The generation principle and mathematical models of a novel cosine gear drive

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1 The generation principle and mathematical models of 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 , China b 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 2007 Available 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. It takes the zero line of the cosine curve as the pitch circle, a period of the curve as a tooth space, and the amplitude of the curve as the tooth addendum. The generation principle of the cosine gear is described. The mathematical models, including the 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 simulation is carried out. A few characteristics, such as the contact and bending stresses, the sliding coefficient and the contact ratio, of this new drive are analyzed. A comparison study of these characteristics with the involute gear drive was also carried out in this work. The results confirm that the cosine gear drive has lower sliding coefficients and the contact and bending stresses of the cosine gear are reduced in comparison with the involute gear. Ó 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. At present, there are three types of tooth profiles: involute, cycloid and circular-arc, which have been used in gear drives to fulfil different application requirements [1 4]. The most commonly used tooth profile is the involute due to its advantages of simplicity for manufacture, mesh in line contact, constancy of pressure angle, and insensitivity 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 complicated curve and surface can be realized [5 9]. Therefore, new tooth profiles can be applied to improve the transmission performance. Litvin et al. [10] investigated and compared two versions of face-gear drives based on * Corresponding author. Tel.: ; fax: address: s.luo@hotmail.com (S. Luo) X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi: /j.mechmachtheory 转载

2 1544 S. Luo et al. / Mechanism and Machine Theory 43 (2008) application of a spur pinion. The first version was a spur involute pinion and the other version was a pinion that 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 contact stresses. Tsay and Fong [11,12] studied a helical gear drive whose profiles consist of involute and circulararc. The tooth surfaces of this gearing contact with each other at every instant at one point instead of one line. The bearing contact of the gear tooth surface is localized and the centre of the bearing contact moves along the tooth surface. Thus, this helical gear drive is insensitive to centre distance variation and gear axes misalignment. Komori et al. [13] developed a gear with logic tooth profiles which have zero relative curvature at many contact points. The gear therefore has higher durability and strength than involute gear. However, the logic tooth profile can be applied only to a helical gear due to point contact. Zhang et al. [14] presented a double involute gear. The tooth profile of the gear consists of two involute curves, which are linked by a transition curve and form a ladder shape of tooth. Ariga and Nagata [15] used a specific cutter with combined circulararc and involute tooth profiles to generate a new type of Wildhaber Novikov gear. This particular tooth profile can solve the problem of conventional W N gear profile, that is, the profile sensitivity to centre distance variations. Fong et al. [16] proposed a mathematical model for parametric tooth profile of spur gear using a giving equation of line of action. According to this model, it will be easier to manipulate the line of action when multi-segment tooth profile curves are used. The mathematical model can enhance the freedom of tooth profile design by combining the simple curves into the line of action. Kapelevich [17] presented a method of research and design of gears with asymmetric teeth that enables to increase load capacity, reduce weight, size and vibration level. Francesco and Marini [18] consider a low pressure angle profile for the drive side and a high 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 for tooth profile generation. Litvin and Tsay [19] applied the vector analysis, differential geometry, matrix transformation and meshing equation to develop mathematical models for describing tooth profiles and their geometric properties. Litvin et al. [20] proposed a basic algorithm for analysis and synthesis of gear drives based on replacement of the instantaneous line of contact of tooth surfaces by point contact. In this approach, design, generation and simulation of the meshing and contact of gear drives with favorable bearing contact and reduced noise are investigated. Chang and Tsay [21] also proposed a method for determining the complete mathematical model of non-circular gear tooth profiles, which is manufactured with shaper cutters, based on the inverse mechanism relation and the equation of motion. Tsai and Tsai [22] proposed a method of designing high-contact-ratio spur gears using quadratic parametric tooth profiles for the shorter addendum without undercut. Lee et al. [23] studied 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 variable motion transmission involving band drives, tape drives and time belts with a minimum slack. Yildirim and Munro [25] introduced a systematic approach to the design of tooth profile relief of both low and high-contact-ratio spur gears and its effects on transmission error and tooth loads. Chen and Tsai [26,27] described rack cutters with circular-arc profile teeth to generate elliptical gears which rotate about one of their foci. A mathematical model for the elliptical gears with circular-arc teeth was developed according to gear theory. Additionally, they presented 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 inherent characteristics of skew gears and design special gears. Danieli and Mundo [29] presented a new methodology which greatly increases the contact ratio between the teeth of non-circular gears, using a constant pressure angle for any given tooth. Mundo [30] presented a new concept of epicyclical gear train which is able to generate a variable gear ratio law. The basic mechanical configuration of the epicyclical gear train consists of three non-circular 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 this work. Particularly, the pitch circle of the pinion lies on the zero line of cosine curve; the tooth space takes a period of the curve; and the tooth addendum is decided by the amplitude of the curve. As shown in Fig. 1, the cosine 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 than that of involute gear.

3 S. Luo et al. / Mechanism and Machine Theory 43 (2008) Fig. 1. Cosine tooth profile and involute tooth profile. The remainder of this paper is organized into five sections. In Section 2, the generation principle of the cosine gear is studied. The mathematical models: the equation of the conjugate tooth profile and the equation of the line of action are deduced in Sections 3 and 4, respectively. Section 5 presents an example of the cosine gear drive in solid model for computerized simulation of its meshing process. A few characteristics of the new drive are analyzed and are compared with involute gears in this section. Finally, a conclusive summary of this study is given in Section Generation of the cosine tooth profile A tooth profile is the geometry of a tooth, which determines kinemical and dynamical properties of a gear drive. One of the reasons for widespread application of the involute gears in industry is due to their simple geometry. An involute is a curve traced by the end of a string as it unwinds from a base circle [31]. Unlike the involute tooth profile, the cosine tooth profile proposed in this work is generated by using the following methods: (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 a cosine curve taking the pitch curve as the baseline, as shown in Fig. 2. InFig. 2a, two coordinate systems are Fig. 2. Generation principle of the cosine tooth profile.

4 1546 S. Luo et al. / Mechanism and Machine Theory 43 (2008) used: a Cartesian coordinate system R 1 (X 1,O 1,Y 1 ) and a natural coordinate system R(X,O,Y). In the natural coordinate system, the pitch circle is taken as the X axis which can be expanded to a line as shown in Fig. 2b. Here, r 1 denotes the radius of the pitch curve of the cosine gear, and h represents the angle between the position 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. Then equation of the cosine curve can be expressed in the natural coordinate system as y ¼ h cosðbxþ From Fig. 2a, the polar equation of a point M on the cosine tooth profile can be expressed as q ¼ r 1 þ h cosðbxþ According to Fig. 2a, the following equation can be obtained. x ¼ r 1 h Substituting it into Eq. (2) gives q ¼ r 1 þ h cosðbr 1 hþ The period of the cosine profile, denoted as T, with respect to the angle h can be expressed as T ¼ 2p=br 1 Supposing that Z 1 is the tooth number of the gear, then the tooth space, i.e. the period of the cosine tooth profile, can also be represented by T ¼ 2p=Z 1 Eqs. (4) and (5) give Z 1 ¼ br 1 Substituting Eq. (6) into Eq. (3), the following equation can be obtained: q ¼ r 1 þ h cosðz 1 hþ Assuming that m is the modulus of the cosine gear, the radius of pitch circle can be then expressed as r 1 = mz 1 /2. Substituting it into Eq. (7), the cosine tooth profile can be represented in the Cartesian coordinate system by the following equations: ( x 1 ¼ mz 1 þ h cosðz 2 1hÞ sinðhþ y 1 ¼ mz 1 þ h cosðz ð8þ 2 1hÞ cosðhþ ð1þ ð2þ ð3þ ð4þ ð5þ ð6þ ð7þ 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 a family 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 for the envelope s existence. Simplified approaches to the solution of these problems have also been developed by Litvin [2] in the theory of gearing. As shown in Fig. 3, the notations C 1 and C 2 are respectively used for the generating and generated surfaces. P is the pitch point; and P 1 is the intersection point of the normal line of point M 1 on tooth profile C 1 with the pitch circle. If the normal line of point M on tooth profile C 1 gets across the pitch point P, then point M is a contact point between the two mating tooth profiles. In order to make M 1 become a contact point, the tooth profile C 1 must move to the dashed position after rotating an angle u 1, whilst point P 1 arrives at point P. Supposing that b is the angle between the tangent at point M 1 (x 1,y 1 ) on tooth profile C 1 and axis X 1, the following equation can be obtained.

5 S. Luo et al. / Mechanism and Machine Theory 43 (2008) Fig. 3. Conjugate tooth profile. b ¼ arctan dy 1 dx 1 Substituting Eq. (8) into Eq. (9) gives b ¼ arctan ½mZ 1=2 þ h cosðz 1 hþš tan h hz 1 sinðz 1 hþ ½mZ 1 =2 þ h cosðz 1 hþš hz 1 tan h sinðz 1 hþ In addition, making a perpendicular from point O 1 to line M 1 P 1 and supposing w is the angle between the perpendicular and line O 1 P 1, the following equations can be obtained according to Fig. 3. ( cos w ¼ x 1 cos bþy 1 sin b r 1 ð11þ u 1 ¼ p b w 2 Substituting Eq. (8) into Eq. (11) gives 1 u 1 ¼ arcsin ½mZ 1 þ 2h cosðz 1 hþš sinðh þ bþ b mz 1 In the following discussion, coordinate systems R 1 (X 1,O 1,Y 1 ), R 2 (X 2,O 2,Y 2 ) and R(X,O,Y) are designated as shown in Fig. 4. Coordinate system R is a fixed coordinate system whose origin O coincides with the pitch point P, while coordinate systems R 1 and R 2 are moving coordinate systems rigidly connected with gear 1 and gear 2, respectively. The position vector r M1 of contact point M on the tooth profile C 1 can be expressed in coordinate system R 1 as follows: r M1 ¼ x 1 i 1 þ y 1 j 1 According to the concept of relative motion, if a point moves along a given path of contact, its trace, as described by two gears rotating about each axis, will become the tooth profile of each gear. In addition, there should be a special relationship between the position of the contact point and the rotation displacement of the driving gear to obtain the conjugate tooth profiles [22]. From kinematics, the conjugate tooth profile C 2 can be deemed as the envelope of the tooth profile C 1 in coordinate system R 2 [2 4]. Using the coordinate transformation from R 1 to R 2 [16,19], the position vector of contact point M on tooth profile C 2 can be expressed as ð9þ ð10þ ð12þ ð13þ

6 1548 S. Luo et al. / Mechanism and Machine Theory 43 (2008) Fig. 4. Coordinate systems used for meshing analysis. r M2 ¼ M 12 r M1 þ r 12 ð14þ where r 10 and r 12 denote the position vectors of origin O 1 with respect to coordinate system R and R 2 respectively; r 02 represents the position vector of origin O with respect to coordinate system R 2 ; M 02 and M 12 denote the coordinate transformation matrices from coordinate system R to R 1 and R 2 respectively. Moreover, the following relationships exist: r 12 ¼ M 02 r 10 þ r 02 r 10 ¼ 0 r 1 r 02 ¼ r 2 sin u 2 r 2 cos u 2 M 02 ¼ cos u 2 sin u 2 sin u 2 cos u 2 M 12 ¼ cosðu 1 þ u 2 Þ sinðu 1 þ u 2 Þ sinðu 1 þ u 2 Þ cosðu 1 þ u 2 Þ Substituting above equations into Eq. (14) gives r M2 ¼ cosðu 1 þ u 2 Þ sinðu 1 þ u 2 Þ x1 sinðu 1 þ u 2 Þ cosðu 1 þ u 2 Þ y 1 þ a sin u 2 a cos u 2 or x 2 ¼ x 1 cosðu 1 þ u 2 Þ y 1 sinðu 1 þ u 2 Þþasin u 2 y 2 ¼ x 1 sinðu 1 þ u 2 Þþy 1 cosðu 1 þ u 2 Þ a cos u 2 ð15þ where a denotes the central distance between the centres of gears 1 and gear 2, a = r 1 + r 2. Substituting Eq. (8) into Eq. (15), equations of conjugate tooth profile C 2 in the Cartesian coordinate system can be expressed as

7 ( x 2 ¼ 1 mz 2 1 þ h cosðz 1 hþ sin h 1 þ 1 u1 i þ a sin u 1 i y 2 ¼ 1 mz 2 1 þ h cosðz 1 hþ cos h 1 þ 1 u1 a cos u 1 where i represents the gear ratio, i = u 1 /u 2. S. Luo et al. / Mechanism and Machine Theory 43 (2008) i i ð16þ 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 contact points between two mating tooth profiles in the fixed coordinate system [16]. According to the meshing theory [2 4], transforming the position vector r M1 of contact point M from coordinate system R 1 (X 1,O 1,Y 1 )to R(X,O,Y), the equation of the line of action between the mating tooth profiles C 1 and C 2 can be expressed as follows: r M ¼ M 10 r M1 þ r 10 where r M denotes the position vector of contact point M in coordinate system R; M 10 is the coordinate transformation matrix from coordinate system R 1 to R, which can be expressed as M 10 ¼ cos u 1 sin u 1 sin u 1 cos u 1 ð17þ Hence, Eq. (17) can be rewritten as r M ¼ cos u 1 sin u 1 x1 0 þ sin u 1 cos u 1 r 1 y 1 ð18þ Substituting Eq. (8) into Eq. (18), the equation of the line of action in the Cartesian coordinate system can be expressed as ( x ¼ 1 mz 2 1 þ h cosðz 1 hþ sinðh u1 Þ y ¼ 1 mz 2 1 þ h cosðz 1 hþ cosðh u1 Þ 1 mz ð19þ Evaluation of the cosine drive In this section, a study of a set of configurations, where the tooth profiles can be visualized, is presented. In addition, a further evaluation of the newly designed gears is made. Data compiled from industrial experience and laboratory experiments reveal two high stress areas that generally are the sources for gear failures. These two areas are root fillet and the contact surface of the teeth [33]. Based on the above concerns, both bending and contact stresses of the cosine gears are analyzed. Two gear characteristics, sliding coefficient and contact ratio, are also discussed and a comparison study of all these characteristics with the involute gears are carried out Solid model and computerized simulation Before the tooth profiles can be determined, it is necessary to specify a few parameters for the cosine spur gear drive of the case: (i) the modulus m is 3 mm; (ii) the tooth number Z 1 is 13; (iii) the tooth number Z 2 is 27; and (iv) the tooth addendum h is 3 mm. The meshing surfaces can be constructed from Eqs. (8) and (16), and the constraints of continuity Eqs. (10), (12) and (19). By using solid modelling software Pro/E and a parametric 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 contact conditions, a computerized simulation of the meshing process was carried out by using the mechanism module of Pro/E. The following behaviors were observed from the simulation:

8 1550 S. Luo et al. / Mechanism and Machine Theory 43 (2008) 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 gear ratio 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] Contact and bending stresses This section presents the analysis of contact and bending stresses of a cosine gear drive, and a comparison with those of an involute gear drive. The stress results presented in this paper are obtained by using the FE program ANSYS. The numerical computations have been performed for the cosine gear drive with the following design parameters: Z 1 =25,Z 2 = 40, m = 3 mm. The torque applied to the pinion was 98.8 N m. Properties of materials are: l = 0.29, E = 204 GPa. The involute gear drive uses the same set of specifications for comparison. Two models of contacting teeth based on the real geometry of the pinion and its conjugate profile have been developed by Pro/E and then transferred to ANSYS for stress analysis. The finite element models that consist of the loaded teeth and the neighboring teeth are shown in Fig. 7. Two sides of each model sufficiently far from the fillet are chosen to justify the rigid constraints applied along the boundaries. A large enough part of the wheel below the tooth is chosen for the fixed boundary. This model was constructed using 8-node isoparametric elements. There are 1064 elements and 3080 nodes in each transverse section. Two options related to the contact problem, small sliding and no friction, have been selected. Fig. 8 shows the contour plot of Von-Mises stresses. 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.

9 S. Luo et al. / Mechanism and Machine Theory 43 (2008) Fig. 7. Models applied for finite element analysis. Fig. 8. Stress distribution of cosine gear drive. Fig. 9. Stress distribution of involute gear drive.

10 1552 S. Luo et al. / Mechanism and Machine Theory 43 (2008) analyzed. The bending stresses obtained in the fillet of the contacting tooth side are considered as tension stresses, and those in the fillet of the opposite tooth side are considered as compression stresses [34]. The numerical 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 involute gear; (ii) the tension bending stress of the cosine gear is 33.86% less than that of the involute gear, and the compression bending stress is reduced by 34.33% in comparison with the involute gear. Therefore, application of the cosine tooth profile can reduce both contact and bending stresses Sliding coefficient Sliding coefficient is a measure of the sliding action during the meshing process. A lower coefficient will have greater transmission efficiency due to the less friction. The sliding coefficient is the ratio of the relative sliding velocity to the velocity of the contact point on each tooth profile while the gears are in mesh and the 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, and point H is the intersection point of the normal n n with the central line O 1 O 2. According to Ref. [35], the sliding coefficients of the cosine gear and driven gear, U 1 and U 2, can be expressed as Table 1 Maximum bending and contact stresses (Unit: MPa) Gears Contact stress Bending stress (tension) Bending stress (compression) Cosine gear Involute gear Fig. 10. Relative sliding of the cosine gear drive.

11 8 >< U 1 ¼ 1 >: U 2 ¼ 1 r 2 b r 1 i þb 21 r 1þb r 2 b i 12 where b ¼ PH, i 12 ¼ 1 i 21 ¼ r 2 r 1. According to Eq. (19), the slope, k, of the normal n n can be expressed as k ¼ dx dy ¼ dx=dh mz1 dy=dh ¼ þ h cosðz 2 1hÞ ð1 u 0 1 Þ cosðh u 1 Þ hz 1 sinðh u 1 Þ sinðz 1 hþ mz 1 þ h cosðz 2 1hÞ ð1 u 0 1 Þ sinðh u 1 Þþhz 1 cosðh u 1 Þ sinðz 1 hþ where u 0 1 is the derivative of u 1 with respect to h, which can be obtained as follows from Eqs. (12) and (10): u 0 1 ¼ du mz1 1 dh ¼ þ h cosðz 2 1hÞ ð1 þ b 0 q Þ cosðh þ bþ 2hz 1 sinðz 1 hþ sinðh þ bþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 2 z 2 1 mz 1 þ h cosðz b 0 ð22þ 2 2 1hÞ sin 2 ðh þ bþ b 0 ¼ db mz1 2 ¼ þ h cosðz 1hÞ sec 2 h 2h 2 z 2 1 sin2 ðz 1 hþ sec 2 h þ h 2 z 3 1 tan h½sin2 ðz 1 hþ sinðz 1 hþ cosðz 1 hþš dh mz 1 þ h cosðz 2 2 1hÞ hz1 tan h sinðz 1 hþ þ hz2 mz 1 1 þ h cosðz 2 1hÞ ½sinðz1 hþþtan 2 h cosðz 1 hþš mz 1 þ h cosðz 2 ð23þ 2 1hÞ hz1 tan h sinðz 1 hþ As shown in Fig. 10, the vertical coordinate, b, of the point H in coordinate system R(X,P,Y) can be expressed as b ¼ kx k þ y k where (x k,y k ) denote the coordinates of the contact point K in coordinate system R(X,P,Y). Substituting Eqs. (19) and (21) into Eq. (24) gives mz 1 2 b ¼ þ h cosðz 1hÞ 2 ð1 u0 1Þ sinðh u1þ cosðh u Þ hz mz þ h cosðz 2 1hÞ sin 2 ðh u 1 Þ sinðz 1 hþ mz 1 þ h cosðz 2 1hÞ ð1 u 0 1 Þ sinðh u 1 Þþhz 1 cosðh u 1 Þ sinðz 1 hþ þ 1 2 mz 1 þ h cosðz 1 hþ cosðh u 1 Þ 1 2 mz 1 The sliding coefficients, U 1 and U 2, 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 sliding coefficients of the two gears according to Eqs. (20) and (25). Fig. 11 shows the simulation results of the sliding coefficients of the cosine gear drive with design parameters: Z 1 = 35, Z 2 = 75, m = 3 mm. The sliding coefficients of the involute gear drive under the same parameters are also drawn in Fig. 11 according to Yan et al. [36]. The following conclusions can be made from these numerical results: (i) the sliding coefficients of cosine gear drive is much less than that of involute gear drive, (ii) the variation of the sliding coefficients of cosine gear drive is much smaller than that of involute gear drive. Therefore, application of cosine tooth profile helps to reduce the sliding coefficient and improves the transmission performance Contact ratio S. Luo et al. / Mechanism and Machine Theory 43 (2008) The contact ratio of a gear drive is defined as the number of teeth being in mesh simultaneously, or the rotating angle of the gear between the starting and the end points of contact divided by the angle between every two teeth, which equals to 2p divided by the number of teeth [37]. The contact ratio should be greater than 1 in a gear drive for smooth transmission. As shown in Fig. 12, the contact ratio of cosine gear drive can be expressed as ð20þ ð21þ ð24þ ð25þ

12 1554 S. Luo et al. / Mechanism and Machine Theory 43 (2008) 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 Z 1ðu b u c Þ 2p where u b and u c are the values of rotation angle u 1 as x = x b and x = x c, respectively. These values can be calculated by Eq. (19). ð26þ

13 S. Luo et al. / Mechanism and Machine Theory 43 (2008) Table 2 Contact ratio of the cosine and the involute gear drive Tooth number Z 1 Tooth number Z 2 Modulus m Cosine gear drive Involute gear drive mm mm mm 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% less than that of involute gear drive. However, such cosine gear drives can be applied in field of gear pump since less 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 profiles that are capable of bearing high loads is still needed. The cosine gear drive proposed in this work is an attempt on this direction. The generation principle of the cosine tooth profile and the mathematical models of the cosine gear drive deduced from the meshing theory have been presented. The theoretical analysis and numerical calculation shows that the cosine gear drive has lower sliding coefficients, and the contact and bending stresses are reduced in comparison with the involute gear drive according to simulation results of the example FE model. Kinematical simulation of the meshing process of a given example also demonstrates that the mating tooth surfaces of the new drive are in mesh in line contact. Although the contact ratio of the proposed drive is less than that of the conventional involute gear drive, the preliminary results show that the cosine gear drive 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 other assembly 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 the authors. 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 , the Provincial Natural Science Foundation of Hunan under Grant No. 06JJ10008 and the Program for New Century Excellent Talents in University. These financial supports are gratefully acknowledged. The authors also sincerely appreciate the comments and modification suggestions made by the editors and anonymous referees. 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) [2] F.L. Litvin, Theory of Gearing, NASA Publication, Washington, [3] F.L. Litvin, A. Fuentes, Gear Geometry and Applied Theory, second ed., Cambridge University Press, New York, [4] D.B. Dooner, A.A. Seireg, The Kinematic Geometry of Gearing: A Concurrent Engineering Approach, John Wiley & Sons, New York, [5] A. Ishibashi, H. Yoshino, Design and manufacture of gear cutting tools and gears with an arbitrary profile, JSME International Journal 30 (265) (1987) [6] S.M. Vijayakar, B. Sarkar, D.R. Houser, Gear tooth profile determination from arbitrary rack geometry, Gear Technology 5 (6) (1988) [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) [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)

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