediction Vaishya and Singh [192]summarized various strategies calculating friction forces andmoments Kar and Mohanty [193] proposed formulae offriction forces and moments based on time-varying contactlength for helical gear pairs +e general expressions forfrictional excitation are shown as follows

Ff μnFmeshnε (4a)

Mf μnFmeshnεnζ (4b)

in which μ is the friction coefficient Fmesh is the normaltooth meshing load ε is the sign function ζ is the arm of thefriction moment

+e normal tooth load is the contact force along theLOA which has a close relationship with the stiffness anddeflections of the mating teeth+e general expression of thenormal force in Ref [178] is illustrated by

Fmesh ki(t)n δ(t) minus εi(t)( 1113857 + Ci(t)n _δ(t) minus _εi(t)1113872 1113873 (5)

in which ki(t) is the varying mesh stiffness Ci(t) is thevarying damping coefficient δ(t) minus εi(t) is the displacementalong the LOA direction _δ(t) minus _εi(t) is the first derivation ofthe displacement

+e direction reversal is a nonlinear factor and of greatimportance in predicting frictional excitations+e formulaedetermining this nonlinear factor are summarized in Table 2As can be seen from equations (4a) and (4b) besides thedirection reversal friction forces are the product of thenormal tooth load and frictional coefficient Friction mo-ments are the product of friction forces and moment armsFormulae for moment arms based on the gear geometry aredescribed in Table 3 +e coefficient of the frictional exci-tation is complicated and discussed in the following para-graph in detail

As reviewed by Martin [194] the coefficient of frictionforce was mainly estimated empirically before the year of2000 such as by Benedict and Kelley [195] Kelly andLemanski [196] Johnson and Spence [197] and Rebbechiet al [198] Four different frictional coefficient formulaecorresponding to the Coulomb friction [180] empiricallyestimated friction [195] smoothed Coulomb friction [199]and thermal non-Newtonian elastohydrodynamic lubricant(EHL) [200 201] were then adopted into a spur-gear dy-namic model by He et al [202] +e difference betweenmotions along the OLOA and TE curves predicted by thesemodels is not significant A mixed EHL frictional coefficientformula [203] is accurate enough to predict the frictionalexcitation Han et al [204] proposed helical gear frictionforces and torque formulae considering nonuniform loaddistribution and the time-varying friction coefficient Pre-dictions indicated that effects of nonuniform load distri-bution were more significant than the variance of the frictioncoefficient In addition it has been proved that [173 177] theexact form of the friction law is of secondary importance andthe dominant factor is the reversal of the sliding directions atthe pitch point Afterwards researchers understand that aconstant friction coefficient is acceptable in gear dynamicanalysis and whine noise prediction Most scholars adopt theCoulomb friction model [205ndash210] and a user-definedconstant friction coefficient However Liu et al [211] ob-served that the variance of the frictional coefficient shouldnot be neglected

Among the reviewed frictional research studies a vastmajority of researchers adopted a Coulomb friction modelassumed a uniform distributed friction coefficient along theOLOA and changed its value within an acceptable range tostudy the effects of the frictional excitations on gear vi-bration and noise Some investigators utilized Benedict and


Kellyrsquos empirical formulae to calculate the coefficient offrictional excitation +e empirical formulae indicate thatthe coefficient of frictional excitation is a function of manyparameters such as sliding and rolling velocities surfaceroughness and load distribution +ese formulae are moreaccurate than the constant coefficient but being based onexperimental results they are expensive and not generalBased on experimentally validated simulations Xursquos EHLformulae are accurate general and convenient Howeverthese formulae are widely used only in the area of gearsystem efficiency the application of these formulae in geardynamic vibration and noise areas should be more ex-tensive in the future Meanwhile a mixed EHL formula iswidely utilized in gear dynamic analysis +e frictionalcoefficient formulae from the abovementioned publicationscan be classified into four groups (listed in Table 4) Cou-lomb formula empirical formula EHL formula and mixedEHL formula Here the ISO formula is cited for comparison

33 Strategies to Reduce Frictional Excitations Although thefrictional coefficient is of secondary importance when cal-culating the frictional excitations it is the most importantterm for reducing frictional excitation From Table 4 onecan learn that many parameters such as gear surfaceroughness and property of the lubricant oil have greatinfluence on the coefficient of frictional excitation +is isbecause the gear whine noise increases with the surfaceroughness [214 215] Moreover as described by Huang et al[212] a smoother surface leads to a smaller frictional ex-citation +erefore by improving the surface finish of thegear teeth the level of the gear whine noise induced byfrictional excitations would be reduced significantly [174]Frictional excitation also has a close relationship with the

viscosity of the lubricant oil With a decrease in the viscosityof the lubricant oil the frictional excitations decrease sig-nificantly [174] Sufficient lubricant oil would decrease theviscosity of the oil and hence reduce the frictional excitationsfor the gear whine noise However a possible alternativecause of noise in a spur gearbox is an overgenerous oilsupply if oil is trapped in the roots of the meshing teeth[5ndash7] If the oil cannot escape fast through the backlash gapit will be expelled forcibly axially from the tooth roots and atonce-per-tooth frequency can affect the end walls of the gearcase +is phenomenon induces another well-known an-noying noise gear rattle noise

4 Shuttling Excitation

In the meshing process of helical gears the centroid of themeshing force axially shifts back and forth along the toothface width [124] which induces time-varying bearing forcesand dynamic moments In addition angular misalignmentsmay move the action position of the meshing force to oneend of the face +e angular misalignments are also loaddependent which means that the action position of themeshing force would change as the load varies Shuttlingeffect due to the helical angle acts along the LOA directionwith a frequency equal to the gear mesh frequency Shuttlingeffect due to angular misalignments results in a twistingmoment that may push the gears to an aligned position

Borner and Houser [171] quantitatively evaluated theinfluence of shuttling excitation and reported that theamplitude of the bearing forces induced by shuttling exci-tation might be three times as those induced by TE Houseret al [221] suggested that shuttling was one kind of exci-tation for gear whine noise Nishino [124] proposed anexcitation model in which both TE and shuttling excitation

Table 2 Sign functions for frictional excitations

Sign functions Reference

εi (minus V21

rarr(Mi)n y

rarrV21

rarr(Mi)) plusmn y

rarr

[173 177]V

21

rarr(Mi) [Ω01nl1i +Ω02nl2i + (Ω02 +Ω01)nηin sin βb]

sgn[AP minus Ωpnrbpt + λn floor(Ωpnrbptλ)][178]

sgn(f1) tanminus 1((αpinRpnΩp minus αginRgnΩg)VR)

sgn(f2) sgn[ζg(t)n(Ωg + _θg(t)) minus ζp(t)n(Ωp + _θp(t))] [175 176 189]sgn[minus xnΩp + (xg minus x)nΩg] [181]sgn[ωpnρpi(t) minus ωgnρgi(t)] [190]+e definition and value of each coefficient in the formula can be found in the corresponding reference

Table 3 Formula of frictional moment arms

Formula of frictional moment arms Referenceζp l1i minus ηin sin βb

ζg (Rb1 + Rb2)n tan αp

[173]

xpi(t) LXA + (n minus i)nλ + Ωpnrbpt minus λn floor(Ωpnrbptλ)

xgi(t) LYC + inλ minus Ωpnrbpt + λn floor(Ωpnrbptλ)[178 180 186 190]

ζ1(t) (Rb1 + Rb2)n tan α minusR2

a1 minus R2b2

1113969+ Rb1nω1nt1 + Rb1nθ1

ζ2(t) (Rb1 + Rb2)n tan α minus ζ1(t)[189]

+e definition and value of each coefficient in the formula can be found in the corresponding reference


were considered After evaluating the individual contribu-tions of TE and shuttling excitation the authors reached theconclusion that shuttling excitation had a significant in-fluence on the total response of the housing plates From thepublications by Eritenel and Parker [216 217] one can learnthat shuttling motions can excite the twist mode of thegeared system According to the simulation by Palermo et al[218 219] shuttling excitation due to angular misalignmentand helix angle both contribute to varying bearing forcesthat finally induce gear whine noise Teja et al [220] cal-culated the bearing forces induced by shuttling forces andTE for helical gears +e authors concluded that when TEwas minimized by microgeometry modification shuttlingexcitation becomes the dominant excitation for the gearwhine noise

Shuttling excitation is inherent for helical gear pairs withwide faces As described by Eritenel and Parker [216 217]for spur gears with misalignments and aligned helical gearsthe shuttling excitation generates fluctuating moments witha mean value For helical gears with misalignments thefluctuating moments for gear and pinion are quite differentHelical angle and amplitude of misalignment both influencethe strength of the shuttling excitation+e distance betweensupporting bearings also has a significant influence on thestrength of the shuttling excitation [220] the longer thedistance the weaker the excitation In addition unequallength on either side of the gear increases the shuttlingexcitation Teja et al [220] also observed that helical gearswith a higher contact ratio introduce stronger shuttlingexcitation

Despite the fact that shuttling may be a nonnegligibleexcitation for the gear whine noise in the axial directionthere are a limited number of research studies focusing onshuttling excitation and its effect on gear whine noiseShuttling is a 3D issue so two-dimensional contact modelsof gear pairs cannot calculate the shuttling excitation De-tailed 3D contact models for gear pairs enable the inclusionof nonuniform load distribution and misalignments andthus are suitable for predicting shuttling excitation Ana-lytical formulae for shuttling forces and moments inreviewed articles are summarized in Table 5 Meanwhilestrategies for decreasing the shuttling excitations are difficultto illustrate Modelling strategies and the influence ofshuttling excitation on gear whine noise should be inves-tigated further Approaches reducing shuttling excitations

should also be proposed and verified as soon as possible +erelationship between shuttling excitation TE and frictionalexcitation should be studied further

5 Lightweight Gears

Most of the strategies relating to gear whine noise reductionare gear tooth micromodification as mentioned in Section 2However with a precisely designed optimum profile toothmeshing remains a vibration generator Using lightweightgears originated in the aerospace industry and is recentlyextending to the vehicle industry Because of the stricterregulation on emissions in common fuel vehicles and thepopulation of the electric vehicles lightweight gears are priorto solid steel gears in the gearbox +erefore the moderngear design should meet the need of weight reductionwithout compromising the requirements of NVH perfor-mance and reliability From this point of view severalstrategies have been proposed which include the thin-rimmed gears the material removal through holes and theadoption of new materials

Common lightweight strategies are based on materialremoval such as a thinner rim or manufacturing holes andsluts in the gear body As mentioned by Li [222] althoughthere is an increasing trend of adopting lightweight gearsdesign problems related to the dynamic strength and vi-bration of such gears are not fully solved yet From thesurvey by Li one can learn that the thickness of the rim andweb significantly influences the bending modes and naturalvibration behavior of the lightweight gears Researchers[223ndash225] point out that due to the nonuniform mass andstiffness distribution along the gear blank the lightweightgears with holes produce additional harmonic componentsat a lower frequency range +e additional order of the staticTE would increase the risk of exciting the rest of the systemSystem responses to the excitation due to the holes in theweb are comparable to that of the mesh excitation +elightweight gears with holes are more compliant than thesolid gears so the contact loss phenomenon would be easedHowever as described by Shweiki et al [224] the peak-to-peak magnitude of the static TE and the magnitude ofdynamic TE for the lightweight gears are larger than that ofsolid ones +e modelling strategies of the thin-rimmedgears and lightweight gears with holes are complicated thancommon solid gears as described by Guilbert et al [226 227]

Table 4 Formula of the friction force coefficient

Formulae Model type Referenceμ μnsgn(Vs) Coulomb [176ndash184 187ndash190 192 193 199 205ndash210]μ 00127[50(50 minus S)]log[317(10)8WprimevVsV

2r ]

10 Empirical [191 195]

μ 00099n(1((1 minus S)45))log[(35 times 108w)η0VSV2T(Rp + Rg)2]

Mix lubricant condition(experimentally) [174 196]

μ ef(SRPhv0 S)Pb2h |SR|b3V

b6e V

b70 Rb8

f(SR Ph v0 S) b1 + b4|SR|Phlog(v0)10 + b5e

minus |SR|Ph log(v0 )

10 + b9eS EHL [185 200 201 211 212]

μML (1 minus fa)nμBL + fanμEL

μBL 008 sim 013μEL μ by XuHai

Mixed EHL condition(analytically) [203]

μ 0325[VsVrvk]minus 025 (ISO) [213]+e definition and value of each coefficient in the formula can be found in the corresponding reference


and Shweiki et al [225] respectively Only continuousflexible models can capture the particular dynamic prop-erties of lightweight gears Guilbert et al [227] developed amortar-based methodology that can connect 1D grids and3D finite elements Based on this approach thin-rimmedgears can be precisely modelled in dynamic models In orderto calculate the load distribution and static TE of thelightweight gears precisely a complex numerical model isrequired as demonstrated by Shweiki et al [225] +eremaining issue is the development of simulation modelsable to predict the impact of different design parameters ongear vibration behavior Hou et al [228] numerically andquantitatively investigated the impact of the rim and webthickness on static TE meshing forces bearing forces andhousing vibration Gear rim and web thickness have a greatinfluence on both the static and dynamic responses of thegeared system of an electric vehicle Optimization design oflightweight gears would at most reduce 685 percentage ofthe dynamic meshing force and 667 percentage of thehousing vibration compared to solid gears

Lightweight gears by material removal have manydrawbacks such as the introduction of additional mechanicalexcitations more harmonic components greater fluctua-tions of mesh stiffness and static TE severe stress concen-tration lower load capacity and higher magnitude ofdeformation Meanwhile thinner walls and induced holessimply do not reduce the vibration transmission from thetoothed ring to other parts of the drive +erefore thedynamic responses and vibration level of these gearedsystems show no significant improvement compared to thesolid ones

Researchers develop innovative lightweight strategiesthat can maintain the load-carrying capability of the gearsand improve the dynamic performance by the exploitationof new materials Bimetallic gears and hybrid composite-metal gears are the most popular approaches With steel ringand aluminum on the hub region bimetallic gears [229] canprovide up to 40 percentage of weight reduction without a

decrease in the performance of the gears Ring thickness hasa great influence on the static performance of gears such asroot stress time-varying mesh stiffness and static TE but hasan ignorable influence on dynamic responses However inanother publication [230] a hybrid gear with a carbon fiber-reinforced polymer web is superior to that with an alumi-num alloy web Composite materials are initially used inrotorcraft transmission and then are widely used in vehiclesin recent years [231] for their outstanding damping prop-erty Incorporating composite materials with metals into thestructure of a gear has many promising benefits such asweight reduction and vibration dispassion Carbon fiber-reinforced polymer material [232] is attractive for its ca-pability of reducing vibrations and high damping capacityBoth experimental [233ndash235] and numerical [236] surveysshowed that there are significant improvements in bothweight reduction and NVH performance when adoptinghybrid composite-metal gears From the experiments con-ducted by Handschuh et al [233] the hybrid gear pairsgenerally reduce both the vibration level and the soundpressure level of the gearbox due to their higher dampingproperties From the report by Handschuh et al [234] ahybrid-hybrid gear pair exhibits the lowest vibrations andnoise level at high-speed ranges while the hybrid geardriving the steel gear pair exhibits the highest vibrationsFrom the experiments conducted by Laberge et al [235] thevibration and orbit magnitudes are comparable to steelgears Catera et al [236] compared the numerically calcu-lated static TE of the thin-rimmed steel gear pair and thecomposite-steel hybrid one +e peak-to-peak and meanvalue of the static TE of the hybrid gear pair are lower thanthe thin-rimmed steel one which indicates an enhancementin NVH performance for hybrid gears Innovative webstructures alter the vibration propagation path which startsfrom the gear mesh through the gear body and shaft to thehousing Filling powders [237] and particle damper [238]with optimum diameter to the holes of the lightweight gearscan effectively reduce the vibration since the holes are a vital

Table 5 Formulae for shuttling forces and moments

Formulae ReferenceMres 1113936

ni1 Mi

Mi qmn 1113938xi2

xi1

xn(1 + an(xb))dx[171]

Mθ 1113936mq1 Eqn exp(jnφq)nZqn cos βb [124]

Mp Ktnc + FnAp

Mg Ktnc + FnAg

Ap (c minus tep)n cosφ + bn sinφAg minus (c minus teg)n cosφ + (B minus b)n sinφ

[216 217]

sp1 FCFWa

FC

sp2

n 01 αgt α01

n 04((α minus α01) (α05 minus α01)) + 01 α01 ge αgt α05


n 09 αle α09

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

[218 219]

Mz Mbz + Maz

Mbz ([1113936iFbi lowast xi + 1113936iFbi lowast (fw minus xi)]2)

Maz ([1113936iFai lowastyi + 1113936iFai lowast (La minus yi)]2)

[220]



place of vibration transfer As described by Xiao et al [237]tungsten alloy powder is the best and the steel alloy and leadalloy are much better than that of magnesium alloy andaluminum alloy powder Ramadani et al [239] replaced thesolid structure of the gear body by a lattice structure toreduce the total weight of the gear and to reduce thepropagation of the vibration generated by the gear teeth+eaddition of the polymer matrix to the cellular latticestructure may reduce the vibration significantly Generallythe hybrid gear has three parts steel rim composite web andsteel hub respectively +ere are two composite-steel in-terfaces in a common hybrid gear which increase the op-portunity of stress concentration To minimize the stressconcentration phenomenon Gauntt et al [240] adoptedcomposite shafts and a sinusoidal interface Meanwhile theaccuracy of the FE model of the hybrid composite-metalgears strongly relies on the property estimations for inter-faces and composite materials +e complicated property ofthe composites and interfaces poses difficulty in establishingpredictive models for hybrid gears [236 241ndash243] +ere aretwo different joining technologies for metal-composite gearsnamely interface fitting and adhesive bonding respectivelyCooley and Parker [15] established FEmodels with ply-by-plyand homogenized webs for hybrid composite-steel gears inwhich the interface is an adhesive bounding one Catera et al[242] adopted an adhesive bounding interface in their mul-tiscale model of the hybrid metal-composite gears +e FEmodels [243] of the hybrid gear with an adhesive boundinginterface do not accurately predict the higher orders of themodes Catera et al [244] experimentally and numericallyinvestigated two different joining technologies for metal-composite gears namely interface fitting and adhesivebonding Both techniques are prior to the lightweight steelgears in terms of gear natural frequencies and dampingproperties A hybrid metal-composite gear with the variablethickness web [245] is proposed to explore its dynamicproperties in which a film adhesive bounding method isadopted at the metal-composite interfaces

Both material removal and composite-metal gears meetthe requirement of weight reduction However the per-formance of lightweight gears by material removal in termsof load-carrying capacity and static and dynamic responsesis not satisfactory Hybrid composite-metal gears can re-duce vibration not only by improving the gear bodydamping property but also by modifying the vibrationpropagation path However there are still several chal-lenges to be faced +e manufacturing procedure of thehybrid gears is more complicated than solid and materialremoval gears +e properties of composite materials andhybrid interfaces are key parameters in establishing pre-dictive models which need further investigation +emodelling of the composite materials is at the microscalewhile the modelling of the steel parts of the gear is at themacroscale leaving the overall model of the hybrid gear atmultiscales Further effort should be devoted to the fasterand more accurate modelling strategies of the hybrid gears+e author believes that innovative web structures in termsof modifying vibration propagation would be popular inthe coming years

6 Summary and Prospects

61 Summary +e authors reviewed more than 200 pub-lished papers related to the excitations of gear whine noise+e proportion of each excitation in the reviewed articles iscalculated as can be seen in Figure 4 A considerable numberof investigators considered TE as the primary excitation+erefore there are numerous papers focusing on predic-tion measurement and optimization of TE A few re-searchers have mentioned that frictional excitation betweenmating gears is the secondary excitation of gear whine noiseLimited studies are related to shuttling excitation

+emain content of this article is summarized as follows+ree kinds of excitations for gear whine noise are

summarized transmission error along the LOA directionfrictional excitations along the OLOA direction and shut-tling excitations along the axial direction Transmissionerror is the primary excitation of gear whine noise frictionalexcitations are the secondary excitations and shuttlingexcitation is less important than the aforementioned twokinds of excitations

Transmission error consists of two parts deflections anddeviations Gear teeth deflections and variance of meshingstiffness contribute to the deflection part of TEManufacturing errors intentional modifications and mis-alignments due to supporting structures contribute to thedeviation part of TE

+ere are four sets of harmonics for TE in the frequencydomain mesh frequency harmonics two sets of rotationalharmonics related to the shaft rotating frequency andfundamental harmonic of the meshing gear pairs Amongthe four sets mentioned above the tooth meshing harmonicsare the strongest excitation the rotational ones are thesecond strongest and the one associated with fundamentalharmonics is the weakest

+e deflection part of TE is generally predicted usinganalytical numerical or combined analytical and numericalmethods +e advantages and disadvantages of these threemethods are discussed Among these methods the analyticalmethods requiring many assumptions are efficient but notaccurate the numerical approach is precise but not efficientand the combined methods offer a quick and accurate so-lution +e analytical formulae for predicting TE are sum-marized in Table 1 From this table researchers who are newto this field can learn about many TE formulae withoutreading so many articles

2383

374

7243

Static transmission errorFrictional excitationsShuttling excitations

Figure 4 Gear whine noise excitation distribution


+e deviation part of TE is generally measured using aspecially designed test rig Transmission error measuredunder light-load and slow-speed conditions mainly consistsof the manufacturing error and intentional modificationsTransmission error measured under low-speed and varying-load conditions consists of deflections and deviations Dy-namic TE measured under varying speed and load conditionsalso consists of dynamic effects +e test rigs and instrumentsmeasuring TE under different operating conditions aresummarized Benefits and drawbacks of each measurementapproaches are discussed Methods based on encoders areaccurate while the boundary condition of the tested gears isquite different from that in a real gearbox Methods based onvibrometers offer an opportunity to measure TE in a realgearbox However there should be holes in the housing platesfor laser access which poses a challenge for gearbox lubri-cation Approaches based on accelerations require precisecalibration tight misalignment and adequate mountingspace In spite of that the output still requires a second in-tegration to obtain the TE

A reduction in the primary excitation directly leads to areduction in the level of the gear whine noise +e mostpopular way to reduce TE is through microgeometry mod-ifications Tip and root relieves are widely applied to spur-gearpairs to reduce TE For helical gears bias modificationmay bebeneficial

For frictional excitations it is clear that the friction forcesare the main excitation of the gear whine noise along theOLOA direction +e friction forces and moments may be-come dominant excitations of the gear whine noise when theTE isminimized viamicrogeometrymodificationsMeanwhilethe influences of frictional excitations on motions along LOAand axial directions are difficult to illustrate

Frictional excitations are in general predicted analyticallyNormal meshing forces sign function frictional coefficientsand moment arms are essential parameters for predictingfrictional excitations Among them the sign function is thedominant factor and the coefficient is of secondary impor-tance Analytical formulae for sign function frictional coef-ficient and moment arms are summarized in Tables 2ndash4

Improving surface roughness and providing sufficientlubricant oil can reduce frictional excitations Meanwhile anovergenerous supply of lubricant oil will increase the chanceof impacts and induce rattle noise

Shuttling is an inherent excitation for helical gears spurgears with misalignment should take shuttling into con-sideration as well Parameters influencing the strength ofshuttling excitation are summarized Helical angle andamplitude of misalignment both have a close relationshipwith shuttling excitation +e distance between gears andbearings has a significant influence on the strength of theshuttling excitation Analytical formulae for predicting theshuttling excitations are summarized in Table 5

Lightweight gears based on material removal andadoption of new materials are popular in automotivetransmission lately +e advantages and disadvantages ofthese gears are summarized +e author also points out theremaining challenges for lightweight gears in terms ofmanufacturing and modelling strategies

62 Prospects Based on the topics covered in this reviewpaper some essential existing problems in the reportedresearch work are presented and some prospects for futureresearch directions are suggested

Current TE predictionmethods model two isolated gearsunder quasistatic condition assuming rigid shaft andbearings and ignoring housing plates A system-level modelcontaining multiple pairs of gears flexible shafts bearingsand housing plates should be adopted in predicting TE andother kinds of excitations

Once manufactured and mounted any individual gearpair has a unique TE When measuring TE the tested gearpairs are mounted on two isolated shafts +e boundaryconditions of the gears are extremely different from the onesthe gear pairs are subject to in a real gearbox +us themeasured TE is quite different from the TE of the gear pair inthe gearbox

Microgeometry modification does reduce the gear whinenoise to a certain extent However the amounts and theeffects of modification both have a limitation especially forgears with a small module +erefore it is necessary topropose new strategies to reduce TE and hence gear whinenoise Using magnetic gears gears with a small module orgears with thin and tall teeth might be an effective strategy

+e effect of frictional excitations on motions along theOLOA is straightforward while its influence on motionsalong LOA and axial directions is difficult to illustrate whichrequires further study

+ere are two strategies to reduce the frictional excita-tions improving the surface roughness and providing suf-ficient lubricant oil However improving the surfaceroughness is expensive and providing sufficient lubricant oilmay induce gear rattle noise Other approaches to decreasethe frictional excitations should be proposed

A limited number of papers focus on the prediction andreduction of shuttling excitation +is does not meanshuttling excitation is not important on the contraryshuttling excitation may be dominant when TE and fric-tional excitations are minimized Strategies for minimizingshuttling excitation should be proposed as soon as possible

Reducing excitations is not the only way to minimizegear whine noise Modifying vibration transmission pathsand optimizing housing plates are also effective means toreduce radiated noise Hybrid composite-metal gears have agreat potential on vibration and noise reduction of thegearbox

Conflicts of Interest

+e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

Acknowledgments

+is research was supported by the National Key Researchand Development Program-Research on Application ofVibration Noise and Post-Processing of Medium PowerAgricultural Diesel Engine (Grant no 2016YFD0700704B)


Innovative Research Team Development Program of theMinistry of Education of China (IRT_17R83) and 111Project (B17034)

References

[1] S +eodossiades M De la Cruz and H Rahnejat ldquoPre-diction of airborne radiated noise from lightly loaded lu-bricated meshing gear teethrdquo Applied Acoustics vol 100pp 79ndash86 2015

[2] R Brancati E Rocca D Siano and M Viscardi ldquoExperi-mental vibro-acoustic analysis of the gear rattle induced bymulti-harmonic excitationrdquo Proceedings of the Institution ofMechanical Engineers Part D Journal of Automobile Engi-neering vol 232 no 6 pp 785ndash796 2017

[3] R Brancati E Rocca and D Lauria ldquoFeasibility study of theHilbert transform in detecting the gear rattle phenomenon ofautomotive transmissionsrdquo Journal of Vibration and Con-trol vol 24 no 12 pp 2631ndash2641 2017

[4] A Fernandez-Del-Rincon A Diez-Ibarbia andS +eodossiades ldquoGear transmission rattle assessment ofmeshing forces under hydrodynamic lubricationrdquo AppliedAcoustics vol 144 pp 85ndash95 2019

[5] R Brancati E Rocca and R Russo ldquoAn analysis of theautomotive driveline dynamic behaviour focusing on theinfluence of the oil squeeze effect on the idle rattle phe-nomenonrdquo Journal of Sound and Vibration vol 303 no 3ndash5pp 858ndash872 2007

[6] C Gill-Jeong ldquoAnalysis of the nonlinear behavior of gearpairs considering hydrodynamic lubrication and slidingfrictionrdquo Journal of Mechanical Science and Technologyvol 23 no 8 pp 2125ndash2137 2009

[7] R Russo R Brancati and E Rocca ldquoExperimental inves-tigations about the influence of oil lubricant between teethon the gear rattle phenomenonrdquo Journal of Sound and Vi-bration vol 321 no 3ndash5 pp 647ndash661 2009

[8] R Singh H Xie and R J Comparin ldquoAnalysis of auto-motive neutral grear rattlerdquo Journal of Sound and Vibrationvol 131 no 2 pp 177ndash196 1989

[9] J Zhang D Liu and H Yu ldquoExperimental and numericalanalysis for the transmission gear rattle in a power-splithybrid electric vehiclerdquo International Journal of VehicleDesign vol 74 no 1 pp 1ndash18 2017

[10] S L Harris ldquoDynamic loads on the teeth of spur gearsrdquoProceedings of the Institution of Mechanical Engineersvol 172 no 1 pp 87ndash112 1958

[11] H N Ozguven and D R Houser ldquoMathematical modelsused in gear dynamicsmdasha reviewrdquo Journal of Sound andVibration vol 121 no 3 pp 383ndash411 1988

[12] J Wang R Li and X Peng ldquoSurvey of nonlinear vibration ofgear transmission systemsrdquo Applied Mechanics Reviewsvol 56 no 3 pp 309ndash329 2003

[13] Y Lei J Lin M J Zuo and Z He ldquoCondition monitoringand fault diagnosis of planetary gearboxes a reviewrdquoMeasurement vol 48 pp 292ndash305 2014

[14] X Liang M J Zuo and Z Feng ldquoDynamic modeling ofgearbox faults a reviewrdquo Mechanical Systems and SignalProcessing vol 98 pp 852ndash876 2018

[15] C G Cooley and R G Parker ldquoA review of planetary andepicyclic gear dynamics and vibrations researchrdquo AppliedMechanics Reviews vol 66 no 4 p 40804 2014

[16] R G Munro ldquoPaper 10 effect of geometrical errors on thetransmission of motion between gearsrdquo Proceedings of the

Institution of Mechanical Engineers Conference Proceedingsvol 184 no 15 pp 79ndash84 1969

[17] H K Kohler A Pratt and A M +ompson ldquoPaper 14dynamics and noise of parallel-axis gearingrdquo Proceedings ofthe Institution of Mechanical Engineers Conference Pro-ceedings vol 184 no 15 pp 111ndash121 1969

[18] D R Houser F B Oswald M J Valco R J Drago andJ W Lenski ldquoComparison of transmission error predictionswith noise measurements for several spur and helical gearsrdquoin Proceedings of the 30th Joint Propulsion ConferenceIndianapolis IN USA June 1994

[19] D Palmer and R G Munro ldquoMeasurements of transmissionerror vibration and noise in spur gearsrdquo in Proceedings of the1995 British Gear Association Technical Congress BirminghamUK November 1995

[20] M Akerblom Gearbox Noise Correlation with TransmissionError and Influence of Bearing Preload Royal Institute ofTechnology Stockholm Sweden 2008

[21] R W Gregory S L Harris and R G Munro ldquoDynamicbehaviour of spur gearsrdquo Proceedings of the Institution ofMechanical Engineers vol 178 no 1 pp 207ndash218 1963

[22] D B Welbourn ldquoFundamental knowledge of gear noise asurveyrdquo in Noise and Vibrations of Engines and Transmis-sions pp 9ndash14 Cranfield Institute of Technology CranfieldUK 1979

[23] H N Ozguven and D R Houser ldquoDynamic analysis of highspeed gears by using loaded static transmission errorrdquoJournal of Sound and Vibration vol 125 no 1 pp 71ndash831988

[24] R G Munro ldquoA review of the theory and measurement ofgear transmission errorrdquo Gearbox Noise and Vibrationpp 3ndash10 Marcel Dekker New York NY USA 1990

[25] J D Smith Gear Noise and Vibration Marcel Dekker NewYork NY USA 2nd edition 2003

[26] P Velex and M Ajmi ldquoOn the modelling of excitations ingeared systems by transmission errorsrdquo Journal of Sound andVibration vol 290 no 3ndash5 pp 882ndash909 2006

[27] P Davoli C Gorla F Rosa F Rossi and G BonildquoTransmission error and noise emission of spur gears atheoretical and experimental approachrdquo in Proceedings of the2007 International Design Engineering Technical Conferencesand Computers and Information in Engineering ConferenceVolume 7 10th International Power Transmission andGearing Conference pp 443ndash449 Las Vegas NV USA 2007

[28] A Palermo D Mundo A S Lentini R Hadjit P Mas andW Desmet ldquoGear noise evaluation through multibody TE-based simulationsrdquo in Proceedings of ISMA-InternationalConference on Noise and Vibration Engineering pp 3033ndash3046 Leuven Belgium 2010

[29] W D Mark ldquoAnalysis of the vibratory excitation of gearsystems basic theoryrdquoe Journal of the Acoustical Society ofAmerica vol 63 no 5 pp 1409ndash1430 1978

[30] W D Mark ldquoAnalysis of the vibratory excitation of gearsystems II Tooth error representations approximationsand applicationrdquo e Journal of the Acoustical Society ofAmerica vol 66 no 6 pp 1758ndash1787 1979

[31] W D Mark Performance-Based Gear Metrology Kinematic-Transmission-Error Computation and Diagnosis John Wileyamp Sons Chichester UK 2012

[32] W DMark ldquoTooth-meshing-harmonic static-transmission-error amplitudes of helical gearsrdquo Mechanical Systems andSignal Processing vol 98 pp 506ndash533 2018

[33] H Walker ldquoGear tooth deflection and profile modificationrdquoEngineer vol 166 pp 409ndash412 1938


[34] CWebereDeformation of Loaded Gears and the Effect oneir Load Carrying Capacity British Dept of Scientific andIndustrial Research London UK 1949

[35] K Umezawa T Suzuki and T Sato ldquoVibration of powertransmission helical gears Approximate equation of stiff-nessrdquo Transactions of the Japan Society of Mechanical En-gineers Series C vol 51 no 469 pp 2316ndash2323 1985

[36] Y Cai ldquoSimulation on the rotational vibration of helicalgears in consideration of the tooth separation phenomenon(a new stiffness function of helical involute tooth pair)rdquoJournal of Mechanical Design vol 117 no 3 pp 460ndash4691995

[37] T Kiekbusch and I Howard ldquoA common formula for thecombined torsional mesh stiffness of spur gearsrdquo in Pro-ceedings of the 5th Australasian Congress on Applied Me-chanics pp 710ndash716 Brisbane Australia 2007

[38] V Skrickij and M Bogdevicius ldquoVehicle gearbox dynamicscentre distance influence on mesh stiffness and spur geardynamicsrdquo Transport vol 25 no 3 pp 278ndash286 2010

[39] Z Chen and Y Shao ldquoMesh stiffness calculation of a spurgear pair with tooth profile modification and tooth rootcrackrdquo Mechanism and Machine eory vol 62 pp 63ndash742013

[40] A F D Rincon F Viadero M Iglesias P Garcıa A De-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and Ma-chine eory vol 61 pp 30ndash58 2013

[41] N L Pedersen and M F Joslashrgensen ldquoOn gear tooth stiffnessevaluationrdquo Computers amp Structures vol 135 no 3pp 109ndash117 2014

[42] Z Wan H Cao Y Zi W He and Z He ldquoAn improvedtime-varying mesh stiffness algorithm and dynamic mod-eling of gear-rotor system with tooth root crackrdquo Engi-neering Failure Analysis vol 42 pp 157ndash177 2014

[43] X Gu P Velex P Sainsot and J Bruyere ldquoAnalytical in-vestigations on the mesh stiffness function of solid spur andhelical gearsrdquo Journal of Mechanical Design vol 137 no 6p 63301 2015

[44] L Chang G Liu and L Wu ldquoA robust model for deter-mining the mesh stiffness of cylindrical gearsrdquo Mechanismand Machine eory vol 87 pp 93ndash114 2015

[45] Z Wan H Cao Y Zi W He and Y Chen ldquoMesh stiffnesscalculation using an accumulated integral potential energymethod and dynamic analysis of helical gearsrdquo Mechanismand Machine eory vol 92 pp 447ndash463 2015

[46] C G Cooley C Liu X Dai and R G Parker ldquoGear toothmesh stiffness a comparison of calculation approachesrdquoMechanism andMachineeory vol 105 pp 540ndash553 2016

[47] H Ma M Feng Z Li R Feng and B Wen ldquoTime-varyingmesh characteristics of a spur gear pair considering the tip-fillet and frictionrdquo Meccanica vol 52 no 7 pp 1695ndash17092017

[48] C I Park and J M Lee ldquoExperimental investigation of theeffect of lead errors on helical gear and bearing vibrationtransmission characteristicsrdquo KSME International Journalvol 16 no 11 pp 1395ndash1403 2002

[49] A Fernandez-del-Rincon M Iglesias A De-Juan A Diez-Ibarbia P Garcıa and F Viadero ldquoGear transmission dy-namics effects of index and run out errorsrdquo AppliedAcoustics vol 108 no 1-2 pp 63ndash83 2016

[50] D Talbot A Sun and A Kahraman ldquoImpact of toothindexing errors on dynamic factors of spur gears experi-ments and model simulationsrdquo Journal of Mechanical De-sign vol 138 no 9 p 93302 2016

[51] E Rigaud and D Barday ldquoModelling and analysis of statictransmission error Effect of wheel body deformation andinteractions between adjacent loaded teethrdquo in Proceedingsof the 4th World Congress on Gearing and Power Trans-mission pp 1961ndash1972 Paris France 1999

[52] S Li ldquoEffects of machining errors assembly errors and toothmodifications on loading capacity load-sharing ratio andtransmission error of a pair of spur gearsrdquo Mechanism andMachine eory vol 42 no 6 pp 698ndash726 2007

[53] S Li ldquoFinite element analyses for contact strength andbending strength of a pair of spur gears with machiningerrors assembly errors and tooth modificationsrdquo Mecha-nism and Machine eory vol 42 no 1 pp 88ndash114 2007

[54] Y Guo T Eritenel T M Ericson and R G Parker ldquoVibro-acoustic propagation of gear dynamics in a gear-bearing-housing systemrdquo Journal of Sound and Vibration vol 333no 22 pp 5762ndash5785 2014

[55] H Jiang Y Shao C K Mechefske and X Chen ldquo+e in-fluence of meshmisalignment on the dynamic characteristicsof helical gears including sliding frictionrdquo Journal of Me-chanical Science and Technology vol 29 no 11 pp 4563ndash4573 2015

[56] S Li ldquoEffects of misalignment error tooth modifications andtransmitted torque on tooth engagements of a pair of spurgearsrdquoMechanism andMachineeory vol 83 pp 125ndash1362015

[57] A Saxena A Parey and M Chouksey ldquoEffect of shaftmisalignment and friction force on time varying meshstiffness of spur gear pairrdquo Engineering Failure Analysisvol 49 pp 79ndash91 2015

[58] Y Guo S Lambert R Wallen R Errichello and J Kellerldquo+eoretical and experimental study on gear-couplingcontact and loads considering misalignment torque andfriction influencesrdquoMechanism andMachineeory vol 98pp 242ndash262 2016

[59] M S Tavakoli and D R Houser ldquoOptimum profile mod-ifications for the minimization of static transmission errorsof spur gearsrdquo Journal of Mechanisms Transmissions andAutomation in Design vol 108 no 1 pp 86ndash94 1986

[60] F L Litvin A Fuentes I Gonzalez-Perez L CarvenaliK Kawasaki and R F Handschuh ldquoModified involutehelical gears computerized design simulation of meshingand stress analysisrdquo Computer Methods in Applied Me-chanics and Engineering vol 192 no 33-34 pp 3619ndash36552003

[61] D R Houser and J Harianto ldquo+e effect of micro-geometryand load on helical gear noise excitationsrdquo in Proceedings ofthe 2005 Noise and Vibration Conference and ExhibitionTraverse City MI USA 2005

[62] E N Mohamad M Komori H Murakami A Kubo andS Fang ldquoEffect of convex tooth flank form deviation on thecharacteristics of transmission error of gears consideringelastic deformationrdquo Journal of Mechanical Design vol 132no 10 Article ID 101005 2010

[63] A Artoni M Guiggiani A Kahraman and J HariantoldquoRobust optimization of cylindrical gear tooth surfacemodifications within ranges of torque and misalignmentsrdquoJournal of Mechanical Design vol 135 no 12 Article ID121005 2013

[64] J Jiang and Z Fang ldquoDesign and analysis of modified cy-lindrical gears with a higher-order transmission errorrdquoMechanism and Machine eory vol 88 pp 141ndash152 2015


[65] W Yu and C K Mechefske ldquoAnalytical modeling of spurgear corner contact effectsrdquo Mechanism and Machine e-ory vol 96 pp 146ndash164 2016

[66] B Yu and K-l Ting ldquoCompensated conjugation and geartooth modification designrdquo Journal of Mechanical Designvol 138 no 7 p 73301 2016

[67] W D Mark ldquoContributions to the vibratory excitation ofgear systems from periodic undulations on tooth runningsurfacesrdquo e Journal of the Acoustical Society of Americavol 91 no 1 pp 166ndash186 1992

[68] J Yoo K Kang J Huh and C Choi ldquo+e mechanism andsolution of harmonic gear whine noise in automotivetransmission systemsrdquo in Proceedings of the ASME 2007International Design Engineering Technical Conferences andComputers and Information in Engineering ConferenceVolume 7 10th International Power Transmission andGearing Conference pp 569ndash575 Las Vegas NV USA 2007

[69] S Sundar R Singh K Jayasankaran and S Kim ldquoEffect ofthe tooth surface waviness on the dynamics and structure-borne noise of a spur gear pairrdquo SAE International Journal ofPassenger CarsmdashMechanical Systems vol 6 no 2pp 1087ndash1093 2013

[70] S Jolivet S Mezghani J Isselin and M El Mansori ldquoEx-perimental and numerical study of tooth finishing processescontribution to gear noiserdquo Tribology International vol 102pp 436ndash443 2016

[71] G Bonori and F Pellicano ldquoNon-smooth dynamics of spurgears with manufacturing errorsrdquo Journal of Sound andVibration vol 306 no 1-2 pp 271ndash283 2007

[72] M Inalpolat M Handschuh and A Kahraman ldquoInfluenceof indexing errors on dynamic response of spur gear pairsrdquoMechanical Systems and Signal Processing vol 60-61pp 391ndash405 2015

[73] S A Hambric A D Hanford M R ShepherdR L Campbell and E C Smith Rotorcraft TransmissionNoise Path Model Including Distributed Fluid Film BearingImpedance Modeling NASA Washington DC USA 2010

[74] W D Mark C P Reagor and D R McPherson ldquoAssessingthe role of plastic deformation in gear-health monitoring byprecision measurement of failed gearsrdquo Mechanical Systemsand Signal Processing vol 21 no 1 pp 177ndash192 2007

[75] W D Mark and C P Reagor ldquoStatic-transmission-errorvibratory-excitation contributions from plastically deformedgear teeth caused by tooth bending-fatigue damagerdquo Me-chanical Systems and Signal Processing vol 21 no 2pp 885ndash905 2007

[76] W D Mark H Lee R Patrick and J D Coker ldquoA simplefrequency-domain algorithm for early detection of damagedgear teethrdquo Mechanical Systems and Signal Processingvol 24 no 8 pp 2807ndash2823 2010

[77] W D Mark ldquoTime-synchronous-averaging of gear-mesh-ing-vibration transducer responses for elimination of har-monic contributions from the mating gear and the gearpairrdquo Mechanical Systems and Signal Processing vol 62-63pp 21ndash29 2015

[78] W D Mark ldquoAnalytical approximations to damaged geartooth transmission-error contributions for gear-healthmonitoringrdquo Proceedings of the Institution of MechanicalEngineers Part C Journal of Mechanical Engineering Sciencevol 230 no 7-8 pp 1157ndash1182 2016

[79] W D Mark A C Isaacson and M E Wagner ldquoTrans-mission-error frequency-domain-behavior of failing gearsrdquoMechanical Systems and Signal Processing vol 115pp 102ndash119 2019

[80] A Kahraman and G W Blankenship ldquoEffect of involute tiprelief on dynamic response of spur gear pairsrdquo Journal ofMechanical Design vol 121 no 2 pp 313ndash315 1999

[81] M Kubur A Kahraman D M Zini and K KienzleldquoDynamic analysis of a multi-shaft helical gear transmissionby finite elements model and experimentrdquo Journal of Vi-bration and Acoustics vol 126 no 3 pp 398ndash406 2004

[82] C I Park ldquoMulti-objective optimization of the tooth surfacein helical gears using design of experiment and the responsesurface methodrdquo Journal of Mechanical Science and Tech-nology vol 24 no 3 pp 823ndash829 2010

[83] C Zhou Z Wang and S Chen ldquoCoupling of the 2Dmicrotopography of tooth surface and transmission errorrdquoJournal of Mechanical Science and Technology vol 32 no 2pp 723ndash730 2018

[84] R W Cornell ldquoCompliance and stress sensitivity of spurgear teethrdquo Journal of Mechanical Design vol 103 no 2pp 447ndash459 1981

[85] H Lin and R L Huston Dynamic Loading on Parallel ShaftGears NASA Washington DC USA 1986

[86] H-H Lin R L Huston and J J Coy ldquoOn dynamic loads inparallel shaft transmissions part I-modelling and analysisrdquoJournal of Mechanisms Transmissions and Automation inDesign vol 110 no 2 pp 221ndash225 1988

[87] C-H Liou H H Lin F B Oswald and D P TownsendldquoEffect of contact ratio on spur gear dynamic load with notooth profile modificationsrdquo Journal of Mechanical Designvol 118 no 3 pp 439ndash443 1996

[88] K Y Yoon and S S Rao ldquoDynamic load analysis of spurgears using a new tooth profilerdquo Journal of MechanicalDesign vol 118 no 1 pp 1ndash6 1996

[89] R G Munro L Morrish and D Palmer ldquoGear transmissionerror outside the normal path of contact due to corner andtop contactrdquo Proceedings of the Institution of MechanicalEngineers Part C Journal of Mechanical Engineering Sciencevol 213 no 4 pp 389ndash400 1999

[90] H H Lin J Wang F B Oswald and J J Coy ldquoEffect ofextended tooth contact on the modeling of spur geartransmissionsrdquo in Proceedings of the 29th Joint PropulsionConference and Exhibit Joint Propulsion ConferencesMonterey CA USA 1993

[91] Y Terauchi and K Nagamura ldquoStudy on deflection of spurgear teeth 1st report calculation of tooth deflection by two-dimensional elastic theoryrdquo Bulletin of JSME vol 23 no 184pp 1682ndash1688 1980

[92] Y Terauchi and K Nagamura ldquoStudy on deflection of spurgear teeth 2nd report calculation of tooth deflection for spurgears with various tooth profilesrdquo Bulletin of JSME vol 24no 188 pp 447ndash452 1981

[93] C W MacGregor ldquoDeflection of a long helical gear toothrdquoMechanical Engineering vol 57 pp 225ndash227 1935

[94] E J Wellauer and A Seireg ldquoBending strength of gear teethby cantilever-plate theoryrdquo Journal of Engineering for In-dustry vol 82 no 3 pp 213ndash220 1960

[95] T F Conry and A Seireg ldquoA mathematical programmingmethod for design of elastic bodies in contactrdquo Journal ofApplied Mechanics vol 38 no 2 pp 387ndash392 1971

[96] T F Conry and A Seireg ldquoA mathematical programmingtechnique for the evaluation of load distribution and optimalmodifications for gear systemsrdquo Journal of Engineering forIndustry vol 95 no 4 pp 1115ndash1122 1973

[97] J Harianto and D R Houser ldquoA methodology for obtainingoptimum gear tooth micro-topographies for noise and stressminimization over a broad operating torque rangerdquo in


Proceedings of the 2007 International Design EngineeringTechnical Conferences and Computers and Information inEngineering Conference Volume 7 10th International PowerTransmission and Gearing Conference pp 289ndash303 LasVegas NV USA 2007

[98] Q Zhang J Kang W Dong and S Lyu ldquoA study on toothmodification and radiation noise of a manual transaxlerdquoInternational Journal of Precision Engineering andManufacturing vol 13 no 6 pp 1013ndash1020 2012

[99] H S Kwon A Kahraman H K Lee and H S Suh ldquoAnautomated design search for single and double-planetplanetary gear setsrdquo Journal of Mechanical Design vol 136no 6 p 61004 2014

[100] B Yuan S Chang G Liu L Chang and L Liu ldquoOptimi-zation of bias modification and dynamic behavior analysis ofhelical gear systemrdquo Advances in Mechanical Engineeringvol 9 no 11 Article ID 168781401773325 2017

[101] L Vedmar and B Henriksson ldquoA general approach fordetermining dynamic forces in spur gearsrdquo Journal of Me-chanical Design vol 120 no 4 pp 593ndash598 1998

[102] J Wang and I Howard ldquoFinite element analysis of highcontact ratio spur gears in meshrdquo Journal of Tribologyvol 127 no 3 pp 469ndash483 2005

[103] A Andersson and L Vedmar ldquoA dynamic model to de-termine vibrations in involute helical gearsrdquo Journal ofSound and Vibration vol 260 no 2 pp 195ndash212 2003

[104] T Lin H Ou and R Li ldquoA finite element method for 3Dstatic and dynamic contactimpact analysis of gear drivesrdquoComputer Methods in Applied Mechanics and Engineeringvol 196 no 9ndash12 pp 1716ndash1728 2007

[105] K Mao ldquoAn approach for powertrain gear transmissionerror prediction using the non-linear finite elementmethodrdquo Proceedings of the Institution of Mechanical En-gineers Part D Journal of Automobile Engineering vol 220no 10 pp 1455ndash1463 2006

[106] Y A Tesfahunegn F Rosa and C Gorla ldquo+e effects of theshape of tooth profile modifications on the transmissionerror bending and contact stress of spur gearsrdquo Proceedingsof the Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 224 no 8 pp 1749ndash1758 2010

[107] Y-j Wu J-j Wang and Q-k Han ldquoContact finite elementmethod for dynamic meshing characteristics analysis ofcontinuous engaged gear drivesrdquo Journal of MechanicalScience and Technology vol 26 no 6 pp 1671ndash1685 2012

[108] J S-S Wu S-L Xu Y-T Lin W-H Chen and Y-L Laildquo3D contact analysis of conjugate spur gears by a completemating processrdquo Journal of Mechanical Science and Tech-nology vol 27 no 12 pp 3787ndash3795 2013

[109] X Deng L Hua and X Han ldquoResearch on the design andmodification of asymmetric spur gearrdquo MathematicalProblems in Engineering vol 2015 Article ID 89725713 pages 2015

[110] N K Raghuwanshi and A Parey ldquoEffect of back-side contactonmesh stiffness of spur gear pair by finite element methodrdquoProcedia Engineering vol 173 pp 1538ndash1543 2017

[111] N Wang X Li K Wang Q Zeng and X Shen ldquoA novelaxial modification and simulation analysis of involute spurgearrdquo Strojniski VestnikmdashJournal of Mechanical Engineeringvol 63 no 12 2017

[112] M Barbieri A Zippo and F Pellicano ldquoAdaptive grid-sizefinite element modeling of helical gear pairsrdquo Mechanismand Machine eory vol 82 pp 17ndash32 2014

[113] Y Wang Y Liu W Tang and P Liu ldquoParametric finiteelement modeling and tooth contact analysis of spur andhelical gears including profile and lead modificationsrdquo En-gineering Computations vol 34 no 8 pp 2877ndash2898 2017

[114] J Brauer ldquoA general finite element model of involute gearsrdquoFinite Elements in Analysis and Design vol 40 no 13-14pp 1857ndash1872 2004

[115] K Jao Huang and H Wei Su ldquoApproaches to parametricelement constructions and dynamic analyses of spurhelicalgears including modifications and undercuttingrdquo FiniteElements in Analysis and Design vol 46 no 12 pp 1106ndash1113 2010

[116] J Wei W Sun and L Wang ldquoEffects of flank deviation onload distributions for helical gearrdquo Journal of MechanicalScience and Technology vol 25 no 7 pp 1781ndash1789 2011

[117] Z He T Lin T Luo T Deng and Q Hu ldquoParametricmodeling and contact analysis of helical gears with modi-ficationsrdquo Journal of Mechanical Science and Technologyvol 30 no 11 pp 4859ndash4867 2016

[118] X Li C Li B Chen and D Liang ldquoDesign and investigationof a cycloid helical gear driverdquo Journal of Mechanical Scienceand Technology vol 31 no 9 pp 4329ndash4336 2017

[119] T Lin and Z He ldquoAnalytical method for coupled trans-mission error of helical gear system with machining errorsassembly errors and tooth modificationsrdquo Mechanical Sys-tems and Signal Processing vol 91 pp 167ndash182 2017

[120] I Gonzalez-Perez and A Fuentes-Aznar ldquoImplementationof a finite element model for stress analysis of gear drivesbased on multi-point constraintsrdquo Mechanism and Machineeory vol 117 pp 35ndash47 2017

[121] S Li ldquoGear contact model and loaded tooth contact analysisof a three-dimensional thin-rimmed gearrdquo Journal of Me-chanical Design vol 124 no 3 pp 511ndash517 2002

[122] V Simon ldquoLoad and stress distributions in spur and helicalgearsrdquo Journal of Mechanisms Transmissions and Auto-mation in Design vol 110 no 2 pp 197ndash202 1988

[123] V Simon ldquoOptimal tooth modifications for spur and helicalgearsrdquo Journal of Mechanisms Transmissions and Auto-mation in Design vol 111 no 4 pp 611ndash615 1989

[124] T Nishino ldquoVibration analysis of the helical gear systemusing the integrated excitation modelrdquo Journal of AdvancedMechanical Design Systems andManufacturing vol 1 no 4pp 541ndash552 2007

[125] J S Kang and Y-S Choi ldquoOptimization of helix angle forhelical gear systemrdquo Journal of Mechanical Science andTechnology vol 22 no 12 pp 2393ndash2402 2008

[126] E N Mohamad M Komori H Murakami A Kubo andS Fang ldquoAnalysis of general characteristics of transmissionerror of gears with convex modification of tooth flank formconsidering elastic deformation under loadrdquo Journal ofMechanical Design vol 131 no 6 p 61015 2009

[127] R Guilbault C Gosselin and L Cloutier ldquoExpress model forload sharing and stress analysis in helical gearsrdquo Journal ofMechanical Design vol 127 no 6 pp 1161ndash1172 2005

[128] Y Miyoshi K Tobisawa and K Saiki ldquoComposite analysismethod of tooth contact load distribution of helical gearrdquo inProceedings of the International Design Engineering TechnicalConferences and Computers and Information in EngineeringConference Volume 7 10th International Power Transmis-sion and Gearing Conference pp 173ndash180 Las Vegas NVUSA 2007

[129] P Velex J Bruyere and D R Houser ldquoSome analyticalresults on transmission errors in narrow-faced spur and


helical gears influence of profile modificationsrdquo Journal ofMechanical Design vol 133 no 3 p 31010 2011

[130] M Umeyama M Kato and K Inoue ldquoEffects of gear di-mensions and tooth surface modifications on the loadedtransmission error of a helical gear pairrdquo Journal of Me-chanical Design vol 120 no 1 pp 119ndash125 1998

[131] S S Rao and K Y Yoon ldquoMinimization of transmissionerror in helical gearsrdquo Proceedings of the Institution ofMechanical Engineers Part C Journal of Mechanical Engi-neering Science vol 215 no 4 pp 447ndash459 2001

[132] J A Korta and D Mundo ldquoMulti-objective micro-geometryoptimization of gear tooth supported by response surfacemethodologyrdquo Mechanism and Machine eory vol 109pp 278ndash295 2017

[133] M Akerblom Gear Noise and VibrationmdashA LiteratureSurvey KTH Royal Institute of Technology StockholmSweden 2001

[134] R W Gregory S L Harris and R G Munro ldquoA method ofmeasuring transmission error in spur gears of 1 1 ratiordquoJournal of Scientific Instruments vol 40 no 1 pp 5ndash9 1963

[135] D R Houser and G W Blankenship ldquoMethods for mea-suring gear transmission error under load and at operatingspeedsrdquo in Proceedings of the International Off-Highway andPowerplant Congress and Exposition Warrendale PA USA1989

[136] S Sasaoka ldquoMeasurement technique for loaded geartransmission errorrdquo in Proceedings of the 1997 InternationalCongress and Exposition Detroit MI USA 1997

[137] C Gosselin T Guertin D Remond and Y Jean ldquoSimu-lation and experimental measurement of the transmissionerror of real hypoid gears under loadrdquo Journal of MechanicalDesign vol 122 no 1 p 109 2000

[138] R G Munro D Palmer and L Morrish ldquoAn experimentalmethod to measure gear tooth stiffness throughout andbeyond the path of contactrdquo Proceedings of the Institution ofMechanical Engineers Part C Journal of Mechanical Engi-neering Science vol 215 no 7 pp 793ndash803 2001

[139] S Kurokawa Y Ariura Y Matsukawa and T Doi ldquoEval-uation of gear engagement accuracy by transmission errorwith sub-microradian resolutionrdquo International Journal ofSurface Science and Engineering vol 3 no 3 pp 160ndash1772009

[140] J H Yoon B J Choi I H Yang and J E Oh ldquoDeflectiontest and transmission error measurement to identify hypoidgear whine noiserdquo International Journal of AutomotiveTechnology vol 12 no 1 pp 59ndash66 2011

[141] S Park S Kim and J-H Choi ldquoGear fault diagnosis usingtransmission error and ensemble empirical mode decom-positionrdquo Mechanical Systems and Signal Processingvol 108 pp 262ndash275 2018

[142] M R Kang and A Kahraman ldquoMeasurement of vibratorymotions of gears supported by compliant shaftsrdquoMechanicalSystems and Signal Processing vol 29 pp 391ndash403 2012

[143] D Remond and J Mahfoudh ldquoFrom transmission errormeasurements to angular sampling in rotating machineswith discrete geometryrdquo Shock and Vibration vol 12 no 2pp 149ndash161 2005

[144] A Palermo L Britte K Janssens D Mundo andW Desmet ldquo+e measurement of gear transmission error asan NVH indicator theoretical discussion and industrialapplication via low-cost digital encoders to an all-electricvehicle gearboxrdquo Mechanical Systems and Signal Processingvol 110 pp 368ndash389 2018

[145] R White and V Palan ldquoMeasurement of transmission errorusing rotational laser vibrometersrdquo in Proceedings of theInternational Design Engineering Technical Conferences andComputers and Information in Engineering ConferenceVolume 7 10th International Power Transmission andGearing Conference pp 527ndash545 Las Vegas NV USA 2007

[146] V K Tamminana A Kahraman and S Vijayakar ldquoA studyof the relationship between the dynamic factors and thedynamic transmission error of spur gear pairsrdquo Journal ofMechanical Design vol 129 no 1 pp 75ndash84 2007

[147] C Brecher C Gorgels J Hesse and M Hellmann ldquoDy-namic transmission error measurements of a drive trainrdquoProduction Engineering vol 5 no 3 pp 321ndash327 2011

[148] B Anichowski A Kahraman and D Talbot ldquoDynamictransmission error measurements from spur gear pairshaving tooth indexing errorsrdquo in Proceedings of the Inter-national Design Engineering Technical Conferences andComputers and Information in Engineering ConferenceVolume 10 2017 ASME International Power Transmissionand Gearing Conference Cleveland OH USA 2017

[149] H Guo J Zhang and H Yu ldquoDynamic modelling andparametric optimization of a full hybrid transmissionrdquoProceedings of the Institution of Mechanical Engineers PartK Journal of Multi-Body Dynamics vol 233 no 1 pp 17ndash292019

[150] Y Teranchi H Nanano andM Nohara ldquoOn the effect of thetooth profile modification on the dynamic load and thesound level of the spur gearrdquo Bulletin of JSME vol 25no 207 pp 1474ndash1481 1982

[151] H Ma J Yang R Song S Zhang and B Wen ldquoEffects of tiprelief on vibration responses of a geared rotor systemrdquoProceedings of the Institution of Mechanical Engineers PartC Journal of Mechanical Engineering Science vol 228 no 7pp 1132ndash1154 2014

[152] H H Lin D P Townsend and F B Oswald ldquoProfilemodification to minimize spur gear dynamic loadingrdquo inProceedings of the Design Engineering Technical ConferenceOrlando FL USA 1987

[153] H H Lin F B Oswald and D P Townsend ldquoDynamicloading of spur gears with linear or parabolic tooth profilemodificationsrdquo Mechanism and Machine eory vol 29no 8 pp 1115ndash1129 1994

[154] G Bonori M Barbieri and F Pellicano ldquoOptimum profilemodifications of spur gears by means of genetic algorithmsrdquoJournal of Sound and Vibration vol 313 no 3ndash5 pp 603ndash616 2008

[155] M Barbieri G Bonori and F Pellicano ldquoCorrigendum tooptimum profile modifications of spur gears by means ofgenetic algorithmsrdquo Journal of Sound and Vibrationvol 331 no 21 pp 4825ndash4829 2012

[156] Y-j Wu J-j Wang and Q-k Han ldquoStaticdynamic contactFEA and experimental study for tooth profile modification ofhelical gearsrdquo Journal of Mechanical Science and Technologyvol 26 no 5 pp 1409ndash1417 2012

[157] N Yaldirim and R G Munro ldquoA systematic approach toprofile relief design of low and high contact ratio spur gearsrdquoProceedings of the Institution of Mechanical Engineers PartC Journal of Mechanical Engineering Science vol 213 no 6pp 551ndash562 1999

[158] N Yaldirim and R G Munro ldquoA new type of profile relieffor high contact ratio spur gearsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 213 no 6 pp 563ndash568 1999


[159] P Wagaj and A Kahraman ldquoImpact of tooth profilemodifications on the transmission error excitation of helicalgear pairsrdquo in Proceedings of ESDA2002 6th BiennialConference on Engineering Systems Design and AnalysisIstanbul Turkey 2002

[160] S Oh S Oh J Kang I Lee and S Lyu ldquoA study onmodeling and optimization of tooth microgeometry for ahelical gear pairrdquo International Journal of Precision Engi-neering and Manufacturing vol 14 no 3 pp 423ndash427 2013

[161] C Jia Z Fang and Y Zhang ldquoTopography of modifiedsurfaces based on compensated conjugation for the mini-mization of transmission errors of cylindrical gearsrdquoMechanism and Machineeory vol 116 pp 145ndash161 2017

[162] D Ghribi J Bruyere P Velex M Octrue and M HaddarldquoA contribution to the design of robust profile modificationsin spur and helical gears by combining analytical results andnumerical simulationsrdquo Journal of Mechanical Designvol 134 no 6 p 61011 2012

[163] J Bruyere and P Velex ldquoDerivation of optimum profilemodifications in narrow-faced spur and helical gears using aperturbation methodrdquo Journal of Mechanical Designvol 135 no 7 p 71009 2013

[164] J Bruyere and P Velex ldquoA simplified multi-objectiveanalysis of optimum profile modifications in spur and helicalgearsrdquo Mechanism and Machine eory vol 80 pp 70ndash832014

[165] J Bruyere X Gu and P Velex ldquoOn the analytical definitionof profile modifications minimising transmission errorvariations in narrow-faced spur helical gearsrdquo Mechanismand Machine eory vol 92 pp 257ndash272 2015

[166] A Fernandez M Iglesias A De-Juan P GarcıaR Sancibrian and F Viadero ldquoGear transmission dynamiceffects of tooth profile deviations and support flexibilityrdquoApplied Acoustics vol 77 no 3 pp 138ndash149 2014

[167] H Liu C Zhang C L Xiang and C Wang ldquoTooth profilemodification based on lateral- torsional-rocking couplednonlinear dynamic model of gear systemrdquo Mechanism andMachine eory vol 105 pp 606ndash619 2016

[168] S S Ghosh and G Chakraborty ldquoOn optimal tooth profilemodification for reduction of vibration and noise in spurgear pairsrdquo Mechanism and Machine eory vol 105pp 145ndash163 2016

[169] H Ma X Pang R Feng and B Wen ldquoEvaluation of op-timum profile modification curves of profile shifted spurgears based on vibration responsesrdquoMechanical Systems andSignal Processing vol 70-71 pp 1131ndash1149 2016

[170] H Iida A Tamura and Y Yamada ldquoVibrational charac-teristics of friction between gear teethrdquo Bulletin of JSMEvol 28 no 241 pp 1512ndash1519 1985

[171] J Borner and D R Houser ldquoFriction and bending momentsas gear noise excitationsrdquo in Proceedings of the InternationalOff-Highway and Powerplant Congress and ExpositionIndianapolis IN USA 1996

[172] D Hochmann Friction Force Excitations in Spur and HelicalInvolute Parallel Axis Gearing +e Ohio State UniversityColumbus OH USA 1997

[173] P Velex and V Cahouet ldquoExperimental and numericalinvestigations on the influence of tooth friction in spur andhelical gear dynamicsrdquo Journal of Mechanical Designvol 122 no 4 pp 515ndash522 2000

[174] D R Houser M Vaishya and J D Sorenson ldquoVibro-acousticeffects of friction in gears an experimental investigationrdquo inProceedings of the Noise and Vibration Conference andExposition Traverse City MI USA 2001

[175] M Vaishya and R Singh ldquoAnalysis of periodically varyinggear mesh systems with coulomb friction using Floquettheoryrdquo Journal of Sound and Vibration vol 243 no 3pp 525ndash545 2001

[176] M Vaishya and R Singh ldquoSliding friction-induced non-linearity and parametric effects in gear dynamicsrdquo Journal ofSound and Vibration vol 248 no 4 pp 671ndash694 2001

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

[178] R Gunda and R Singh ldquoDynamic analysis of sliding frictionin a gear pairrdquo in Proceedings of the Design EngineeringTechnical Conferences and Computers and Information inEngineering Conference Volume 4 9th International PowerTransmission and Gearing Conference Parts A and Bpp 2ndash6 Chicago IL USA 2003

[179] O Lundvall N Stromberg and A Klarbring ldquoA flexiblemulti-body approach for frictional contact in spur gearsrdquoJournal of Sound and Vibration vol 278 no 3 pp 479ndash4992004

[180] S He R Gunda and R Singh ldquoEffect of sliding friction onthe dynamics of spur gear pair with realistic time-varyingstiffnessrdquo Journal of Sound and Vibration vol 301 no 3ndash5pp 927ndash949 2007

[181] S He R Gunda and R Singh ldquoInclusion of sliding frictionin contact dynamics model for helical gearsrdquo Journal ofMechanical Design vol 129 no 1 pp 48ndash57 2007

[182] S He and R Singh ldquoAnalytical study of helical gear dy-namics with sliding friction using Floquet theoryrdquo in Pro-ceedings of the International Design Engineering TechnicalConferences and Computers and Information in EngineeringConference Volume 7 10th International Power Transmis-sion and Gearing Conference pp 433ndash442 Las Vegas NVUSA 2007

[183] R Singh A Lake V Asnani and S He ldquoVibro-acousticmodel of a geared system including friction excitationrdquo inProceedings of the 2007 Inter-Noise and Noise-Con Congressand Conference pp 2921ndash2930 Istanbul Turkey 2007

[184] S He and R Singh ldquoDynamic transmission error predictionof helical gear pair under sliding friction using Floquettheoryrdquo Journal of Mechanical Design vol 130 no 5p 52603 2008

[185] A Kahraman J Lim and H Ding ldquoA dynamic model of aspur gear pair with frictionrdquo in Proceedings of the 12thIFToMM World Congress Besancon France 2007

[186] S He R Singh and G Pavic ldquoEffect of sliding friction ongear noise based on a refined vibro-acoustic formulationrdquoNoise Control Engineering Journal vol 56 no 3 pp 164ndash175 2008

[187] G Liu and R G Parker ldquoImpact of tooth friction and itsbending effect on gear dynamicsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1039ndash1063 2009

[188] S Chen J Tang C Luo and Q Wang ldquoNonlinear dynamiccharacteristics of geared rotor bearing systems with dynamicbacklash and frictionrdquo Mechanism and Machine eoryvol 46 no 4 pp 466ndash478 2011

[189] J Wang J Zheng and A Yang ldquoAn analytical study ofbifurcation and chaos in a spur gear pair with sliding fric-tionrdquo Procedia Engineering vol 31 pp 563ndash570 2012

[190] K F Brethee J Gao F Gu and A D Ball ldquoAnalysis offrictional effects on the dynamic response of gear systemsand the implications for diagnosticsrdquo in Proceedings of the21st International Conference on Automation andComputing Glasgow UK 2015


[191] H Jiang and F Liu ldquoDynamic modeling and analysis of spurgears considering friction-vibration interactionsrdquo Journal ofthe Brazilian Society of Mechanical Sciences and Engineeringvol 39 no 12 pp 4911ndash4920 2017

[192] M Vaishya and R Singh ldquoStrategies for modeling friction ingear dynamicsrdquo Journal of Mechanical Design vol 125 no 2pp 383ndash393 2003

[193] C Kar and A R Mohanty ldquoAn algorithm for determinationof time-varying frictional force and torque in a helical gearsystemrdquo Mechanism and Machine eory vol 42 no 4pp 482ndash496 2007

[194] K F Martin ldquoA review of friction predictions in gear teethrdquoWear vol 49 no 2 pp 201ndash238 1978

[195] G H Benedict and B W Kelley ldquoInstantaneous coefficientsof gear tooth frictionrdquo ASLE Transactions vol 4 no 1pp 59ndash70 1961

[196] B W Kelley and A J Lemanski ldquoPaper 11 lubrication ofinvolute gearingrdquo Proceedings of the Institution of Me-chanical Engineers Conference Proceedings vol 182 no 1pp 173ndash184 1967

[197] K L Johnson and D I Spence ldquoDetermination of gear toothfriction by disc machinerdquo Tribology International vol 24no 5 pp 269ndash275 1991

[198] B Rebbechi F B Oswald and D P Townsend ldquoMea-surement of gear tooth dynamic frictionrdquo in Proceedings ofthe 7th International Power Transmission and GearingConference San Diego CA USA 1996

[199] C Duan and R Singh ldquoDynamics of a 3DOF torsionalsystem with a dry friction controlled pathrdquo Journal of Soundand Vibration vol 289 no 4-5 pp 657ndash688 2006

[200] H Xu Development of a Generalized Mechanical EfficiencyPrediction Methodology for Gear Pairs +e Ohio StateUniversity Columbus OH USA 2005

[201] H Xu A Kahraman N E Anderson and D G MaddockldquoPrediction of mechanical efficiency of parallel-axis gearpairsrdquo Journal of Mechanical Design vol 129 no 1pp 58ndash68 2007

[202] S He S Cho and R Singh ldquoPrediction of dynamic frictionforces in spur gears using alternate sliding friction formu-lationsrdquo Journal of Sound and Vibration vol 309 no 3ndash5pp 843ndash851 2008

[203] S Hou J Wei A Zhang T C Lim and C Zhang ldquoStudy ofdynamic model of helicalherringbone planetary gear systemwith friction excitationrdquo Journal of Computational andNonlinear Dynamics vol 13 no 12 p 121007 2018

[204] L Han W Niu D Zhang and F Wang ldquoAn improvedalgorithm for calculating friction force and torque in in-volute helical gearsrdquoMathematical Problems in Engineeringvol 2013 Article ID 575302 13 pages 2013

[205] L Wang Z Chen Y Shao and X Wang ldquoDynamic featuresof gear system with tooth crack and tooth surface sliding dueto speed variationrdquo in Proceedings of the International DesignEngineering Technical Conferences and Computers and In-formation in Engineering Conference Volume 1 24th Con-ference on Mechanical Vibration and Noise Parts A and Bpp 171ndash175 Chicago IL USA 2012

[206] H Jiang Y Shao and C K Mechefske ldquoDynamic char-acteristics of helical gears under sliding friction with spallingdefectrdquo Engineering Failure Analysis vol 39 pp 92ndash1072014

[207] A Guerine A El Hami L Walha T Fakhfakh andM Haddar ldquoA polynomial chaos method for the analysis ofthe dynamic behavior of uncertain gear friction systemrdquo

European Journal of MechanicsmdashASolids vol 59 pp 76ndash842016

[208] F Liu H Jiang S Liu and X Yu ldquoDynamic behavioranalysis of spur gears with constant amp variable excitationsconsidering sliding friction influencerdquo Journal of MechanicalScience and Technology vol 30 no 12 pp 5363ndash5370 2016

[209] A Guerine A E Hami L Walha T Fakhfakh andM Haddar ldquoDynamic response of a spur gear system withuncertain friction coefficientrdquo Advances in EngineeringSoftware vol 120 pp 45ndash54 2018

[210] I Howard S Jia and J Wang ldquo+e dynamic modelling of aspur gear in mesh including friction and a crackrdquo Me-chanical Systems and Signal Processing vol 15 no 5pp 831ndash853 2001

[211] C Liu D Qin and Y Liao ldquoDynamic model of variablespeed process for herringbone gears including friction cal-culated by variable friction coefficientrdquo Journal of Me-chanical Design vol 136 no 4 p 41006 2014

[212] K Huang Y Xiong TWang andQ Chen ldquoResearch on thedynamic response of high-contact-ratio spur gears influ-enced by surface roughness under EHL conditionrdquo AppliedSurface Science vol 392 pp 8ndash18 2017

[213] ISO Calculation of Scuffing Load Capacity of CylindricalBevel and Hypoid Gears ISO Geneva Switzerland 2000

[214] S Kim and R Singh ldquoGear surface roughness induced noiseprediction based on a linear time-varying model with slidingfrictionrdquo Journal of Vibration and Control vol 13 no 7pp 1045ndash1063 2007

[215] K Jayasankaran Structure-borne Noise Model of a Spur GearPair with Surface Undulation and Sliding Friction as Exci-tations +e Ohio State University Columbus OH USA2010

[216] T Eritenel and R G Parker ldquoAn investigation of tooth meshnonlinearity and partial contact loss in gear pairs using alumped-parameter modelrdquoMechanism andMachineeoryvol 56 pp 28ndash51 2012

[217] T Eritenel and R G Parker ldquo+ree-dimensional nonlinearvibration of gear pairsrdquo Journal of Sound and Vibrationvol 331 no 15 pp 3628ndash3648 2012

[218] A Palermo D Mundo R Hadjit P Mas and W DesmetldquoMultibody modelling of shuttling excitation in spur andhelical geared transmissionsrdquo in Proceedings of ISMA 2012-USD 2012 pp 4005ndash4016 Leuven Belgium 2012

[219] A Palermo D Mundo R Hadjit and W Desmet ldquoMul-tibody element for spur and helical gear meshing based ondetailed three-dimensional contact calculationsrdquo Mecha-nism and Machine eory vol 62 pp 13ndash30 2013

[220] R Teja T R Milind R C Glover and S SonawaneldquoDynamic analysis of helical gear pair due to TE andshuttling moment excitationsrdquo in Proceedings of the Noiseand Vibration Conference and Exhibition Warrendale PAUSA 2017

[221] D R Houser J Harianto and Y Ueda ldquoDetermining thesource of gear whine noiserdquo Gear Solutions pp 17ndash22 2004

[222] S Li ldquoExperimental investigation and FEM analysis ofresonance frequency behavior of three-dimensional thin-walled spur gears with a power-circulating test rigrdquo Mech-anism and Machineeory vol 43 no 8 pp 934ndash963 2008

[223] A Toso F van Wermeskerken N Cappellini andG Heirman On the Effect of Lightweight Gear Blank To-pology on Transmission Dynamics [1])year)+0^(^name(following-sibling[2])volume)+0^(^name(following-sibling[3])publisher-loc)+0^(^name(following-sibling[4])publisher-name)=0


^[fish 0volOn] Vol $^content-markup(following-sib-lingvolume)gt ASME New York NY USA 2015

[224] S Shweiki A Palermo and D Mundo ldquoA study on thedynamic behaviour of lightweight gearsrdquo Shock and Vi-bration vol 2017 Article ID 7982170 12 pages 2017

[225] S Shweiki A Rezayat T Tamarozzi and D MundoldquoTransmission error and strain analysis of lightweight gearsby using a hybrid FE-analytical gear contact modelrdquo Me-chanical Systems and Signal Processing vol 123 pp 573ndash5902019

[226] B Guilbert P Velex D Dureisseix and P Cutuli ldquoAmortar-based mesh interface for hybrid finite-elementlumped-parameter gear dynamic models-applications tothin-rimmed geared systemsrdquo Journal of Mechanical Designvol 138 no 12 Article ID 123301 2016

[227] B Guilbert P Velex D Dureisseix and P Cutuli ldquoModularhybrid models to simulate the static and dynamic behaviourof high-speed thin-rimmed gearsrdquo Journal of Sound andVibration vol 438 pp 353ndash380 2019

[228] L Hou Y Lei Y Fu and J Hu ldquoEffects of lightweight gearblank on noise vibration and harshness for electric drivesystem in electric vehiclesrdquo Proceedings of the Institution ofMechanical Engineers Part K Journal of Multi-Body Dy-namics vol 234 no 3 pp 447ndash464 2020

[229] T G Yılmaz O Dogan and F Karpat ldquoA comparativenumerical study of forged bi-metal gears bending strengthand dynamic responserdquo Mechanism and Machine eoryvol 141 pp 117ndash135 2019

[230] F Karpat T G Yılmaz O Dogan and O C Kalay ldquoStressand mesh stiffness evaluation of bimaterial spur gearsrdquo inProceedings of the ASME 2019 International MechanicalEngineering Congress and Exposition Salt Lake City UTUSA 2019

[231] J-H Bae K-C Jung S-H Yoo S-H Chang M Kim andT Lim ldquoDesign and fabrication of a metal-composite hybridwheel with a friction damping layer for enhancement of ridecomfortrdquo Composite Structures vol 133 pp 576ndash584 2015

[232] A Treviso B Van Genechten D Mundo and M TournourldquoDamping in composite materials properties and modelsrdquoComposites Part B Engineering vol 78 pp 144ndash152 2015

[233] R F Handschuh G D Roberts R R SinnamonD B Stringer B D Dykas and L W Kohlman ldquoHybridgear preliminary resultsmdashapplication of composites to dy-namic mechanical componentsrdquo Gear Technology vol 302013

[234] R F Handschuh K E LaBerge S DeLuca and R PelagalliVibration and Operational Characteristics of a Composite-Steel (Hybrid) Gear NASA Washington DC USA 2014

[235] K E Laberge R F Handschuh G Roberts and S +orpldquoPerformance investigation of a full-scale hybrid compositebull gearrdquo in Proceedings of the 72nd AHS 2016 Forum WestPalm Beach FL USA 2016

[236] P G Catera D Mundo A Treviso et al ldquoOn the design andsimulation of hybrid metal-composite gearsrdquo AppliedComposite Materials vol 26 no 3 pp 817ndash833 2019

[237] W Xiao Y Huang H Jiang and L Jin ldquoEffect of powdermaterial on vibration reduction of gear system in centrifugalfieldrdquo Powder Technology vol 294 pp 146ndash158 2016

[238] W Xiao J Li S Wang and X Fang ldquoStudy on vibrationsuppression based on particle damping in centrifugal field ofgear transmissionrdquo Journal of Sound and Vibration vol 366pp 62ndash80 2016

[239] R Ramadani A Belsak M Kegl J Predan and S PehanldquoTopology optimization based design of lightweight and low

vibration gear bodiesrdquo International Journal of SimulationModelling vol 17 no 1 pp 92ndash104 2018

[240] S Gauntt R Campbell and S Mcintyre ldquoDesign optimi-zation of a hybrid spur gearrdquo Vertical Flight Society Forumvol 75 2019

[241] N Contartese P G Catera and D Mundo ldquoStatic meshstiffness decomposition in hybrid metal-composite spurgearsrdquo Advances in Mechanism andMachine Science vol 73pp 977ndash985 2019

[242] P G Catera F Gagliardi D Mundo L De NapoliA Matveeva and L Farkas ldquoMulti-scale modeling of triaxialbraided composites for FE-based modal analysis of hybridmetal-composite gearsrdquo Composite Structures vol 182pp 116ndash123 2017

[243] S M Gauntt and R L Campbell ldquoCharacterization of ahybrid (steel-composite) gear with various composite ma-terials and layupsrdquo in Proceedings of the AIAA Scitech 2019Forum San Diego CA USA 2019

[244] P G Catera D Mundo F Gagliardi and A Treviso ldquoAcomparative analysis of adhesive bonding and interferencefitting as joining technologies for hybrid metal-compositegear manufacturingrdquo International Journal on InteractiveDesign and Manufacturing (IJIDeM) vol 14 no 2pp 535ndash550 2020

[245] K E LaBerge J P Johnston R F Handschuh andG D Roberts ldquoEvaluation of a variable thickness hybridcomposite bull gearrdquo in Proceedings of the AHS International74th Annual Forum amp Technology Display Phoenix AZUSA 2018


In geared systems if the loads transmitted by the gearswere constant the geometry of the gears were perfect andthe motions of the gears were smooth there would not beany vibration However in real conditions the profiles ofgears are imperfect because of manufacturing errors andintentional modifications In addition the teeth will deflectsignificantly when they are subjected to transmitted torqueMoreover supporting structures of gears are also deform-able when subjected to operation loads which induces in-evitable misalignments Deviations of the real gear profilefrom the ideal involute one deflections of teeth underoperating conditions and inevitable misalignments allcontribute to periodic displacement which induces varyingmeshing forces along the line of action (LOA) +e frictionforces between gear surfaces which act along the off-line ofaction (OLOA) change directions before and after the teethpasses through the pitch point For helical gears and spurgears with severe misalignments the centroid of the meshingforces shifts back and forth axially along the tooth facewidth +e displacement along the LOA and the frictionforces along the OLOA together with the axially back-and-forth shifts of the centroid of the meshing forces change theamplitudes action positions and directions of the meshingforces between mating gears +e oscillating meshing forcesare transmitted to supporting bearings by shafts resulting invarying bearing forces +e varying bearing forces vibrategearbox-housing plates which finally radiate undesired gearwhine noise

+e complexity of gear whine noise has inspired a largenumber of researchers to study this issue Dating back to1958 Harris [10] noticed the gear whine noise phenomenonand investigated the effect of the transmission error (TE) ongear whine noise Because of computer technology break-throughs in 1980s a large number of studies on topics re-lated to gear whine noise were conducted Coming into the21st century the number of the publications in this areashows an exponential growth trend as shown in Figure 1Many outstanding researchers published review papers re-lated to gear vibration and noise such as gear system dy-namic models [11] nonlinear dynamics of gear-drivensystems [12] condition monitoring and fault diagnosis[13 14] and planetary dynamics and vibration [15] re-spectively Conversely so far no one has made a compre-hensive summary on the sources of gear whine noise+erefore there is an urgent need for a systematic literaturereview on the sources of gear whine noise

During the generation of gear whine noise excitations ofgear meshes act as sources shafts and bearings as vibrationtransfer paths and the gearbox-housing plates as the re-ceiver +e reduction of gear whine noise can be reachedonly via the reduction of the amplitudes of the housing platevibrations Both the strength of the sources and the prop-agation property of the transfer paths influence the mag-nitudes of the vibrations of the housing plates In this reviewthe authors focus on the sources of gear whine noise

+is paper aims to summarize the excitations of gearwhine noise from the literature classify these excitationsaccording to the three action directions of the meshingforces present methods for predicting these excitations

conclude strategies to reduce these excitations and proposenew possibilities for gear whine noise reduction at its source+e remaining part of this paper is organized as follows InSection 2 the excitation along the LOA direction namelyTE is summarized the definition of TE harmonic contri-butions from TE strategies to calculate and measure TE andapproaches to reduce TE are also described In Section 3excitations along the OLOA are reviewed+e approaches toevaluate the frictional excitations and strategies to reducethem are presented In Section 4 shuttling excitations in theaxial direction and the properties of shuttling excitation aresummarized +e potential of vibration reduction forlightweight gears is discussed in Section 5 Section 6 gives asummary +e authors also point out essential existingproblems in reported research work and describe prospectsof future research directions regarding excitations of gearwhine noise

2 Transmission Error

If the profiles of gears were geometrically perfect the teethon gears were perfectly rigid and correctly spaced and thesupporting structures were rigid and accurately installedthere would be no variance inmeshing forces whenmeshingwhich results in no vibration being generated In reality thisideal scenario does not occur for a variety of reasons such asmanufacturing imperfections intentional modificationsand inevitable teeth deflections All of these imperfectionscontribute to a periodic displacement namely TE whichwas first defined by Harris [10] As stated by Munro [16] theperiodic variation of TE induces periodic advancement andretardation of the driven gear while the gears are rotating Inaddition if the rotation speed coincides with the frequencyof one component of TE then resonance will occur +eperiodic motions of the driven gear and the potential res-onance may give rise to large dynamic meshing loads andhigh noise levels Transmission error is therefore a primarysource of gear noise and vibration Since these two studies

100

50

18

35

71

101

0ndash1980 1981ndash1999

Year2000ndash2009 2010ndash2019

Publ

icat

ion

num

ber

Figure 1 Histogram of research papers on gear vibration andnoise





TE(θ) θgminus

Rbp

Rbg

θp (1)

TE(d) Rbgθg

minus Rbpθp (2)




TE(l) W

KM(l)+

1113936jη(p)

Kj (l) + η(g)

Kj (l)

KM(l) (3a)


δKM(l)

KminusM


δKM(l)

KminusM

1113888 1113889

minus 1

(3b)








Op

Og

Rbp

Rbg

θp

θg

Line of action

Pitch line


α















Sideband






















7i1(ci(ai + ε)) [91 92]

x wk + εk + P

wk 1113936Ni1 akiFi


[94ndash96 121]





[124ndash128]







Ff μnFmeshnε (4a)


















εi (minus V21

rarr(Mi)n y

rarrV21

rarr(Mi)) plusmn y

rarr

[173 177]V

21








[173]




a1 minus R2b2

1113969+ Rb1nω1nt1 + Rb1nθ1








5 Lightweight Gears





2r ]





b6e V

b70 Rb8



10 + b9eS EHL [185 200 201 211 212]












ni1 Mi

Mi qmn 1113938xi2

xi1

xn(1 + an(xb))dx[171]


Mp Ktnc + FnAp

Mg Ktnc + FnAg


[216 217]

sp1 FCFWa

FC

sp2

n 01 αgt α01



n 09 αle α09

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

[218 219]

Mz Mbz + Maz



[220]












2383

374

7243





















Acknowledgments




References







































































































































































































































































TE(θ) θgminus

Rbp

Rbg

θp (1)

TE(d) Rbgθg

minus Rbpθp (2)




TE(l) W

KM(l)+

1113936jη(p)

Kj (l) + η(g)

Kj (l)

KM(l) (3a)


δKM(l)

KminusM


δKM(l)

KminusM

1113888 1113889

minus 1

(3b)








Op

Og

Rbp

Rbg

θp

θg

Line of action

Pitch line


α















Sideband






















7i1(ci(ai + ε)) [91 92]

x wk + εk + P

wk 1113936Ni1 akiFi


[94ndash96 121]





[124ndash128]







Ff μnFmeshnε (4a)


















εi (minus V21

rarr(Mi)n y

rarrV21

rarr(Mi)) plusmn y

rarr

[173 177]V

21








[173]




a1 minus R2b2

1113969+ Rb1nω1nt1 + Rb1nθ1








5 Lightweight Gears





2r ]





b6e V

b70 Rb8



10 + b9eS EHL [185 200 201 211 212]












ni1 Mi

Mi qmn 1113938xi2

xi1

xn(1 + an(xb))dx[171]


Mp Ktnc + FnAp

Mg Ktnc + FnAg


[216 217]

sp1 FCFWa

FC

sp2

n 01 αgt α01



n 09 αle α09

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

[218 219]

Mz Mbz + Maz



[220]












2383

374

7243





















Acknowledgments




References
















































































































































































































































































Sideband






















7i1(ci(ai + ε)) [91 92]

x wk + εk + P

wk 1113936Ni1 akiFi


[94ndash96 121]





[124ndash128]







Ff μnFmeshnε (4a)


















εi (minus V21

rarr(Mi)n y

rarrV21

rarr(Mi)) plusmn y

rarr

[173 177]V

21








[173]




a1 minus R2b2

1113969+ Rb1nω1nt1 + Rb1nθ1








5 Lightweight Gears





2r ]





b6e V

b70 Rb8



10 + b9eS EHL [185 200 201 211 212]












ni1 Mi

Mi qmn 1113938xi2

xi1

xn(1 + an(xb))dx[171]


Mp Ktnc + FnAp

Mg Ktnc + FnAg


[216 217]

sp1 FCFWa

FC

sp2

n 01 αgt α01



n 09 αle α09

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

[218 219]

Mz Mbz + Maz



[220]












2383

374

7243





















Acknowledgments




References























































































































































































































































































7i1(ci(ai + ε)) [91 92]

x wk + εk + P

wk 1113936Ni1 akiFi


[94ndash96 121]





[124ndash128]







Ff μnFmeshnε (4a)


















εi (minus V21

rarr(Mi)n y

rarrV21

rarr(Mi)) plusmn y

rarr

[173 177]V

21








[173]




a1 minus R2b2

1113969+ Rb1nω1nt1 + Rb1nθ1








5 Lightweight Gears





2r ]





b6e V

b70 Rb8



10 + b9eS EHL [185 200 201 211 212]












ni1 Mi

Mi qmn 1113938xi2

xi1

xn(1 + an(xb))dx[171]


Mp Ktnc + FnAp

Mg Ktnc + FnAg


[216 217]

sp1 FCFWa

FC

sp2

n 01 αgt α01



n 09 αle α09

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

[218 219]

Mz Mbz + Maz



[220]












2383

374

7243





















Acknowledgments




References







































































































































































































































































Ff μnFmeshnε (4a)


















εi (minus V21

rarr(Mi)n y

rarrV21

rarr(Mi)) plusmn y

rarr

[173 177]V

21








[173]




a1 minus R2b2

1113969+ Rb1nω1nt1 + Rb1nθ1








5 Lightweight Gears





2r ]





b6e V

b70 Rb8



10 + b9eS EHL [185 200 201 211 212]












ni1 Mi

Mi qmn 1113938xi2

xi1

xn(1 + an(xb))dx[171]


Mp Ktnc + FnAp

Mg Ktnc + FnAg


[216 217]

sp1 FCFWa

FC

sp2

n 01 αgt α01



n 09 αle α09

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

[218 219]

Mz Mbz + Maz



[220]












2383

374

7243





















Acknowledgments




References












































































































































































































































































εi (minus V21

rarr(Mi)n y

rarrV21

rarr(Mi)) plusmn y

rarr

[173 177]V

21








[173]




a1 minus R2b2

1113969+ Rb1nω1nt1 + Rb1nθ1








5 Lightweight Gears





2r ]





b6e V

b70 Rb8



10 + b9eS EHL [185 200 201 211 212]












ni1 Mi

Mi qmn 1113938xi2

xi1

xn(1 + an(xb))dx[171]


Mp Ktnc + FnAp

Mg Ktnc + FnAg


[216 217]

sp1 FCFWa

FC

sp2

n 01 αgt α01



n 09 αle α09

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

[218 219]

Mz Mbz + Maz



[220]












2383

374

7243





















Acknowledgments




References








































































































































































































































































5 Lightweight Gears





2r ]





b6e V

b70 Rb8



10 + b9eS EHL [185 200 201 211 212]












ni1 Mi

Mi qmn 1113938xi2

xi1

xn(1 + an(xb))dx[171]


Mp Ktnc + FnAp

Mg Ktnc + FnAg


[216 217]

sp1 FCFWa

FC

sp2

n 01 αgt α01



n 09 αle α09

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

[218 219]

Mz Mbz + Maz



[220]












2383

374

7243





















Acknowledgments




References










































































































































































































































































ni1 Mi

Mi qmn 1113938xi2

xi1

xn(1 + an(xb))dx[171]


Mp Ktnc + FnAp

Mg Ktnc + FnAg


[216 217]

sp1 FCFWa

FC

sp2

n 01 αgt α01



n 09 αle α09

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

[218 219]

Mz Mbz + Maz



[220]












2383

374

7243





















Acknowledgments




References













































































































































































































































































2383

374

7243





















Acknowledgments




References





















































































































































































































































































Acknowledgments




References





































































































































































































































































References




































































































































































































































































ReviewArticledownloads.hindawi.com/journals/sv/2020/9834939.pdf · 2020-08-06 · ReviewArticle...

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