Modeling and solving for transverse vibration of gear with variational thickness

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J.Cent.South Univ.(2013、20：2124 2133DoI：10.1007/sl1771-013-l7l6-3Modeling and solving for transverse vibration ofgear with variational thicknessQIN Hui.bin(秦慧斌) -，LO Ming( 明) ，SHE Yin-zhu(佘银柱) ，WANG Shi-ying(王时英) ，LI Xiang-peng(李向鹏)鱼Springer1.College of Mechanical Engineering，Taiyuan University of Technology，Taiyuan 030024，China；2.School of Mechanical Engineering＆ Automation，North University of China，Taiyuan 03005 1，China◎ Central South University Press and Springer-Verlag Berlin Heidelberg 20 1 3Abstract：A analyzed model of gear with wheel hub.web and rim was derived from the Mindlin moderate plate theory.The gear wasdivided into three annular segments along the locations of the step variations.Traverse displacement.rotation angle.shear force andflexura1 moment were equa1 to ensure the continuity along the interface of the wheel hub.web and rim segments.The governingdiflferential equations for harmonic vibration of annular segments were derived to solve the gear vibration problem.The influence ofhole to diameter ratios，segment thickness ratios，segment 1ocation ratios，Poisson ratin on the vibration behavior of stepped circularMindlin disk were calculated，tableted and ploted.Comparisons were made with the frequencies arising from the presented method，finite elements method.and structure moda1 experiment.The result correlation among these three ways is very good.The largesterror for al1 frequencies is 5.46%.and 1ess than 5% for most frequencies。

Key words：gear with variational thickness；Mindlin moderately plate theory；transverse and flexural vibration model；resonantfrequencies of vibrationl IntrOductiOnUltrasonic machining can enhance cuting eficiency，extend tool1ife.improve surface quality in contrast to theconventional machining[1].It is imperative to researchultrasonic gear machining and bring it into practice，according to ultrasonic vibration machining merits andits application status. The cuting performance ofultrasonic gear machining such as ultrasonic hobbing，lapping，electrochemical finishing and honing dependsprimarily on the resonant vibration characteristic of theacoustic vibration system.The gear is not only an objectmachined，but also the main component of vibrationsystem in ultrasonic gear machining[2-3].When theultrasonic system is designed，the gear is usually reducedto a annular uniform disk or stepped plate with diameterof reference circle of the gear.It is important to makegear vibrate in the transverse mode with only circle nodallines，a symmetrical mode to ensure the homogeneity ofultrasonic gear machining.The dynamic characteristicsand the natural frequency of the gear are critical to thevibration system design。

The gear has numerous applications in machine，automobile.aerospace and wind turbine industry，and itcan be simplified as thin，moderate，thick an ular disk，orthick.walled cylinders according to the ratios of thick todiameter 4-5].Different plate theories have beenapplied in studying the vibration of circular and annularplates with variable thickness，ranging from the classicalthin plate theory，to the shear deformable plate theory to3-D elasticity theory [6].The Rayleigh Ritz method isone of the most popular methods that have been used tostudy the vibration of circular plates with thicknessvariations f7].Other methods that are used in this areainclude finite element method.analytical method anddiferential quadrature method81。

Some scholars presented an investigatio13on thevibration behavior of circular Mindlin plates withstep.wise thickn ess variations.The influence of the platethickn ess ratios， the step thickn ess ratios and thelocations of the step variations on the frequencyparameters were examined9-11].Though these studiesmake contributions to understand vibration behavior ofthe gears with hub，web and rim.However，manyproblems about the vibration behavior of the gears havenot been solved yet。

LI[1 2implemented a fundamental study onFoundation item：Proiect(5O975l91)supposed by the National Natural Science Foundation of China；Project(201 13027)supposed by the OutstandingInnovation Project of Shanxi Province Foundation for Graduate StudentReceived date：2012 05 14；Accepted date：2012-09-1lCorresponding author：L0 Ming，Profesor，PhD；Tel：86-351-6010399；E-mail：lvming###tyut.edu.cnJ.Cent.South Univ.(2013)20：2124-2133 2125resonant frequency behavior of three.dimensional，thin-waled spur gears from experimental tests and finiteelement analyses.Solution in theory was not broughtforward in it.The examined objects were thin.walledspur gears，rather than moderate thick gears that arewidely application in transmission.YAU et al131 andVAIDYANATHAN et al[14]developed the approach ofstudying the deflection of gear teeth including sheardeformation using the Rayleigh-Ritz energy method。

The shear plate mode1 could replace the finite elementmodel since it is more computationaly e cient and itsresults are accurate enough for most engineeringpurposes.SHA0 [1 5]and LI et al[1 6]establishedvibration model of the gears with uniform thicknessbased on Mindlins moderate plate theory.Among theseresearchers，only few gave a complete description of thegears with hub，web and rim。

The primary obiective of this work is to establishthe vibration model and solve exact vibration frequenciesfor stepped gears in ultrasonic machining.The govemingdifferential equations for harmonic vibration of annularsegments were derived and an analytical method basedon the domain decomposition technique was developedto solve the gear vibration problem.The influences ofhole diameter ratios，the hub，web and rim thickn essratios.the segments locations and materia1 Poisson ratiora1fb1on the vibration behavior of the gears with hub，web andrim were highlighted.The finite element method modalanalysis and experiment modal analysis were carried out。

The correlation between these results is consistent。

2 Theoretical vibration analyses2.1 Physical model and analytical modelCertain efrect factors such as keyway，cast fillet。

machining chamfer and hole designed for less weightwere omitted in vibration model of straight toothed spurgear and helical-spur gear.Gear was simplified tostructure combined by medium thick annular plates withthe outer diameters of reference circle.Gears can besubstituted for analytical models shown in Fig.1.If d4 isused to express the gear reference circle diameter，and dl，a2 and a3 are used to express the diameters of center hole，hub and web，respectively.tl，t2 and t3 are used to expressthe thickn ess of hub，web and rim，respectively.Theannular disk kinetic equations were deduced from thethree-dimensional equations of elasticity，referred to apolar coordinate system shown in Fig.2.where r and 0are the radial and circumferential coordinates。

respectively.t.a and b are the thickn ess inner holediameter and outer diameter of annular disk，respectively。

I. ± ! .I(c)Fig.1 Physical models and analytical models of gear with stepped thickness：(a)Gear with hub，web and rim；(b)Gear with hub andweb；(c)Gear with uniform thickness2126 J. Cent.South Univ.(2013)20：2124-2133x/Fig.2 Geometry and polar coordinate system of moderate thickannular diskThe gear with hub，web and rim shown in Fig.1(a1may be taken as assembly bodv gathered by threeannular disks shown in Fig.3.In the gear vibration.hubannular disk and web annular disk should keep thedisplacement.strength and flexural moment consistenton the cylindrical surface with diameter d2 and height t2。

At the same time，annular web disk and annular rim diskshould keep displacement，strength and flexural momentconsistent on the cylindrical surface with diameter d3 andheight t2.When t3t2t1.the gear with hub.web and rimdegenerates to gear with hub and web shown in Fig.1(b)，and may be taken as assembly body gathered bv t、voannular disks shown in Fig.3(a)and Fig.3(b).In thegear vibration，annular hub disk and annular web disklI 。 。

(a1(b)0 Fig.3 Domain annular disks decomposed for gear with hubweb and rim：(a)Annular disk of hub；(b)Annular disk of web(c)Annular disk of rimshould keep displacement，strength and flexura1 momentconsistent on the cylindrical surface with diameter d2 andheight t2.W hen t3t2tl，the gear with hub，web and rimdegenerates to gear with unifornl thickness shown inFig.1(c)and may be taken as an annular disk with innerdiameter d，outer diameter de and high t 。

2.2 Derivation of vibration equationsThe problem at hand is to determine the naturalfrequencies of the gear with the hub.web and rim.Thegear can be considered three isotropic，elastic annulardisks as shown in Fig.3.The Mindlin first order sheardeformable plate theory was employed to study thisproblem.Using the polar coordinate system shown inFig.2，the equations of motion for the i-th segment maybe expressed as f17]鲨 - ：8r 80 2Mo- p，38r 80 - 12型8r盟00 譬 0t 2f- 2 i(1)8twhere卢1，2，3，F10.5 ，r20.5d3，r30.5幽，w istraverse displacement， is rotation angle along radialaxis of annular disk and is rotation angle alongcircumferentia1 of the midsurface of the i-th annular disk，r and 0 are the radial and circumferential coordinates，Pis the density，ti is the thickness of the i-th annular disk。

the stress resultants for shear force and bending moment， ， M；， and ： are given by鲁 8wi] Gti l( i 1 c3wi]攀or 刻 l l d Jl 警 - where E，G and are elastic modulus，the shear modulusand Poisson ratio， respectively， G：E/[2(1 )]，D尸E /[12(1-/ )]，and K 12/n is the shear corectionfactor required in the Mindlin plate theory.In polarcoordinates，again omitting a time factor e ，if Eq.(2)were substituted in Eq.(1)，the resulting annular diskJ.Cent.South Univ.(20131 20：2124-2133 2127displacement equations of motion would be thosecorresponding to Eq.(3).The displacement fields of thef-th segment may be expressed as functions of threepotentialswi， ，%。

w/Ⅵwhere ， w21 and are mode shape functions ofthef.th annular disk。

( ) [ ( ) -Si ]～， ( ) [ ( ) -Si ]( pti t2 c (R )[(R- ) 4c - ])( )。 1( ) (R )-[( - ) 4( ) ] )( 2.3 Boundary conditions and solution process ofvibration equationsWhen the i-th segment annular disk represents freevibration，Ⅵl， and should satisfyI[V ( ) ] 0，( 1，2，3；j1，2)[V ( ) ]Hi0 (4)where the Laplacian operator in polar coordinates isgivenbyv：： 旦 2。 a ，2 a 2In view of the modesappropriate solutions of Eq。

be derived as[17]of transverse vibration，the(4)for the i-th segment canI-[ lm 。 !) im 口1 ]c0s [ ( r) (6r)]cosmO (5)l [ 日 (%r) iHmrm(翰r)]sinmOwhere (：1，2，，oo)is the number of nodal diameters，， B ， ， ，锅 and 8；3(i1，2，3)arethe unknown constants that may be determined using thefree boundary conditions at the edge and the matchingconditions at the interface between hub.web and rimanular disks， and are the first and secondkind Bessel functions of order m，respectively。

It is important to make the gear vibrate in thetransverse mode with only circle nodal lines fi.e.，modeshape with no nodal diameter，m0、in ultrasonic gearmachining，which can ensure machining homogeneity ofthe gear tooth surface.That is， (r， 三0.Equation(3)may simplify asw ( - ) ( - )： (cr2i～1) 州a(6)The boundary conditions of a Mindlin annular diskcan be expressed asQr'0 for free surface (7)M：w 0 for simply supported surface (8)w 0 for clamped surface (9)To satisfy the thickness variation and ensure thecontinuity along the interface of the wheel hub，web andrim segments， the essential and natural interfaceconditions between the two segments must hold asfolows：w i： w 1 ；(10)(11)(12)(13)The displacement fields and stress resultants of thef.th segment can be expressed in terms of the arbitraryconstants A：and ，(i-1，2，3；产1，2)，as detailed inRe[18].In view of Eqs.(5)and(6)and implementingthe boun dary conditions of the annular disk along thecircumferentia1 surface and interface conditions betweensegments，a homogeneous system of equations for thegear with hub，web and rim can be derived as肠f 4 104f×1 (14)where K is the matrix of f4f1×(4f) elements foraxisymmetfic vibration mode f 01.i1 for the gearwith uniform thickness，i2 for the gear with hub andweb，and i3 for the gear with hub，web and rim，respectivel is the vector of unknown coefficients.Theangular frequency∞ of the gear is evaluated by settingthe determinant of K in Eq.f141 to be zero，and thensolving the characteristic equation by a root finding2128 J.Cent.South Univ.(2013)20：2124-2133algorithm.The eigenvalue problem can be solved byusing the computational software such as Matlab。

3 Numerical solving results and discussionMatlab201 1Ra was employed in this work tocompute the eigenvalues as governed by Eq.(14).In thesearch for the eigenvalue an iterative process wasdeveloped.Firstly，it sets initial and terminate values forand small increments of c[)were subsequently added tothe initial value.The determinants of K were evaluatedaccordingly.Once the sign of the determ inant K changes，solve the value of .In the search，the calculationstopped with a given tolerance between the value of I/q。

In this work，the tolerance was set to be l×101。.09makes l/q 1 x 1 0 。be taken as a numerical precisionsolution.The first tw o sequence frequencies 1，2solution graph of the gear with d145，d272，d31 08，180，h45，t29，t318ram is depicted in Fig.4。

Mechanica1 properties of aloy-steel 45 listed in Tlable 1were employed in the calculation。

星暑专∽ 20- 1- 23740y01.679×104y00 4 8 12 l6 20Axisymmetric transverse fiexuralfree vibration frequency，f/kHzFig.4 Relationship between solution errors A and transversefree vibration frequency(m0； 1，2)Table 1 Mechanical properties of gear materials3.1 Gears with hub.web and rimrl dl/d4，r2 di/&， 3 d3/d4(tl/d4，rlt2/tl，r2t3/tl(15)f161where r1 .r2 and r3 are hole diameter，hub diameterand web diameter to re：ference circle diameter ratios，respectively；r1 and r2 are web thickness ratios and rimthickness ratios.respectively； is thickness of hub toreference circle diameter ratio；r1 ，r7 and r shouldkeep rl r2 3 ；rl and r2 should keep rlr2。

The gear vibrates in the transverse mode with onlyone or tw o circle noda1 lines.which is necessary inultrasonic gear machining.The rim thickn ess，webthickn ess and hole diameter increase when gear referencediameter and hub thickn ess are not changed.The lowertw o sequence resonant frequencies of the gears with hub，web and rim were examined in detail.The emphasis is onthe calculated lower two sequence mode shapes and theirrespective resonant frequencies with the influence ofhole to diameter ratios rl。，segment thickn ess ratios f1and r2，segment location ratios r2 and r3"，materialPoisson ratios on the vibration behavior of the gearswith hub，web and rim.In order to observe the variationof the frequencies more clearly， the results werecalculated.tableted in Tables 2 and 3 and plotted in Figs。

5-6.The reference diameter，hub thickness are fixed at以l 80 mm and t145 mm，respectively。

Table 2 Transverse free vibration frequencies ofgears with hub，web and rim(t218，，j 1/4，m0，nl，2)Table 3 Transverse free vibration frequency ofgears with hub，web and rim(t227，rt 1/4，m0，nl，2)J.Cent.South Univ.(2013)20：2124-2133 212924204Rim thickness ratios ofgear withhub，web and rim，2-2：：Firs.：tse：q：：-u：：：en：：c。e。fr e qu en c y .. 。.. - 。 ： l1. .. ll-..1.. 1. 1l.1ll1.... ... 0.4 0.5 0.6 0.7 0.8 0.9 1.ORim thickness ratios of gear withhub，web and rim，"g2Rim thickness ratios of gear w1thhub，web and rim，2"2Fig.5 Relationship between rim thickness ratio and transversefree vibration frequency of gears with hub，web and rim( 1/3，ra 0.6，r3 0.8，m0，nl，2)：(a)2"10.2；(b)r10.4；(c)2"lo.6Tables 2 and 3 present the exact transverse freevibration frequency of the first two modes for the gearswith hub.web and rim.The rim thickness is set to be 1 8InIrl to 45 min with step.up 9 mm.The hole and hub，web were fixed at d145 mm， 108 mlTl and d3144min.Two web thickn ess parameters sets were examined，i.e.。t218 mm for Table 2 and t227 mm for Table 3。

Rim thickness ratios ofgear withhub，web and rim， 2(b) ...... . 。：J ：：1.。

～ 。。。。。。 1.1l 。。。。。 -- 0.27- 0.30- ◆- 0330.4 0.5 0.6 0.7 0.8 0.9 1.0Rim thickness ratios of gear withhub，web and rim， 2Rim thickness ratios of gear withhub，web and rim，2"2Fig.6 Relationship betw een rim thickn ess ratio and transversefree vibration frequency of gears with hub，web and rim( 1/2， o.6， 0.8，mO，nl，2)：(a)rl0.2；(b)2"10.4(e)2"10.6Tl1ree material sets shown in 1址)lel were considered.i.e。

0-27，0.30，0.33.The frequencies decrease as the rimthickness and materia1 Poisson ratio increases due to theeffect Of transverse shear deformation and rotary inertia。

On the other hand，the frequencies increase as the webthickn ess increases from 1 8 tO 27 mm.The exactvibration solutions in Tables 2 and 3 are highly valuableN u 0Il0芦口00 I10-l - 00JIn ∞I。 善 扫 0E a鲁 sIxv如 加N u 裔uII。n1 出口。-苗JqIA Q占 -BJ )( 岛 ∞Ia s暑扫u量0量 uA∞-XvN -uI10拿口蠹 I10～ 叠 - 器 I时 )( ∞ ∞1IJE-扫 0-扛oII营 ∞ 《山- 0口 暑口 I10- ->0占 -g 0 ∞10 ∞II1w扫 0I扫0gII ∞ 《N 山-u 0Il0 口 I10-l -g )(0 H 0∞.10 ∞口砖扫 0-日og -II ∞)(《2l30as benchmark values to check the validity and accuracvor印proximate numerica1 methods for the vibration ofthe gears with hub， web and rim.Especially， webthickness exerts greater effects on frequency than othersparameters。

The curves in Figs.5 and 6 start from rim thicknessratio o4 and end at r21.0 and there are four samplepoints on each curve.The frequency厂increases as theweb thickness ratio z-1 Increases from 0.2 to 0.6.and itdecreases as the material Poisson ratio/2 increases from0.27 to 0，33.Figure 5(c)and Figure 6(c)show that thesecond sequence frequency for the gears increases firstas rim thickn ess ratio r2 varies from 0.4 to 0.6 and thendecreases as varies from 0.6 to I.0.The secondsequence trequencyfdecreases as the rim thickness ratiore Increases due to the effect of transverse sheardetbrmation and rotary inertia with z-10I2. 0.4.Thetrequency/increases as the hole to diameter ratio rincreases from 1/3 to 1/2.It is found that rim and webthicknesses have greater effects on the second structuralfrequencies than the first structural frequencies of thegears with hub，rim and web。

ent.south Univ.(2013)20：2124-21333.2 Gears with hub and webFigure 7 shows the variation of the first twosequence frequencies versus the web thickn ess ratios fortransverse free vibration of the gears with hub and web。

The web location ratio was fixed at 0.6.The hole todiameter ratios were set to be ，1。1/4 for Fig. 7(a1，巧 1/3 for Fig.7(b)and 1/2 for Fig.7(c)，respectively.瞻 6 thickness ratio of the web thickness tDthe hub thickn ess varies from 0.2 to 0.5 and there aretour sample points on each curve in Fig. 7.It is observedthat the frequencies for all cases in Fig. 7 increasemonotonicaly as the web thickness ratio increases. 0nthe other hand，the frequencies for al cases in Fig. 7decrease as the materia1 Poisson ratio increases from0.27 to 0.33.However， the first sequence freauencvcurves with 0.27，0.30 are superposed into a samecurve，as there is little diference between them. One canfind that the first and second sequence frequenciesIncrease as the hole to diameter ratios rl increasesfrom 1/4 to 1/2，further， the second frequency increasesmore pronounced than the first frequency。

3.3 Gears with uniform thicknessFigure 8 shows the relationship between the firsttwo sequence frequencies and thickn ess ratio varyingfrom 0.2 to 0.5，which takes into account the hole todiameter ratios 1/4， 1/3reference thickn ess ratios，and 1/2.For this ran ge ofthe first sequence frequenciesincrease with respect to increasing thickn ess ratio oftheircorresponding reference plates.Figure 8(a)shows thatthe second sequence frequency increases with varyingWeb thickness ratios of gear with and web， fWeb thickness ratios of gear with and web， r弓曼3 。 ' - .- I：s-- ：s：.·： · l、. 。

：-.-.---， seq uen嘶 。、，Web thickness ratios of gear with and web， lFig·7 Relationship between web thickness ratio transverse freevibration frequency of gears with hub and web( 0， nl，2)：Ca)rl 1/4，r2 0.6；(b) 1/3，r2 ：0.6；(c) 1/2， r2 0.6from 0.2 to 0.5.Figure 8(b)shows that the secondsequence frequency Increases initially with varyingfrom 0.2 to 0.3 and then decreases in the consideredrange ofd varying from 0.3 to 0.5.Figure 8(c)shows thatthe second sequence frequency decreases with varyingfrom 0.2 to 0.5.It is observed that the frequencies for a1cases in Fig.8 decrease as malcerial Poisson ratioIncreases from 0.27 to 0.33.On the other hand. thetirst sequence frequency curves with p0.27.0.30 areN I 0 叮 I10口 H -0。国-对n 0- Q∞1 ∞ 对扫 0扛0吕 ∽ 《N墨 u口 叮∞国 0 lJ ->0血矗-E拿 0 0∞BAsI时扫 0-H0吕骞 )(《N u 。 0 H - o占 -皇 0∞矗As口盘 u暑 吕lII 《J.Cent.South Univ.(20131 20：2124-2133 2131Thickness ratio of gear with uniform thicknessThickn ess ratio ofgear with uniform thicknessThickness ratio of gear with uniform thickness。

Fig.8 Relationship between thickness ratio and transverse freevibration frequency of gears with uniform thickness(m 0， 1，2)：(a)rl 1/4；(b)，i 1/3；(c)，i 1/2superposed into a same curve，between them.One can findas there is little differencethat the first and secondsequence frequencies increase as the hole to diameterratios rl increases from 1/4 to 1/2，further，the secondfrequency increases more pronounced than the firstfrequency。

4 Modal analysis and experiment validationModal analysis determines the fundamentalvibration mode shapes and corresponding frequencies.Inthis work，Ansys codes were utilized to represent gearvibration frequency and mode shape.Twenty-noded solidelement Solid95 was used in vibration analysis of thegears.The material constants used in the calculationswere consistent with the work in Section 3.Mechanicalproperties of gear materials alloy-steel 45 listed in Table lwere employed in the preprocessor.Nodes on the insideand outside surface of the gears are set as free boundaryconditions when the FEM analysis was conducted.Thefirst tw o natural frequencies for仃ansverse free vibrationand their vibration mode shapes were extracted for thesegear configurations using the block Lanczos method。

Using the postprocessors.in bom the finite elementprogram and the experimental test software package，toexamine the mode shape results. row of Table 4 liststhe first two natural frequencies.The description of themode for three gear configurations is shown in Fig.9。

Both the theoretical and experimental modalanalyses for the model were performed for the frequencyranging from 1 kHz to 70 kHz.A comparison studybetween the analytical and finite element results wasshown in Table 4.It can be seen that the vibrationfrequencies are in good agreement(most less than 5%diference)with the three methods，thereby verifying thecorrectness ofthe analytical results。

An experimental verifcation was performed on thegears shown in Fig.1 0 in order to achieve the first twosequence frequencies for transverse free vibration。

Piezoelectric acceleration sensors CA-YD-1 25 wereconnected with electric charge amplifier YE5850B inorder to measure vibration signals of gear.The excitationwas fixed at one point，and the responses were picked upthrough symmetrical three different points all over theouter surface of the gears，therefore，acceleration sensorswere glued on the web surface with 502 adhesive pasternsymmetrically along the circumferentia1 direction.Figure1l shows the modal experiment apparatus for the gears。

To ensure that none of the resonant frequencies is missedand to get the best data quality for each mode，themeasurements were perform ed with the excitation in thetransverse direction.The transverse free vibration wasexcitated by hammering the web of gear with ham er。

To get better resolution，the whole frequency range wasdivided into seven sections r 1 0 kHz each)for thetransverse excitation. The first two sequence freevibration frequencies measured are listed in fE row ofTable 4。

One of the purposes of experimental moda1 analysisis to verify the theoretical results.The comparison of thefrequency results is listed in Table 4.知，UA and denotefrequency in Mindlin theory，Ansys softw are and modalexperiment respectively. denotes frequency could notbe traced.because of limitation in the measurementequipment，modal tests could not be conducted at suchN . ， u口 0 I - 0 -时l1 0∞I >呐口 扫 0 .I I宣 ∞- 《v 10t1口 L0- .1 - 尝 -舟.I )( Q∞.1 ∞II订.I 扫0 II吕 ∞ 《N 由 u口0t1口0毒Il0 ，j ～》0JBJ - o∞JaAsIIBJ 0Lu g吕 sH)(《J.Cent.South Univ.(20131 20：2124-2133 2133high frequencies.The experimental results were treatedas a true estimation for the analytical mode1.The relativeerrors in Tab1e 4 were calculated as - and- .The eror results in Table 4 show that theagreement among the Mindlin theory analysis，finiteelement analysis and modal experimental results is verygood.The largest eror is 5.46%.and for most of themodes.the errors are less than 5.0%。

5 Conclusions1)The proposed analytical method may be used toobtain exact vibration frequencies and mode shape forthe gears with stepped thickness. There is closeagreement between theoretical，FEM calculations andmodal experimental results.The largest eror for allfrequencies is 5.46%. and less than 5% for mostfrequencies。

2、Rim and web thickn esses have greater effect onthe natural frequencies of the gears than other parameters。

Wleb position.gear face width and material Poisson ratioalso affect structural frequencies of the gear，but theeffects of these parameters are not so greater than hole todiameter ratio。

3、The vibration analytical mode1 of the gear withsymmetrical holes designed for less mass will bedescribed in a subsequent paper.Future eforts will bedirected towards predicting the frequency and modalshape of the gear and machining tool by combining withdifferent materia1。

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