Research ArticleGeometric and Meshing Properties of Conjugate Curves forGear Transmission

Dong Liang Bingkui Chen Yane Gao Shuai Peng and Siling Qin

State Key Laboratory of Mechanical Transmission Chongqing University Chongqing 400044 China

Correspondence should be addressed to Bingkui Chen bkchencqueducn

Received 23 June 2014 Revised 13 November 2014 Accepted 16 November 2014 Published 7 December 2014

Academic Editor Evangelos J Sapountzakis

Copyright copy 2014 Dong Liang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Conjugate curves have been put forward previously by authors for gear transmission Comparedwith traditional conjugate surfacesthe conjugate curves havemore flexibility and diversity in aspects of gear design and generation To further extend its application inpower transmission the geometric andmeshing properties of conjugate curves are discussed in this paper Firstly general principledescriptions of conjugate curves for arbitrary axial position are introduced Secondly geometric analysis of conjugate curves iscarried out based on differential geometry including tangent and normal in arbitrary contact direction characteristic point andcurvature relationships Then meshing properties of conjugate curves are further revealed According to a given plane or spatialcurve the uniqueness of conjugated curve under different contact angle conditions is discussedMeshing commonality of conjugatecurves is also demonstrated in terms of a class of spiral curves contacting in the given direction for various gear axes Finally aconclusive summary of this study is given

1 Introduction

The theory of plane or space curves and surfaces in the three-dimensional Euclidean space forms the basis for developmentof differential geometry [1 2] And its application in geartransmission namely the geometry theory of conjugatesurfaces has been widely applied to conventional gear drive[3ndash5] Working performance of conjugate surfaces affectsgreatly the overall power andmotion properties of gear driveThe mathematical principle geometrical design and charac-teristic analysis about conjugate surfaces were developed bymany scholars

Litvin et al [6] proposed systematic methodology tostudymathematicalmodel of conjugate surfaces and analyzedthe geometrical and meshing characteristics based on dif-ferential geometry Chen [7] investigated surface geometryof spatial gear pairs and discussed general property fromthe practical point of view Li [8] described spatial geometrymodeling of conjugate surfaces The specific application inengineering was also introduced Di Puccio et al [9] putforward a rather general formulation for generation andcurvature analysis of conjugate surfaces via the alternativeformulation of the theory of gearing In [10] Wu and Luo

studied a geometric theory of conjugate tooth surfaces andderived curvatures equations in terms of the limit functionsof the first kind and considering themating surfaces subjectedto relative screwmotion with constant translational and rota-tional velocities Ito and Takahashi [11] analyzed curvaturesin hypoid gears starting from a classical differential geometrypoint of view but then introducing kinematic relationshipsBy employing the theory of screws Dooner [12] provided thethird law of gearing and formulated the limiting relationshipbetween the radii of curvature of conjugate surfaces whichis valid only for the reference pitch surfaces Duan et al [13]presented the conjugate principle and basic characteristic ofBertrand conjugate surfaces And Chen et al [14] researchedthe geometric properties of moulding conjugate surfaces

However in some cases the higher overload require-ments are difficult to meet in existing conjugate surfacesThe convex-to-convex tooth profile is more common incontact pattern and it has low contact strength In additionalthere is larger sliding between general tooth surfaces whichleads to the low transmission efficiency Many studies havebeen carried out to develop various concepts design andanalysis approaches toward these problems [15ndash21] Generallyspeaking the surface and curve are both common elements

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 484802 12 pageshttpdxdoiorg1011552014484802

2 Mathematical Problems in Engineering

in nature Compared with general surfaces the contactbetween curves has more flexibility and diversity The relatedinvestigations on conjugate curves have been carried outby the authors and basic meshing principle and theoreticalapplications for gear transmission have been studied [22ndash25]To further reveal general property of conjugate curves andextend the application in gear transmission the geometricand meshing properties of conjugate curves are discussed inthis paper

The remainder of this paper is organized into foursections In the following section the principle descriptionsof conjugate curves for arbitrary axial position are intro-duced Based on differential geometry geometric analysis ofconjugate curves is carried out including tangent and nor-mal in arbitrary contact direction characteristic point andcurvature relationships in the next section The subsequentsection reveals meshing properties of conjugate curves theuniqueness of conjugated curves and meshing commonalityof conjugate curves And a conclusive summary of this studyis given in the last section

2 Principle Descriptions of Conjugate Curvesfor Arbitrary Axial Position

Conjugate curves are described as two smooth curves thatalways keep continuous and tangent contact with each otherin given contact direction under motion law Particularly theprinciple of conjugate curves was proposed only for parallel-axes gears in [22] and its procedure graph is displayed inFigure 1 However for arbitrary axial position the principledescriptions of conjugate curves are studied

As shown in Figure 2 119878(119874minus119909 119910 119911) and 119878119901(119874119901minus119909119901 119910119901 119911119901)are the fixed coordinate systems Conjugate curves Γ1 and Γ2are defined using coordinate systems 1198781(1198741 minus 1199091 1199101 1199111) and1198782(1198742 minus 1199092 1199102 1199112) which are connected to pinion 1 and gear2 respectively Point 119875 is the contact point

The transformationmatrix from coordinate systems 1198781 to1198782 can be expressed as

M21 =[[[

[

cos1206011 cos1206012 minus sin1206011 sin1206012 cosΣ minus sin1206011 cos1206012 minus cos1206011 sin1206012 cosΣ minus sin1206012 sinΣ minus119886 cos1206012cos1206011 sin1206012 + sin1206011 cos1206012 cosΣ minus sin1206011 sin1206012 + cos1206011 cos1206012 cosΣ cos1206012 sinΣ minus119886 sin1206012

minus sin1206011 sinΣ minus cos1206011 sinΣ cosΣ 0

0 0 0 1

]]]

]

(1)

where1206011 and1206012 are the angular displacements of pinion 1 andgear 2 respectively 119886 is center distance andΣ is angle betweenthe axes of mating gears

The parallel axis intersecting axis and crossed axis geartransmission can be realized separately by adjusting thecenter distance 119886 and angleΣ Consider that the general curveΓ1 is represented in parametric form as

Γ1 r1 = 1199091 (120579) i1 + 1199101 (120579) j1 + 1199111 (120579) k1 (2)

where 120579 is curve parameter i1 j1 and k1 are the unit vectorsof coordinate system 1198781

The relative velocity of point 119875(1) of pinion 1 with respectto point 119875(2) of gear 2 is calculated as

120592(12)

1= [minus1199101 (1 + 11989421 cosΣ) minus 119911111989421 cos1206011 sinΣ

minus11988611989421 sin1206011 cosΣ] i1

+ [1199091 (1 + 11989421 cosΣ) + 119911111989421 sin1206011 sinΣ

minus11988611989421 cos1206011 cosΣ] j1

+ 11989421 sinΣ (1199091 cos1206011 minus 1199101 sin1206011 minus 119886) k1

(3)

where 11989421 is the transmission ratio and there exists 1206012 = 119894211206011Based on the curve trihedron established in [22] the

normal vector to contact point in arbitrary direction ofcontact angle is expressed as

n119899 = (11990511198991205731199091

+ 11990521198991205741199091

) i1 + (11990511198991205731199101

+ 11990521198991205741199101

) j1

+ (11990511198991205731199111

+ 11990521198991205741199111

) k1(4)

where 1199051 and 1199052 are the coefficients indicating the projectionof contact angle 1205720 on the direction of principal normal andbinormal vectors respectively

Moreover meshing equation along given contact direc-tion is derived as

119880 cos1206011 minus 119881 sin1206011 = 119882 (5)

where119880 = 11989421 sinΣ (11989911989911991111199091 minus 11989911989911990911199111) minus 11989421 cosΣ1198991198991199101119886

119881 = minus11989421 sinΣ (11989911989911991011199111 minus 11989911989911991111199101) + 11989421 cosΣ1198991198991199091119886

119882 = (1 + 11989421 cosΣ) (11989911989911990911199101 minus 11989911989911991011199091) + 119899119899119911111988611989421 sinΣ

(6)

Using transformation relation r2 = M21r1 and meshingequation simultaneously according to original curve Γ1 theequation of conjugated curve Γ2 is derived as (7) r2 is positionvector of point 119875 in 1198782 Considering transformation matrixfrom 1198781 to 119878 the equation of line of action can also beobtained

The comparison analysis of general principles of con-jugate surfaces and conjugate curves is introduced As theexisting theoretical basis of gear geometry the conjugatesurfaces are widely used in the design and generation of toothsurfaces of gears Usually as shown in Figure 3 conjugatedsurface 2 can be obtained based on original surface 1 and thegiven motion law Then the mating tooth surfaces of gear aregenerated [3]

However the conjugate curves are described as twosmooth curves that always keep continuous and tangent

Mathematical Problems in Engineering 3

Conjugated curve

Demonstration of necessary and

sufficient conditions

Given original curve Γ1 r1 = r1(120579)

Contact angle

Normal vectoralong contact angle

nn = u0(1205720) middot + 1199590(1205720) middot

Relative velocityat contact point

= minus

Meshing equation along given contact position

Φ = nn middot = 0

Γ2r2(120579) = M21r1(120579)

Φ = nn middot = 0

1205720

120574120573 119959

119959

119959

1199591 1199592

Figure 1 Principle procedure graph of conjugate curves

x aP

x2

x1

xp1206012

y2

yp

1206011

y1

y

120596(2)

120596(1)

O O1

O2 Opzp z2

z z1

Σ

Figure 2 Coordinate systems for arbitrary axial position

contact with each other in the given contact direction undermotion law Conjugated curve 2 can be derived accordingto original curve 1 designated contact direction and givenmotion law Then the tubular meshing tooth surfaces inher-iting meshing characteristics of the conjugate curves aregenerated based on equidistant-enveloping method [22 23]The generated tooth profiles specially can be diversity ifchoosing the different equidistant orientation and positionThe theory of conjugate curves is simply represented in

Original surface 1

Motion law

Conjugated surface 2

Figure 3 Theory of conjugate surfaces

Figure 4 and three different contact models of tooth profilesare also displayed in Figure 5

1199092 = 1199091 [cos1206011 cos (119894211206011) minus sin1206011 sin (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 cos (119894211206011) minus cos1206011 sin (119894211206011) cosΣ]

minus 1199111 sin (119894211206011) sinΣ minus 119886 cos (119894211206011)

1199102 = 1199091 [cos1206011 sin (119894211206011) + sin1206011 cos (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 sin (119894211206011) + cos1206011 cos (119894211206011) cosΣ]

+ 1199111 cos (119894211206011) sinΣ minus 119886 sin (119894211206011)

1199112 = minus1199091 sin1206011 sinΣ minus 1199101 cos1206011 sinΣ + 1199111 cosΣ

119880 cos1206011 minus 119881 sin1206011 = 119882(7)

According to the aforementioned descriptions the fol-lowing analysis conclusions can be got as follows

(1) The contact element is surface for the theory ofconjugate surfaces while curve is the contact elementfor the theory of conjugate curves

(2) The tooth surfaces of conjugate-surface gear are gen-erated based on the surface But the tooth surfacesof conjugate-curve gear are developed by means ofthe spatial curve The relation between the curve andsurface is revealed

(3) Conjugated surface 2 is unique for original surface1 and the meshing pair of conjugate surfaces is alsounique However conjugated curve 2 is unique inthe arbitrary designated contact direction for originalcurve 1 and the mating conjugate curves are diversityif choosing the various contact direction

(4) The carrier and form of curves are varied comparedto that of surface The ideal conjugate curves can beobtained according to the selection of original curveand determination of contact direction The differentmeshing tooth surfaces containing the characteristicsof conjugate curves can be generated

4 Mathematical Problems in Engineering

d

Normal plane

d

Designated contact direction

Equidistant curveSphere surface

Tooth tip circle

Tooth root circle

Step 1 Step 2 Step 3 Step 4

Enveloping surface of sphere family

Γ998400

Γ

Γ

998400

120573

120574120572

Figure 4 Theory of conjugate curves

Pbull

h 1

h 2

(a)

Pbull

h 1

(b)

Pbull

h 1

h 2

(c)

Figure 5 Three different contact models of gear (a) convex-convex contact (b) convex-plane contact (c) convex-concave contact

3 Geometric Characteristic Analysis ofConjugate Curves

31 Tangent and Normal to Conjugate Curves in ArbitraryContact Direction The concept of a tangent to the conjugatecurves at contact point is similar to that of general curvebased on the so-called limiting positions of rays [3] Considera set of rays that are drawn through a curve point 119872 andits neighboring points 119872119894 (119894 = 1 2 119899) As points 119872119894approach point 119872 all rays come to some limit position Inthe case shown in Figure 6 there are two limiting rays withcoinciding lines of actionThese two rays form the tangent tothe curve at point119872 and it is identified as a regular point ofthe curve A tangentT exists only at a regular point of a curveA curve point where the tangent T does not exist or is equalto zero is identified as a singular point

Then tangent T is determined and it has

T r120579 = 1199091015840(120579) i1 + 119910

1015840(120579) j1 + 119911

1015840(120579) k1 (8)

For conjugate curves Γ1 and Γ2 the tangentsT1 andT2 areexpressed by r1120579 r2120579 respectively

The normal n119899 to conjugate curves at contact point inarbitrary direction of contact angle is perpendicular to thetangent to the mating curves There is an infinite number

Mbull

bull

bull

bull bull

Space curve

Tangent (two limiting rays)

Rayszi

M2

M3

M4

M1

xi

yi

Figure 6 Illustration of rays and tangent

of normals n belonging to normal plane at contact point119872For instance vector n119899 is one of the set of curve normals inFigure 7 Three orthogonal vectors can be determined thetangent vector 120572 principal normal 120573 and binormal 120574

Mathematical Problems in Engineering 5

M bull

bullM

Osculating plane

Normal plane

Γ

1205720

1205720

t1

t1

t2

t2

nn

nn

120574

120574

120573

120573

120572

120572

Figure 7 Normal to conjugate curves in arbitrary contact direction

The principal normal vector 120573 and binormal 120574 can becalculated respectively as

120573 =

119909120579120579 (1199102

120579+ 1199112

120579) minus 119909120579 (119910120579119910120579120579 + 119911120579119911120579120579)

(1199092

120579+ 1199102

120579+ 1199112

120579)2

i1

+

119910120579120579 (1199092

120579+ 1199112

120579) minus 119910120579 (119909120579119909120579120579 + 119911120579119911120579120579)

(1199092

120579+ 1199102

120579+ 1199112

120579)2

j1

+

119911120579120579 (1199092

120579+ 1199102

120579) minus 119911120579 (119909120579119909120579120579 + 119910120579119910120579120579)

(1199092

120579+ 1199102

120579+ 1199112

120579)2

k1

120574 =119910120579119911120579120579 minus 119911120579119910120579120579

(1199092

120579+ 1199102

120579+ 1199112

120579)32

i1 minus119909120579119911120579120579 minus 119911120579119909120579120579

(1199092

120579+ 1199102

120579+ 1199112

120579)32

j1

+119909120579119910120579120579 minus 119910120579119909120579120579

(1199092

120579+ 1199102

120579+ 1199112

120579)32

k1

(9)

The normal vector to contact point in arbitrary directionof contact angle can be represented as a linear combination ofprincipal normal and binormal vectors and it has n119899 = 1199051120573+1199052120574 Here contact angle 1205720 is defined as the angle betweenarbitrary normal vector n119899 and binormal vector 120574

32 Characteristic Point Given the curve Γ1 adding amotionparameter 120582 the family of curves Γ119894 is obtained [2] Theenveloping curve Γ2 can be generated as depicted in Figure 8if satisfying the following conditions (1) curve Γ1 is smoothand regular curve (2) At every instant 119905 Γ1 and Γ2 touch eachother in contact point119872 (3) Each point of curve Γ2 is alsoa contact point at a unique instant 119905 Obviously it is the setof different contact points (4) The relation between curvesΓ1 and Γ2 can be rendered quite symmetrical and each maybe said to be conjugated for the other if their domains aresuitably restricted

According to (2) family of curves Γ119894 can be expressed as

Γ119894 r1 = r1 (120579 120582) (10)

1205720 Γ1

Γ1

Γ1

Γ2nn

120573

120572

120574

Figure 8 Envelope of a family of curves in given contact direction

And if the enveloping curve exists it has

r1 = r1 (120579 (120582) 120582) = r1 (120582) (11)

For arbitrary curve in family of curves Γ119894 its tangentvector is written as

r1120579 =120597r1120597120579 (12)

The calculation for enveloping curve is

r1 =120597r1120597120579

119889120579

119889120582+120597r1120597120582= r1120579

119889120579

119889120582+ r1120582 (13)

There is common tangent between both curves at contactpoint so that it has r1120579r1 Then

r1120579 times r1 = 0 (14)

Substituting (12) and (13) into (14) yields the result as

r1120579 times r1 = r1120579 times (120597r1120597120579

119889120579

119889120582+120597r1120597120582) = r1120579 times r1120582 = 0 (15)

that is

r1120579 times r1120582 =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

i1 j1 k11205971199091

120597120579

1205971199101

120597120579

1205971199111

120597120579

1205971199091

120597120582

1205971199101

120597120582

1205971199111

120597120582

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (16)

So the mathematical formula of enveloping curve offamily of curves Γ119894 is derived as

r1 = r1 (120579 120582)

r1120579 times r1120582 = Φ (120579 120582) = 0(17)

Compared with the solution of conjugated curve inSection 2 (17) has the same meaning with (7) Usually eachfamily of surfaces and its enveloping surface are not generallytangent at one point but along a curve which is called thecharacteristic line on the surface Actually it is also the contactcurve for both surfaces proposed above The envelopingsurface could be used as the set of family of single parametercharacteristic lines [3] Similarly each family of curves and itsenveloping curve are tangent with point contact and the set

6 Mathematical Problems in Engineering

of contact points which are called characteristic points finallyforms the target curve

According to aforementioned discussion we can get suchfollowing conditions

(1) The necessary and sufficient conditions for contactbetween arbitrary curve of family of curves Γ119894 and itsenveloping curve are

Φ (120579 120582) = r1120579 times r1120582 = 0 (18)

and the generation formula of enveloping curve is

r1 = r1 (120579 120582)

r1120579 times r1120582 = Φ (120579 120582) = 0(19)

(2) The characteristic point which is also the conjugatecontact point with generated enveloping curve exists on eachcurve of family of curves Γ119894 It has

r1 = r1 (120579 1205820)

r1120579 times r11205820

= Φ (120579 1205820) = 0

(20)

where 1205820 is the constant

33 Curvature Relationships It is well known that the curva-ture is of great importance indicator for evaluating geometriccharacteristic of curve and it reflects the bending degreeat one point of curve As the parameter of arc length 119904 isintroduced to the derivation of curve curvature As shownin Figure 91198720 is arbitrary given point of curve Γ and1198721 isits neighboring curve point The correspondence parametersof arc length are 1199040 and 1199040 + Δ119904 respectively Δ120593 is angleformed by the tangents taken at the given and neighboringcurve pointsΔ119904 is arc length between the neighboring pointsThen the change rate of tangents when point1198720 turns the arclength Δ119904 can be described as the curvature of curve Γ at thispoint

The curvature 1205810 of a spatial curve is determined as

1205810 = limΔ119904rarr0

10038161003816100381610038161003816100381610038161003816

Δ120593

Δ119904

10038161003816100381610038161003816100381610038161003816

(21)

where the subscript ldquo0rdquo in 1205810 indicates that the curvature isconsidered for a small piece of the curve located in the oscu-lating plane Based on differential geometry the curvature 1205810can be calculated with formula 1205810 = |r119904 times r119904119904|

To derive equations for determination of the curvecurvature 1205810 and curve torsion 120591 the parameter 119904 of arclength and curve parameter 120579 are considered to be related byfunction 119904(120579) Based on (2) the curve to be discussed can berepresented as r1(119904(120579)) Differentiation of this vector functionyields

r1120579 = r119904119889119904

119889120579 (22)

bull

bull

zi

xi

yi

Γ

(s0)

Δ120593(s0 + Δs)

r(s0 + Δs)r(s0)

M0

M1

120572

120572

Figure 9 Curve curvature analysis

Here 119889119904119889120579 = |r1120579| because |r119904| = 1 Moreover it has

r1120579120579 = r119904119904 (119889119904

119889120579)

2

+ r119904 (1198892119904

1198891205792)

r1120579120579120579 = r119904119904119904 (119889119904

119889120579)

3

+ 3r119904119904 (119889119904

119889120579)(1198892119904

1198891205792) + r119904 (

1198893119904

1198891205793)

(23)

Substituting (22) and (23) into the expression of curvature1205810 we obtain the simplified result

r119904 times r119904119904 =r1120579 times r1120579120579(119889119904119889120579)

3=r1120579 times r11205791205791003816100381610038161003816r11205791003816100381610038161003816

3

1205810 =

1003816100381610038161003816r1120579 times r11205791205791003816100381610038161003816

1003816100381610038161003816r11205791003816100381610038161003816

3

= [(11990911205791199101120579120579 minus 11990911205791205791199101120579)2+ (11990911205791199111120579120579 minus 11990911205791205791199111120579)

2

+ (11991011205791199111120579120579 minus 11991011205791205791199111120579)2]12

times ((1199092

1120579+ 1199102

1120579+ 1199112

1120579)32

)

minus1

(24)

For the analysis of torsion of curve the formula 120591 =(r119904 r119904119904 r119904119904119904)12058120 yields

120591 =(r1120579 times r1120579120579) sdot r1120579120579120579(r1120579 times r1120579120579)

2 (25)

4 Meshing Characteristics ofConjugate Curves

41 Uniqueness of Conjugated Curve The conjugated curve isgenerated in terms of the meshing equation and transforma-tion matrixes among different coordinate systems accordingto a given original curve The normal vector n119899 to conjugatecurves at contact point in arbitrary direction of contact angle

Mathematical Problems in Engineering 7

M

z

x

y

Δ120572

120572

1205721

1205722

Γ1

Γ21Γ22

120573

120574

Figure 10 Two conjugated curves for different contact angles 1205721 1205722

can be determined in the previous study Particularly fromthe expression of normal vector n119899 we can identify thatcoefficients 1199051 and 1199052 indicate the projection of contact angle1205720 on the direction of principal normal and binormal vectorsrespectively

Given the different coefficients 1199051 and 1199052 if the originalcurve is determined the corresponding conjugated curvecan be represented in various results Then the further studyon the relationship between obtained conjugated curves iscarried out Given the contact angles 1205721 1205722 (where 1205722 = 1205721 +Δ120572) in Figure 10 respectively based on principle descriptionsof conjugate curves in Section 2 the conjugated curves Γ2119894(119894 = 1 2) which are separately related to given conditions areexpressed as

119909(119894)

2= 1199091 [cos120601

(119894)

1cos (11989421120601

(119894)

1) minus sin120601(119894)

1sin (11989421120601

(119894)

1) cosΣ]

+ 1199101 [minus sin120601(119894)

1cos (11989421120601

(119894)

1)

minus cos120601(119894)1sin (11989421120601

(119894)

1) cosΣ]

minus 1199111 sin (11989421120601(119894)

1) sinΣ minus 119886 cos (11989421120601

(119894)

1)

119910(119894)

2= 1199091 [cos120601

(119894)

1sin (11989421120601

(119894)

1) + sin120601(119894)

1cos (11989421120601

(119894)

1) cosΣ]

+ 1199101 [minus sin120601(119894)

1sin (11989421120601

(119894)

1)

+ cos120601(119894)1cos (11989421120601

(119894)

1) cosΣ]

+ 1199111 cos (11989421120601(119894)

1) sinΣ minus 119886 sin (11989421120601

(119894)

1)

119911(119894)

2= minus1199091 sin120601

(119894)

1sinΣ minus 1199101 cos120601

(119894)

1sinΣ + 1199111 cosΣ

120601(119894)

1= arcsin 119882

(119894)

radic119880(119894)2+ 119881(119894)

2

minus arctan 119881(119894)

119880(119894)

(26)

Considering that conjugated curve Γ21 rotates aboutcentral contact point119872 with angle Δ120572 it has this expressionafter the coordinate transformation as

119909(1)1015840

2= 119909(1)

2cosΔ120572 + 119910(1)

2sinΔ120572

119910(1)1015840

2= minus 119909

(1)

2sinΔ120572 + 119910(1)

2cosΔ120572

119911(1)1015840

2= 119911(1)

2

(27)

Substituting (26) into (27) the derived result is written as

119909(1)1015840

2= (11990911198601 + 11991011198611 minus 11991111198621) cosΔ120572

+ (11990911198602 + 11991011198612 + 11991111198622) sinΔ120572

minus 119886 cos (Δ120572 minus 11989421120601(1)

1)

119910(1)1015840

2= (minus11990911198601 minus 11991011198611 + 11991111198621) sinΔ120572

minus (11990911198602 minus 11991011198612 minus 11991111198622) cosΔ120572

minus 119886 sin (Δ120572 minus 11989421120601(1)

1)

119911(1)1015840

2= minus1199091 sin120601

(1)

1sinΣ minus 1199101 cos120601

(1)

1sinΣ + 1199111 cosΣ

(28)

where1198601 = cos120601

(1)

1cos (11989421120601

(1)

1) minus sin120601(1)

1sin (11989421120601

(1)

1) cosΣ

1198602 = cos120601(1)

1sin (11989421120601

(1)

1) + sin120601(1)

1cos (11989421120601

(1)

1) cosΣ

1198611 = minus sin120601(1)

1cos (11989421120601

(1)

1) minus cos120601(1)

1sin (11989421120601

(1)

1) cosΣ

1198612 = minus sin120601(1)

1sin (11989421120601

(1)

1) + cos120601(1)

1cos (11989421120601

(1)

1) cosΣ

1198621 = sin (11989421120601(1)

1) sinΣ

1198622 = cos (11989421120601(1)

1) sinΣ

(29)

A comparison about the expressions of Γ22 and Γ101584021

between (26) and (28) is worked It can be concluded fromthe simplified results that the two conjugated curves Γ21 andΓ22 are the same and unique curves while the difference isthat Γ22 is the rotational curve of Γ21 about contact point119872 under given angle Δ120572 For verifying above conclusionthe mathematical example for parallel axis gearing based oncylindrical spiral curve is introduced and its parametric formis represented as

r1 = 119877 cos 120579i1 + 119877 sin 120579j1 + 119901120579k1 (30)

where 119877 is the radius of pitch circle 120579 is spiral curve param-eter and 119901 is helix parameter Through the aforementionedprinciple of conjugate curves the equation of conjugatedcurve is derived as

1199092 = (119877 minus 119886) cos (11989421120579)

1199102 = minus (119877 minus 119886) sin (11989421120579)

1199112 = 119901120579

(31)

8 Mathematical Problems in Engineering

75 80 85 90 95

010

2030

40

0

10

20

30

40

50

Given curve

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(a)

85 90

010

20

0

10

20

30

40

50

Given curve

minus20

minus10

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(b)

75 80 85 90 95

0

10

20

30

40

50

Given curve

minus50

minus40

minus30

minus20

minus10

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(c)

Figure 11 Uniqueness of conjugated curve at any contact position (a) beginning position (b) middle position (c) end position

A simplified meshing model of conjugate curves alongvarious contact angles for gear drive is developed by meansof Matlab software as displayed in Figure 11 according to thedesignated parameters shown in Table 1

From Figure 11 the generated conjugated curves aredifferent if the selected contact angles are various When thegear pair rotates with fixed angular velocity the movementof conjugate curves with 20 degrees rotation 30 degreesrotation and 40 degrees rotation can be described respec-tively The engagement point is still fixed on the commonpoint during the whole meshing Throughout this processcontact point changes gradually in the axial direction and theline of action is always a straight line which is parallel with

gear axes The characteristics of conjugate curves providemore flexibility for the design of gear transmission We canchoose the optimized design result which is suitable forpractical application in terms of appropriate parameters

42 Meshing Commonality of Conjugate Curves To analyzeintrinsic nature of conjugate curves a class of spiral curvescontacting in the given direction for various gear axes isdiscussed

421 Gear Pair with Parallel Axis Assuming that parameters119886 = 0 and Σ = 0 the gear pair with parallel axis can beobtained as displayed in Figure 12 A calculation procedure

Mathematical Problems in Engineering 9

Cylindrical spiral curve Conjugated cylindrical spiral curve

x1

y1

x2

y2

z1z2

(a)

Conjugated curveGiven curve

Line of action

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

(b)

Figure 12 Conjugate spiral curves for parallel axis gear drive

Table 1 Design parameters for conjugate curves

Parameters ValuesRadius of pitch circle Rmm 24Tooth number of pinion 1 1198851 6Tooth number of gear 2 1198852 30Transmission ratio 11989421 5Module119898119899mm 4Helix parameter 119901 286478Central distance 119886mm 144Tooth width 119861mm 50Parameter of spiral curve 120579rad 0sim10472Parameter 1199051 minus05Parameter 1199052 minus0866

of designated spiral curve expressed in (30) for generalcylindrical gears is carried out and the conjugated curve isderived as (31)

Here

1198771 = 119877 minus 119886 1205791 = 11989421120579 (32)

Then (31) can be written as

1199092 = 1198771 cos 1205791

1199102 = minus1198771 sin 1205791

1199112 = 119901120579

(33)

It is consistent with general form of cylindrical spiralcurve in parallel axis gear drive

422 Gear Pair with Intersecting Axis Assuming that param-eters 119886 = 0 and Σ = 0 the gear pair with intersecting axis canbe obtained as shown in Figure 13The selected conical spiralcurve for bevel gears is expressed in coordinate system 1198781 andits parametric form is represented as

r1 = (1198771119891 minus 119888120579) cos 120579i1 + (1198771119891 minus 119888120579) sin 120579j1 + 119901120579k1 (34)

where 1198771119891 is the radius of pitch circle and 120579 and 119901 are alsothe spiral curve and helix parameters respectively It has thecorrelation 119888 = 119901 tan 120575 120575 is the conical angle

Through a series of calculation on the basis of principle ofconjugate curves its conjugated curve is expressed as

1199092 = 1199091 [cos1206011 cos (119894211206011) minus sin1206011 sin (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 cos (119894211206011) minus cos1206011 sin (119894211206011) cosΣ]

minus 1199111 sin (119894211206011) sinΣ

1199102 = 1199091 [cos1206011 sin (119894211206011) + sin1206011 cos (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 sin (119894211206011) + cos1206011 cos (119894211206011) cosΣ]

+ 1199111 cos (119894211206011) sinΣ

1199112 = minus1199091 cos1206011 sinΣ minus 1199101 sin1206011 sinΣ + 1199111 cosΣ

119880 cos1206011 minus 119881 sin1206011 = 119882(35)

10 Mathematical Problems in Engineering

Conical spiral curve

Conjugated conical spiral curve

z1 z2

x1y1x2

y2

1205751

1205752

Rf1

Rf2

O1

O2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

Conjugated curveGiven curve

Line of action

(b)

Figure 13 Conjugate spiral curves for intersecting axis gear drive

Furthermore substituting (34) into (35) the simplifiedresult with parameter 120579 is represented as

1199092 = [1198771119891 minus (119888 + 119901) 120579] cos (11989421120579 + 2120579 + Σ)

1199102 = [1198771119891 minus (119888 + 119901) 120579] sin (11989421120579 + 2120579 + Σ)

1199112 = (minus119888 sinΣ + 119901 cosΣ) 120579 minus 1198771119891 sinΣ

(36)

Here

1198881 = 119888 + 119901 1205792 = 11989421120579 + 2120579 + Σ

1199011 = minus119888 sinΣ + 119901 cosΣ(37)

Then (36) can be written as

1199092 = (1198771119891 minus 1198881120579) cos 1205792

1199102 = (1198771119891 minus 1198881120579) sin 1205792

1199112 = 1199011120579 minus 1198771119891 sinΣ

(38)

Conjugated curve attached to gear 2 is got according toabove derivation Obviously it is also a conical spiral curve inform

423 Gear Pair with CrossedAxis Assuming that parameters119886 = 0 and Σ = 0 it gets the gear pair with crossed axis whichis depicted in Figure 14 The general spiral curve selectedfor worm gear drive is given and it has the same expression

with (30) Based on the principle of conjugate curves itsconjugated curve is derived as

1199092 =radic1199012sin2Σ + (119877 minus 119886)2 cos [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199102 =radic1199012sin2Σ + (119877 minus 119886)2 sin [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199112 = 119901120579 cosΣ(39)

Here

1198772 =radic1199012sin2Σ + (119877 minus 119886)2

1205793 = 11989421120579 + arctan(119877 minus 119886

119901 sinΣ)

(40)

Then (39) can be written as

1199092 = 1198772 cos 1205793

1199102 = 1198772 sin 1205793

1199112 = 119901120579 cosΣ

(41)

It also represents the similar relationship with givenoriginal curve in form

From what are discussed above we can conclude thatthe conjugated curves corresponding to the given spiralcurves for various gear axes are still the spiral curves Itcan be written as the same expression form with the given

Mathematical Problems in Engineering 11

Worm spiral curve

Conjugated worm spiral curve

x1

y1

x2

y2

O1

O2

z1

z2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

Conjugated curveGiven curve

Line of action

x (mm)y (mm)

z(m

m)

(b)

Figure 14 Conjugate spiral curves for crossed axis gear drive

conditions It is called meshing commonality of conjugatecurves Moreover based on the principle of conjugate curvesfor any chosen contact curve the corresponding same typeof conjugated curve in the direction of arbitrary contact anglecan be identified Freedomof selecting contact curve suggeststhe possibility of optimal conjugation design

According to the aforementioned research it offers aversatile way for high-performance profile design (1) fewteeth number and large module may be obtained withouttooth undercutting The problems of limited installationspace and lightweight design can be solved (2)Three differentcontact models of tooth profiles can be established basedon the proposed study Particularly the special meshing ofgenerated convex and concave tooth profiles makes relativeradius of curvature of the contact point longer and increasesthe contact strength The load capacity and useful life hasbeen evidently improved (3) The tooth surfaces mesh inpoint contact along the conjugate curves and contain thetransmission properties of conjugate curves Due to theselection and generation processes the approximate purerolling contact between mating tooth surfaces maybe occursThe transmission efficiency will be improved

The further studies on strength property manufacturingkey technology and performance experiment of conjugate-curve gear drive will be carried out And the excellent trans-mission performance of gearing is expected to be obtained onthe basis of theoretical and experimental investigations

5 Conclusions

(1) The principle descriptions of conjugate curves forarbitrary axial position are introducedWith the aid of

given plane or space curve generation principle andmathematical model of conjugated curve are devel-oped It can be applied to parallel axis intersectingaxis and crossed axis gear drive respectively

(2) Based on differential geometry geometric charac-teristic analysis of conjugate curves is carried outThe tangent and normal to conjugate curves inarbitrary contact direction are discussed and generalcalculation methods are provided The envelopingcondition of family of conjugate curves with singleparameter is analyzed and characteristic point isdetermined Curvature and torsion relationships ofconjugate curves are also derived

(3) Meshing properties of conjugate curves are furtherrevealed According to a given plane or spatial curvethe variation law of meshing function is analyzedThe uniqueness of conjugated curves is discussedTheconjugated curves are always the same and uniquecurves for any given curve under different contactangles while the difference is that there exists therotational angle relationship with contact point 119872among generated curves

(4) Meshing commonality of conjugate curves is demon-strated in terms of a class of spiral curves contactingin the given direction for various gear axes For anychosen contact curve under various gear axes theconjugated curve has the same type and it can bewritten as the same expression form

(5) The proposed theory lays the foundation for design ofnew types of gear drive The further study on conju-gate curves and applications in gear transmission will

12 Mathematical Problems in Engineering

be carried out Excellent transmission performance ofgearing is expected to be obtained

Conflict of Interests

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

Acknowledgments

This project is supported by National Key Technology RampDProgram of the Twelfth Five-year Plan of China (Grantno 2013BAF01B04) National Natural Science Foundation ofChina (Grant no 51205425) and Science and TechnologyProject of Chongqing City Board of Education (Grant noKJ122202) The authors also sincerely appreciate the com-ments and modification suggestions made by the editors andanonymous referees

References

[1] S S Chern W H Chen and K S Lam Lectures on DifferentialGeometry World Scientific Singapore 2006

[2] P G Ciarlet and T-T Li Differential Geometry Theory andApplications World Scientific Singapore 2007

[3] F L Litvin and A Fuentes Gear Geometry and Applied TheoryCambridge University Press New York NY USA 2nd edition2004

[4] S P Radzevich Theory of Gearing Kinetics Geometry andSynthesis CRC Press Boca Raton Fla USA 2012

[5] D B Dooner Kinematic Geometry of Gearing John Wiley ampSons 2nd edition 2012

[6] F L Litvin A Fuentes A Demenego D Vecchiato and Q FanldquoNewdevelopments in the design and generation of gear drivesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 215 no 7 pp747ndash757 2001

[7] C H Chen Theory of Conjugate Surfaces and Its ApplicationsScience and Technology Press Beijing China 2008

[8] G X Li Spatial Geometry Modeling and Its Application inEngineering Higher Education Press Beijing China 2007

[9] F Di Puccio M Gabiccini and M Guiggiani ldquoGeneration andcurvature analysis of conjugate surfaces via a new approachrdquoMechanism and Machine Theory vol 41 no 4 pp 382ndash4042006

[10] D R Wu and J S Luo A Geometric Theory of Conjugate ToothSurface World Scientific Publishing Singapore 1992

[11] N Ito and K Takahashi ldquoDifferential geometrical conditions ofhypoid gears with conjugate tooth surfacesrdquo ASME Journal ofMechanical Design vol 122 no 3 pp 323ndash330 2000

[12] D B Dooner ldquoOn the three laws of gearingrdquo Journal ofMechanical Design Transactions of the ASME vol 124 no 4pp 733ndash744 2002

[13] Z Duan H Chen and J Liu ldquoPrinciple and transmissiontechnology of Bertrand conjugate surfacesrdquo Chinese Journal ofMechanical Engineering vol 19 no 4 pp 600ndash604 2006

[14] H Chen Z Duan J Liu and H Wu ldquoResearch on basicprinciple of moulding-surface conjugationrdquo Mechanism andMachine Theory vol 43 no 7 pp 791ndash811 2008

[15] 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

[16] C F Chen and C B Tsay ldquoTooth profile design for themanufacture of helical gear sets with small numbers of teethrdquoInternational Journal ofMachine Tools andManufacture vol 45no 12-13 pp 1531ndash1541 2005

[17] R Imin and M Geni ldquoStress analysis of gear meshing impactbased on SPH methodrdquoMathematical Problems in Engineeringvol 2014 Article ID 328216 7 pages 2014

[18] D Park and A Kahraman ldquoA surface wear model for hypoidgear pairsrdquoWear vol 267 no 9-10 pp 1595ndash1604 2009

[19] JWang L Hou SM Luo and R YWu ldquoActive design of toothprofiles using parabolic curve as the line of actionrdquoMechanismand Machine Theory vol 67 no 9 pp 47ndash63 2013

[20] Q B Wang P Hu Y M Zhang et al ldquoA model to determinemesh characteristics in a gear pair with tooth profile errorrdquoAdvances in Mechanical Engineering vol 2014 Article ID751476 10 pages 2014

[21] C F Chen and C B Tsay ldquoComputerized tooth profile gen-eration and analysis of characteristics of elliptical gears withcircular arc teethrdquo Journal of Materials Processing Technologyvol 148 no 2 pp 226ndash234 2004

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

[23] B K Chen Y E Gao and D Liang ldquoThe generation principleof tooth surfaces of conjugate-curve gear transmissionrdquo Journalof Mechanical Engineering vol 50 no 3 pp 18ndash24 2014

[24] D Liang B K Chen and Y E Gao ldquoThe generation principleand mathematical model of a new involute-helix gear driverdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 227 no 12 pp2834ndash2843 2013

[25] B Chen D Liang and Z Li ldquoA study on geometry designof spiral bevel gears based on conjugate curvesrdquo InternationalJournal of Precision Engineering and Manufacturing vol 15 no3 pp 477ndash482 2014

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

in nature Compared with general surfaces the contactbetween curves has more flexibility and diversity The relatedinvestigations on conjugate curves have been carried outby the authors and basic meshing principle and theoreticalapplications for gear transmission have been studied [22ndash25]To further reveal general property of conjugate curves andextend the application in gear transmission the geometricand meshing properties of conjugate curves are discussed inthis paper

The remainder of this paper is organized into foursections In the following section the principle descriptionsof conjugate curves for arbitrary axial position are intro-duced Based on differential geometry geometric analysis ofconjugate curves is carried out including tangent and nor-mal in arbitrary contact direction characteristic point andcurvature relationships in the next section The subsequentsection reveals meshing properties of conjugate curves theuniqueness of conjugated curves and meshing commonalityof conjugate curves And a conclusive summary of this studyis given in the last section

2 Principle Descriptions of Conjugate Curvesfor Arbitrary Axial Position

Conjugate curves are described as two smooth curves thatalways keep continuous and tangent contact with each otherin given contact direction under motion law Particularly theprinciple of conjugate curves was proposed only for parallel-axes gears in [22] and its procedure graph is displayed inFigure 1 However for arbitrary axial position the principledescriptions of conjugate curves are studied

As shown in Figure 2 119878(119874minus119909 119910 119911) and 119878119901(119874119901minus119909119901 119910119901 119911119901)are the fixed coordinate systems Conjugate curves Γ1 and Γ2are defined using coordinate systems 1198781(1198741 minus 1199091 1199101 1199111) and1198782(1198742 minus 1199092 1199102 1199112) which are connected to pinion 1 and gear2 respectively Point 119875 is the contact point

The transformationmatrix from coordinate systems 1198781 to1198782 can be expressed as

M21 =[[[

[

cos1206011 cos1206012 minus sin1206011 sin1206012 cosΣ minus sin1206011 cos1206012 minus cos1206011 sin1206012 cosΣ minus sin1206012 sinΣ minus119886 cos1206012cos1206011 sin1206012 + sin1206011 cos1206012 cosΣ minus sin1206011 sin1206012 + cos1206011 cos1206012 cosΣ cos1206012 sinΣ minus119886 sin1206012

minus sin1206011 sinΣ minus cos1206011 sinΣ cosΣ 0

0 0 0 1

]]]

]

(1)

where1206011 and1206012 are the angular displacements of pinion 1 andgear 2 respectively 119886 is center distance andΣ is angle betweenthe axes of mating gears

The parallel axis intersecting axis and crossed axis geartransmission can be realized separately by adjusting thecenter distance 119886 and angleΣ Consider that the general curveΓ1 is represented in parametric form as

Γ1 r1 = 1199091 (120579) i1 + 1199101 (120579) j1 + 1199111 (120579) k1 (2)

where 120579 is curve parameter i1 j1 and k1 are the unit vectorsof coordinate system 1198781

The relative velocity of point 119875(1) of pinion 1 with respectto point 119875(2) of gear 2 is calculated as

120592(12)

1= [minus1199101 (1 + 11989421 cosΣ) minus 119911111989421 cos1206011 sinΣ

minus11988611989421 sin1206011 cosΣ] i1

+ [1199091 (1 + 11989421 cosΣ) + 119911111989421 sin1206011 sinΣ

minus11988611989421 cos1206011 cosΣ] j1

+ 11989421 sinΣ (1199091 cos1206011 minus 1199101 sin1206011 minus 119886) k1

(3)

where 11989421 is the transmission ratio and there exists 1206012 = 119894211206011Based on the curve trihedron established in [22] the

normal vector to contact point in arbitrary direction ofcontact angle is expressed as

n119899 = (11990511198991205731199091

+ 11990521198991205741199091

) i1 + (11990511198991205731199101

+ 11990521198991205741199101

) j1

+ (11990511198991205731199111

+ 11990521198991205741199111

) k1(4)

where 1199051 and 1199052 are the coefficients indicating the projectionof contact angle 1205720 on the direction of principal normal andbinormal vectors respectively

Moreover meshing equation along given contact direc-tion is derived as

119880 cos1206011 minus 119881 sin1206011 = 119882 (5)

where119880 = 11989421 sinΣ (11989911989911991111199091 minus 11989911989911990911199111) minus 11989421 cosΣ1198991198991199101119886

119881 = minus11989421 sinΣ (11989911989911991011199111 minus 11989911989911991111199101) + 11989421 cosΣ1198991198991199091119886

119882 = (1 + 11989421 cosΣ) (11989911989911990911199101 minus 11989911989911991011199091) + 119899119899119911111988611989421 sinΣ

(6)

Using transformation relation r2 = M21r1 and meshingequation simultaneously according to original curve Γ1 theequation of conjugated curve Γ2 is derived as (7) r2 is positionvector of point 119875 in 1198782 Considering transformation matrixfrom 1198781 to 119878 the equation of line of action can also beobtained

The comparison analysis of general principles of con-jugate surfaces and conjugate curves is introduced As theexisting theoretical basis of gear geometry the conjugatesurfaces are widely used in the design and generation of toothsurfaces of gears Usually as shown in Figure 3 conjugatedsurface 2 can be obtained based on original surface 1 and thegiven motion law Then the mating tooth surfaces of gear aregenerated [3]

However the conjugate curves are described as twosmooth curves that always keep continuous and tangent

Mathematical Problems in Engineering 3

Conjugated curve

Demonstration of necessary and

sufficient conditions

Given original curve Γ1 r1 = r1(120579)

Contact angle

Normal vectoralong contact angle

nn = u0(1205720) middot + 1199590(1205720) middot

Relative velocityat contact point

= minus

Meshing equation along given contact position

Φ = nn middot = 0

Γ2r2(120579) = M21r1(120579)

Φ = nn middot = 0

1205720

120574120573 119959

119959

119959

1199591 1199592

Figure 1 Principle procedure graph of conjugate curves

x aP

x2

x1

xp1206012

y2

yp

1206011

y1

y

120596(2)

120596(1)

O O1

O2 Opzp z2

z z1

Σ

Figure 2 Coordinate systems for arbitrary axial position

contact with each other in the given contact direction undermotion law Conjugated curve 2 can be derived accordingto original curve 1 designated contact direction and givenmotion law Then the tubular meshing tooth surfaces inher-iting meshing characteristics of the conjugate curves aregenerated based on equidistant-enveloping method [22 23]The generated tooth profiles specially can be diversity ifchoosing the different equidistant orientation and positionThe theory of conjugate curves is simply represented in

Original surface 1

Motion law

Conjugated surface 2

Figure 3 Theory of conjugate surfaces

Figure 4 and three different contact models of tooth profilesare also displayed in Figure 5

1199092 = 1199091 [cos1206011 cos (119894211206011) minus sin1206011 sin (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 cos (119894211206011) minus cos1206011 sin (119894211206011) cosΣ]

minus 1199111 sin (119894211206011) sinΣ minus 119886 cos (119894211206011)

1199102 = 1199091 [cos1206011 sin (119894211206011) + sin1206011 cos (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 sin (119894211206011) + cos1206011 cos (119894211206011) cosΣ]

+ 1199111 cos (119894211206011) sinΣ minus 119886 sin (119894211206011)

1199112 = minus1199091 sin1206011 sinΣ minus 1199101 cos1206011 sinΣ + 1199111 cosΣ

119880 cos1206011 minus 119881 sin1206011 = 119882(7)

According to the aforementioned descriptions the fol-lowing analysis conclusions can be got as follows

(1) The contact element is surface for the theory ofconjugate surfaces while curve is the contact elementfor the theory of conjugate curves

(2) The tooth surfaces of conjugate-surface gear are gen-erated based on the surface But the tooth surfacesof conjugate-curve gear are developed by means ofthe spatial curve The relation between the curve andsurface is revealed

(3) Conjugated surface 2 is unique for original surface1 and the meshing pair of conjugate surfaces is alsounique However conjugated curve 2 is unique inthe arbitrary designated contact direction for originalcurve 1 and the mating conjugate curves are diversityif choosing the various contact direction

(4) The carrier and form of curves are varied comparedto that of surface The ideal conjugate curves can beobtained according to the selection of original curveand determination of contact direction The differentmeshing tooth surfaces containing the characteristicsof conjugate curves can be generated

4 Mathematical Problems in Engineering

d

Normal plane

d

Designated contact direction

Equidistant curveSphere surface

Tooth tip circle

Tooth root circle

Step 1 Step 2 Step 3 Step 4

Enveloping surface of sphere family

Γ998400

Γ

Γ

998400

120573

120574120572

Figure 4 Theory of conjugate curves

Pbull

h 1

h 2

(a)

Pbull

h 1

(b)

Pbull

h 1

h 2

(c)

Figure 5 Three different contact models of gear (a) convex-convex contact (b) convex-plane contact (c) convex-concave contact

3 Geometric Characteristic Analysis ofConjugate Curves

31 Tangent and Normal to Conjugate Curves in ArbitraryContact Direction The concept of a tangent to the conjugatecurves at contact point is similar to that of general curvebased on the so-called limiting positions of rays [3] Considera set of rays that are drawn through a curve point 119872 andits neighboring points 119872119894 (119894 = 1 2 119899) As points 119872119894approach point 119872 all rays come to some limit position Inthe case shown in Figure 6 there are two limiting rays withcoinciding lines of actionThese two rays form the tangent tothe curve at point119872 and it is identified as a regular point ofthe curve A tangentT exists only at a regular point of a curveA curve point where the tangent T does not exist or is equalto zero is identified as a singular point

Then tangent T is determined and it has

T r120579 = 1199091015840(120579) i1 + 119910

1015840(120579) j1 + 119911

1015840(120579) k1 (8)

For conjugate curves Γ1 and Γ2 the tangentsT1 andT2 areexpressed by r1120579 r2120579 respectively

The normal n119899 to conjugate curves at contact point inarbitrary direction of contact angle is perpendicular to thetangent to the mating curves There is an infinite number

Mbull

bull

bull

bull bull

Space curve

Tangent (two limiting rays)

Rayszi

M2

M3

M4

M1

xi

yi

Figure 6 Illustration of rays and tangent

of normals n belonging to normal plane at contact point119872For instance vector n119899 is one of the set of curve normals inFigure 7 Three orthogonal vectors can be determined thetangent vector 120572 principal normal 120573 and binormal 120574

Mathematical Problems in Engineering 5

M bull

bullM

Osculating plane

Normal plane

Γ

1205720

1205720

t1

t1

t2

t2

nn

nn

120574

120574

120573

120573

120572

120572

Figure 7 Normal to conjugate curves in arbitrary contact direction

The principal normal vector 120573 and binormal 120574 can becalculated respectively as

120573 =

119909120579120579 (1199102

120579+ 1199112

120579) minus 119909120579 (119910120579119910120579120579 + 119911120579119911120579120579)

(1199092

120579+ 1199102

120579+ 1199112

120579)2

i1

+

119910120579120579 (1199092

120579+ 1199112

120579) minus 119910120579 (119909120579119909120579120579 + 119911120579119911120579120579)

(1199092

120579+ 1199102

120579+ 1199112

120579)2

j1

+

119911120579120579 (1199092

120579+ 1199102

120579) minus 119911120579 (119909120579119909120579120579 + 119910120579119910120579120579)

(1199092

120579+ 1199102

120579+ 1199112

120579)2

k1

120574 =119910120579119911120579120579 minus 119911120579119910120579120579

(1199092

120579+ 1199102

120579+ 1199112

120579)32

i1 minus119909120579119911120579120579 minus 119911120579119909120579120579

(1199092

120579+ 1199102

120579+ 1199112

120579)32

j1

+119909120579119910120579120579 minus 119910120579119909120579120579

(1199092

120579+ 1199102

120579+ 1199112

120579)32

k1

(9)

The normal vector to contact point in arbitrary directionof contact angle can be represented as a linear combination ofprincipal normal and binormal vectors and it has n119899 = 1199051120573+1199052120574 Here contact angle 1205720 is defined as the angle betweenarbitrary normal vector n119899 and binormal vector 120574

32 Characteristic Point Given the curve Γ1 adding amotionparameter 120582 the family of curves Γ119894 is obtained [2] Theenveloping curve Γ2 can be generated as depicted in Figure 8if satisfying the following conditions (1) curve Γ1 is smoothand regular curve (2) At every instant 119905 Γ1 and Γ2 touch eachother in contact point119872 (3) Each point of curve Γ2 is alsoa contact point at a unique instant 119905 Obviously it is the setof different contact points (4) The relation between curvesΓ1 and Γ2 can be rendered quite symmetrical and each maybe said to be conjugated for the other if their domains aresuitably restricted

According to (2) family of curves Γ119894 can be expressed as

Γ119894 r1 = r1 (120579 120582) (10)

1205720 Γ1

Γ1

Γ1

Γ2nn

120573

120572

120574

Figure 8 Envelope of a family of curves in given contact direction

And if the enveloping curve exists it has

r1 = r1 (120579 (120582) 120582) = r1 (120582) (11)

For arbitrary curve in family of curves Γ119894 its tangentvector is written as

r1120579 =120597r1120597120579 (12)

The calculation for enveloping curve is

r1 =120597r1120597120579

119889120579

119889120582+120597r1120597120582= r1120579

119889120579

119889120582+ r1120582 (13)

There is common tangent between both curves at contactpoint so that it has r1120579r1 Then

r1120579 times r1 = 0 (14)

Substituting (12) and (13) into (14) yields the result as

r1120579 times r1 = r1120579 times (120597r1120597120579

119889120579

119889120582+120597r1120597120582) = r1120579 times r1120582 = 0 (15)

that is

r1120579 times r1120582 =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

i1 j1 k11205971199091

120597120579

1205971199101

120597120579

1205971199111

120597120579

1205971199091

120597120582

1205971199101

120597120582

1205971199111

120597120582

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (16)

So the mathematical formula of enveloping curve offamily of curves Γ119894 is derived as

r1 = r1 (120579 120582)

r1120579 times r1120582 = Φ (120579 120582) = 0(17)

Compared with the solution of conjugated curve inSection 2 (17) has the same meaning with (7) Usually eachfamily of surfaces and its enveloping surface are not generallytangent at one point but along a curve which is called thecharacteristic line on the surface Actually it is also the contactcurve for both surfaces proposed above The envelopingsurface could be used as the set of family of single parametercharacteristic lines [3] Similarly each family of curves and itsenveloping curve are tangent with point contact and the set

6 Mathematical Problems in Engineering

of contact points which are called characteristic points finallyforms the target curve

According to aforementioned discussion we can get suchfollowing conditions

(1) The necessary and sufficient conditions for contactbetween arbitrary curve of family of curves Γ119894 and itsenveloping curve are

Φ (120579 120582) = r1120579 times r1120582 = 0 (18)

and the generation formula of enveloping curve is

r1 = r1 (120579 120582)

r1120579 times r1120582 = Φ (120579 120582) = 0(19)

(2) The characteristic point which is also the conjugatecontact point with generated enveloping curve exists on eachcurve of family of curves Γ119894 It has

r1 = r1 (120579 1205820)

r1120579 times r11205820

= Φ (120579 1205820) = 0

(20)

where 1205820 is the constant

33 Curvature Relationships It is well known that the curva-ture is of great importance indicator for evaluating geometriccharacteristic of curve and it reflects the bending degreeat one point of curve As the parameter of arc length 119904 isintroduced to the derivation of curve curvature As shownin Figure 91198720 is arbitrary given point of curve Γ and1198721 isits neighboring curve point The correspondence parametersof arc length are 1199040 and 1199040 + Δ119904 respectively Δ120593 is angleformed by the tangents taken at the given and neighboringcurve pointsΔ119904 is arc length between the neighboring pointsThen the change rate of tangents when point1198720 turns the arclength Δ119904 can be described as the curvature of curve Γ at thispoint

The curvature 1205810 of a spatial curve is determined as

1205810 = limΔ119904rarr0

10038161003816100381610038161003816100381610038161003816

Δ120593

Δ119904

10038161003816100381610038161003816100381610038161003816

(21)

where the subscript ldquo0rdquo in 1205810 indicates that the curvature isconsidered for a small piece of the curve located in the oscu-lating plane Based on differential geometry the curvature 1205810can be calculated with formula 1205810 = |r119904 times r119904119904|

To derive equations for determination of the curvecurvature 1205810 and curve torsion 120591 the parameter 119904 of arclength and curve parameter 120579 are considered to be related byfunction 119904(120579) Based on (2) the curve to be discussed can berepresented as r1(119904(120579)) Differentiation of this vector functionyields

r1120579 = r119904119889119904

119889120579 (22)

bull

bull

zi

xi

yi

Γ

(s0)

Δ120593(s0 + Δs)

r(s0 + Δs)r(s0)

M0

M1

120572

120572

Figure 9 Curve curvature analysis

Here 119889119904119889120579 = |r1120579| because |r119904| = 1 Moreover it has

r1120579120579 = r119904119904 (119889119904

119889120579)

2

+ r119904 (1198892119904

1198891205792)

r1120579120579120579 = r119904119904119904 (119889119904

119889120579)

3

+ 3r119904119904 (119889119904

119889120579)(1198892119904

1198891205792) + r119904 (

1198893119904

1198891205793)

(23)

Substituting (22) and (23) into the expression of curvature1205810 we obtain the simplified result

r119904 times r119904119904 =r1120579 times r1120579120579(119889119904119889120579)

3=r1120579 times r11205791205791003816100381610038161003816r11205791003816100381610038161003816

3

1205810 =

1003816100381610038161003816r1120579 times r11205791205791003816100381610038161003816

1003816100381610038161003816r11205791003816100381610038161003816

3

= [(11990911205791199101120579120579 minus 11990911205791205791199101120579)2+ (11990911205791199111120579120579 minus 11990911205791205791199111120579)

2

+ (11991011205791199111120579120579 minus 11991011205791205791199111120579)2]12

times ((1199092

1120579+ 1199102

1120579+ 1199112

1120579)32

)

minus1

(24)

For the analysis of torsion of curve the formula 120591 =(r119904 r119904119904 r119904119904119904)12058120 yields

120591 =(r1120579 times r1120579120579) sdot r1120579120579120579(r1120579 times r1120579120579)

2 (25)

4 Meshing Characteristics ofConjugate Curves

41 Uniqueness of Conjugated Curve The conjugated curve isgenerated in terms of the meshing equation and transforma-tion matrixes among different coordinate systems accordingto a given original curve The normal vector n119899 to conjugatecurves at contact point in arbitrary direction of contact angle

Mathematical Problems in Engineering 7

M

z

x

y

Δ120572

120572

1205721

1205722

Γ1

Γ21Γ22

120573

120574

Figure 10 Two conjugated curves for different contact angles 1205721 1205722

can be determined in the previous study Particularly fromthe expression of normal vector n119899 we can identify thatcoefficients 1199051 and 1199052 indicate the projection of contact angle1205720 on the direction of principal normal and binormal vectorsrespectively

Given the different coefficients 1199051 and 1199052 if the originalcurve is determined the corresponding conjugated curvecan be represented in various results Then the further studyon the relationship between obtained conjugated curves iscarried out Given the contact angles 1205721 1205722 (where 1205722 = 1205721 +Δ120572) in Figure 10 respectively based on principle descriptionsof conjugate curves in Section 2 the conjugated curves Γ2119894(119894 = 1 2) which are separately related to given conditions areexpressed as

119909(119894)

2= 1199091 [cos120601

(119894)

1cos (11989421120601

(119894)

1) minus sin120601(119894)

1sin (11989421120601

(119894)

1) cosΣ]

+ 1199101 [minus sin120601(119894)

1cos (11989421120601

(119894)

1)

minus cos120601(119894)1sin (11989421120601

(119894)

1) cosΣ]

minus 1199111 sin (11989421120601(119894)

1) sinΣ minus 119886 cos (11989421120601

(119894)

1)

119910(119894)

2= 1199091 [cos120601

(119894)

1sin (11989421120601

(119894)

1) + sin120601(119894)

1cos (11989421120601

(119894)

1) cosΣ]

+ 1199101 [minus sin120601(119894)

1sin (11989421120601

(119894)

1)

+ cos120601(119894)1cos (11989421120601

(119894)

1) cosΣ]

+ 1199111 cos (11989421120601(119894)

1) sinΣ minus 119886 sin (11989421120601

(119894)

1)

119911(119894)

2= minus1199091 sin120601

(119894)

1sinΣ minus 1199101 cos120601

(119894)

1sinΣ + 1199111 cosΣ

120601(119894)

1= arcsin 119882

(119894)

radic119880(119894)2+ 119881(119894)

2

minus arctan 119881(119894)

119880(119894)

(26)

Considering that conjugated curve Γ21 rotates aboutcentral contact point119872 with angle Δ120572 it has this expressionafter the coordinate transformation as

119909(1)1015840

2= 119909(1)

2cosΔ120572 + 119910(1)

2sinΔ120572

119910(1)1015840

2= minus 119909

(1)

2sinΔ120572 + 119910(1)

2cosΔ120572

119911(1)1015840

2= 119911(1)

2

(27)

Substituting (26) into (27) the derived result is written as

119909(1)1015840

2= (11990911198601 + 11991011198611 minus 11991111198621) cosΔ120572

+ (11990911198602 + 11991011198612 + 11991111198622) sinΔ120572

minus 119886 cos (Δ120572 minus 11989421120601(1)

1)

119910(1)1015840

2= (minus11990911198601 minus 11991011198611 + 11991111198621) sinΔ120572

minus (11990911198602 minus 11991011198612 minus 11991111198622) cosΔ120572

minus 119886 sin (Δ120572 minus 11989421120601(1)

1)

119911(1)1015840

2= minus1199091 sin120601

(1)

1sinΣ minus 1199101 cos120601

(1)

1sinΣ + 1199111 cosΣ

(28)

where1198601 = cos120601

(1)

1cos (11989421120601

(1)

1) minus sin120601(1)

1sin (11989421120601

(1)

1) cosΣ

1198602 = cos120601(1)

1sin (11989421120601

(1)

1) + sin120601(1)

1cos (11989421120601

(1)

1) cosΣ

1198611 = minus sin120601(1)

1cos (11989421120601

(1)

1) minus cos120601(1)

1sin (11989421120601

(1)

1) cosΣ

1198612 = minus sin120601(1)

1sin (11989421120601

(1)

1) + cos120601(1)

1cos (11989421120601

(1)

1) cosΣ

1198621 = sin (11989421120601(1)

1) sinΣ

1198622 = cos (11989421120601(1)

1) sinΣ

(29)

A comparison about the expressions of Γ22 and Γ101584021

between (26) and (28) is worked It can be concluded fromthe simplified results that the two conjugated curves Γ21 andΓ22 are the same and unique curves while the difference isthat Γ22 is the rotational curve of Γ21 about contact point119872 under given angle Δ120572 For verifying above conclusionthe mathematical example for parallel axis gearing based oncylindrical spiral curve is introduced and its parametric formis represented as

r1 = 119877 cos 120579i1 + 119877 sin 120579j1 + 119901120579k1 (30)

where 119877 is the radius of pitch circle 120579 is spiral curve param-eter and 119901 is helix parameter Through the aforementionedprinciple of conjugate curves the equation of conjugatedcurve is derived as

1199092 = (119877 minus 119886) cos (11989421120579)

1199102 = minus (119877 minus 119886) sin (11989421120579)

1199112 = 119901120579

(31)

8 Mathematical Problems in Engineering

75 80 85 90 95

010

2030

40

0

10

20

30

40

50

Given curve

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(a)

85 90

010

20

0

10

20

30

40

50

Given curve

minus20

minus10

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(b)

75 80 85 90 95

0

10

20

30

40

50

Given curve

minus50

minus40

minus30

minus20

minus10

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(c)

Figure 11 Uniqueness of conjugated curve at any contact position (a) beginning position (b) middle position (c) end position

A simplified meshing model of conjugate curves alongvarious contact angles for gear drive is developed by meansof Matlab software as displayed in Figure 11 according to thedesignated parameters shown in Table 1

From Figure 11 the generated conjugated curves aredifferent if the selected contact angles are various When thegear pair rotates with fixed angular velocity the movementof conjugate curves with 20 degrees rotation 30 degreesrotation and 40 degrees rotation can be described respec-tively The engagement point is still fixed on the commonpoint during the whole meshing Throughout this processcontact point changes gradually in the axial direction and theline of action is always a straight line which is parallel with

gear axes The characteristics of conjugate curves providemore flexibility for the design of gear transmission We canchoose the optimized design result which is suitable forpractical application in terms of appropriate parameters

42 Meshing Commonality of Conjugate Curves To analyzeintrinsic nature of conjugate curves a class of spiral curvescontacting in the given direction for various gear axes isdiscussed

421 Gear Pair with Parallel Axis Assuming that parameters119886 = 0 and Σ = 0 the gear pair with parallel axis can beobtained as displayed in Figure 12 A calculation procedure

Mathematical Problems in Engineering 9

Cylindrical spiral curve Conjugated cylindrical spiral curve

x1

y1

x2

y2

z1z2

(a)

Conjugated curveGiven curve

Line of action

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

(b)

Figure 12 Conjugate spiral curves for parallel axis gear drive

Table 1 Design parameters for conjugate curves

Parameters ValuesRadius of pitch circle Rmm 24Tooth number of pinion 1 1198851 6Tooth number of gear 2 1198852 30Transmission ratio 11989421 5Module119898119899mm 4Helix parameter 119901 286478Central distance 119886mm 144Tooth width 119861mm 50Parameter of spiral curve 120579rad 0sim10472Parameter 1199051 minus05Parameter 1199052 minus0866

of designated spiral curve expressed in (30) for generalcylindrical gears is carried out and the conjugated curve isderived as (31)

Here

1198771 = 119877 minus 119886 1205791 = 11989421120579 (32)

Then (31) can be written as

1199092 = 1198771 cos 1205791

1199102 = minus1198771 sin 1205791

1199112 = 119901120579

(33)

It is consistent with general form of cylindrical spiralcurve in parallel axis gear drive

422 Gear Pair with Intersecting Axis Assuming that param-eters 119886 = 0 and Σ = 0 the gear pair with intersecting axis canbe obtained as shown in Figure 13The selected conical spiralcurve for bevel gears is expressed in coordinate system 1198781 andits parametric form is represented as

r1 = (1198771119891 minus 119888120579) cos 120579i1 + (1198771119891 minus 119888120579) sin 120579j1 + 119901120579k1 (34)

where 1198771119891 is the radius of pitch circle and 120579 and 119901 are alsothe spiral curve and helix parameters respectively It has thecorrelation 119888 = 119901 tan 120575 120575 is the conical angle

Through a series of calculation on the basis of principle ofconjugate curves its conjugated curve is expressed as

1199092 = 1199091 [cos1206011 cos (119894211206011) minus sin1206011 sin (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 cos (119894211206011) minus cos1206011 sin (119894211206011) cosΣ]

minus 1199111 sin (119894211206011) sinΣ

1199102 = 1199091 [cos1206011 sin (119894211206011) + sin1206011 cos (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 sin (119894211206011) + cos1206011 cos (119894211206011) cosΣ]

+ 1199111 cos (119894211206011) sinΣ

1199112 = minus1199091 cos1206011 sinΣ minus 1199101 sin1206011 sinΣ + 1199111 cosΣ

119880 cos1206011 minus 119881 sin1206011 = 119882(35)

10 Mathematical Problems in Engineering

Conical spiral curve

Conjugated conical spiral curve

z1 z2

x1y1x2

y2

1205751

1205752

Rf1

Rf2

O1

O2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

Conjugated curveGiven curve

Line of action

(b)

Figure 13 Conjugate spiral curves for intersecting axis gear drive

Furthermore substituting (34) into (35) the simplifiedresult with parameter 120579 is represented as

1199092 = [1198771119891 minus (119888 + 119901) 120579] cos (11989421120579 + 2120579 + Σ)

1199102 = [1198771119891 minus (119888 + 119901) 120579] sin (11989421120579 + 2120579 + Σ)

1199112 = (minus119888 sinΣ + 119901 cosΣ) 120579 minus 1198771119891 sinΣ

(36)

Here

1198881 = 119888 + 119901 1205792 = 11989421120579 + 2120579 + Σ

1199011 = minus119888 sinΣ + 119901 cosΣ(37)

Then (36) can be written as

1199092 = (1198771119891 minus 1198881120579) cos 1205792

1199102 = (1198771119891 minus 1198881120579) sin 1205792

1199112 = 1199011120579 minus 1198771119891 sinΣ

(38)

Conjugated curve attached to gear 2 is got according toabove derivation Obviously it is also a conical spiral curve inform

423 Gear Pair with CrossedAxis Assuming that parameters119886 = 0 and Σ = 0 it gets the gear pair with crossed axis whichis depicted in Figure 14 The general spiral curve selectedfor worm gear drive is given and it has the same expression

with (30) Based on the principle of conjugate curves itsconjugated curve is derived as

1199092 =radic1199012sin2Σ + (119877 minus 119886)2 cos [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199102 =radic1199012sin2Σ + (119877 minus 119886)2 sin [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199112 = 119901120579 cosΣ(39)

Here

1198772 =radic1199012sin2Σ + (119877 minus 119886)2

1205793 = 11989421120579 + arctan(119877 minus 119886

119901 sinΣ)

(40)

Then (39) can be written as

1199092 = 1198772 cos 1205793

1199102 = 1198772 sin 1205793

1199112 = 119901120579 cosΣ

(41)

It also represents the similar relationship with givenoriginal curve in form

From what are discussed above we can conclude thatthe conjugated curves corresponding to the given spiralcurves for various gear axes are still the spiral curves Itcan be written as the same expression form with the given

Mathematical Problems in Engineering 11

Worm spiral curve

Conjugated worm spiral curve

x1

y1

x2

y2

O1

O2

z1

z2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

Conjugated curveGiven curve

Line of action

x (mm)y (mm)

z(m

m)

(b)

Figure 14 Conjugate spiral curves for crossed axis gear drive

conditions It is called meshing commonality of conjugatecurves Moreover based on the principle of conjugate curvesfor any chosen contact curve the corresponding same typeof conjugated curve in the direction of arbitrary contact anglecan be identified Freedomof selecting contact curve suggeststhe possibility of optimal conjugation design

According to the aforementioned research it offers aversatile way for high-performance profile design (1) fewteeth number and large module may be obtained withouttooth undercutting The problems of limited installationspace and lightweight design can be solved (2)Three differentcontact models of tooth profiles can be established basedon the proposed study Particularly the special meshing ofgenerated convex and concave tooth profiles makes relativeradius of curvature of the contact point longer and increasesthe contact strength The load capacity and useful life hasbeen evidently improved (3) The tooth surfaces mesh inpoint contact along the conjugate curves and contain thetransmission properties of conjugate curves Due to theselection and generation processes the approximate purerolling contact between mating tooth surfaces maybe occursThe transmission efficiency will be improved

The further studies on strength property manufacturingkey technology and performance experiment of conjugate-curve gear drive will be carried out And the excellent trans-mission performance of gearing is expected to be obtained onthe basis of theoretical and experimental investigations

5 Conclusions

(1) The principle descriptions of conjugate curves forarbitrary axial position are introducedWith the aid of

given plane or space curve generation principle andmathematical model of conjugated curve are devel-oped It can be applied to parallel axis intersectingaxis and crossed axis gear drive respectively

(2) Based on differential geometry geometric charac-teristic analysis of conjugate curves is carried outThe tangent and normal to conjugate curves inarbitrary contact direction are discussed and generalcalculation methods are provided The envelopingcondition of family of conjugate curves with singleparameter is analyzed and characteristic point isdetermined Curvature and torsion relationships ofconjugate curves are also derived

(3) Meshing properties of conjugate curves are furtherrevealed According to a given plane or spatial curvethe variation law of meshing function is analyzedThe uniqueness of conjugated curves is discussedTheconjugated curves are always the same and uniquecurves for any given curve under different contactangles while the difference is that there exists therotational angle relationship with contact point 119872among generated curves

(4) Meshing commonality of conjugate curves is demon-strated in terms of a class of spiral curves contactingin the given direction for various gear axes For anychosen contact curve under various gear axes theconjugated curve has the same type and it can bewritten as the same expression form

(5) The proposed theory lays the foundation for design ofnew types of gear drive The further study on conju-gate curves and applications in gear transmission will

12 Mathematical Problems in Engineering

be carried out Excellent transmission performance ofgearing is expected to be obtained

Conflict of Interests

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

Acknowledgments

This project is supported by National Key Technology RampDProgram of the Twelfth Five-year Plan of China (Grantno 2013BAF01B04) National Natural Science Foundation ofChina (Grant no 51205425) and Science and TechnologyProject of Chongqing City Board of Education (Grant noKJ122202) The authors also sincerely appreciate the com-ments and modification suggestions made by the editors andanonymous referees

References

[1] S S Chern W H Chen and K S Lam Lectures on DifferentialGeometry World Scientific Singapore 2006

[2] P G Ciarlet and T-T Li Differential Geometry Theory andApplications World Scientific Singapore 2007

[3] F L Litvin and A Fuentes Gear Geometry and Applied TheoryCambridge University Press New York NY USA 2nd edition2004

[4] S P Radzevich Theory of Gearing Kinetics Geometry andSynthesis CRC Press Boca Raton Fla USA 2012

[5] D B Dooner Kinematic Geometry of Gearing John Wiley ampSons 2nd edition 2012

[6] F L Litvin A Fuentes A Demenego D Vecchiato and Q FanldquoNewdevelopments in the design and generation of gear drivesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 215 no 7 pp747ndash757 2001

[7] C H Chen Theory of Conjugate Surfaces and Its ApplicationsScience and Technology Press Beijing China 2008

[8] G X Li Spatial Geometry Modeling and Its Application inEngineering Higher Education Press Beijing China 2007

[9] F Di Puccio M Gabiccini and M Guiggiani ldquoGeneration andcurvature analysis of conjugate surfaces via a new approachrdquoMechanism and Machine Theory vol 41 no 4 pp 382ndash4042006

[10] D R Wu and J S Luo A Geometric Theory of Conjugate ToothSurface World Scientific Publishing Singapore 1992

[11] N Ito and K Takahashi ldquoDifferential geometrical conditions ofhypoid gears with conjugate tooth surfacesrdquo ASME Journal ofMechanical Design vol 122 no 3 pp 323ndash330 2000

[12] D B Dooner ldquoOn the three laws of gearingrdquo Journal ofMechanical Design Transactions of the ASME vol 124 no 4pp 733ndash744 2002

[13] Z Duan H Chen and J Liu ldquoPrinciple and transmissiontechnology of Bertrand conjugate surfacesrdquo Chinese Journal ofMechanical Engineering vol 19 no 4 pp 600ndash604 2006

[14] H Chen Z Duan J Liu and H Wu ldquoResearch on basicprinciple of moulding-surface conjugationrdquo Mechanism andMachine Theory vol 43 no 7 pp 791ndash811 2008

[15] 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

[16] C F Chen and C B Tsay ldquoTooth profile design for themanufacture of helical gear sets with small numbers of teethrdquoInternational Journal ofMachine Tools andManufacture vol 45no 12-13 pp 1531ndash1541 2005

[17] R Imin and M Geni ldquoStress analysis of gear meshing impactbased on SPH methodrdquoMathematical Problems in Engineeringvol 2014 Article ID 328216 7 pages 2014

[18] D Park and A Kahraman ldquoA surface wear model for hypoidgear pairsrdquoWear vol 267 no 9-10 pp 1595ndash1604 2009

[19] JWang L Hou SM Luo and R YWu ldquoActive design of toothprofiles using parabolic curve as the line of actionrdquoMechanismand Machine Theory vol 67 no 9 pp 47ndash63 2013

[20] Q B Wang P Hu Y M Zhang et al ldquoA model to determinemesh characteristics in a gear pair with tooth profile errorrdquoAdvances in Mechanical Engineering vol 2014 Article ID751476 10 pages 2014

[21] C F Chen and C B Tsay ldquoComputerized tooth profile gen-eration and analysis of characteristics of elliptical gears withcircular arc teethrdquo Journal of Materials Processing Technologyvol 148 no 2 pp 226ndash234 2004

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

[23] B K Chen Y E Gao and D Liang ldquoThe generation principleof tooth surfaces of conjugate-curve gear transmissionrdquo Journalof Mechanical Engineering vol 50 no 3 pp 18ndash24 2014

[24] D Liang B K Chen and Y E Gao ldquoThe generation principleand mathematical model of a new involute-helix gear driverdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 227 no 12 pp2834ndash2843 2013

[25] B Chen D Liang and Z Li ldquoA study on geometry designof spiral bevel gears based on conjugate curvesrdquo InternationalJournal of Precision Engineering and Manufacturing vol 15 no3 pp 477ndash482 2014

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Conjugated curve

Demonstration of necessary and

sufficient conditions

Given original curve Γ1 r1 = r1(120579)

Contact angle

Normal vectoralong contact angle

nn = u0(1205720) middot + 1199590(1205720) middot

Relative velocityat contact point

= minus

Meshing equation along given contact position

Φ = nn middot = 0

Γ2r2(120579) = M21r1(120579)

Φ = nn middot = 0

1205720

120574120573 119959

119959

119959

1199591 1199592

Figure 1 Principle procedure graph of conjugate curves

x aP

x2

x1

xp1206012

y2

yp

1206011

y1

y

120596(2)

120596(1)

O O1

O2 Opzp z2

z z1

Σ

Figure 2 Coordinate systems for arbitrary axial position

contact with each other in the given contact direction undermotion law Conjugated curve 2 can be derived accordingto original curve 1 designated contact direction and givenmotion law Then the tubular meshing tooth surfaces inher-iting meshing characteristics of the conjugate curves aregenerated based on equidistant-enveloping method [22 23]The generated tooth profiles specially can be diversity ifchoosing the different equidistant orientation and positionThe theory of conjugate curves is simply represented in

Original surface 1

Motion law

Conjugated surface 2

Figure 3 Theory of conjugate surfaces

Figure 4 and three different contact models of tooth profilesare also displayed in Figure 5

1199092 = 1199091 [cos1206011 cos (119894211206011) minus sin1206011 sin (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 cos (119894211206011) minus cos1206011 sin (119894211206011) cosΣ]

minus 1199111 sin (119894211206011) sinΣ minus 119886 cos (119894211206011)

1199102 = 1199091 [cos1206011 sin (119894211206011) + sin1206011 cos (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 sin (119894211206011) + cos1206011 cos (119894211206011) cosΣ]

+ 1199111 cos (119894211206011) sinΣ minus 119886 sin (119894211206011)

1199112 = minus1199091 sin1206011 sinΣ minus 1199101 cos1206011 sinΣ + 1199111 cosΣ

119880 cos1206011 minus 119881 sin1206011 = 119882(7)

According to the aforementioned descriptions the fol-lowing analysis conclusions can be got as follows

(1) The contact element is surface for the theory ofconjugate surfaces while curve is the contact elementfor the theory of conjugate curves

(2) The tooth surfaces of conjugate-surface gear are gen-erated based on the surface But the tooth surfacesof conjugate-curve gear are developed by means ofthe spatial curve The relation between the curve andsurface is revealed

(3) Conjugated surface 2 is unique for original surface1 and the meshing pair of conjugate surfaces is alsounique However conjugated curve 2 is unique inthe arbitrary designated contact direction for originalcurve 1 and the mating conjugate curves are diversityif choosing the various contact direction

(4) The carrier and form of curves are varied comparedto that of surface The ideal conjugate curves can beobtained according to the selection of original curveand determination of contact direction The differentmeshing tooth surfaces containing the characteristicsof conjugate curves can be generated

4 Mathematical Problems in Engineering

d

Normal plane

d

Designated contact direction

Equidistant curveSphere surface

Tooth tip circle

Tooth root circle

Step 1 Step 2 Step 3 Step 4

Enveloping surface of sphere family

Γ998400

Γ

Γ

998400

120573

120574120572

Figure 4 Theory of conjugate curves

Pbull

h 1

h 2

(a)

Pbull

h 1

(b)

Pbull

h 1

h 2

(c)

Figure 5 Three different contact models of gear (a) convex-convex contact (b) convex-plane contact (c) convex-concave contact

3 Geometric Characteristic Analysis ofConjugate Curves

31 Tangent and Normal to Conjugate Curves in ArbitraryContact Direction The concept of a tangent to the conjugatecurves at contact point is similar to that of general curvebased on the so-called limiting positions of rays [3] Considera set of rays that are drawn through a curve point 119872 andits neighboring points 119872119894 (119894 = 1 2 119899) As points 119872119894approach point 119872 all rays come to some limit position Inthe case shown in Figure 6 there are two limiting rays withcoinciding lines of actionThese two rays form the tangent tothe curve at point119872 and it is identified as a regular point ofthe curve A tangentT exists only at a regular point of a curveA curve point where the tangent T does not exist or is equalto zero is identified as a singular point

Then tangent T is determined and it has

T r120579 = 1199091015840(120579) i1 + 119910

1015840(120579) j1 + 119911

1015840(120579) k1 (8)

For conjugate curves Γ1 and Γ2 the tangentsT1 andT2 areexpressed by r1120579 r2120579 respectively

The normal n119899 to conjugate curves at contact point inarbitrary direction of contact angle is perpendicular to thetangent to the mating curves There is an infinite number

Mbull

bull

bull

bull bull

Space curve

Tangent (two limiting rays)

Rayszi

M2

M3

M4

M1

xi

yi

Figure 6 Illustration of rays and tangent

of normals n belonging to normal plane at contact point119872For instance vector n119899 is one of the set of curve normals inFigure 7 Three orthogonal vectors can be determined thetangent vector 120572 principal normal 120573 and binormal 120574

Mathematical Problems in Engineering 5

M bull

bullM

Osculating plane

Normal plane

Γ

1205720

1205720

t1

t1

t2

t2

nn

nn

120574

120574

120573

120573

120572

120572

Figure 7 Normal to conjugate curves in arbitrary contact direction

The principal normal vector 120573 and binormal 120574 can becalculated respectively as

120573 =

119909120579120579 (1199102

120579+ 1199112

120579) minus 119909120579 (119910120579119910120579120579 + 119911120579119911120579120579)

(1199092

120579+ 1199102

120579+ 1199112

120579)2

i1

+

119910120579120579 (1199092

120579+ 1199112

120579) minus 119910120579 (119909120579119909120579120579 + 119911120579119911120579120579)

(1199092

120579+ 1199102

120579+ 1199112

120579)2

j1

+

119911120579120579 (1199092

120579+ 1199102

120579) minus 119911120579 (119909120579119909120579120579 + 119910120579119910120579120579)

(1199092

120579+ 1199102

120579+ 1199112

120579)2

k1

120574 =119910120579119911120579120579 minus 119911120579119910120579120579

(1199092

120579+ 1199102

120579+ 1199112

120579)32

i1 minus119909120579119911120579120579 minus 119911120579119909120579120579

(1199092

120579+ 1199102

120579+ 1199112

120579)32

j1

+119909120579119910120579120579 minus 119910120579119909120579120579

(1199092

120579+ 1199102

120579+ 1199112

120579)32

k1

(9)

The normal vector to contact point in arbitrary directionof contact angle can be represented as a linear combination ofprincipal normal and binormal vectors and it has n119899 = 1199051120573+1199052120574 Here contact angle 1205720 is defined as the angle betweenarbitrary normal vector n119899 and binormal vector 120574

32 Characteristic Point Given the curve Γ1 adding amotionparameter 120582 the family of curves Γ119894 is obtained [2] Theenveloping curve Γ2 can be generated as depicted in Figure 8if satisfying the following conditions (1) curve Γ1 is smoothand regular curve (2) At every instant 119905 Γ1 and Γ2 touch eachother in contact point119872 (3) Each point of curve Γ2 is alsoa contact point at a unique instant 119905 Obviously it is the setof different contact points (4) The relation between curvesΓ1 and Γ2 can be rendered quite symmetrical and each maybe said to be conjugated for the other if their domains aresuitably restricted

According to (2) family of curves Γ119894 can be expressed as

Γ119894 r1 = r1 (120579 120582) (10)

1205720 Γ1

Γ1

Γ1

Γ2nn

120573

120572

120574

Figure 8 Envelope of a family of curves in given contact direction

And if the enveloping curve exists it has

r1 = r1 (120579 (120582) 120582) = r1 (120582) (11)

For arbitrary curve in family of curves Γ119894 its tangentvector is written as

r1120579 =120597r1120597120579 (12)

The calculation for enveloping curve is

r1 =120597r1120597120579

119889120579

119889120582+120597r1120597120582= r1120579

119889120579

119889120582+ r1120582 (13)

There is common tangent between both curves at contactpoint so that it has r1120579r1 Then

r1120579 times r1 = 0 (14)

Substituting (12) and (13) into (14) yields the result as

r1120579 times r1 = r1120579 times (120597r1120597120579

119889120579

119889120582+120597r1120597120582) = r1120579 times r1120582 = 0 (15)

that is

r1120579 times r1120582 =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

i1 j1 k11205971199091

120597120579

1205971199101

120597120579

1205971199111

120597120579

1205971199091

120597120582

1205971199101

120597120582

1205971199111

120597120582

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (16)

So the mathematical formula of enveloping curve offamily of curves Γ119894 is derived as

r1 = r1 (120579 120582)

r1120579 times r1120582 = Φ (120579 120582) = 0(17)

Compared with the solution of conjugated curve inSection 2 (17) has the same meaning with (7) Usually eachfamily of surfaces and its enveloping surface are not generallytangent at one point but along a curve which is called thecharacteristic line on the surface Actually it is also the contactcurve for both surfaces proposed above The envelopingsurface could be used as the set of family of single parametercharacteristic lines [3] Similarly each family of curves and itsenveloping curve are tangent with point contact and the set

6 Mathematical Problems in Engineering

of contact points which are called characteristic points finallyforms the target curve

According to aforementioned discussion we can get suchfollowing conditions

(1) The necessary and sufficient conditions for contactbetween arbitrary curve of family of curves Γ119894 and itsenveloping curve are

Φ (120579 120582) = r1120579 times r1120582 = 0 (18)

and the generation formula of enveloping curve is

r1 = r1 (120579 120582)

r1120579 times r1120582 = Φ (120579 120582) = 0(19)

(2) The characteristic point which is also the conjugatecontact point with generated enveloping curve exists on eachcurve of family of curves Γ119894 It has

r1 = r1 (120579 1205820)

r1120579 times r11205820

= Φ (120579 1205820) = 0

(20)

where 1205820 is the constant

33 Curvature Relationships It is well known that the curva-ture is of great importance indicator for evaluating geometriccharacteristic of curve and it reflects the bending degreeat one point of curve As the parameter of arc length 119904 isintroduced to the derivation of curve curvature As shownin Figure 91198720 is arbitrary given point of curve Γ and1198721 isits neighboring curve point The correspondence parametersof arc length are 1199040 and 1199040 + Δ119904 respectively Δ120593 is angleformed by the tangents taken at the given and neighboringcurve pointsΔ119904 is arc length between the neighboring pointsThen the change rate of tangents when point1198720 turns the arclength Δ119904 can be described as the curvature of curve Γ at thispoint

The curvature 1205810 of a spatial curve is determined as

1205810 = limΔ119904rarr0

10038161003816100381610038161003816100381610038161003816

Δ120593

Δ119904

10038161003816100381610038161003816100381610038161003816

(21)

where the subscript ldquo0rdquo in 1205810 indicates that the curvature isconsidered for a small piece of the curve located in the oscu-lating plane Based on differential geometry the curvature 1205810can be calculated with formula 1205810 = |r119904 times r119904119904|

To derive equations for determination of the curvecurvature 1205810 and curve torsion 120591 the parameter 119904 of arclength and curve parameter 120579 are considered to be related byfunction 119904(120579) Based on (2) the curve to be discussed can berepresented as r1(119904(120579)) Differentiation of this vector functionyields

r1120579 = r119904119889119904

119889120579 (22)

bull

bull

zi

xi

yi

Γ

(s0)

Δ120593(s0 + Δs)

r(s0 + Δs)r(s0)

M0

M1

120572

120572

Figure 9 Curve curvature analysis

Here 119889119904119889120579 = |r1120579| because |r119904| = 1 Moreover it has

r1120579120579 = r119904119904 (119889119904

119889120579)

2

+ r119904 (1198892119904

1198891205792)

r1120579120579120579 = r119904119904119904 (119889119904

119889120579)

3

+ 3r119904119904 (119889119904

119889120579)(1198892119904

1198891205792) + r119904 (

1198893119904

1198891205793)

(23)

Substituting (22) and (23) into the expression of curvature1205810 we obtain the simplified result

r119904 times r119904119904 =r1120579 times r1120579120579(119889119904119889120579)

3=r1120579 times r11205791205791003816100381610038161003816r11205791003816100381610038161003816

3

1205810 =

1003816100381610038161003816r1120579 times r11205791205791003816100381610038161003816

1003816100381610038161003816r11205791003816100381610038161003816

3

= [(11990911205791199101120579120579 minus 11990911205791205791199101120579)2+ (11990911205791199111120579120579 minus 11990911205791205791199111120579)

2

+ (11991011205791199111120579120579 minus 11991011205791205791199111120579)2]12

times ((1199092

1120579+ 1199102

1120579+ 1199112

1120579)32

)

minus1

(24)

For the analysis of torsion of curve the formula 120591 =(r119904 r119904119904 r119904119904119904)12058120 yields

120591 =(r1120579 times r1120579120579) sdot r1120579120579120579(r1120579 times r1120579120579)

2 (25)

4 Meshing Characteristics ofConjugate Curves

41 Uniqueness of Conjugated Curve The conjugated curve isgenerated in terms of the meshing equation and transforma-tion matrixes among different coordinate systems accordingto a given original curve The normal vector n119899 to conjugatecurves at contact point in arbitrary direction of contact angle

Mathematical Problems in Engineering 7

M

z

x

y

Δ120572

120572

1205721

1205722

Γ1

Γ21Γ22

120573

120574

Figure 10 Two conjugated curves for different contact angles 1205721 1205722

can be determined in the previous study Particularly fromthe expression of normal vector n119899 we can identify thatcoefficients 1199051 and 1199052 indicate the projection of contact angle1205720 on the direction of principal normal and binormal vectorsrespectively

Given the different coefficients 1199051 and 1199052 if the originalcurve is determined the corresponding conjugated curvecan be represented in various results Then the further studyon the relationship between obtained conjugated curves iscarried out Given the contact angles 1205721 1205722 (where 1205722 = 1205721 +Δ120572) in Figure 10 respectively based on principle descriptionsof conjugate curves in Section 2 the conjugated curves Γ2119894(119894 = 1 2) which are separately related to given conditions areexpressed as

119909(119894)

2= 1199091 [cos120601

(119894)

1cos (11989421120601

(119894)

1) minus sin120601(119894)

1sin (11989421120601

(119894)

1) cosΣ]

+ 1199101 [minus sin120601(119894)

1cos (11989421120601

(119894)

1)

minus cos120601(119894)1sin (11989421120601

(119894)

1) cosΣ]

minus 1199111 sin (11989421120601(119894)

1) sinΣ minus 119886 cos (11989421120601

(119894)

1)

119910(119894)

2= 1199091 [cos120601

(119894)

1sin (11989421120601

(119894)

1) + sin120601(119894)

1cos (11989421120601

(119894)

1) cosΣ]

+ 1199101 [minus sin120601(119894)

1sin (11989421120601

(119894)

1)

+ cos120601(119894)1cos (11989421120601

(119894)

1) cosΣ]

+ 1199111 cos (11989421120601(119894)

1) sinΣ minus 119886 sin (11989421120601

(119894)

1)

119911(119894)

2= minus1199091 sin120601

(119894)

1sinΣ minus 1199101 cos120601

(119894)

1sinΣ + 1199111 cosΣ

120601(119894)

1= arcsin 119882

(119894)

radic119880(119894)2+ 119881(119894)

2

minus arctan 119881(119894)

119880(119894)

(26)

Considering that conjugated curve Γ21 rotates aboutcentral contact point119872 with angle Δ120572 it has this expressionafter the coordinate transformation as

119909(1)1015840

2= 119909(1)

2cosΔ120572 + 119910(1)

2sinΔ120572

119910(1)1015840

2= minus 119909

(1)

2sinΔ120572 + 119910(1)

2cosΔ120572

119911(1)1015840

2= 119911(1)

2

(27)

Substituting (26) into (27) the derived result is written as

119909(1)1015840

2= (11990911198601 + 11991011198611 minus 11991111198621) cosΔ120572

+ (11990911198602 + 11991011198612 + 11991111198622) sinΔ120572

minus 119886 cos (Δ120572 minus 11989421120601(1)

1)

119910(1)1015840

2= (minus11990911198601 minus 11991011198611 + 11991111198621) sinΔ120572

minus (11990911198602 minus 11991011198612 minus 11991111198622) cosΔ120572

minus 119886 sin (Δ120572 minus 11989421120601(1)

1)

119911(1)1015840

2= minus1199091 sin120601

(1)

1sinΣ minus 1199101 cos120601

(1)

1sinΣ + 1199111 cosΣ

(28)

where1198601 = cos120601

(1)

1cos (11989421120601

(1)

1) minus sin120601(1)

1sin (11989421120601

(1)

1) cosΣ

1198602 = cos120601(1)

1sin (11989421120601

(1)

1) + sin120601(1)

1cos (11989421120601

(1)

1) cosΣ

1198611 = minus sin120601(1)

1cos (11989421120601

(1)

1) minus cos120601(1)

1sin (11989421120601

(1)

1) cosΣ

1198612 = minus sin120601(1)

1sin (11989421120601

(1)

1) + cos120601(1)

1cos (11989421120601

(1)

1) cosΣ

1198621 = sin (11989421120601(1)

1) sinΣ

1198622 = cos (11989421120601(1)

1) sinΣ

(29)

A comparison about the expressions of Γ22 and Γ101584021

between (26) and (28) is worked It can be concluded fromthe simplified results that the two conjugated curves Γ21 andΓ22 are the same and unique curves while the difference isthat Γ22 is the rotational curve of Γ21 about contact point119872 under given angle Δ120572 For verifying above conclusionthe mathematical example for parallel axis gearing based oncylindrical spiral curve is introduced and its parametric formis represented as

r1 = 119877 cos 120579i1 + 119877 sin 120579j1 + 119901120579k1 (30)

where 119877 is the radius of pitch circle 120579 is spiral curve param-eter and 119901 is helix parameter Through the aforementionedprinciple of conjugate curves the equation of conjugatedcurve is derived as

1199092 = (119877 minus 119886) cos (11989421120579)

1199102 = minus (119877 minus 119886) sin (11989421120579)

1199112 = 119901120579

(31)

8 Mathematical Problems in Engineering

75 80 85 90 95

010

2030

40

0

10

20

30

40

50

Given curve

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(a)

85 90

010

20

0

10

20

30

40

50

Given curve

minus20

minus10

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(b)

75 80 85 90 95

0

10

20

30

40

50

Given curve

minus50

minus40

minus30

minus20

minus10

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(c)

Figure 11 Uniqueness of conjugated curve at any contact position (a) beginning position (b) middle position (c) end position

A simplified meshing model of conjugate curves alongvarious contact angles for gear drive is developed by meansof Matlab software as displayed in Figure 11 according to thedesignated parameters shown in Table 1

From Figure 11 the generated conjugated curves aredifferent if the selected contact angles are various When thegear pair rotates with fixed angular velocity the movementof conjugate curves with 20 degrees rotation 30 degreesrotation and 40 degrees rotation can be described respec-tively The engagement point is still fixed on the commonpoint during the whole meshing Throughout this processcontact point changes gradually in the axial direction and theline of action is always a straight line which is parallel with

gear axes The characteristics of conjugate curves providemore flexibility for the design of gear transmission We canchoose the optimized design result which is suitable forpractical application in terms of appropriate parameters

42 Meshing Commonality of Conjugate Curves To analyzeintrinsic nature of conjugate curves a class of spiral curvescontacting in the given direction for various gear axes isdiscussed

421 Gear Pair with Parallel Axis Assuming that parameters119886 = 0 and Σ = 0 the gear pair with parallel axis can beobtained as displayed in Figure 12 A calculation procedure

Mathematical Problems in Engineering 9

Cylindrical spiral curve Conjugated cylindrical spiral curve

x1

y1

x2

y2

z1z2

(a)

Conjugated curveGiven curve

Line of action

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

(b)

Figure 12 Conjugate spiral curves for parallel axis gear drive

Table 1 Design parameters for conjugate curves

Parameters ValuesRadius of pitch circle Rmm 24Tooth number of pinion 1 1198851 6Tooth number of gear 2 1198852 30Transmission ratio 11989421 5Module119898119899mm 4Helix parameter 119901 286478Central distance 119886mm 144Tooth width 119861mm 50Parameter of spiral curve 120579rad 0sim10472Parameter 1199051 minus05Parameter 1199052 minus0866

of designated spiral curve expressed in (30) for generalcylindrical gears is carried out and the conjugated curve isderived as (31)

Here

1198771 = 119877 minus 119886 1205791 = 11989421120579 (32)

Then (31) can be written as

1199092 = 1198771 cos 1205791

1199102 = minus1198771 sin 1205791

1199112 = 119901120579

(33)

It is consistent with general form of cylindrical spiralcurve in parallel axis gear drive

422 Gear Pair with Intersecting Axis Assuming that param-eters 119886 = 0 and Σ = 0 the gear pair with intersecting axis canbe obtained as shown in Figure 13The selected conical spiralcurve for bevel gears is expressed in coordinate system 1198781 andits parametric form is represented as

r1 = (1198771119891 minus 119888120579) cos 120579i1 + (1198771119891 minus 119888120579) sin 120579j1 + 119901120579k1 (34)

where 1198771119891 is the radius of pitch circle and 120579 and 119901 are alsothe spiral curve and helix parameters respectively It has thecorrelation 119888 = 119901 tan 120575 120575 is the conical angle

Through a series of calculation on the basis of principle ofconjugate curves its conjugated curve is expressed as

1199092 = 1199091 [cos1206011 cos (119894211206011) minus sin1206011 sin (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 cos (119894211206011) minus cos1206011 sin (119894211206011) cosΣ]

minus 1199111 sin (119894211206011) sinΣ

1199102 = 1199091 [cos1206011 sin (119894211206011) + sin1206011 cos (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 sin (119894211206011) + cos1206011 cos (119894211206011) cosΣ]

+ 1199111 cos (119894211206011) sinΣ

1199112 = minus1199091 cos1206011 sinΣ minus 1199101 sin1206011 sinΣ + 1199111 cosΣ

119880 cos1206011 minus 119881 sin1206011 = 119882(35)

10 Mathematical Problems in Engineering

Conical spiral curve

Conjugated conical spiral curve

z1 z2

x1y1x2

y2

1205751

1205752

Rf1

Rf2

O1

O2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

Conjugated curveGiven curve

Line of action

(b)

Figure 13 Conjugate spiral curves for intersecting axis gear drive

Furthermore substituting (34) into (35) the simplifiedresult with parameter 120579 is represented as

1199092 = [1198771119891 minus (119888 + 119901) 120579] cos (11989421120579 + 2120579 + Σ)

1199102 = [1198771119891 minus (119888 + 119901) 120579] sin (11989421120579 + 2120579 + Σ)

1199112 = (minus119888 sinΣ + 119901 cosΣ) 120579 minus 1198771119891 sinΣ

(36)

Here

1198881 = 119888 + 119901 1205792 = 11989421120579 + 2120579 + Σ

1199011 = minus119888 sinΣ + 119901 cosΣ(37)

Then (36) can be written as

1199092 = (1198771119891 minus 1198881120579) cos 1205792

1199102 = (1198771119891 minus 1198881120579) sin 1205792

1199112 = 1199011120579 minus 1198771119891 sinΣ

(38)

Conjugated curve attached to gear 2 is got according toabove derivation Obviously it is also a conical spiral curve inform

423 Gear Pair with CrossedAxis Assuming that parameters119886 = 0 and Σ = 0 it gets the gear pair with crossed axis whichis depicted in Figure 14 The general spiral curve selectedfor worm gear drive is given and it has the same expression

with (30) Based on the principle of conjugate curves itsconjugated curve is derived as

1199092 =radic1199012sin2Σ + (119877 minus 119886)2 cos [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199102 =radic1199012sin2Σ + (119877 minus 119886)2 sin [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199112 = 119901120579 cosΣ(39)

Here

1198772 =radic1199012sin2Σ + (119877 minus 119886)2

1205793 = 11989421120579 + arctan(119877 minus 119886

119901 sinΣ)

(40)

Then (39) can be written as

1199092 = 1198772 cos 1205793

1199102 = 1198772 sin 1205793

1199112 = 119901120579 cosΣ

(41)

It also represents the similar relationship with givenoriginal curve in form

From what are discussed above we can conclude thatthe conjugated curves corresponding to the given spiralcurves for various gear axes are still the spiral curves Itcan be written as the same expression form with the given

Mathematical Problems in Engineering 11

Worm spiral curve

Conjugated worm spiral curve

x1

y1

x2

y2

O1

O2

z1

z2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

Conjugated curveGiven curve

Line of action

x (mm)y (mm)

z(m

m)

(b)

Figure 14 Conjugate spiral curves for crossed axis gear drive

conditions It is called meshing commonality of conjugatecurves Moreover based on the principle of conjugate curvesfor any chosen contact curve the corresponding same typeof conjugated curve in the direction of arbitrary contact anglecan be identified Freedomof selecting contact curve suggeststhe possibility of optimal conjugation design

According to the aforementioned research it offers aversatile way for high-performance profile design (1) fewteeth number and large module may be obtained withouttooth undercutting The problems of limited installationspace and lightweight design can be solved (2)Three differentcontact models of tooth profiles can be established basedon the proposed study Particularly the special meshing ofgenerated convex and concave tooth profiles makes relativeradius of curvature of the contact point longer and increasesthe contact strength The load capacity and useful life hasbeen evidently improved (3) The tooth surfaces mesh inpoint contact along the conjugate curves and contain thetransmission properties of conjugate curves Due to theselection and generation processes the approximate purerolling contact between mating tooth surfaces maybe occursThe transmission efficiency will be improved

The further studies on strength property manufacturingkey technology and performance experiment of conjugate-curve gear drive will be carried out And the excellent trans-mission performance of gearing is expected to be obtained onthe basis of theoretical and experimental investigations

5 Conclusions

(1) The principle descriptions of conjugate curves forarbitrary axial position are introducedWith the aid of

given plane or space curve generation principle andmathematical model of conjugated curve are devel-oped It can be applied to parallel axis intersectingaxis and crossed axis gear drive respectively

(2) Based on differential geometry geometric charac-teristic analysis of conjugate curves is carried outThe tangent and normal to conjugate curves inarbitrary contact direction are discussed and generalcalculation methods are provided The envelopingcondition of family of conjugate curves with singleparameter is analyzed and characteristic point isdetermined Curvature and torsion relationships ofconjugate curves are also derived

(3) Meshing properties of conjugate curves are furtherrevealed According to a given plane or spatial curvethe variation law of meshing function is analyzedThe uniqueness of conjugated curves is discussedTheconjugated curves are always the same and uniquecurves for any given curve under different contactangles while the difference is that there exists therotational angle relationship with contact point 119872among generated curves

(4) Meshing commonality of conjugate curves is demon-strated in terms of a class of spiral curves contactingin the given direction for various gear axes For anychosen contact curve under various gear axes theconjugated curve has the same type and it can bewritten as the same expression form

(5) The proposed theory lays the foundation for design ofnew types of gear drive The further study on conju-gate curves and applications in gear transmission will

12 Mathematical Problems in Engineering

be carried out Excellent transmission performance ofgearing is expected to be obtained

Conflict of Interests

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

Acknowledgments

This project is supported by National Key Technology RampDProgram of the Twelfth Five-year Plan of China (Grantno 2013BAF01B04) National Natural Science Foundation ofChina (Grant no 51205425) and Science and TechnologyProject of Chongqing City Board of Education (Grant noKJ122202) The authors also sincerely appreciate the com-ments and modification suggestions made by the editors andanonymous referees

References

[1] S S Chern W H Chen and K S Lam Lectures on DifferentialGeometry World Scientific Singapore 2006

[2] P G Ciarlet and T-T Li Differential Geometry Theory andApplications World Scientific Singapore 2007

[3] F L Litvin and A Fuentes Gear Geometry and Applied TheoryCambridge University Press New York NY USA 2nd edition2004

[4] S P Radzevich Theory of Gearing Kinetics Geometry andSynthesis CRC Press Boca Raton Fla USA 2012

[5] D B Dooner Kinematic Geometry of Gearing John Wiley ampSons 2nd edition 2012

[6] F L Litvin A Fuentes A Demenego D Vecchiato and Q FanldquoNewdevelopments in the design and generation of gear drivesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 215 no 7 pp747ndash757 2001

[7] C H Chen Theory of Conjugate Surfaces and Its ApplicationsScience and Technology Press Beijing China 2008

[8] G X Li Spatial Geometry Modeling and Its Application inEngineering Higher Education Press Beijing China 2007

[9] F Di Puccio M Gabiccini and M Guiggiani ldquoGeneration andcurvature analysis of conjugate surfaces via a new approachrdquoMechanism and Machine Theory vol 41 no 4 pp 382ndash4042006

[10] D R Wu and J S Luo A Geometric Theory of Conjugate ToothSurface World Scientific Publishing Singapore 1992

[11] N Ito and K Takahashi ldquoDifferential geometrical conditions ofhypoid gears with conjugate tooth surfacesrdquo ASME Journal ofMechanical Design vol 122 no 3 pp 323ndash330 2000

[12] D B Dooner ldquoOn the three laws of gearingrdquo Journal ofMechanical Design Transactions of the ASME vol 124 no 4pp 733ndash744 2002

[13] Z Duan H Chen and J Liu ldquoPrinciple and transmissiontechnology of Bertrand conjugate surfacesrdquo Chinese Journal ofMechanical Engineering vol 19 no 4 pp 600ndash604 2006

[14] H Chen Z Duan J Liu and H Wu ldquoResearch on basicprinciple of moulding-surface conjugationrdquo Mechanism andMachine Theory vol 43 no 7 pp 791ndash811 2008

[15] 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

[16] C F Chen and C B Tsay ldquoTooth profile design for themanufacture of helical gear sets with small numbers of teethrdquoInternational Journal ofMachine Tools andManufacture vol 45no 12-13 pp 1531ndash1541 2005

[17] R Imin and M Geni ldquoStress analysis of gear meshing impactbased on SPH methodrdquoMathematical Problems in Engineeringvol 2014 Article ID 328216 7 pages 2014

[18] D Park and A Kahraman ldquoA surface wear model for hypoidgear pairsrdquoWear vol 267 no 9-10 pp 1595ndash1604 2009

[19] JWang L Hou SM Luo and R YWu ldquoActive design of toothprofiles using parabolic curve as the line of actionrdquoMechanismand Machine Theory vol 67 no 9 pp 47ndash63 2013

[20] Q B Wang P Hu Y M Zhang et al ldquoA model to determinemesh characteristics in a gear pair with tooth profile errorrdquoAdvances in Mechanical Engineering vol 2014 Article ID751476 10 pages 2014

[21] C F Chen and C B Tsay ldquoComputerized tooth profile gen-eration and analysis of characteristics of elliptical gears withcircular arc teethrdquo Journal of Materials Processing Technologyvol 148 no 2 pp 226ndash234 2004

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

[23] B K Chen Y E Gao and D Liang ldquoThe generation principleof tooth surfaces of conjugate-curve gear transmissionrdquo Journalof Mechanical Engineering vol 50 no 3 pp 18ndash24 2014

[24] D Liang B K Chen and Y E Gao ldquoThe generation principleand mathematical model of a new involute-helix gear driverdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 227 no 12 pp2834ndash2843 2013

[25] B Chen D Liang and Z Li ldquoA study on geometry designof spiral bevel gears based on conjugate curvesrdquo InternationalJournal of Precision Engineering and Manufacturing vol 15 no3 pp 477ndash482 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

d

Normal plane

d

Designated contact direction

Equidistant curveSphere surface

Tooth tip circle

Tooth root circle

Step 1 Step 2 Step 3 Step 4

Enveloping surface of sphere family

Γ998400

Γ

Γ

998400

120573

120574120572

Figure 4 Theory of conjugate curves

Pbull

h 1

h 2

(a)

Pbull

h 1

(b)

Pbull

h 1

h 2

(c)

Figure 5 Three different contact models of gear (a) convex-convex contact (b) convex-plane contact (c) convex-concave contact

3 Geometric Characteristic Analysis ofConjugate Curves

31 Tangent and Normal to Conjugate Curves in ArbitraryContact Direction The concept of a tangent to the conjugatecurves at contact point is similar to that of general curvebased on the so-called limiting positions of rays [3] Considera set of rays that are drawn through a curve point 119872 andits neighboring points 119872119894 (119894 = 1 2 119899) As points 119872119894approach point 119872 all rays come to some limit position Inthe case shown in Figure 6 there are two limiting rays withcoinciding lines of actionThese two rays form the tangent tothe curve at point119872 and it is identified as a regular point ofthe curve A tangentT exists only at a regular point of a curveA curve point where the tangent T does not exist or is equalto zero is identified as a singular point

Then tangent T is determined and it has

T r120579 = 1199091015840(120579) i1 + 119910

1015840(120579) j1 + 119911

1015840(120579) k1 (8)

For conjugate curves Γ1 and Γ2 the tangentsT1 andT2 areexpressed by r1120579 r2120579 respectively

The normal n119899 to conjugate curves at contact point inarbitrary direction of contact angle is perpendicular to thetangent to the mating curves There is an infinite number

Mbull

bull

bull

bull bull

Space curve

Tangent (two limiting rays)

Rayszi

M2

M3

M4

M1

xi

yi

Figure 6 Illustration of rays and tangent

of normals n belonging to normal plane at contact point119872For instance vector n119899 is one of the set of curve normals inFigure 7 Three orthogonal vectors can be determined thetangent vector 120572 principal normal 120573 and binormal 120574

Mathematical Problems in Engineering 5

M bull

bullM

Osculating plane

Normal plane

Γ

1205720

1205720

t1

t1

t2

t2

nn

nn

120574

120574

120573

120573

120572

120572

Figure 7 Normal to conjugate curves in arbitrary contact direction

The principal normal vector 120573 and binormal 120574 can becalculated respectively as

120573 =

119909120579120579 (1199102

120579+ 1199112

120579) minus 119909120579 (119910120579119910120579120579 + 119911120579119911120579120579)

(1199092

120579+ 1199102

120579+ 1199112

120579)2

i1

+

119910120579120579 (1199092

120579+ 1199112

120579) minus 119910120579 (119909120579119909120579120579 + 119911120579119911120579120579)

(1199092

120579+ 1199102

120579+ 1199112

120579)2

j1

+

119911120579120579 (1199092

120579+ 1199102

120579) minus 119911120579 (119909120579119909120579120579 + 119910120579119910120579120579)

(1199092

120579+ 1199102

120579+ 1199112

120579)2

k1

120574 =119910120579119911120579120579 minus 119911120579119910120579120579

(1199092

120579+ 1199102

120579+ 1199112

120579)32

i1 minus119909120579119911120579120579 minus 119911120579119909120579120579

(1199092

120579+ 1199102

120579+ 1199112

120579)32

j1

+119909120579119910120579120579 minus 119910120579119909120579120579

(1199092

120579+ 1199102

120579+ 1199112

120579)32

k1

(9)

The normal vector to contact point in arbitrary directionof contact angle can be represented as a linear combination ofprincipal normal and binormal vectors and it has n119899 = 1199051120573+1199052120574 Here contact angle 1205720 is defined as the angle betweenarbitrary normal vector n119899 and binormal vector 120574

32 Characteristic Point Given the curve Γ1 adding amotionparameter 120582 the family of curves Γ119894 is obtained [2] Theenveloping curve Γ2 can be generated as depicted in Figure 8if satisfying the following conditions (1) curve Γ1 is smoothand regular curve (2) At every instant 119905 Γ1 and Γ2 touch eachother in contact point119872 (3) Each point of curve Γ2 is alsoa contact point at a unique instant 119905 Obviously it is the setof different contact points (4) The relation between curvesΓ1 and Γ2 can be rendered quite symmetrical and each maybe said to be conjugated for the other if their domains aresuitably restricted

According to (2) family of curves Γ119894 can be expressed as

Γ119894 r1 = r1 (120579 120582) (10)

1205720 Γ1

Γ1

Γ1

Γ2nn

120573

120572

120574

Figure 8 Envelope of a family of curves in given contact direction

And if the enveloping curve exists it has

r1 = r1 (120579 (120582) 120582) = r1 (120582) (11)

For arbitrary curve in family of curves Γ119894 its tangentvector is written as

r1120579 =120597r1120597120579 (12)

The calculation for enveloping curve is

r1 =120597r1120597120579

119889120579

119889120582+120597r1120597120582= r1120579

119889120579

119889120582+ r1120582 (13)

There is common tangent between both curves at contactpoint so that it has r1120579r1 Then

r1120579 times r1 = 0 (14)

Substituting (12) and (13) into (14) yields the result as

r1120579 times r1 = r1120579 times (120597r1120597120579

119889120579

119889120582+120597r1120597120582) = r1120579 times r1120582 = 0 (15)

that is

r1120579 times r1120582 =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

i1 j1 k11205971199091

120597120579

1205971199101

120597120579

1205971199111

120597120579

1205971199091

120597120582

1205971199101

120597120582

1205971199111

120597120582

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (16)

So the mathematical formula of enveloping curve offamily of curves Γ119894 is derived as

r1 = r1 (120579 120582)

r1120579 times r1120582 = Φ (120579 120582) = 0(17)

Compared with the solution of conjugated curve inSection 2 (17) has the same meaning with (7) Usually eachfamily of surfaces and its enveloping surface are not generallytangent at one point but along a curve which is called thecharacteristic line on the surface Actually it is also the contactcurve for both surfaces proposed above The envelopingsurface could be used as the set of family of single parametercharacteristic lines [3] Similarly each family of curves and itsenveloping curve are tangent with point contact and the set

6 Mathematical Problems in Engineering

of contact points which are called characteristic points finallyforms the target curve

According to aforementioned discussion we can get suchfollowing conditions

(1) The necessary and sufficient conditions for contactbetween arbitrary curve of family of curves Γ119894 and itsenveloping curve are

Φ (120579 120582) = r1120579 times r1120582 = 0 (18)

and the generation formula of enveloping curve is

r1 = r1 (120579 120582)

r1120579 times r1120582 = Φ (120579 120582) = 0(19)

(2) The characteristic point which is also the conjugatecontact point with generated enveloping curve exists on eachcurve of family of curves Γ119894 It has

r1 = r1 (120579 1205820)

r1120579 times r11205820

= Φ (120579 1205820) = 0

(20)

where 1205820 is the constant

33 Curvature Relationships It is well known that the curva-ture is of great importance indicator for evaluating geometriccharacteristic of curve and it reflects the bending degreeat one point of curve As the parameter of arc length 119904 isintroduced to the derivation of curve curvature As shownin Figure 91198720 is arbitrary given point of curve Γ and1198721 isits neighboring curve point The correspondence parametersof arc length are 1199040 and 1199040 + Δ119904 respectively Δ120593 is angleformed by the tangents taken at the given and neighboringcurve pointsΔ119904 is arc length between the neighboring pointsThen the change rate of tangents when point1198720 turns the arclength Δ119904 can be described as the curvature of curve Γ at thispoint

The curvature 1205810 of a spatial curve is determined as

1205810 = limΔ119904rarr0

10038161003816100381610038161003816100381610038161003816

Δ120593

Δ119904

10038161003816100381610038161003816100381610038161003816

(21)

where the subscript ldquo0rdquo in 1205810 indicates that the curvature isconsidered for a small piece of the curve located in the oscu-lating plane Based on differential geometry the curvature 1205810can be calculated with formula 1205810 = |r119904 times r119904119904|

To derive equations for determination of the curvecurvature 1205810 and curve torsion 120591 the parameter 119904 of arclength and curve parameter 120579 are considered to be related byfunction 119904(120579) Based on (2) the curve to be discussed can berepresented as r1(119904(120579)) Differentiation of this vector functionyields

r1120579 = r119904119889119904

119889120579 (22)

bull

bull

zi

xi

yi

Γ

(s0)

Δ120593(s0 + Δs)

r(s0 + Δs)r(s0)

M0

M1

120572

120572

Figure 9 Curve curvature analysis

Here 119889119904119889120579 = |r1120579| because |r119904| = 1 Moreover it has

r1120579120579 = r119904119904 (119889119904

119889120579)

2

+ r119904 (1198892119904

1198891205792)

r1120579120579120579 = r119904119904119904 (119889119904

119889120579)

3

+ 3r119904119904 (119889119904

119889120579)(1198892119904

1198891205792) + r119904 (

1198893119904

1198891205793)

(23)

Substituting (22) and (23) into the expression of curvature1205810 we obtain the simplified result

r119904 times r119904119904 =r1120579 times r1120579120579(119889119904119889120579)

3=r1120579 times r11205791205791003816100381610038161003816r11205791003816100381610038161003816

3

1205810 =

1003816100381610038161003816r1120579 times r11205791205791003816100381610038161003816

1003816100381610038161003816r11205791003816100381610038161003816

3

= [(11990911205791199101120579120579 minus 11990911205791205791199101120579)2+ (11990911205791199111120579120579 minus 11990911205791205791199111120579)

2

+ (11991011205791199111120579120579 minus 11991011205791205791199111120579)2]12

times ((1199092

1120579+ 1199102

1120579+ 1199112

1120579)32

)

minus1

(24)

For the analysis of torsion of curve the formula 120591 =(r119904 r119904119904 r119904119904119904)12058120 yields

120591 =(r1120579 times r1120579120579) sdot r1120579120579120579(r1120579 times r1120579120579)

2 (25)

4 Meshing Characteristics ofConjugate Curves

41 Uniqueness of Conjugated Curve The conjugated curve isgenerated in terms of the meshing equation and transforma-tion matrixes among different coordinate systems accordingto a given original curve The normal vector n119899 to conjugatecurves at contact point in arbitrary direction of contact angle

Mathematical Problems in Engineering 7

M

z

x

y

Δ120572

120572

1205721

1205722

Γ1

Γ21Γ22

120573

120574

Figure 10 Two conjugated curves for different contact angles 1205721 1205722

can be determined in the previous study Particularly fromthe expression of normal vector n119899 we can identify thatcoefficients 1199051 and 1199052 indicate the projection of contact angle1205720 on the direction of principal normal and binormal vectorsrespectively

Given the different coefficients 1199051 and 1199052 if the originalcurve is determined the corresponding conjugated curvecan be represented in various results Then the further studyon the relationship between obtained conjugated curves iscarried out Given the contact angles 1205721 1205722 (where 1205722 = 1205721 +Δ120572) in Figure 10 respectively based on principle descriptionsof conjugate curves in Section 2 the conjugated curves Γ2119894(119894 = 1 2) which are separately related to given conditions areexpressed as

119909(119894)

2= 1199091 [cos120601

(119894)

1cos (11989421120601

(119894)

1) minus sin120601(119894)

1sin (11989421120601

(119894)

1) cosΣ]

+ 1199101 [minus sin120601(119894)

1cos (11989421120601

(119894)

1)

minus cos120601(119894)1sin (11989421120601

(119894)

1) cosΣ]

minus 1199111 sin (11989421120601(119894)

1) sinΣ minus 119886 cos (11989421120601

(119894)

1)

119910(119894)

2= 1199091 [cos120601

(119894)

1sin (11989421120601

(119894)

1) + sin120601(119894)

1cos (11989421120601

(119894)

1) cosΣ]

+ 1199101 [minus sin120601(119894)

1sin (11989421120601

(119894)

1)

+ cos120601(119894)1cos (11989421120601

(119894)

1) cosΣ]

+ 1199111 cos (11989421120601(119894)

1) sinΣ minus 119886 sin (11989421120601

(119894)

1)

119911(119894)

2= minus1199091 sin120601

(119894)

1sinΣ minus 1199101 cos120601

(119894)

1sinΣ + 1199111 cosΣ

120601(119894)

1= arcsin 119882

(119894)

radic119880(119894)2+ 119881(119894)

2

minus arctan 119881(119894)

119880(119894)

(26)

Considering that conjugated curve Γ21 rotates aboutcentral contact point119872 with angle Δ120572 it has this expressionafter the coordinate transformation as

119909(1)1015840

2= 119909(1)

2cosΔ120572 + 119910(1)

2sinΔ120572

119910(1)1015840

2= minus 119909

(1)

2sinΔ120572 + 119910(1)

2cosΔ120572

119911(1)1015840

2= 119911(1)

2

(27)

Substituting (26) into (27) the derived result is written as

119909(1)1015840

2= (11990911198601 + 11991011198611 minus 11991111198621) cosΔ120572

+ (11990911198602 + 11991011198612 + 11991111198622) sinΔ120572

minus 119886 cos (Δ120572 minus 11989421120601(1)

1)

119910(1)1015840

2= (minus11990911198601 minus 11991011198611 + 11991111198621) sinΔ120572

minus (11990911198602 minus 11991011198612 minus 11991111198622) cosΔ120572

minus 119886 sin (Δ120572 minus 11989421120601(1)

1)

119911(1)1015840

2= minus1199091 sin120601

(1)

1sinΣ minus 1199101 cos120601

(1)

1sinΣ + 1199111 cosΣ

(28)

where1198601 = cos120601

(1)

1cos (11989421120601

(1)

1) minus sin120601(1)

1sin (11989421120601

(1)

1) cosΣ

1198602 = cos120601(1)

1sin (11989421120601

(1)

1) + sin120601(1)

1cos (11989421120601

(1)

1) cosΣ

1198611 = minus sin120601(1)

1cos (11989421120601

(1)

1) minus cos120601(1)

1sin (11989421120601

(1)

1) cosΣ

1198612 = minus sin120601(1)

1sin (11989421120601

(1)

1) + cos120601(1)

1cos (11989421120601

(1)

1) cosΣ

1198621 = sin (11989421120601(1)

1) sinΣ

1198622 = cos (11989421120601(1)

1) sinΣ

(29)

A comparison about the expressions of Γ22 and Γ101584021

between (26) and (28) is worked It can be concluded fromthe simplified results that the two conjugated curves Γ21 andΓ22 are the same and unique curves while the difference isthat Γ22 is the rotational curve of Γ21 about contact point119872 under given angle Δ120572 For verifying above conclusionthe mathematical example for parallel axis gearing based oncylindrical spiral curve is introduced and its parametric formis represented as

r1 = 119877 cos 120579i1 + 119877 sin 120579j1 + 119901120579k1 (30)

where 119877 is the radius of pitch circle 120579 is spiral curve param-eter and 119901 is helix parameter Through the aforementionedprinciple of conjugate curves the equation of conjugatedcurve is derived as

1199092 = (119877 minus 119886) cos (11989421120579)

1199102 = minus (119877 minus 119886) sin (11989421120579)

1199112 = 119901120579

(31)

8 Mathematical Problems in Engineering

75 80 85 90 95

010

2030

40

0

10

20

30

40

50

Given curve

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(a)

85 90

010

20

0

10

20

30

40

50

Given curve

minus20

minus10

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(b)

75 80 85 90 95

0

10

20

30

40

50

Given curve

minus50

minus40

minus30

minus20

minus10

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(c)

Figure 11 Uniqueness of conjugated curve at any contact position (a) beginning position (b) middle position (c) end position

A simplified meshing model of conjugate curves alongvarious contact angles for gear drive is developed by meansof Matlab software as displayed in Figure 11 according to thedesignated parameters shown in Table 1

From Figure 11 the generated conjugated curves aredifferent if the selected contact angles are various When thegear pair rotates with fixed angular velocity the movementof conjugate curves with 20 degrees rotation 30 degreesrotation and 40 degrees rotation can be described respec-tively The engagement point is still fixed on the commonpoint during the whole meshing Throughout this processcontact point changes gradually in the axial direction and theline of action is always a straight line which is parallel with

gear axes The characteristics of conjugate curves providemore flexibility for the design of gear transmission We canchoose the optimized design result which is suitable forpractical application in terms of appropriate parameters

42 Meshing Commonality of Conjugate Curves To analyzeintrinsic nature of conjugate curves a class of spiral curvescontacting in the given direction for various gear axes isdiscussed

421 Gear Pair with Parallel Axis Assuming that parameters119886 = 0 and Σ = 0 the gear pair with parallel axis can beobtained as displayed in Figure 12 A calculation procedure

Mathematical Problems in Engineering 9

Cylindrical spiral curve Conjugated cylindrical spiral curve

x1

y1

x2

y2

z1z2

(a)

Conjugated curveGiven curve

Line of action

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

(b)

Figure 12 Conjugate spiral curves for parallel axis gear drive

Table 1 Design parameters for conjugate curves

Parameters ValuesRadius of pitch circle Rmm 24Tooth number of pinion 1 1198851 6Tooth number of gear 2 1198852 30Transmission ratio 11989421 5Module119898119899mm 4Helix parameter 119901 286478Central distance 119886mm 144Tooth width 119861mm 50Parameter of spiral curve 120579rad 0sim10472Parameter 1199051 minus05Parameter 1199052 minus0866

of designated spiral curve expressed in (30) for generalcylindrical gears is carried out and the conjugated curve isderived as (31)

Here

1198771 = 119877 minus 119886 1205791 = 11989421120579 (32)

Then (31) can be written as

1199092 = 1198771 cos 1205791

1199102 = minus1198771 sin 1205791

1199112 = 119901120579

(33)

It is consistent with general form of cylindrical spiralcurve in parallel axis gear drive

422 Gear Pair with Intersecting Axis Assuming that param-eters 119886 = 0 and Σ = 0 the gear pair with intersecting axis canbe obtained as shown in Figure 13The selected conical spiralcurve for bevel gears is expressed in coordinate system 1198781 andits parametric form is represented as

r1 = (1198771119891 minus 119888120579) cos 120579i1 + (1198771119891 minus 119888120579) sin 120579j1 + 119901120579k1 (34)

where 1198771119891 is the radius of pitch circle and 120579 and 119901 are alsothe spiral curve and helix parameters respectively It has thecorrelation 119888 = 119901 tan 120575 120575 is the conical angle

Through a series of calculation on the basis of principle ofconjugate curves its conjugated curve is expressed as

1199092 = 1199091 [cos1206011 cos (119894211206011) minus sin1206011 sin (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 cos (119894211206011) minus cos1206011 sin (119894211206011) cosΣ]

minus 1199111 sin (119894211206011) sinΣ

1199102 = 1199091 [cos1206011 sin (119894211206011) + sin1206011 cos (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 sin (119894211206011) + cos1206011 cos (119894211206011) cosΣ]

+ 1199111 cos (119894211206011) sinΣ

1199112 = minus1199091 cos1206011 sinΣ minus 1199101 sin1206011 sinΣ + 1199111 cosΣ

119880 cos1206011 minus 119881 sin1206011 = 119882(35)

10 Mathematical Problems in Engineering

Conical spiral curve

Conjugated conical spiral curve

z1 z2

x1y1x2

y2

1205751

1205752

Rf1

Rf2

O1

O2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

Conjugated curveGiven curve

Line of action

(b)

Figure 13 Conjugate spiral curves for intersecting axis gear drive

Furthermore substituting (34) into (35) the simplifiedresult with parameter 120579 is represented as

1199092 = [1198771119891 minus (119888 + 119901) 120579] cos (11989421120579 + 2120579 + Σ)

1199102 = [1198771119891 minus (119888 + 119901) 120579] sin (11989421120579 + 2120579 + Σ)

1199112 = (minus119888 sinΣ + 119901 cosΣ) 120579 minus 1198771119891 sinΣ

(36)

Here

1198881 = 119888 + 119901 1205792 = 11989421120579 + 2120579 + Σ

1199011 = minus119888 sinΣ + 119901 cosΣ(37)

Then (36) can be written as

1199092 = (1198771119891 minus 1198881120579) cos 1205792

1199102 = (1198771119891 minus 1198881120579) sin 1205792

1199112 = 1199011120579 minus 1198771119891 sinΣ

(38)

Conjugated curve attached to gear 2 is got according toabove derivation Obviously it is also a conical spiral curve inform

423 Gear Pair with CrossedAxis Assuming that parameters119886 = 0 and Σ = 0 it gets the gear pair with crossed axis whichis depicted in Figure 14 The general spiral curve selectedfor worm gear drive is given and it has the same expression

with (30) Based on the principle of conjugate curves itsconjugated curve is derived as

1199092 =radic1199012sin2Σ + (119877 minus 119886)2 cos [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199102 =radic1199012sin2Σ + (119877 minus 119886)2 sin [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199112 = 119901120579 cosΣ(39)

Here

1198772 =radic1199012sin2Σ + (119877 minus 119886)2

1205793 = 11989421120579 + arctan(119877 minus 119886

119901 sinΣ)

(40)

Then (39) can be written as

1199092 = 1198772 cos 1205793

1199102 = 1198772 sin 1205793

1199112 = 119901120579 cosΣ

(41)

It also represents the similar relationship with givenoriginal curve in form

From what are discussed above we can conclude thatthe conjugated curves corresponding to the given spiralcurves for various gear axes are still the spiral curves Itcan be written as the same expression form with the given

Mathematical Problems in Engineering 11

Worm spiral curve

Conjugated worm spiral curve

x1

y1

x2

y2

O1

O2

z1

z2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

Conjugated curveGiven curve

Line of action

x (mm)y (mm)

z(m

m)

(b)

Figure 14 Conjugate spiral curves for crossed axis gear drive

conditions It is called meshing commonality of conjugatecurves Moreover based on the principle of conjugate curvesfor any chosen contact curve the corresponding same typeof conjugated curve in the direction of arbitrary contact anglecan be identified Freedomof selecting contact curve suggeststhe possibility of optimal conjugation design

According to the aforementioned research it offers aversatile way for high-performance profile design (1) fewteeth number and large module may be obtained withouttooth undercutting The problems of limited installationspace and lightweight design can be solved (2)Three differentcontact models of tooth profiles can be established basedon the proposed study Particularly the special meshing ofgenerated convex and concave tooth profiles makes relativeradius of curvature of the contact point longer and increasesthe contact strength The load capacity and useful life hasbeen evidently improved (3) The tooth surfaces mesh inpoint contact along the conjugate curves and contain thetransmission properties of conjugate curves Due to theselection and generation processes the approximate purerolling contact between mating tooth surfaces maybe occursThe transmission efficiency will be improved

The further studies on strength property manufacturingkey technology and performance experiment of conjugate-curve gear drive will be carried out And the excellent trans-mission performance of gearing is expected to be obtained onthe basis of theoretical and experimental investigations

5 Conclusions

(1) The principle descriptions of conjugate curves forarbitrary axial position are introducedWith the aid of

given plane or space curve generation principle andmathematical model of conjugated curve are devel-oped It can be applied to parallel axis intersectingaxis and crossed axis gear drive respectively

(2) Based on differential geometry geometric charac-teristic analysis of conjugate curves is carried outThe tangent and normal to conjugate curves inarbitrary contact direction are discussed and generalcalculation methods are provided The envelopingcondition of family of conjugate curves with singleparameter is analyzed and characteristic point isdetermined Curvature and torsion relationships ofconjugate curves are also derived

(3) Meshing properties of conjugate curves are furtherrevealed According to a given plane or spatial curvethe variation law of meshing function is analyzedThe uniqueness of conjugated curves is discussedTheconjugated curves are always the same and uniquecurves for any given curve under different contactangles while the difference is that there exists therotational angle relationship with contact point 119872among generated curves

(4) Meshing commonality of conjugate curves is demon-strated in terms of a class of spiral curves contactingin the given direction for various gear axes For anychosen contact curve under various gear axes theconjugated curve has the same type and it can bewritten as the same expression form

(5) The proposed theory lays the foundation for design ofnew types of gear drive The further study on conju-gate curves and applications in gear transmission will

12 Mathematical Problems in Engineering

be carried out Excellent transmission performance ofgearing is expected to be obtained

Conflict of Interests

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

Acknowledgments

This project is supported by National Key Technology RampDProgram of the Twelfth Five-year Plan of China (Grantno 2013BAF01B04) National Natural Science Foundation ofChina (Grant no 51205425) and Science and TechnologyProject of Chongqing City Board of Education (Grant noKJ122202) The authors also sincerely appreciate the com-ments and modification suggestions made by the editors andanonymous referees

References

[1] S S Chern W H Chen and K S Lam Lectures on DifferentialGeometry World Scientific Singapore 2006

[2] P G Ciarlet and T-T Li Differential Geometry Theory andApplications World Scientific Singapore 2007

[3] F L Litvin and A Fuentes Gear Geometry and Applied TheoryCambridge University Press New York NY USA 2nd edition2004

[4] S P Radzevich Theory of Gearing Kinetics Geometry andSynthesis CRC Press Boca Raton Fla USA 2012

[5] D B Dooner Kinematic Geometry of Gearing John Wiley ampSons 2nd edition 2012

[6] F L Litvin A Fuentes A Demenego D Vecchiato and Q FanldquoNewdevelopments in the design and generation of gear drivesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 215 no 7 pp747ndash757 2001

[7] C H Chen Theory of Conjugate Surfaces and Its ApplicationsScience and Technology Press Beijing China 2008

[8] G X Li Spatial Geometry Modeling and Its Application inEngineering Higher Education Press Beijing China 2007

[9] F Di Puccio M Gabiccini and M Guiggiani ldquoGeneration andcurvature analysis of conjugate surfaces via a new approachrdquoMechanism and Machine Theory vol 41 no 4 pp 382ndash4042006

[10] D R Wu and J S Luo A Geometric Theory of Conjugate ToothSurface World Scientific Publishing Singapore 1992

[11] N Ito and K Takahashi ldquoDifferential geometrical conditions ofhypoid gears with conjugate tooth surfacesrdquo ASME Journal ofMechanical Design vol 122 no 3 pp 323ndash330 2000

[12] D B Dooner ldquoOn the three laws of gearingrdquo Journal ofMechanical Design Transactions of the ASME vol 124 no 4pp 733ndash744 2002

[13] Z Duan H Chen and J Liu ldquoPrinciple and transmissiontechnology of Bertrand conjugate surfacesrdquo Chinese Journal ofMechanical Engineering vol 19 no 4 pp 600ndash604 2006

[14] H Chen Z Duan J Liu and H Wu ldquoResearch on basicprinciple of moulding-surface conjugationrdquo Mechanism andMachine Theory vol 43 no 7 pp 791ndash811 2008

[15] 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

[16] C F Chen and C B Tsay ldquoTooth profile design for themanufacture of helical gear sets with small numbers of teethrdquoInternational Journal ofMachine Tools andManufacture vol 45no 12-13 pp 1531ndash1541 2005

[17] R Imin and M Geni ldquoStress analysis of gear meshing impactbased on SPH methodrdquoMathematical Problems in Engineeringvol 2014 Article ID 328216 7 pages 2014

[18] D Park and A Kahraman ldquoA surface wear model for hypoidgear pairsrdquoWear vol 267 no 9-10 pp 1595ndash1604 2009

[19] JWang L Hou SM Luo and R YWu ldquoActive design of toothprofiles using parabolic curve as the line of actionrdquoMechanismand Machine Theory vol 67 no 9 pp 47ndash63 2013

[20] Q B Wang P Hu Y M Zhang et al ldquoA model to determinemesh characteristics in a gear pair with tooth profile errorrdquoAdvances in Mechanical Engineering vol 2014 Article ID751476 10 pages 2014

[21] C F Chen and C B Tsay ldquoComputerized tooth profile gen-eration and analysis of characteristics of elliptical gears withcircular arc teethrdquo Journal of Materials Processing Technologyvol 148 no 2 pp 226ndash234 2004

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

[23] B K Chen Y E Gao and D Liang ldquoThe generation principleof tooth surfaces of conjugate-curve gear transmissionrdquo Journalof Mechanical Engineering vol 50 no 3 pp 18ndash24 2014

[24] D Liang B K Chen and Y E Gao ldquoThe generation principleand mathematical model of a new involute-helix gear driverdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 227 no 12 pp2834ndash2843 2013

[25] B Chen D Liang and Z Li ldquoA study on geometry designof spiral bevel gears based on conjugate curvesrdquo InternationalJournal of Precision Engineering and Manufacturing vol 15 no3 pp 477ndash482 2014

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

M bull

bullM

Osculating plane

Normal plane

Γ

1205720

1205720

t1

t1

t2

t2

nn

nn

120574

120574

120573

120573

120572

120572

Figure 7 Normal to conjugate curves in arbitrary contact direction

The principal normal vector 120573 and binormal 120574 can becalculated respectively as

120573 =

119909120579120579 (1199102

120579+ 1199112

120579) minus 119909120579 (119910120579119910120579120579 + 119911120579119911120579120579)

(1199092

120579+ 1199102

120579+ 1199112

120579)2

i1

+

119910120579120579 (1199092

120579+ 1199112

120579) minus 119910120579 (119909120579119909120579120579 + 119911120579119911120579120579)

(1199092

120579+ 1199102

120579+ 1199112

120579)2

j1

+

119911120579120579 (1199092

120579+ 1199102

120579) minus 119911120579 (119909120579119909120579120579 + 119910120579119910120579120579)

(1199092

120579+ 1199102

120579+ 1199112

120579)2

k1

120574 =119910120579119911120579120579 minus 119911120579119910120579120579

(1199092

120579+ 1199102

120579+ 1199112

120579)32

i1 minus119909120579119911120579120579 minus 119911120579119909120579120579

(1199092

120579+ 1199102

120579+ 1199112

120579)32

j1

+119909120579119910120579120579 minus 119910120579119909120579120579

(1199092

120579+ 1199102

120579+ 1199112

120579)32

k1

(9)

The normal vector to contact point in arbitrary directionof contact angle can be represented as a linear combination ofprincipal normal and binormal vectors and it has n119899 = 1199051120573+1199052120574 Here contact angle 1205720 is defined as the angle betweenarbitrary normal vector n119899 and binormal vector 120574

32 Characteristic Point Given the curve Γ1 adding amotionparameter 120582 the family of curves Γ119894 is obtained [2] Theenveloping curve Γ2 can be generated as depicted in Figure 8if satisfying the following conditions (1) curve Γ1 is smoothand regular curve (2) At every instant 119905 Γ1 and Γ2 touch eachother in contact point119872 (3) Each point of curve Γ2 is alsoa contact point at a unique instant 119905 Obviously it is the setof different contact points (4) The relation between curvesΓ1 and Γ2 can be rendered quite symmetrical and each maybe said to be conjugated for the other if their domains aresuitably restricted

According to (2) family of curves Γ119894 can be expressed as

Γ119894 r1 = r1 (120579 120582) (10)

1205720 Γ1

Γ1

Γ1

Γ2nn

120573

120572

120574

Figure 8 Envelope of a family of curves in given contact direction

And if the enveloping curve exists it has

r1 = r1 (120579 (120582) 120582) = r1 (120582) (11)

For arbitrary curve in family of curves Γ119894 its tangentvector is written as

r1120579 =120597r1120597120579 (12)

The calculation for enveloping curve is

r1 =120597r1120597120579

119889120579

119889120582+120597r1120597120582= r1120579

119889120579

119889120582+ r1120582 (13)

There is common tangent between both curves at contactpoint so that it has r1120579r1 Then

r1120579 times r1 = 0 (14)

Substituting (12) and (13) into (14) yields the result as

r1120579 times r1 = r1120579 times (120597r1120597120579

119889120579

119889120582+120597r1120597120582) = r1120579 times r1120582 = 0 (15)

that is

r1120579 times r1120582 =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

i1 j1 k11205971199091

120597120579

1205971199101

120597120579

1205971199111

120597120579

1205971199091

120597120582

1205971199101

120597120582

1205971199111

120597120582

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (16)

So the mathematical formula of enveloping curve offamily of curves Γ119894 is derived as

r1 = r1 (120579 120582)

r1120579 times r1120582 = Φ (120579 120582) = 0(17)

Compared with the solution of conjugated curve inSection 2 (17) has the same meaning with (7) Usually eachfamily of surfaces and its enveloping surface are not generallytangent at one point but along a curve which is called thecharacteristic line on the surface Actually it is also the contactcurve for both surfaces proposed above The envelopingsurface could be used as the set of family of single parametercharacteristic lines [3] Similarly each family of curves and itsenveloping curve are tangent with point contact and the set

6 Mathematical Problems in Engineering

of contact points which are called characteristic points finallyforms the target curve

According to aforementioned discussion we can get suchfollowing conditions

(1) The necessary and sufficient conditions for contactbetween arbitrary curve of family of curves Γ119894 and itsenveloping curve are

Φ (120579 120582) = r1120579 times r1120582 = 0 (18)

and the generation formula of enveloping curve is

r1 = r1 (120579 120582)

r1120579 times r1120582 = Φ (120579 120582) = 0(19)

(2) The characteristic point which is also the conjugatecontact point with generated enveloping curve exists on eachcurve of family of curves Γ119894 It has

r1 = r1 (120579 1205820)

r1120579 times r11205820

= Φ (120579 1205820) = 0

(20)

where 1205820 is the constant

33 Curvature Relationships It is well known that the curva-ture is of great importance indicator for evaluating geometriccharacteristic of curve and it reflects the bending degreeat one point of curve As the parameter of arc length 119904 isintroduced to the derivation of curve curvature As shownin Figure 91198720 is arbitrary given point of curve Γ and1198721 isits neighboring curve point The correspondence parametersof arc length are 1199040 and 1199040 + Δ119904 respectively Δ120593 is angleformed by the tangents taken at the given and neighboringcurve pointsΔ119904 is arc length between the neighboring pointsThen the change rate of tangents when point1198720 turns the arclength Δ119904 can be described as the curvature of curve Γ at thispoint

The curvature 1205810 of a spatial curve is determined as

1205810 = limΔ119904rarr0

10038161003816100381610038161003816100381610038161003816

Δ120593

Δ119904

10038161003816100381610038161003816100381610038161003816

(21)

where the subscript ldquo0rdquo in 1205810 indicates that the curvature isconsidered for a small piece of the curve located in the oscu-lating plane Based on differential geometry the curvature 1205810can be calculated with formula 1205810 = |r119904 times r119904119904|

To derive equations for determination of the curvecurvature 1205810 and curve torsion 120591 the parameter 119904 of arclength and curve parameter 120579 are considered to be related byfunction 119904(120579) Based on (2) the curve to be discussed can berepresented as r1(119904(120579)) Differentiation of this vector functionyields

r1120579 = r119904119889119904

119889120579 (22)

bull

bull

zi

xi

yi

Γ

(s0)

Δ120593(s0 + Δs)

r(s0 + Δs)r(s0)

M0

M1

120572

120572

Figure 9 Curve curvature analysis

Here 119889119904119889120579 = |r1120579| because |r119904| = 1 Moreover it has

r1120579120579 = r119904119904 (119889119904

119889120579)

2

+ r119904 (1198892119904

1198891205792)

r1120579120579120579 = r119904119904119904 (119889119904

119889120579)

3

+ 3r119904119904 (119889119904

119889120579)(1198892119904

1198891205792) + r119904 (

1198893119904

1198891205793)

(23)

Substituting (22) and (23) into the expression of curvature1205810 we obtain the simplified result

r119904 times r119904119904 =r1120579 times r1120579120579(119889119904119889120579)

3=r1120579 times r11205791205791003816100381610038161003816r11205791003816100381610038161003816

3

1205810 =

1003816100381610038161003816r1120579 times r11205791205791003816100381610038161003816

1003816100381610038161003816r11205791003816100381610038161003816

3

= [(11990911205791199101120579120579 minus 11990911205791205791199101120579)2+ (11990911205791199111120579120579 minus 11990911205791205791199111120579)

2

+ (11991011205791199111120579120579 minus 11991011205791205791199111120579)2]12

times ((1199092

1120579+ 1199102

1120579+ 1199112

1120579)32

)

minus1

(24)

For the analysis of torsion of curve the formula 120591 =(r119904 r119904119904 r119904119904119904)12058120 yields

120591 =(r1120579 times r1120579120579) sdot r1120579120579120579(r1120579 times r1120579120579)

2 (25)

4 Meshing Characteristics ofConjugate Curves

41 Uniqueness of Conjugated Curve The conjugated curve isgenerated in terms of the meshing equation and transforma-tion matrixes among different coordinate systems accordingto a given original curve The normal vector n119899 to conjugatecurves at contact point in arbitrary direction of contact angle

Mathematical Problems in Engineering 7

M

z

x

y

Δ120572

120572

1205721

1205722

Γ1

Γ21Γ22

120573

120574

Figure 10 Two conjugated curves for different contact angles 1205721 1205722

can be determined in the previous study Particularly fromthe expression of normal vector n119899 we can identify thatcoefficients 1199051 and 1199052 indicate the projection of contact angle1205720 on the direction of principal normal and binormal vectorsrespectively

Given the different coefficients 1199051 and 1199052 if the originalcurve is determined the corresponding conjugated curvecan be represented in various results Then the further studyon the relationship between obtained conjugated curves iscarried out Given the contact angles 1205721 1205722 (where 1205722 = 1205721 +Δ120572) in Figure 10 respectively based on principle descriptionsof conjugate curves in Section 2 the conjugated curves Γ2119894(119894 = 1 2) which are separately related to given conditions areexpressed as

119909(119894)

2= 1199091 [cos120601

(119894)

1cos (11989421120601

(119894)

1) minus sin120601(119894)

1sin (11989421120601

(119894)

1) cosΣ]

+ 1199101 [minus sin120601(119894)

1cos (11989421120601

(119894)

1)

minus cos120601(119894)1sin (11989421120601

(119894)

1) cosΣ]

minus 1199111 sin (11989421120601(119894)

1) sinΣ minus 119886 cos (11989421120601

(119894)

1)

119910(119894)

2= 1199091 [cos120601

(119894)

1sin (11989421120601

(119894)

1) + sin120601(119894)

1cos (11989421120601

(119894)

1) cosΣ]

+ 1199101 [minus sin120601(119894)

1sin (11989421120601

(119894)

1)

+ cos120601(119894)1cos (11989421120601

(119894)

1) cosΣ]

+ 1199111 cos (11989421120601(119894)

1) sinΣ minus 119886 sin (11989421120601

(119894)

1)

119911(119894)

2= minus1199091 sin120601

(119894)

1sinΣ minus 1199101 cos120601

(119894)

1sinΣ + 1199111 cosΣ

120601(119894)

1= arcsin 119882

(119894)

radic119880(119894)2+ 119881(119894)

2

minus arctan 119881(119894)

119880(119894)

(26)

Considering that conjugated curve Γ21 rotates aboutcentral contact point119872 with angle Δ120572 it has this expressionafter the coordinate transformation as

119909(1)1015840

2= 119909(1)

2cosΔ120572 + 119910(1)

2sinΔ120572

119910(1)1015840

2= minus 119909

(1)

2sinΔ120572 + 119910(1)

2cosΔ120572

119911(1)1015840

2= 119911(1)

2

(27)

Substituting (26) into (27) the derived result is written as

119909(1)1015840

2= (11990911198601 + 11991011198611 minus 11991111198621) cosΔ120572

+ (11990911198602 + 11991011198612 + 11991111198622) sinΔ120572

minus 119886 cos (Δ120572 minus 11989421120601(1)

1)

119910(1)1015840

2= (minus11990911198601 minus 11991011198611 + 11991111198621) sinΔ120572

minus (11990911198602 minus 11991011198612 minus 11991111198622) cosΔ120572

minus 119886 sin (Δ120572 minus 11989421120601(1)

1)

119911(1)1015840

2= minus1199091 sin120601

(1)

1sinΣ minus 1199101 cos120601

(1)

1sinΣ + 1199111 cosΣ

(28)

where1198601 = cos120601

(1)

1cos (11989421120601

(1)

1) minus sin120601(1)

1sin (11989421120601

(1)

1) cosΣ

1198602 = cos120601(1)

1sin (11989421120601

(1)

1) + sin120601(1)

1cos (11989421120601

(1)

1) cosΣ

1198611 = minus sin120601(1)

1cos (11989421120601

(1)

1) minus cos120601(1)

1sin (11989421120601

(1)

1) cosΣ

1198612 = minus sin120601(1)

1sin (11989421120601

(1)

1) + cos120601(1)

1cos (11989421120601

(1)

1) cosΣ

1198621 = sin (11989421120601(1)

1) sinΣ

1198622 = cos (11989421120601(1)

1) sinΣ

(29)

A comparison about the expressions of Γ22 and Γ101584021

between (26) and (28) is worked It can be concluded fromthe simplified results that the two conjugated curves Γ21 andΓ22 are the same and unique curves while the difference isthat Γ22 is the rotational curve of Γ21 about contact point119872 under given angle Δ120572 For verifying above conclusionthe mathematical example for parallel axis gearing based oncylindrical spiral curve is introduced and its parametric formis represented as

r1 = 119877 cos 120579i1 + 119877 sin 120579j1 + 119901120579k1 (30)

where 119877 is the radius of pitch circle 120579 is spiral curve param-eter and 119901 is helix parameter Through the aforementionedprinciple of conjugate curves the equation of conjugatedcurve is derived as

1199092 = (119877 minus 119886) cos (11989421120579)

1199102 = minus (119877 minus 119886) sin (11989421120579)

1199112 = 119901120579

(31)

8 Mathematical Problems in Engineering

75 80 85 90 95

010

2030

40

0

10

20

30

40

50

Given curve

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(a)

85 90

010

20

0

10

20

30

40

50

Given curve

minus20

minus10

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(b)

75 80 85 90 95

0

10

20

30

40

50

Given curve

minus50

minus40

minus30

minus20

minus10

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(c)

Figure 11 Uniqueness of conjugated curve at any contact position (a) beginning position (b) middle position (c) end position

A simplified meshing model of conjugate curves alongvarious contact angles for gear drive is developed by meansof Matlab software as displayed in Figure 11 according to thedesignated parameters shown in Table 1

From Figure 11 the generated conjugated curves aredifferent if the selected contact angles are various When thegear pair rotates with fixed angular velocity the movementof conjugate curves with 20 degrees rotation 30 degreesrotation and 40 degrees rotation can be described respec-tively The engagement point is still fixed on the commonpoint during the whole meshing Throughout this processcontact point changes gradually in the axial direction and theline of action is always a straight line which is parallel with

gear axes The characteristics of conjugate curves providemore flexibility for the design of gear transmission We canchoose the optimized design result which is suitable forpractical application in terms of appropriate parameters

42 Meshing Commonality of Conjugate Curves To analyzeintrinsic nature of conjugate curves a class of spiral curvescontacting in the given direction for various gear axes isdiscussed

421 Gear Pair with Parallel Axis Assuming that parameters119886 = 0 and Σ = 0 the gear pair with parallel axis can beobtained as displayed in Figure 12 A calculation procedure

Mathematical Problems in Engineering 9

Cylindrical spiral curve Conjugated cylindrical spiral curve

x1

y1

x2

y2

z1z2

(a)

Conjugated curveGiven curve

Line of action

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

(b)

Figure 12 Conjugate spiral curves for parallel axis gear drive

Table 1 Design parameters for conjugate curves

Parameters ValuesRadius of pitch circle Rmm 24Tooth number of pinion 1 1198851 6Tooth number of gear 2 1198852 30Transmission ratio 11989421 5Module119898119899mm 4Helix parameter 119901 286478Central distance 119886mm 144Tooth width 119861mm 50Parameter of spiral curve 120579rad 0sim10472Parameter 1199051 minus05Parameter 1199052 minus0866

of designated spiral curve expressed in (30) for generalcylindrical gears is carried out and the conjugated curve isderived as (31)

Here

1198771 = 119877 minus 119886 1205791 = 11989421120579 (32)

Then (31) can be written as

1199092 = 1198771 cos 1205791

1199102 = minus1198771 sin 1205791

1199112 = 119901120579

(33)

It is consistent with general form of cylindrical spiralcurve in parallel axis gear drive

422 Gear Pair with Intersecting Axis Assuming that param-eters 119886 = 0 and Σ = 0 the gear pair with intersecting axis canbe obtained as shown in Figure 13The selected conical spiralcurve for bevel gears is expressed in coordinate system 1198781 andits parametric form is represented as

r1 = (1198771119891 minus 119888120579) cos 120579i1 + (1198771119891 minus 119888120579) sin 120579j1 + 119901120579k1 (34)

where 1198771119891 is the radius of pitch circle and 120579 and 119901 are alsothe spiral curve and helix parameters respectively It has thecorrelation 119888 = 119901 tan 120575 120575 is the conical angle

Through a series of calculation on the basis of principle ofconjugate curves its conjugated curve is expressed as

1199092 = 1199091 [cos1206011 cos (119894211206011) minus sin1206011 sin (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 cos (119894211206011) minus cos1206011 sin (119894211206011) cosΣ]

minus 1199111 sin (119894211206011) sinΣ

1199102 = 1199091 [cos1206011 sin (119894211206011) + sin1206011 cos (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 sin (119894211206011) + cos1206011 cos (119894211206011) cosΣ]

+ 1199111 cos (119894211206011) sinΣ

1199112 = minus1199091 cos1206011 sinΣ minus 1199101 sin1206011 sinΣ + 1199111 cosΣ

119880 cos1206011 minus 119881 sin1206011 = 119882(35)

10 Mathematical Problems in Engineering

Conical spiral curve

Conjugated conical spiral curve

z1 z2

x1y1x2

y2

1205751

1205752

Rf1

Rf2

O1

O2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

Conjugated curveGiven curve

Line of action

(b)

Figure 13 Conjugate spiral curves for intersecting axis gear drive

Furthermore substituting (34) into (35) the simplifiedresult with parameter 120579 is represented as

1199092 = [1198771119891 minus (119888 + 119901) 120579] cos (11989421120579 + 2120579 + Σ)

1199102 = [1198771119891 minus (119888 + 119901) 120579] sin (11989421120579 + 2120579 + Σ)

1199112 = (minus119888 sinΣ + 119901 cosΣ) 120579 minus 1198771119891 sinΣ

(36)

Here

1198881 = 119888 + 119901 1205792 = 11989421120579 + 2120579 + Σ

1199011 = minus119888 sinΣ + 119901 cosΣ(37)

Then (36) can be written as

1199092 = (1198771119891 minus 1198881120579) cos 1205792

1199102 = (1198771119891 minus 1198881120579) sin 1205792

1199112 = 1199011120579 minus 1198771119891 sinΣ

(38)

Conjugated curve attached to gear 2 is got according toabove derivation Obviously it is also a conical spiral curve inform

423 Gear Pair with CrossedAxis Assuming that parameters119886 = 0 and Σ = 0 it gets the gear pair with crossed axis whichis depicted in Figure 14 The general spiral curve selectedfor worm gear drive is given and it has the same expression

with (30) Based on the principle of conjugate curves itsconjugated curve is derived as

1199092 =radic1199012sin2Σ + (119877 minus 119886)2 cos [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199102 =radic1199012sin2Σ + (119877 minus 119886)2 sin [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199112 = 119901120579 cosΣ(39)

Here

1198772 =radic1199012sin2Σ + (119877 minus 119886)2

1205793 = 11989421120579 + arctan(119877 minus 119886

119901 sinΣ)

(40)

Then (39) can be written as

1199092 = 1198772 cos 1205793

1199102 = 1198772 sin 1205793

1199112 = 119901120579 cosΣ

(41)

It also represents the similar relationship with givenoriginal curve in form

From what are discussed above we can conclude thatthe conjugated curves corresponding to the given spiralcurves for various gear axes are still the spiral curves Itcan be written as the same expression form with the given

Mathematical Problems in Engineering 11

Worm spiral curve

Conjugated worm spiral curve

x1

y1

x2

y2

O1

O2

z1

z2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

Conjugated curveGiven curve

Line of action

x (mm)y (mm)

z(m

m)

(b)

Figure 14 Conjugate spiral curves for crossed axis gear drive

conditions It is called meshing commonality of conjugatecurves Moreover based on the principle of conjugate curvesfor any chosen contact curve the corresponding same typeof conjugated curve in the direction of arbitrary contact anglecan be identified Freedomof selecting contact curve suggeststhe possibility of optimal conjugation design

According to the aforementioned research it offers aversatile way for high-performance profile design (1) fewteeth number and large module may be obtained withouttooth undercutting The problems of limited installationspace and lightweight design can be solved (2)Three differentcontact models of tooth profiles can be established basedon the proposed study Particularly the special meshing ofgenerated convex and concave tooth profiles makes relativeradius of curvature of the contact point longer and increasesthe contact strength The load capacity and useful life hasbeen evidently improved (3) The tooth surfaces mesh inpoint contact along the conjugate curves and contain thetransmission properties of conjugate curves Due to theselection and generation processes the approximate purerolling contact between mating tooth surfaces maybe occursThe transmission efficiency will be improved

The further studies on strength property manufacturingkey technology and performance experiment of conjugate-curve gear drive will be carried out And the excellent trans-mission performance of gearing is expected to be obtained onthe basis of theoretical and experimental investigations

5 Conclusions

(1) The principle descriptions of conjugate curves forarbitrary axial position are introducedWith the aid of

given plane or space curve generation principle andmathematical model of conjugated curve are devel-oped It can be applied to parallel axis intersectingaxis and crossed axis gear drive respectively

(2) Based on differential geometry geometric charac-teristic analysis of conjugate curves is carried outThe tangent and normal to conjugate curves inarbitrary contact direction are discussed and generalcalculation methods are provided The envelopingcondition of family of conjugate curves with singleparameter is analyzed and characteristic point isdetermined Curvature and torsion relationships ofconjugate curves are also derived

(3) Meshing properties of conjugate curves are furtherrevealed According to a given plane or spatial curvethe variation law of meshing function is analyzedThe uniqueness of conjugated curves is discussedTheconjugated curves are always the same and uniquecurves for any given curve under different contactangles while the difference is that there exists therotational angle relationship with contact point 119872among generated curves

(4) Meshing commonality of conjugate curves is demon-strated in terms of a class of spiral curves contactingin the given direction for various gear axes For anychosen contact curve under various gear axes theconjugated curve has the same type and it can bewritten as the same expression form

(5) The proposed theory lays the foundation for design ofnew types of gear drive The further study on conju-gate curves and applications in gear transmission will

12 Mathematical Problems in Engineering

be carried out Excellent transmission performance ofgearing is expected to be obtained

Conflict of Interests

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

Acknowledgments

This project is supported by National Key Technology RampDProgram of the Twelfth Five-year Plan of China (Grantno 2013BAF01B04) National Natural Science Foundation ofChina (Grant no 51205425) and Science and TechnologyProject of Chongqing City Board of Education (Grant noKJ122202) The authors also sincerely appreciate the com-ments and modification suggestions made by the editors andanonymous referees

References

[1] S S Chern W H Chen and K S Lam Lectures on DifferentialGeometry World Scientific Singapore 2006

[2] P G Ciarlet and T-T Li Differential Geometry Theory andApplications World Scientific Singapore 2007

[3] F L Litvin and A Fuentes Gear Geometry and Applied TheoryCambridge University Press New York NY USA 2nd edition2004

[4] S P Radzevich Theory of Gearing Kinetics Geometry andSynthesis CRC Press Boca Raton Fla USA 2012

[5] D B Dooner Kinematic Geometry of Gearing John Wiley ampSons 2nd edition 2012

[6] F L Litvin A Fuentes A Demenego D Vecchiato and Q FanldquoNewdevelopments in the design and generation of gear drivesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 215 no 7 pp747ndash757 2001

[7] C H Chen Theory of Conjugate Surfaces and Its ApplicationsScience and Technology Press Beijing China 2008

[8] G X Li Spatial Geometry Modeling and Its Application inEngineering Higher Education Press Beijing China 2007

[9] F Di Puccio M Gabiccini and M Guiggiani ldquoGeneration andcurvature analysis of conjugate surfaces via a new approachrdquoMechanism and Machine Theory vol 41 no 4 pp 382ndash4042006

[10] D R Wu and J S Luo A Geometric Theory of Conjugate ToothSurface World Scientific Publishing Singapore 1992

[11] N Ito and K Takahashi ldquoDifferential geometrical conditions ofhypoid gears with conjugate tooth surfacesrdquo ASME Journal ofMechanical Design vol 122 no 3 pp 323ndash330 2000

[12] D B Dooner ldquoOn the three laws of gearingrdquo Journal ofMechanical Design Transactions of the ASME vol 124 no 4pp 733ndash744 2002

[13] Z Duan H Chen and J Liu ldquoPrinciple and transmissiontechnology of Bertrand conjugate surfacesrdquo Chinese Journal ofMechanical Engineering vol 19 no 4 pp 600ndash604 2006

[14] H Chen Z Duan J Liu and H Wu ldquoResearch on basicprinciple of moulding-surface conjugationrdquo Mechanism andMachine Theory vol 43 no 7 pp 791ndash811 2008

[15] 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

[16] C F Chen and C B Tsay ldquoTooth profile design for themanufacture of helical gear sets with small numbers of teethrdquoInternational Journal ofMachine Tools andManufacture vol 45no 12-13 pp 1531ndash1541 2005

[17] R Imin and M Geni ldquoStress analysis of gear meshing impactbased on SPH methodrdquoMathematical Problems in Engineeringvol 2014 Article ID 328216 7 pages 2014

[18] D Park and A Kahraman ldquoA surface wear model for hypoidgear pairsrdquoWear vol 267 no 9-10 pp 1595ndash1604 2009

[19] JWang L Hou SM Luo and R YWu ldquoActive design of toothprofiles using parabolic curve as the line of actionrdquoMechanismand Machine Theory vol 67 no 9 pp 47ndash63 2013

[20] Q B Wang P Hu Y M Zhang et al ldquoA model to determinemesh characteristics in a gear pair with tooth profile errorrdquoAdvances in Mechanical Engineering vol 2014 Article ID751476 10 pages 2014

[21] C F Chen and C B Tsay ldquoComputerized tooth profile gen-eration and analysis of characteristics of elliptical gears withcircular arc teethrdquo Journal of Materials Processing Technologyvol 148 no 2 pp 226ndash234 2004

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

[23] B K Chen Y E Gao and D Liang ldquoThe generation principleof tooth surfaces of conjugate-curve gear transmissionrdquo Journalof Mechanical Engineering vol 50 no 3 pp 18ndash24 2014

[24] D Liang B K Chen and Y E Gao ldquoThe generation principleand mathematical model of a new involute-helix gear driverdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 227 no 12 pp2834ndash2843 2013

[25] B Chen D Liang and Z Li ldquoA study on geometry designof spiral bevel gears based on conjugate curvesrdquo InternationalJournal of Precision Engineering and Manufacturing vol 15 no3 pp 477ndash482 2014

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Mathematical Problems in Engineering

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6 Mathematical Problems in Engineering

of contact points which are called characteristic points finallyforms the target curve

According to aforementioned discussion we can get suchfollowing conditions

(1) The necessary and sufficient conditions for contactbetween arbitrary curve of family of curves Γ119894 and itsenveloping curve are

Φ (120579 120582) = r1120579 times r1120582 = 0 (18)

and the generation formula of enveloping curve is

r1 = r1 (120579 120582)

r1120579 times r1120582 = Φ (120579 120582) = 0(19)

(2) The characteristic point which is also the conjugatecontact point with generated enveloping curve exists on eachcurve of family of curves Γ119894 It has

r1 = r1 (120579 1205820)

r1120579 times r11205820

= Φ (120579 1205820) = 0

(20)

where 1205820 is the constant

33 Curvature Relationships It is well known that the curva-ture is of great importance indicator for evaluating geometriccharacteristic of curve and it reflects the bending degreeat one point of curve As the parameter of arc length 119904 isintroduced to the derivation of curve curvature As shownin Figure 91198720 is arbitrary given point of curve Γ and1198721 isits neighboring curve point The correspondence parametersof arc length are 1199040 and 1199040 + Δ119904 respectively Δ120593 is angleformed by the tangents taken at the given and neighboringcurve pointsΔ119904 is arc length between the neighboring pointsThen the change rate of tangents when point1198720 turns the arclength Δ119904 can be described as the curvature of curve Γ at thispoint

The curvature 1205810 of a spatial curve is determined as

1205810 = limΔ119904rarr0

10038161003816100381610038161003816100381610038161003816

Δ120593

Δ119904

10038161003816100381610038161003816100381610038161003816

(21)

where the subscript ldquo0rdquo in 1205810 indicates that the curvature isconsidered for a small piece of the curve located in the oscu-lating plane Based on differential geometry the curvature 1205810can be calculated with formula 1205810 = |r119904 times r119904119904|

To derive equations for determination of the curvecurvature 1205810 and curve torsion 120591 the parameter 119904 of arclength and curve parameter 120579 are considered to be related byfunction 119904(120579) Based on (2) the curve to be discussed can berepresented as r1(119904(120579)) Differentiation of this vector functionyields

r1120579 = r119904119889119904

119889120579 (22)

bull

bull

zi

xi

yi

Γ

(s0)

Δ120593(s0 + Δs)

r(s0 + Δs)r(s0)

M0

M1

120572

120572

Figure 9 Curve curvature analysis

Here 119889119904119889120579 = |r1120579| because |r119904| = 1 Moreover it has

r1120579120579 = r119904119904 (119889119904

119889120579)

2

+ r119904 (1198892119904

1198891205792)

r1120579120579120579 = r119904119904119904 (119889119904

119889120579)

3

+ 3r119904119904 (119889119904

119889120579)(1198892119904

1198891205792) + r119904 (

1198893119904

1198891205793)

(23)

Substituting (22) and (23) into the expression of curvature1205810 we obtain the simplified result

r119904 times r119904119904 =r1120579 times r1120579120579(119889119904119889120579)

3=r1120579 times r11205791205791003816100381610038161003816r11205791003816100381610038161003816

3

1205810 =

1003816100381610038161003816r1120579 times r11205791205791003816100381610038161003816

1003816100381610038161003816r11205791003816100381610038161003816

3

= [(11990911205791199101120579120579 minus 11990911205791205791199101120579)2+ (11990911205791199111120579120579 minus 11990911205791205791199111120579)

2

+ (11991011205791199111120579120579 minus 11991011205791205791199111120579)2]12

times ((1199092

1120579+ 1199102

1120579+ 1199112

1120579)32

)

minus1

(24)

For the analysis of torsion of curve the formula 120591 =(r119904 r119904119904 r119904119904119904)12058120 yields

120591 =(r1120579 times r1120579120579) sdot r1120579120579120579(r1120579 times r1120579120579)

2 (25)

4 Meshing Characteristics ofConjugate Curves

41 Uniqueness of Conjugated Curve The conjugated curve isgenerated in terms of the meshing equation and transforma-tion matrixes among different coordinate systems accordingto a given original curve The normal vector n119899 to conjugatecurves at contact point in arbitrary direction of contact angle

Mathematical Problems in Engineering 7

M

z

x

y

Δ120572

120572

1205721

1205722

Γ1

Γ21Γ22

120573

120574

Figure 10 Two conjugated curves for different contact angles 1205721 1205722

can be determined in the previous study Particularly fromthe expression of normal vector n119899 we can identify thatcoefficients 1199051 and 1199052 indicate the projection of contact angle1205720 on the direction of principal normal and binormal vectorsrespectively

Given the different coefficients 1199051 and 1199052 if the originalcurve is determined the corresponding conjugated curvecan be represented in various results Then the further studyon the relationship between obtained conjugated curves iscarried out Given the contact angles 1205721 1205722 (where 1205722 = 1205721 +Δ120572) in Figure 10 respectively based on principle descriptionsof conjugate curves in Section 2 the conjugated curves Γ2119894(119894 = 1 2) which are separately related to given conditions areexpressed as

119909(119894)

2= 1199091 [cos120601

(119894)

1cos (11989421120601

(119894)

1) minus sin120601(119894)

1sin (11989421120601

(119894)

1) cosΣ]

+ 1199101 [minus sin120601(119894)

1cos (11989421120601

(119894)

1)

minus cos120601(119894)1sin (11989421120601

(119894)

1) cosΣ]

minus 1199111 sin (11989421120601(119894)

1) sinΣ minus 119886 cos (11989421120601

(119894)

1)

119910(119894)

2= 1199091 [cos120601

(119894)

1sin (11989421120601

(119894)

1) + sin120601(119894)

1cos (11989421120601

(119894)

1) cosΣ]

+ 1199101 [minus sin120601(119894)

1sin (11989421120601

(119894)

1)

+ cos120601(119894)1cos (11989421120601

(119894)

1) cosΣ]

+ 1199111 cos (11989421120601(119894)

1) sinΣ minus 119886 sin (11989421120601

(119894)

1)

119911(119894)

2= minus1199091 sin120601

(119894)

1sinΣ minus 1199101 cos120601

(119894)

1sinΣ + 1199111 cosΣ

120601(119894)

1= arcsin 119882

(119894)

radic119880(119894)2+ 119881(119894)

2

minus arctan 119881(119894)

119880(119894)

(26)

Considering that conjugated curve Γ21 rotates aboutcentral contact point119872 with angle Δ120572 it has this expressionafter the coordinate transformation as

119909(1)1015840

2= 119909(1)

2cosΔ120572 + 119910(1)

2sinΔ120572

119910(1)1015840

2= minus 119909

(1)

2sinΔ120572 + 119910(1)

2cosΔ120572

119911(1)1015840

2= 119911(1)

2

(27)

Substituting (26) into (27) the derived result is written as

119909(1)1015840

2= (11990911198601 + 11991011198611 minus 11991111198621) cosΔ120572

+ (11990911198602 + 11991011198612 + 11991111198622) sinΔ120572

minus 119886 cos (Δ120572 minus 11989421120601(1)

1)

119910(1)1015840

2= (minus11990911198601 minus 11991011198611 + 11991111198621) sinΔ120572

minus (11990911198602 minus 11991011198612 minus 11991111198622) cosΔ120572

minus 119886 sin (Δ120572 minus 11989421120601(1)

1)

119911(1)1015840

2= minus1199091 sin120601

(1)

1sinΣ minus 1199101 cos120601

(1)

1sinΣ + 1199111 cosΣ

(28)

where1198601 = cos120601

(1)

1cos (11989421120601

(1)

1) minus sin120601(1)

1sin (11989421120601

(1)

1) cosΣ

1198602 = cos120601(1)

1sin (11989421120601

(1)

1) + sin120601(1)

1cos (11989421120601

(1)

1) cosΣ

1198611 = minus sin120601(1)

1cos (11989421120601

(1)

1) minus cos120601(1)

1sin (11989421120601

(1)

1) cosΣ

1198612 = minus sin120601(1)

1sin (11989421120601

(1)

1) + cos120601(1)

1cos (11989421120601

(1)

1) cosΣ

1198621 = sin (11989421120601(1)

1) sinΣ

1198622 = cos (11989421120601(1)

1) sinΣ

(29)

A comparison about the expressions of Γ22 and Γ101584021

between (26) and (28) is worked It can be concluded fromthe simplified results that the two conjugated curves Γ21 andΓ22 are the same and unique curves while the difference isthat Γ22 is the rotational curve of Γ21 about contact point119872 under given angle Δ120572 For verifying above conclusionthe mathematical example for parallel axis gearing based oncylindrical spiral curve is introduced and its parametric formis represented as

r1 = 119877 cos 120579i1 + 119877 sin 120579j1 + 119901120579k1 (30)

where 119877 is the radius of pitch circle 120579 is spiral curve param-eter and 119901 is helix parameter Through the aforementionedprinciple of conjugate curves the equation of conjugatedcurve is derived as

1199092 = (119877 minus 119886) cos (11989421120579)

1199102 = minus (119877 minus 119886) sin (11989421120579)

1199112 = 119901120579

(31)

8 Mathematical Problems in Engineering

75 80 85 90 95

010

2030

40

0

10

20

30

40

50

Given curve

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(a)

85 90

010

20

0

10

20

30

40

50

Given curve

minus20

minus10

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(b)

75 80 85 90 95

0

10

20

30

40

50

Given curve

minus50

minus40

minus30

minus20

minus10

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(c)

Figure 11 Uniqueness of conjugated curve at any contact position (a) beginning position (b) middle position (c) end position

A simplified meshing model of conjugate curves alongvarious contact angles for gear drive is developed by meansof Matlab software as displayed in Figure 11 according to thedesignated parameters shown in Table 1

From Figure 11 the generated conjugated curves aredifferent if the selected contact angles are various When thegear pair rotates with fixed angular velocity the movementof conjugate curves with 20 degrees rotation 30 degreesrotation and 40 degrees rotation can be described respec-tively The engagement point is still fixed on the commonpoint during the whole meshing Throughout this processcontact point changes gradually in the axial direction and theline of action is always a straight line which is parallel with

gear axes The characteristics of conjugate curves providemore flexibility for the design of gear transmission We canchoose the optimized design result which is suitable forpractical application in terms of appropriate parameters

42 Meshing Commonality of Conjugate Curves To analyzeintrinsic nature of conjugate curves a class of spiral curvescontacting in the given direction for various gear axes isdiscussed

421 Gear Pair with Parallel Axis Assuming that parameters119886 = 0 and Σ = 0 the gear pair with parallel axis can beobtained as displayed in Figure 12 A calculation procedure

Mathematical Problems in Engineering 9

Cylindrical spiral curve Conjugated cylindrical spiral curve

x1

y1

x2

y2

z1z2

(a)

Conjugated curveGiven curve

Line of action

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

(b)

Figure 12 Conjugate spiral curves for parallel axis gear drive

Table 1 Design parameters for conjugate curves

Parameters ValuesRadius of pitch circle Rmm 24Tooth number of pinion 1 1198851 6Tooth number of gear 2 1198852 30Transmission ratio 11989421 5Module119898119899mm 4Helix parameter 119901 286478Central distance 119886mm 144Tooth width 119861mm 50Parameter of spiral curve 120579rad 0sim10472Parameter 1199051 minus05Parameter 1199052 minus0866

of designated spiral curve expressed in (30) for generalcylindrical gears is carried out and the conjugated curve isderived as (31)

Here

1198771 = 119877 minus 119886 1205791 = 11989421120579 (32)

Then (31) can be written as

1199092 = 1198771 cos 1205791

1199102 = minus1198771 sin 1205791

1199112 = 119901120579

(33)

It is consistent with general form of cylindrical spiralcurve in parallel axis gear drive

422 Gear Pair with Intersecting Axis Assuming that param-eters 119886 = 0 and Σ = 0 the gear pair with intersecting axis canbe obtained as shown in Figure 13The selected conical spiralcurve for bevel gears is expressed in coordinate system 1198781 andits parametric form is represented as

r1 = (1198771119891 minus 119888120579) cos 120579i1 + (1198771119891 minus 119888120579) sin 120579j1 + 119901120579k1 (34)

where 1198771119891 is the radius of pitch circle and 120579 and 119901 are alsothe spiral curve and helix parameters respectively It has thecorrelation 119888 = 119901 tan 120575 120575 is the conical angle

Through a series of calculation on the basis of principle ofconjugate curves its conjugated curve is expressed as

1199092 = 1199091 [cos1206011 cos (119894211206011) minus sin1206011 sin (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 cos (119894211206011) minus cos1206011 sin (119894211206011) cosΣ]

minus 1199111 sin (119894211206011) sinΣ

1199102 = 1199091 [cos1206011 sin (119894211206011) + sin1206011 cos (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 sin (119894211206011) + cos1206011 cos (119894211206011) cosΣ]

+ 1199111 cos (119894211206011) sinΣ

1199112 = minus1199091 cos1206011 sinΣ minus 1199101 sin1206011 sinΣ + 1199111 cosΣ

119880 cos1206011 minus 119881 sin1206011 = 119882(35)

10 Mathematical Problems in Engineering

Conical spiral curve

Conjugated conical spiral curve

z1 z2

x1y1x2

y2

1205751

1205752

Rf1

Rf2

O1

O2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

Conjugated curveGiven curve

Line of action

(b)

Figure 13 Conjugate spiral curves for intersecting axis gear drive

Furthermore substituting (34) into (35) the simplifiedresult with parameter 120579 is represented as

1199092 = [1198771119891 minus (119888 + 119901) 120579] cos (11989421120579 + 2120579 + Σ)

1199102 = [1198771119891 minus (119888 + 119901) 120579] sin (11989421120579 + 2120579 + Σ)

1199112 = (minus119888 sinΣ + 119901 cosΣ) 120579 minus 1198771119891 sinΣ

(36)

Here

1198881 = 119888 + 119901 1205792 = 11989421120579 + 2120579 + Σ

1199011 = minus119888 sinΣ + 119901 cosΣ(37)

Then (36) can be written as

1199092 = (1198771119891 minus 1198881120579) cos 1205792

1199102 = (1198771119891 minus 1198881120579) sin 1205792

1199112 = 1199011120579 minus 1198771119891 sinΣ

(38)

Conjugated curve attached to gear 2 is got according toabove derivation Obviously it is also a conical spiral curve inform

423 Gear Pair with CrossedAxis Assuming that parameters119886 = 0 and Σ = 0 it gets the gear pair with crossed axis whichis depicted in Figure 14 The general spiral curve selectedfor worm gear drive is given and it has the same expression

with (30) Based on the principle of conjugate curves itsconjugated curve is derived as

1199092 =radic1199012sin2Σ + (119877 minus 119886)2 cos [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199102 =radic1199012sin2Σ + (119877 minus 119886)2 sin [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199112 = 119901120579 cosΣ(39)

Here

1198772 =radic1199012sin2Σ + (119877 minus 119886)2

1205793 = 11989421120579 + arctan(119877 minus 119886

119901 sinΣ)

(40)

Then (39) can be written as

1199092 = 1198772 cos 1205793

1199102 = 1198772 sin 1205793

1199112 = 119901120579 cosΣ

(41)

It also represents the similar relationship with givenoriginal curve in form

From what are discussed above we can conclude thatthe conjugated curves corresponding to the given spiralcurves for various gear axes are still the spiral curves Itcan be written as the same expression form with the given

Mathematical Problems in Engineering 11

Worm spiral curve

Conjugated worm spiral curve

x1

y1

x2

y2

O1

O2

z1

z2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

Conjugated curveGiven curve

Line of action

x (mm)y (mm)

z(m

m)

(b)

Figure 14 Conjugate spiral curves for crossed axis gear drive

conditions It is called meshing commonality of conjugatecurves Moreover based on the principle of conjugate curvesfor any chosen contact curve the corresponding same typeof conjugated curve in the direction of arbitrary contact anglecan be identified Freedomof selecting contact curve suggeststhe possibility of optimal conjugation design

According to the aforementioned research it offers aversatile way for high-performance profile design (1) fewteeth number and large module may be obtained withouttooth undercutting The problems of limited installationspace and lightweight design can be solved (2)Three differentcontact models of tooth profiles can be established basedon the proposed study Particularly the special meshing ofgenerated convex and concave tooth profiles makes relativeradius of curvature of the contact point longer and increasesthe contact strength The load capacity and useful life hasbeen evidently improved (3) The tooth surfaces mesh inpoint contact along the conjugate curves and contain thetransmission properties of conjugate curves Due to theselection and generation processes the approximate purerolling contact between mating tooth surfaces maybe occursThe transmission efficiency will be improved

The further studies on strength property manufacturingkey technology and performance experiment of conjugate-curve gear drive will be carried out And the excellent trans-mission performance of gearing is expected to be obtained onthe basis of theoretical and experimental investigations

5 Conclusions

(1) The principle descriptions of conjugate curves forarbitrary axial position are introducedWith the aid of

given plane or space curve generation principle andmathematical model of conjugated curve are devel-oped It can be applied to parallel axis intersectingaxis and crossed axis gear drive respectively

(2) Based on differential geometry geometric charac-teristic analysis of conjugate curves is carried outThe tangent and normal to conjugate curves inarbitrary contact direction are discussed and generalcalculation methods are provided The envelopingcondition of family of conjugate curves with singleparameter is analyzed and characteristic point isdetermined Curvature and torsion relationships ofconjugate curves are also derived

(3) Meshing properties of conjugate curves are furtherrevealed According to a given plane or spatial curvethe variation law of meshing function is analyzedThe uniqueness of conjugated curves is discussedTheconjugated curves are always the same and uniquecurves for any given curve under different contactangles while the difference is that there exists therotational angle relationship with contact point 119872among generated curves

(4) Meshing commonality of conjugate curves is demon-strated in terms of a class of spiral curves contactingin the given direction for various gear axes For anychosen contact curve under various gear axes theconjugated curve has the same type and it can bewritten as the same expression form

(5) The proposed theory lays the foundation for design ofnew types of gear drive The further study on conju-gate curves and applications in gear transmission will

12 Mathematical Problems in Engineering

be carried out Excellent transmission performance ofgearing is expected to be obtained

Conflict of Interests

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

Acknowledgments

This project is supported by National Key Technology RampDProgram of the Twelfth Five-year Plan of China (Grantno 2013BAF01B04) National Natural Science Foundation ofChina (Grant no 51205425) and Science and TechnologyProject of Chongqing City Board of Education (Grant noKJ122202) The authors also sincerely appreciate the com-ments and modification suggestions made by the editors andanonymous referees

References

[1] S S Chern W H Chen and K S Lam Lectures on DifferentialGeometry World Scientific Singapore 2006

[2] P G Ciarlet and T-T Li Differential Geometry Theory andApplications World Scientific Singapore 2007

[3] F L Litvin and A Fuentes Gear Geometry and Applied TheoryCambridge University Press New York NY USA 2nd edition2004

[4] S P Radzevich Theory of Gearing Kinetics Geometry andSynthesis CRC Press Boca Raton Fla USA 2012

[5] D B Dooner Kinematic Geometry of Gearing John Wiley ampSons 2nd edition 2012

[6] F L Litvin A Fuentes A Demenego D Vecchiato and Q FanldquoNewdevelopments in the design and generation of gear drivesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 215 no 7 pp747ndash757 2001

[7] C H Chen Theory of Conjugate Surfaces and Its ApplicationsScience and Technology Press Beijing China 2008

[8] G X Li Spatial Geometry Modeling and Its Application inEngineering Higher Education Press Beijing China 2007

[9] F Di Puccio M Gabiccini and M Guiggiani ldquoGeneration andcurvature analysis of conjugate surfaces via a new approachrdquoMechanism and Machine Theory vol 41 no 4 pp 382ndash4042006

[10] D R Wu and J S Luo A Geometric Theory of Conjugate ToothSurface World Scientific Publishing Singapore 1992

[11] N Ito and K Takahashi ldquoDifferential geometrical conditions ofhypoid gears with conjugate tooth surfacesrdquo ASME Journal ofMechanical Design vol 122 no 3 pp 323ndash330 2000

[12] D B Dooner ldquoOn the three laws of gearingrdquo Journal ofMechanical Design Transactions of the ASME vol 124 no 4pp 733ndash744 2002

[13] Z Duan H Chen and J Liu ldquoPrinciple and transmissiontechnology of Bertrand conjugate surfacesrdquo Chinese Journal ofMechanical Engineering vol 19 no 4 pp 600ndash604 2006

[14] H Chen Z Duan J Liu and H Wu ldquoResearch on basicprinciple of moulding-surface conjugationrdquo Mechanism andMachine Theory vol 43 no 7 pp 791ndash811 2008

[15] 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

[16] C F Chen and C B Tsay ldquoTooth profile design for themanufacture of helical gear sets with small numbers of teethrdquoInternational Journal ofMachine Tools andManufacture vol 45no 12-13 pp 1531ndash1541 2005

[17] R Imin and M Geni ldquoStress analysis of gear meshing impactbased on SPH methodrdquoMathematical Problems in Engineeringvol 2014 Article ID 328216 7 pages 2014

[18] D Park and A Kahraman ldquoA surface wear model for hypoidgear pairsrdquoWear vol 267 no 9-10 pp 1595ndash1604 2009

[19] JWang L Hou SM Luo and R YWu ldquoActive design of toothprofiles using parabolic curve as the line of actionrdquoMechanismand Machine Theory vol 67 no 9 pp 47ndash63 2013

[20] Q B Wang P Hu Y M Zhang et al ldquoA model to determinemesh characteristics in a gear pair with tooth profile errorrdquoAdvances in Mechanical Engineering vol 2014 Article ID751476 10 pages 2014

[21] C F Chen and C B Tsay ldquoComputerized tooth profile gen-eration and analysis of characteristics of elliptical gears withcircular arc teethrdquo Journal of Materials Processing Technologyvol 148 no 2 pp 226ndash234 2004

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

[23] B K Chen Y E Gao and D Liang ldquoThe generation principleof tooth surfaces of conjugate-curve gear transmissionrdquo Journalof Mechanical Engineering vol 50 no 3 pp 18ndash24 2014

[24] D Liang B K Chen and Y E Gao ldquoThe generation principleand mathematical model of a new involute-helix gear driverdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 227 no 12 pp2834ndash2843 2013

[25] B Chen D Liang and Z Li ldquoA study on geometry designof spiral bevel gears based on conjugate curvesrdquo InternationalJournal of Precision Engineering and Manufacturing vol 15 no3 pp 477ndash482 2014

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

M

z

x

y

Δ120572

120572

1205721

1205722

Γ1

Γ21Γ22

120573

120574

Figure 10 Two conjugated curves for different contact angles 1205721 1205722

can be determined in the previous study Particularly fromthe expression of normal vector n119899 we can identify thatcoefficients 1199051 and 1199052 indicate the projection of contact angle1205720 on the direction of principal normal and binormal vectorsrespectively

Given the different coefficients 1199051 and 1199052 if the originalcurve is determined the corresponding conjugated curvecan be represented in various results Then the further studyon the relationship between obtained conjugated curves iscarried out Given the contact angles 1205721 1205722 (where 1205722 = 1205721 +Δ120572) in Figure 10 respectively based on principle descriptionsof conjugate curves in Section 2 the conjugated curves Γ2119894(119894 = 1 2) which are separately related to given conditions areexpressed as

119909(119894)

2= 1199091 [cos120601

(119894)

1cos (11989421120601

(119894)

1) minus sin120601(119894)

1sin (11989421120601

(119894)

1) cosΣ]

+ 1199101 [minus sin120601(119894)

1cos (11989421120601

(119894)

1)

minus cos120601(119894)1sin (11989421120601

(119894)

1) cosΣ]

minus 1199111 sin (11989421120601(119894)

1) sinΣ minus 119886 cos (11989421120601

(119894)

1)

119910(119894)

2= 1199091 [cos120601

(119894)

1sin (11989421120601

(119894)

1) + sin120601(119894)

1cos (11989421120601

(119894)

1) cosΣ]

+ 1199101 [minus sin120601(119894)

1sin (11989421120601

(119894)

1)

+ cos120601(119894)1cos (11989421120601

(119894)

1) cosΣ]

+ 1199111 cos (11989421120601(119894)

1) sinΣ minus 119886 sin (11989421120601

(119894)

1)

119911(119894)

2= minus1199091 sin120601

(119894)

1sinΣ minus 1199101 cos120601

(119894)

1sinΣ + 1199111 cosΣ

120601(119894)

1= arcsin 119882

(119894)

radic119880(119894)2+ 119881(119894)

2

minus arctan 119881(119894)

119880(119894)

(26)

Considering that conjugated curve Γ21 rotates aboutcentral contact point119872 with angle Δ120572 it has this expressionafter the coordinate transformation as

119909(1)1015840

2= 119909(1)

2cosΔ120572 + 119910(1)

2sinΔ120572

119910(1)1015840

2= minus 119909

(1)

2sinΔ120572 + 119910(1)

2cosΔ120572

119911(1)1015840

2= 119911(1)

2

(27)

Substituting (26) into (27) the derived result is written as

119909(1)1015840

2= (11990911198601 + 11991011198611 minus 11991111198621) cosΔ120572

+ (11990911198602 + 11991011198612 + 11991111198622) sinΔ120572

minus 119886 cos (Δ120572 minus 11989421120601(1)

1)

119910(1)1015840

2= (minus11990911198601 minus 11991011198611 + 11991111198621) sinΔ120572

minus (11990911198602 minus 11991011198612 minus 11991111198622) cosΔ120572

minus 119886 sin (Δ120572 minus 11989421120601(1)

1)

119911(1)1015840

2= minus1199091 sin120601

(1)

1sinΣ minus 1199101 cos120601

(1)

1sinΣ + 1199111 cosΣ

(28)

where1198601 = cos120601

(1)

1cos (11989421120601

(1)

1) minus sin120601(1)

1sin (11989421120601

(1)

1) cosΣ

1198602 = cos120601(1)

1sin (11989421120601

(1)

1) + sin120601(1)

1cos (11989421120601

(1)

1) cosΣ

1198611 = minus sin120601(1)

1cos (11989421120601

(1)

1) minus cos120601(1)

1sin (11989421120601

(1)

1) cosΣ

1198612 = minus sin120601(1)

1sin (11989421120601

(1)

1) + cos120601(1)

1cos (11989421120601

(1)

1) cosΣ

1198621 = sin (11989421120601(1)

1) sinΣ

1198622 = cos (11989421120601(1)

1) sinΣ

(29)

A comparison about the expressions of Γ22 and Γ101584021

between (26) and (28) is worked It can be concluded fromthe simplified results that the two conjugated curves Γ21 andΓ22 are the same and unique curves while the difference isthat Γ22 is the rotational curve of Γ21 about contact point119872 under given angle Δ120572 For verifying above conclusionthe mathematical example for parallel axis gearing based oncylindrical spiral curve is introduced and its parametric formis represented as

r1 = 119877 cos 120579i1 + 119877 sin 120579j1 + 119901120579k1 (30)

where 119877 is the radius of pitch circle 120579 is spiral curve param-eter and 119901 is helix parameter Through the aforementionedprinciple of conjugate curves the equation of conjugatedcurve is derived as

1199092 = (119877 minus 119886) cos (11989421120579)

1199102 = minus (119877 minus 119886) sin (11989421120579)

1199112 = 119901120579

(31)

8 Mathematical Problems in Engineering

75 80 85 90 95

010

2030

40

0

10

20

30

40

50

Given curve

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(a)

85 90

010

20

0

10

20

30

40

50

Given curve

minus20

minus10

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(b)

75 80 85 90 95

0

10

20

30

40

50

Given curve

minus50

minus40

minus30

minus20

minus10

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(c)

Figure 11 Uniqueness of conjugated curve at any contact position (a) beginning position (b) middle position (c) end position

A simplified meshing model of conjugate curves alongvarious contact angles for gear drive is developed by meansof Matlab software as displayed in Figure 11 according to thedesignated parameters shown in Table 1

From Figure 11 the generated conjugated curves aredifferent if the selected contact angles are various When thegear pair rotates with fixed angular velocity the movementof conjugate curves with 20 degrees rotation 30 degreesrotation and 40 degrees rotation can be described respec-tively The engagement point is still fixed on the commonpoint during the whole meshing Throughout this processcontact point changes gradually in the axial direction and theline of action is always a straight line which is parallel with

gear axes The characteristics of conjugate curves providemore flexibility for the design of gear transmission We canchoose the optimized design result which is suitable forpractical application in terms of appropriate parameters

42 Meshing Commonality of Conjugate Curves To analyzeintrinsic nature of conjugate curves a class of spiral curvescontacting in the given direction for various gear axes isdiscussed

421 Gear Pair with Parallel Axis Assuming that parameters119886 = 0 and Σ = 0 the gear pair with parallel axis can beobtained as displayed in Figure 12 A calculation procedure

Mathematical Problems in Engineering 9

Cylindrical spiral curve Conjugated cylindrical spiral curve

x1

y1

x2

y2

z1z2

(a)

Conjugated curveGiven curve

Line of action

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

(b)

Figure 12 Conjugate spiral curves for parallel axis gear drive

Table 1 Design parameters for conjugate curves

Parameters ValuesRadius of pitch circle Rmm 24Tooth number of pinion 1 1198851 6Tooth number of gear 2 1198852 30Transmission ratio 11989421 5Module119898119899mm 4Helix parameter 119901 286478Central distance 119886mm 144Tooth width 119861mm 50Parameter of spiral curve 120579rad 0sim10472Parameter 1199051 minus05Parameter 1199052 minus0866

of designated spiral curve expressed in (30) for generalcylindrical gears is carried out and the conjugated curve isderived as (31)

Here

1198771 = 119877 minus 119886 1205791 = 11989421120579 (32)

Then (31) can be written as

1199092 = 1198771 cos 1205791

1199102 = minus1198771 sin 1205791

1199112 = 119901120579

(33)

It is consistent with general form of cylindrical spiralcurve in parallel axis gear drive

422 Gear Pair with Intersecting Axis Assuming that param-eters 119886 = 0 and Σ = 0 the gear pair with intersecting axis canbe obtained as shown in Figure 13The selected conical spiralcurve for bevel gears is expressed in coordinate system 1198781 andits parametric form is represented as

r1 = (1198771119891 minus 119888120579) cos 120579i1 + (1198771119891 minus 119888120579) sin 120579j1 + 119901120579k1 (34)

where 1198771119891 is the radius of pitch circle and 120579 and 119901 are alsothe spiral curve and helix parameters respectively It has thecorrelation 119888 = 119901 tan 120575 120575 is the conical angle

Through a series of calculation on the basis of principle ofconjugate curves its conjugated curve is expressed as

1199092 = 1199091 [cos1206011 cos (119894211206011) minus sin1206011 sin (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 cos (119894211206011) minus cos1206011 sin (119894211206011) cosΣ]

minus 1199111 sin (119894211206011) sinΣ

1199102 = 1199091 [cos1206011 sin (119894211206011) + sin1206011 cos (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 sin (119894211206011) + cos1206011 cos (119894211206011) cosΣ]

+ 1199111 cos (119894211206011) sinΣ

1199112 = minus1199091 cos1206011 sinΣ minus 1199101 sin1206011 sinΣ + 1199111 cosΣ

119880 cos1206011 minus 119881 sin1206011 = 119882(35)

10 Mathematical Problems in Engineering

Conical spiral curve

Conjugated conical spiral curve

z1 z2

x1y1x2

y2

1205751

1205752

Rf1

Rf2

O1

O2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

Conjugated curveGiven curve

Line of action

(b)

Figure 13 Conjugate spiral curves for intersecting axis gear drive

Furthermore substituting (34) into (35) the simplifiedresult with parameter 120579 is represented as

1199092 = [1198771119891 minus (119888 + 119901) 120579] cos (11989421120579 + 2120579 + Σ)

1199102 = [1198771119891 minus (119888 + 119901) 120579] sin (11989421120579 + 2120579 + Σ)

1199112 = (minus119888 sinΣ + 119901 cosΣ) 120579 minus 1198771119891 sinΣ

(36)

Here

1198881 = 119888 + 119901 1205792 = 11989421120579 + 2120579 + Σ

1199011 = minus119888 sinΣ + 119901 cosΣ(37)

Then (36) can be written as

1199092 = (1198771119891 minus 1198881120579) cos 1205792

1199102 = (1198771119891 minus 1198881120579) sin 1205792

1199112 = 1199011120579 minus 1198771119891 sinΣ

(38)

Conjugated curve attached to gear 2 is got according toabove derivation Obviously it is also a conical spiral curve inform

423 Gear Pair with CrossedAxis Assuming that parameters119886 = 0 and Σ = 0 it gets the gear pair with crossed axis whichis depicted in Figure 14 The general spiral curve selectedfor worm gear drive is given and it has the same expression

with (30) Based on the principle of conjugate curves itsconjugated curve is derived as

1199092 =radic1199012sin2Σ + (119877 minus 119886)2 cos [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199102 =radic1199012sin2Σ + (119877 minus 119886)2 sin [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199112 = 119901120579 cosΣ(39)

Here

1198772 =radic1199012sin2Σ + (119877 minus 119886)2

1205793 = 11989421120579 + arctan(119877 minus 119886

119901 sinΣ)

(40)

Then (39) can be written as

1199092 = 1198772 cos 1205793

1199102 = 1198772 sin 1205793

1199112 = 119901120579 cosΣ

(41)

It also represents the similar relationship with givenoriginal curve in form

From what are discussed above we can conclude thatthe conjugated curves corresponding to the given spiralcurves for various gear axes are still the spiral curves Itcan be written as the same expression form with the given

Mathematical Problems in Engineering 11

Worm spiral curve

Conjugated worm spiral curve

x1

y1

x2

y2

O1

O2

z1

z2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

Conjugated curveGiven curve

Line of action

x (mm)y (mm)

z(m

m)

(b)

Figure 14 Conjugate spiral curves for crossed axis gear drive

conditions It is called meshing commonality of conjugatecurves Moreover based on the principle of conjugate curvesfor any chosen contact curve the corresponding same typeof conjugated curve in the direction of arbitrary contact anglecan be identified Freedomof selecting contact curve suggeststhe possibility of optimal conjugation design

According to the aforementioned research it offers aversatile way for high-performance profile design (1) fewteeth number and large module may be obtained withouttooth undercutting The problems of limited installationspace and lightweight design can be solved (2)Three differentcontact models of tooth profiles can be established basedon the proposed study Particularly the special meshing ofgenerated convex and concave tooth profiles makes relativeradius of curvature of the contact point longer and increasesthe contact strength The load capacity and useful life hasbeen evidently improved (3) The tooth surfaces mesh inpoint contact along the conjugate curves and contain thetransmission properties of conjugate curves Due to theselection and generation processes the approximate purerolling contact between mating tooth surfaces maybe occursThe transmission efficiency will be improved

The further studies on strength property manufacturingkey technology and performance experiment of conjugate-curve gear drive will be carried out And the excellent trans-mission performance of gearing is expected to be obtained onthe basis of theoretical and experimental investigations

5 Conclusions

(1) The principle descriptions of conjugate curves forarbitrary axial position are introducedWith the aid of

given plane or space curve generation principle andmathematical model of conjugated curve are devel-oped It can be applied to parallel axis intersectingaxis and crossed axis gear drive respectively

(2) Based on differential geometry geometric charac-teristic analysis of conjugate curves is carried outThe tangent and normal to conjugate curves inarbitrary contact direction are discussed and generalcalculation methods are provided The envelopingcondition of family of conjugate curves with singleparameter is analyzed and characteristic point isdetermined Curvature and torsion relationships ofconjugate curves are also derived

(3) Meshing properties of conjugate curves are furtherrevealed According to a given plane or spatial curvethe variation law of meshing function is analyzedThe uniqueness of conjugated curves is discussedTheconjugated curves are always the same and uniquecurves for any given curve under different contactangles while the difference is that there exists therotational angle relationship with contact point 119872among generated curves

(4) Meshing commonality of conjugate curves is demon-strated in terms of a class of spiral curves contactingin the given direction for various gear axes For anychosen contact curve under various gear axes theconjugated curve has the same type and it can bewritten as the same expression form

(5) The proposed theory lays the foundation for design ofnew types of gear drive The further study on conju-gate curves and applications in gear transmission will

12 Mathematical Problems in Engineering

be carried out Excellent transmission performance ofgearing is expected to be obtained

Conflict of Interests

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

Acknowledgments

This project is supported by National Key Technology RampDProgram of the Twelfth Five-year Plan of China (Grantno 2013BAF01B04) National Natural Science Foundation ofChina (Grant no 51205425) and Science and TechnologyProject of Chongqing City Board of Education (Grant noKJ122202) The authors also sincerely appreciate the com-ments and modification suggestions made by the editors andanonymous referees

References

[1] S S Chern W H Chen and K S Lam Lectures on DifferentialGeometry World Scientific Singapore 2006

[2] P G Ciarlet and T-T Li Differential Geometry Theory andApplications World Scientific Singapore 2007

[3] F L Litvin and A Fuentes Gear Geometry and Applied TheoryCambridge University Press New York NY USA 2nd edition2004

[4] S P Radzevich Theory of Gearing Kinetics Geometry andSynthesis CRC Press Boca Raton Fla USA 2012

[5] D B Dooner Kinematic Geometry of Gearing John Wiley ampSons 2nd edition 2012

[6] F L Litvin A Fuentes A Demenego D Vecchiato and Q FanldquoNewdevelopments in the design and generation of gear drivesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 215 no 7 pp747ndash757 2001

[7] C H Chen Theory of Conjugate Surfaces and Its ApplicationsScience and Technology Press Beijing China 2008

[8] G X Li Spatial Geometry Modeling and Its Application inEngineering Higher Education Press Beijing China 2007

[9] F Di Puccio M Gabiccini and M Guiggiani ldquoGeneration andcurvature analysis of conjugate surfaces via a new approachrdquoMechanism and Machine Theory vol 41 no 4 pp 382ndash4042006

[10] D R Wu and J S Luo A Geometric Theory of Conjugate ToothSurface World Scientific Publishing Singapore 1992

[11] N Ito and K Takahashi ldquoDifferential geometrical conditions ofhypoid gears with conjugate tooth surfacesrdquo ASME Journal ofMechanical Design vol 122 no 3 pp 323ndash330 2000

[12] D B Dooner ldquoOn the three laws of gearingrdquo Journal ofMechanical Design Transactions of the ASME vol 124 no 4pp 733ndash744 2002

[13] Z Duan H Chen and J Liu ldquoPrinciple and transmissiontechnology of Bertrand conjugate surfacesrdquo Chinese Journal ofMechanical Engineering vol 19 no 4 pp 600ndash604 2006

[14] H Chen Z Duan J Liu and H Wu ldquoResearch on basicprinciple of moulding-surface conjugationrdquo Mechanism andMachine Theory vol 43 no 7 pp 791ndash811 2008

[15] 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

[16] C F Chen and C B Tsay ldquoTooth profile design for themanufacture of helical gear sets with small numbers of teethrdquoInternational Journal ofMachine Tools andManufacture vol 45no 12-13 pp 1531ndash1541 2005

[17] R Imin and M Geni ldquoStress analysis of gear meshing impactbased on SPH methodrdquoMathematical Problems in Engineeringvol 2014 Article ID 328216 7 pages 2014

[18] D Park and A Kahraman ldquoA surface wear model for hypoidgear pairsrdquoWear vol 267 no 9-10 pp 1595ndash1604 2009

[19] JWang L Hou SM Luo and R YWu ldquoActive design of toothprofiles using parabolic curve as the line of actionrdquoMechanismand Machine Theory vol 67 no 9 pp 47ndash63 2013

[20] Q B Wang P Hu Y M Zhang et al ldquoA model to determinemesh characteristics in a gear pair with tooth profile errorrdquoAdvances in Mechanical Engineering vol 2014 Article ID751476 10 pages 2014

[21] C F Chen and C B Tsay ldquoComputerized tooth profile gen-eration and analysis of characteristics of elliptical gears withcircular arc teethrdquo Journal of Materials Processing Technologyvol 148 no 2 pp 226ndash234 2004

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

[23] B K Chen Y E Gao and D Liang ldquoThe generation principleof tooth surfaces of conjugate-curve gear transmissionrdquo Journalof Mechanical Engineering vol 50 no 3 pp 18ndash24 2014

[24] D Liang B K Chen and Y E Gao ldquoThe generation principleand mathematical model of a new involute-helix gear driverdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 227 no 12 pp2834ndash2843 2013

[25] B Chen D Liang and Z Li ldquoA study on geometry designof spiral bevel gears based on conjugate curvesrdquo InternationalJournal of Precision Engineering and Manufacturing vol 15 no3 pp 477ndash482 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

75 80 85 90 95

010

2030

40

0

10

20

30

40

50

Given curve

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(a)

85 90

010

20

0

10

20

30

40

50

Given curve

minus20

minus10

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(b)

75 80 85 90 95

0

10

20

30

40

50

Given curve

minus50

minus40

minus30

minus20

minus10

Conjugate curve(20∘ contact angle)

Conjugate curve(30∘ contact angle)

Conjugate curve(40∘ contact angle)

(c)

Figure 11 Uniqueness of conjugated curve at any contact position (a) beginning position (b) middle position (c) end position

A simplified meshing model of conjugate curves alongvarious contact angles for gear drive is developed by meansof Matlab software as displayed in Figure 11 according to thedesignated parameters shown in Table 1

From Figure 11 the generated conjugated curves aredifferent if the selected contact angles are various When thegear pair rotates with fixed angular velocity the movementof conjugate curves with 20 degrees rotation 30 degreesrotation and 40 degrees rotation can be described respec-tively The engagement point is still fixed on the commonpoint during the whole meshing Throughout this processcontact point changes gradually in the axial direction and theline of action is always a straight line which is parallel with

gear axes The characteristics of conjugate curves providemore flexibility for the design of gear transmission We canchoose the optimized design result which is suitable forpractical application in terms of appropriate parameters

42 Meshing Commonality of Conjugate Curves To analyzeintrinsic nature of conjugate curves a class of spiral curvescontacting in the given direction for various gear axes isdiscussed

421 Gear Pair with Parallel Axis Assuming that parameters119886 = 0 and Σ = 0 the gear pair with parallel axis can beobtained as displayed in Figure 12 A calculation procedure

Mathematical Problems in Engineering 9

Cylindrical spiral curve Conjugated cylindrical spiral curve

x1

y1

x2

y2

z1z2

(a)

Conjugated curveGiven curve

Line of action

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

(b)

Figure 12 Conjugate spiral curves for parallel axis gear drive

Table 1 Design parameters for conjugate curves

Parameters ValuesRadius of pitch circle Rmm 24Tooth number of pinion 1 1198851 6Tooth number of gear 2 1198852 30Transmission ratio 11989421 5Module119898119899mm 4Helix parameter 119901 286478Central distance 119886mm 144Tooth width 119861mm 50Parameter of spiral curve 120579rad 0sim10472Parameter 1199051 minus05Parameter 1199052 minus0866

of designated spiral curve expressed in (30) for generalcylindrical gears is carried out and the conjugated curve isderived as (31)

Here

1198771 = 119877 minus 119886 1205791 = 11989421120579 (32)

Then (31) can be written as

1199092 = 1198771 cos 1205791

1199102 = minus1198771 sin 1205791

1199112 = 119901120579

(33)

It is consistent with general form of cylindrical spiralcurve in parallel axis gear drive

422 Gear Pair with Intersecting Axis Assuming that param-eters 119886 = 0 and Σ = 0 the gear pair with intersecting axis canbe obtained as shown in Figure 13The selected conical spiralcurve for bevel gears is expressed in coordinate system 1198781 andits parametric form is represented as

r1 = (1198771119891 minus 119888120579) cos 120579i1 + (1198771119891 minus 119888120579) sin 120579j1 + 119901120579k1 (34)

where 1198771119891 is the radius of pitch circle and 120579 and 119901 are alsothe spiral curve and helix parameters respectively It has thecorrelation 119888 = 119901 tan 120575 120575 is the conical angle

Through a series of calculation on the basis of principle ofconjugate curves its conjugated curve is expressed as

1199092 = 1199091 [cos1206011 cos (119894211206011) minus sin1206011 sin (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 cos (119894211206011) minus cos1206011 sin (119894211206011) cosΣ]

minus 1199111 sin (119894211206011) sinΣ

1199102 = 1199091 [cos1206011 sin (119894211206011) + sin1206011 cos (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 sin (119894211206011) + cos1206011 cos (119894211206011) cosΣ]

+ 1199111 cos (119894211206011) sinΣ

1199112 = minus1199091 cos1206011 sinΣ minus 1199101 sin1206011 sinΣ + 1199111 cosΣ

119880 cos1206011 minus 119881 sin1206011 = 119882(35)

10 Mathematical Problems in Engineering

Conical spiral curve

Conjugated conical spiral curve

z1 z2

x1y1x2

y2

1205751

1205752

Rf1

Rf2

O1

O2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

Conjugated curveGiven curve

Line of action

(b)

Figure 13 Conjugate spiral curves for intersecting axis gear drive

Furthermore substituting (34) into (35) the simplifiedresult with parameter 120579 is represented as

1199092 = [1198771119891 minus (119888 + 119901) 120579] cos (11989421120579 + 2120579 + Σ)

1199102 = [1198771119891 minus (119888 + 119901) 120579] sin (11989421120579 + 2120579 + Σ)

1199112 = (minus119888 sinΣ + 119901 cosΣ) 120579 minus 1198771119891 sinΣ

(36)

Here

1198881 = 119888 + 119901 1205792 = 11989421120579 + 2120579 + Σ

1199011 = minus119888 sinΣ + 119901 cosΣ(37)

Then (36) can be written as

1199092 = (1198771119891 minus 1198881120579) cos 1205792

1199102 = (1198771119891 minus 1198881120579) sin 1205792

1199112 = 1199011120579 minus 1198771119891 sinΣ

(38)

Conjugated curve attached to gear 2 is got according toabove derivation Obviously it is also a conical spiral curve inform

423 Gear Pair with CrossedAxis Assuming that parameters119886 = 0 and Σ = 0 it gets the gear pair with crossed axis whichis depicted in Figure 14 The general spiral curve selectedfor worm gear drive is given and it has the same expression

with (30) Based on the principle of conjugate curves itsconjugated curve is derived as

1199092 =radic1199012sin2Σ + (119877 minus 119886)2 cos [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199102 =radic1199012sin2Σ + (119877 minus 119886)2 sin [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199112 = 119901120579 cosΣ(39)

Here

1198772 =radic1199012sin2Σ + (119877 minus 119886)2

1205793 = 11989421120579 + arctan(119877 minus 119886

119901 sinΣ)

(40)

Then (39) can be written as

1199092 = 1198772 cos 1205793

1199102 = 1198772 sin 1205793

1199112 = 119901120579 cosΣ

(41)

It also represents the similar relationship with givenoriginal curve in form

From what are discussed above we can conclude thatthe conjugated curves corresponding to the given spiralcurves for various gear axes are still the spiral curves Itcan be written as the same expression form with the given

Mathematical Problems in Engineering 11

Worm spiral curve

Conjugated worm spiral curve

x1

y1

x2

y2

O1

O2

z1

z2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

Conjugated curveGiven curve

Line of action

x (mm)y (mm)

z(m

m)

(b)

Figure 14 Conjugate spiral curves for crossed axis gear drive

conditions It is called meshing commonality of conjugatecurves Moreover based on the principle of conjugate curvesfor any chosen contact curve the corresponding same typeof conjugated curve in the direction of arbitrary contact anglecan be identified Freedomof selecting contact curve suggeststhe possibility of optimal conjugation design

According to the aforementioned research it offers aversatile way for high-performance profile design (1) fewteeth number and large module may be obtained withouttooth undercutting The problems of limited installationspace and lightweight design can be solved (2)Three differentcontact models of tooth profiles can be established basedon the proposed study Particularly the special meshing ofgenerated convex and concave tooth profiles makes relativeradius of curvature of the contact point longer and increasesthe contact strength The load capacity and useful life hasbeen evidently improved (3) The tooth surfaces mesh inpoint contact along the conjugate curves and contain thetransmission properties of conjugate curves Due to theselection and generation processes the approximate purerolling contact between mating tooth surfaces maybe occursThe transmission efficiency will be improved

The further studies on strength property manufacturingkey technology and performance experiment of conjugate-curve gear drive will be carried out And the excellent trans-mission performance of gearing is expected to be obtained onthe basis of theoretical and experimental investigations

5 Conclusions

(1) The principle descriptions of conjugate curves forarbitrary axial position are introducedWith the aid of

given plane or space curve generation principle andmathematical model of conjugated curve are devel-oped It can be applied to parallel axis intersectingaxis and crossed axis gear drive respectively

(2) Based on differential geometry geometric charac-teristic analysis of conjugate curves is carried outThe tangent and normal to conjugate curves inarbitrary contact direction are discussed and generalcalculation methods are provided The envelopingcondition of family of conjugate curves with singleparameter is analyzed and characteristic point isdetermined Curvature and torsion relationships ofconjugate curves are also derived

(3) Meshing properties of conjugate curves are furtherrevealed According to a given plane or spatial curvethe variation law of meshing function is analyzedThe uniqueness of conjugated curves is discussedTheconjugated curves are always the same and uniquecurves for any given curve under different contactangles while the difference is that there exists therotational angle relationship with contact point 119872among generated curves

(4) Meshing commonality of conjugate curves is demon-strated in terms of a class of spiral curves contactingin the given direction for various gear axes For anychosen contact curve under various gear axes theconjugated curve has the same type and it can bewritten as the same expression form

(5) The proposed theory lays the foundation for design ofnew types of gear drive The further study on conju-gate curves and applications in gear transmission will

12 Mathematical Problems in Engineering

be carried out Excellent transmission performance ofgearing is expected to be obtained

Conflict of Interests

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

Acknowledgments

This project is supported by National Key Technology RampDProgram of the Twelfth Five-year Plan of China (Grantno 2013BAF01B04) National Natural Science Foundation ofChina (Grant no 51205425) and Science and TechnologyProject of Chongqing City Board of Education (Grant noKJ122202) The authors also sincerely appreciate the com-ments and modification suggestions made by the editors andanonymous referees

References

[1] S S Chern W H Chen and K S Lam Lectures on DifferentialGeometry World Scientific Singapore 2006

[2] P G Ciarlet and T-T Li Differential Geometry Theory andApplications World Scientific Singapore 2007

[3] F L Litvin and A Fuentes Gear Geometry and Applied TheoryCambridge University Press New York NY USA 2nd edition2004

[4] S P Radzevich Theory of Gearing Kinetics Geometry andSynthesis CRC Press Boca Raton Fla USA 2012

[5] D B Dooner Kinematic Geometry of Gearing John Wiley ampSons 2nd edition 2012

[6] F L Litvin A Fuentes A Demenego D Vecchiato and Q FanldquoNewdevelopments in the design and generation of gear drivesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 215 no 7 pp747ndash757 2001

[7] C H Chen Theory of Conjugate Surfaces and Its ApplicationsScience and Technology Press Beijing China 2008

[8] G X Li Spatial Geometry Modeling and Its Application inEngineering Higher Education Press Beijing China 2007

[9] F Di Puccio M Gabiccini and M Guiggiani ldquoGeneration andcurvature analysis of conjugate surfaces via a new approachrdquoMechanism and Machine Theory vol 41 no 4 pp 382ndash4042006

[10] D R Wu and J S Luo A Geometric Theory of Conjugate ToothSurface World Scientific Publishing Singapore 1992

[11] N Ito and K Takahashi ldquoDifferential geometrical conditions ofhypoid gears with conjugate tooth surfacesrdquo ASME Journal ofMechanical Design vol 122 no 3 pp 323ndash330 2000

[12] D B Dooner ldquoOn the three laws of gearingrdquo Journal ofMechanical Design Transactions of the ASME vol 124 no 4pp 733ndash744 2002

[13] Z Duan H Chen and J Liu ldquoPrinciple and transmissiontechnology of Bertrand conjugate surfacesrdquo Chinese Journal ofMechanical Engineering vol 19 no 4 pp 600ndash604 2006

[14] H Chen Z Duan J Liu and H Wu ldquoResearch on basicprinciple of moulding-surface conjugationrdquo Mechanism andMachine Theory vol 43 no 7 pp 791ndash811 2008

[15] 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

[16] C F Chen and C B Tsay ldquoTooth profile design for themanufacture of helical gear sets with small numbers of teethrdquoInternational Journal ofMachine Tools andManufacture vol 45no 12-13 pp 1531ndash1541 2005

[17] R Imin and M Geni ldquoStress analysis of gear meshing impactbased on SPH methodrdquoMathematical Problems in Engineeringvol 2014 Article ID 328216 7 pages 2014

[18] D Park and A Kahraman ldquoA surface wear model for hypoidgear pairsrdquoWear vol 267 no 9-10 pp 1595ndash1604 2009

[19] JWang L Hou SM Luo and R YWu ldquoActive design of toothprofiles using parabolic curve as the line of actionrdquoMechanismand Machine Theory vol 67 no 9 pp 47ndash63 2013

[20] Q B Wang P Hu Y M Zhang et al ldquoA model to determinemesh characteristics in a gear pair with tooth profile errorrdquoAdvances in Mechanical Engineering vol 2014 Article ID751476 10 pages 2014

[21] C F Chen and C B Tsay ldquoComputerized tooth profile gen-eration and analysis of characteristics of elliptical gears withcircular arc teethrdquo Journal of Materials Processing Technologyvol 148 no 2 pp 226ndash234 2004

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

[23] B K Chen Y E Gao and D Liang ldquoThe generation principleof tooth surfaces of conjugate-curve gear transmissionrdquo Journalof Mechanical Engineering vol 50 no 3 pp 18ndash24 2014

[24] D Liang B K Chen and Y E Gao ldquoThe generation principleand mathematical model of a new involute-helix gear driverdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 227 no 12 pp2834ndash2843 2013

[25] B Chen D Liang and Z Li ldquoA study on geometry designof spiral bevel gears based on conjugate curvesrdquo InternationalJournal of Precision Engineering and Manufacturing vol 15 no3 pp 477ndash482 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Cylindrical spiral curve Conjugated cylindrical spiral curve

x1

y1

x2

y2

z1z2

(a)

Conjugated curveGiven curve

Line of action

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

(b)

Figure 12 Conjugate spiral curves for parallel axis gear drive

Table 1 Design parameters for conjugate curves

Parameters ValuesRadius of pitch circle Rmm 24Tooth number of pinion 1 1198851 6Tooth number of gear 2 1198852 30Transmission ratio 11989421 5Module119898119899mm 4Helix parameter 119901 286478Central distance 119886mm 144Tooth width 119861mm 50Parameter of spiral curve 120579rad 0sim10472Parameter 1199051 minus05Parameter 1199052 minus0866

of designated spiral curve expressed in (30) for generalcylindrical gears is carried out and the conjugated curve isderived as (31)

Here

1198771 = 119877 minus 119886 1205791 = 11989421120579 (32)

Then (31) can be written as

1199092 = 1198771 cos 1205791

1199102 = minus1198771 sin 1205791

1199112 = 119901120579

(33)

It is consistent with general form of cylindrical spiralcurve in parallel axis gear drive

422 Gear Pair with Intersecting Axis Assuming that param-eters 119886 = 0 and Σ = 0 the gear pair with intersecting axis canbe obtained as shown in Figure 13The selected conical spiralcurve for bevel gears is expressed in coordinate system 1198781 andits parametric form is represented as

r1 = (1198771119891 minus 119888120579) cos 120579i1 + (1198771119891 minus 119888120579) sin 120579j1 + 119901120579k1 (34)

where 1198771119891 is the radius of pitch circle and 120579 and 119901 are alsothe spiral curve and helix parameters respectively It has thecorrelation 119888 = 119901 tan 120575 120575 is the conical angle

Through a series of calculation on the basis of principle ofconjugate curves its conjugated curve is expressed as

1199092 = 1199091 [cos1206011 cos (119894211206011) minus sin1206011 sin (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 cos (119894211206011) minus cos1206011 sin (119894211206011) cosΣ]

minus 1199111 sin (119894211206011) sinΣ

1199102 = 1199091 [cos1206011 sin (119894211206011) + sin1206011 cos (119894211206011) cosΣ]

+ 1199101 [minus sin1206011 sin (119894211206011) + cos1206011 cos (119894211206011) cosΣ]

+ 1199111 cos (119894211206011) sinΣ

1199112 = minus1199091 cos1206011 sinΣ minus 1199101 sin1206011 sinΣ + 1199111 cosΣ

119880 cos1206011 minus 119881 sin1206011 = 119882(35)

10 Mathematical Problems in Engineering

Conical spiral curve

Conjugated conical spiral curve

z1 z2

x1y1x2

y2

1205751

1205752

Rf1

Rf2

O1

O2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

Conjugated curveGiven curve

Line of action

(b)

Figure 13 Conjugate spiral curves for intersecting axis gear drive

Furthermore substituting (34) into (35) the simplifiedresult with parameter 120579 is represented as

1199092 = [1198771119891 minus (119888 + 119901) 120579] cos (11989421120579 + 2120579 + Σ)

1199102 = [1198771119891 minus (119888 + 119901) 120579] sin (11989421120579 + 2120579 + Σ)

1199112 = (minus119888 sinΣ + 119901 cosΣ) 120579 minus 1198771119891 sinΣ

(36)

Here

1198881 = 119888 + 119901 1205792 = 11989421120579 + 2120579 + Σ

1199011 = minus119888 sinΣ + 119901 cosΣ(37)

Then (36) can be written as

1199092 = (1198771119891 minus 1198881120579) cos 1205792

1199102 = (1198771119891 minus 1198881120579) sin 1205792

1199112 = 1199011120579 minus 1198771119891 sinΣ

(38)

Conjugated curve attached to gear 2 is got according toabove derivation Obviously it is also a conical spiral curve inform

423 Gear Pair with CrossedAxis Assuming that parameters119886 = 0 and Σ = 0 it gets the gear pair with crossed axis whichis depicted in Figure 14 The general spiral curve selectedfor worm gear drive is given and it has the same expression

with (30) Based on the principle of conjugate curves itsconjugated curve is derived as

1199092 =radic1199012sin2Σ + (119877 minus 119886)2 cos [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199102 =radic1199012sin2Σ + (119877 minus 119886)2 sin [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199112 = 119901120579 cosΣ(39)

Here

1198772 =radic1199012sin2Σ + (119877 minus 119886)2

1205793 = 11989421120579 + arctan(119877 minus 119886

119901 sinΣ)

(40)

Then (39) can be written as

1199092 = 1198772 cos 1205793

1199102 = 1198772 sin 1205793

1199112 = 119901120579 cosΣ

(41)

It also represents the similar relationship with givenoriginal curve in form

From what are discussed above we can conclude thatthe conjugated curves corresponding to the given spiralcurves for various gear axes are still the spiral curves Itcan be written as the same expression form with the given

Mathematical Problems in Engineering 11

Worm spiral curve

Conjugated worm spiral curve

x1

y1

x2

y2

O1

O2

z1

z2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

Conjugated curveGiven curve

Line of action

x (mm)y (mm)

z(m

m)

(b)

Figure 14 Conjugate spiral curves for crossed axis gear drive

conditions It is called meshing commonality of conjugatecurves Moreover based on the principle of conjugate curvesfor any chosen contact curve the corresponding same typeof conjugated curve in the direction of arbitrary contact anglecan be identified Freedomof selecting contact curve suggeststhe possibility of optimal conjugation design

According to the aforementioned research it offers aversatile way for high-performance profile design (1) fewteeth number and large module may be obtained withouttooth undercutting The problems of limited installationspace and lightweight design can be solved (2)Three differentcontact models of tooth profiles can be established basedon the proposed study Particularly the special meshing ofgenerated convex and concave tooth profiles makes relativeradius of curvature of the contact point longer and increasesthe contact strength The load capacity and useful life hasbeen evidently improved (3) The tooth surfaces mesh inpoint contact along the conjugate curves and contain thetransmission properties of conjugate curves Due to theselection and generation processes the approximate purerolling contact between mating tooth surfaces maybe occursThe transmission efficiency will be improved

The further studies on strength property manufacturingkey technology and performance experiment of conjugate-curve gear drive will be carried out And the excellent trans-mission performance of gearing is expected to be obtained onthe basis of theoretical and experimental investigations

5 Conclusions

(1) The principle descriptions of conjugate curves forarbitrary axial position are introducedWith the aid of

given plane or space curve generation principle andmathematical model of conjugated curve are devel-oped It can be applied to parallel axis intersectingaxis and crossed axis gear drive respectively

(2) Based on differential geometry geometric charac-teristic analysis of conjugate curves is carried outThe tangent and normal to conjugate curves inarbitrary contact direction are discussed and generalcalculation methods are provided The envelopingcondition of family of conjugate curves with singleparameter is analyzed and characteristic point isdetermined Curvature and torsion relationships ofconjugate curves are also derived

(3) Meshing properties of conjugate curves are furtherrevealed According to a given plane or spatial curvethe variation law of meshing function is analyzedThe uniqueness of conjugated curves is discussedTheconjugated curves are always the same and uniquecurves for any given curve under different contactangles while the difference is that there exists therotational angle relationship with contact point 119872among generated curves

(4) Meshing commonality of conjugate curves is demon-strated in terms of a class of spiral curves contactingin the given direction for various gear axes For anychosen contact curve under various gear axes theconjugated curve has the same type and it can bewritten as the same expression form

(5) The proposed theory lays the foundation for design ofnew types of gear drive The further study on conju-gate curves and applications in gear transmission will

12 Mathematical Problems in Engineering

be carried out Excellent transmission performance ofgearing is expected to be obtained

Conflict of Interests

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

Acknowledgments

This project is supported by National Key Technology RampDProgram of the Twelfth Five-year Plan of China (Grantno 2013BAF01B04) National Natural Science Foundation ofChina (Grant no 51205425) and Science and TechnologyProject of Chongqing City Board of Education (Grant noKJ122202) The authors also sincerely appreciate the com-ments and modification suggestions made by the editors andanonymous referees

References

[1] S S Chern W H Chen and K S Lam Lectures on DifferentialGeometry World Scientific Singapore 2006

[2] P G Ciarlet and T-T Li Differential Geometry Theory andApplications World Scientific Singapore 2007

[3] F L Litvin and A Fuentes Gear Geometry and Applied TheoryCambridge University Press New York NY USA 2nd edition2004

[4] S P Radzevich Theory of Gearing Kinetics Geometry andSynthesis CRC Press Boca Raton Fla USA 2012

[5] D B Dooner Kinematic Geometry of Gearing John Wiley ampSons 2nd edition 2012

[6] F L Litvin A Fuentes A Demenego D Vecchiato and Q FanldquoNewdevelopments in the design and generation of gear drivesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 215 no 7 pp747ndash757 2001

[7] C H Chen Theory of Conjugate Surfaces and Its ApplicationsScience and Technology Press Beijing China 2008

[8] G X Li Spatial Geometry Modeling and Its Application inEngineering Higher Education Press Beijing China 2007

[9] F Di Puccio M Gabiccini and M Guiggiani ldquoGeneration andcurvature analysis of conjugate surfaces via a new approachrdquoMechanism and Machine Theory vol 41 no 4 pp 382ndash4042006

[10] D R Wu and J S Luo A Geometric Theory of Conjugate ToothSurface World Scientific Publishing Singapore 1992

[11] N Ito and K Takahashi ldquoDifferential geometrical conditions ofhypoid gears with conjugate tooth surfacesrdquo ASME Journal ofMechanical Design vol 122 no 3 pp 323ndash330 2000

[12] D B Dooner ldquoOn the three laws of gearingrdquo Journal ofMechanical Design Transactions of the ASME vol 124 no 4pp 733ndash744 2002

[13] Z Duan H Chen and J Liu ldquoPrinciple and transmissiontechnology of Bertrand conjugate surfacesrdquo Chinese Journal ofMechanical Engineering vol 19 no 4 pp 600ndash604 2006

[14] H Chen Z Duan J Liu and H Wu ldquoResearch on basicprinciple of moulding-surface conjugationrdquo Mechanism andMachine Theory vol 43 no 7 pp 791ndash811 2008

[15] 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

[16] C F Chen and C B Tsay ldquoTooth profile design for themanufacture of helical gear sets with small numbers of teethrdquoInternational Journal ofMachine Tools andManufacture vol 45no 12-13 pp 1531ndash1541 2005

[17] R Imin and M Geni ldquoStress analysis of gear meshing impactbased on SPH methodrdquoMathematical Problems in Engineeringvol 2014 Article ID 328216 7 pages 2014

[18] D Park and A Kahraman ldquoA surface wear model for hypoidgear pairsrdquoWear vol 267 no 9-10 pp 1595ndash1604 2009

[19] JWang L Hou SM Luo and R YWu ldquoActive design of toothprofiles using parabolic curve as the line of actionrdquoMechanismand Machine Theory vol 67 no 9 pp 47ndash63 2013

[20] Q B Wang P Hu Y M Zhang et al ldquoA model to determinemesh characteristics in a gear pair with tooth profile errorrdquoAdvances in Mechanical Engineering vol 2014 Article ID751476 10 pages 2014

[21] C F Chen and C B Tsay ldquoComputerized tooth profile gen-eration and analysis of characteristics of elliptical gears withcircular arc teethrdquo Journal of Materials Processing Technologyvol 148 no 2 pp 226ndash234 2004

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

[23] B K Chen Y E Gao and D Liang ldquoThe generation principleof tooth surfaces of conjugate-curve gear transmissionrdquo Journalof Mechanical Engineering vol 50 no 3 pp 18ndash24 2014

[24] D Liang B K Chen and Y E Gao ldquoThe generation principleand mathematical model of a new involute-helix gear driverdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 227 no 12 pp2834ndash2843 2013

[25] B Chen D Liang and Z Li ldquoA study on geometry designof spiral bevel gears based on conjugate curvesrdquo InternationalJournal of Precision Engineering and Manufacturing vol 15 no3 pp 477ndash482 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Conical spiral curve

Conjugated conical spiral curve

z1 z2

x1y1x2

y2

1205751

1205752

Rf1

Rf2

O1

O2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

x (mm)y (mm)

z(m

m)

Conjugated curveGiven curve

Line of action

(b)

Figure 13 Conjugate spiral curves for intersecting axis gear drive

Furthermore substituting (34) into (35) the simplifiedresult with parameter 120579 is represented as

1199092 = [1198771119891 minus (119888 + 119901) 120579] cos (11989421120579 + 2120579 + Σ)

1199102 = [1198771119891 minus (119888 + 119901) 120579] sin (11989421120579 + 2120579 + Σ)

1199112 = (minus119888 sinΣ + 119901 cosΣ) 120579 minus 1198771119891 sinΣ

(36)

Here

1198881 = 119888 + 119901 1205792 = 11989421120579 + 2120579 + Σ

1199011 = minus119888 sinΣ + 119901 cosΣ(37)

Then (36) can be written as

1199092 = (1198771119891 minus 1198881120579) cos 1205792

1199102 = (1198771119891 minus 1198881120579) sin 1205792

1199112 = 1199011120579 minus 1198771119891 sinΣ

(38)

Conjugated curve attached to gear 2 is got according toabove derivation Obviously it is also a conical spiral curve inform

423 Gear Pair with CrossedAxis Assuming that parameters119886 = 0 and Σ = 0 it gets the gear pair with crossed axis whichis depicted in Figure 14 The general spiral curve selectedfor worm gear drive is given and it has the same expression

with (30) Based on the principle of conjugate curves itsconjugated curve is derived as

1199092 =radic1199012sin2Σ + (119877 minus 119886)2 cos [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199102 =radic1199012sin2Σ + (119877 minus 119886)2 sin [11989421120579 + arctan(

119877 minus 119886

119901 sinΣ)]

1199112 = 119901120579 cosΣ(39)

Here

1198772 =radic1199012sin2Σ + (119877 minus 119886)2

1205793 = 11989421120579 + arctan(119877 minus 119886

119901 sinΣ)

(40)

Then (39) can be written as

1199092 = 1198772 cos 1205793

1199102 = 1198772 sin 1205793

1199112 = 119901120579 cosΣ

(41)

It also represents the similar relationship with givenoriginal curve in form

From what are discussed above we can conclude thatthe conjugated curves corresponding to the given spiralcurves for various gear axes are still the spiral curves Itcan be written as the same expression form with the given

Mathematical Problems in Engineering 11

Worm spiral curve

Conjugated worm spiral curve

x1

y1

x2

y2

O1

O2

z1

z2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

Conjugated curveGiven curve

Line of action

x (mm)y (mm)

z(m

m)

(b)

Figure 14 Conjugate spiral curves for crossed axis gear drive

conditions It is called meshing commonality of conjugatecurves Moreover based on the principle of conjugate curvesfor any chosen contact curve the corresponding same typeof conjugated curve in the direction of arbitrary contact anglecan be identified Freedomof selecting contact curve suggeststhe possibility of optimal conjugation design

According to the aforementioned research it offers aversatile way for high-performance profile design (1) fewteeth number and large module may be obtained withouttooth undercutting The problems of limited installationspace and lightweight design can be solved (2)Three differentcontact models of tooth profiles can be established basedon the proposed study Particularly the special meshing ofgenerated convex and concave tooth profiles makes relativeradius of curvature of the contact point longer and increasesthe contact strength The load capacity and useful life hasbeen evidently improved (3) The tooth surfaces mesh inpoint contact along the conjugate curves and contain thetransmission properties of conjugate curves Due to theselection and generation processes the approximate purerolling contact between mating tooth surfaces maybe occursThe transmission efficiency will be improved

The further studies on strength property manufacturingkey technology and performance experiment of conjugate-curve gear drive will be carried out And the excellent trans-mission performance of gearing is expected to be obtained onthe basis of theoretical and experimental investigations

5 Conclusions

(1) The principle descriptions of conjugate curves forarbitrary axial position are introducedWith the aid of

given plane or space curve generation principle andmathematical model of conjugated curve are devel-oped It can be applied to parallel axis intersectingaxis and crossed axis gear drive respectively

(2) Based on differential geometry geometric charac-teristic analysis of conjugate curves is carried outThe tangent and normal to conjugate curves inarbitrary contact direction are discussed and generalcalculation methods are provided The envelopingcondition of family of conjugate curves with singleparameter is analyzed and characteristic point isdetermined Curvature and torsion relationships ofconjugate curves are also derived

(3) Meshing properties of conjugate curves are furtherrevealed According to a given plane or spatial curvethe variation law of meshing function is analyzedThe uniqueness of conjugated curves is discussedTheconjugated curves are always the same and uniquecurves for any given curve under different contactangles while the difference is that there exists therotational angle relationship with contact point 119872among generated curves

(4) Meshing commonality of conjugate curves is demon-strated in terms of a class of spiral curves contactingin the given direction for various gear axes For anychosen contact curve under various gear axes theconjugated curve has the same type and it can bewritten as the same expression form

(5) The proposed theory lays the foundation for design ofnew types of gear drive The further study on conju-gate curves and applications in gear transmission will

12 Mathematical Problems in Engineering

be carried out Excellent transmission performance ofgearing is expected to be obtained

Conflict of Interests

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

Acknowledgments

This project is supported by National Key Technology RampDProgram of the Twelfth Five-year Plan of China (Grantno 2013BAF01B04) National Natural Science Foundation ofChina (Grant no 51205425) and Science and TechnologyProject of Chongqing City Board of Education (Grant noKJ122202) The authors also sincerely appreciate the com-ments and modification suggestions made by the editors andanonymous referees

References

[1] S S Chern W H Chen and K S Lam Lectures on DifferentialGeometry World Scientific Singapore 2006

[2] P G Ciarlet and T-T Li Differential Geometry Theory andApplications World Scientific Singapore 2007

[3] F L Litvin and A Fuentes Gear Geometry and Applied TheoryCambridge University Press New York NY USA 2nd edition2004

[4] S P Radzevich Theory of Gearing Kinetics Geometry andSynthesis CRC Press Boca Raton Fla USA 2012

[5] D B Dooner Kinematic Geometry of Gearing John Wiley ampSons 2nd edition 2012

[6] F L Litvin A Fuentes A Demenego D Vecchiato and Q FanldquoNewdevelopments in the design and generation of gear drivesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 215 no 7 pp747ndash757 2001

[7] C H Chen Theory of Conjugate Surfaces and Its ApplicationsScience and Technology Press Beijing China 2008

[8] G X Li Spatial Geometry Modeling and Its Application inEngineering Higher Education Press Beijing China 2007

[9] F Di Puccio M Gabiccini and M Guiggiani ldquoGeneration andcurvature analysis of conjugate surfaces via a new approachrdquoMechanism and Machine Theory vol 41 no 4 pp 382ndash4042006

[10] D R Wu and J S Luo A Geometric Theory of Conjugate ToothSurface World Scientific Publishing Singapore 1992

[11] N Ito and K Takahashi ldquoDifferential geometrical conditions ofhypoid gears with conjugate tooth surfacesrdquo ASME Journal ofMechanical Design vol 122 no 3 pp 323ndash330 2000

[12] D B Dooner ldquoOn the three laws of gearingrdquo Journal ofMechanical Design Transactions of the ASME vol 124 no 4pp 733ndash744 2002

[13] Z Duan H Chen and J Liu ldquoPrinciple and transmissiontechnology of Bertrand conjugate surfacesrdquo Chinese Journal ofMechanical Engineering vol 19 no 4 pp 600ndash604 2006

[14] H Chen Z Duan J Liu and H Wu ldquoResearch on basicprinciple of moulding-surface conjugationrdquo Mechanism andMachine Theory vol 43 no 7 pp 791ndash811 2008

[15] 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

[16] C F Chen and C B Tsay ldquoTooth profile design for themanufacture of helical gear sets with small numbers of teethrdquoInternational Journal ofMachine Tools andManufacture vol 45no 12-13 pp 1531ndash1541 2005

[17] R Imin and M Geni ldquoStress analysis of gear meshing impactbased on SPH methodrdquoMathematical Problems in Engineeringvol 2014 Article ID 328216 7 pages 2014

[18] D Park and A Kahraman ldquoA surface wear model for hypoidgear pairsrdquoWear vol 267 no 9-10 pp 1595ndash1604 2009

[19] JWang L Hou SM Luo and R YWu ldquoActive design of toothprofiles using parabolic curve as the line of actionrdquoMechanismand Machine Theory vol 67 no 9 pp 47ndash63 2013

[20] Q B Wang P Hu Y M Zhang et al ldquoA model to determinemesh characteristics in a gear pair with tooth profile errorrdquoAdvances in Mechanical Engineering vol 2014 Article ID751476 10 pages 2014

[21] C F Chen and C B Tsay ldquoComputerized tooth profile gen-eration and analysis of characteristics of elliptical gears withcircular arc teethrdquo Journal of Materials Processing Technologyvol 148 no 2 pp 226ndash234 2004

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

[23] B K Chen Y E Gao and D Liang ldquoThe generation principleof tooth surfaces of conjugate-curve gear transmissionrdquo Journalof Mechanical Engineering vol 50 no 3 pp 18ndash24 2014

[24] D Liang B K Chen and Y E Gao ldquoThe generation principleand mathematical model of a new involute-helix gear driverdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 227 no 12 pp2834ndash2843 2013

[25] B Chen D Liang and Z Li ldquoA study on geometry designof spiral bevel gears based on conjugate curvesrdquo InternationalJournal of Precision Engineering and Manufacturing vol 15 no3 pp 477ndash482 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Worm spiral curve

Conjugated worm spiral curve

x1

y1

x2

y2

O1

O2

z1

z2

(a)

30

25

20

15

10

5

0

minus5 05

10 15 15 10 5

Conjugated curveGiven curve

Line of action

x (mm)y (mm)

z(m

m)

(b)

Figure 14 Conjugate spiral curves for crossed axis gear drive

conditions It is called meshing commonality of conjugatecurves Moreover based on the principle of conjugate curvesfor any chosen contact curve the corresponding same typeof conjugated curve in the direction of arbitrary contact anglecan be identified Freedomof selecting contact curve suggeststhe possibility of optimal conjugation design

According to the aforementioned research it offers aversatile way for high-performance profile design (1) fewteeth number and large module may be obtained withouttooth undercutting The problems of limited installationspace and lightweight design can be solved (2)Three differentcontact models of tooth profiles can be established basedon the proposed study Particularly the special meshing ofgenerated convex and concave tooth profiles makes relativeradius of curvature of the contact point longer and increasesthe contact strength The load capacity and useful life hasbeen evidently improved (3) The tooth surfaces mesh inpoint contact along the conjugate curves and contain thetransmission properties of conjugate curves Due to theselection and generation processes the approximate purerolling contact between mating tooth surfaces maybe occursThe transmission efficiency will be improved

The further studies on strength property manufacturingkey technology and performance experiment of conjugate-curve gear drive will be carried out And the excellent trans-mission performance of gearing is expected to be obtained onthe basis of theoretical and experimental investigations

5 Conclusions

(1) The principle descriptions of conjugate curves forarbitrary axial position are introducedWith the aid of

given plane or space curve generation principle andmathematical model of conjugated curve are devel-oped It can be applied to parallel axis intersectingaxis and crossed axis gear drive respectively

(2) Based on differential geometry geometric charac-teristic analysis of conjugate curves is carried outThe tangent and normal to conjugate curves inarbitrary contact direction are discussed and generalcalculation methods are provided The envelopingcondition of family of conjugate curves with singleparameter is analyzed and characteristic point isdetermined Curvature and torsion relationships ofconjugate curves are also derived

(3) Meshing properties of conjugate curves are furtherrevealed According to a given plane or spatial curvethe variation law of meshing function is analyzedThe uniqueness of conjugated curves is discussedTheconjugated curves are always the same and uniquecurves for any given curve under different contactangles while the difference is that there exists therotational angle relationship with contact point 119872among generated curves

(4) Meshing commonality of conjugate curves is demon-strated in terms of a class of spiral curves contactingin the given direction for various gear axes For anychosen contact curve under various gear axes theconjugated curve has the same type and it can bewritten as the same expression form

(5) The proposed theory lays the foundation for design ofnew types of gear drive The further study on conju-gate curves and applications in gear transmission will

12 Mathematical Problems in Engineering

be carried out Excellent transmission performance ofgearing is expected to be obtained

Conflict of Interests

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

Acknowledgments

This project is supported by National Key Technology RampDProgram of the Twelfth Five-year Plan of China (Grantno 2013BAF01B04) National Natural Science Foundation ofChina (Grant no 51205425) and Science and TechnologyProject of Chongqing City Board of Education (Grant noKJ122202) The authors also sincerely appreciate the com-ments and modification suggestions made by the editors andanonymous referees

References

[1] S S Chern W H Chen and K S Lam Lectures on DifferentialGeometry World Scientific Singapore 2006

[2] P G Ciarlet and T-T Li Differential Geometry Theory andApplications World Scientific Singapore 2007

[3] F L Litvin and A Fuentes Gear Geometry and Applied TheoryCambridge University Press New York NY USA 2nd edition2004

[4] S P Radzevich Theory of Gearing Kinetics Geometry andSynthesis CRC Press Boca Raton Fla USA 2012

[5] D B Dooner Kinematic Geometry of Gearing John Wiley ampSons 2nd edition 2012

[6] F L Litvin A Fuentes A Demenego D Vecchiato and Q FanldquoNewdevelopments in the design and generation of gear drivesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 215 no 7 pp747ndash757 2001

[7] C H Chen Theory of Conjugate Surfaces and Its ApplicationsScience and Technology Press Beijing China 2008

[8] G X Li Spatial Geometry Modeling and Its Application inEngineering Higher Education Press Beijing China 2007

[9] F Di Puccio M Gabiccini and M Guiggiani ldquoGeneration andcurvature analysis of conjugate surfaces via a new approachrdquoMechanism and Machine Theory vol 41 no 4 pp 382ndash4042006

[10] D R Wu and J S Luo A Geometric Theory of Conjugate ToothSurface World Scientific Publishing Singapore 1992

[11] N Ito and K Takahashi ldquoDifferential geometrical conditions ofhypoid gears with conjugate tooth surfacesrdquo ASME Journal ofMechanical Design vol 122 no 3 pp 323ndash330 2000

[12] D B Dooner ldquoOn the three laws of gearingrdquo Journal ofMechanical Design Transactions of the ASME vol 124 no 4pp 733ndash744 2002

[13] Z Duan H Chen and J Liu ldquoPrinciple and transmissiontechnology of Bertrand conjugate surfacesrdquo Chinese Journal ofMechanical Engineering vol 19 no 4 pp 600ndash604 2006

[14] H Chen Z Duan J Liu and H Wu ldquoResearch on basicprinciple of moulding-surface conjugationrdquo Mechanism andMachine Theory vol 43 no 7 pp 791ndash811 2008

[15] 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

[16] C F Chen and C B Tsay ldquoTooth profile design for themanufacture of helical gear sets with small numbers of teethrdquoInternational Journal ofMachine Tools andManufacture vol 45no 12-13 pp 1531ndash1541 2005

[17] R Imin and M Geni ldquoStress analysis of gear meshing impactbased on SPH methodrdquoMathematical Problems in Engineeringvol 2014 Article ID 328216 7 pages 2014

[18] D Park and A Kahraman ldquoA surface wear model for hypoidgear pairsrdquoWear vol 267 no 9-10 pp 1595ndash1604 2009

[19] JWang L Hou SM Luo and R YWu ldquoActive design of toothprofiles using parabolic curve as the line of actionrdquoMechanismand Machine Theory vol 67 no 9 pp 47ndash63 2013

[20] Q B Wang P Hu Y M Zhang et al ldquoA model to determinemesh characteristics in a gear pair with tooth profile errorrdquoAdvances in Mechanical Engineering vol 2014 Article ID751476 10 pages 2014

[21] C F Chen and C B Tsay ldquoComputerized tooth profile gen-eration and analysis of characteristics of elliptical gears withcircular arc teethrdquo Journal of Materials Processing Technologyvol 148 no 2 pp 226ndash234 2004

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

[23] B K Chen Y E Gao and D Liang ldquoThe generation principleof tooth surfaces of conjugate-curve gear transmissionrdquo Journalof Mechanical Engineering vol 50 no 3 pp 18ndash24 2014

[24] D Liang B K Chen and Y E Gao ldquoThe generation principleand mathematical model of a new involute-helix gear driverdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 227 no 12 pp2834ndash2843 2013

[25] B Chen D Liang and Z Li ldquoA study on geometry designof spiral bevel gears based on conjugate curvesrdquo InternationalJournal of Precision Engineering and Manufacturing vol 15 no3 pp 477ndash482 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

be carried out Excellent transmission performance ofgearing is expected to be obtained

Conflict of Interests

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

Acknowledgments

This project is supported by National Key Technology RampDProgram of the Twelfth Five-year Plan of China (Grantno 2013BAF01B04) National Natural Science Foundation ofChina (Grant no 51205425) and Science and TechnologyProject of Chongqing City Board of Education (Grant noKJ122202) The authors also sincerely appreciate the com-ments and modification suggestions made by the editors andanonymous referees

References

[1] S S Chern W H Chen and K S Lam Lectures on DifferentialGeometry World Scientific Singapore 2006

[2] P G Ciarlet and T-T Li Differential Geometry Theory andApplications World Scientific Singapore 2007

[3] F L Litvin and A Fuentes Gear Geometry and Applied TheoryCambridge University Press New York NY USA 2nd edition2004

[4] S P Radzevich Theory of Gearing Kinetics Geometry andSynthesis CRC Press Boca Raton Fla USA 2012

[5] D B Dooner Kinematic Geometry of Gearing John Wiley ampSons 2nd edition 2012

[6] F L Litvin A Fuentes A Demenego D Vecchiato and Q FanldquoNewdevelopments in the design and generation of gear drivesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 215 no 7 pp747ndash757 2001

[7] C H Chen Theory of Conjugate Surfaces and Its ApplicationsScience and Technology Press Beijing China 2008

[8] G X Li Spatial Geometry Modeling and Its Application inEngineering Higher Education Press Beijing China 2007

[9] F Di Puccio M Gabiccini and M Guiggiani ldquoGeneration andcurvature analysis of conjugate surfaces via a new approachrdquoMechanism and Machine Theory vol 41 no 4 pp 382ndash4042006

[10] D R Wu and J S Luo A Geometric Theory of Conjugate ToothSurface World Scientific Publishing Singapore 1992

[11] N Ito and K Takahashi ldquoDifferential geometrical conditions ofhypoid gears with conjugate tooth surfacesrdquo ASME Journal ofMechanical Design vol 122 no 3 pp 323ndash330 2000

[12] D B Dooner ldquoOn the three laws of gearingrdquo Journal ofMechanical Design Transactions of the ASME vol 124 no 4pp 733ndash744 2002

[13] Z Duan H Chen and J Liu ldquoPrinciple and transmissiontechnology of Bertrand conjugate surfacesrdquo Chinese Journal ofMechanical Engineering vol 19 no 4 pp 600ndash604 2006

[14] H Chen Z Duan J Liu and H Wu ldquoResearch on basicprinciple of moulding-surface conjugationrdquo Mechanism andMachine Theory vol 43 no 7 pp 791ndash811 2008

[15] 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

[16] C F Chen and C B Tsay ldquoTooth profile design for themanufacture of helical gear sets with small numbers of teethrdquoInternational Journal ofMachine Tools andManufacture vol 45no 12-13 pp 1531ndash1541 2005

[17] R Imin and M Geni ldquoStress analysis of gear meshing impactbased on SPH methodrdquoMathematical Problems in Engineeringvol 2014 Article ID 328216 7 pages 2014

[18] D Park and A Kahraman ldquoA surface wear model for hypoidgear pairsrdquoWear vol 267 no 9-10 pp 1595ndash1604 2009

[19] JWang L Hou SM Luo and R YWu ldquoActive design of toothprofiles using parabolic curve as the line of actionrdquoMechanismand Machine Theory vol 67 no 9 pp 47ndash63 2013

[20] Q B Wang P Hu Y M Zhang et al ldquoA model to determinemesh characteristics in a gear pair with tooth profile errorrdquoAdvances in Mechanical Engineering vol 2014 Article ID751476 10 pages 2014

[21] C F Chen and C B Tsay ldquoComputerized tooth profile gen-eration and analysis of characteristics of elliptical gears withcircular arc teethrdquo Journal of Materials Processing Technologyvol 148 no 2 pp 226ndash234 2004

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

[23] B K Chen Y E Gao and D Liang ldquoThe generation principleof tooth surfaces of conjugate-curve gear transmissionrdquo Journalof Mechanical Engineering vol 50 no 3 pp 18ndash24 2014

[24] D Liang B K Chen and Y E Gao ldquoThe generation principleand mathematical model of a new involute-helix gear driverdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 227 no 12 pp2834ndash2843 2013

[25] B Chen D Liang and Z Li ldquoA study on geometry designof spiral bevel gears based on conjugate curvesrdquo InternationalJournal of Precision Engineering and Manufacturing vol 15 no3 pp 477ndash482 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of