research article research on the design and modification

Research ArticleResearch on the Design and Modification ofAsymmetric Spur Gear

Xiaohe Deng Lin Hua and Xinghui Han

School of Automotive Engineering Hubei Key Laboratory of Advanced Technology of Automotive PartsWuhan University of Technology Wuhan 430070 China

Correspondence should be addressed to Lin Hua lhuawhuteducn

Received 9 June 2014 Accepted 17 October 2014

Academic Editor Davide Spinello

Copyright copy 2015 Xiaohe Deng 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 design method for the geometric shape and modification of asymmetric spur gear was proposed in which the geometric shapeand modification of the gear can be obtained directly according to the rack-cutter profile In the geometric design process of thegear a rack-cutter with different pressure angles and fillet radius in the driving side and coast side was selected and the generatedasymmetric spur gear profiles also had different pressure angles and fillets accordingly In the modification design of the gear thepressure angle modification of rack-cutter was conducted firstly and then the corresponding modified involute gear profile wasobtained The geometric model of spur gears was developed using computer-aided design and the meshing process was analyzedusing finite element simulation method Furthermore the transmission error and load sharing ratio of unmodified and modifiedasymmetric spur gears were investigated Research results showed that the proposed gear design method was feasible and desiredspur gear can be obtained through one time rapid machining by the method Asymmetric spur gear with better transmissioncharacteristic can be obtained via involute modification

1 Introduction

Gear drives are one of the most widely used modes in powertransmission system with the advantages of large power highefficiency and long service life The involute tooth profile hasvarious advantages such as good transmission performanceconvenient installation and low cost [1 2] Compared withtraditional involute spur gears asymmetric involute gearshave larger carrying capacity smaller volume lighter weightand better dynamic characteristics [3] So the asymmetricspur gear has significant potential applications in the fields ofheavy load and high-speed gear transmission system In thegear transmission process it is not possible to have thematinggear enter contact in pure involute position because therewould be sudden interference corresponding to the elasticdeflection and the corner of the tooth tipwould gouge into themating surface [1] Real meshing tooth profile of asymmetricspur gears will deviate from the ideal location under loadand will cause transmission error and the change of loaddistributionThe transmission error changes over timewhichcauses displacement excitation in the gear dynamic model

and has great influence on the dynamics of meshing gears[4ndash6] Therefore it is necessary to improve the meshingperformance of asymmetric spur gears by the reasonabledesign and modification

Until now many researches have been carried out on thedesign and modification of symmetric gear Litvin et al [7]proposed a modified involute helical gear that matches witha double-crowned pinion and simulated the meshing pro-cess between profile-crowned and double-crowned helicoidsFaggioni et al [8] presented a global optimization methodthat focused on reducing gear vibration by means of profilemodifications Barbieri et al [9] used genetic algorithms toresearch the optimum profile Zhang et al [10] performedcomputer-aided design based on the contact conditions ofconjugated gears and the tooth end relief with helix by finiteelement analysis (FEA) Gonzalez-Perez et al [11] appliedanalytical approach in the stress analysis for gear drivewith localized bearing contact based on the Hertz theoryHuang and Su [12] presented various approaches aboutelement constructions and dynamic analyses for involutespurhelical gears where meshed elements are constructed

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 897257 13 pageshttpdxdoiorg1011552015897257

2 Mathematical Problems in Engineering

directly without CAD geometric model The gear profileequations were described via C programming language Theeffects of modification coefficients and helical angles on thetransientmeshing performance of the gears were investigatedusing the method of explicit dynamic FEA from the perspec-tive of energy by Hu et al [13] A qualitative analysis aboutthe effects of flank deviation on load distribution for helicalgear was performed using FEM by Wei et al [14] Wu etal [15] proposed a tooth profile modification approach forthe helical pairs and the static contact process was analyzedusing FEA and experimental tests Li [16] applied 3D FEMto conduct surface contact stress and root bending stresscalculations for a pair of spur gears considering machiningerrors assembly errors and tooth modifications Simon [17]researched on the influence of tooth modifications inducedby machine tool setting and head-cutter profile variationson tooth contact characteristics in face-hobbed spiral bevelgears Chen and Shao [18] proposed a general analyticalmeshing stiffness model to establish the relationship betweenthe gear tooth errors and the total mesh stiffness whereinload sharing among different tooth pairs duringmeshing andstatic transmission errors under loadwere considered Sankarand Nataraj [19] proposed a modification design approachfor increasing the tooth strength in spur gear Wang et al[20] proposed a method to simulate residual stress field ofplanetary gear Li and Kahraman [21] proposed a tridynamicsmodel for spur gear pairs The proposed model couples amixed elastohydrodynamic lubrication model of a spur gearpair with a transverse torsional dynamicmodel Kolivand andKahraman [22] defined new general concept based on theconstruction of a surface of roll angle to simplify the task oflocating instantaneous contact lines of any type of gearingWith the surface of roll angle defined instantaneous contactlines are determined by a novel approach based on an analogyto parallel-axes gears Eritenel and Parker [23] provided ananalytical solution for the nonlinear vibration of gear pairsthat exhibit partial and total contact loss

Moreover more and more attention has been paid toasymmetric gear recently Litvin et al [3] proposed themodified geometric model of an asymmetric spur gear drivewhich integrated an involute gear and a double crownedpinion and simulated the meshing and contact process ofthe spur gears with proposed geometric parameters usingcomputer-aided design method Alipiev [24] proposed a newgeometric design method for the symmetric and asymmetricinvolute meshing in which the gear load sharing ratio is equalto its expected value Pedersen [25] observed the use of twonew standard cutting tools for the asymmetric gears since theshape of the gear profile influences its bending strength

In the gear design process the traditional method onlygives basic parameters of gear structure and it is difficultto obtain precise model for the actual generated gear Inthis paper the equations of gear tooth profile were deducedbased on a given rack-cutter and the accurate geometricmodel was established in the design process In the traditionalmachining process of modified gear a standard gear withoutmodification is generated at first and themodification is con-ducted subsequently In order to enhance the manufacturingefficiency the pressure angle modification is carried out on

E

D

A

B

C

12057211205722

l1l2

1205791

1205792 1205881 1205882

Q1

Q3

Q4

Q2

y1

x1

Oc

Od

O1

hlowasta1m

hlowasta1m

clowast1 m

Figure 1 Gear rack-cutter normal profile and tool coordinatesystem

the rack-cutter so the desired modified spur gear can beobtained through one time rapid machining

Based on the proposed gear design and modificationmethod a FEM of asymmetric involute gear was establishedwhere computer-aided designs were adopted to establish thegeometrymodel of the spur gears finally themeshing processwas simulated

The present paper mainly consists of the following threeaspects

(1) The equations of a new rack-cutter profile for asym-metric gear were established The pressure anglemodification in rack-cutter design was proposed andthe driving side of rack-cutter profile equations wasdeduced The given rack-cutter profile has differentpressure angles and fillet radius in the driving sideand coast side The generated asymmetric spur gearprofiles have different pressure angles and filletsaccordingly (Section 2)

(2) According to conjugate theory the equation of asym-metric spur gear was derived by the forming rack-cutter tool Computer-aided design of spur gears withproposed geometric parameters and the simulationof their meshing were conducted using the proposedmethods (Section 3)

(3) The transmission error and the load sharing ratioof generated gears were studied The transmissionprinciples of asymmetric spur gear were discussed(Section 4)

2 Geometric Characteristics ofthe Rack-Cutter Profile

In order to obtain the precise gear model the geometriccharacteristic of the corresponding rack-cutter profile isrequired The given rack-cutter has different pressure anglesand fillet radius in the driving side and coast side

21 The Rack-Cutter Normal Profile The typical rack-cutterconsists of straight lines and tip circular fillet which cangenerate involutes and fillets of typical involute tooth respec-tively As depicted in Figure 1 the tool coordinate system 119878

1

is rigidly fixed to the gear rack-cutter and the coordinate axis

Mathematical Problems in Engineering 3

1199091is the reference line of rack-cutterThe basic normal profile

of rack-cutter consists of four segments driving side flankstraight line 119861119862 coast side flank straight line 119863119864 drivingside flank tip fillet curve 119860119861 and coast side flank tip filletcurve 119860119863 The gear tool possesses different gradients oneither side of the rack-cutter straight tooth profile and hasthe same addendum coefficient and tip clearance coefficientThe profile of the rack-cutter in 119878

1can be described with the

following equations

(i) Coast side flank straight line 119863119864 is

1199091= minus 1198971sin1205721minus 1205881cos1205721

1199101= 1198971cos1205721minus ℎlowast

1198861119898

0 le 1198971le

2ℎlowast

1198861119898

cos1205721

(1)

where 1198761is an arbitrary point on line 119863119864 119897

1is a

variable parameter about the length of straight line1198631198761 1205721is the pressure angle representing the slope

of line 119863119864 1205881is the radius of fillet curve 119860119863 ℎlowast

1198861is

the addendum coefficient factor and 119898 is the normalmodule

(ii) Driving side flank straight line 119861119862 is

1199091= 1198972sin1205722+ 1205882cos1205722


1198861119898

0 le 1198972le

2ℎlowast

1198861119898

cos1205722

(2)

where point 1198762is an arbitrary point on line 119861119862 119897

2

is the variable parameter about the length of straightline 119861119876

2 1205722is the pressure angle representing the

slope of line 119861119862 and 1205882is the radius of fillet curve

119860119861(iii) Coast side flank cutter fillet curve 119860119863 is

1199091= minus 120588

1sin 1205791

1199101= minus 120588

1(cos 120579

1minus sin120572

1) minus ℎlowast

1198861119898

0 le 1205791le

120587

2minus 1205721

(3)

where 1205791is the angle of

1198601198741198881198763and119876

3is an arbitrary

point on the fillet curve 119860119863(iv) Driving side flank rack-cutter fillet curve 119860119861 is

1199091= 1205882sin 1205792

1199101= minus 120588

2(cos 120579

2minus sin120572

2) minus ℎlowast

1198861119898

0 le 1205792le

120587

2minus 1205722

(4)


1198601198741198881198764and119876

4is an arbitrary

point on the fillet curve 119860119861

E

D

A

B

C

F

1205722

hlowasta1m

Od1205881 1205882

Oc

y11205723

flowast2 m

O1 Q2

Q4

Q5

l2

l3

hlowasta1m

clowast1 m

x1

Figure 2 Gear rack-cutter normal profile with pressure anglemodification

The radii of fillet curve 1205881and 1205882meet

1205881=

119888lowast

1119898

(1 minus sin1205721)

1205882=

119888lowast

1119898


(5)

where 119888lowast

1is the tip clearance coefficient

The tangential vector 997888119905 to the generated shape in the 119878

1

is

997888119905 =

1198891199091

119889V9978881198941+

1198891199101

119889V9978881198951 (6)

The unit normal vector in 1198781is

997888119899119899=

119905 times9978881198961

1003816100381610038161003816100381610038161003816119905 times

9978881198961

1003816100381610038161003816100381610038161003816

9978881198961= [0 0 1] (7)

The unit normal vectors of each profile in 1198781can be

derived as

997888997888997888997888119899119899 119863119864

= [

[

cos1205721

minus sin1205721

0

]

]

997888997888997888997888119899119899 119861119862

= [

[

minus cos1205722

minus sin1205722

0

]

]

997888997888997888997888119899119899 119860119863

= [

[

minus sin 1205791

minus cos 1205791

0

]

]

997888997888997888997888119899119899 119860119861

= [

[

sin 1205792

minus cos 1205792

0

]

]

(8)

where the subscripts 119863119864 119861119862 119860119863 and 119860119861 represent thecorresponding line and curves

22 The Equations of Pressure Angle Modification The con-ventional design of asymmetric spur gear is based on theapplication of involute profiles [1] If the lines in the tooltip have different slopes the generated gear will be modifiedcorrespondingly Thus the driving side of rack-cutter willhave two pressure angles and the generated gear will havetwo segments of involute profile on the driving side It is easyand feasible to produce such a rack-cutter

The pressure angle modification can be represented bytwo parameters in the tool coordinate the modified angle1205723and the modified length 119891

lowast

2119898 as shown in Figure 2 The


H

L

Original involute

Modified involute

Gear tip

MH0

Pj

ra

rj

O2

profile

profile

Figure 3 Schematic diagram of a single tooth profile

parameter 119891lowast

2is called modified coefficient The modified

new gear profiles are shown in Figure 3 119871 is the maximummodified quantity 119867 is the maximum modified length 119867 isthe distance between the modified point 119872 and the gear tipwhich should be within the range of 0 sim 119867

0

The equations of modified tool profiles are listed asfollows

(i) Driving side flank straight line 119861119865 is

1199091= 1198972sin1205722+ 1205882cos1205722


1198861119898

0 le 1198972le

2ℎlowast

1198861119898 minus 119891

lowast

2119898

cos1205722

(9)

where 119891lowast

2is the modification coefficient 120572

3is the

pressure angle after modificationThe unit normal vector 997888997888997888

119899119899 119861119865

of line 119861119865 meets

997888997888997888119899119899 119861119865

= [

[

minus cos1205722

minus sin1205722

0

]

]

(10)


1199091= 1198973sin1205723+ 1205882cos1205722+

ℎlowast

1198861119898

sin1205722

1199101= 1198973cos1205723+ ℎlowast

1198861119898 minus 119891

lowast

2119898

0 le 1198973le

ℎlowast

1198861119898 minus 119891

lowast

2119898

cos1205723

(11)

The unit normal vector 997888997888997888997888119899119899 119865119862


997888997888997888997888119899119899 119865119862

= [

[

minus cos1205723

minus sin1205723

0

]

]

(12)

3 The Gear Profiles Equations

The gear profile equation is the foundation of the parametermodeling Due to the application of coordinate transforma-tion we can deduce the generated gear profiles from the equa-tion of rack-cutter profile based on the kinematic relation and

O

P

y1 yj

1

n

y2O1

120596 O2 Oj

r1

rj

120601

z1

x2

x1

xj

z2 zj

s = rj120601

Figure 4 Coordination systems transformations

conjugate theory After the pressure angle modification ofrack-cutter the generated gear will be modified with involutecurve

31 The Coordinate Transformation Coordinate systems 1198781

1198782 and 119878

119895were rigidly fixed to the rack-cutter the spur gear

to be generated and the frame (see Figure 4) [7] A point 119873in coordinate system 119878

2was represented by position vector

1198742119873 =

9978881199032 A vector in coordinate system 119878

1was signed

as 9978881199031 120601 is the rotate angle of gear blank The coordinate

transformation from 1198781to 1198782was based on the matrix

equation9978881199032= 11987221

9978881199031 (13)

The transformation matrix 11987221can be determined by

11987221

= [

[

cos120601 sin120601 0 119903119895(sin120601 minus 120601 cos120601)

minus sin120601 cos120601 0 119903119895(cos120601 + 120601 sin120601)

0 0 1 0

]

]

(14)

In this way we can obtain the equation of gear profile fromthat of rack-cutter profile as follows

1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)

(15)

32 Meshing Equations Suppose that the rack moves at avelocity of 997888V

1and the gear rotates at an angular velocity of 997888120596

(Figure 4) The instantaneous center of rotation 119874 is locatedon line 119874

2119874 that is perpendicular to 997888V

1 The location of 119874

satisfies the vector equation

997888V1=

997888120596 times

9978889978889978881198742119874 (16)

The relative motion of centrode is pure rolling around 119874 andthe displacements of rack and the rotate angle of gear blank 120601

can be correlated by

119904 = 119903119895120601 (17)

where 119903119895is the pitch circle of gear


The vector radius of a moving point 119875 in coordinates 1198781is

9978881199031= 1199091

9978881198941+ 1199101

9978881198951 (18)

The sliding velocity at point 119875 is

997888V119899=

997888V1minus

997888V2 (19)

where 997888V2is the velocity of point 119875 in coordinates 119878

2given by

997888V2=

997888120596 times

9978881199031+

99788899788899788899788811987411198742times

997888120596 (20)

where

997888V1= minus V

1

9978881198941

= minus1205961199031

9978881198941

997888120596 = 120596

9978881198961

99788899788899788899788811987411198742= 1199031198951206019978881198941minus 119903119895

9978881198951

(21)

From (8) through (20) we can derive

997888V119899= 120596 [119910

1

9978881198941+ (119903119895120601 minus 1199091)9978881198951] (22)

Any meshing point 119875 on the gear tooth profile needs tomeet the following conditions the relative motion velocityvector of the point should be located in a common tangentof the two tooth profiles along a direction namely

997888120593 =

997888119899119899sdot997888V119899=

9978880 (23)

where997888119899119899is the unit normal vector ofmeshing point (equation

(8)) and 997888V119899is the relative velocity vector

According to the normal vector of different line segmentsand the relative velocity in (23) meshing equation can beobtained by

119863119864 1199091sin1205721+ 1199101cos1205721minus 119903119895120601 sin120572

1= 0

119861119862 1199091sin1205722minus 1199101cos1205722minus 119903119895120601 sin120572

2= 0

119860119863 1199091cos 1205791minus 1199101sin 1205791minus 119903119895120601 sin 120579

1= 0

119860119861 1199091cos 1205792+ 1199101sin 1205792minus 119903119895120601 sin 120579

2= 0

(24)

33 Conjugate Gear Profile Equations The tooth shape of thegenerated gear without modification is represented by (15)and (24) Each segment of gear profile curves is listed asfollows

(i) Coast side flank involute gear profile is

1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)

1199091sin1205721+ 1199101cos1205721minus 119903119895120601 sin120572

1= 0

(25)

(ii) Driving side flank involute gear profile is

1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)

1199091sin1205722minus 1199101cos1205722minus 119903119895120601 sin120572

2= 0

(26)

(iii) Coast side flank gear fillet curve is

1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)

1199091cos 1205791minus 1199101sin 1205791minus 119903119895120601 sin 120579

1= 0

(27)

(iv) Driving side flank gear fillet curve is

1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)

1199091cos 1205792+ 1199101sin 1205792minus 119903119895120601 sin 120579

2= 0

(28)

We can deduce the general equation of asymmetricinvolute gears profile with double pressure angles in 119878

2from

(1)ndash(4) and (25)ndash(28) as follows


1199092= (minus119897

1sin1205721minus 1205881cos1205721minus 119903119895120601) cos120601

+ (1198971cos1205721minus ℎlowast

1198861119898 + 119903119895) sin120601

1199102= (1198971sin1205721+ 1205881cos1205721+ 119903119895120601) sin120601


1198861119898 + 119903119895) cos120601

120601 = minus(1198971+ 1205881cos1205721sin1205721minus ℎlowast

1198861119898 cos120572

1)

(119903119895sin1205721)

0 le 1198971le

2ℎlowast

1198861119898

cos1205721

(29)


1199092= (1198972sin1205722+ 1205882cos1205722minus 119903119895120601) cos120601


1198861119898 + 119903119895) sin120601

1199102= minus (119897

2sin1205722+ 1205882cos1205722minus 119903119895120601) sin120601


1198861119898 + 119903119895) cos120601

120601 =(1198972+ 1205882cos1205722sin1205722minus ℎlowast

1198861119898 cos120572

2)

(119903119895sin1205722)

0 le 1198972le

2ℎlowast

1198861119898

cos1205722

(30)



1199092= (minus120588

1sin 1205791minus 119903119895120601) cos120601

+ (minus1205881(cos 120579

1minus sin120572

1) minus ℎlowast

1198861119898 + 119903119895) sin120601

1199102= (1205881sin 1205791+ 119903119895120601) sin120601

+ (minus1205881(cos 120579

1minus sin120572

1) minus ℎlowast

1198861119898 + 119903119895) cos120601

120601 =(1205881sin1205721sin 1205791minus ℎlowast

1198861119898 sin 120579

1)

(119903119895sin 1205791)

0 le 1205791le

120587

2minus 1205721

(31)


1199092= (1205882sin 1205792minus 119903119895120601) cos120601

+ (minus1205882(cos 120579

2minus sin120572

2) minus ℎlowast

1198861119898 + 119903119895) sin120601

1199102= minus (120588

2sin 1205792minus 119903119895120601) sin120601

+ (minus1205882(cos 120579

2minus sin120572

2) minus ℎlowast

1198861119898 + 119903119895) cos120601

120601 = minus(1205882sin1205722sin 1205792minus ℎlowast

1198861119898 sin 120579

2)

(119903119895sin 1205792)

0 le 1205792le

120587

2minus 1205722

(32)

When the modification is taken into consideration themodified profile of driving side can be derived as follows

In Figure 2 themeshing process can be described by (33)according to (10) and (22)-(23)

minus1199101cos1205722+ 1199091sin1205722minus 119903119895120601 sin120572

2= 0 (33)

From (10) and (33) we can deduce the gear profile equationof profile 119861119865

1199092= (1198972sin1205722+ 1205882cos1205722minus 119903119895120601) cos120601


1198861119898 + 119903119895) sin120601

1199102= minus (119897

2sin1205722+ 1205882cos1205722minus 119903119895120601) sin120601


1198861119898 + 119903119895) cos120601


1198861119898 cos120572

2)

(119903119895sin1205722)

(34)

where 0 le 1198972le (2ℎlowast

1198861119898 minus 119891

lowast

2119898) cos120572

2

Similarly the deduced meshing equation of profile 119865119862 is


3= 0 (35)

The corresponding gear profiles satisfy the following equa-tions

1199092= (1198973sin1205723+ 1205882cos1205722+

ℎlowast

1198861119898

sin1205722

minus 119903119895120601) cos120601

+ (1198973cos1205723+ ℎlowast

1198861119898 minus 119891

lowast

2119898 + 119903119895) sin120601

1199102= minus (119897

3sin1205723+ 1205882cos1205722+

ℎlowast

1198861119898

sin1205722

minus 119903119895120601) sin120601

+ (1198973cos1205723+ ℎlowast

1198861119898 minus 119891

lowast

2119898 + 119903119895) cos120601

120601 = (1198973+ ℎlowast

1198861119898 cos120572

3minus 119891lowast

2119898 cos120572

3

+1205882cos1205722sin1205723+

ℎlowast

1198861119898

sin1205722

sin1205723)

times (119903119895sin1205723)minus1

(36)

where 0 le 1198973le (ℎlowast

1198861119898 minus 119891

lowast

2119898) cos120572

3

34 Simulation of Meshing and Contact The gear profilewith and without modification was generated according tothe derived equations in Section 33 and the correspondingfinite element models were also established for subsequentnumerical simulation

The solid models are constructed with the followingdesign parameters The number of the teeth 119885

1= 1198852

= 19normalmodule119898 = 287mm face width 119887 = 16mm normalpressure angle of coast side flank 120572

1= 20∘ normal pressure

angle of driving side flank 1205722= 2269

∘ addendum coefficientℎlowast

1198861= 1002 and tip clearance coefficient 119888

lowast

1= 0364

The solid model was discretized with 8-node hexahedronisoparametric elements The material was 20CrMnTi steelwith the properties of Youngrsquos modulus 119864 = 20711989011Pa andPoissonrsquos ratio 120583 = 025 The motor power 119875 = 1641 kwthe rated speed 119899

1of driving gear was 3000 rmin and the

applied torque 119879 = 5225Nsdotm according to the equation119879 = 60119875(2120587119899

1)

The considerations for the finite element models are asfollows

(1) The 3D models are adopted due to the differencesof tooth surface contact stress along the tooth widthdirection Since the overlap ratio of gear pair is 1463it is suitable to have the solid model with five gearteeth and inner ring The simplification is sufficientand highly efficient for the simulationThe two-toothgeometric model is shown in Figure 5(a)

(2) Gear meshing process is a dynamic problem that themeshing position changes with the change of time Sodynamic meshing simulation can be discrete into aseries of quasistatic processes After establishing thecorresponding quasistaticmathematical model usingthe existing computer technology fulfills the iterativecalculation repeatedly to get suitable results Finiteelement models of the gear drive can be automatically


(a) (b)

(c)

X

Y

Z

(d)

Figure 5 Two teeth gear models for simulation (a) the solid model (b) the finite element model (c) contact pair and (d) the whole modelwith boundary conditions and applied force

obtained for any meshing position Through rotatingthe gear drive we can achieve the simulation in thewhole meshing cycle

(3) The gear solid model was discretized with 8-nodehexahedron isoparametric elements as shown inFigure 5(b)

(4) The tooth contact is performed by contact elements Acontact pair consists of a target surface and a contactsurface One contact pair is shown in Figure 5(c)

(5) Boundary conditions were imposed automatically ineach cylindrical coordinate system The whole fiveteeth mating gear finite element models are shown inFigure 5(d)

(6) Nodes of the inner rim of driven gear were fixedby restricting all degrees of freedom nodes of theinner rim of driving gear are fixed by restrictingthe radial and axial degree of freedom and releasingthe rotational degree of freedom the transmissiontorque was exerted to each node in the form ofcircumferential force The relationship between the

number of nodes in the driving gear rim 119899number andthe radius of rim 119903rim meets the equation

119865 =119879

119899number times 119903rim (37)

Assumption of load distribution in the contact area is notrequired since the contact algorithm of the general computerprogram is used to get the contact area and stresses by appli-cation of the torque to the driving gearwhile the driven gear isconsidered at rest Both finite element models of driving anddriven gears are rotated to simulate whole meshing cycle andnonlinear contact analysis was accomplished by the contactalgorithm in the finite element program

Through the numerical simulation we can predict thecharacteristic of the gear drive comprehensively and canreveal the principle of gear mating in the design process

4 Calculation Results and Discussions

The purposes of modification are to improve meshing per-formance and to realize load distribution balance The twoaspects can be evaluated by transmission errors and loadsharing ratio


The parameter 119891 (Figure 3) was defined as the modifiedratio

119891 =119891lowast

2

ℎlowast

1198861

=119867

1198670

(38)

The parameter Δ120572 was defined as the modified anglemeeting

Δ120572 = 1205723minus 1205722 (39)

The pitch point 119875119895where the gears began to mate with each

other was defined as the valid starting point The drivengear rotated counter-clockwise was defined as the positivedirection and the clockwise direction was chosen as thenegative direction

The theoretic angle at the start of active profile (SAP) is1465∘ and theoretic angle at the end of active profile (EAP)

is minus1465∘ according to the formula [26]Therefore the actual

mating angle of one gear tooth should be within the rangeof 0∘ to 2929186

∘ The theoretic angles of the lowest pointof single tooth contact (LPSTC) and the highest point forsingle tooth contact (HPSTC) are 43

∘ and minus43∘ calculated

by formula respectively [2 26] The investigators chose therotation angles of symmetric spur gear pairs ranging from 15

∘

to minus165∘ with an interval of 05∘

41 Influence of the Transmission Errors Transmission error(TE) is one of the most important factors causing gearnoise and vibration TE refers to the difference between thedisplacements of gears in reality and those in theory TE iscalculated as the relative displacement at the pitch point inthe line of action between the two engaged gears using smallangle approximations for any relative rotations as shown in

TE = 1199062minus 1199061 (40)

where 1199062and 1199061are the displacements of engaged gears at the

pitch point in the line of action The resulting transmissionerror is specified as a distanceThe displacement of each gearcan be obtained by the simulation results

411 The Transmission Errors under Different Relief LengthsWhen the torque (119879 = 5225N sdotm) was applied the modifiedangleΔ120572 = 05

∘ and the modified ratio values119891were 13 1223 and 1 The TE curves of driven gear in a meshing cycleunder different relief lengths are shown in Figure 6 TE peakamplitudes and percentage increments of double toothmesh-ing area compared with single tooth meshing area for theunmodified and modified gears under different relief lengthsare given in Table 1 Through analysis based on Figure 6the following results can be drawn

(1) Mating zones 119860 and 119862 are the single tooth meshingareas and mating zone 119861 is double tooth meshingarea In the unmodifiedmodel there exist convex linein zone 119861 and concave lines in zones 119860 and 119862 Thegear has lower mating stiffness in zone 119861 thereforethe amplitude of TE in zone 119861 is greater than that inzones 119860 and 119862 The lowest point in zones 119860 and 119862 is

0

0005

001

0015

002

0025

003

0035

004

Rotation angle (deg)

Tran

smiss

ion

erro

r (m

m)

AB

C

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19

Unmodified Δ120572 = 05∘ f = 23Δ120572 = 05∘ f = 1Δ120572 = 05∘ f = 13

Δ120572 = 05∘ f = 12

Figure 6 The transmission errors for driven gear under differentrelief lengths Δ120572 = 05

∘

called the negative TE peak point The TE amplitudein zones 119860 and 119862 is 39782 lower than in zone 119861

(Table 1)(2) Compared with the unmodified model the relief

length is a critical factor for determining TE curveshape When 119891 = 13 zone 119861 has wider space Zones119860 and 119862 have same concave shape as the unmodifiedmodel but have a sharp point at the lowest positionWhen 119891 = 12 zone 119861 also has wider range thanthe unmodified model The curves in zones 119860 and119862 are rough and uneven When 119891 = 23 there isobvious difference between single tooth meshing anddouble tooth meshing areas The curves in the zones119860 and 119862 also change from concave to convex shapeand its range decreases with increasing zone 119861 When119891 = 1 the relief starts from the pitch point sozone 119861 changes from a line to a point The curve inzones119860 and119862 changes from concave to convex shapedramatically

(3) The relief length has great effect on the TE amplitudeespecially the double tooth meshing areas When119891 = 13 the curve in zones 119860 and 119862 has sharpernegative peak but even amplitude compared with theunmodified model The negative peak amplitude inzones119860 and119862 is 657885120583m 400133 lower than thepeak amplitude in zone119861When119891 = 12 the negativepeak amplitude in zones 119860 and 119862 is 1164246 120583m1607 higher than the peak amplitude in zone 119861When 119891 = 23 the peak amplitude in zones 119860 and119862 is 1571421 120583m which is 5598 higher than thepeak amplitude in zone 119861 When 119891 = 1 the negativepeak amplitude in zones 119860 and 119862 is 2489898 120583m1426177 higher than the peak amplitude in zone 119861

(Table 1)(4) The TE peak amplitude of modified model in zone 119861

changes little when compared with the unmodifiedone with the increasing modified factor 119891 The TEpeak amplitude of the modified model in zones 119860


Table 1 TE peak amplitudes and percentage increments in double tooth meshing area when compared with single tooth meshing area of theunmodified and modified gears under different relief lengths

Modification dimensions TE peak amplitude in zones Aand C (120583m)

TE peak amplitude in zone B(120583m)

Percentage of TE peakincrement ()

Unmodified gear 63684 1057557 minus39782Δ120572 = 05∘ 119891 = 13 657885 1096719 minus400133Δ120572 = 05∘ 119891 = 12 1164246 1003023 1607371Δ120572 = 05∘ 119891 = 23 1571421 1007415 5598547Δ120572 = 05∘ 119891 = 1 2489898 1026264 1426177

and 119862 increases as the relief factor 119891 increases Itwas found that the relief factor 119891 = 12 is relativelyoptimal to make the curve of TE level and smoothWhen 119891 = 13 or 119891 = 23 the relief lengthis sufficient and when 119891 = 1 the relief length isexcessive

412 The Transmission Errors under Different Relief PressureAngles When the applied torque was 5225Nsdotm the modi-fied angle values Δ120572 were 03

∘ 05∘ or 07∘ and the modified

ratio values 119891 were 13 12 23 or 1 The TE curves of drivengear in a meshing cycle under different relief pressure anglesare shown in Figure 7 TE peak amplitudes of modified gearsunder different relief lengths are given in Table 2 Throughthe analysis based on Figure 7 the following results can beobtained

(1) The relief pressure angle has great effect on thevariation range of the TE when the modified ratiois a constant As the Δ120572 increases the value of TEincreases

(2) When 119891 = 13 the maximum value of TE insingle tooth meshing area increases with increasingΔ120572 The negative peak amplitudes in double toothmeshing area change very little (Table 2)Thewidth ofdouble tooth meshing areas becomes narrower withincreasing Δ120572 When 119891 = 12 the shape of curvein double tooth meshing areas changes from concavewith two peaks into convex with one peak The peakamplitudes increase with increasing Δ120572 When 119891 =

23 or 119891 = 1 the TE values increase with increasingΔ120572

(3) The modification length is the main factor thatdetermines the shape of the transmission error curveAppropriate modification length can make the geartransmission error curve in the process of alternatingsingle and double teeth smooth thereby reducingthe meshing dynamics Modification angle is themain factor influencing the peak transmission errorSuitable pressure angle can make the transmissionerror fluctuation decrease In present study it wasfound that the TE curve would be level and smoothwhen 119891 = 12 120572 = 05

∘ for the assumed torque

42 Influence of the Contact Force

421 The Load Sharing Ratio under Different Relief LengthsDue to the material having flexibility the deformed teethcan rotate bend and shear and the tooth flanks have beenflattened at the contact points In this situation the load variesoutside the normal contact path of asymmetric spur gearsThe curves of tooth load sharing ratio can clearly show thecomplex variation from the tooth running in to runningout of contact The circumference force and radial force ofmeshing gear satisfy the following equations respectively

119865119905=

119879

119903119895

(41)

119865119903= 119865119905tan120572 (42)

The theoretic total meshing force is given by

119865119899= radic1198652119903+ 1198652

119905 (43)

where 119903119895

= 1198981199112 Then the total meshing force 119865119899can

be determined to be 2075N by substituting the involvedparameters A single gear meshing force 119865sn can be obtainedby simulation results The load sharing ratio is given by

119870 =119865sn119865119899

(44)

Taking one gear tooth for consideration in thewholemeshingcycle distribution of the contact force under different relieflengths is given in Figure 8 Through analysis based onFigure 8 the following results can be obtained

(1) The load sharing ratio is 100 in single toothmeshingarea and 35ndash46 in double tooth meshing areasof unmodified model There is a sharp drop at theLPSTC and the HPSTC

(2) The load sharing ratio in double tooth meshing areasis 0ndash100 The slopes of the line increase and thespace becomes narrower with the increasing relieflength in double tooth meshing areas

(3) When the rotation angles are within the range of15∘ndash96∘ and minus95∘ndash15∘ the change amplitude of loadsharing ratio decreases as 119891 increases When therotation angle is in the range of 96∘ndash4∘ and from minus4∘to 96∘ the change amplitude of load sharing ratioincreases as 119891 increases


0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)

minus17 minus13 minus9 minus5 minus1 3 7 11 15 19

Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)

Figure 7 The transmission errors for driven gear under relief pressure angles (a) 119891 = 13 (b) 119891 = 12 (c) 119891 = 23 and (d) 119891 = 1

0

02

04

06

08

1

12


Load

shar

ing

ratio

Unmodified

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19

Δ120572 = 05∘ f = 23Δ120572 = 05∘ f = 1Δ120572 = 05∘ f = 13

Δ120572 = 05∘ f = 12

Figure 8 The load sharing ratio against different relief lengths input at Δ120572 = 05


0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)

Figure 9 The load sharing ratio against different relief lengths and relief angles input (a) 119891 = 13 (b) 119891 = 12 (c) 119891 = 23 and (d) 119891 = 1

Table 2 TE peak amplitudes of the modified gears under different relief lengths (120583m)

Single tooth meshing area Double tooth meshing areaΔ120572 = 03∘ Δ120572 = 05∘ Δ120572 = 07∘ Δ120572 = 03∘ Δ120572 = 05∘ Δ120572 = 07∘

119891 = 13 1091595 1096719 1096902 654042 657885 654591119891 = 12 1111725 1164246 116937 892308 1003023 1278804119891 = 23 1002108 1007415 1015467 1217865 1571421 1910337119891 = 1 1023519 1026264 1048041 1757715 2489898 321714


422The Contact Force under Different Relief Pressure AnglesWhen the applied torque was 52250Nm the modified anglevaluesΔ120572were 03∘ 05∘ or 07∘ and themodified ratio values119891 were 13 12 23 or 1 Through analysis based on Figure 9the following results can be obtained

(1) When the rotation angles are in the range of 15∘ndash96∘and from minus95∘ to 15∘ the amplitude of load sharingratio declines as 119891 increases When the rotationangles are in the range of 96∘ndash4∘ and minus4∘ndash96∘ thechange amplitude of load sharing ratio increases as 119891

increases(2) The load sharing ratio is 100 in single toothmeshing

area The slope of the line increases and the spacerange becomes narrower with the increasing relieflength in double tooth meshing areas

(3) Length of modification and modification angle deter-mines the shape of the curve of load distribution andproportion In the analysis of load distribution 119891 =

12 120572 = 05∘ is reasonable

5 Conclusions

Based on the research performed the following conclusionscan be drawn

(1) A design method for the geometric profile of asym-metric spur gear was proposed in which the geomet-ric profile of the gear can be obtained directly accord-ing to the rack-cutter profile in the design processThegeometric equations of rack-cutter were conductedand the corresponding equations of generated gearwere deduced based on the conjugated theory

(2) A design method for the modification of asymmetricspur gear was proposed in which the modificationof the gear can be obtained via the pressure anglemodification of the rack-cutter The geometric equa-tions of the modified driving side of rack-cutter wereconducted and the corresponding equations of thegenerated gear were deduced based on the conjugatedtheory

(3) The designed and modified gear models can beadopted to carry out the subsequent mating analysisThe research results indicated that the involute mod-ification of the generated gear contributed to bettertransmission performance

Nomenclature

119898 Metric module mm119911 Tooth number120572119894(119894 = 1 2 3) Pressure angle for asymmetric spur gear

drive (119894 = 1) for coast side profile (119894 = 2)

for driving side profile (119894 = 3) formodified driving side profile

120588119894(119894 = 1 2) Radius of rack-cutter fillet (119894 = 1) for coast

side profile (119894 = 2) for driving side profileℎlowast

1198861 Addendum coefficient

119888lowast

1 Tip clearance coefficient

119864 End point of coast side straight line119863 Start point of coast side straight line119862 End point of driving side straight line119861 Start point of driving side straight line119860 Bottom point of rack-cutter fillet curve119874119894(119894 = 119888 119889) Center point of rack-cutter fillet (119894 = 119888) for

coast side profile (119894 = 119889) for driving sideprofile

1198761 1198762 1198765 Arbitrary point in straight line119863119864 119861119862 119865119862

1198763 1198764 Arbitrary point in rack-cutter fillet curve

119860119863 119860119861

1198971 1198972 1198973 Length of straight line 119863119876

1 1198611198762 1198651198765

1205791 1205792 Angle of

1198601198741198891198763119860

1198741198881198764

119891lowast

2 Modified coefficient

119891 Modified ratioΔ120572 Modified angle119878119894(119894 = 1 2 119895) Coordinate systems (119894 = 1) for rack-cutter

(119894 = 2) for conjugate gear (119894 = 119895) for fixedframe

119903119895 Radius of pitch cylinder of gear

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the Natural Science Foun-dation of China (no 51105287) and the Key Research andDevelopment Project of New Product and New Technologyof Hubei Province (no 2012BAA08003) for the support givento this research

References

[1] J D SmithGearNoise andVibrationMarcelDekkerNewYorkNY USA 2003

[2] F L Litvin Gear Geometry and Applied Theory CambridgeUniversity Press Cambridge UK 2nd edition 2004

[3] F L Litvin Q Lian and A L Kapelevich ldquoAsymmetricmodified spur gear drives reduction of noise localization ofcontact simulation of meshing and stress analysisrdquo ComputerMethods in Applied Mechanics and Engineering vol 188 no 1pp 363ndash390 2000

[4] C Song C Zhu T C Lim and T Peng ldquoParametric analysisof gear mesh and dynamic response of loaded helical beveloidtransmission with small shaft anglerdquo Journal of MechanicalDesign vol 134 no 8 Article ID 084501 2012

[5] J Y Tang Z H Hu L J Wu and S Y Chen ldquoEffect of statictransmission error on dynamic responses of spiral bevel gearsrdquoJournal of Central South University vol 20 no 3 pp 640ndash6472013

[6] 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


[7] F L Litvin A Fuentes I Gonzalez-Perez L Carvenali KKawasaki and R F Handschuh ldquoModified involute helicalgears computerized design simulation of meshing and stressanalysisrdquo Computer Methods in Applied Mechanics and Engi-neering vol 192 no 33-34 pp 3619ndash3655 2003

[8] M Faggioni F S Samani G Bertacchi and F PellicanoldquoDynamic optimization of spur gearsrdquoMechanism andMachineTheory vol 46 no 4 pp 544ndash557 2011

[9] M Barbieri G Bonori and F Pellicano ldquoOptimum profilemodifications of spur gears by means of genetic algorithmsrdquoJournal of Sound and Vibration vol 331 no 21 pp 4825ndash48292012

[10] 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

[11] I Gonzalez-Perez J L Iserte and A Fuentes ldquoImplementationof Hertz theory and validation of a finite element model forstress analysis of gear drives with localized bearing contactrdquoMechanism and Machine Theory vol 46 no 6 pp 765ndash7832011

[12] K J Huang and H W Su ldquoApproaches to parametric ele-ment constructions and dynamic analyses of spurhelical gearsincluding modifications and undercuttingrdquo Finite Elements inAnalysis and Design vol 46 no 12 pp 1106ndash1113 2010

[13] Y Hu Y Shao Z Chen and M J Zuo ldquoTransient meshingperformance of gears with different modification coefficientsand helical angles using explicit dynamic FEArdquo MechanicalSystems and Signal Processing vol 25 no 5 pp 1786ndash1802 2011

[14] J Wei W Sun and L Wang ldquoEffects of flank deviation on loaddistributions for helical gearrdquo Journal of Mechanical Science andTechnology vol 25 no 7 pp 1781ndash1789 2011

[15] Y-J Wu J-J Wang and Q-K Han ldquoStaticdynamic contactFEA and experimental study for tooth profile modification ofhelical gearsrdquo Journal ofMechanical Science and Technology vol26 no 5 pp 1409ndash1417 2012

[16] S T Li ldquoFinite element analyses for contact strength andbending strength of a pair of spur gears with machiningerrors assembly errors and tooth modificationsrdquo Mechanismand Machine Theory vol 42 no 1 pp 88ndash114 2007

[17] V V Simon ldquoInfluence of tooth modifications on tooth contactin face-hobbed spiral bevel gearsrdquo Mechanism and MachineTheory vol 46 no 12 pp 1980ndash1998 2011

[18] Z Chen and Y Shao ldquoMesh stiffness calculation of a spurgear pair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013

[19] S Sankar and M Nataraj ldquoProfile modification-a designapproach for increasing the tooth strength in spur gearrdquoInternational Journal of Advanced Manufacturing Technologyvol 55 no 1ndash4 pp 1ndash10 2011

[20] J Wang Y Wang and Z Huo ldquoFinite element residual stressanalysis of planetary gear toothrdquo Advances in MechanicalEngineering vol 2013 Article ID 761957 12 pages 2013

[21] S Li and A Kahraman ldquoA tribo-dynamic model of a spur gearpairrdquo Journal of Sound and Vibration vol 332 no 20 pp 4963ndash4978 2013

[22] M Kolivand and A Kahraman ldquoA general approach to locateinstantaneous contact lines of gears using surface of roll anglerdquoJournal of Mechanical Design vol 133 no 1 Article ID 0145032011

[23] T Eritenel and R G Parker ldquoNonlinear vibration of gears withtooth surfacemodificationsrdquo Journal of Vibration and Acousticsvol 135 no 5 Article ID 051005 2013

[24] O Alipiev ldquoGeometric design of involute spur gear drives withsymmetric and asymmetric teeth using the Realized PotentialMethodrdquoMechanism andMachineTheory vol 46 no 1 pp 10ndash32 2011

[25] N L Pedersen ldquoImproving bending stress in spur gears usingasymmetric gears and shape optimizationrdquo Mechanism andMachine Theory vol 45 no 11 pp 1707ndash1720 2010

[26] S-C Hwang J-H Lee D-H Lee S-H Han and K-H LeeldquoContact stress analysis for a pair of mating gearsrdquoMathemati-cal and Computer Modelling vol 57 no 1-2 pp 40ndash49 2013

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


directly without CAD geometric model The gear profileequations were described via C programming language Theeffects of modification coefficients and helical angles on thetransientmeshing performance of the gears were investigatedusing the method of explicit dynamic FEA from the perspec-tive of energy by Hu et al [13] A qualitative analysis aboutthe effects of flank deviation on load distribution for helicalgear was performed using FEM by Wei et al [14] Wu etal [15] proposed a tooth profile modification approach forthe helical pairs and the static contact process was analyzedusing FEA and experimental tests Li [16] applied 3D FEMto conduct surface contact stress and root bending stresscalculations for a pair of spur gears considering machiningerrors assembly errors and tooth modifications Simon [17]researched on the influence of tooth modifications inducedby machine tool setting and head-cutter profile variationson tooth contact characteristics in face-hobbed spiral bevelgears Chen and Shao [18] proposed a general analyticalmeshing stiffness model to establish the relationship betweenthe gear tooth errors and the total mesh stiffness whereinload sharing among different tooth pairs duringmeshing andstatic transmission errors under loadwere considered Sankarand Nataraj [19] proposed a modification design approachfor increasing the tooth strength in spur gear Wang et al[20] proposed a method to simulate residual stress field ofplanetary gear Li and Kahraman [21] proposed a tridynamicsmodel for spur gear pairs The proposed model couples amixed elastohydrodynamic lubrication model of a spur gearpair with a transverse torsional dynamicmodel Kolivand andKahraman [22] defined new general concept based on theconstruction of a surface of roll angle to simplify the task oflocating instantaneous contact lines of any type of gearingWith the surface of roll angle defined instantaneous contactlines are determined by a novel approach based on an analogyto parallel-axes gears Eritenel and Parker [23] provided ananalytical solution for the nonlinear vibration of gear pairsthat exhibit partial and total contact loss

Moreover more and more attention has been paid toasymmetric gear recently Litvin et al [3] proposed themodified geometric model of an asymmetric spur gear drivewhich integrated an involute gear and a double crownedpinion and simulated the meshing and contact process ofthe spur gears with proposed geometric parameters usingcomputer-aided design method Alipiev [24] proposed a newgeometric design method for the symmetric and asymmetricinvolute meshing in which the gear load sharing ratio is equalto its expected value Pedersen [25] observed the use of twonew standard cutting tools for the asymmetric gears since theshape of the gear profile influences its bending strength

In the gear design process the traditional method onlygives basic parameters of gear structure and it is difficultto obtain precise model for the actual generated gear Inthis paper the equations of gear tooth profile were deducedbased on a given rack-cutter and the accurate geometricmodel was established in the design process In the traditionalmachining process of modified gear a standard gear withoutmodification is generated at first and themodification is con-ducted subsequently In order to enhance the manufacturingefficiency the pressure angle modification is carried out on

E

D

A

B

C

12057211205722

l1l2

1205791

1205792 1205881 1205882

Q1

Q3

Q4

Q2

y1

x1

Oc

Od

O1

hlowasta1m

hlowasta1m

clowast1 m

Figure 1 Gear rack-cutter normal profile and tool coordinatesystem

the rack-cutter so the desired modified spur gear can beobtained through one time rapid machining

Based on the proposed gear design and modificationmethod a FEM of asymmetric involute gear was establishedwhere computer-aided designs were adopted to establish thegeometrymodel of the spur gears finally themeshing processwas simulated

The present paper mainly consists of the following threeaspects

(1) The equations of a new rack-cutter profile for asym-metric gear were established The pressure anglemodification in rack-cutter design was proposed andthe driving side of rack-cutter profile equations wasdeduced The given rack-cutter profile has differentpressure angles and fillet radius in the driving sideand coast side The generated asymmetric spur gearprofiles have different pressure angles and filletsaccordingly (Section 2)

(2) According to conjugate theory the equation of asym-metric spur gear was derived by the forming rack-cutter tool Computer-aided design of spur gears withproposed geometric parameters and the simulationof their meshing were conducted using the proposedmethods (Section 3)

(3) The transmission error and the load sharing ratioof generated gears were studied The transmissionprinciples of asymmetric spur gear were discussed(Section 4)

2 Geometric Characteristics ofthe Rack-Cutter Profile

In order to obtain the precise gear model the geometriccharacteristic of the corresponding rack-cutter profile isrequired The given rack-cutter has different pressure anglesand fillet radius in the driving side and coast side

21 The Rack-Cutter Normal Profile The typical rack-cutterconsists of straight lines and tip circular fillet which cangenerate involutes and fillets of typical involute tooth respec-tively As depicted in Figure 1 the tool coordinate system 119878

1

is rigidly fixed to the gear rack-cutter and the coordinate axis





following equations




1198861119898

0 le 1198971le

2ℎlowast

1198861119898

cos1205721

(1)


1is a



1198861is



1199091= 1198972sin1205722+ 1205882cos1205722


1198861119898

0 le 1198972le

2ℎlowast

1198861119898

cos1205722

(2)


2





1199091= minus 120588

1sin 1205791

1199101= minus 120588

1(cos 120579

1minus sin120572

1) minus ℎlowast

1198861119898

0 le 1205791le

120587

2minus 1205721

(3)


1198601198741198881198763and119876

3is an arbitrary


1199091= 1205882sin 1205792

1199101= minus 120588

2(cos 120579

2minus sin120572

2) minus ℎlowast

1198861119898

0 le 1205792le

120587

2minus 1205722

(4)


1198601198741198881198764and119876

4is an arbitrary


E

D

A

B

C

F

1205722

hlowasta1m

Od1205881 1205882

Oc

y11205723

flowast2 m

O1 Q2

Q4

Q5

l2

l3

hlowasta1m

clowast1 m

x1



1205881=

119888lowast

1119898


1205882=

119888lowast

1119898


(5)

where 119888lowast



1

is

997888119905 =

1198891199091

119889V9978881198941+

1198891199101

119889V9978881198951 (6)


997888119899119899=

119905 times9978881198961

1003816100381610038161003816100381610038161003816119905 times

9978881198961

1003816100381610038161003816100381610038161003816

9978881198961= [0 0 1] (7)


derived as

997888997888997888997888119899119899 119863119864

= [

[

cos1205721

minus sin1205721

0

]

]

997888997888997888997888119899119899 119861119862

= [

[

minus cos1205722

minus sin1205722

0

]

]

997888997888997888997888119899119899 119860119863

= [

[

minus sin 1205791

minus cos 1205791

0

]

]

997888997888997888997888119899119899 119860119861

= [

[

sin 1205792

minus cos 1205792

0

]

]

(8)




lowast



H

L

Original involute

Modified involute

Gear tip

MH0

Pj

ra

rj

O2

profile

profile





0



1199091= 1198972sin1205722+ 1205882cos1205722


1198861119898

0 le 1198972le

2ℎlowast

1198861119898 minus 119891

lowast

2119898

cos1205722

(9)

where 119891lowast


3is the


119899119899 119861119865


997888997888997888119899119899 119861119865

= [

[

minus cos1205722

minus sin1205722

0

]

]

(10)


1199091= 1198973sin1205723+ 1205882cos1205722+

ℎlowast

1198861119898

sin1205722

1199101= 1198973cos1205723+ ℎlowast

1198861119898 minus 119891

lowast

2119898

0 le 1198973le

ℎlowast

1198861119898 minus 119891

lowast

2119898

cos1205723

(11)



997888997888997888997888119899119899 119865119862

= [

[

minus cos1205723

minus sin1205723

0

]

]

(12)



O

P

y1 yj

1

n

y2O1

120596 O2 Oj

r1

rj

120601

z1

x2

x1

xj

z2 zj

s = rj120601




1198782 and 119878




1198742119873 =


1was signed



equation9978881199032= 11987221

9978881199031 (13)


11987221

= [

[



0 0 1 0

]

]

(14)


1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)

(15)







997888V1=

997888120596 times

9978889978889978881198742119874 (16)



119904 = 119903119895120601 (17)




9978881199031= 1199091

9978881198941+ 1199101

9978881198951 (18)


997888V119899=

997888V1minus

997888V2 (19)


2given by

997888V2=

997888120596 times

9978881199031+

99788899788899788899788811987411198742times

997888120596 (20)

where

997888V1= minus V

1

9978881198941

= minus1205961199031

9978881198941

997888120596 = 120596

9978881198961

99788899788899788899788811987411198742= 1199031198951206019978881198941minus 119903119895

9978881198951

(21)


997888V119899= 120596 [119910

1

9978881198941+ (119903119895120601 minus 1199091)9978881198951] (22)


997888120593 =

997888119899119899sdot997888V119899=

9978880 (23)




119863119864 1199091sin1205721+ 1199101cos1205721minus 119903119895120601 sin120572

1= 0


2= 0


1= 0

119860119861 1199091cos 1205792+ 1199101sin 1205792minus 119903119895120601 sin 120579

2= 0

(24)



1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)

1199091sin1205721+ 1199101cos1205721minus 119903119895120601 sin120572

1= 0

(25)


1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)


2= 0

(26)


1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)


1= 0

(27)


1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)

1199091cos 1205792+ 1199101sin 1205792minus 119903119895120601 sin 120579

2= 0

(28)


2from



1199092= (minus119897



1198861119898 + 119903119895) sin120601

1199102= (1198971sin1205721+ 1205881cos1205721+ 119903119895120601) sin120601


1198861119898 + 119903119895) cos120601


1198861119898 cos120572

1)

(119903119895sin1205721)

0 le 1198971le

2ℎlowast

1198861119898

cos1205721

(29)


1199092= (1198972sin1205722+ 1205882cos1205722minus 119903119895120601) cos120601


1198861119898 + 119903119895) sin120601

1199102= minus (119897

2sin1205722+ 1205882cos1205722minus 119903119895120601) sin120601


1198861119898 + 119903119895) cos120601


1198861119898 cos120572

2)

(119903119895sin1205722)

0 le 1198972le

2ℎlowast

1198861119898

cos1205722

(30)



1199092= (minus120588

1sin 1205791minus 119903119895120601) cos120601

+ (minus1205881(cos 120579

1minus sin120572

1) minus ℎlowast

1198861119898 + 119903119895) sin120601

1199102= (1205881sin 1205791+ 119903119895120601) sin120601

+ (minus1205881(cos 120579

1minus sin120572

1) minus ℎlowast

1198861119898 + 119903119895) cos120601


1198861119898 sin 120579

1)

(119903119895sin 1205791)

0 le 1205791le

120587

2minus 1205721

(31)


1199092= (1205882sin 1205792minus 119903119895120601) cos120601

+ (minus1205882(cos 120579

2minus sin120572

2) minus ℎlowast

1198861119898 + 119903119895) sin120601

1199102= minus (120588

2sin 1205792minus 119903119895120601) sin120601

+ (minus1205882(cos 120579

2minus sin120572

2) minus ℎlowast

1198861119898 + 119903119895) cos120601


1198861119898 sin 120579

2)

(119903119895sin 1205792)

0 le 1205792le

120587

2minus 1205722

(32)




2= 0 (33)


1199092= (1198972sin1205722+ 1205882cos1205722minus 119903119895120601) cos120601


1198861119898 + 119903119895) sin120601

1199102= minus (119897

2sin1205722+ 1205882cos1205722minus 119903119895120601) sin120601


1198861119898 + 119903119895) cos120601


1198861119898 cos120572

2)

(119903119895sin1205722)

(34)


1198861119898 minus 119891

lowast

2119898) cos120572

2



3= 0 (35)


1199092= (1198973sin1205723+ 1205882cos1205722+

ℎlowast

1198861119898

sin1205722

minus 119903119895120601) cos120601

+ (1198973cos1205723+ ℎlowast

1198861119898 minus 119891

lowast

2119898 + 119903119895) sin120601

1199102= minus (119897

3sin1205723+ 1205882cos1205722+

ℎlowast

1198861119898

sin1205722

minus 119903119895120601) sin120601

+ (1198973cos1205723+ ℎlowast

1198861119898 minus 119891

lowast

2119898 + 119903119895) cos120601

120601 = (1198973+ ℎlowast

1198861119898 cos120572

3minus 119891lowast

2119898 cos120572

3

+1205882cos1205722sin1205723+

ℎlowast

1198861119898

sin1205722

sin1205723)

times (119903119895sin1205723)minus1

(36)


1198861119898 minus 119891

lowast

2119898) cos120572

3



1= 1198852






lowast

1= 0364




1)





(a) (b)

(c)

X

Y

Z

(d)








119865 =119879








119891 =119891lowast

2

ℎlowast

1198861

=119867

1198670

(38)


Δ120572 = 1205723minus 1205722 (39)









∘



TE = 1199062minus 1199061 (40)






0

0005

001

0015

002

0025

003

0035

004


Tran

smiss

ion

erro

r (m

m)

AB

C

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19


Δ120572 = 05∘ f = 12


∘
























119865119905=

119879

119903119895

(41)

119865119903= 119865119905tan120572 (42)


119865119899= radic1198652119903+ 1198652

119905 (43)

where 119903119895



119870 =119865sn119865119899

(44)






0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)


0

02

04

06

08

1

12


Load

shar

ing

ratio

Unmodified

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19

Δ120572 = 05∘ f = 23Δ120572 = 05∘ f = 1Δ120572 = 05∘ f = 13

Δ120572 = 05∘ f = 12



0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)




119891 = 13 1091595 1096719 1096902 654042 657885 654591119891 = 12 1111725 1164246 116937 892308 1003023 1278804119891 = 23 1002108 1007415 1015467 1217865 1571421 1910337119891 = 1 1023519 1026264 1048041 1757715 2489898 321714








5 Conclusions





Nomenclature







119888lowast






119860119863 119860119861


1 1198611198762 1198651198765

1205791 1205792 Angle of

1198601198741198891198763119860

1198741198881198764

119891lowast







Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













following equations




1198861119898

0 le 1198971le

2ℎlowast

1198861119898

cos1205721

(1)


1is a



1198861is



1199091= 1198972sin1205722+ 1205882cos1205722


1198861119898

0 le 1198972le

2ℎlowast

1198861119898

cos1205722

(2)


2





1199091= minus 120588

1sin 1205791

1199101= minus 120588

1(cos 120579

1minus sin120572

1) minus ℎlowast

1198861119898

0 le 1205791le

120587

2minus 1205721

(3)


1198601198741198881198763and119876

3is an arbitrary


1199091= 1205882sin 1205792

1199101= minus 120588

2(cos 120579

2minus sin120572

2) minus ℎlowast

1198861119898

0 le 1205792le

120587

2minus 1205722

(4)


1198601198741198881198764and119876

4is an arbitrary


E

D

A

B

C

F

1205722

hlowasta1m

Od1205881 1205882

Oc

y11205723

flowast2 m

O1 Q2

Q4

Q5

l2

l3

hlowasta1m

clowast1 m

x1



1205881=

119888lowast

1119898


1205882=

119888lowast

1119898


(5)

where 119888lowast



1

is

997888119905 =

1198891199091

119889V9978881198941+

1198891199101

119889V9978881198951 (6)


997888119899119899=

119905 times9978881198961

1003816100381610038161003816100381610038161003816119905 times

9978881198961

1003816100381610038161003816100381610038161003816

9978881198961= [0 0 1] (7)


derived as

997888997888997888997888119899119899 119863119864

= [

[

cos1205721

minus sin1205721

0

]

]

997888997888997888997888119899119899 119861119862

= [

[

minus cos1205722

minus sin1205722

0

]

]

997888997888997888997888119899119899 119860119863

= [

[

minus sin 1205791

minus cos 1205791

0

]

]

997888997888997888997888119899119899 119860119861

= [

[

sin 1205792

minus cos 1205792

0

]

]

(8)




lowast



H

L

Original involute

Modified involute

Gear tip

MH0

Pj

ra

rj

O2

profile

profile





0



1199091= 1198972sin1205722+ 1205882cos1205722


1198861119898

0 le 1198972le

2ℎlowast

1198861119898 minus 119891

lowast

2119898

cos1205722

(9)

where 119891lowast


3is the


119899119899 119861119865


997888997888997888119899119899 119861119865

= [

[

minus cos1205722

minus sin1205722

0

]

]

(10)


1199091= 1198973sin1205723+ 1205882cos1205722+

ℎlowast

1198861119898

sin1205722

1199101= 1198973cos1205723+ ℎlowast

1198861119898 minus 119891

lowast

2119898

0 le 1198973le

ℎlowast

1198861119898 minus 119891

lowast

2119898

cos1205723

(11)



997888997888997888997888119899119899 119865119862

= [

[

minus cos1205723

minus sin1205723

0

]

]

(12)



O

P

y1 yj

1

n

y2O1

120596 O2 Oj

r1

rj

120601

z1

x2

x1

xj

z2 zj

s = rj120601




1198782 and 119878




1198742119873 =


1was signed



equation9978881199032= 11987221

9978881199031 (13)


11987221

= [

[



0 0 1 0

]

]

(14)


1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)

(15)







997888V1=

997888120596 times

9978889978889978881198742119874 (16)



119904 = 119903119895120601 (17)




9978881199031= 1199091

9978881198941+ 1199101

9978881198951 (18)


997888V119899=

997888V1minus

997888V2 (19)


2given by

997888V2=

997888120596 times

9978881199031+

99788899788899788899788811987411198742times

997888120596 (20)

where

997888V1= minus V

1

9978881198941

= minus1205961199031

9978881198941

997888120596 = 120596

9978881198961

99788899788899788899788811987411198742= 1199031198951206019978881198941minus 119903119895

9978881198951

(21)


997888V119899= 120596 [119910

1

9978881198941+ (119903119895120601 minus 1199091)9978881198951] (22)


997888120593 =

997888119899119899sdot997888V119899=

9978880 (23)




119863119864 1199091sin1205721+ 1199101cos1205721minus 119903119895120601 sin120572

1= 0


2= 0


1= 0

119860119861 1199091cos 1205792+ 1199101sin 1205792minus 119903119895120601 sin 120579

2= 0

(24)



1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)

1199091sin1205721+ 1199101cos1205721minus 119903119895120601 sin120572

1= 0

(25)


1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)


2= 0

(26)


1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)


1= 0

(27)


1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)

1199091cos 1205792+ 1199101sin 1205792minus 119903119895120601 sin 120579

2= 0

(28)


2from



1199092= (minus119897



1198861119898 + 119903119895) sin120601

1199102= (1198971sin1205721+ 1205881cos1205721+ 119903119895120601) sin120601


1198861119898 + 119903119895) cos120601


1198861119898 cos120572

1)

(119903119895sin1205721)

0 le 1198971le

2ℎlowast

1198861119898

cos1205721

(29)


1199092= (1198972sin1205722+ 1205882cos1205722minus 119903119895120601) cos120601


1198861119898 + 119903119895) sin120601

1199102= minus (119897

2sin1205722+ 1205882cos1205722minus 119903119895120601) sin120601


1198861119898 + 119903119895) cos120601


1198861119898 cos120572

2)

(119903119895sin1205722)

0 le 1198972le

2ℎlowast

1198861119898

cos1205722

(30)



1199092= (minus120588

1sin 1205791minus 119903119895120601) cos120601

+ (minus1205881(cos 120579

1minus sin120572

1) minus ℎlowast

1198861119898 + 119903119895) sin120601

1199102= (1205881sin 1205791+ 119903119895120601) sin120601

+ (minus1205881(cos 120579

1minus sin120572

1) minus ℎlowast

1198861119898 + 119903119895) cos120601


1198861119898 sin 120579

1)

(119903119895sin 1205791)

0 le 1205791le

120587

2minus 1205721

(31)


1199092= (1205882sin 1205792minus 119903119895120601) cos120601

+ (minus1205882(cos 120579

2minus sin120572

2) minus ℎlowast

1198861119898 + 119903119895) sin120601

1199102= minus (120588

2sin 1205792minus 119903119895120601) sin120601

+ (minus1205882(cos 120579

2minus sin120572

2) minus ℎlowast

1198861119898 + 119903119895) cos120601


1198861119898 sin 120579

2)

(119903119895sin 1205792)

0 le 1205792le

120587

2minus 1205722

(32)




2= 0 (33)


1199092= (1198972sin1205722+ 1205882cos1205722minus 119903119895120601) cos120601


1198861119898 + 119903119895) sin120601

1199102= minus (119897

2sin1205722+ 1205882cos1205722minus 119903119895120601) sin120601


1198861119898 + 119903119895) cos120601


1198861119898 cos120572

2)

(119903119895sin1205722)

(34)


1198861119898 minus 119891

lowast

2119898) cos120572

2



3= 0 (35)


1199092= (1198973sin1205723+ 1205882cos1205722+

ℎlowast

1198861119898

sin1205722

minus 119903119895120601) cos120601

+ (1198973cos1205723+ ℎlowast

1198861119898 minus 119891

lowast

2119898 + 119903119895) sin120601

1199102= minus (119897

3sin1205723+ 1205882cos1205722+

ℎlowast

1198861119898

sin1205722

minus 119903119895120601) sin120601

+ (1198973cos1205723+ ℎlowast

1198861119898 minus 119891

lowast

2119898 + 119903119895) cos120601

120601 = (1198973+ ℎlowast

1198861119898 cos120572

3minus 119891lowast

2119898 cos120572

3

+1205882cos1205722sin1205723+

ℎlowast

1198861119898

sin1205722

sin1205723)

times (119903119895sin1205723)minus1

(36)


1198861119898 minus 119891

lowast

2119898) cos120572

3



1= 1198852






lowast

1= 0364




1)





(a) (b)

(c)

X

Y

Z

(d)








119865 =119879








119891 =119891lowast

2

ℎlowast

1198861

=119867

1198670

(38)


Δ120572 = 1205723minus 1205722 (39)









∘



TE = 1199062minus 1199061 (40)






0

0005

001

0015

002

0025

003

0035

004


Tran

smiss

ion

erro

r (m

m)

AB

C

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19


Δ120572 = 05∘ f = 12


∘
























119865119905=

119879

119903119895

(41)

119865119903= 119865119905tan120572 (42)


119865119899= radic1198652119903+ 1198652

119905 (43)

where 119903119895



119870 =119865sn119865119899

(44)






0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)


0

02

04

06

08

1

12


Load

shar

ing

ratio

Unmodified

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19

Δ120572 = 05∘ f = 23Δ120572 = 05∘ f = 1Δ120572 = 05∘ f = 13

Δ120572 = 05∘ f = 12



0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)




119891 = 13 1091595 1096719 1096902 654042 657885 654591119891 = 12 1111725 1164246 116937 892308 1003023 1278804119891 = 23 1002108 1007415 1015467 1217865 1571421 1910337119891 = 1 1023519 1026264 1048041 1757715 2489898 321714








5 Conclusions





Nomenclature







119888lowast






119860119863 119860119861


1 1198611198762 1198651198765

1205791 1205792 Angle of

1198601198741198891198763119860

1198741198881198764

119891lowast







Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










H

L

Original involute

Modified involute

Gear tip

MH0

Pj

ra

rj

O2

profile

profile





0



1199091= 1198972sin1205722+ 1205882cos1205722


1198861119898

0 le 1198972le

2ℎlowast

1198861119898 minus 119891

lowast

2119898

cos1205722

(9)

where 119891lowast


3is the


119899119899 119861119865


997888997888997888119899119899 119861119865

= [

[

minus cos1205722

minus sin1205722

0

]

]

(10)


1199091= 1198973sin1205723+ 1205882cos1205722+

ℎlowast

1198861119898

sin1205722

1199101= 1198973cos1205723+ ℎlowast

1198861119898 minus 119891

lowast

2119898

0 le 1198973le

ℎlowast

1198861119898 minus 119891

lowast

2119898

cos1205723

(11)



997888997888997888997888119899119899 119865119862

= [

[

minus cos1205723

minus sin1205723

0

]

]

(12)



O

P

y1 yj

1

n

y2O1

120596 O2 Oj

r1

rj

120601

z1

x2

x1

xj

z2 zj

s = rj120601




1198782 and 119878




1198742119873 =


1was signed



equation9978881199032= 11987221

9978881199031 (13)


11987221

= [

[



0 0 1 0

]

]

(14)


1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)

(15)







997888V1=

997888120596 times

9978889978889978881198742119874 (16)



119904 = 119903119895120601 (17)




9978881199031= 1199091

9978881198941+ 1199101

9978881198951 (18)


997888V119899=

997888V1minus

997888V2 (19)


2given by

997888V2=

997888120596 times

9978881199031+

99788899788899788899788811987411198742times

997888120596 (20)

where

997888V1= minus V

1

9978881198941

= minus1205961199031

9978881198941

997888120596 = 120596

9978881198961

99788899788899788899788811987411198742= 1199031198951206019978881198941minus 119903119895

9978881198951

(21)


997888V119899= 120596 [119910

1

9978881198941+ (119903119895120601 minus 1199091)9978881198951] (22)


997888120593 =

997888119899119899sdot997888V119899=

9978880 (23)




119863119864 1199091sin1205721+ 1199101cos1205721minus 119903119895120601 sin120572

1= 0


2= 0


1= 0

119860119861 1199091cos 1205792+ 1199101sin 1205792minus 119903119895120601 sin 120579

2= 0

(24)



1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)

1199091sin1205721+ 1199101cos1205721minus 119903119895120601 sin120572

1= 0

(25)


1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)


2= 0

(26)


1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)


1= 0

(27)


1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)

1199091cos 1205792+ 1199101sin 1205792minus 119903119895120601 sin 120579

2= 0

(28)


2from



1199092= (minus119897



1198861119898 + 119903119895) sin120601

1199102= (1198971sin1205721+ 1205881cos1205721+ 119903119895120601) sin120601


1198861119898 + 119903119895) cos120601


1198861119898 cos120572

1)

(119903119895sin1205721)

0 le 1198971le

2ℎlowast

1198861119898

cos1205721

(29)


1199092= (1198972sin1205722+ 1205882cos1205722minus 119903119895120601) cos120601


1198861119898 + 119903119895) sin120601

1199102= minus (119897

2sin1205722+ 1205882cos1205722minus 119903119895120601) sin120601


1198861119898 + 119903119895) cos120601


1198861119898 cos120572

2)

(119903119895sin1205722)

0 le 1198972le

2ℎlowast

1198861119898

cos1205722

(30)



1199092= (minus120588

1sin 1205791minus 119903119895120601) cos120601

+ (minus1205881(cos 120579

1minus sin120572

1) minus ℎlowast

1198861119898 + 119903119895) sin120601

1199102= (1205881sin 1205791+ 119903119895120601) sin120601

+ (minus1205881(cos 120579

1minus sin120572

1) minus ℎlowast

1198861119898 + 119903119895) cos120601


1198861119898 sin 120579

1)

(119903119895sin 1205791)

0 le 1205791le

120587

2minus 1205721

(31)


1199092= (1205882sin 1205792minus 119903119895120601) cos120601

+ (minus1205882(cos 120579

2minus sin120572

2) minus ℎlowast

1198861119898 + 119903119895) sin120601

1199102= minus (120588

2sin 1205792minus 119903119895120601) sin120601

+ (minus1205882(cos 120579

2minus sin120572

2) minus ℎlowast

1198861119898 + 119903119895) cos120601


1198861119898 sin 120579

2)

(119903119895sin 1205792)

0 le 1205792le

120587

2minus 1205722

(32)




2= 0 (33)


1199092= (1198972sin1205722+ 1205882cos1205722minus 119903119895120601) cos120601


1198861119898 + 119903119895) sin120601

1199102= minus (119897

2sin1205722+ 1205882cos1205722minus 119903119895120601) sin120601


1198861119898 + 119903119895) cos120601


1198861119898 cos120572

2)

(119903119895sin1205722)

(34)


1198861119898 minus 119891

lowast

2119898) cos120572

2



3= 0 (35)


1199092= (1198973sin1205723+ 1205882cos1205722+

ℎlowast

1198861119898

sin1205722

minus 119903119895120601) cos120601

+ (1198973cos1205723+ ℎlowast

1198861119898 minus 119891

lowast

2119898 + 119903119895) sin120601

1199102= minus (119897

3sin1205723+ 1205882cos1205722+

ℎlowast

1198861119898

sin1205722

minus 119903119895120601) sin120601

+ (1198973cos1205723+ ℎlowast

1198861119898 minus 119891

lowast

2119898 + 119903119895) cos120601

120601 = (1198973+ ℎlowast

1198861119898 cos120572

3minus 119891lowast

2119898 cos120572

3

+1205882cos1205722sin1205723+

ℎlowast

1198861119898

sin1205722

sin1205723)

times (119903119895sin1205723)minus1

(36)


1198861119898 minus 119891

lowast

2119898) cos120572

3



1= 1198852






lowast

1= 0364




1)





(a) (b)

(c)

X

Y

Z

(d)








119865 =119879








119891 =119891lowast

2

ℎlowast

1198861

=119867

1198670

(38)


Δ120572 = 1205723minus 1205722 (39)









∘



TE = 1199062minus 1199061 (40)






0

0005

001

0015

002

0025

003

0035

004


Tran

smiss

ion

erro

r (m

m)

AB

C

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19


Δ120572 = 05∘ f = 12


∘
























119865119905=

119879

119903119895

(41)

119865119903= 119865119905tan120572 (42)


119865119899= radic1198652119903+ 1198652

119905 (43)

where 119903119895



119870 =119865sn119865119899

(44)






0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)


0

02

04

06

08

1

12


Load

shar

ing

ratio

Unmodified

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19

Δ120572 = 05∘ f = 23Δ120572 = 05∘ f = 1Δ120572 = 05∘ f = 13

Δ120572 = 05∘ f = 12



0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)




119891 = 13 1091595 1096719 1096902 654042 657885 654591119891 = 12 1111725 1164246 116937 892308 1003023 1278804119891 = 23 1002108 1007415 1015467 1217865 1571421 1910337119891 = 1 1023519 1026264 1048041 1757715 2489898 321714








5 Conclusions





Nomenclature







119888lowast






119860119863 119860119861


1 1198611198762 1198651198765

1205791 1205792 Angle of

1198601198741198891198763119860

1198741198881198764

119891lowast







Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











9978881199031= 1199091

9978881198941+ 1199101

9978881198951 (18)


997888V119899=

997888V1minus

997888V2 (19)


2given by

997888V2=

997888120596 times

9978881199031+

99788899788899788899788811987411198742times

997888120596 (20)

where

997888V1= minus V

1

9978881198941

= minus1205961199031

9978881198941

997888120596 = 120596

9978881198961

99788899788899788899788811987411198742= 1199031198951206019978881198941minus 119903119895

9978881198951

(21)


997888V119899= 120596 [119910

1

9978881198941+ (119903119895120601 minus 1199091)9978881198951] (22)


997888120593 =

997888119899119899sdot997888V119899=

9978880 (23)




119863119864 1199091sin1205721+ 1199101cos1205721minus 119903119895120601 sin120572

1= 0


2= 0


1= 0

119860119861 1199091cos 1205792+ 1199101sin 1205792minus 119903119895120601 sin 120579

2= 0

(24)



1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)

1199091sin1205721+ 1199101cos1205721minus 119903119895120601 sin120572

1= 0

(25)


1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)


2= 0

(26)


1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)


1= 0

(27)


1199092= 1199091cos120601 + 119910

1sin120601 + 119903

119895(sin120601 minus 120601 cos120601)

1199102= minus1199091sin120601 + 119910

1cos120601 + 119903

119895(cos120601 + 120601 sin120601)

1199091cos 1205792+ 1199101sin 1205792minus 119903119895120601 sin 120579

2= 0

(28)


2from



1199092= (minus119897



1198861119898 + 119903119895) sin120601

1199102= (1198971sin1205721+ 1205881cos1205721+ 119903119895120601) sin120601


1198861119898 + 119903119895) cos120601


1198861119898 cos120572

1)

(119903119895sin1205721)

0 le 1198971le

2ℎlowast

1198861119898

cos1205721

(29)


1199092= (1198972sin1205722+ 1205882cos1205722minus 119903119895120601) cos120601


1198861119898 + 119903119895) sin120601

1199102= minus (119897

2sin1205722+ 1205882cos1205722minus 119903119895120601) sin120601


1198861119898 + 119903119895) cos120601


1198861119898 cos120572

2)

(119903119895sin1205722)

0 le 1198972le

2ℎlowast

1198861119898

cos1205722

(30)



1199092= (minus120588

1sin 1205791minus 119903119895120601) cos120601

+ (minus1205881(cos 120579

1minus sin120572

1) minus ℎlowast

1198861119898 + 119903119895) sin120601

1199102= (1205881sin 1205791+ 119903119895120601) sin120601

+ (minus1205881(cos 120579

1minus sin120572

1) minus ℎlowast

1198861119898 + 119903119895) cos120601


1198861119898 sin 120579

1)

(119903119895sin 1205791)

0 le 1205791le

120587

2minus 1205721

(31)


1199092= (1205882sin 1205792minus 119903119895120601) cos120601

+ (minus1205882(cos 120579

2minus sin120572

2) minus ℎlowast

1198861119898 + 119903119895) sin120601

1199102= minus (120588

2sin 1205792minus 119903119895120601) sin120601

+ (minus1205882(cos 120579

2minus sin120572

2) minus ℎlowast

1198861119898 + 119903119895) cos120601


1198861119898 sin 120579

2)

(119903119895sin 1205792)

0 le 1205792le

120587

2minus 1205722

(32)




2= 0 (33)


1199092= (1198972sin1205722+ 1205882cos1205722minus 119903119895120601) cos120601


1198861119898 + 119903119895) sin120601

1199102= minus (119897

2sin1205722+ 1205882cos1205722minus 119903119895120601) sin120601


1198861119898 + 119903119895) cos120601


1198861119898 cos120572

2)

(119903119895sin1205722)

(34)


1198861119898 minus 119891

lowast

2119898) cos120572

2



3= 0 (35)


1199092= (1198973sin1205723+ 1205882cos1205722+

ℎlowast

1198861119898

sin1205722

minus 119903119895120601) cos120601

+ (1198973cos1205723+ ℎlowast

1198861119898 minus 119891

lowast

2119898 + 119903119895) sin120601

1199102= minus (119897

3sin1205723+ 1205882cos1205722+

ℎlowast

1198861119898

sin1205722

minus 119903119895120601) sin120601

+ (1198973cos1205723+ ℎlowast

1198861119898 minus 119891

lowast

2119898 + 119903119895) cos120601

120601 = (1198973+ ℎlowast

1198861119898 cos120572

3minus 119891lowast

2119898 cos120572

3

+1205882cos1205722sin1205723+

ℎlowast

1198861119898

sin1205722

sin1205723)

times (119903119895sin1205723)minus1

(36)


1198861119898 minus 119891

lowast

2119898) cos120572

3



1= 1198852






lowast

1= 0364




1)





(a) (b)

(c)

X

Y

Z

(d)








119865 =119879








119891 =119891lowast

2

ℎlowast

1198861

=119867

1198670

(38)


Δ120572 = 1205723minus 1205722 (39)









∘



TE = 1199062minus 1199061 (40)






0

0005

001

0015

002

0025

003

0035

004


Tran

smiss

ion

erro

r (m

m)

AB

C

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19


Δ120572 = 05∘ f = 12


∘
























119865119905=

119879

119903119895

(41)

119865119903= 119865119905tan120572 (42)


119865119899= radic1198652119903+ 1198652

119905 (43)

where 119903119895



119870 =119865sn119865119899

(44)






0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)


0

02

04

06

08

1

12


Load

shar

ing

ratio

Unmodified

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19

Δ120572 = 05∘ f = 23Δ120572 = 05∘ f = 1Δ120572 = 05∘ f = 13

Δ120572 = 05∘ f = 12



0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)




119891 = 13 1091595 1096719 1096902 654042 657885 654591119891 = 12 1111725 1164246 116937 892308 1003023 1278804119891 = 23 1002108 1007415 1015467 1217865 1571421 1910337119891 = 1 1023519 1026264 1048041 1757715 2489898 321714








5 Conclusions





Nomenclature







119888lowast






119860119863 119860119861


1 1198611198762 1198651198765

1205791 1205792 Angle of

1198601198741198891198763119860

1198741198881198764

119891lowast







Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199092= (minus120588

1sin 1205791minus 119903119895120601) cos120601

+ (minus1205881(cos 120579

1minus sin120572

1) minus ℎlowast

1198861119898 + 119903119895) sin120601

1199102= (1205881sin 1205791+ 119903119895120601) sin120601

+ (minus1205881(cos 120579

1minus sin120572

1) minus ℎlowast

1198861119898 + 119903119895) cos120601


1198861119898 sin 120579

1)

(119903119895sin 1205791)

0 le 1205791le

120587

2minus 1205721

(31)


1199092= (1205882sin 1205792minus 119903119895120601) cos120601

+ (minus1205882(cos 120579

2minus sin120572

2) minus ℎlowast

1198861119898 + 119903119895) sin120601

1199102= minus (120588

2sin 1205792minus 119903119895120601) sin120601

+ (minus1205882(cos 120579

2minus sin120572

2) minus ℎlowast

1198861119898 + 119903119895) cos120601


1198861119898 sin 120579

2)

(119903119895sin 1205792)

0 le 1205792le

120587

2minus 1205722

(32)




2= 0 (33)


1199092= (1198972sin1205722+ 1205882cos1205722minus 119903119895120601) cos120601


1198861119898 + 119903119895) sin120601

1199102= minus (119897

2sin1205722+ 1205882cos1205722minus 119903119895120601) sin120601


1198861119898 + 119903119895) cos120601


1198861119898 cos120572

2)

(119903119895sin1205722)

(34)


1198861119898 minus 119891

lowast

2119898) cos120572

2



3= 0 (35)


1199092= (1198973sin1205723+ 1205882cos1205722+

ℎlowast

1198861119898

sin1205722

minus 119903119895120601) cos120601

+ (1198973cos1205723+ ℎlowast

1198861119898 minus 119891

lowast

2119898 + 119903119895) sin120601

1199102= minus (119897

3sin1205723+ 1205882cos1205722+

ℎlowast

1198861119898

sin1205722

minus 119903119895120601) sin120601

+ (1198973cos1205723+ ℎlowast

1198861119898 minus 119891

lowast

2119898 + 119903119895) cos120601

120601 = (1198973+ ℎlowast

1198861119898 cos120572

3minus 119891lowast

2119898 cos120572

3

+1205882cos1205722sin1205723+

ℎlowast

1198861119898

sin1205722

sin1205723)

times (119903119895sin1205723)minus1

(36)


1198861119898 minus 119891

lowast

2119898) cos120572

3



1= 1198852






lowast

1= 0364




1)





(a) (b)

(c)

X

Y

Z

(d)








119865 =119879








119891 =119891lowast

2

ℎlowast

1198861

=119867

1198670

(38)


Δ120572 = 1205723minus 1205722 (39)









∘



TE = 1199062minus 1199061 (40)






0

0005

001

0015

002

0025

003

0035

004


Tran

smiss

ion

erro

r (m

m)

AB

C

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19


Δ120572 = 05∘ f = 12


∘
























119865119905=

119879

119903119895

(41)

119865119903= 119865119905tan120572 (42)


119865119899= radic1198652119903+ 1198652

119905 (43)

where 119903119895



119870 =119865sn119865119899

(44)






0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)


0

02

04

06

08

1

12


Load

shar

ing

ratio

Unmodified

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19

Δ120572 = 05∘ f = 23Δ120572 = 05∘ f = 1Δ120572 = 05∘ f = 13

Δ120572 = 05∘ f = 12



0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)




119891 = 13 1091595 1096719 1096902 654042 657885 654591119891 = 12 1111725 1164246 116937 892308 1003023 1278804119891 = 23 1002108 1007415 1015467 1217865 1571421 1910337119891 = 1 1023519 1026264 1048041 1757715 2489898 321714








5 Conclusions





Nomenclature







119888lowast






119860119863 119860119861


1 1198611198762 1198651198765

1205791 1205792 Angle of

1198601198741198891198763119860

1198741198881198764

119891lowast







Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c)

X

Y

Z

(d)








119865 =119879








119891 =119891lowast

2

ℎlowast

1198861

=119867

1198670

(38)


Δ120572 = 1205723minus 1205722 (39)









∘



TE = 1199062minus 1199061 (40)






0

0005

001

0015

002

0025

003

0035

004


Tran

smiss

ion

erro

r (m

m)

AB

C

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19


Δ120572 = 05∘ f = 12


∘
























119865119905=

119879

119903119895

(41)

119865119903= 119865119905tan120572 (42)


119865119899= radic1198652119903+ 1198652

119905 (43)

where 119903119895



119870 =119865sn119865119899

(44)






0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)


0

02

04

06

08

1

12


Load

shar

ing

ratio

Unmodified

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19

Δ120572 = 05∘ f = 23Δ120572 = 05∘ f = 1Δ120572 = 05∘ f = 13

Δ120572 = 05∘ f = 12



0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)




119891 = 13 1091595 1096719 1096902 654042 657885 654591119891 = 12 1111725 1164246 116937 892308 1003023 1278804119891 = 23 1002108 1007415 1015467 1217865 1571421 1910337119891 = 1 1023519 1026264 1048041 1757715 2489898 321714








5 Conclusions





Nomenclature







119888lowast






119860119863 119860119861


1 1198611198762 1198651198765

1205791 1205792 Angle of

1198601198741198891198763119860

1198741198881198764

119891lowast







Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119891 =119891lowast

2

ℎlowast

1198861

=119867

1198670

(38)


Δ120572 = 1205723minus 1205722 (39)









∘



TE = 1199062minus 1199061 (40)






0

0005

001

0015

002

0025

003

0035

004


Tran

smiss

ion

erro

r (m

m)

AB

C

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19


Δ120572 = 05∘ f = 12


∘
























119865119905=

119879

119903119895

(41)

119865119903= 119865119905tan120572 (42)


119865119899= radic1198652119903+ 1198652

119905 (43)

where 119903119895



119870 =119865sn119865119899

(44)






0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)


0

02

04

06

08

1

12


Load

shar

ing

ratio

Unmodified

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19

Δ120572 = 05∘ f = 23Δ120572 = 05∘ f = 1Δ120572 = 05∘ f = 13

Δ120572 = 05∘ f = 12



0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)




119891 = 13 1091595 1096719 1096902 654042 657885 654591119891 = 12 1111725 1164246 116937 892308 1003023 1278804119891 = 23 1002108 1007415 1015467 1217865 1571421 1910337119891 = 1 1023519 1026264 1048041 1757715 2489898 321714








5 Conclusions





Nomenclature







119888lowast






119860119863 119860119861


1 1198611198762 1198651198765

1205791 1205792 Angle of

1198601198741198891198763119860

1198741198881198764

119891lowast







Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























119865119905=

119879

119903119895

(41)

119865119903= 119865119905tan120572 (42)


119865119899= radic1198652119903+ 1198652

119905 (43)

where 119903119895



119870 =119865sn119865119899

(44)






0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)


0

02

04

06

08

1

12


Load

shar

ing

ratio

Unmodified

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19

Δ120572 = 05∘ f = 23Δ120572 = 05∘ f = 1Δ120572 = 05∘ f = 13

Δ120572 = 05∘ f = 12



0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)




119891 = 13 1091595 1096719 1096902 654042 657885 654591119891 = 12 1111725 1164246 116937 892308 1003023 1278804119891 = 23 1002108 1007415 1015467 1217865 1571421 1910337119891 = 1 1023519 1026264 1048041 1757715 2489898 321714








5 Conclusions





Nomenclature







119888lowast






119860119863 119860119861


1 1198611198762 1198651198765

1205791 1205792 Angle of

1198601198741198891198763119860

1198741198881198764

119891lowast







Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

0005

001

0015

002

0025

003


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

001

002

003

004

005

006


Tran

smiss

ion

erro

r (m

m)


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)


0

02

04

06

08

1

12


Load

shar

ing

ratio

Unmodified

minus17

minus15

minus13

minus11 minus9

minus7

minus5

minus3

minus1 1 3 5 7 9 11 13 15 17 19

Δ120572 = 05∘ f = 23Δ120572 = 05∘ f = 1Δ120572 = 05∘ f = 13

Δ120572 = 05∘ f = 12



0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)




119891 = 13 1091595 1096719 1096902 654042 657885 654591119891 = 12 1111725 1164246 116937 892308 1003023 1278804119891 = 23 1002108 1007415 1015467 1217865 1571421 1910337119891 = 1 1023519 1026264 1048041 1757715 2489898 321714








5 Conclusions





Nomenclature







119888lowast






119860119863 119860119861


1 1198611198762 1198651198765

1205791 1205792 Angle of

1198601198741198891198763119860

1198741198881198764

119891lowast







Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 13

Δ120572 = 05∘ f = 13Δ120572 = 07∘ f = 13

(a)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 12

Δ120572 = 05∘ f = 12Δ120572 = 07∘ f = 12

(b)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 23

Δ120572 = 05∘ f = 23Δ120572 = 07∘ f = 23

(c)

0

02

04

06

08

1

12


Load

shar

ing

ratio


Δ120572 = 03∘ f = 1

Δ120572 = 05∘ f = 1Δ120572 = 07∘ f = 1

(d)




119891 = 13 1091595 1096719 1096902 654042 657885 654591119891 = 12 1111725 1164246 116937 892308 1003023 1278804119891 = 23 1002108 1007415 1015467 1217865 1571421 1910337119891 = 1 1023519 1026264 1048041 1757715 2489898 321714








5 Conclusions





Nomenclature







119888lowast






119860119863 119860119861


1 1198611198762 1198651198765

1205791 1205792 Angle of

1198601198741198891198763119860

1198741198881198764

119891lowast







Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















5 Conclusions





Nomenclature







119888lowast






119860119863 119860119861


1 1198611198762 1198651198765

1205791 1205792 Angle of

1198601198741198891198763119860

1198741198881198764

119891lowast







Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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