The generation principle and mathematical models of a

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The generation principle and mathematical modelsof a novel cosine gear drive

Shanming Luo a,*, Yue Wu b, Jian Wang a

a School of Mechanical Engineering, Hunan University of Science and Technology, Taoyuan Road, Xiangtan 411201, Chinab School of Engineering, Computer Science and Mathematics, University of Exeter, Exeter EX4 4QF, UK

Received 19 November 2006; received in revised form 11 December 2007; accepted 26 December 2007Available online 14 February 2008

Abstract

A novel cosine gear drive is presented in this paper. The pinion of the drive utilizes a cosine curve as the tooth profile. Ittakes the zero line of the cosine curve as the pitch circle, a period of the curve as a tooth space, and the amplitude of thecurve as the tooth addendum. The generation principle of the cosine gear is described. The mathematical models, includingthe equation of the cosine tooth profile, the equation of the conjugate tooth profile and the equation of the line of action,are established based on the meshing theory. An example drive in solid model is presented and its computerized simulationis carried out. A few characteristics, such as the contact and bending stresses, the sliding coefficient and the contact ratio, ofthis new drive are analyzed. A comparison study of these characteristics with the involute gear drive was also carried out inthis work. The results confirm that the cosine gear drive has lower sliding coefficients and the contact and bending stressesof the cosine gear are reduced in comparison with the involute gear.� 2008 Elsevier Ltd. All rights reserved.

Keywords: Gear drive; Cosine gear; Tooth profile; Meshing theory; Mathematical model

1. Introduction

Tooth profile of gear is a fundamental element to determine transmission performance of a gear drive. Atpresent, there are three types of tooth profiles: involute, cycloid and circular-arc, which have been used in geardrives to fulfil different application requirements [1–4]. The most commonly used tooth profile is the involutedue to its advantages of simplicity for manufacture, mesh in line contact, constancy of pressure angle, andinsensitivity to central distance variation. However, the involute tooth profile suffers from low load capacity,relatively poor lubrication and proneness to interference [1].

With the developments of computerized numerical control technology, the manufacture of complicatedcurve and surface can be realized [5–9]. Therefore, new tooth profiles can be applied to improve the transmis-sion performance. Litvin et al. [10] investigated and compared two versions of face-gear drives based on

0094-114X/$ - see front matter � 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mechmachtheory.2007.12.007

* Corresponding author. Tel.: +86 732 8290480; fax: +86 732 8290509.E-mail address: [email protected] (S. Luo).

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1544 S. Luo et al. / Mechanism and Machine Theory 43 (2008) 1543–1556

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application of a spur pinion. The first version was a spur involute pinion and the other version was a pinionthat is conjugated to a parabolic rack-cutter. The following advantages of the new design were reported: (i)existence of a longitudinal bearing contact, (ii) avoidance of edge contact, and (iii) reduction of contactstresses. Tsay and Fong [11,12] studied a helical gear drive whose profiles consist of involute and circular-arc. The tooth surfaces of this gearing contact with each other at every instant at one point instead of oneline. The bearing contact of the gear tooth surface is localized and the centre of the bearing contact movesalong the tooth surface. Thus, this helical gear drive is insensitive to centre distance variation and gear axesmisalignment. Komori et al. [13] developed a gear with logic tooth profiles which have zero relative curvatureat many contact points. The gear therefore has higher durability and strength than involute gear. However, thelogic tooth profile can be applied only to a helical gear due to point contact. Zhang et al. [14] presented a dou-ble involute gear. The tooth profile of the gear consists of two involute curves, which are linked by a transitioncurve and form a ladder shape of tooth. Ariga and Nagata [15] used a specific cutter with combined circular-arc and involute tooth profiles to generate a new type of Wildhaber–Novikov gear. This particular tooth pro-file can solve the problem of conventional W–N gear profile, that is, the profile sensitivity to centre distancevariations. Fong et al. [16] proposed a mathematical model for parametric tooth profile of spur gear using agiving equation of line of action. According to this model, it will be easier to manipulate the line of actionwhen multi-segment tooth profile curves are used. The mathematical model can enhance the freedom of toothprofile design by combining the simple curves into the line of action. Kapelevich [17] presented a method ofresearch and design of gears with asymmetric teeth that enables to increase load capacity, reduce weight, sizeand vibration level. Francesco and Marini [18] consider a low pressure angle profile for the drive side and ahigh pressure angle profile for the coast side teeth. Such an approach enables to decrease the bending stresses,and keeps contact stresses on the same level as for symmetric teeth with equal pressure angle.

In recent decades, a large amount of literature is also available showing the mechanisms and methods fortooth profile generation. Litvin and Tsay [19] applied the vector analysis, differential geometry, matrix transfor-mation and meshing equation to develop mathematical models for describing tooth profiles and their geometricproperties. Litvin et al. [20] proposed a basic algorithm for analysis and synthesis of gear drives based on replace-ment of the instantaneous line of contact of tooth surfaces by point contact. In this approach, design, generationand simulation of the meshing and contact of gear drives with favorable bearing contact and reduced noise areinvestigated. Chang and Tsay [21] also proposed a method for determining the complete mathematical model ofnon-circular gear tooth profiles, which is manufactured with shaper cutters, based on the inverse mechanismrelation and the equation of motion. Tsai and Tsai [22] proposed a method of designing high-contact-ratio spurgears using quadratic parametric tooth profiles for the shorter addendum without undercut. Lee et al. [23] stud-ied the effects of linear profile modification on the dynamic tooth load and stress for high-contact-ratio gearing.Freudenstein and Chen [24] developed variable-ratio chain drives, and applied them to bicycles and variablemotion transmission involving band drives, tape drives and time belts with a minimum slack. Yildirim and Mun-ro [25] introduced a systematic approach to the design of tooth profile relief of both low and high-contact-ratiospur gears and its effects on transmission error and tooth loads. Chen and Tsai [26,27] described rack cutters withcircular-arc profile teeth to generate elliptical gears which rotate about one of their foci. A mathematical modelfor the elliptical gears with circular-arc teeth was developed according to gear theory. Additionally, theypresented a complete mathematical model of a helical gear set with small number of teeth. Tsai and Jehng[28] presented a generalized mathematical model of skew gears, which can be used to investigate the inherentcharacteristics of skew gears and design special gears. Danieli and Mundo [29] presented a new methodologywhich greatly increases the contact ratio between the teeth of non-circular gears, using a constant pressure anglefor any given tooth. Mundo [30] presented a new concept of epicyclical gear train which is able to generate a var-iable gear ratio law. The basic mechanical configuration of the epicyclical gear train consists of three non-circu-lar gears in a typical planetary arrangement, in which all pitch lines are variable-radius curves.

Based on the aforementioned researches, a novel gear drive with a cosine tooth profile is proposed in thiswork. Particularly, the pitch circle of the pinion lies on the zero line of cosine curve; the tooth space takes aperiod of the curve; and the tooth addendum is decided by the amplitude of the curve. As shown in Fig. 1, thecosine tooth profile appears very close to the involute tooth profile in the area near or above the pitch circle,i.e. the part of addendum. However, in area of dedendum, the tooth thickness of cosine gear is greater thanthat of involute gear.

Fig. 1. Cosine tooth profile and involute tooth profile.

S. Luo et al. / Mechanism and Machine Theory 43 (2008) 1543–1556 1545


The remainder of this paper is organized into five sections. In Section 2, the generation principle of thecosine gear is studied. The mathematical models: the equation of the conjugate tooth profile and the equationof the line of action are deduced in Sections 3 and 4, respectively. Section 5 presents an example of the cosinegear drive in solid model for computerized simulation of its meshing process. A few characteristics of the newdrive are analyzed and are compared with involute gears in this section. Finally, a conclusive summary of thisstudy is given in Section 6.

2. Generation of the cosine tooth profile

A tooth profile is the geometry of a tooth, which determines kinemical and dynamical properties of a geardrive. One of the reasons for widespread application of the involute gears in industry is due to their simplegeometry. An involute is a curve traced by the end of a string as it unwinds from a base circle [31]. Unlikethe involute tooth profile, the cosine tooth profile proposed in this work is generated by using the followingmethods: (i) the zero line of cosine curve is taken as the pitch circle; (ii) a period of the curve as a tooth space;and (iii) the amplitude of the curve as the tooth addendum. That is, the tooth profile of the cosine gear is acosine curve taking the pitch curve as the baseline, as shown in Fig. 2. In Fig. 2a, two coordinate systems are

Fig. 2. Generation principle of the cosine tooth profile.



used: a Cartesian coordinate system R1(X1,O1,Y1) and a natural coordinate system R(X,O,Y). In the naturalcoordinate system, the pitch circle is taken as the X axis which can be expanded to a line as shown in Fig. 2b.Here, r1 denotes the radius of the pitch curve of the cosine gear, and h represents the angle between the posi-tion vector of point M and Y axis.

In Fig. 2b, it is assumed that h is the amplitude of the cosine curve, and 2p/b is the period of the curve. Thenequation of the cosine curve can be expressed in the natural coordinate system as

y ¼ h cosðbxÞ ð1Þ

From Fig. 2a, the polar equation of a point M on the cosine tooth profile can be expressed as

q ¼ r1 þ h cosðbxÞ ð2Þ

According to Fig. 2a, the following equation can be obtained.

x ¼ r1h

Substituting it into Eq. (2) gives

q ¼ r1 þ h cosðbr1hÞ ð3Þ

The period of the cosine profile, denoted as T, with respect to the angle h can be expressed as

T ¼ 2p=br1 ð4Þ

Supposing that Z1 is the tooth number of the gear, then the tooth space, i.e. the period of the cosine toothprofile, can also be represented by

T ¼ 2p=Z1 ð5Þ

Eqs. (4) and (5) give

Z1 ¼ br1 ð6Þ

Substituting Eq. (6) into Eq. (3), the following equation can be obtained:

q ¼ r1 þ h cosðZ1hÞ ð7Þ

Assuming that m is the modulus of the cosine gear, the radius of pitch circle can be then expressed asr1 = mZ1/2. Substituting it into Eq. (7), the cosine tooth profile can be represented in the Cartesian coordinatesystem by the following equations:

x1 ¼ mZ1

2þ h cosðZ1hÞ

� �sinðhÞ

y1 ¼ mZ1

2þ h cosðZ1hÞ

� �cosðhÞ

(ð8Þ

3. Equation of the conjugate tooth profile

The generation of the conjugate tooth surfaces in line contact is based on the concept of the envelope to afamily of surfaces. This topic is related to differential geometry and to the theory of gearing. Zalgaller’s book[32] significantly contributes to the theory of envelopes and covers the necessary and sufficient conditions forthe envelope’s existence. Simplified approaches to the solution of these problems have also been developed byLitvin [2] in the theory of gearing.

As shown in Fig. 3, the notations C1 and C2 are respectively used for the generating and generated surfaces.P is the pitch point; and P1 is the intersection point of the normal line of point M1 on tooth profile C1 with thepitch circle. If the normal line of point M on tooth profile C1 gets across the pitch point P, then point M is acontact point between the two mating tooth profiles. In order to make M1 become a contact point, the toothprofile C1 must move to the dashed position after rotating an angle u1, whilst point P1 arrives at point P. Sup-posing that b is the angle between the tangent at point M1(x1,y1) on tooth profile C1 and axis X1, the followingequation can be obtained.

Fig. 3. Conjugate tooth profile.



b ¼ arctandy1

dx1

ð9Þ

Substituting Eq. (8) into Eq. (9) gives

b ¼ arctan�½mZ1=2þ h cosðZ1hÞ� tan h� hZ1 sinðZ1hÞ½mZ1=2þ h cosðZ1hÞ� � hZ1 tan h sinðZ1hÞ

ð10Þ

In addition, making a perpendicular from point O1 to line M1P1 and supposing w is the angle between theperpendicular and line O1P1, the following equations can be obtained according to Fig. 3.

cos w ¼ x1 cos bþy1 sin br1

u1 ¼ p2� b� w

(ð11Þ

Substituting Eq. (8) into Eq. (11) gives

u1 ¼ arcsin1

mZ1

½mZ1 þ 2h cosðZ1hÞ� sinðhþ bÞ� �

� b ð12Þ

In the following discussion, coordinate systems R1(X1,O1,Y1), R2(X2,O2,Y2) and R(X,O,Y) are designated asshown in Fig. 4. Coordinate system R is a fixed coordinate system whose origin O coincides with the pitchpoint P, while coordinate systems R1 and R2 are moving coordinate systems rigidly connected with gear 1and gear 2, respectively.

The position vector rM1 of contact point M on the tooth profile C1 can be expressed in coordinate system R1

as follows:

rM1 ¼ x1i1 þ y1j1 ð13Þ
According to the concept of relative motion, if a point moves along a given path of contact, its trace, asdescribed by two gears rotating about each axis, will become the tooth profile of each gear. In addition, thereshould be a special relationship between the position of the contact point and the rotation displacement of thedriving gear to obtain the conjugate tooth profiles [22].
From kinematics, the conjugate tooth profile C2 can be deemed as the envelope of the tooth profile C1 incoordinate system R2 [2–4]. Using the coordinate transformation from R1 to R2 [16,19], the position vector ofcontact point M on tooth profile C2 can be expressed as

Fig. 4. Coordinate systems used for meshing analysis.



rM2 ¼M12rM1 þ r12 ð14Þ

where r10 and r12 denote the position vectors of origin O1 with respect to coordinate system R and R2 respec-tively; r02 represents the position vector of origin O with respect to coordinate system R2; M02 and M12 denotethe coordinate transformation matrices from coordinate system R to R1 and R2 respectively. Moreover, thefollowing relationships exist:

r12 ¼M02r10 þ r02

r10 ¼0

�r1

� �

r02 ¼r2 sin u2

�r2 cos u2

� �

M02 ¼cos u2 � sin u2

sin u2 cos u2

� �

M12 ¼cosðu1 þ u2Þ � sinðu1 þ u2Þsinðu1 þ u2Þ cosðu1 þ u2Þ

� �

Substituting above equations into Eq. (14) gives

rM2 ¼cosðu1 þ u2Þ � sinðu1 þ u2Þsinðu1 þ u2Þ cosðu1 þ u2Þ

� �x1

y1

� �þ

a sin u2

�a cos u2

� �

or

x2 ¼ x1 cosðu1 þ u2Þ � y1 sinðu1 þ u2Þ þ a sin u2

y2 ¼ x1 sinðu1 þ u2Þ þ y1 cosðu1 þ u2Þ � a cos u2

�ð15Þ

where a denotes the central distance between the centres of gears 1 and gear 2, a = r1 + r2.Substituting Eq. (8) into Eq. (15), equations of conjugate tooth profile C2 in the Cartesian coordinate

system can be expressed as



x2 ¼ 12mZ1 þ h cosðZ1hÞ

� �sin h� 1þ 1

i

� u1

� �þ a sin u1

i

y2 ¼ 12mZ1 þ h cosðZ1hÞ

� �cos h� 1þ 1

i

� u1

� �� a cos u1

i

(ð16Þ

where i represents the gear ratio, i = u1/u2.

4. Equation of the line of action

The line of action, which passes through the pitch point, is defined as the set of the instantaneous contactpoints between two mating tooth profiles in the fixed coordinate system [16]. According to the meshing theory[2–4], transforming the position vector rM1 of contact point M from coordinate system R1(X1,O1,Y1) toR(X,O,Y), the equation of the line of action between the mating tooth profiles C1 and C2 can be expressedas follows:

rM ¼M10rM1 þ r10 ð17Þ

where rM denotes the position vector of contact point M in coordinate system R; M10 is the coordinate trans-formation matrix from coordinate system R1 to R, which can be expressed as

M10 ¼cos u1 � sin u1

sin u1 cos u1

� �

Hence, Eq. (17) can be rewritten as

rM ¼cos u1 � sin u1

sin u1 cos u1

�x1

y1

�þ

0

�r1

�ð18Þ

Substituting Eq. (8) into Eq. (18), the equation of the line of action in the Cartesian coordinate system can beexpressed as

x ¼ 12mZ1 þ h cosðZ1hÞ

� �sinðh� u1Þ

y ¼ 12mZ1 þ h cosðZ1hÞ

� �cosðh� u1Þ � 1

2mZ1

(ð19Þ

5. Evaluation of the cosine drive

In this section, a study of a set of configurations, where the tooth profiles can be visualized, is presented. Inaddition, a further evaluation of the newly designed gears is made. Data compiled from industrial experienceand laboratory experiments reveal two high stress areas that generally are the sources for gear failures. Thesetwo areas are root fillet and the contact surface of the teeth [33]. Based on the above concerns, both bendingand contact stresses of the cosine gears are analyzed. Two gear characteristics, sliding coefficient and contactratio, are also discussed and a comparison study of all these characteristics with the involute gears are carriedout.

5.1. Solid model and computerized simulation

Before the tooth profiles can be determined, it is necessary to specify a few parameters for the cosine spurgear drive of the case: (i) the modulus m is 3 mm; (ii) the tooth number Z1 is 13; (iii) the tooth number Z2 is 27;and (iv) the tooth addendum h is 3 mm. The meshing surfaces can be constructed from Eqs. (8) and (16), andthe constraints of continuity Eqs. (10), (12) and (19). By using solid modelling software Pro/E and a paramet-ric method, a three-dimensional model of the cosine gear drive was established as shown in Fig. 5.

In order to demonstrate the feasibility of the cosine gear drive and evaluate its meshing and contactconditions, a computerized simulation of the meshing process was carried out by using the mechanism moduleof Pro/E. The following behaviors were observed from the simulation:

Fig. 5. Three-dimensional solid model of the cosine gear drive.



(i) the cosine gear drive was able to transmit rotation between the two mating gears with a constant gearratio and continuous transmission;

(ii) no meshing interference between the couple tooth surfaces was found in the meshing process;(iii) the mating tooth surfaces were in mesh in line contact as shown in Fig. 6, which is similar to the involute

gear drives [4,16].

5.2. Contact and bending stresses

This section presents the analysis of contact and bending stresses of a cosine gear drive, and a comparisonwith those of an involute gear drive. The stress results presented in this paper are obtained by using the FEprogram ANSYS. The numerical computations have been performed for the cosine gear drive with the follow-ing design parameters: Z1 = 25, Z2 = 40, m = 3 mm. The torque applied to the pinion was 98.8 N m. Proper-ties of materials are: l = 0.29, E = 204 GPa. The involute gear drive uses the same set of specifications forcomparison.

Two models of contacting teeth based on the real geometry of the pinion and its conjugate profile have beendeveloped by Pro/E and then transferred to ANSYS for stress analysis. The finite element models that consistof the loaded teeth and the neighboring teeth are shown in Fig. 7. Two sides of each model sufficiently far fromthe fillet are chosen to justify the rigid constraints applied along the boundaries. A large enough part of thewheel below the tooth is chosen for the fixed boundary. This model was constructed using 8-node isoparamet-ric elements. There are 1064 elements and 3080 nodes in each transverse section. Two options related to thecontact problem, small sliding and no friction, have been selected. Fig. 8 shows the contour plot of Von-Misesstresses. Under the same parameters, stress distribution of an involute gear drive as shown in Fig. 9 was also

Fig. 6. Contact condition of the cosine gear drive.

Fig. 7. Models applied for finite element analysis.

Fig. 8. Stress distribution of cosine gear drive.

Fig. 9. Stress distribution of involute gear drive.





analyzed. The bending stresses obtained in the fillet of the contacting tooth side are considered as tensionstresses, and those in the fillet of the opposite tooth side are considered as compression stresses [34]. Thenumerical results are listed in Table 1.

According to the numerical results obtained, the following conclusions to the specified gears can be drawn:(i) the maximum contact stress of the cosine gear is reduced by about 22.59% in comparison with the involutegear; (ii) the tension bending stress of the cosine gear is 33.86% less than that of the involute gear, and thecompression bending stress is reduced by 34.33% in comparison with the involute gear. Therefore, applicationof the cosine tooth profile can reduce both contact and bending stresses.

5.3. Sliding coefficient

Sliding coefficient is a measure of the sliding action during the meshing process. A lower coefficient willhave greater transmission efficiency due to the less friction. The sliding coefficient is the ratio of the relativesliding velocity to the velocity of the contact point on each tooth profile while the gears are in mesh andthe relative sliding velocity is the difference between the velocities at the contact point [1,4,35].

As shown in Fig. 10, line n–n represents the normal of the line of contact c–c at the contact point K, andpoint H is the intersection point of the normal n–n with the central line O1O2. According to Ref. [35], the slid-ing coefficients of the cosine gear and driven gear, U1 and U2, can be expressed as

Fig. 10. Relative sliding of the cosine gear drive.

Table 1Maximum bending and contact stresses (Unit: MPa)

Gears Contact stress Bending stress (tension) Bending stress (compression)

Cosine gear 520.10 116.16 136.87Involute gear 671.87 175.64 208.43



U 1 ¼ 1� r2�br1þb

� i21

U 2 ¼ 1� r1þbr2�b

� i12

8><>: ð20Þ

where b ¼ PH , i12 ¼ 1i21¼ r2

r1.

According to Eq. (19), the slope, k, of the normal n–n can be expressed as

k ¼ � dxdy¼ dx=dh

dy=dh¼

mz1

2þ h cosðz1hÞ

� �ð1� u01Þ cosðh� u1Þ � hz1 sinðh� u1Þ sinðz1hÞ

mz1

2þ h cosðz1hÞ

� �ð1� u01Þ sinðh� u1Þ þ hz1 cosðh� u1Þ sinðz1hÞ

ð21Þ

where u01 is the derivative of u1 with respect to h, which can be obtained as follows from Eqs. (12) and (10):

u01 ¼du1

dh¼

mz1

2þ h cosðz1hÞ

� �ð1þ b0Þ cosðhþ bÞ � 2hz1 sinðz1hÞ sinðhþ bÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

m2z21 � mz1

2þ h cosðz1hÞ

� �2sin2ðhþ bÞ

q � b0 ð22Þ

b0 ¼ dbdh¼� mz1

2þ h cosðz1hÞ

� �sec2 h� 2h2z2

1 sin2ðz1hÞ sec2 hþ h2z31 tan h½sin2ðz1hÞ � sinðz1hÞ cosðz1hÞ�

mz1

2þ h cosðz1hÞ

� �� hz1 tan h sinðz1hÞ

� �2

þhz2

1mz1

2þ h cosðz1hÞ

� �½sinðz1hÞ þ tan2 h cosðz1hÞ�

mz1

2þ h cosðz1hÞ

� �� hz1 tan h sinðz1hÞ

� �2ð23Þ

As shown in Fig. 10, the vertical coordinate, b, of the point H in coordinate system R(X,P,Y) can be expressedas

b ¼ �kxk þ yk ð24Þ

where (xk,yk) denote the coordinates of the contact point K in coordinate system R(X,P,Y).Substituting Eqs. (19) and (21) into Eq. (24) gives

b ¼ �mz1

2þ h cosðz1hÞ

� �2ð1� u01Þ sinðh� u1Þ cosðh� u1Þ � hz1mz1

2þ h cosðz1hÞ

� �sin2ðh� u1Þ sinðz1hÞ

mz1

2þ h cosðz1hÞ

� �ð1� u01Þ sinðh� u1Þ þ hz1 cosðh� u1Þ sinðz1hÞ

þ 1

2mZ1 þ h cosðZ1hÞ

� �cosðh� u1Þ �

1

2mZ1

ð25Þ

The sliding coefficients, U1 and U2, of the cosine gear drive can be therefore obtained by substituting Eq. (25)into Eq. (20).

The computational procedure has been implemented in Matlab 5.0 and has been used to carry out the slid-ing coefficients of the two gears according to Eqs. (20) and (25). Fig. 11 shows the simulation results of thesliding coefficients of the cosine gear drive with design parameters: Z1 = 35, Z2 = 75, m = 3 mm. The slidingcoefficients of the involute gear drive under the same parameters are also drawn in Fig. 11 according to Yanet al. [36]. The following conclusions can be made from these numerical results: (i) the sliding coefficients ofcosine gear drive is much less than that of involute gear drive, (ii) the variation of the sliding coefficients ofcosine gear drive is much smaller than that of involute gear drive. Therefore, application of cosine tooth pro-file helps to reduce the sliding coefficient and improves the transmission performance.

5.4. Contact ratio

The contact ratio of a gear drive is defined as the number of teeth being in mesh simultaneously, or therotating angle of the gear between the starting and the end points of contact divided by the angle betweenevery two teeth, which equals to 2p divided by the number of teeth [37]. The contact ratio should be greaterthan 1 in a gear drive for smooth transmission. As shown in Fig. 12, the contact ratio of cosine gear drive canbe expressed as

Fig. 11. Sliding coefficients against the radius at the contact point of: (a) the cosine gear; (b) the conjugate gear.

Fig. 12. Contact ratio of the cosine gear drive.



e ¼ Z1ðub � ucÞ2p

ð26Þ

where ub and uc are the values of rotation angle u1 as x = xb and x = xc, respectively. These values can becalculated by Eq. (19).

Table 2Contact ratio of the cosine and the involute gear drive

Tooth number Z1 Tooth number Z2 Modulus m Cosine gear drive Involute gear drive

15 32 3 mm 1.2642 1.574517 40 3 mm 1.2826 1.614221 60 3 mm 1.3402 1.6769



The computation of three sets of gear configurations as shown in Table 2 was carried out by using Matlab.The contact ratios, as the computation results, of both cosine and involute gear drives are also listed in Table 2.According to Table 2, the contact ratio of cosine gear drive ranges from 1.26 to 1.35, which is about 20% lessthan that of involute gear drive. However, such cosine gear drives can be applied in field of gear pump sinceless contact ratio is advantageous to ease the trapping phenomenon [38–40].

6. Conclusions

Due to various limitations on load capacity of common used gear profiles, looking for new gear profilesthat are capable of bearing high loads is still needed. The cosine gear drive proposed in this work is an attempton this direction. The generation principle of the cosine tooth profile and the mathematical models of thecosine gear drive deduced from the meshing theory have been presented. The theoretical analysis and numer-ical calculation shows that the cosine gear drive has lower sliding coefficients, and the contact and bendingstresses are reduced in comparison with the involute gear drive according to simulation results of the exampleFE model. Kinematical simulation of the meshing process of a given example also demonstrates that the mat-ing tooth surfaces of the new drive are in mesh in line contact. Although the contact ratio of the proposeddrive is less than that of the conventional involute gear drive, the preliminary results show that the cosine geardrive is feasible and could have a promising future in application in the field of gear pumps.

The related investigations on this gear drive, which include: (i) the sensitivity to misalignments and otherassembly errors, (ii) the effect of contact ratio on strength of the gears, (iii) experiments of prototypes, and (iv)manufacturing method for mass production, are being carried out or would be the next step of work by theauthors. Efforts putting this drive forward into practical application are also needed in the near future.

Acknowledgements

The work was supported by the National Natural Science Foundation of China under Grant No. 50575071,the Provincial Natural Science Foundation of Hunan under Grant No. 06JJ10008 and the Program for NewCentury Excellent Talents in University. These financial supports are gratefully acknowledged. The authorsalso sincerely appreciate the comments and modification suggestions made by the editors and anonymousreferees.

References

[1] T. Yeh, D.C.H. Yang, S.H. Tong, Design of new tooth profile for high-load capacity gears, Mechanism and Machine Theory 36 (10)(2001) 1105–1120.

[2] F.L. Litvin, Theory of Gearing, NASA Publication, Washington, 1989.[3] F.L. Litvin, A. Fuentes, Gear Geometry and Applied Theory, second ed., Cambridge University Press, New York, 2004.[4] D.B. Dooner, A.A. Seireg, The Kinematic Geometry of Gearing: A Concurrent Engineering Approach, John Wiley & Sons, New

York, 1995.[5] A. Ishibashi, H. Yoshino, Design and manufacture of gear cutting tools and gears with an arbitrary profile, JSME International

Journal 30 (265) (1987) 1159–1166.[6] S.M. Vijayakar, B. Sarkar, D.R. Houser, Gear tooth profile determination from arbitrary rack geometry, Gear Technology 5 (6)

(1988) 18–23.[7] H. Yoshino, M. Shao, A. Ishibashi, Design and manufacture of pinion cutters for finishing gears with an arbitrary profile, JSME

International Journal, Series 3: Vibration, Control Engineering, Engineering for Industry 35 (2) (1992) 313–319.[8] A.Q. Wang, X.Y. Feng, X. Ai, Study on generation of specific gear and simulation of machining process, Chinese Journal of

Mechanical Engineering 39 (1) (2003) 143–148.



[9] X.Y. Feng, H. Li, X. Ai, CNC flexible generating specific gear tooth profile based on standard involute gear hob, Chinese Journal ofMechanical Engineering 17 (3) (2004) 377–380 (English Edition).

[10] F.F. Litvin, A. Fuentes, C. Zanzi, M. Pontiggia, Design, generation, and stress analysis of two versions of geometry of face geardrives, Mechanism and Machine Theory 37 (10) (2002) 1179–1211.

[11] C.B. Tsay, Z.H. Fong, Teeth contact analysis for helical gears with pinion circular-arc teeth and gear involute shaped teeth, Journal ofMechanism, Transmission, and Automation in Design 111 (2) (1989) 278–284.

[12] C.B. Tsay, Z.H. Fong, Computer simulation and stress analysis of helical gear with pinion circular arc teeth and gear involute teeth,Mechanism and Machine Theory 26 (2) (1991) 145–154.

[13] T. Komori, Y. Ariga, S. Nagata, New gears profile having zero relative curvature at many contact points (logix tooth profile), Journalof Mechanisms, Transmissions, and Automation in Design 112 (3) (1990) 430–436.

[14] G.H. Zhang, H.B. Xu, H. Long, Double involute gear with ladder-shape teeth, Chinese Journal of Mechanical Engineering 31 (6)(1995) 47–52.

[15] Y. Ariga, S. Nagata, Load capacity of a new W–N gear with basic rack of combined circular and involute profile, Journal ofMechanisms, Transmissions, and Automation in Design 107 (4) (1985) 565–572.

[16] Z.H. Fong, T.W. Chiang, C.W. Tsai, Mathematical model for parametric tooth profile of spur gear using line of action, Mathematicaland Computer Modelling 36 (4–5) (2002) 603–614.

[17] A. Kapelevich, Geometry and design of involute spur gears with asymmetric teeth, Mechanism and Machine Theory 35 (1) (2000)117–130.

[18] G.D. Francesco, S. Marini, Structural analysis of asymmetrical teeth: reduction of size and weight, Gear Technology 14 (5) (1997) 47–51.

[19] F.L. Litvin, C.B. Tsay, Helical gears with circular arc teeth: simulation of conditions of meshing and bearing contact, Journal ofMechanisms, Transmissions, and Automation in Design 107 (4) (1985) 556–564.

[20] F.L. Litvin, A. Fuentes, A. Demenego, D. Vecchiato, New developments in the design and generation of gear drives, Proceedings ofthe Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 215 (7) (2001) 747–757.

[21] S.L. Chang, C.B. Tsay, Computerized tooth profile generation and undercut analysis of noncircular gears manufactured with shapercutters, ASME Journal of Mechanical Design 120 (1) (1998) 92–99.

[22] M.H. Tsai, Y.C. Tsai, Design of high-contact-ratio spur gears using quadratic parametric tooth profile, Mechanism and MachineTheory 33 (5) (1998) 551–564.

[23] C. Lee, H.H. Lin, F.B. Oswald, D.P. Townsend, Influence of linear profile modification and loading conditions on the dynamic toothload and stress of high-contact-ratio spur gears, ASME Journal of Mechanical Design 113 (4) (1991) 473–480.

[24] F. Freudenstein, C.K. Chen, Variable-ratio chain drives with noncircular sprockets and minimum slack theory and application,ASME Journal of Mechanical Design 113 (3) (1991) 253–262.

[25] N. Yildirim, R.G. Munro, A systematic approach to profile relief design of low and high contact ratio spur gears, Proceedings of theInstitution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 213 (6) (1999) 551–562.

[26] C.F. Chen, C.B. Tsay, Computerized tooth profile generation and analysis of characteristics of elliptical gears with circular arc teeth,Journal of Materials Processing Technology 148 (2) (2004) 226–234.

[27] C.F. Chen, C.B. Tsay, Tooth profile design for the manufacture of helical gear sets with small numbers of teeth, International Journalof Machine Tools and Manufacture 45 (12–13) (2005) 1531–1541.

[28] Y.C. Tsai, W.K. Jehng, A kinematics parametric method to generate new tooth profiles of gear sets with skew axes, Mechanism andMachine Theory 34 (6) (1999) 857–876.

[29] G.A. Danieli, D. Mundo, New developments in variable radius gears using constant pressure angle teeth, Mechanism and MachineTheory 40 (2) (2005) 203–217.

[30] D. Mundo, Geometric design of a planetary gear train with noncircular gears, Mechanism and Machine Theory 41 (4) (2006) 456–472.[31] S. Barone, Gear Geometric design by B-spline curve fitting and sweep surface modelling, Engineering with Computers 17 (1) (2001)

66–74.[32] V.A. Zalgaller, Theory of Envelopes, Nauka, Moscow, 1975.[33] J. Lu, F.L. Litvin, J.S. Chen, Load share and finite element stress analysis for double circular-arc helical gears, Mathematical and

Computer Modelling 21 (10) (1995) 13–30.[34] F.L. Litvin, Q. Lian, A.L. Kapelevich, Asymmetric modified spur gear drives: reduction of noise, localization of contact, simulation

of meshing and stress analysis, Computer Method in Applied Mechanics and Engineering 188 (1–3) (2000) 363–390.[35] L.R. Hu, Spatial Meshing Theory and its Application, China Coal Industry Publishing House, Beijing, 1987.[36] S.T. Yan, C.B. Hu, Z.X. Wu, X.G. Xu, The calculating method of sliding wear on tooth outline, Machine Design and Manufacturing

Engineering 30 (4) (2001) 9–10.[37] J.I. Pedrero, M. Artes, J.C. Garcıa-Prada, Determination of the addendum modification factors for gears with pre-established contact

ratio, Mechanism and Machine Theory 31 (7) (1996) 937–945.[38] C.K. Chen, S.C. Yang, Geometric modelling for cylindrical and helical gear pumps with circular arc teeth, Proceedings of the

Institution of Mechanical Engineers, Part G, Journal of Aerospace Engineering 214 (4) (2000) 509–617.[39] Z.M. Liu, D.H. Hou, X.C. Wang, et al., Reverse design and analyses of compound tooth profile of gear pump, Journal of Mechanical

Transmission 24 (1) (2000) 13–15.[40] J.K. Cui, S. Qin, B. Wen, Analysis of meshing properties for the straight conjugate internal gear pair, Journal of Mechanical

Transmission 28 (6) (2004) 12–15.

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