Elastic Analysis Of Load Distribution in Wide Faced Spur Gears

7/27/2019 Elastic Analysis Of Load Distribution in Wide Faced Spur Gears

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ELASTICANALYSISOFLOADDISTRMUTTON-INWIDE-FAM SPURGEARS

by

J. H. STEWARD

.1

A thesis submitted tothe University of Newcastle upon Tyneforthe Degree of Doctor of Philosophy

January 1989r, !=tIJU(Ab I LE UN I VF-Wi IIYLI HRARY----------------------------089 54049 I'S-----------------------------rho-6L 44

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ABSTRACTThe load distribution across the contact line(s) of spur gearsis essential for the gear designer to be able to accurately stressgears for a given application.Existing gear standards (eg BS 436, AGMA218 DIN 3990) use a thinslice (2D) model of the meshing gear teeth to estimate the contactline load distribution. This approach clearly fails to model properlyteeth subjected to mal-distributed loads, since the buttressing effectof adjacent tooth sections tends to flatten the load distribution.Non-linear tooth modifications such as crowning and some forms of leadcorrection are also inadequately modelled.This thesis sets out the theory for a 3D elastic model of wide-faced- spur gears that has been implemented on a micro-computer. Therequired 3D contact line influence coefficients for standard form zeromodification spur gears with 18 to 100 teeth have been determined byFinite Element analysis. These theoretical values have been comparedwith results from experiments carried out on a complete large module

(18. Omm) wide-faced spur gear. The effect of the various elementalgear errors (eg pitch, profile, lead) and profile modifications havebeen investigated using the 3D computer model; the results comparedwith results predicted by the existing gear design standards.The existing gear standards use 2D tooth compliance values up to50% less than those obtained in this work, largely due to inadequatemodelling of the gear body compliance, which is most significant ingear wheels. Comparison of 3D tooth compliance values shows a largediscrepancy between author's results again due to inadequate modelling

of the gear body.

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ACKNOWLEDGEMENTS

I would like to thank the following people for their continuedhelp in this project; Mr DA Hofmann, Director of Design Unit,University of Newcastle upon Tyne, for setting up the SERC researchproject that forms the basis of this thesis aell as for histechnical assistance; my tutor, Mr JA Pennell for his enthusiasticand highly constructive technical guidance throughout the project; MrC Woodford of The University Computing Department for his invaluablehelp in the use of the University mainframe facilities; the remainingmembersof Design Unit, especially Mr ME Normanfor his expertise inthe use of micro-computers and finally many thanks to Mrs J Macleanfor her meticulous and prompt typing of the thesis.

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DECLARATIONThis thesis consists of the original work of the author exceptwhere specific reference is made to the work of others, and has notbeen previously submitted for any other degree or qualification.

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CONTENTSCHAPTER INTRODUCTION1.1 Background1.2 Notation

CHAPTER MATHEMATICALODELLING F THE MESHINGCONDITIONS ETWEENSPURGEARS2.1 A Three Dimensional Elastic Model of Spur Gears

2.1.1 Elementary Theory of Tooth Meshing2.1.2 Load Intensity Solution at any Arbitrary Position2.1.3 Engagement Outside the Kinematically Defined Phase of Mesh2.2 Modelling of Tooth Contact Compliance and Stresses

2.2.1 Introduction2.2.2 2D Contact Compliance of an Elastic Half Space2.2.3 Finite Element Modelling of Contact Compliance and Stresses2.2.4 Comparison of Contact Deflection Results with Published Data2.2.5 Investigation of F. E. Surface Loading Discrepancy2.2.6 Modelling the Contact Compliance and Stresses in a Gear Tooth2.2.7 Contact Compliance Near the Tip/Root of the Tooth

2.3 Modelling of Tooth Centre-Line/Gear Body Compliance2.3.1 Introduction2.3.2 Two Dimensional Finite Element Modelling of Gear Teeth2.3.3 A Two Dimensional Study of Gear Body Compliance2.3.4 Three Dimensional Finite Element Model-ling of Tooth Centre-Line Compliance2.3.5 Modelling of Adjacent Tooth Compliance

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2.4 Curve Fitting of Tooth Centre-Line/Gear Body Compliance Data2.4.1 Introduction2.4.2 Curve Fitting of Loaded Tooth Deflection2.4.3 Curve Fitting of Adjacent Tooth Deflections

CHAPTER EXPERIMENTALNVESTIGATIONOF SPURGEARCOMPLIANCE3.1 Objectives of the Experiment3.2 Design of the Test Rig3.3 Method of Loading the Test Gear3.4 Measurement of the Static Deflections3.5 Measurement of the Tooth Root Strains/Stresses3.6 Calculation of Tooth Root Stresses from Surface Strains3.7 Finite Element Model of the Test Gear3.8 Comparison of Calculated and Experimental Results

3.8.1 Shaft Deflection Results3.8.2 Tooth Bending Deflection Gsults3.8.3 Tooth Contact Deflection Results3.8.4 Tensile Root Bending Stress Results3.9 Experimental Error

3.9.1 Tooth Bending Deflection Experimental Error3.9.2 Tensile Root Bending Stress Experimental Error3.10 Conclusions

CHAPTER COMPARISONF SPURGEARCOMPLIANCE ITH EXISTINGDATA4.1 Introduction4.2 Theoretical Tooth Compliance at Reference Diameter

4.2.1 Equations Used to Determine Tooth Compliance4.2.2 Comparison of Theoretical Single Tooth Stiffness Results

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4.2.3 Significance of Tooth Stiffness Values4.3 Two Dimensional Finite Element Model of Seager's Test Gear

CHAPTER EFFECTOF MANUFACTURINGRRORS NDPROFILEMODIFICATIONONLOAD DISTRIBUTION5.1 Introduction5.2 Effect of Helix Angle Errors5.3 Effect of Profile Errors5.4 Effect of Pitch Errors5.5 Calculation of Contact Stress %7'.According to BS/ISO/DIN Standards It in the Presence of Pitch Errors

5.5.1 Contact Stress at Pitch Circle5.5.2 Contact Stress away from Pitch Circle5.6 Effect of Tooth Profile Modifications

5.6.1 Tip/Root Relief5.6.2 Crowning5.6.3 End Relief

GiAPTER6 LIMITATIONSOF THE EXISTING SPURGEARANALYSISAND AREASFORFUTURESTUDY6.1 Limitations of the Existing Spur Gear Analysis6.2 Areas for Future Study

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Annr-Rinirt-rr

2.1 Program SPURDISTor the Elastic Analysis of Spur Gears2.1.3 Calculation of Loading Diameter, dy, and Initial ToothClearance, ct, Outside Theoretical Phase of Mesh2.3.2 Surface Co-ordinates of a Spur Gear with AddendumModification2.3.4 Plots of F. E. Net Tooth Centre-Line Deflection Results2.4.2 Optimising Routine for Curve Fitting of F. E. Tooth DeflectionData3.8.4 Test Gear Tooth Root Strain Gauge Results and CalculatedStresses

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CHAPTERINTRODUCTION1.1 Background

The design of gears of appropriate size and reliability for aparticular application requires a method for predicting both thesurface stresses between the meshing gear teeth and the bendingstresses in the tooth root. These are relatively easily determinedonce the load distribution across the contact line of the meshing gearteeth is known.

The methods of analysis currently used in practical gear designare usually based on one of the modern gear rating standards such asBS[B31, ISON51, DINED31, AGMAEA81.n these analyses, the contact andbending stresses are first calculated for idealised "perfect 11 gears(without errors of any form). The values so calculated are thenmodified (for "real" gears) by introducing various load distributionfactors to allow for the effec, of inaccurate manufacturing andalignment errors. All adopt a fundamentally similar approach tocalculating contact and root bending stress and so BS 436: PART 3: 1986will be looked at as an example.The mechanism for pitting failure is not fully understood, so thata rigorous failure analysis is not possible. However, Hertziancontact stress Or'h is an adequate criterion for comparing surfacefatigue strength. The contact stress is calculated by considering themating gear tooth surfaces as equivalent to two elastic cylindrical

bodies in contact with an elliptical pressure distribution firsttreated by Hertz in 1895EH81. Using the notation shown in Fig. 1.1the contact stress at the reference diameter for a pair of perfectgears is given by:

a- ____ (1.1)'1 2tr(i) V

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or a-40 11U1--7

Where the three Z factors respectively take account of tooth geometry;elastic material properties ind transverse contact ratio, whichdetermines how the load is shared between adjacent teeth in mesh. Thelength of contact line varies continuously through the mesh cycle inhelical gears and, due to elastic interaction between the teeth, theload is not distributed uniformly 'along the line of contact. It is alargely empirical factor which is intended to allow for theseeffects.

/7/

Fig. 1.1, Notation for Contact Stress: BS 436

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In BS 436 the nominal root bending stress is cal cul ated - at theoutermost point of single tooth contact, treating the gear tooth as asimple cantilever in bending subjected to the tangential component ofthe tooth load. The critical section at the tooth root is assumed tobe defined by the 30" tangent as shown in Fig 1.2. There are alsoshear and compressive loads on the tooth root, but these are ofsecondary importance and in BS 436 are allowed for in 't he stressconcentration factor. Using the notation shown in Fig. 1.2 the nominalbending stress for the cantilever at the critical section is givenby:

lzr--= LF, C-05lel% P (1.3)F6 5L C.5or-whence, introducing a stress concentatfon factor Ys, the peak notchstress at the 3(f tangent is given by: -Y4

hF

Fiq. 1.2 Notation for Bending Stress: BS 436

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3. The stiffness values employed in the 2D analyses'are generally toogreat dueto inaccurate modelling of the gear body compliance.4. The face and transverse loads distribution factors K11,,, nd 1; areassumed multiplicative but there is no evidence to suggest thattheir effects are independent.

Several attempts have been made to overcome these shortcommings.The elastic deflection of each tooth at'any point along the contactline actually depends on the loads at ALL the other points (not asassumed in this "slice model" which is only valid for the contactcompliance). It is thus possible (with the use of a, computer) todetermine the contact load distribution by solving a system of nearlinear equations of compatibility of deformation for points along thecontact lines of engaging gear teeth in which the deflection at anypoint is obtained by integrating the the effects of loads appliedanywhere along the contact line and also, to a much lesser extent, onthe adjacent meshing teeth.

Since the teeth are such a complex geometrical shape, no simpleanalytical solution has so far been developed for the contact linecompliance of any given set of gear teeth. Several approximationshave consequently been suggested for modelling the tooth compliance,generally involving splitting the total deflection into (assumedindependent) deflections. The influence coefficients of the contactline have previously been split up into Hertzian or local contactdeflections, bending/shear of the gear tooth, deflections of the gearbodies, and finally, deflections of the shafts, bearings, casingsetc.

In 1949 Weber [W4-

W71obtained expressions for the toothcontact deflection by integrating the two dimensional stress functionequations derived by Hertz. Tooth deflections were obtained byequating the strain energy due to the applied bending moment, shearforce and normal force to the work of deformation. The gear bodyconsidered by modelling it as a semi-infinite plane loaded by thetooth root bending moment, shear force and normal force, again byusing a strain energy method.KagawaEK11 put forward a theory for calculating the gear tooth

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compliance based on the formulae for a beam on an elastic foundationmodified by a torsional compliance along the tooth facewidth. Thisprovided a basis for several models developed in the 1960's.In 1963 Hayashi and Sayama[H21measured the bending deflectionsof a 240mmwide, 8. Ommmodule rack tooth. The experimental resultswere correlated with deflection equations for a thin cantilever platesimilar to those developed by Kagawa EK11. This work was continued byUmezawa [U1 - U81 who developed a finite difference solution fortapered rack shaped cantilever plates. Local contact deflections were

calculated from a 'point' load. Deflections of the gear body were notincluded in the analysis, and consequently the tooth compliancewas underestimated.Schmidt [SI, S21 used equations of the same orm as Weber andBanaschek's tooth stiffness formulae (derived from Kagawals formulae)to obtain the combinedstiffness of the meshing pinion and wheel. Theequation constants were slightly modified to allow for the additionalflexibility of the wheel only. This at least acknowledged hat thewheel has a signifTicantly greater flexibility than the pinion.However, the additional compliance was still based on theWeber/Banaschek emi-infinite plane assumption for the gear wheel bodyand was thus inaccurate. The contact deflections were treated as atwo-dimensional Hertzian compliance.Tobe[T41, treated a helical gear as a rack shaped plate encastreat the root. The deflections were determined by Finite Elementanalysis. The deformation of the root was again based on an analysisof forces and moments applied to a half -space. The contactdeflections were approximated by using Lundberg's empirical formulae[L51.Conry and Seirig [C3, C41 defined the flexibility of the tooth as atorsional and bending compliance by considering it a cantilever plate.The increased compliance near each end of the gear teeth was takeninto account by using the mirror or moment'image law[JI]. The dataobtained from Jaramillo[JI] for an infinitely long plate was used as abasis for the plate model. It can be shown that under uniform loadthe ends of the teeth then behave as though the tooth were infinitelywide, i. e. the tooth stiffness is over estimated. Deflection of the

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an elastic 'half space is valid for a gear tooth surfaceespecially at the tip and root.c. Examine the magnitude and effects of gear body deformation,

2. Incorporate 1. into a 3D mesh capable of:a. Analysing multi-tooth contact at all phases of mesh includingwhen teeth are not in contact until under load and areelastically deformed (outside the kinematically defined phase

of mesh).b. Correctly take account of arbitrary tooth and gearmisalignments (pitch, profile, lead error; profilemodifications).3. Develop and implement 1. and 2. on a micro-computer. The program isto be more easily used as an improved analysis of load distributionthan existing gear design standards and also operate as a standalone gear load distribution analysis program.4. Investigate and report the shortcommings of the existing geardesign standards (BS, DIN, ISO, AGMA).

In order to properly represent the gear body distortion, theauthor has found it necessary to model the complete gear, includingthe adjacent shaft, during the Finite Element Analysis, (Section2.3). Separate 2D studies have shown that the analytical solution forthe contact compliance, (Section 2.2), is a valid approximation to thecompression between the tooth surface and centre-line.A test rig has been constructed to load an 18 tooth pinion and theF. E. /analytical gear modelling of contact line deflections and rootstressing validated (see Chapter 3).The stiffness data has been incorporated in a micro-computerelastic analysis program for spur gears, and Chapter 5 reports aninvestigation, using this program, into the effect of manufacturingerrors and profile modifications on face and transverse loaddistribution. The results are compared with predictions based on the

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1.2 NotationA[il Weighting coefficient at the lilth gauss integration pointB Offset of tool tip origin from tool axisCa Height of tip reliefCb Height of end reliefCc Height of crowningC[l .. 71 Coefficients for curve fitting tooth centre-line deflection.FELAI Coefficients for curve fitting the 'Master curve ' deflectionCGEL. 31 Coefficients for curve fitting the 'end effect' deflectionD Contact surface diameterE Youngs Modulus (209 E3 N/mm2 or steel)FW) Centre-line deflection 'master curve' functionFn Total tooth force along base tangentFt Tangential tooth force acting along reference di ameterFte Tantential tooth force at OPSTCFp6u Equivalent mesh misalignmentG(z, zF) Bending deflection end effect functionK(z, zF) Contact line deflection influence functionKA BS436 application factorKD Relative (contact) surface diameter of curvatureKii, KFI Transverse load distribuflon factorsKup KFO Face load distribution factorsKLoad Ratio of peak specific load to nominal specific loadKV BS436 dynamic factorKtb[z, zF] Tooth bending deflection influence functionKtc[z] Tooth contact compliance -influence functionK(?- Tooth root stress influence functionSr-W BS 436 normal tooth chord thickness at critical sectionX X coordinateY Y coordinateYF BS 436 Form factorYS BS 436 stress correction factorY., Running in allowanceZ Z coordinateZB Pinion single contact factorh BS 436 elasticity factorZN BS 436 zone factorZP BS 436 helix angle factorZZ BS 436 contact ratio factor

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u Gear ratiox Addendummodification coefficientz Number of teeth, Axial coordinate of tooth deflectionzF Axial coordinate of applied point (Gauss) loadZI Iz - zFj

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additional Subscriptsa Tip diameterb Base diameterf Root diametery Meshing pointA Start of mesh, adjacent toothE End of meshNon-dimensionalAbbreviationsDTC Double tooth contactEN Engineering number for material specificationF. E. Finite elementB. E Boundary elementW Hardness, VickersIPSTC Inner point of single tooth contactLVDT Linearly variable differential transformerLOA Line of actionLOC Line of contactOPSTC Outer point of single tooth contactSAP Start of active profileSTC Single tooth contactmu Microns (1. OE-6 metres)2D Two dimensional3D Three dimensional

* ** **

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due to 'rigid' shaft bending, torsion, shear deformations.Shaft deflections only vary slightlywith change in loaddistribution and so can be considered independent of loaddistribution to a first approximation.ct Initial gap between perfect teeth due to being outside ofthe theoretical phase of engagement. This is importantonly in multi-tooth engagement but has been included herefor completeness.

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Local tooth contactcompression between surfaceand tooth centre-line

Deflection of tooth centre-linerelative to the shaft

Fig. 2.2 Components of Tooth Contact Line Deflection

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linear function of the load intensity w(zF) so that the contactcompliance is not a constant but decreases with w(zF). For a"negative" w(zF) Oe no tooth contact at zF the "compliance" isevidently infinite.Equations 2.6 and 2.7 are thus non-linear, appreciably so if theloadi ng is such that no contact occurs over part of the facewidth.This necessitates an iterative solution. An initial estimate of w(zF)is made to allow calculation of the contact compliance. Equations 2.6and 2.7 are then solved to give an improved estimate of w(zF). This isthen used to re-calculate the contact compliances. The process isrepeated until convergance is reached. Loss of contact at any point zis indicated by a negative value of w(zF). For these points, thecontact compliance is progressively increased so that the convergenceprocess is smooth.Contact is assumedto occur on up' to- three pairs, of teeth. Thi stakes account of all practical spur gears including the so-called, highcontact ratio MR) spur gears (most spur gears contact on a maximumof two pairs at any one instant). Note that tooth -deflection due toloads applied to an adjacent tooth (especially significant in gear

wheels) must be included.2.1.2 Load Intensity Solution at any Arbitrary PositionSection 2.1.1showed how the load distribution at specific points ofintegration along the contact line(s) could be obtained. Of speci alinterest are the loads at each end of the gear teeth. Onemethod ofobtaining the load intensity at any point would be to use a largenumber of Gauss points in the initial solution and then interpolatebut this would be very inefficient.

Hayashi 1H11 suggests using the relationship from eqnRearranging, we obtain the following:

AA - K. LL il I- Ss-'+ LT i- atit, Ie

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whi ch can be solved for w(zF) by iteration since Ktc is a knownfunction of w(zF). This method has proved to be unsatisfactory 'andgives unreasonable variations forloading between Gauss points.A better method is to proceed as follows:

VJLLIThe tooth contactfunction of the load atfor the loadWtCl An in

Yt t.,L LI (2.9)Kt: ELI

compliance, Ktc, at any point z is only aZ. Equation 2.9 can be solved by iteratingitial estimate of the load must be made.The tooth bending deflection function-, Ktb, is known for loadsapplied at the Gauss points but nevertheless describe-s the mid-toothdeflection at any point across the facewidth. Tooth bendingcompliance increases rapidly near the ends of the teeth and at the endof the tooth are extrapolated values so are prone to greater error.

This is why Hayashi's method giyes spurious results. The methodproposed is based on the physical* argument that even for severe mal-distribution of load the tooth bending deflection will be "smooth".The bending deflection at the point of interest is accordinglycalculated by spline fitting the (already known) Gauss-point values,NOT by calculating new values for Ktb in eqn 2.8. This interpolateddeflection is then inserted into equation 2.9 instead of that givenby the integral.2.1.3 Engagement Outside the Kinematically Defined Phase of Mesh

For perfect, infinitely stiff, gears the limit of phase ofengagement is easily determined from geometric ' considerations. Forreal elastic gears, however, contact occurs Outside the theoreticallimits of phase of mesh. An example of the effect of this new limitof phase of mesh is in load sharing between adjacent contacting teeth.The loads may or may not be shared between two teeth depending on thedifference between adjacent pitches. Consequently, peak contactstresses may at the inner point of single tooth contact due to the lowrelative radius of curvature (KD), see Fig. 2.3

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aHv [N/mn]CNIm 23KD[mm]1000110 0A

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Relative Diameter, KO

Fi_q. 2.3 Effect of Pitch Error on Peak Contact Stress

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75The -equation. of compatibility for tooth contact (equation 2-4)contains a term, ct, for the initial separation of the two perfect,unloaded teeth. Within the kinematically defined phase of contact ctis zero Oe there is no gap). Where such a tooth pair are justengaging or disengaging contact between the teeth (outside the normalphase angles) can only be between the involute flanks of one gear andthe tip (corner) of the mating "flank". Fig. 2.4 shows two gear teethcoming into mesh.Contact will be on the pinion flank along a line tangential to thepinion base diameter, dbl, and passing through the wheel flank corner,A. The loading diameter, dyl used to calculate the pinion bendingand contact compliances is thaikintersection between this base tangentline and involute profile. Appendix 2.13 gives the equation forcalculating ct and dyl. The wheel compliances are assumed to be thosecorresponting to no mal tip contact, although the actual contactcompliance will th us be under-estimated due to the "non-Hertzian"nature of corner contacts (see 2.2.7 below).

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Fi g. 2.6 Surface Pressure Distribution with Hertzian ContactCondi ti oni'-On the surface of the elastic half space,. z = 0, we obtain:

, (2.13)17- 'PO = -1

cr- +2yQc (2.14)The strains are given by:

4 e- I /Ee ik-LM - (2.15)V (O-Z 4:. CZ = ilr- I o-zl i-') - v( 14 v) r, 2 (2.16)

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Whence, inserting equations (2.10) to (2.12) into (2.16) gives:

CLZ 41 (2.17)

The contact deflection is obtained by integration, giving:FZ = Paj VI"=-0"6 ( (2.18)Wf

Substitutinq for po from (2.12) and assuming h' >> 1 (typically h, isof order 30 at the tooth, centre-line) gives:

U In (2 W) - 3/1 LiO (2.19)This expression will be valid for reasonably uniform load distributionaiay from tooth tip/root/end effects (for which the Hertzian solutionis not valid).

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Note:The deflections appear to converge to a value approximately 5%greater than , the analytical solution. Although the mesh wasprogressively refined, there is clearly still some region in whichthe F. E. approximation is inad* quate., The edge of the ellipticalpressure distribution produces very high stress gradients and seemsthe most likely source of error. This potential source of error isconsidered later in this section. The corner-node model is 20% toocompliant for h' = 30 (approximate gear tooth reference depth). Themidside node model converges to 5% error (acceptable) by the time h'= 30, provided that the loaded element has a width of 2b.The stress modelling is less accurate than the deflection modellingnear the surface as would be expected. The stresses do not convergeto the correct value at the peak stress depth (0.8b) unless three ormore 'surface' elements are used to model the applied load.

2.2.4 Comparison of Contact Deflection Results with Published DataFig. 2.8 plots a total of four different theoretical contactdeflection curves according to various published theories.Westergaard [H. 8] solves for the contact deflection by usingstress functions and obtains the same formula as developed here (validfor large h').Johnson [H. 51 quotes Boussinesq's use of potential functions toobtain the deflection of an elastic half space. He again calculatesthe strains and deflections from the known stress equations butobtains different results. Nikpar and Gohar [H. 91 obtain anotherdeflection formula from integrating L'Ure's stress equations, [L. 61.

Note:- The analytical solution developed in this work, eqn (2.19);Westegaard stress function solution and the F. E. deflection results

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all converge to the samevalue below the surface (h' > 3).- Johnson's formula and that of Nikpar and Gohar are clearlyincorrect.2.2.5 Investigation of F. E. Surface Loading Discrepancy

If the F. E. model had converged to the correct solution thesurface loading output should have exactly correspond to the appliedelliptical pressure distribution. In an attempt to compensate forthe inaccuracy the surface stresses were subtracted from the requiredsurface loading. The F. E. model was then re-analysed with this errorload applied to the surface (by itself) with the total load appliedconstrained to remain at 200NImm. Figs. 2.12 and 2.13 show theresult of superposing these "corrections" on the original results forthe surface load distribution and contact deflection. The modifiedloading has given a slightly better deflection curve but has not fullyexplained the 2.5% discrepancy. Very fine mesh modelling of the edgeof the loading surface would probably produce better results but isnot justified for the increase in accuracy.

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2.2.6 Modelling the Contact Compliance and Stresses of a Gear ToothIn sections 2.2.2 and 2.2.3, analytical and F. E. sol ut ions forHertzian contact deflection were developed. Most previous authorsassume that these results are valid for a gear tooth flank away fromtip/root/end effects. In order to verify this assumption a 2D finiteel ement study of the contact deflections in a typical gear tooth wascarried out. The tooth geometry was that used by Winter CW131or hisstrain gauge experiments.

Numberof teeth z: 14Pressure angle C4 : 20Module mn : 10.0mmAddendummodifications x: 0.0Cutter addendum hao : 1.2mnTool tip radius rao : 0.304mnTip diameter da : 160mm

Three loading diameters were considered corresponding (wherepossible) to Winter's loading diameters (Fig. 2.14). The root bendingstress results could then be compared with Winter's strain gaugedata. The three loading diameters (dy = 157.9,153.6 and 131.94mm)modelled a 'root' loading; a typical OPSTC loading and a 'tip'loading.

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supported end. Approximately half a shaft diameter has been modelledeither side of the gear to ensure that local stress concentrations atthe support points and the shaft/gear body section change can beadequately modelled by the relatively coarse mesh in these regions.The tooth centre-line compliance as defined above is assumed notto change appreciably if the shaft mounted configuration is altered(to say, an overhung configuration). The main changes will be those-due the different "shaft" deflections, not the local "gear body"deflections associated with transfer of,, the tooth load to the shaft.A representative shaft diameter, ds must however be used in the F. E.model. Fig. 2.21 shows how the chosen shaft diameter, ds, andgear reference diameter, d, varied with the number of teeth, z. Fora 4: 1 reduction and a pinion with 20 teeth, the shaft diameter wouldtypically be 16 modules. Torsional strength is proportional todiameter cubed, requiring a wheel shaft diameter of about 25 modules.The shaft diameters shown in Fig. 2.21 are thus close to those thatwould be used in practice for each tooth number.

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As shown the adjacent tooth deflections are found to be 1argel yindependent of the loading diameter, dy. By Maxwell's reciprocalth eo em. the deflection at point 2 due to a load applied at point 1equals the deflection at point 1 due to the same load applied at point2, viz.

=7ZII "i-I, (2.21)If the adjacent tooth deflection is independent of loadingdiameter it follows that the deflection on both adjacent Oepreceeding and succeeding) teeth must be equal. Fig. 2.24 showvariation of the non-dimensional adjacent tooth deflection across thefacewidth for various tooth numbers. Because these deflections arelargely due to rotation of the tooth root they do not vary much withchange in axial load position, zF. Gear wheels show the most'pronounced deflection.For a 100 tooth wheel the deflection of the adjacent teeth is 43%of the peak deflection of the loaded tooth. Because the curves aremuch 1`1 tter, (more convective, ) the total adjacent tooth deflection

due to a complete distributed load is a far greater proportion ofthe loaded tooth deflection.2.4 Curve Fitting of Tooth Centre- line/Gear Body Compliance Data2.4.1 Introducti on

The displacements due to tooth bending/gear body deformation havebeen calculated by F. E. analysis. Since these numerical values of thedi spl acements are evaluated at a finite number of pairs of (z, zF),we must devi se a means of interpolating the displacements. Anapproximate analytical equation for the deflection must be built upfor arbitrary values of (z, zF). This is because it impractical tosol ve for the deflections at arbitrary points using a mico-computerusing the 3D finiteetement anal9ses reported in previous chapters. Itis desirable to have the final load distribution program on a microfor portability and ease of use.

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Using this method, the tooth deflections at 0 and 1.5 modulesf rom th e end of the tooth were found to be 0.76 and 1.13 times thedeflections obtained by finite element analysis, indicating that themom'ent image method does not accurately represent tooth end effects.jt can be shown that it models the tooth as though it was continuous,with mirror image load distributions on successive sections of width"b".For thi s reason -an empiri cal equati on f or the end ef f ect f uncti onG(z, zF) given in equation (2.26) has been developed such that:

orzF) = CD-1cC51[ (2.27)For each tooth number analysed bly F. M. (19,25% 409 100) and eachloading height, (-1.0, -0.5,0,0.5,1.0mn), the optimum coefficientswere determined for CF[11.. CF[41 and CG[11.. CG[31 using the previouslydeveloped minimising routine.

Fi g. 2.29 shows results from the optimising program. Note thatonly the coefficients CGE11 o CG[31 are optimised, the master curvecoef fi ci ents, CFEi I havi ng previ ousl y been fi tted. The deflectionsare given in non-dimensional form by multiplying by E*mn

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The Finite Element model is simply supported at sections A&B andtorsionally restrained at section B. The extra shaft deflection inthe test gear due to M1, M2 has been taken into account when comparingthe two sets of deflection results.3.8 Comparison of Calculated and Experimental Results3.8.1 Shaft Deflection Results

As a preamble to the measurement/gear body deflections were examined.simple engineering calculations osufficiently accurate. All the testsdiameter loading (dy = 324mm), butconsidered with the distance from the50,30 and 10mm.

of tooth deflection, the shaftThis also served to verify thatf the shaft deflections werewere carried out for a referencefour separate load cases weretorque restrained end, zF = 130,The quantities of interest in this investigation were thevertical deflections of the gear shaft/gear center-line ( ie in the

direction of loading) and the shaft rotation (twist) again at theshaft centre-line. Simple beam theory predicts no horizontaldeflection. Three sources of information of these quantities areavailable viz:- simple engineering beam theory- the results of the F.E. analysis- measurements taken on the surface of the test gearExamination of the F.E. results showed that although there wasdistortion of each shaft and gear cross-section, the verticaldeflections at the ends of vertical diameters were very close to thatof the centre-line. Rotation and horizontal deflections of the shaftcentre could, likewise, be estimated with sufficient accuracy bytaking horizontal measurements at the ends of vertical diameters.

Note 1. The shaft rotated relative to the measuring frame underloading, (lifting up support M3 in Fig. 3.3) so that theshaft centre-line moved vertically at this end relativeto the frame. This required a correction to be applied

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to any measured deflections. The measured value for theshaft twist between sections A and B, has been used tocorrect the readings. Calculated values of this twistbased on the root diameter for the gear section agree towithin 5%.2. In the plane of loading, tooth 10 rotated by approximately10% less than tooth 15, demonstrating significantdistortion of the gear body. This was confirmed by theF. E. analysis3. Distortion of the shaft diameter at the measuring framesupports was of the order O. mu.

Calculation ExperimentalFig. 3.7 Determination of Shaft Deflection, J7-at any Section, z5 ---

90

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4.2.3 Significance of Tooth Stiffness ValuesDepending on the type of analysis being carried out, a differenttooth stiffness will describe most accurately the behaviour of the

qear mesh in a simplified 2D mesh model. A full 3D elastic/plasticmodel will always describe- the behaviour correctly but May beprohibitively expensive in manpower and computing time.For static stressing of gear teeth during STC contact, (e. g. OPSTC

used in BS[B3], ISO method BEI51, DINED31), by definition only onetooth pair is in contact. The single tooth stiffness must be used inthis case based on the absolute compliance of the contact line andcalculated for the correc-t-pF-ase of mesh.Note: BS, ISO, DIN use the so called mesh stiffness cr for toothstressing which is typically about 40% greater than cl; AGMA correctlyuses cl, although the AGMA standard makes only a relatively crudeapproximation for the stiffness.

For estimating load sharing during double tooth contact it is therelative deflections of the two meshing gear pairs that determinestheir behaviour. In this case either the relative tooth compliancemay be used, or the absolute compliance along with the compliance ofthe adjacent tooth. --77. -Fe- 2D analysis assumes that each tooth isonly deflected by its own tooth loads (as in the BS, ISO, DINcalculation of K14,c) then the relative tooth compliance cl must beused. The fact that most authors have implicitly calculated therelative compliance (by ignoring the gear body compliance) means thatthe stiffness values used in the standards for this purpose arereasonably correct. A better approach, however, is to recognise thatloads on both meshing tooth pairs influence the deflections of each,and use the correct absolute compliances of the 'loaded' and adjacenttooth to determine how it is shared. Static load distribution of gearsin the presence of errors is dealt with in Chapter 5.

For dynamic/vibration analysis of gears it is the inertial forcesthat are of prime concern. If the gear were to vibrate torsionally asa rigid body, use of the absolute contact line compliance would

beappropriate. However, the gear is not rigid so the effectivecompliance will depend on where the inertial mass of the gear is

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From Fig. 5.2 it is clear that the BS/ISO/DIN standardsoverestimate 'r,Load. This is probably because they use the meshstiffness cy rather than the single tooth (pair) stiffness c, inequation 5.1,5.2. At OPSTC,use of c' is clearly logical ( at leastfor spur gears) and as shown in Fig. 5.2, this gives much betteragreement with the results from SPURDIST.

Fig. 5.3 shows the effect of gear accuracy, grade and size(module) on the values of KLpad obtained from SPURDIST. Equations 5.1and 5.2 show that for geometrically similar gears, the load factor isa function of the ratio given in equation 5.4 (only c' is independentof module).For gears with a fixed permissible root bending stress, theallowable specific load, and hence the elastic deflection decreasesin proportion to the module, whereas for small module gears, themanufacturing tolerances/errors tend towards constant values. Theratio in equation 5.4 is thus always higher for small module gears,which thus have an inherently worse load factor kLoad.

5.3 Effect of Profile ErrorsAccording to BS 436 the profile tolerance limit is defined by twoparallel involutes between which the actual tooth profile must lie. Inaddition, no positive profile deviations may occur outside the centralworking third of the flank (Fig. 5.4).

DIN 3962 defines two parameters for profile tolerances; a profileform error ff (akin to a base pitch error); and a profile angle errorf (Fig 5.4). For the purposes of this investigatIon only theerfect oi a profile error is considered, the effect of pitch errorsare dealt with in the following section.

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page 38 DIN 3960

Erc111,- IrI. -. -- z)imcuon of chen food -Teit range L

Figure 40. Flank deviation$Test diagram and 3UrVOYof deviation$

Profile Tooth trace - Generator* Total profile error Ft Total alignmenterror Fp Total generant error F1* Profile angle error /me Tooth alignment arrar As G*nerant angle error INg* Profile form error It Longitudinal form error fj Generant form error fig

Test range Profile test range L. Tooth trace test range L, Generator test range Lg68 Averaging actual involute Averaging actual tooth trace Averaging actual generator

AA. AA' Nominal profiles Nominal tooth traces Nominal generatorwhich envelope the actual flank

813'. S"S, Actual Involute$ I Actual heficeg Actual generatorwhich envelope the actual flank

CV. C'C' Nominal profiles I Nominal tooth traces Nominal generatorwhich cut the actual generators or tooth traces at the starting and finishing point, respectively. of the test range

GearMeasurement 59B5DesignprofilePositive irrutA /--- o---

/'-Design profileNegativek%fB

ActLk-Arofile

Profile tolerame

Design rofileAAhialprofiletositiveUmt-,,

NegativeinitMeanworlinndelepftCentralhirdofworking epth

itI Peryussible positive eparhn fromdesign rofilewithn the centralhirdol the not exceedingdl=* given ninetatie 4Figure Sa. Tolerance zon* of tooth profile error Figure 5b. Control of positive departures from designprofile

Fig. 5.4 Definition of Profile Errors AccordLng to BS436:Pt. 2: 1970 andfff-3W

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For geometrically similar gears just having STCcontact throughoutmesh we havefpa = 2.009mn Cmul (5.8)

Consider the ISO specific load formula for a perfect longitudinalload distribution (K,, = 1-0) and STC.

L' =zz IN. (5.9)Hes. !E Ljbmsubstituting for 7j and KWA. ives,

0

Equation 5.10 reduces to:

,=i -'2 4 r--1% CNQI (5.11)where -X,. -= runn,., j

Comparedwith equation 5.8 the actual pitch errors giving STCare62% greater than those predicted by ISO. This is mainly due to theuse of the meshstiffness ce where STCoccurs.Fig. 5.9 shows the effect of fp,& on STCregion.

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5.5 Calculation of Contact Stresso, in the Presence of Pitch Errors.According to BS/I70/ I5.5.1 Contact Stress at Pitch Circle

According to BS 436: Pt-3: 1986, -ISO/DP 6336 and DIN 3990, thecontact stressr, -, at the pitch circle for a gear pair, with perfectlongitudinal load distribution is given by:

G7 (U+I)K" _Zk ZE ZEbJ, -CAWhere Z,, accounts for flank curvature and 'Z- takes account ofmaterial properties. The effect of pitch, (and'--Profile) errors isaccounted for by 7-j: 046

Effect of ZtAccording to ISO/DP 6336 accounts for the influence oftransverse contact ratio on specific surface loading.

(5.13)

K accounts for load sharing between adjacent teeth and for spurgears is:


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5.6 Tooth Profile Modifications5.6.1 Tip/Root Relief

Like pitch and profile errors, tip and root relief are 2D effectsgoverning load sharing between meshing gear pairs. BS 436: Pt. 2: 1970specifies upper limits of relief of Ca = 0.2mm applied linearly overno more than 0.6mn of the flank. BS 436: Pt. 3: 1986 recommends thefollowing:

Height of relief C., z P+ O-OLf: z/b t &. 1applied in any combination to the pinion and wheel.17i The pitch error fee may be +ve or -ve and with a specific load ofji3 z! 20mn x cos20 the net tooth tip relief is:

rn n r., LAIEquation (5.16) is valid for the 18/54 example of any module.Tip/root relief is applied to spur gears to reduce noisegenerated by the difference in mesh stiffness between STC and OTC(c'and cy). For a small pinion (less than 20 teeth) it is desirable toreduce the specific load near the root diameter loading (smallequivalent diameter of curvature). It is strongly recommended thattip/root relief be confined to the wheel tip wherever possible. Fig.5.11 shows the 10. Omnmodule 18/54 mesh with wheel tip relief.

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Note:- Crowning of a perfectly aligned gear increases the peak specificload (KLoad > 1.0).- Less accurate gears require more crowning to reduce thepeak specific load. For very inaccurate gears a load factor ofKLoad -- 1.7 appears to be the lowest attainable.- Accurate gears require only a small amount of crowning. (Hence thecross-over point on the graphs, approximately BS grade 8-9).

5.6.3 End ReliefEnd relief is a crude form of cftwning and inherently produces aworse load distribution due to the flank discontinuity. As the typeof profile modification is a function of the gear cutting, machine usedand, after running-in, the end relieved and crowned gear will possessnear identical fatigue strength, only one analysis has been carriedout to BS 436 recommendation. This is for a, BS grade 8 gear fpy

34mu; C6,= 18.5mu; t, /2 = 5.625mm. This is. shown on Fig. 5.12. ,*****

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CHAPTERLIMITATIONSOF THE EXISTING SPURGEARANALYSISANDAREAS R

6.1 Limitations of the Existing Spur Gear AnalysisA three dimensional elastic model has been developed for

calculating the load distribution, transmission error and' contactstresses in wide faced spur gears, and a computer program SPURDISThas been written to implement the model on a micro-computer. Elementalgear errors, tooth profile modificalfons, and the compliance of the two5hafts (a function of the mounting configuration) and the phase ofmeshare all input variables.The contact line compliances of the loaded teeth (includingadjacent tooth coupling compliances) are automatically calculated forwide faced solid spur gears 18


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2. Narrow faced gearsThe compliance data generated has been for wide faced spur gearswhere the effects of increased flexibility of each end of -the gearteeth do not overlap. In narrow face gears, (for example, as'used inthe automotive industry), the tooth compliance will be largelydependent on the overlapping effects of the tooth ends.

A variation of Jaramillo's moment-image method U11 could be usedto model this effect but as it has already been shown to be in errorfor wide faced gears its validity to narrow face gears must bequestioned. The model developed in this thesis will analyse narrowfaced gears but its accuracy has not been verified.

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6. Plasti'c deformation of the gear flank and tooth rootThese two effects lead to 'bedding in' of the teeth and improvedcontact conditions. Existing standards, (and the model developed inthis thesis), although allowing for 'normal' running in wear, areotherwise based on elastic theory and are unable to accommodateplastic deformation or work hardening of the gear teeth. A substantialeffort would be involved in developing a full elastic/plastic toothmeshing model. All the influence coefficients would become functionsof the applied load and load history (unless tooth root plasticity isneglected or approximated by an equivalent 'elastic' compliance).There are areas of study in gear technology that would improve gearrating far more readily than a full elastic/plastic model at present,particularly in view of the move away from the use of soft or throughhardened gears towards surface-hardened types in which plasticdeformation is unacceptable.7. Movement of the Line of Contact

Gear teeth that have profile modifications or manufacturingerrors are not tru, Iy of involute form, and thus do not make contactexactly in the theoretical base tangent plane. The same is true ofperfect involute gear teeth under load.The analyses reported only allow for gross departures from the'theoretical' planje of contact (eg during tip contact) but does notallow for the small departures caused, away from the tooth tips, byprofile errors, modifications or elastic distortions.A further step would be to redefine the contact points each timethe loads and deformations were calculated but it would be necessaryto recalculate the stiffness matrix and this would be too time

consuming.6.2 Areas for Further StudyThe shortcomings of the spur gear analysis listed in section6.1 provide the basis for much further research into the understandingof spur gear stress analysis. The existing program SPURDISTcould bedirectly linked to commercially available design standards software,

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(DUISO, DU436END, to give an improved overall gear design package.Such features as an automatic output of the optimum lead or profilecorrections are immediate possibilities. The complete synthesis ofoptimum - gear geometry and mounting configurations are furtherpossibilities, but present a formidable task which is possiblyinsoluble.The general influence coefficient theory and loaddistribution/stress. analysis approach used in this thesis can beapplied to all types of gearing:

a) Helical gears: These have an oblique line of contact dnd atransverse section that is rotated about the gear axis but are inprincipal the same as spur gears. The influence coefficients are notsymmetric about zF = b/2 due to the different stiffness of the twoangled tooth ends. The peak tooth root stress will probably be in thenormal NOT ransverse plane except near the ends of the tooth (planestress). The 20 contact deflection formula will be less accuratealong an oblique contact line with continuously changing curvature andspecific load even for perfectly aligned gears.The theory is not really applicable to crossed helicals asgeometrically they have point contact and 'are rarely used for torquetransmission purposes.

b) Worm gears: These also exhibit line contact and so are wellsuited to analysis by the influence coefficient method. Worm gearshave sliding contact and the frictional forces may have asignificantly greater effect on the tooth deflection and contactstress field. Movement of the line of contact under load is wellknown, and must be clearly be included in any analysis of wormcontacts.C) Bevel gears: Again the line contact present means that theinfluence coefficient method can be readily applied. The majority ofbevel gears have an overhung mounting configuration potentiallyproducing very bad load distributions although this is normallyimproved by crowning. An accurate study of the bearing and shaftcompliances would be a prerequisite of a meaningful 3D elastic modelof bevel gears.

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d) Spiral Bevel, Hypoid, Spiroid gears: Spiral bevel gears are tobevel gears what helicals are to spurs, W'I Ilk &W S (ICNO"eds and quieter operation at 'higher pitch line velocities due?oa

more constant length of line of contact. In hypoid and spiroid gearsthe shaft axes are offset as the pinion starts to resemble a taperedworm gear and sliding contact occurs. The influence coefficientapproach is applicable to all three gear types with the potentialproblem areas analagous to bevels and worms respectively.

000

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: Part 3: 1986 Method for Calculation of Contact and RootBending Stress Limitations for Metallic Involute GearsIBU Buckingham E,. : "Analytical-Mechanics of Gears"McGraw-Hill 1949EB51 British Steel Corperation : "Iron and Steel Specifications",, 1986IC11 Cardou A., Tordion G.V., : "Calculation of Spur Gear ToothFLexibility bythe Complex Potential Method"Trans. ASME, Vol. 107, Mar. 85[C21 Chabert G., :" Evaluation of Stresses and Deflection of SpurGear Teeth Under Strain"C. E.T. I. M. Senlis. Fr. ASME72-PTG-27 Oct. 1972, p. 16[C31 Conry T. F., Seireg A., : "A Mathematic Programming Method forthe Design of Elastic Bodies in Contact"Jnl. App. Mechs., Trans ASME93E(1), 1971, pp. 387-392EC41 Conry T. F., Seireg A., : "A Mathematical Programming Techniquefor the Evaluation of Load Distribution and OptimalModification in Gear Systems"Trans. ASME, Jnl. Engg. for Ind. (Nov. 1973)EC51 Cornell R.W,,,, : "Compliance and Stress Sensitivity of Spur GearTeeth"ASME, Jnl. Mech. Des., Vol. 103, April 1981[C61 Case J., Chilver A. H., : "Strength of Materials and Structures"Edward Arnold 1978ID11 Davies W.J., "Some Design Considerations AffectingPerformance and Reliability of High-Duty Spur and HelicalGearing for Aircraft"Proc. Int. Conf. Gearing, I. Mech. E. London (1958), pp. 82-90[D21 Davis A. W., : "Marine Reduction Gearing"Proc. Inst. Mech. E. 170 (16) 1956, pp. 477-498


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[D31 DIN 3990 Calculation of Load Capacity of Spur, Helical andBevel Gears[041 Dolan T. J., BroghammerE.L., "A Photoelastic Study in Gear

-Tooth Fillets",Univ. of Illonois Engrg Expt. Station Bulletin No. 335,1942ED51 Oubbel, : "Tascenbuch fur den MaskinenbaullSpringer"Verlag, 1974ED61 DIN 3961 : "Accuracy of Cylindrical Gears: General Bases" 1978CD71 DU436 Computer program implementing BS436 gear stress analysisstandard [B31. ,.,,esign Unit, University of Newcastle uponTyneED81 Dobrovolsky V. : "Machine Elements"Mir Publishers, Moscow 1977[D91 DIN 3960 :,!, Concepts and Parameters Associated With CylindricalGears and Gear Pairs With Involute Teeth" 1980ED10] DIN 3962 : "Accuracy of Cylindrical Gears: Tolerances forIndividual Errors" 1978[Ell Elkholy A.H., : -"Tooth Load-,Sharing in High-Contact Ratio SpurGears"Pratt & Whitney Canada Inc., Ontario[FI1 Fujio H., Aida T., "Varying Thickness Plate (Greens and

F. E. Analysis)"Preprint JSMENo. 710-6 (1968-4), p. 113Preprint JSMENo. 192 (1971-4), p. 161, (in Jap. )EF21 Fujita K., : Trans. JSME(1960-3) 26 (163) p. 430, (in Jap)EF31 Fujita K., Obata F., Miyanishi K., : "Gear Tooth StressCalculation Method for Heavily Crowned Gears"


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Bull. JSME(1974) 17 (104), pp. 264-272[F41 Fujita K., : "Heavily Crowned Gears"Machinist (In Jap. ), (1967-12) 11 (12), p. 24[F51 Furrow R.W., Mabie H. H., "The Measurement of StaticDeformation of-Spur Gear Teeth"Jnl. Mechanisms, 11970, pp. 147-168[G11 GromannM.B., : "Load Distribution in Gear Teeth"Russ. Engg. Jnl. 1967,5,

j,pp. 44-47

EG21 Geick K., : "Technical Formulae"Geick-Verlag 1982[H11 Hayashi K., : "Load Distribution on Contact Line of HelicalGear Teeth", (Part 1: Fundamental Concept)1963 Bull. JSME6 (22), pp. 336-343EH21 Hayashi K., SayamaT., : "Load Distribution on the Contact Lineof Helical Gear Teeth". (Part 2: Gears of Large

Facewidth)1963 Bull. JSME6 (22), pp. 344-353[H31 Hayashi K., : "Analysis of Loading in Cylindrical Gears"Proc.,,,JSMESemi Internat. Symp., Tokyo 1967, pp. 181-188EH41 Hayashi K., : Trans. JSME(1962-3) 28 (193), p. 1093 (In Jap. )[H51 Hayashi K., SayamaT., Trans. JSME(1962-3) 28 (193), p. 1102(In Jap. )[H61 Hayashi K., Sayama T., "Load Distribution in Marine ReductionGears"Bull. JSME(1962) 5 (18)p pp. 351-359[H71 Holl D.L., : "Cantilever Plate with Concentrated Edge Load"Jnl. Appl. Mechs., Trans. ASME59 (1937), pp. A8-10[H81 Hertz K., : "Gessamelte Werke, " vol. I, Leipzig, 1895.


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[H91 Hirt M.C. O., : "Stresses in-Spur Gear Teeth and Their Strengthas Influenced by Fillet Radius"English translation of Phd Thesis by John Maddock, 1976Published by AGMAEH101 Heywood R.B., : "Tensile Fillet Stresses in Loaded Projections"Trans. Inst. Mech. Engrs., 1948[Hlll Huber M. T., Fuchs S., : "Spannungsverteilung ben Beruhrung"Physik. Zeitschr XV, 1914CH121 Hofmann D.A., "The Importance of the New Standards BS436(1986) and DIN 3990 (1986) for Gear Design in the UK"TransIMarE(TM), Vol. 99, Paper 14,1987[Ill Inoue K., Tobe T., : "Longitudinal Load Distribution of HelicalGears"Tohoku University, Sendai, JapanE121 Ishikawa J., : Trans. JSME(1951) 17 (59) p. 103 (In Jap. )[131 Ishikawa J., : Bull. T. I. T. (3) 1957 (Tokyo Inst. Technol. )E141 Ishikawa J., On the elastic problem of cantilever subject tocombined bending and twisting"15 (90) 1972E151 ISO/DP 6336 Gear Loading PARTS , II, III ANDIV.[Jll Jaramillo T, J., "Deflections and Moments due to aConcentrated Load on a Cantilever Plate of Infinite Length"

Jnl. Appl. Mechs. 17,1950 Trans. ASME72[J21 Johnson K.L., : 11100Years of Hertz Contact"Proc. Inst. Mech. Engrs., Vol 196,1982EKll Kagawa T., : "Deflection and Momentdue to a Concentrated EdgeLoad on a Cantilever Plate of Finite Length"Proc. llth Jap. Nat. Congress of Applied Mechanics'1961


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Jnl. Appl. Mechs. Jrans'ASME 67, (1945), pp. 78-86EP21 Premilhat A., Tordion G.V., : "An Improved Determination of theElastic Compliance of a Spur Gear Tooth Acted. on by aConcentrated Load"ASME, Jnl. Eng. Ind. (1974)EP31 PAFEC "Data Preparation User Manual""Theory and Results"PAFEC1982[R11 Rao A. C., : "On Load Sharing by Pairs of Gear Teeth"J. Inst. Eng. (India) 58 Pt. ME (1978) pp. 265-267ER21 Rademacher J., : "Ermittlung von Lastverteilungsfaktoren fuerStirnradgetriebe"Industrie-Anzeiger 69 (17), (1967) pp. 31-34[R31 Richardson H. H., : "Static and Dynamic Load, Stress andDeflection Cycles in Spur Gear Systems"

-ScD. Thesis, MIT (1958)ER41 Roark R.J., Young W.C., : "Formulas for Stress and Strain"McGraw-Hill 1975ES11 Schmidt G., % : "Berechnu*ng der Walzpressung SchragverzahnterStirnrader unter Berucksichtigung der Lastverteilung"Diss. T. U. Munchen 1973ES21 Schmidt G.R., "Optimum Tooth Profile Correction of HelicalGears"ASMEPaper 80-C2/Det-110[S31 Seager D.L., : "SomeElastic Effects in Helical Gear Teeth"PhD Thesis Cambridge Univ. (1976)ES41 Seager D.L., "Tooth Loading and Static Behaviour of HelicalGears"Trans. ASLE 13 (1), 1970, pp. 66-77


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P11 Umezawa K., "Meshing- Test on Helical Gears under LoadTransmission"3rd Report, Bull. JSME (1974 Oct. ), 17 (112) pp. 1348-1355

EP21 UmezawaK., Bull., JSME (1969-70) 12 (53) p. 12041U31 UmezawaK., Bull.. -JSME(1971-1) 15 (79) p. 116IU41 UmezawaK., Bull. JSME(1972-12) 15 (90), p. 1632[U51 Umezawa K., "The meshing test on Helical Gears under LoadTransmission" 2nd Report :- "The Approximate Formula for

Bending Moment Distribution, of Gear Teeth"(1972-3) Bull. JSME, 6 (92) pp. 407-413EU61 UmezawaK., Ishikawa, J., "Deflection due to Contact BetweenGear Teeth With Finite Width"Bull. JSME(1972-73) 16 (97), pp. 1085-1093EU71 UmezawaK., : "Thick Plate Theory,, Finite Length"Trans. JSME(1969-2) 35 (270) p. 423 (In Jap. )EU81 UmezawaK., : "Practical Equation for Deflections and Stresses"Trans. JSME(1970-71) 36 (288) p. 1385, (In Jap. )1V11 Van Zandt R. P. : "Beam Strength of Spur Gears"Quart. *Trans. SAE (1952), pp. 352-357EV21 Vathayanon B., : Diss. (1965) Univ. Illinois, Urbanar, Ill.EV31 Vedmar L., : "On the Design of External Involute Helical Gears"Lund Inst. Tech. (Sweden)[W11 Walker H., : "Gear Tooth Deflection and Profile Modification"The Engineer 166 (1938)[W21 Walker H., : "Helical Gears"The Engineer 1946 (i) July 12, pp. 24-26 182 (ii) July19, pp. 46-48 (iii) July 26, pp. 70-71


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APPENDIX2.1

PROGRAMPURDIST,ORTHE ELASTICANALYSIS OF SPURGEARSAppendix 2.1 explains the computer program SPURDIST fordetermining the load distribution, contact stress and transmissionerror in any pair of meshing real spur gears. The elastic equationsused are set out in Section 2.1.Fig. 1 shows the menu hierarchy and Fig. 2a summaryof the maincommands. The program defines a master tooth (tooth 3) which is usedfor specifying the phase of mesh (see Fig. 4 procedure PHASE). Upto two teeth either side of the master tooth may be in mesh at any onetime. A tooth is assumed to be potentially in mesh if the tooth tipis within 0.25 base pitches of the 'mating' tooth surface (ie touchingwhen in,side the geometrically defined phase of mesh). Elemental gearerror datafiles are manipulated using procedure ERROR. Errordatafiles are permanently stored on disk.The solution is obtained and output by calling procedure ANALYSE(Fig. 4) which calls CALCMATFig. 5) and LOADDIST Fig. 6). CALCMATgenerates the tooth compliance, matrix (not including the loaddependent contact compliance). ' The tooth centre-line deflectioncoefficients C[l.. 71 (for 10 to 1,00 teeth at = -2, -1,0,1,2modules) are permanently stored on datafile COEFDATA. Thecoefficients have been calculated by fitting natural splines to theF. E. result coefficients in two 'directions'LOADDIST completes the compliance matrix, by adding in theestimated contact deflection (Function CONDEFN), inverting thecompliance matrix and solving for the load distribution. Each contactline has 8 gauss points (2 point integration over successive quartersof the facewidth) giving a 25 x 25 augmented compliance matrix for 3teeth in contact. The solution is iterated until all gauss pointloads have converged to values greater than -wbm/1000. When negativeloads are predicted (i. e. teeth are separated), the contact complianceis multiplied by 10 at each iteration. This reduces the estimatedgauss load in a smooth process until it approximates to zero or


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C(earE(rror

Commands for Spur Gear'Load DistributionProgram, Version 04.26-10-87

Prompts for gear tooth geometry independent of phase of meshReturns the error file menu for handling the elemental geartooqh errors and pr,ofi'le modificaitons (pitch, profile,barrelling, crowning, tip, root and end relief)Wake : Creates-a new error datafileMist : Lists an existing error'datafil'e on the screenVill : Destroys. an error datafileP(rint : Lists an existing error datafile on the printerS(elect Assigns an error dataf ile to each tooth of the 5

possible engaging tooth meshes (maximum of 3 at anyphase of mesh) --P(inion : prompts for pinion tooth number 1-5Wheel : prompts for wheel tooth number 1-5

Q(uit Returns to main program menuS(haft

P(hase

Prompts for shaft deflections at the gauss integration pointsPrompts for the phase of mesh and displays which tooth pairsare potentially in mesh

A(nalyse If all mandatory input data is present the mesh is analysed andthe results output to the printer

Uncrement A(nalyse is performed for seven phases of mesh corresponding topoints A, B, C, D and the 3 midpoints on the roll angle graph

(Uit Exits the program

Fig. 2


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STARTI

Print the A, B, C, Dphase anglesF-Input v3

N

Calc adjacent toothphase angles tof: i4L-L-

Calc initial toothclearance ct for 5tooth meshesPROCEDURECLEARANCE

Print Yqct,-d -doryl, y2

Set order ofcompliance matrixmat-ord

Cal CL boYBal c ;' Sy[ : :s

ylf r =th-meshesEND

for calculating the tooth clearance ct dealt with in Appendix 2.1.3

Fig 4. Spur Gear Load Distribution Progra Procedure PHASE


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START1

FOR k: = 1 TO numzcon DoFOR i: = 1 TO numcoeff DOCalc each bending defncoefficient ci atphase yy us ngPROCEDURENATSPLINEI

Calc bendin influencefunctions & loadaNtrix totkompliance

con N

FOR i: =1 TO 4 DOCalc adj tooth coeffsacL as f(l/z) using

Copy tooth 1 compliancesub-matrix intotooth 2 sub-matrix

Calc adj influencefuctions & load off-*diagonal complianceI

sub-matrices

I

(Fig2.3 shows 3 tooth zconmesh compliance matrix)


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converges too a stable positive load. The tooth centre-line deflectionis fitted with natural splines to obtain the end tooth centre-linedeflections. The specific load is calculated at the end of eachmeshing tooth from equation 2.8.

** ***

*4


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Geometry of Tooth Clearance, Lt, outside the Geometrically DefinedTh-ase of ge-sh--


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Nowconsider the gear pair:X 11 Son (A2.1.3.9)

j4t (A2.1.3.10)2

(A2.1.3.11)

AiN = V (A2.1.3.12)4-a; ( ) (A2.1.3.13)C,0.7 (A2.1.3.14)

where suffix lyp2l refers to the tip of pinion relative to the wheelaxis.The equation of the tangent to the wheel base circle can now becalculated.


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ak involute .gent to wheel base circle

Fig 2. Wheel Contact Point Outside the Geometrically Defined Phaase -ofFfitshReferring to Fig. 2 we have

195'.e) (A2.1.3.15)4mThe equation of tangent line 2 is:

(A2.1.3.16)Transforming this to polar co-ordinates gives:

n112. = ALZ (A2.1.3. '1.7)LOSE Gs x -d2AesA xja Iz


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For the wheel involute profile, line 1:

A2.1.3.18)

+ Inv C'L - (A2.1.3.19)

The wheel mesh diameter, dy2, can be solved iteratively. Thetooth clearance, ct, is the distance along the assumed base tangentgiven by:

,cf: (A2.1.3.20)

*****


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*du

Fig. 2 Generating Rack Geometry and Trochoid/Gear AxesNow consider the generating rack given in Fig. 2. The origin ofthe tool tip radius is offset by:

kaAa - rao) 4-AAi %Z rAO (A2.3.2.10)

The co-ordinates of the trochoid are defined relative to thetooth centre-line which is offset from the trochoid axis by where:

(A2.3.2.11)r

IA-


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The ' polar co-ordinates of the trochoid are given by (9 and rf(equation A2.3.2.10) where:

8'": W. 3.2.12)

Rectangular co-ordinates are obtained using equation A2.3.2.6.Note that the trochoid co-ordinat6s are not obtained directly from thetrochoid radius rf but by inputting rt, the radius of the tool ti p,origin.Calculation of Midside Node Co-ordinates

If midside node co-ordinatesare omitted from the F. E. inputthe elements are assumed to havestraight sides, (shown dotted).To ensure a correct trochoidprofile the midside node co-ordinates must be calculated.The midpoint is defined as apoint on the trochoid of equaldistance from points 1 and 2.1,2 Corner NiM Midside Noi

Equations given for thetrochoid are not closed form sothe midpoint is determined byiteration between the two calcu-lated corner node co-ordinates.Fig. 3 F. E. Midside Node Calculation

The flow diagram for the iteration is given in Fig. 4. Theprogram successively divides the trochoid in half and detects whichsection the midpoint must lie until further subdivision produces noimprovement in midpoint co-ordinates.*****


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Fig 4. Trochoid Midpoint Coordinate Routine


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c c c c cE E E E EIn Ln In C) C)C%j N- cv In C)

-* C-i -dIt It 11LL- U- LL- U- LL-44 t4 kt m qCD a) 0 a) 00 o 0 L) 0c c c c cco (a m m co41 4.0 41 41 41to co 0 0 0)

C3 C3 C3 C3 C3a) 0) m a) mc c c c c

-0 -0 -0 -0 -0al m m (0 coo o o 0 0

(a (1) co m (13x x x x x

.4 0 0 U'l

vivi

0vi

Ln00m

tAa)ca4JU)

Ln

0

LAci

ciC Ln Co Ln CP ulvi 44 Pei Pei ej ci


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C, c c c c9 r= E E f=-Ln Ln Ln C) CDlli Lrl C)

-* cl 1611 it itU- LL- U- LL- LL-w " R tj R(1) Q (D (1) (1)0 L) 0 U uc c c c c(a m ca m m4-0 4j 4J 41 41co (1) V) U) (PFE C3 cl C3 C3a) cr) CD m CY)c c c c c(a ea (a I'D m0- 0 0 0 0

m eox x x x x

P. m 0

kml

cl

Lr.4J00

Pei0

LA

C! coCYLrl.Z

0

LndC

0o Co N o Ln 4? r" %


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.c c c c cS r= a E ELn Ln C) nCj Ln C)CD CS C, 611 if 11 11LL- L U- U- LL.

CD 0 (D CD 00 0 0 0 0c c c C: c:1 m (a co (a co4-P p 41 41 +jw CO (D (n COC3 C3 C3 M 0a) (7) cr cr) mC c c c c0 0 0 0 0

-i -i -i -i -i-0 --d --1 -. 0 __ox x x x x

tr.

o

LrI

4-a

Ul

0

tncs

ci 0ci,rIIIIIIIIC) Ln ul C5 Ln 0 in Lnt1i 4 r4i Pei (I; 0i -d


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Ln Ln Ln CD C)C%j P- CNJLn C)6 C:, cli 6-LL- LL- LL. LL. LL-w

" ?4 "N

a) (D a (1) a)C:ca co co co CD41W 4JCO 41W +jW 4JO)

C3 C3m CY) m a)ccC, c

M -0 -0 -0(5 m (in (00000--f --0 --0 --0 --0('0 (U (U m (0xxxxx-IC -< -< -< --r-4 D> Ei e

.. lw Etr.

LM --0

.-. (1)

Cl co rl- 10 Ln 1.


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c c c cE E E E aIn In In 0 clC-, F, C'i In C)

C; C; CIf It ItL4 14(D a) 0 (1) (1)0 0 0 0 ()c c c c cco (a co (a OfA.P 41 4-P 41 4J

-10 W 19

-0 -WM 1=1 C3 M MCY) 07 07 M Mc c c c ccc (a (a cc co0 0 0 0 a

x x x x Y.

0- El 0Ln-dC0

Lf!ul

C!

Ln

%r

r-Ic

Ln00mc0Ln

cC! coJtoLn is

C!0cld

.C1

0

-0

c

L

+000h-4.1tuZ

(U0

LULi-

tz0 Ln 0 In 0 Ln .1LA 4 Pei pli ri ciulu*3* '14,9ui2l, 14110eulle-nueo Ll400J. 29N


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c cr= , r= F- 2 r=tn tn c> ciN Ln 0

LL- LL- U- LL-Kp , hi hi m t4C) 0l 0 (D 0t) ,0 0 u ur -C C C C(0 (0 , (1) , (0 0

c2 IE c3M M C MC C C C

M M0 0 0 0

x x x x

IdLrItiri1LALn

Lr'iC3

CY)c0

uclfli 4J(j)

Lnci00


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APPENDIX2.4.2OPTIMISATION ROUTINE FOR CURVE FITTING OF FINITE ELEMENTTOOTHDEFLECTION-DATA

To minimise the error between the F. E. tooth deflection resultsand the mathematical fitting function the coefficients of F( ) andG( ) were optimised by computer. The method used was a modifiedgradient method.

Elastic Analysis Of Load Distribution in Wide Faced Spur Gears

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