Generation and Meshing Analysis of a New Type of Double ...

Research ArticleGeneration and Meshing Analysis of a New Type of DoubleHelical Gear Transmission

Dong Liang 12 Tianhong Luo1 and Rulong Tan3

1School of Mechanotronics amp Vehicle Engineering Chongqing Jiaotong University No 66 Xuefu Street Nanrsquoan DistrictChongqing 400074 China2The Key Laboratory of Expressway Construction Machinery of Shanxi Province Changrsquoan UniversityMiddle-Section of Nanrsquoer Huan Road Xirsquoan Shanxi 710064 China3The State Key Laboratory of Mechanical Transmission Chongqing University No 174 Shazheng Street Shapingba DistrictChongqing 400030 China

Correspondence should be addressed to Dong Liang cqjtuliangdong me163com

Received 6 March 2018 Revised 3 June 2018 Accepted 11 June 2018 Published 13 September 2018

Academic Editor Francesco Franco

Copyright copy 2018 Dong Liang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A study on the double helical gear transmission with curve element constructed tooth pairs is carried out in this paper Generationand mathematical model of tooth profiles are proposed based on the geometric relationship Tooth profiles equations are derivedconsidering the developed equidistance-enveloping approach Design of tooth profiles with point contact is conducted andnumerical example is illustrated And the solid models of double helical gear pair with curve element constructed tooth pairsare established Computerized design and meshing simulation are also put forward Furthermore stress analysis of the gear pair isconducted and gear prototypes are manufactured using CNC machining technology Further tooth contact analysis dynamic andexperimental studies on transmission properties of gear prototypes will be carried out

1 Introduction

New types of gear drive are currently research focus forhigh transmission performance requirements Various the-ories design and analysis approaches were provided byresearchers Utilizing the face-milling cutter Fuentes et al[1] put forward general geometric descriptions of circular arcgears with curvilinear teeth Generation process of gear pairwas given Meshing simulation and tooth contact analysiswere also carried out Considering the intermediary helicoidas basic generation using the established circular arc profilein axle plane Dudas [2] presented the mathematical modeland motion simulation of a worm gear drive Based onthe theory of gear kinematics Zimmer et al [3] derived amathematical framework to calculate geometric relationshipsof arbitrary gear types and the general algorithms wereillustrated Kahraman [4] developed a family of torsionaldynamic models of compound gear sets to predict the freevibration characteristics under different kinematic configura-tions resulting in different speed ratios The compound gear

sets considered consist of two planes of single- or double-planet gear sets connected by a straight long planet Kahra-man [5] also provided a model to predict load-dependent(mechanical) power losses of spur gear pairs based on anelastohydrodynamic lubrication (EHL) model Simon [6] putforward a new type of cylindrical worm gear driveThewormis ground by a grinding wheel of a double arc profile and theobtained worm profile is concave The teeth of the gear areprocessed by a hob whose generator surface corresponds tothe worm surface Simon [7] also developed the design andmanufacturing methods of the hob for processing a wormgear with circular axial profile Bahk et al [8] investigatedthe impact of tooth profile modification on spur planetarygear vibrationThe analytical model was proposed to capturethe excitation from tooth profile modifications at the sun-planet and ring-planet meshes Song et al [9] designed theconjugated straight-line internal gear pairs for fluid powergear machines The conjugated straight-line internal gearpair includes a pinion with straight-line profile and aninternal gear with profile conjugated to the pinion profile

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 1539849 10 pageshttpsdoiorg10115520181539849

2 Mathematical Problems in Engineering

Huang et al [10] proposed an internal mesh planetary gearwith small tooth number difference (PGSTD) and dynamiccharacteristics analysis was carried out Lin [11] provided anewnon-circular bevel gear based on the combination of cammechanism and non-circular bevel gear

For double helical gear drive it has currently attractedmore attentions in the field of low speed and heavy loadmachinery Sondkar et al [12] established the dynamicmodelof double helical planetary gear pair A linear time-invariantform was developed and further studies on dynamic charac-teristics were carried out Zhang et al [13] investigated thecomputerized design and simulation of meshing of modifieddouble circular arc helical gears by tooth end relief withhelix The proposed theory was illustrated with numericalexamples which confirm the advantages of the gear drivesof the modified geometry According to the characteristicof tooth profile modification Wang et al [14] presented themethod of three-segmentmodification for pinion profileTheoptimization results of tooth profile and axial modificationwithout consideration of axes error can reduce the loadedtransmission error Kawasaki et al [15 16] proposed a man-ufacturing method of double helical gears using a multiaxiscontrol and multitasking machine tool The pitch errorstooth thickness runout profile lead and surface roughnessof manufactured double helical gears were also measuredOtherwise the relationship between the tool wear and lifetime of the end mill was made clear And the manufacturingmethod was applied to the gears for a double helical gearpump

To obtain the better transmission performance and tomeet the higher strength requirements a new double helicalgear transmissionwith curve element constructed tooth pairsis studied in this paper Generation principle and mathe-matical model of this gear drive are provided Numericalexample is illustrated in terms of the established tooth profilesform Computerized design and meshing simulation are alsoput forward Furthermore stress analysis of the gear pair isconducted and gear prototypes are manufactured using CNCmachining technology

2 Generation and Mathematical Model ofTooth Profiles

21 Contact Principle Contact curves of tooth profiles aredefined with three parts spatial helix curve D1 spatial helixcurve D2 and circular arc curve D3 The spatial helix curve D1and the spatial helix curve D2 are symmetric curves relative tothe middle section of tooth width The circular arc curve D3is the smooth transition curve connecting the curves D1 andD2

The spatial helix curvesD1 andD2 are expressed as follows

Γ119894

119909119894 = 119877119903 cos 120579119903119894119910119894 = 119877119903 sin 120579119903119894119911119894 = 119901119903120579119903119894

(119894 = 1 2) (1)

where Rr is cylinder helix radius 120579ri (i=1 2) is spatial helixcurve parameter and pr is helix parameter

A

O

Spatial helix curve Γ1

Spatial helix curve Γ2T2T1

K1

K2

N1N2

Figure 1 Geometric relationship

Transition circular arc curve D3 is derived by mathematicapproach and its location plane needs to be first determinedAs shown in Figure 1 T1 and T2 are the tangent at points K1andK2 on the spatial helix curves D1 and D2 respectivelyThetwo lines are given to the A point The normal lines at pointsK1 andK2 are expressed usingN1N2The two lines are givento the O point

According to (1) its derivative can be calculated as

1199091015840119894 = minus119877119903 sin 1205791199031198941199101015840119894 = 119877119903 cos 1205791199031198941199111015840119894 = 119901119903

(119894 = 1 2)

(2)

and at point K1 we have the coordinate results (xk1 yk1 zk1)and (11990910158401198961 11991010158401198961 11991110158401198961) So the equation of tangent T1 is shownas

119909 minus 119909119896111990910158401198961

= 119910 minus 119910119896111991010158401198961= 119911 minus 119911119896111991110158401198961 (3)

Similarly the equation of tangent T2 is expressed as

119909 minus 119909119896211990910158401198962

= 119910 minus 119910119896211991010158401198962= 119911 minus 119911119896211991110158401198962 (4)

With simultaneous equations (3) and (4) the value ofpoint A can be obtained Obviously normal lines N1 N2 aredetermined based on the above tangent expressions

Normal line N1

(119909 minus 1199091198961) 11990910158401198961 + (119910 minus 1199101198961) 11991010158401198961 + (119911 minus 1199111198961) 11991110158401198961 = 0 (5)

Normal line N2


In simultaneous equations (5) and (6) the family of point Ocan be calculated and the point meeting the actual require-ments will be further determined by the following conditions

A plane can be generated at three points in space Asshown in Figure 2 suppose that the intersection point O isthe circle center and the circular arc curve passing points K1and K2 is drawn and it should satisfy the constant tangentsT1 and T2

Mathematical Problems in Engineering 3

A

O

Circular arc curve

T1

T2

K2

K1

Figure 2 Determination of circular arc curve

The normal vector at the location plane of circulararc curve can be represented using the developed tangentconditions as

1199051 = (119909119860 minus 1199091198961) 119894 + (119910119860 minus 1199101198961) 119895 + (119911119860 minus 1199111198961) 1198961199052 = (119909119860 minus 1199091198962) 119894 + (119910119860 minus 1199101198962) 119895 + (119911119860 minus 1199111198962) 119896119899 = 1199051 times 1199052

(7)

Using the normal condition it also can be written as


(8)

Considering that the results of (7) and (8) are simplifiedto unit vectors it should meet the expression 119899 ∙ 1198991015840 = 1Then the point O (xO yO zO) can be obtained in terms ofaforementioned conclusion

The location plane of circular arc curve is provided as

119899 [(119909 minus 1199091198961) 119894 + (119910 minus 1199101198961) 119895 + (119911 minus 1199111198961) 119896] = 0 (9)

We can find that the angle between plane y1O1z1 and thelocation plane yOz of circular arc curve is 120593 as displayed in

Figure 3 It can be calculated through the normal vectors ofthe plane y1O1z1 and the plane yOz and it has

cos120593 = 119899 ∙119873|119899| ∙ |119873| (10)

The circular arc curve in coordinate system S(O-xyz) isexpressed as

119909 = 0119910 = minus119903 cos 120575119911 = 119903 sin 120575

(11)

where r is the radius of circular arc and 120575 is in the rangeminus arctan(11991111989621199101198962) le 120575 le arctan(11991111989621199101198962) Further based onthe coordinate transformation the equation of circular arccurve D3 is written as

1199093 = 119903 cos 120575 sin120593 + 1199091198741199103 = minus119903 cos 120575 cos120593 + 1199101198741199113 = 119903 sin 120575

(12)

According to the developed equidistance-envelopingapproach [17 18] the tubular meshing tooth profiles aregenerated and general equations of tooth profiles for spatialhelix curve D1 spatial helix curve D2 and circular arc curveD3 are

Σ119894

119909Σ119894 = 119877119903 cos 120579119903119894 + ℎ (11989901198991199091 + cos1205931 cos1205721)119910Σ119894 = 119877119903 sin 120579119903119894 + ℎ (11989901198991199101 + cos1205931 sin1205721)119911Σ119894 = 119901119903120579119903119894 + ℎ (11989901198991199111 + sin1205931)(minus119877119903 sin 120579119903119894 + ℎ11989901198991199091

1015840) cos1205721 + (119877119903 cos 120579119903119894 + ℎ119899011989911991011015840) sin1205721

+ (119901119903 + ℎ119899011989911991111015840) tan1205931 = 0

(119894 = 1 2)

(13)

and

Σ3

119909Σ3 = 119903 cos 120575 sin120593 + 119909119874 + ℎ (11989901198991199091 + cos1205931 cos1205721)119910Σ3 = minus119903 cos 120575 cos120593 + 119910119874 + ℎ (11989901198991199101 + cos1205931 sin1205721)119911Σ3 = 119903 sin 120575 + ℎ (11989901198991199111 + sin1205931)(minus119903 sin 120575 sin120593 + ℎ11989901198991199091

1015840) cos1205721 + (119903 sin 120575 cos120593 + ℎ119899011989911991011015840) sin1205721

+119903 cos 120575 tan1205931 = 0

(14)

where h is the given equidistance with respect to the contactcurves 11989901198991199091 11989901198991199101 11989901198991199111 are the unit components of normal

vector And sphere expression in (13) and (14) is written using(ℎ cos1205931 cos1205721 ℎ cos1205931 sin1205721 ℎ sin1205931)


y

x

zO

K1

K2

z3

y3

x3

O1

(a) Position relationship between plane y1O1z1 and planeyOz

y

z

O

r

K1

K2

(b) Circular arc curve on plane yOz

Figure 3 Equation derivation of circular arc curve



50

40

30

20

10

0

minus10

minus20

minus30

minus40

minus50 7580

8590

95

Axi

s z (m

m)

Axis y (mm)

Axis x (mm)

Figure 4 Engagement of spatial helix conjugate curves

Parameters such as 119877119903 = 12119898119898 1198851 = 6 1198852 = 30 119886 =72119898119898 11989421 = 02 119901119903 = 286479 and 0 le 120579119903 le 10472119903119886119889are given where Rr is cylinder helix radius Z1 and Z2 are thenumber of teeth for pinion and gear a is central distance i21is gear ratio pr is helix parameter and 120579r is curve parameterangle We obtain the engagement drawing results shown inFigures 4 and 5 using MATLAB software

22 Design of Tooth Profiles In this paper normal sectionof convex-to-concave tooth profiles is provided The mainengagement region is set to the parabolic curves and basicparametric design of tooth profiles is given in Table 1 Mean-while the scheme of tooth profiles is depicted in Figure 6

The left or right tooth profiles of double helical gear drivecontact in point which is also the common tangent point

50

40

30

20

10

0

minus10

minus20

minus30

minus40

minus50 7580

8590

95

Axi

s z (m

m)

Axis y (mm)

Axis x (mm)

Tubular surface Σ2

Tubular surface Σ1

Figure 5 Engagement of tubular surfaces generated by MATLAB

The double helical gear drive with curve element constructedtooth pairs is generated and it can be expressed in Figure 7

For general conjugate surface theory tooth profilesmainly are generated based on the given surface 1 and theestablished meshing relationship And the only generatedsurface 2 is fixed However for conjugate curve theory toothprofiles inheriting meshing characteristics of the conjugatecurves are generated based on the given curve 1 The gener-ated tooth profiles can be diversity if choosing the differentcurve form and contact position Comparisons of two kindsof generation theory are shown in Figure 8

Compared with the existing gears the new type of gearpair may have the following characteristics (1) few teethnumber and large module may be obtained without toothundercutting (2) The special meshing of generated convex


Table 1 Parametric design of normal section of tooth profiles

Parametric design Symbols Pinion tooth profile Gear tooth profileNormal modulus 119898n

Pressure angle 120572 18∘ sim36∘ 18∘ sim36∘Tooth height h1 h2 (13sim15)mn (134sim154)mn

Radius of tooth profile curve 120588a 120588f (13sim15)mn (134sim154)mn

Circle centre movement distance ea ef 0 001mn

Circle centre offset distance la lf 059mn 055mn

Distance linking contact point and pitch curve hk 063mn 063mn

Tooth thick at contact point Sa Sf 154mn 154mn

Tooth space at contact point 120596119886119896 120596119891119896 16mn 154mn

Tooth crack j 0 005mn

Addendum chamfer angle of gear tooth 120574e 45∘

Addendum chamfer height of gear tooth he 016mn

Table 2 Mathematical parameters for generating the gear pair

Parameters Symbols ValuesRadius of pitch circle of pinion (mm) 1198771199031 185Radius of pitch circle of gear (mm) 1198771199032 925Normal module (mm) 119898n 6Pinion tooth number Z1 6Gear tooth number Z2 30Pressure angle (∘) 120572120572 25Gear ratio 11989421 5Tooth profile radius of pinion (mm) 120588119886 9Tooth profile radius of gear (mm) 120588119891 99Helix parameter P 286478Pinion equidistant distance (mm) 119863ℎ1 9Gear equidistant distance (mm) 119863ℎ2 99Tooth width (mm) B 40Curve parameter range (rad) 120579 0sim1048

and concave tooth profiles makes relative radius of curvatureof the contact point longer and increases the contact strengthThe load capacity will be improved (3) The tooth profilesmesh in point contact along the conjugate curves andthe approximate pure rolling contact between mating toothsurfaces may occur

23 Numerical Example Numerical example is illustratedaccording to the developed tooth profiles equationsThroughthe generation idea shown in Figure 9 we establish the solidmodels of double helical gear pair in terms of mathematicalparameters in Table 2 And the results are displayed inFigure 10

Computerized engagement motion of tooth profiles issimulated and the results show that gear pair rotates witha fixed transmission ratio and continuous motion For theaxial direction tooth profiles mesh in point contact andthere is no engagement interference during the mated gearpair

3 Meshing Analysis of Tooth Profiles

31 Force Analysis Force conditions of tooth profiles in thenormal direction are analyzed in Figure 11 and they aresummarized as follows

(1) Circumferential force Fti is

119865119905119894 =2119879119894119889119894

(119894 = 119901119894119899119894119900119899 1 gear 2) (15)

(2) Axial force Fai is

119865119886119894 = 119865119905119894 tan120573 (119894 = 119901119894119899119894119900119899 1 gear 2) (16)

(3) Radial force Fri is

119865119903119894 =119865119905119894 tan120572119899cos120573 (119894 = 119901119894119899119894119900119899 1 gear 2) (17)


K

a

Sa

ℎk

la

ℎ1

ℎa1

ℎf1

(a) Pinion tooth profile

wfk

f

e

ℎk

ef

ℎf2

Of

ℎ2

lf ℎa2

(b) Gear tooth profile

Figure 6 Normal section of tooth profiles

Conjugate contact curves

Contact pointConjugate tooth profiles

Figure 7Double helical gearswith curve element constructed toothpairs

Then the normal force in name is expressed as follows

119865119899119894 =119865119905119894

cos120572119905 cos120573119887(119894 = 119901119894119899119894119900119899 1 gear 2) (18)

where T is the moment of force 120573 is helix angle and 120573b is thehelix angle in pitch circle 120572t is pressure angle in end and 120572nis pressure angle in normal

32 Stress Evaluation Using FEA Stress analysis of teeth ofgear pair is carried out and ANSYS software is utilized toanalyze the process To get ideal results of contact positionand to guarantee of calculation precision we take one side oftooth profiles as the object

Considering the actual engagement conditions gear pairis plotted with hexahedron unit Solid185 using Hypermesh

Given Surface 1 Fixed Surface 2

Meshing relationship

Given motion conditions

(a) Conjugate surface theory

Given Curve 1

Conjugate Curve 2



Tooth Surface 1

Tooth Surface 2

Equidistance-envelopingapproach


(b) Conjugate curve theory

Figure 8 Comparisons of two kinds of generation theory

Contactcurve

Tubularsurfaces

Toothprofiles

Using MATLAB Soware

Surfacefunction

Auxiliaryfunction

Solidmodels

Using Pro E SowareDatepo int s

Figure 9 Generation idea of solid models

software The network should be densely drawn due to stressconvergence and contact position change The unit size ofcontact area is set as 02mm Supposing that tooth profilesof the pinion and gear are the contact surfaces and targetsurface respectively we adopt the Contact 173 and Targe 170forms as the contact units for them In addition engagementaction is assumed as standard and the contact stiffnessfactor in normal section is 1 Pin connection and multipointconstraint are respectively applied to the engagement pairThe gear inner surface unit and its rotational center arefixed Hertz model is applied to analysis process of contactstress Using the extended Lagrange algorithm and Mpc 184constraint unit to the analysis process finite element modelof conjugate pair is established in Figure 12 In this process20CrMnTi material whose properties are v=025 E=207GPais selected

Torque applied to the gear is 1250Nmand angular velocityapplied to the pinion is 085rads Finite element analysisresults of tooth profiles are obtained in Figure 13Thegear pairshows point contact characteristic on the surface of contact


Figure 10 Solid models of double helical gear pair

Fni

Fni

FtiFti

t

n

b

Fri

Fri

Fai

Fai

K2

K1

di

Ti

Figure 11 Force analysis of tooth profiles

unit Because the applied torque is bigger it will produceapproximate two-point contact condition which conformsto the characteristic of the parabola Figure 13(a) shows thecontact stress resultThe contact deformation of tooth profilesis obtained in Figure 13(b) The equivalent stress conditionis shown in Figure 13(c) and the equivalent stress will notchange suddenly when the gear pair meshes in normal

Figure 14 shows the maximum contact stress of toothprofiles during meshing process The maximum contactstress is 763406MPa and the minimum contact stress is476689MPa According to the presented results the teethmesh at the time of 01s and contact position locates on the

Figure 12 Finite element model of single gear teeth

edge In the process of 02ssim04s time gear pair meshes withdouble teeth And the single tooth is in mesh after the time05s while at the time 12s the other teeth are in the state ofedge contact When the time is 14s the teeth begin to engageout

Figure 15 shows the maximum contact deformation oftooth profiles during meshing process The maximum valueis 00353mm and the minimum value is 00064mm Themaximum and minimum deformation positions locate at theengagement in and out of gear pair

4 Gear Prototype

Milling method is used to achieve the tooth profiles ofgear pair Utilizing the CNC (Computer Numerical Con-trol) machining center DMU60 (manufactured by GermanyDMG) the program of machining codes is developed with


(a) Contact stress (b) Contact deformation

(c) Equivalent stress

Figure 13 FEA results of tooth profiles

800

750

700

650

600

550

500

450

02 04 06 08 10 12 14

Time (s)

Max

Con

tact

Stre

ss (M

Pa)

Figure 14 Maximum contact stress during meshing process

the ball milling cutter for generating the complex toothsurfaces The processing of the gear pair is conducted andthe generated gear pair is depicted in Figure 16 Further

004

003

002

001

000

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15

Time (s)

Max

cont

act d

efor

mat

ion

(mm

)

Figure 15 Maximum contact deformation during meshing process

experiments on transmission properties of gear prototypeswill be carried out


(a) DMU60 machining center (b) The pinion

(c) The gear

Figure 16 Gear prototype

5 Conclusions

(1) Research on the double helical gear transmission withcurve element constructed tooth pairs is carried out Thecontact curves of tooth profiles include three parts spatialhelix curve D1 spatial helix curve D2 and circular arccurve D3 Generation principle and theoretical mathematicalmodel of tooth profiles are presented based on the geometricrelationship General tooth profiles equations are also derivedin terms of equidistance-enveloping approach

(2) Parametric design of tooth profiles is conducted andnumerical example is illustrated according to the developedtooth profiles equations Solid models of double helicalgear pair in terms of given parameters are establishedComputerized engagement of tooth profiles is also sim-ulated The results show that gear pair rotates with afixed transmission ratio and continuous motion For theaxial direction tooth profiles mesh in point contact andthere is no engagement interference during the mated gearpair

(3) Force conditions of tooth profiles in the normaldirection are analyzed and the normal force in name isexpressed using circumferential force axial force and radialforce Stress analysis of tooth profiles of gear pair is carried outbased on ANSYS software During the meshing process themaximum contact stress value is 7634MPa The maximumcontact deformation value of contact surface is 00352mm

(4) Gear prototypes are manufactured using CNCmachining technology The further tooth contact analysis

and dynamic and experimental studies on transmissionproperties of gear prototypes will be carried out

Data Availability

Thedata generated or analyzed during this study are includedin this submitted article and the current study data are alsoavailable from the corresponding author Dong Liang uponreasonable request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research is supported by the National Natural ScienceFoundation of China (GrantNo 51605049) and FundamentalResearch and Frontier Exploration Program of ChongqingCity (Grant No cstc2018jcyjAX0029)Their financial supportis grateful acknowledged

References

[1] A Fuentes R Ruiz-Orzaez and I Gonzalez-Perez ldquoCom-puterized design simulation of meshing and finite elementanalysis of two types of geometry of curvilinear cylindricalgearsrdquo Computer Methods Applied Mechanics and Engineeringvol 272 pp 321ndash339 2014


[2] L Dudas ldquoModelling and simulation of a new worm gear drivehaving point-like contactrdquo Engineering with Computers vol 29no 3 pp 251ndash272 2013

[3] M Zimmer M Otto and K Stahl ldquoCalculation of arbitrarytooth shapes to support gear designrdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 231 no 11 pp 2075ndash2079 2017

[4] A Kahraman ldquoFree torsional vibration characteristics of com-pound planetary gear setsrdquo Mechanism and Machine Theoryvol 36 no 8 pp 953ndash971 2001

[5] S Li and A Kahraman ldquoPrediction of spur gear mechanicalpower losses using a transient elastohydrodynamic lubricationmodelrdquo Tribology Transactions vol 53 no 4 pp 554ndash563 2010

[6] V Simon ldquoA new worm gear drive with ground double arcprofilerdquoMechanism andMachineTheory vol 29 no 3 pp 407ndash414 1994

[7] V Simon ldquoHob for worm gear manufacturing with circularprofilerdquo The International Journal of Machine Tools and Man-ufacture vol 33 no 4 pp 615ndash625 1993

[8] C-J Bahk and R G Parker ldquoAnalytical investigation of toothprofile modification effects on planetary gear dynamicsrdquoMech-anism and Machine Theory vol 70 pp 298ndash319 2013

[9] W Song H Zhou and Y Zhao ldquoDesign of the conjugatedstraight-line internal gear pairs for fluid power gear machinesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 227 no 8 pp1776ndash1790 2013

[10] C Huang J-X Wang K Xiao M Li and J-Y Li ldquoDynamiccharacteristics analysis and experimental research on a newtype planetary gear apparatus with small tooth number differ-encerdquo Journal of Mechanical Science and Technology vol 27 no5 pp 1233ndash1244 2013

[11] C Lin L Zhang and Z Zhang ldquoTransmission theory and toothsurface solution of a new type of non-circular bevel gearsrdquoJournal of Mechanical Engineering vol 50 no 13 pp 66ndash722014 (Chinese)

[12] P Sondkar and A Kahraman ldquoA dynamic model of a double-helical planetary gear setrdquoMechanism andMachineTheory vol70 pp 157ndash174 2013

[13] H Zhang L Hua and X H Han ldquoComputerized design andsimulation of meshing of modified double circular-arc helicalgears by tooth end relief with helixrdquo Mechanism and MachineTheory vol 45 no 1 pp 46ndash64 2010

[14] C Wang Z-D Fang H-T Jia and J-H Zhang ldquoModificationoptimization of double helical gearsrdquo Journal of AerospacePower vol 24 no 6 pp 1432ndash1436 2009

[15] K Kawasaki I Tsuji and H Gunbara ldquoManufacturing methodof double helical gears using multi-axis control and multi-taskingmachine toolrdquo in International Gear Conference pp 86ndash95 2014

[16] K Kawasaki I Tsuji and H Gunbara ldquoManufacturing methodof double-helical gears using CNC machining centerrdquo Proceed-ings of the Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 230 no 7-8 pp 1149ndash11562015

[17] B Chen D Liang and Y Gao ldquoThe principle of conjugatecurves for gear transmissionrdquo Journal of Mechanical Engineer-ing vol 50 no 1 pp 130ndash136 2014 (Chinese)

[18] B Chen Y Gao and D Liang ldquoTooth profile generation ofconjugate-curve gearsrdquo Journal of Mechanical Engineering vol50 no 3 pp 18ndash24 2014 (Chinese)

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


Huang et al [10] proposed an internal mesh planetary gearwith small tooth number difference (PGSTD) and dynamiccharacteristics analysis was carried out Lin [11] provided anewnon-circular bevel gear based on the combination of cammechanism and non-circular bevel gear

For double helical gear drive it has currently attractedmore attentions in the field of low speed and heavy loadmachinery Sondkar et al [12] established the dynamicmodelof double helical planetary gear pair A linear time-invariantform was developed and further studies on dynamic charac-teristics were carried out Zhang et al [13] investigated thecomputerized design and simulation of meshing of modifieddouble circular arc helical gears by tooth end relief withhelix The proposed theory was illustrated with numericalexamples which confirm the advantages of the gear drivesof the modified geometry According to the characteristicof tooth profile modification Wang et al [14] presented themethod of three-segmentmodification for pinion profileTheoptimization results of tooth profile and axial modificationwithout consideration of axes error can reduce the loadedtransmission error Kawasaki et al [15 16] proposed a man-ufacturing method of double helical gears using a multiaxiscontrol and multitasking machine tool The pitch errorstooth thickness runout profile lead and surface roughnessof manufactured double helical gears were also measuredOtherwise the relationship between the tool wear and lifetime of the end mill was made clear And the manufacturingmethod was applied to the gears for a double helical gearpump

To obtain the better transmission performance and tomeet the higher strength requirements a new double helicalgear transmissionwith curve element constructed tooth pairsis studied in this paper Generation principle and mathe-matical model of this gear drive are provided Numericalexample is illustrated in terms of the established tooth profilesform Computerized design and meshing simulation are alsoput forward Furthermore stress analysis of the gear pair isconducted and gear prototypes are manufactured using CNCmachining technology

2 Generation and Mathematical Model ofTooth Profiles

21 Contact Principle Contact curves of tooth profiles aredefined with three parts spatial helix curve D1 spatial helixcurve D2 and circular arc curve D3 The spatial helix curve D1and the spatial helix curve D2 are symmetric curves relative tothe middle section of tooth width The circular arc curve D3is the smooth transition curve connecting the curves D1 andD2

The spatial helix curvesD1 andD2 are expressed as follows

Γ119894

119909119894 = 119877119903 cos 120579119903119894119910119894 = 119877119903 sin 120579119903119894119911119894 = 119901119903120579119903119894

(119894 = 1 2) (1)

where Rr is cylinder helix radius 120579ri (i=1 2) is spatial helixcurve parameter and pr is helix parameter

A

O


Spatial helix curve Γ2T2T1

K1

K2

N1N2

Figure 1 Geometric relationship

Transition circular arc curve D3 is derived by mathematicapproach and its location plane needs to be first determinedAs shown in Figure 1 T1 and T2 are the tangent at points K1andK2 on the spatial helix curves D1 and D2 respectivelyThetwo lines are given to the A point The normal lines at pointsK1 andK2 are expressed usingN1N2The two lines are givento the O point

According to (1) its derivative can be calculated as

1199091015840119894 = minus119877119903 sin 1205791199031198941199101015840119894 = 119877119903 cos 1205791199031198941199111015840119894 = 119901119903

(119894 = 1 2)

(2)

and at point K1 we have the coordinate results (xk1 yk1 zk1)and (11990910158401198961 11991010158401198961 11991110158401198961) So the equation of tangent T1 is shownas

119909 minus 119909119896111990910158401198961

= 119910 minus 119910119896111991010158401198961= 119911 minus 119911119896111991110158401198961 (3)

Similarly the equation of tangent T2 is expressed as

119909 minus 119909119896211990910158401198962

= 119910 minus 119910119896211991010158401198962= 119911 minus 119911119896211991110158401198962 (4)

With simultaneous equations (3) and (4) the value ofpoint A can be obtained Obviously normal lines N1 N2 aredetermined based on the above tangent expressions

Normal line N1


Normal line N2


In simultaneous equations (5) and (6) the family of point Ocan be calculated and the point meeting the actual require-ments will be further determined by the following conditions

A plane can be generated at three points in space Asshown in Figure 2 suppose that the intersection point O isthe circle center and the circular arc curve passing points K1and K2 is drawn and it should satisfy the constant tangentsT1 and T2


A

O

Circular arc curve

T1

T2

K2

K1




(7)



(8)






cos120593 = 119899 ∙119873|119899| ∙ |119873| (10)


119909 = 0119910 = minus119903 cos 120575119911 = 119903 sin 120575

(11)



(12)


Σ119894


1015840) cos1205721 + (119877119903 cos 120579119903119894 + ℎ119899011989911991011015840) sin1205721

+ (119901119903 + ℎ119899011989911991111015840) tan1205931 = 0

(119894 = 1 2)

(13)

and

Σ3


1015840) cos1205721 + (119903 sin 120575 cos120593 + ℎ119899011989911991011015840) sin1205721

+119903 cos 120575 tan1205931 = 0

(14)




y

x

zO

K1

K2

z3

y3

x3

O1


y

z

O

r

K1

K2





50

40

30

20

10

0

minus10

minus20

minus30

minus40

minus50 7580

8590

95

Axi

s z (m

m)

Axis y (mm)

Axis x (mm)





50

40

30

20

10

0

minus10

minus20

minus30

minus40

minus50 7580

8590

95

Axi

s z (m

m)

Axis y (mm)

Axis x (mm)

Tubular surface Σ2

Tubular surface Σ1


























119865119905119894 =2119879119894119889119894

(119894 = 119901119894119899119894119900119899 1 gear 2) (15)


119865119886119894 = 119865119905119894 tan120573 (119894 = 119901119894119899119894119900119899 1 gear 2) (16)


119865119903119894 =119865119905119894 tan120572119899cos120573 (119894 = 119901119894119899119894119900119899 1 gear 2) (17)


K

a

Sa

ℎk

la

ℎ1

ℎa1

ℎf1


wfk

f

e

ℎk

ef

ℎf2

Of

ℎ2

lf ℎa2







119865119899119894 =119865119905119894

cos120572119905 cos120573119887(119894 = 119901119894119899119894119900119899 1 gear 2) (18)








Given Curve 1

Conjugate Curve 2



Tooth Surface 1

Tooth Surface 2





Contactcurve

Tubularsurfaces

Toothprofiles

Using MATLAB Soware

Surfacefunction

Auxiliaryfunction

Solidmodels







Fni

Fni

FtiFti

t

n

b

Fri

Fri

Fai

Fai

K2

K1

di

Ti







4 Gear Prototype






800

750

700

650

600

550

500

450

02 04 06 08 10 12 14

Time (s)

Max

Con

tact

Stre

ss (M

Pa)



004

003

002

001

000

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15

Time (s)

Max

cont

act d

efor

mat

ion

(mm

)





(c) The gear


5 Conclusions






Data Availability




Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018









A

O

Circular arc curve

T1

T2

K2

K1




(7)



(8)






cos120593 = 119899 ∙119873|119899| ∙ |119873| (10)


119909 = 0119910 = minus119903 cos 120575119911 = 119903 sin 120575

(11)



(12)


Σ119894


1015840) cos1205721 + (119877119903 cos 120579119903119894 + ℎ119899011989911991011015840) sin1205721

+ (119901119903 + ℎ119899011989911991111015840) tan1205931 = 0

(119894 = 1 2)

(13)

and

Σ3


1015840) cos1205721 + (119903 sin 120575 cos120593 + ℎ119899011989911991011015840) sin1205721

+119903 cos 120575 tan1205931 = 0

(14)




y

x

zO

K1

K2

z3

y3

x3

O1


y

z

O

r

K1

K2





50

40

30

20

10

0

minus10

minus20

minus30

minus40

minus50 7580

8590

95

Axi

s z (m

m)

Axis y (mm)

Axis x (mm)





50

40

30

20

10

0

minus10

minus20

minus30

minus40

minus50 7580

8590

95

Axi

s z (m

m)

Axis y (mm)

Axis x (mm)

Tubular surface Σ2

Tubular surface Σ1


























119865119905119894 =2119879119894119889119894

(119894 = 119901119894119899119894119900119899 1 gear 2) (15)


119865119886119894 = 119865119905119894 tan120573 (119894 = 119901119894119899119894119900119899 1 gear 2) (16)


119865119903119894 =119865119905119894 tan120572119899cos120573 (119894 = 119901119894119899119894119900119899 1 gear 2) (17)


K

a

Sa

ℎk

la

ℎ1

ℎa1

ℎf1


wfk

f

e

ℎk

ef

ℎf2

Of

ℎ2

lf ℎa2







119865119899119894 =119865119905119894

cos120572119905 cos120573119887(119894 = 119901119894119899119894119900119899 1 gear 2) (18)








Given Curve 1

Conjugate Curve 2



Tooth Surface 1

Tooth Surface 2





Contactcurve

Tubularsurfaces

Toothprofiles

Using MATLAB Soware

Surfacefunction

Auxiliaryfunction

Solidmodels







Fni

Fni

FtiFti

t

n

b

Fri

Fri

Fai

Fai

K2

K1

di

Ti







4 Gear Prototype






800

750

700

650

600

550

500

450

02 04 06 08 10 12 14

Time (s)

Max

Con

tact

Stre

ss (M

Pa)



004

003

002

001

000

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15

Time (s)

Max

cont

act d

efor

mat

ion

(mm

)





(c) The gear


5 Conclusions






Data Availability




Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018









y

x

zO

K1

K2

z3

y3

x3

O1


y

z

O

r

K1

K2





50

40

30

20

10

0

minus10

minus20

minus30

minus40

minus50 7580

8590

95

Axi

s z (m

m)

Axis y (mm)

Axis x (mm)





50

40

30

20

10

0

minus10

minus20

minus30

minus40

minus50 7580

8590

95

Axi

s z (m

m)

Axis y (mm)

Axis x (mm)

Tubular surface Σ2

Tubular surface Σ1


























119865119905119894 =2119879119894119889119894

(119894 = 119901119894119899119894119900119899 1 gear 2) (15)


119865119886119894 = 119865119905119894 tan120573 (119894 = 119901119894119899119894119900119899 1 gear 2) (16)


119865119903119894 =119865119905119894 tan120572119899cos120573 (119894 = 119901119894119899119894119900119899 1 gear 2) (17)


K

a

Sa

ℎk

la

ℎ1

ℎa1

ℎf1


wfk

f

e

ℎk

ef

ℎf2

Of

ℎ2

lf ℎa2







119865119899119894 =119865119905119894

cos120572119905 cos120573119887(119894 = 119901119894119899119894119900119899 1 gear 2) (18)








Given Curve 1

Conjugate Curve 2



Tooth Surface 1

Tooth Surface 2





Contactcurve

Tubularsurfaces

Toothprofiles

Using MATLAB Soware

Surfacefunction

Auxiliaryfunction

Solidmodels







Fni

Fni

FtiFti

t

n

b

Fri

Fri

Fai

Fai

K2

K1

di

Ti







4 Gear Prototype






800

750

700

650

600

550

500

450

02 04 06 08 10 12 14

Time (s)

Max

Con

tact

Stre

ss (M

Pa)



004

003

002

001

000

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15

Time (s)

Max

cont

act d

efor

mat

ion

(mm

)





(c) The gear


5 Conclusions






Data Availability




Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018





























119865119905119894 =2119879119894119889119894

(119894 = 119901119894119899119894119900119899 1 gear 2) (15)


119865119886119894 = 119865119905119894 tan120573 (119894 = 119901119894119899119894119900119899 1 gear 2) (16)


119865119903119894 =119865119905119894 tan120572119899cos120573 (119894 = 119901119894119899119894119900119899 1 gear 2) (17)


K

a

Sa

ℎk

la

ℎ1

ℎa1

ℎf1


wfk

f

e

ℎk

ef

ℎf2

Of

ℎ2

lf ℎa2







119865119899119894 =119865119905119894

cos120572119905 cos120573119887(119894 = 119901119894119899119894119900119899 1 gear 2) (18)








Given Curve 1

Conjugate Curve 2



Tooth Surface 1

Tooth Surface 2





Contactcurve

Tubularsurfaces

Toothprofiles

Using MATLAB Soware

Surfacefunction

Auxiliaryfunction

Solidmodels







Fni

Fni

FtiFti

t

n

b

Fri

Fri

Fai

Fai

K2

K1

di

Ti







4 Gear Prototype






800

750

700

650

600

550

500

450

02 04 06 08 10 12 14

Time (s)

Max

Con

tact

Stre

ss (M

Pa)



004

003

002

001

000

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15

Time (s)

Max

cont

act d

efor

mat

ion

(mm

)





(c) The gear


5 Conclusions






Data Availability




Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018









K

a

Sa

ℎk

la

ℎ1

ℎa1

ℎf1


wfk

f

e

ℎk

ef

ℎf2

Of

ℎ2

lf ℎa2







119865119899119894 =119865119905119894

cos120572119905 cos120573119887(119894 = 119901119894119899119894119900119899 1 gear 2) (18)








Given Curve 1

Conjugate Curve 2



Tooth Surface 1

Tooth Surface 2





Contactcurve

Tubularsurfaces

Toothprofiles

Using MATLAB Soware

Surfacefunction

Auxiliaryfunction

Solidmodels







Fni

Fni

FtiFti

t

n

b

Fri

Fri

Fai

Fai

K2

K1

di

Ti







4 Gear Prototype






800

750

700

650

600

550

500

450

02 04 06 08 10 12 14

Time (s)

Max

Con

tact

Stre

ss (M

Pa)



004

003

002

001

000

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15

Time (s)

Max

cont

act d

efor

mat

ion

(mm

)





(c) The gear


5 Conclusions






Data Availability




Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018










Fni

Fni

FtiFti

t

n

b

Fri

Fri

Fai

Fai

K2

K1

di

Ti







4 Gear Prototype






800

750

700

650

600

550

500

450

02 04 06 08 10 12 14

Time (s)

Max

Con

tact

Stre

ss (M

Pa)



004

003

002

001

000

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15

Time (s)

Max

cont

act d

efor

mat

ion

(mm

)





(c) The gear


5 Conclusions






Data Availability




Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018












800

750

700

650

600

550

500

450

02 04 06 08 10 12 14

Time (s)

Max

Con

tact

Stre

ss (M

Pa)



004

003

002

001

000

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15

Time (s)

Max

cont

act d

efor

mat

ion

(mm

)





(c) The gear


5 Conclusions






Data Availability




Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018










(c) The gear


5 Conclusions






Data Availability




Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018

































Journal of












Journal of







Volume 2018






Volume 2018








Generation and Meshing Analysis of a New Type of Double ...

Documents

Transcript of Generation and Meshing Analysis of a New Type of Double ...