USING QUASI-COMPLEMENTARY CROWN GEARS

Nagaoka, Japan

Department of Mechanical and Production Engineering

Niigata University

Niigata, Japan

ABSTRACT In the manufacture of straight bevel gears, there is usually a need for some trial-and-error process by which a good tooth bearing is obtained. In this paper, a new straight bevel gear cutting method in which trial-and-error process is not necessary is proposed. The method can be applied at once without the necessity of generator alteration. Furthermore, each gear and pinion is generated by a newly introducedgquasi-complementary crown gearh. Gears made on an experimental basis showed the expected good tooth bearing. 1. INTRODUCTION The tooth surface of a straight bevel gear is a simple surface as compared with that of a spiral bevel gear; nevertheless, trial-and-error process is still employed to obtain a good tooth bearing in the straight bevel gear manufacturing. The authors think the trial-and-error process actually covers the defect of gear cutting method. The gear cutting method based on the kinematic theory employs the pitch cone of the generated gear and the pitch plane of the complementary crown gear which are arranged to roll with each other. The straight bevel gear cutting method has been developed by Gleason Works [1, 2]. However, the method using the generator such as the Gleason No. 14 straight bevel gear generator, employs the members which are not arranged to roll. As a result, the real tooth surfaces generated in machining are slightly different from the theoretical surfaces and the tooth bearing is unsuitable for practical use when those surfaces are meshed. The unsuitable tooth bearing can be corrected by the trial-and-error process of the skilled worker. In this paper, a new method for cutting straight bevel gears is proposed. The method employs the pitch cone of the generated gear and the pitch cone of a newly introducedgquasi-complementary crown gearhwhich are arranged to roll with each other and keeps the kinematical strictness of gear tooth generation. The pitch cone angle of the quasi-complementary crown gear is designated strictly as 90ß minus the root angle of the gear to be cut. The tool surface of the quasi-complementary crown gear is the plane with pressure angle ¿. The plane tool surface produces the modified tooth surface with depthwise tooth taper. Besides, angle ¿ determines the position of tooth bearing on the generated gear tooth surface. In the previous papers [3, 4], a method using a quasi- complementary crown gear in spiral bevel gear cutting was proposed. The quasi-complementary crown gear can be realized by the Gleason No. 116 spiral bevel gear generator and can generate only the modified tooth surface with depthwise tooth taper. However, the position of tooth bearing was controlled by another technique. The proposed method in this paper can be used in combination with the usual crowning process in pinion machining. The process of this method is the same as the usual process except that the ratio of roll and the pressure angle of the tool are theoretically determined taking the gear generation method and the position of tooth bearing into consideration respectively, as stated above. This method can be applied at once without the necessity of any alteration of the usual generator. 2. QUASI-COMPLEMENTARY CROWN GEAR In general, the complementary crown gear which generates the straight bevel gear has a depthwise tooth taper. Therefore, it is necessary for the tool which simulates the crown gear tooth to move tooth-depthwise, namely, perpendicularly to the cradle plane [5] theoretically. This tool movement is in need of the mechanism of tool inclination to cope with the various root angles of the gear to be cut and complicates the machine, so that the workpiece is set to the Fig. 1 Relations between three gears Fig. 2 Clearance between tooth surfaces machine with more inclination angle than its pitch cone angle in practice to get around the machine complexity. More inclination angle of the workpiece is the root angle Æf of the gear to be generated. However, this additional inclination of workpiece breaks the kinematical strictness of gear tooth generation. In other words, the arrangement that the pitch cone of the workpiece and the pitch plane of the complementary crown gear roll on each other is broken by this workpiece inclination. The broken arrangement is strictly restored by the quasi-complementary crown gear as follows: First, we call the complementary crown gear the crown gear for short. Next, we imagine that the crown gear with the plane tooth surface xc generates both the workpiece and gear-I with a large pitch cone angle at the same time as shown in Fig. 1. The pitch plane of the crown gear lies between the workpiece and gear-I, keeps in contact with each pitch cone, and rolls on each cone [6]. The pitch cone angle of the workpiece is Â0 and the large pitch cone angle of gear-I is ( 90ß-Æf ). The ratio of roll between the crown gear and workpiece is 1/sinÂ0 from the condition of roll. The ratio of roll between the crown gear and gear-I is 1/cosÆf . Therefore, the ratio of roll i between the gear-I and workpiece is: i = cosÆf / sinÂ0 (1) These three gears are conjugate gears, so that gear-I can generate the same gear with pitch cone angle Â0 that the crown gear generates. The gear-I is a straight bevel gear with the tooth surface Fig. 3 Quasi-complementary crown gear and workpiece xT of octoid profile. The tooth surface xT is very close to a plane because the pitch cone angle of gear-I is close to 90ß. Figure 2 shows the surface xT is in contact with the plane tooth surface xc of the crown gear at point Q. There is a gap between xT and xc with the exception of point Q. The gap is very small in the neighborhood of point Q. Therefore, the real plane tool surface x substitutes for xT as the tooth surface of gear-I. A part of the substituted tool plane x in the neighborhood of point Q can generate a conjugate tooth surface in the workpiece. The gap between the tooth surfaces away from point Q is not so large that a part of the tool plane x away from point Q can generate a modified tooth surface. In this study, the gear-I with tool plane x is called the quasi-complementary crown gear, namely, the quasi-crown gear for short, which generates the tooth surface of the straight bevel gear with modified profile and depthwise tooth taper. The characteristics of the quasi-crown gear is as follows: 1. The pitch cone angle is ( 90ß-Æf ). 2. The ratio of roll is cosÆf / sinÂ0. 3. The tool surface is a plane as usual. 4. The generated tooth profile becomes the modified one. 5. The pressure angle of the quasi-crown gear is theoretically determined relating to the position of tooth bearing as state in next section. Figure 3 shows the relation between the quasi-crown gear and workpiece. In the machining of workpiece, the cradle axis is the quasi-crown gear axis, the set angle of the workpiece axis is angle (Â0 - Æf ), and the straight sided tool is employed as usual. However, the setting of the ratio of roll i is only different from the usual setting. From above, it is clear that the quasi-crown gear is realized easily by the employment of an ordinary gear generator such as the Gleason No. 14 straight bevel gear generator which is the most fundamental machine, without any alteration. Fig. 4 Engagement of teeth in transverse plane Fig. 5 Pressure angle ¿ of quasi-complementary crown gear 3. TOOTH BEARING AND PRESSURE ANGLE In the proposed method, a pair of quasi-crown gears is employed instead of a crown gear except the case of miter gear generation. One is for gear generation and the other is for pinion generation. However, the crown gear is a fundamental gear in this method and is very useful for the analysis of the relationship between the position of tooth bearing and the pressure angle of each quasi-crown gear. Figure 4 shows the crown gear tooth, the pinion tooth, and the gear tooth each engage at point Q on the common transverse plane of the bevel gear. The kinematical meaning of point Q is the same as that of Fig. 2. That is, the parts of both the pinion tooth surface and gear tooth surface which are away from point Q are the modified non-conjugate tooth surfaces causing a small transmission error, and the parts of both the tooth surfaces in the neighborhood of point Q are the conjugate surfaces, so that the tooth bearing always appears about point Q when a pair of generated gears are meshed. Thus, the position of point Q on the crown gear tooth surface represents the position of tooth bearing on the meshed tooth surfaces, and besides, relates the tool pressure angle ¿ of each quasi-crown gear as shown Fig. 6 Coordinate systems of each gear in Fig. 5. The tooth surface xc of the crown gear which rotates about its axis by angle Ó is used as the plane tool surface x of the quasi-crown gear which rotates about its axis by angle Õ. The pressure angle ¿ of tool surface x is unknown. ¿ is determined as follows: Referring to Fig. 4, we can introduce the following equation according to the condition of meshing of spur gear kinematics: H = - L sin¿ccos¿c (2) where L is the moving length of tooth surface xc in the transverse plane, ¿c is the pressure angle of the crown gear and H (positive) is the height of point Q from pitch plane S toward addendum. H (negative) is toward dedendum. The length L can be expressed using the mean cone distance Rm and angle Ó: L = RmÓ (3) From Eqs. (2) and (3), the rotation angle Ó of the crown gear is: Ó = - H / (Rm sin¿ccos¿c ) (4) Three coordinate systems shown in Fig. 6 are introduced to analyze the relationship between xc and x. O-XYZ is fixed in space, O-xcyczc is attached to the crown gear and O-xyz is attached to the quasi-crown gear. O-xcyczc and O-xyz rotate by angles Ó and Õ about axis zc and axis z respectively. When both Ó and Õ are zero, axes xc and x agree with axis X, and yc agrees with axis Y which is the pitch surface generator [6]. In Y-Z plane, axis z is inclined by angle Æf which is the root angle of the gear to be cut. In each coordinate system, nc represents the unit surface normal of xc which is the simulated plane of the crown gear tooth surface as shown in Fig. 7. nc is changed in O-XYZ only by angle Ó expressed as a function of H, Rm, and ¿c. n represents the unit surface normal of x which is the real tool plane of the quasi -crown gear as shown in Fig. 8. In O-XYZ, the surface xc attached to O-xcyczc coincides with Fig. 7 Tooth surface of complementary crown gear Fig. 8 Tooth surface of quasi-complementary crown gear the surface x attached to O-xyz, namely, nc coincides with n. Therefore, the following equation is obtained: A(-Æf ) C(Õ) n(¿) = C(Ó) nc (5) where A and C are the coordinate transformation matrices and are as follows: 1 0 0 A(Æf) = m 0 cosÆf -sinÆf n 0 sinÆf cosÆf (6) cosÕ -sinÕ 0 C(Õ) = m sinÕ cosÕ 0 n 0 0 1 Two unknowns ¿ and Õ are determined as the solutions of Eq. (5). The solutions are as follows: ¿ = sin-1 (sinÓcos¿c sinÆf + sin¿c cosÆf) sinÓcos¿c cosÆf - sin¿c sinÆf (7) Õ = tan-1m \\\\\\\\\\\\\\\\\\\ n cosÓcos¿c According to Eqs. (4) and (7), the pressure angle ¿ of the quasi-crown gear is determined directly by H which is the position of tooth bearing. The pressure angle ¿' of mating quasi-crown gear is as follows: ¿' = sin-1(-sinÓcos¿c sinÆf' + sin¿c cosÆf') (8) Table 1 Dimensions of straight bevel gears Table 2 Dimensions of quasi-complementary crown gears where Æf' is the root angle of mating gear. Now, the crown gear is not actually employed in this method, so that the pressure angle ¿c of the crown gear can be omitted from Eqs. (7) and (8), replacing sinÓ with Ó. As a result, the following equation is introduced: aÔ2 + bÔ2 + c = 0 (9) where Ô = sin¿', a = - sinÆf cosÆf , b = sin(Æf -Æf')sin¿, (10) c = sinÆf 'cosÆf' sin2¿ + H sin2(Æf +Æf') / Rm The pressure angle ¿' is determined from Eq. (9) when Æf, Æf', H, Rm and an arbitrary angle ¿ are given. That is, the position of tooth bearing is determined by the designation of ¿' corresponding to ¿. In case of miter gear, for example, Æf = Æf' and H = 0 yield, so that the result of ¿' = ¿ is obtained. This means that, according to this method, the tooth bearing appears on the pitch cone (H = 0) without fail when a pair of gears is generated using the same straight sided tool with an arbitrary pressure angle. 4. AMOUNT OF PROFILE MODIFICATION As stated in section 2, there is a gap between the theoretical tooth Fig. 9 Backlash and kickout surface of gear-I with octoid profile and the tooth surface of the quasi-crown gear (see Fig. 2). The amount of gap becomes that of the profile modification because a small gap directly turns to the generated tooth surface. Furthermore, the amounts become essentially the definite amounts according to the root angle Æf of the gear to be cut. The amount of gap at arbitrary point on x is obtained from the following vector equation: x(u, v) + t n(¿) = xT(U, ³) (11) where U, ³, and u, v are the parameters to express the surface of xT and x respectively, t is the amount of gap or the amount of profile modification, n is the unit surface normal of plane x, and ¿ is the known constant as stated above. The formulation and analysis of the generated tooth surface such as worm gear surface, hypoid gear surface, spiral bevel gear surface, and straight bevel gear surface xT in this study, have been developed by many researchers [7-18] using the tool geometry, machine tool kinematics, differential geometry, theory of conjugate surface and technique of coordinate transformation matrix. 5. EXAMPLE OF MANUFACTURING STRAIGHT BEVEL GEARS The dimensions of the gears manufactured as an example are shown in Table 1. These dimensions are designated according to Gleason method. The root angle of the gear is very large angle 8ß51' and dedendum is twice the length of addendum. If the position of tooth bearing is designated as a point 1 mm apart from the pitch cone toward dedendum, the tooth bearing will appear at the middle of the working depth of the gears. Therefore, the gear manufacturing in two cases is carried out. One is the tooth bearing on the pitch cone (H = 0 mm) and the other is H = -1 mm on the gear tooth (H = +1 mm on the pinion tooth). The corresponding quasi-crown gear dimensions are shown in Table 2. The practical inspections of bevel gears are the check of the position of tooth bearing by the eye-measurement and the check of commonly calledgkickouthwhich is a difference between the backlashes measured in two cases shown in Fig. 9. It is desirable that the amount of kickout is zero. In case of H = 0 mm, the kickout is zero and tooth bearing appears Fig. 10 Amounts of profile modification [Êm ] (without crowning) apart from the middle of tooth depth as arranged previously. In caseof H = -1 mm on the gear tooth (H = +1 mm on the pinion tooth), the kickout is undesirably large value 0.07 mm, but the position of tooth bearing is the middle of tooth depth as arranged. The amount of kickout of the gears generated by this method can be calculated from the amount of profile modification. The amount of modification t is calculated in case of the manufactured gears and the results are shown in Fig.10. The points A, A', B, and B' in Fig. 10 nearly correspond to the points in Fig. 9 respectively. The profile modification causes the additional backlash and two times the amount of modification becomes the amount of kickout. Therefore, in case of H = 0 mm, (AC + B E = ) 10Êm of additional backlash minus (A'C + B'E = ) 10Êm is zero kickout which coincides with experimental result. In case of H = -1 mm on gear tooth and H = +1 mm on pinion, (A 14 + B 24 = ) 38Êm minus (A' 0 + B' 0 = ) 0Êm is 38Êm which is nearly one half of 0.07 mm of experimental result. There are good agreements in two cases. In case of H = -1 mm of both gear and pinion, (A 0 + B 24 = ) 24Êm minus (A' 14 + B' 0 = ) 14Êm is 10Êm, so that twice 10Êm will be the kickout. A pair of gears with H = -1 mm was manufactured, and the position of tooth bearing and the kickout were Fig. 11 Tooth bearing on gear tooth surface (Gear H = -1 mm, Pinion H = -1 mm) checked. Figure 11 shows the tooth bearing which is very close to the previous arrangement. The measured kickout is about 0.015 mm which is slightly smaller than the simulation result. 6. CONCLUSION In this paper, a new method for cutting straight bevel gears was proposed. In this method, the gear and pinion are generated by a correspondingly quasi-complementary crown gear which is a newly introduced tool gear instead of a usual complementary crown gear. The proposed method is a simple one and can be applied at once without any alteration of a usual gear generator or the necessity of trial-and-error process in manufacturing. This method is based on the kinematics of gear generation. Therefore, the ratio of roll, the pressure angle of the tool surface which is a quasi-complementary crown gear tooth surface, the machine setting of workpiece, and the amount of tooth profile modification can be theoretically determined. The usefulness of the proposed method was confirmed by some experiments of gear manufacturing. REFERENCES 1. Gleason Works, 1966,g20ß Straight Bevel Gear System,h Rochester, New York, U.S.A., No. SD4033B. 2. Gleason Works, 1972,gStraight Bevel Gear Design,h Rochester,New York, U.S.A., No. SD3004E. 3. 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S., 1994, "Surface Geometry of Octoid Straight Bevel Gear,hBGA Proceedings of the International Gearing Conference, Newcastle upon Tyne, U.K., pp. 353-358. [Back]