Comput Methods Appl Mech Engrg 200 (2011) 2363–2377 Contents lists available at ScienceDirect Comput Methods Appl Mech Engrg journal homepage: www.elsevier.com/locate/cma Computerized design of advanced straight and skew bevel gears produced by precision forging Alfonso Fuentes a,⇑, Jose L Iserte b, Ignacio Gonzalez-Perez a, Francisco T Sanchez-Marin b a b Department of Mechanical Engineering, Polytechnic University of Cartagena (UPCT), Spain Department of Mechanical Engineering and Construction, Universitat Jaume I, Castellon, Spain a r t i c l e i n f o Article history: Received 28 January 2011 Received in revised form 25 March 2011 Accepted April 2011 Available online 13 April 2011 Keywords: Bevel gears Straight bevel Skew bevel Forging TCA a b s t r a c t The computerized design of advanced straight and skew bevel gears produced by precision forging is proposed Modiﬁcations of the tooth surfaces of one of the members of the gear set are proposed in order to localize the bearing contact and predesign a favorable function of transmission errors The proposed modiﬁcations of the tooth surfaces will be computed by using a modiﬁed imaginary crown-gear and applied in manufacturing through the use of the proper die geometry The geometry of the die is obtained for each member of the gear set from their theoretical geometry obtained considering its generation by the corresponding imaginary crown-gear Two types of surface modiﬁcation, whole and partial crowning, are investigated in order to get the more effective way of surface modiﬁcation of skew and straight bevel gears A favorable function of transmission errors is predesigned to allow low levels of noise and vibration of the gear drive Numerical examples of design of both skew and straight bevel gear drives are included to illustrate the advantages of the proposed geometry Ó 2011 Elsevier B.V All rights reserved Introduction Bevel gears are used to transmit power between intersected axes and are mainly used for automobile differentials These gears are cut or forged from conical blanks and connect shaft axes generally at 90° although designs for different shaft angles can be also provided One of the most extended cutting technology for manufacturing straight bevel gears is the coniﬂexÒ method (coniﬂex is a registered trademark of The Gleason Works, Rochester, USA) This technology takes advantage of the Phoenix free form ﬂexibility and reduces setup time to a minimum [1] Coniﬂex straight bevel gears are cut with a circular cutter with a circumferential blade arrangement Nowadays, in the aim to look for more economical ways of manufacturing bevel gears, cutting technologies might be replaced for forming technologies [2–4] The forging process of gear manufacturing was developed during the 1950s decade for manufacturing bevel gears for automobile differentials, being stimulated by the lack of available gear cutting equipment at that time [5] The obtained precision was sufﬁcient for the automobiles of that period The development of the technology for electric discharge machining of dies for precision forging has allowed the manufacturing and application of forged bevel gears to be extended during the recent years It is based on the ⇑ Corresponding author E-mail address: alfonso.fuentes@upct.es (A Fuentes) 0045-7825/$ - see front matter Ó 2011 Elsevier B.V All rights reserved doi:10.1016/j.cma.2011.04.006 use of electrical discharged to remove material from the workpiece of the die Because the material removal is done point by point, surface modiﬁcations can be applied to the forging die and therefore applied to the tooth surfaces of the manufactured gear Following this idea, the reference geometry for the bevel gears is obtained computationally in order to get the die geometry that will achieve such geometry for the gears Among the different methods of forging, precision forging offers the possibility of obtaining high quality parts, complex geometries and good mechanical and technological properties [3] It allows a better material utilization in comparison to cutting, a reduction of the costs of cutting because of shorter cycle times and new possibilities concerning the tooth surface geometry of the forged gears Precision forging contributes as well to fulﬁll the demand of the production of highly loaded gears because of the ﬁber orientation which is favorable for carrying high oscillating loads [3] In this paper, the computerized design of straight and skew bevel gears with localized bearing contact is proposed, partially based on the ideas proposed by Professor Litvin et al [6] Modiﬁcations of the tooth surfaces of one of the members of the gear set are proposed in order to localize the bearing contact and predesign a favorable function of transmission errors The proposed modiﬁcations of the tooth surfaces will be computed by using a modiﬁed imaginary crown-gear and applied in manufacturing through the use of the proper die geometry The geometry of the die is obtained for each member of the gear set from their theoretical geometry obtained considering its generation by the corresponding 2364 A Fuentes et al / Comput Methods Appl Mech Engrg 200 (2011) 2363–2377 imaginary crown-gear Two types of surface modiﬁcation, whole and partial crowning, are investigated in order to get the more effective way of surface modiﬁcation of skew and straight bevel gears According to typical design practice, the face width of a bevel gear is generally chosen as one third of the outer pitch cone distance, Fw % Basic design of a bevel gear transmission The basic design parameters of a skew bevel gear transmission (considering the straight bevel gear transmission as a particular case of the mentioned above) are the module, m; the number of teeth of pinion and gear, N1 and N2, respectively; the shaft angle, R; the skew angle, b; and the pressure angle ad The gear ratio for bevel gears is given, as for other types of gears, by m12 ¼ x1 N ¼ ; x2 N x2 N ; m21 ¼ ¼ x1 N ð1Þ where x1 and x2 are the angular velocities of pinion and gear, respectively The pitch surfaces for bevel gears are cones The larger end of the pitch cone corresponds to the pitch diameter of the bevel gear Given the module and the number of teeth of pinion and gear, their pitch radii are determined by r p1 ¼ mN1 ; r p2 ¼ mN2 : ð2Þ Eq (2) can be used for straight and skew bevel gears Notice that the skew angle is not considered when determining the pitch radii for skew bevel gears The pitch angles of pinion and gear for any given shaft angle R are determined by c1 ¼ arctan sin R ; cos R þ m12 c2 ¼ arctan sin R : cos R þ m21 ð3Þ As shown in Fig 1, the pitch cones are contained in a sphere of radius Ro, the outer pitch cone distance, determined by Ro ¼ r p1 r p2 ¼ : sin c1 sin c2 Ro : ð6Þ Geometry of the imaginary generating crown-gear The proposed geometry for straight and skew bevel gears is achieved by considering an imaginary generating crown-gear as the theoretical generating tool The generating surfaces of the imaginary crown-gear will be modiﬁed to apply the required surface modiﬁcations to the to-be-generated bevel gear The number of teeth of the theoretical crown gear, Ncg, is given by Ncg ¼ 2Ro ; m ð7Þ where Ro is the outer pitch cone distance The number of teeth of the theoretical crown gear can be a decimal number Fig shows the applied coordinate systems for the theoretical generation of a straight or skew bevel gear by an imaginary crown-gear The crown gear is rotated around axis ycg and the being-generated bevel gear is rotated around axis zi Rotations of the being-generated bevel gear (straight or skew) and the imaginary crown-gear are related by wi ¼ wcg Ncg ; Ni ði ¼ 1; 2Þ; ð8Þ where wi and Ni are the angle of rotation and number of teeth of the pinion (i = 1) or the gear (i = 2), respectively, during their theoretical generation, and wcg is the corresponding angle of rotation of the generating crown-gear Two types of surface modiﬁcations, whole and partial crowning, will be investigated in order to get the more effective way to modify a forged bevel gear Fig shows a bevel gear tooth surface divided in nine zones wherein partial crowning (Fig 3(a)) is applied, or in four zones wherein conventional or whole parabolic crowning ð4Þ The face and root angles of the pinion and gear tooth surfaces, cF and cR, will be determined by cF 1;2 ¼ c1;2 þ m ; Ro cR1;2 ¼ c1;2 À hf m : Ro ð5Þ Here, and hf are the addendum and dedendum coefﬁcients, usually chosen to be 1.0 and 1.25, respectively Fig Pitch cones of bevel gears Fig Applied coordinate systems for theoretical generation of a straight bevel gear by an imaginary crown-gear A Fuentes et al / Comput Methods Appl Mech Engrg 200 (2011) 2363–2377 2365 Fig Areas of proﬁle and longitudinal crowning to be applied to straight and skew bevel gears; (a) partial crowning and (b) whole crowning (Fig 3(b)) is applied With respect to Fig 3(a), representing the application of partial crowning: (i) Zone is an area of the bevel gear tooth surface where proﬁle and longitudinal crowning are not applied (ii) Zones 1, 3, 7, and are areas of crowning in proﬁle and longitudinal directions (iii) Zones and are areas of crowning only in proﬁle direction (iv) Zones and are areas of crowning only in longitudinal direction When whole crowning of the gear tooth surface is applied, only four areas exist provided with crowning in longitudinal and proﬁle directions (Fig 3(b)) Those zones correspond to zones 1, 3, and in Fig 3(a), because areas 2, 4, 5, 6, and (Fig 3(a)) not exist when whole crowning is applied In order to achieve the surface modiﬁcations described above, a modiﬁed imaginary generating crown-gear will be applied for computerized generation of the geometry of the bevel gear 3.1 Geometry of the reference blade proﬁle The geometry of the imaginary generating crown-gear is based on the geometry of a reference blade proﬁle (Fig 4) Both sides of the blade proﬁle will be deﬁned in coordinate system Sc, ﬁxed to the blade, with its origin Oc placed on the middle of the segment Oa Ob , with axis xc directed along the pitch line and the axis yc directed towards the addendum height of the reference blade Auxiliary coordinate systems Sa and Sb (see Fig 4), with origins in Oa and Ob, are rigidly connected to the blade proﬁles that will deﬁne the driving and coast sides of the theoretical crown gear, respectively, and having their origins on the intersection of the pitch line with the respective blade proﬁles The axes ya and yb of coordinate systems Sa and Sb are directed along the reference straight proﬁle of the blade towards the addendum height of the blade The proﬁle of the blade is represented in coordinate systems Sa and Sb (see Fig 4) for left and right sides as 6 ra;b ðuÞ ¼ 6 Æapf ðu À u0 Þ2 u 7 7: ð9Þ Here, u is the blade proﬁle parameter, apf is the parabola coefﬁcient for proﬁle crowning, and u0 is the value of parameter u at the tangency point of the parabolic proﬁle with the corresponding ya or yb axis The upper and lower signs of apf correspond to representation of proﬁle geometry in coordinate systems Sa and Sb for the left and right sides, respectively The following conditions are established in order to apply proﬁle crowning by considering three parts for the active part of the reference blade proﬁle: If u > u0t , then apf ¼ apft and u0 ¼ u0t (area A of zones 1, 2, and in Fig 3(a)) If u u0t and u P u0b , then apf = and u0 = (area B of zones 4, 5, and in Fig 3(a)) If u < u0b , then apf ¼ apfb and u0 ¼ u0b (area C of zones 7, 8, and in Fig 3(a)) Parameters ðapft ; u0t Þ, and ðapfb ; u0b Þ control the crowning and position of areas A and C, respectively, for proﬁle crowning By considering u0t ¼ u0b ¼ and apft ¼ apfb we can take into account a conventional parabolic proﬁle for the reference blade proﬁle Similarly, by considering apft ¼ apfb ¼ we can take into account a conventional straight proﬁle for the reference blade proﬁle Blade proﬁles corresponding to the left and right sides, are represented in coordinate system Sc as Fig Reference blade proﬁle deﬁnition rc ðuÞ ¼ Mca;b ra;b ðuÞ; ð10Þ 2366 A Fuentes et al / Comput Methods Appl Mech Engrg 200 (2011) 2363–2377 where Mca;b cos ad Ç sin a d ¼6 0 Æ sin ad cos ad 0 0 Ç pm 0 3.2 Geometry of the imaginary generating crown-gear 7 7: ð11Þ Here, ad represents the pressure angle of the reference blade proﬁle, and the upper and lower signs correspond to the left and right blade proﬁles By considering Eqs (9)–(11), the reference blade proﬁles are represented in coordinate system Sc as 6 rc ðuÞ ¼ 6 Æapf ðu À u0 Þ2 cos ad Æ u sin ad Ç p4m ðiÞ Àapf ðu À u0 Þ2 sin ad þ u cos ad ðiÞ 7 7: ð12Þ As mentioned above, the upper and lower signs correspond to the left and right blade proﬁles, respectively The following ideas are applied for deﬁnition of the geometry of the generating crown-gear: The reference blade proﬁle is developed over the outer sphere deﬁned by the pitch cones of the to-be-generated pinion and gear, i.e., each point M of the reference blade proﬁle has its corresponding point M0 on the sphere with radius Ro, the outer pitch cone distance, as shown in Fig The pitch plane of the generating crown-gear is deﬁned by the pitch line of the reference blade proﬁle and the center of the sphere The geometry of the imaginary generating crown-gear will be obtained in coordinate system Scg, with origin in the center of the outer sphere and axis zcg containing the origin Oc of the reference blade proﬁle coordinate system Sc, and axes xcg and ycg parallel to axes xc and yc of the reference blade proﬁle, respectively (Fig 5) Fig Towards determination of the geometry of the imaginary generating crown-gear Fig For determination of the geometry of: (a) an imaginary straight generating crown-gear and (b) an imaginary skew generating crown-gear A Fuentes et al / Comput Methods Appl Mech Engrg 200 (2011) 2363–2377 2367 For deﬁnition of an imaginary straight crown-gear generating a straight bevel gear, any given point M0 of the reference blade proﬁle over the outer sphere is projected towards the origin Oh of coordinate system Sh, where Oh coincides with the center of the outer sphere (Fig 6(a)), deﬁning lines of the generating surface of a non-modiﬁed straight crown-gear For deﬁnition of a skew imaginary crown-gear generating a skew bevel gear, the projection point Oh, origin of coordinate system Sh, for any given point M0 , is not the center of the outer sphere but the tangent point with a circle deﬁned on the pitch plane of the generating crown-gear as shown in Fig 6(b), whose radius Rb is given by Rb ¼ Ro sinðbÞ; ð13Þ where b is the skew angle of the bevel gear The skew angle b is considered positive for a right-hand skew bevel gear (as shown in Fig 6(b)) and negative for a left-hand skew bevel gear Fig Towards application of longitudinal crowning For any given point M of the reference blade proﬁle with coorðMÞ ðMÞ dinates xc and yc in coordinate system Sc (see Fig 5), the corresponding point M0 on the outer sphere is deﬁned considering that: (i) point A0 on the outer sphere is obtained considering _ that it is in the pitch plane and (ii) the length of arc Oc A0 is equal ðMÞ to jxc j Point M0 on _ the outer sphere is obtained knowing that the length of arc A0 M measured over the great circle deﬁned by ðMÞ a plane normal to the pitch plane, is equal to jyc j An auxiliary coordinate system Sh is deﬁned for description of the geometry of the imaginary crown-gear for each point M0 of the reference blade proﬁle over the outer sphere Coordinate system Sh has the origin Oh in the center of the outer sphere for a crown-gear generating a straight bevel gear or as mentioned below for generation of skew bevel gears (see Fig 6) Axis yh is parallel to axis yc of the reference blade proﬁle, and axis zh is contained in the pitch plane of the crown gear with direction of the projection of vector Oh M on the pitch plane (Fig 5) A point P(u, h) on the imaginary generating crown-gear tooth surface (Fig 7) is deﬁned by proﬁle parameter u of the blade (that deﬁnes the reference point M on the reference blade proﬁle and corresponding point M0 on the outer sphere) and its longitudinal direction parameter h, measured from Oh on the projection line Oh M (Fig 6) For any given point M0 deﬁned by proﬁle parameter u of the reference blade proﬁle, angles ab and aa can be determined (Fig 5) Angle ab deﬁnes point M0 in coordinate system Sh Then, by considering angle aa and skew angle b, point M0 might be determined in coordinate system Scg (Fig 6) Angles ab and aa are given, for a nonmodiﬁed imaginary crown-gear, by (see Fig 5): _ A0 M yc ðuÞ ab ðuÞ ¼ ¼ ; Ro Ro ð14Þ _ aa ðuÞ ¼ Oc A0 xc ðuÞ ¼ : Ro Ro Fig Coordinate systems applied for bevel gear generation by an imaginary crown-gear ð15Þ 2368 A Fuentes et al / Comput Methods Appl Mech Engrg 200 (2011) 2363–2377 Fig Coordinate systems applied for TCA of bevel gears Longitudinal crowning is applied to the generating surfaces of the imaginary crown-gear by modifying angle aa with Daa, determined by Daa ðhÞ ¼ Æ ald ðh À h0 Þ2 : h ð16Þ Here, ald is the parabola coefﬁcient for longitudinal crowning, h is the longitudinal parameter, deﬁned as mentioned above, and h0 is the value of parameter h where modiﬁcations of the generating surface start By choosing appropriately different values for h0 and ald for the toe and heel areas of the crown-gear generating tooth surface, partial longitudinal crowning can be applied, as shown in Fig The upper sign in Eq (16) is applied for generation of the driving side of the bevel gear (left side) and the lower sign is applied for generation of the coast side of the bevel gear (right side) The modiﬁed angle aa will be denoted as aÃa and is given by Ã a ðu; hÞ a xc ðuÞ ald ðh À h0 Þ2 ¼ aa ðuÞ þ Daa ðhÞ ¼ Æ : Ro h ð17Þ The following conditions have to be observed in Eq (17) in order to provide longitudinal partial crowning to the surfaces of the imaginary generating crown-gear (Fig 7) Three areas will be considered: If h < h0t , then ald ¼ aldt and h0 ¼ h0t (area D of zones 1, 4, and in Fig 3(a)) If h P h0t and h h0h , then ald = (area E of zones 2, 5, and in Fig 3(a)) If h > h0h , then ald ¼ aldh and h0 ¼ h0h (area F of zones 3, 6, and in Fig 3(a)) Parameters ðaldt ; h0t Þ and ðaldh ; h0h Þ control the crowning and position of areas D and F, respectively, for longitudinal crowning By considering h0t ¼ h0h ¼ Ro À F w =2 and aldt ¼ aldh we can take into account a conventional longitudinal parabolic crowned surface for the imaginary crown-gear Similarly, by considering aldt ¼ aldh ¼ we can take into account a non-modiﬁed surface in longitudinal direction for the imaginary generating crown-gear According to the ideas described above, a point P(u, h) is given in coordinate system Sh by (see Fig 5) h sin a ðuÞ b rh ðu; hÞ ¼ 7: h cos ab ðuÞ ð18Þ Considering coordinate transformation from Sh to Scg as shown in Fig 6(b), the generating surfaces of an imaginary skew crown gear are given by rcg ðu; hÞ ¼ Mcgh ðaÃa ðu; hÞÞrh ðu; hÞ; where ð19Þ 2369 A Fuentes et al / Comput Methods Appl Mech Engrg 200 (2011) 2363–2377 Fig 10 Errors of alignments: (a) axial displacement of the pinion DA1, (b) axial displacement of the gear DA2, (c) change of the shaft angle DR and (d) shortest distance between axes DE cosðb À aÃa Þ À sinðb À aÃa Þ Rb cosðb À aÃa Þ 6 Mcgh ðaÃa Þ ¼ 6 sinðb À aÃa Þ 0 cosðb À aÃa Þ 0 @ aÃa dxc =du ; ¼ Ro @u @ ab dyc =du ; ¼ Ro @u 7 7: Ã Rb sinðb À aa Þ @a 2ald ðh À h0 Þh À ald ðh À h0 Þ ; ¼Æ @h h2 @ ab ¼ 0: @h Considering Eqs (18)–(20), Eq (19) can be represented by Rb cosðb À aÃa Þ À h cosðab Þ sinðb À aÃa Þ Step 7 h sinðab Þ rcg ðu; hÞ ¼ 6 R sinðb À aÃ Þ þ h cosða Þ cosðb À aÃ Þ 7: b b a a ð21Þ The derivative of the reference blade proﬁle (Eq (12)) with respect to the proﬁle parameter is obtained as dxc du Æ2apf ðu À u0 Þ cos ad Æ sin ad drc dy 7 ¼ c ¼ À2apf ðu À u0 Þ sin ad þ cos ad 5: du du 0 Step 2 ð26Þ ð27Þ ð28Þ The surface of the imaginary crown-gear is given by Eq (19) The normal will therefore be obtained by ð22Þ Here, the upper and lower signs correspond to the left and right blade proﬁles, respectively The derivatives to the angles aÃa and ab with respect to surface parameters u and h are also needed We recall that the mentioned angles are deﬁned by Eqs (17) and (14), respectively Their derivatives are given by @rh ¼ sin ab 5: @h cos ab Step ð25Þ The position vector of a point P(u, h) of the imaginary crown-gear in coordinate system Sh is given by Eq (18) Its derivatives with respect to surface parameters u and h, used below for determination of the normal to the imaginary crown-gear generating surfaces, are given by @rh dab ¼ h cos ab du 5; @u ab Àh sin ab ddu By considering b = in Eq (21), and therefore considering Rb = 0, the generating surfaces of an imaginary straight crown-gear are obtained in coordinate system Scg For determination of the equation of meshing, the normal to the generating surfaces, represented by Eq (21), is determined by the following steps: Step ð24Þ Ã a ð20Þ ð23Þ Ncg ðu; hÞ ¼ @rcg @rcg Â : @u @h ð29Þ Derivatives @rcg/@u and @rcg/@h are given by @rcg @Mcgh @rh ¼ rh þ Mcgh ; @u @u @u @rcg @Mcgh @rh ¼ rh þ Mcgh : @h @h @h ð30Þ ð31Þ 2370 A Fuentes et al / Comput Methods Appl Mech Engrg 200 (2011) 2363–2377 Table Details of coordinate system transformation from S2 to S1 Matrix Transformation Magnitude Axis Ml2 Mml Mnm Mfn M1f Rotation CCW Rotation CCW Rotation CCW Translation Rotation CCW /2 z2 zl zl zm xm xn N/A zf z1 p R + DR [ÀDE, DA2, ÀDA1]T /1 CCW = Counterclockwise; CW = Clockwise; N/A = Not applicable Table Main design parameters of two forged bevel gear drives Module, m (mm) Number of pinion teeth, N1 Number of gear teeth, N2 Shaft angle, R (deg) Skew angle, b (deg) Pressure angle, an (deg) Drive A (straight) Drive B (skew) 25 36 90° 0° 25° 25 36 90° 10° 25° Here, derivatives @rh/@u and @rh/@h are obtained by Eqs (27) and (28), respectively Derivatives @Mcgh/@u and @Mcgh/@h can be obtained by derivation of Eq (20), considering that angle aÃa ¼ aÃa ðu; hÞ Geometry of straight and skew bevel gears The proposed new geometry of straight and skew bevel gears is obtained by considering their computerized generation by an imaginary crown-gear, whose geometry has been described in the previous section A modiﬁed crown-gear will be used for the theoretical generation of the pinion whereas a non-modiﬁed crown-gear is used for generation of the gear Fig shows the coordinate systems applied for the theoretical generation of a bevel gear (straight or skew) by a crown-gear, and complements those coordinate systems illustrated in Fig Coordinate systems Scg and Si are rigidly connected to the generating imaginary crown-gear and the being generated bevel gear (i = for the pinion and i = for the gear), respectively Coordinate systems Sj, Sk, and Sl are auxiliary coordinate systems Angle ci is the pitch angle of the being-generated gear (Eq (3)) We recall that the imaginary crown-gear generating tooth surfaces are given by Eq (21) The bevel gear tooth surfaces are determined as the envelope of the family of generating crown-gear Table Studied cases of design with design characteristics for tooth surface modiﬁcations of the pinion member of the gear set according to Section Case (Non-modiﬁed) Case (Whole crowned) À1 À1 Case (Partial crowned) apft ¼ 0:0 mm apft ¼ 0:0004 mm apftop ¼ 0:001 mmÀ1 apfb ¼ 0:0 mmÀ1 u0t = 0.0 mm u0b = 0.0 mm h0t = 73.0584 mm h0h = 73.0584 mm aldt ¼ 0:0 mmÀ1 aldh ¼ 0:0 mmÀ1 apfb ¼ 0:0004 mmÀ1 u0t = 0.0 mm u0b = 0.0 mm h0t = 73.0584 mm h0h = 73.0584 mm aldt ¼ 0:0001 mmÀ1 aldh ¼ 0:0001 mmÀ1 apfbottom ¼ 0:0004 mmÀ1 u0t = 0.5517 mm u0b = À1.1034 mm h0t = 64.2984 mm h0h = 81.8184 mm aldtoe ¼ 0:001 mmÀ1 aldheel ¼ 0:001 mmÀ1 Table Studied misaligned conditions Condition a Condition b Condition c Condition d DA1 = 0.0 mm DA2 = 0.0 mm DE = 0.0 mm DR = 0.0 deg DA1 = 0.0 mm DA2 = 0.0 mm DE = 0.05 mm DR = 0.0 deg DA1 = 0.0 mm DA2 = 0.0 mm DE = 0.0 mm D R = 0.5 deg DA1 = 0.1 mm DA2 = 0.0 mm DE = 0.0 mm DR = 0.0 deg tooth surfaces in coordinate system Si, ﬁxed to the pinion (i = 1) or ﬁxed to the gear (i = 2), and represented as (Fig 8) ri ðu; h; wi Þ ¼ Mik ðwi ÞMkj Mjcg ðwcg ðwi ÞÞrcg ðu; hÞ: ð32Þ Here, cos wcg 6 Mjcg ðwcg ðwi ÞÞ ¼ sin wcg cos c i Mkj ¼ À sin ci sin ci 0 cos wi À sin w i Mik ðwi Þ ¼ 0 À sin wcg 0 cos wcg 0 cos c 07 7; 05 ð33Þ 07 7; 05 ð34Þ 0 07 7: 05 0 sin wi cos wi Fig 11 (a) Contact pattern and (b) function of transmission errors for case A1a (straight(A) non-modiﬁed(1) aligned(a) bevel gear drive) ð35Þ 2371 A Fuentes et al / Comput Methods Appl Mech Engrg 200 (2011) 2363–2377 Fig 12 Contact patterns for: (a) case A1b, (b) case A1c, (c) case A1d and (d) functions of transmission errors for previous cases of design Angles wcg and wi are the angles of rotation of the imaginary generating crown-gear and the being-generated bevel gear, related by wcg ðwi Þ ¼ Ni w: Ncg i ð36Þ @ri @Mik ðwi Þ ¼ Mkj Mjcg ðwcg Þrcg ðu; hÞ @wi @wi @Mjcg ðwcg Þ @wcg rcg ðu; hÞ: þ Mik ðwi ÞMkj @wi @wcg Here, The derivation of the bevel gear tooth surfaces is based on the simultaneous consideration of Eq (32) and the equation of meshing, À sin wi @Mik ðwi Þ 6 À cos wi ¼6 @wi cos wi ð37Þ À sin wcg @Mjcg ðwcg Þ 6 ¼6 cos wcg @wcg Eq (37) is represented in differential geometry [7] as @ri @ri @ri ¼ 0: Â Á @u @h @wi ð38Þ 0 0 À cos wcg 0 À sin wcg 0 ð41Þ 07 7; 05 ð43Þ Using Eqs (39) and (40), the equation of meshing is represented ð39Þ where Ncg(u, h) represents the normal to the imaginary generating crown-gear surface represented in coordinate system Scg (Eq (29)), and matrices L are Â matrices, which may be obtained by eliminating the last row and the last column of the corresponding matrices M (Eqs (33)–(35)) Derivative @ri/@ wi in Eq (38) is represented as ð42Þ @wcg Ni ¼ : @wi Ncg Here, @ri @ri ¼ Ni ðu; hÞ ¼ Lik ðwi ÞLkj Ljcg ðwcg ðwi ÞÞNcg ðu; hÞ; Â @u @h 07 7; 05 À sin wi ficg ðu; h; wi Þ ¼ 0: 0 as @Mik ðwi Þ Mkj Mjcg ðwcg Þ @wi # @Mjcg ðwcg Þ @wcg rcg ðu; hÞ ¼ 0: þ Mik ðwi ÞMkj @wcg @wi ficg ðu; h; wi Þ ¼ Ni ðu; hÞ Á ð44Þ 2372 A Fuentes et al / Comput Methods Appl Mech Engrg 200 (2011) 2363–2377 Fig 13 (a) Contact pattern and (b) function of transmission errors for case A2a (straight(A) whole-crowned(2) aligned(a) bevel gear drive) Fig 14 (a) Contact pattern and (b) function of transmission errors for case A3a (straight(A) partial-crowned(3) aligned(a) bevel gear drive) Simultaneous consideration of Eqs (32) and (44) allows determination of the geometry of a straight or skew bevel gear with modiﬁed geometry to be manufacturing by forging Computerized simulation of meshing and contact A new general purpose algorithm for tooth contact analysis (TCA) of gear drives has been developed and applied for tooth contact analysis of forged straight and skew bevel gears It is based on a numerical method that takes into account the positional study of the surfaces and minimization of the distances until contact is achieved A virtual marking compound thickness of 0.0065 mm has been used for determination of the contact patterns for all cases This algorithm for tooth contact analysis does not depend on the precondition that the surfaces are in point contact or the solution of any system of nonlinear equations as the existing approaches, and can be applied for tooth contact analysis of gear drives in point, lineal or edge contact as it will be shown below Alternative algorithms that can be used for tooth contact analysis are found in [7–9] All TCA analyses are conducted under rigidbody assumptions so that no elastic tooth deformation due to actual loading is considered when TCA results are shown 5.1 Applied coordinate systems Fig represents the applied coordinate systems for tooth contact analysis (TCA) of straight and skew bevel gears 5.2 Errors of alignment The errors of alignment considered are: (i) DA1 – the axial displacement of the pinion (Fig 10(a)), (ii) DA2 – the axial displacement of the gear (Fig 10(b)), (iii) DR – the change of the shaft angle R (Fig 10(c)), and (iv) DE – the shortest distance between axes of the pinion and the gear when these axes are not intersected but crossed (Fig 10(d)) The mentioned errors of alignment can also be observed in Fig Coordinate systems S1 and S2 are movable coordinate systems rigidly connected to the pinion and gear, respectively Angles /1 and /2 are the angles of rotation of the pinion and the gear, respectively Table shows details of coordinate transformation from S2 to S1 Transformation Mml is needed if pinion and gear have been generated following the same coordinate transformations, so that one of the members of the gear drive have to be rotated an angle p to face corresponding surfaces for tooth contact analysis 5.3 Discussion of obtained results Two bevel gear drives manufactured by forging, one straight and the other skew, with main design parameters shown in Table will be considered for tooth contact analysis Three cases of design for each gear drive, with design characteristics shown in Table 3, will be considered Those design characteristics correspond to the tooth surface modiﬁcations of the pinion member of the gear set A Fuentes et al / Comput Methods Appl Mech Engrg 200 (2011) 2363–2377 2373 Fig 15 Contact patterns for: (a) case A2b, (b) case A2c, (c) case A2d and (d) functions of transmission errors for previous cases of design The wheel member will be considered non-modiﬁed for all three cases of design Then, four misaligned conditions, shown in Table 4, will be investigated in order to check up the sensitivity of the contact pattern to the appearance of errors of alignment and the obtained function of transmission errors Condition a corresponds to the aligned gear drive Only results of tooth contact analysis concerning the driving side of the bevel gear transmission will be shown below Each investigated gear drive, case of design and misalignment condition will be denoted by three letters As an example, case B2a will correspond to a skew gear drive, with whole crowned surfaces, and aligned conditions (see Tables 2–4) Fig 11 shows the contact pattern and the obtained function of transmission errors for case A1a corresponding to a straight nonmodiﬁed and aligned bevel gear drive The bevel gear drive, under aligned conditions, has no transmission errors, and the contact pattern covers the whole surface of the teeth Fig 12 shows the contact patterns for cases A1b (12(a)), A1c (12(b)), and A1d (12(c)) Fig (12(d)) shows the obtained functions of transmission errors for previous cases of design The shortest distance between axes DE (misaligned condition b) and the change of shaft angle DR (misaligned condition c) not cause high transmission errors for the non-modiﬁed bevel gears However, function of transmission errors is very sensitive to the axial displacement of the pinion (misaligned condition d) and the axial displacement of the gear (not shown in this paper), having in these cases lineal functions of transmission errors that are the source of high levels of noise and vibration of the gear drive In order to absorb the lineal functions of transmission errors caused by errors of alignment in general and the axial displacements of pinion or gear in particular, Designs and (see Table 3) are proposed with whole-crowned and partial-crowned bevel gear tooth surfaces, respectively Figs 13 and 14 shows the contact patterns and the predesigned functions of transmission errors for cases A2a and A3a corresponding to a straight whole-crowned and aligned bevel gear drive (Fig 13) and to a straight partialcrowned and aligned bevel gear drive (Fig 14) Parabolic functions of transmission errors have been predesigned with levels of 8.5 arcsec for whole-crowned surfaces (Design 2) and 5.5 arcsec for partial-crowned surfaces (Design 3) Design (partial-crowning) allows the contact pattern to cover a larger area of the bevel gear contacting surfaces, by creating an area of no modiﬁcation of the tooth surfaces The main advantage of this geometry is that the lower the misalignment is, the bigger the contact pattern is obtained, allowing contact stresses to be reduced Fig 15 shows the contact patterns for cases A2b (15(a)), A2c (15(b)), and A2d (15(c)) Fig (15(d)) shows the obtained functions of transmission errors for previous cases of design For cases of design A2b and A2c, the contact pattern is kept inside de contacting surfaces although for case of design A2d, it is shifted towards the top edge of the wheel, and might cause high contact stresses All functions of transmission errors are obtained with parabolic shape, absorbing efﬁciently the lineal functions of transmission errors caused by errors of alignment for non-modiﬁed bevel gear tooth surfaces Fig 16 shows the contact patterns for cases A3b (16(a)), A3c (16(b)), and A3d (16(c)) Fig (16(d)) shows the obtained functions of transmission errors for previous cases of design For cases of design A3b and A3c, the contact patterns are localized inside the contacting surfaces, avoiding edge contacts, although for case of design A3d the contact pattern is slightly shifted towards the top edge of the pinion The lineal function of transmission errors 2374 A Fuentes et al / Comput Methods Appl Mech Engrg 200 (2011) 2363–2377 Fig 16 Contact patterns for: (a) case A3b, (b) case A3c, (c) case A3d and (d) functions of transmission errors for previous cases of design Fig 17 (a) Contact pattern and (b) function of transmission errors for case B1a (skew(B) non-modiﬁed(1) aligned(a) bevel gear drive) caused by the axial displacement of the pinion (misaligned condition c) is not completely absorbed, so that the partial proﬁle crowning is not working properly for this geometry The skew bevel gear transmission with main design parameters shown in Table is also designed using parameters shown in Table for three different cases of design Fig 17 shows the contact pattern and the obtained function of transmission errors for case B1a corresponding to a skew non-modiﬁed and aligned bevel gear drive The skew bevel gear drive with the proposed geometry, under aligned conditions, has no transmission errors, and the contact pattern covers the whole surface of the teeth as shown in Fig 17(a) Fig 18 shows the contact patterns for cases B1b (18(a)), B1c (18(b)), and B1d (18(c)) Fig (18(d)) shows the obtained functions of transmission errors for previous cases of design All misaligned conditions (from b to d) cause lineal functions of transmission errors The skew bevel gear drive is very sensitive to the change of shaft angle DR (misaligned condition c) and the axial displacement of pinion (misaligned condition d) and the axial displacement of the gear (not shown in this paper) A Fuentes et al / Comput Methods Appl Mech Engrg 200 (2011) 2363–2377 2375 Fig 18 Contact patterns for: (a) case B1b, (b) case B1c, (c) case B1d and (d) functions of transmission errors for previous cases of design Fig 19 (a) Contact pattern and (b) function of transmission errors for case B2a (skew(B) whole-crowned(2) aligned(a) bevel gear drive) In order to absorb those lineal functions of transmission errors caused by errors of alignment for the skew bevel gear drive, Designs and (see Table 3) are proposed also for this transmission, with whole-crowned and partial-crowned bevel gear tooth surfaces, respectively Figs 19 and 20 shows the contact patterns and the predesigned functions of transmission errors for cases B2a and B3a corresponding to a skew whole-crowned and aligned bevel gear drive (Fig 19) and to a skew partial-crowned and aligned bevel gear drive (Fig 20) A parabolic function of transmission errors with maximum level of arcsec has been predesigned for the whole-crowned skew bevel gear drive (Design 2) However, for the case of partial-crowned skew bevel gear drive (Design 3), a function of transmission error of arcsec is obtained taking advantage of an area of non-modiﬁed tooth surface due to partial crowning Again, Design (partial-crowning) allows the contact pattern to cover a larger area of the bevel gear contacting surfaces Fig 21 shows the contact patterns for cases B2b (21(a)), B2c (21(b)), and B2d (21(c)) Fig (21(d)) shows the obtained functions of transmission errors for previous cases of design Although for cases of design B2b and B2c the contact pattern is localized inside the contacting surfaces, avoiding undesirable edge contacts, when an axial displacement of the pinion occurs, the contact pattern is shifted towards the edge of the gear as shown in Fig 21(c) All functions of transmission errors are obtained with parabolic shape, 2376 A Fuentes et al / Comput Methods Appl Mech Engrg 200 (2011) 2363–2377 Fig 20 (a) Contact pattern and (b) function of transmission errors for case B3a (skew(B) partial-crowned(3) aligned(a) bevel gear drive) Fig 21 Contact patterns for: (a) case B2b, (b) case B2c, (c) case B2d and (d) functions of transmission errors for previous cases of design absorbing efﬁciently the lineal functions of transmission errors caused by errors of alignment for non-modiﬁed bevel gear tooth surfaces Fig 22 shows the contact patterns for cases B3b (22(a)), B3c (22(b)), and B3d (22(c)) Fig (22(d)) shows the obtained functions of transmission errors for previous cases of design For this design, the contact patterns are also localized inside the contacting surfaces, avoiding edge contacts, and the predesigned function of transmission errors is able to absorb the lineal functions of transmission errors caused by errors of alignment Conclusions The performed research work allows the following conclusions to be drawn: The computerized design of advanced straight and skew bevel gears produced by precision forging has been developed The developed approach takes into account modiﬁed tooth surfaces for the pinion member of the gear set in order to localize the bearing contact by the use of the proper die geometry A Fuentes et al / Comput Methods Appl Mech Engrg 200 (2011) 2363–2377 2377 Fig 22 Contact patterns for: (a) case B3b, (b) case B3c, (c) case B3d and (d) functions of transmission errors for previous cases of design The geometry of the die for the pinion is computed considering its theoretical generation by a modiﬁed imaginary crown-gear so that the applied surface modiﬁcations are applied directly to the pinion when it is precision forged The wheel member of the gear set remains non-modiﬁed Two types of surface modiﬁcation, whole and partial crowning, are investigated in order to get the more effective way of surface modiﬁcation of skew and straight bevel gears Whole crowning has been probed to be the most effective way of modiﬁcation of straight and skew bevel gear tooth surfaces, keeping the contact localized, and yielding low levels of parabolic functions of transmission errors Acknowledgments The authors express their deep gratitude to the Spanish Ministry of Science and Innovation (MICINN) for the ﬁnancial support of research projects Ref DPI2010-20388-C02-01 (ﬁnanced jointly by FEDER) and DPI2010-20388-C02-02 References [1] H.J Stadtfeld, Straight bevel gears on Phoenix machines using Coniﬂex tool, The Gleason Works, Rochester, New York USA, 2007 [2] K Kawasaki, K Shinma, Design and manufacture of straight bevel gear for precision forging die by direct milling, Mach Sci Technol 12 (2) (2008) 170– 182 [3] E Doege, H Nagele, FE simulation of the precision forging process of bevel gears, CIRP Ann – Manufact Technol 43 (1) (1994) 241–244 [4] Y.J Kim, N.R Chitkara, Determination of preform shape to improve dimensional accuracy of the forged crown gear form in a closed-die forging process, Int J Mech Sci 43 (3) (2001) 853–870 [5] T.A Dean, Z Hu, Net-shape forged gears: the state of the art, Gear Technol (2002) 26–30 [6] F.L Litvin, X Zhao, J Soﬁa, T Barrett, Advanced geometry of skew and straight bevel gears produced by forging, Patent No US 7.191.521, 2007 [7] F.L Litvin, A Fuentes, Gear Geometry and Applied Theory, second ed., Cambridge University Press, New York (USA), 2004 [8] G.I Sheveleva, A.E Volkov, V.I Medvedev, Algorithms for analysis of meshing and contact of spiral bevel gears, Mech Mach Theor 42 (2) (2007) 198–215 [9] A Bracci, M Gabiccini, A Artoni, M Guiggiani, Geometric contact pattern estimation for gear drives, Comput Methods Appl Mech Engrg 198 (17–20) (2009) 1563–1571 [...]... effective way of surface modiﬁcation of skew and straight bevel gears Whole crowning has been probed to be the most effective way of modiﬁcation of straight and skew bevel gear tooth surfaces, keeping the contact localized, and yielding low levels of parabolic functions of transmission errors Acknowledgments The authors express their deep gratitude to the Spanish Ministry of Science and Innovation... whole-crowned and partial-crowned bevel gear tooth surfaces, respectively Figs 19 and 20 shows the contact patterns and the predesigned functions of transmission errors for cases B2a and B3a corresponding to a skew whole-crowned and aligned bevel gear drive (Fig 19) and to a skew partial-crowned and aligned bevel gear drive (Fig 20) A parabolic function of transmission errors with maximum level of 7 arcsec... performed research work allows the following conclusions to be drawn: 1 The computerized
design of advanced straight and skew bevel gears produced by precision forging has been developed The developed approach takes into account modiﬁed tooth surfaces for the pinion member of the gear set in order to localize the bearing contact by the use of the proper die geometry A Fuentes et al / Comput Methods Appl Mech... of noise and vibration of the gear drive In order to absorb the lineal functions of transmission errors caused by errors of alignment in general and the axial displacements of pinion or gear in particular, Designs 2 and 3 (see Table 3) are proposed with whole-crowned and partial-crowned bevel gear tooth surfaces, respectively Figs 13 and 14 shows the contact patterns and the predesigned functions of. .. Litvin, X Zhao, J Soﬁa, T Barrett, Advanced geometry of skew and straight bevel gears produced by forging, Patent No US 7.191.521, 2007 [7] F.L Litvin, A Fuentes, Gear Geometry and Applied Theory, second ed., Cambridge University Press, New York (USA), 2004 [8] G.I Sheveleva, A.E Volkov, V.I Medvedev, Algorithms for analysis of meshing and contact of spiral bevel gears, Mech Mach Theor 42 (2) (2007)... B1b, (b) case B1c, (c) case B1d and (d) functions of transmission errors for previous cases
of design Fig 19 (a) Contact pattern and (b) function of transmission errors for case B2a (skew( B) whole-crowned(2) aligned(a) bevel gear drive) In order to absorb those lineal functions of transmission errors caused by errors of alignment for the skew bevel gear drive, Designs 2 and 3 (see Table 3) are proposed... transmission errors for cases A2a and A3a corresponding to a straight whole-crowned and aligned bevel gear drive (Fig 13) and to a straight partialcrowned and aligned bevel gear drive (Fig 14) Parabolic functions of transmission errors have been predesigned with levels of 8.5 arcsec for whole-crowned surfaces (Design 2) and 5.5 arcsec for partial-crowned surfaces (Design 3) Design 3 (partial-crowning) allows... arcsec has been predesigned for the whole-crowned skew bevel gear drive (Design 2) However, for the case of partial-crowned skew bevel gear drive (Design 3), a function of transmission error of 2 arcsec is obtained taking advantage of an area of non-modiﬁed tooth surface due to partial crowning Again, Design 3 (partial-crowning) allows the contact pattern to cover a larger area of the bevel gear contacting... (MICINN) for the ﬁnancial support of research projects Ref DPI2010-20388-C02-01 (ﬁnanced jointly by FEDER) and DPI2010-20388-C02-02 References [1] H.J Stadtfeld, Straight bevel gears on Phoenix machines using Coniﬂex tool, The Gleason Works, Rochester, New York USA, 2007 [2] K Kawasaki, K Shinma, Design and manufacture of straight bevel gear for precision forging die by direct milling, Mach Sci Technol... previous cases
of design For cases
of design A2b and A2c, the contact pattern is kept inside de contacting surfaces although for case
of design A2d, it is shifted towards the top edge of the wheel, and might cause high contact stresses All functions of transmission errors are obtained with parabolic shape, absorbing efﬁciently the lineal functions of transmission errors caused by errors of alignment