Mechanism and Machine Theory 38 (2003) 213–225 www.elsevier.com/locate/mechmt
New approach for the dimensional synthesis of gear basic surfaces Carlos Garcıa-Masi a a
b
a,*
, Jose I. Pedrero
b
Departamento de Ingenierıa Mec anica, Escuela T ecnica Superior de Ingenierıa Industrial, Universidad Polit ecnica de Cartagena, C/Dr. Fleming, s/n, 30202 Cartagena (Murcia), Spain Departamento de Mec anica, Escuela T ecnica Superior de Ingenieros Industriales, Universidad Nacional de Educaci on a Distancia (UNED), Ciudad Universitaria s/n, 28040, Madrid, Spain Received 3 December 2001; received in revised form 4 October 2002; accepted 21 October 2002
Abstract The authors present a new method for establishing the relations between the primitive surfaces and the teeth basic surfaces in a gear transmission. This new approach allows to make the synthesis not only by considering as known the gear pitch radius, or what is the same, its transverse module, as existing approaches, but also from the normal module in the pitch mean cone or the pinion pitch radius. There are two main factors that affect the size of the gears: the limit pressure angle and the mean tooth curvature or mean radius of the head cutter. This method allows to study the influence of the tooth normal module on the limit pressure angle, and the valid range of mean radius of the head cutter, according to the recommendation of Wildhaber, and their influence on the size of gears. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction The first stage in the design process of a gear transmission consists in the dimensional synthesis of the primitive basic surfaces and the tooth basic surfaces of pinion and gear. From the dimensional synthesis point of view, the geometric parameters can be structured in three categories: i(i) Transmission data: shortest distance between axes, E, and angle between gear shafts, c (ii) Parameters of the primitive basic surfaces: cylindrical (pitch cylinder radii, ri ) or conical (apexes location, di , pitch cone angles, ci , and pitch cone radii, ri ), where subscript i denotes the pinion (i ¼ 1) and the gear (i ¼ 2). *
Corresponding author. Tel.: +34-968-326-434; fax: +34-968-326-449. E-mail address:
[email protected] (C. Garcıa-Masi a).
0094-114X/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 0 2 ) 0 0 1 2 7 - 1
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(iii) Parameters of the teeth basic surfaces: helix or spiral angles in cylindrical or conical transmission, bi , and number of teeth on the teeth basic surfaces, Ni . Some methods for the dimensional synthesis have been described in literature [1–11]. They can be classified into two different groups, according to the design conditions of the transmission: (a) With pre-established location of the pitch point, given the parameters of the primitive basic surface––parameters of category (ii)––of pinion or gear. (b) With location of the pitch point, chosen parameters of the primitive basic surfaces (gear pitch cone radius) and of the teeth basic surfaces (pinion spiral angle and mean tooth curvature or mean radius of the head cutter). The methods of Dudley [1], Litvin [2] and Minkof [3] for hypoid gears, and Goldfarb [4] and Abadjiev [5] for spiroid gears, are included in group (a). Group (b) involves the works of Wildhaber [6–8], Litvin [9,10] and ANSI/AGMA 2005-C96 [11] for hypoid gears, in which both industrial face generation processes, milling and hobbing, are considered. In this work a new approach for the solution of the problem (b) is presented. From established values of the transmission data––geometric parameters of category (i)–– the pinion spiral angle, the mean tooth longitudinal curvature and the gear ratio, the solution is based on a nonlinearequation system. Once the gear ratio is defined, two main factors affect the size of the gears: the limit pressure angle and the mean radius of the head cutter. With this approach the problem may be also treated by considering as known the teeth normal module in the pitch mean cone or the pinion pitch radius, and not only from given values of the gear pitch radius or its traverse module. This treatment allows to study both the influence of the teeth normal module on the limit pressure angle and the range of mean radii of the head cutter valid according to the recommendation of Wildhaber, and consequently their influence on the size of the gears.
2. Background 2.1. Basic pitch surfaces For the model of the primitive basic surfaces, the relations between the surface coordinates and their geometric parameters are established. Considering coordinate systems S1 and S2 rigidly connected to pitch cones 1 and 2, respectively, as represented in Fig. 1, the primitive basic surfaces are described by the equations xi ¼ ui sin ci cos hi yi ¼ ui sin ci sin hi zi ¼ ui cos ci
ði ¼ 1; 2Þ
where ðui ; hi Þ are the surface coordinates. The surface unit normal is represented by
ð1Þ
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215
Fig. 1. Primitive basic surfaces.
ni ¼
Ni ; jNi j
Ni ¼
ori ori ohi oui
ði ¼ 1; 2Þ
ð2Þ
Eqs. (1) and (2) yield T
ni ¼ ½cos ci cos hi ; cos ci sin hi ; sin ci
The relations between the geometric parameters of the transmission and the primitive basic surfaces can be obtained from the necessary and sufficient condition of tangency of the pitch cones [3,5,9,10]. To derive the equations of tangency at the pitch point P , we represent the pitch cones in the fixed coordinate system Sf . The location and orientation of coordinate systems S1 and S2 respect to Sf is shown in Fig. 2. The tangency condition can be expressed as ð1Þ
ð2Þ
ðP Þ
rf ðu1 ; h1 Þ ¼ rf ðu2 ; h2 Þ ¼ rf ð1Þ
ð2Þ
ðP Þ
nf ðh1 Þ ¼ nf ðh2 Þ ¼ nf
ð3Þ ð4Þ
where r is the position vector, n the unit normal and subscript denotes the system the vectors are referred to. The coordinate transformation from S1 and S2 to Sf allows to represent in Sf the ð1Þ ð2Þ position vectors of the pitch cones of pinion and gear, rf and rf , and their respective unit ð1Þ ð2Þ normal, nf and nf , by the following vector functions:
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Fig. 2. Coordinate systems.
" ð1Þ rf
¼ "
ð2Þ rf
¼
# r1 cos h1 ; r1 sin h1 r1 cot c1 d1
" ð1Þ nf ðh1 Þ
¼
# r2 cos h2 þ E r2 sin h2 cos c a2 sin c ; r2 sin h2 sin c þ a2 cos c
cos c1 cos h1 cos c1 sin h1 sin c1 "
ð2Þ nf ðh2 Þ
¼
# ð5Þ
cos c2 cos h2 cos c2 sin h2 cos c þ sin c2 sin c cos c2 sin h2 sin c sin c2 cos c
# ð6Þ
where ri ¼ ui sin ci is the radius of the pitch cone at point P (i ¼ 1; 2), being ai ¼ ri cot ci di , and d1 and d2 the distances represented in Fig. 2. With Eqs. (5) and (6), the system of vector equations (3) and (4) is equivalent to the following equation system: ðP Þ
r1 cos h1 ¼ r2 cos h2 þ E ¼ xf
ð7aÞ ðP Þ
r1 sin h1 ¼ r2 sin h2 cos c a2 sin c ¼ yf
ð7bÞ
ðP Þ
a1 ¼ r2 sin h2 sin c þ a2 cos c ¼ zf
ð7cÞ
ðP Þ
cos c1 cos h1 ¼ cos c2 cos h2 ¼ nxf
ð8aÞ ðP Þ
cos c1 sin h1 ¼ cos c2 sin h2 cos c sin c2 sin c ¼ nyf ðP Þ
sin c1 ¼ cos c2 sin h2 sin c þ sin c2 cos c ¼ nzf
ð8bÞ ð8cÞ
C. Garcıa-Masia, J.I. Pedrero / Mechanism and Machine Theory 38 (2003) 213–225 ðP Þ
ðP Þ
ðP Þ
ðP Þ
217 ðP Þ
ðP Þ
being xf , yf and zf the coordinates of pitch point P in reference system Sf and nxf , nyf and nzf the components of the unit normal at point P . From Eqs. (7) and (8) and taking into account that ð1Þ ð2Þ jnf j ¼ jnf j ¼ 1 are obtained the following nine relations: Relation 1: Eqs. (7a) and (8a), may be written as E ¼ r1 cos h1 r2 cos h2 ðP Þ
cos h1 ¼
nxf ; cos c1
ðP Þ
cos h2 ¼
nxf cos c2
consequently r1 r2 ðP Þ E¼ þ n ¼ r1 cos h1 r2 cos h2 cos c1 cos c2 xf
ð9Þ
Relation 2: Taking into account that a2 ¼ r2 cot c2 d2 and Eqs. (7b) and (8b), we obtain a2 sin c ¼ r2 sin h2 cos c r1 sin h1 d2 ¼ r2 cot c2 r2 sin h2 cot c þ r1
sin h1 sin c
ðP Þ
nyf sin h1 ¼ cos c1 ðP Þ
sin h2 ¼
ðnyf þ sin c2 sin cÞ cos c2 cos c
From these expressions of sin h1 and sin h2 , and taking into account Eq. (9), we obtain after some calculations ðP Þ
d2 ¼
Enyf r2 þ ðP Þ cos c2 sin c2 nxf sin c
ð10Þ
Relation 3: From ai ¼ ri cot ci di (i ¼ 1; 2) and taking into account Eqs. (7c) and (8c), we obtain after some calculations ðP Þ
d1 ¼ r1 cot c1 r2 sin h2 r2 cot c2 cos c þ sin h2 ¼
Enyf cot c r2 cos c þ ðP Þ cos c2 sin c2 nxf
sin c1 þ sin c2 cos c cos c2 sin c
Eliminating sin h2 and taking into account Eq. (9), is derived after some calculations d1 ¼ d1 ¼
E ðP Þ nxf
ðP Þ
ðnyf cot c þ cos c1 cot c1 Þ
r2 cos c2 sin c1
r1 E ðP Þ ðP Þ ðsin c1 nyf cot cÞ cos c1 sin c1 nxf
or ð11Þ
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Relation 4: From Eq. (8a) ðP Þ
nxf cos h2 ¼ cos c2 and replacing cos h2 in Eq. (7a) we obtain r2 ðP Þ ðP Þ xf ¼ E n ¼ r1 cos h1 cos c2 xf
ð12Þ
Relation 5: Dividing Eq. (7b) by Eq. (7a) and Eq. (8b) by Eq. (8a), we obtain ðP Þ
tan h1 ¼
yf
ðP Þ
xf
ðP Þ
¼
nyf
ðP Þ
nxf
This equation, with Eqs. (7b) and (12), yields ! E r2 ðP Þ ðP Þ nyf ¼ r1 sin h1 yf ¼ ðP Þ cos c2 nxf
ð13Þ
Relation 6: From Eq. (7c) and a1 ¼ r1 cot c1 d1 , is derived after some calculations ðP Þ
ðP Þ
zf ¼
nyf r2 sin c1 r2 sin h2 r1 sin h1 cos c E ðP Þ cot c ¼ sin c cos c2 nxf ð1Þ
ð14Þ ð2Þ
Relation 7: Taking into account the condition jnf j ¼ jnf j ¼ 1, the following relations are obtained: ðP Þ
nxf ¼
½cos2 c1 sin2 c ðsin c2 þ sin c1 cos cÞ2 1=2 sin c 2 1=2
¼
½cos2 c2 sin2 c ðsin c1 þ sin c2 cos cÞ sin c
ð15Þ
Relation 8: From Eqs. (8b) and (8c) ðP Þ
nyf ¼
ðsin c2 þ sin c1 cos cÞ sin c
ð16Þ
Relation 9: From Eq. (8c) ðP Þ
nzf ¼ sin c1
ð17Þ
2.2. Tooth longitudinal shapes Any plane containing the tooth longitudinal lines could be taken as a tangency plane for tooth surfaces meshing at pitch point P , since the normal relative velocity is zero, as represented in Fig. 3 V1 cos b1 ¼ V2 cos b2
ð18Þ
The spiral angles in pitch plane bi (i ¼ 1; 2) represent the longitudinal disposition of the teeth basic surfaces at pitch point P . To find bi it is necessary to know the gear ratio,
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Fig. 3. Tooth longitudinal lines.
m12 ¼
x1 r2 cos b2 N2 ¼ ¼ x2 r1 cos b1 N1
ð19Þ
where N1 and N2 are the number of teeth on pinion and gear, respectively. The generatrices of both pitch cones form an angle g that is represented by the equation cos g ¼ sð1Þ sð2Þ
ð20Þ
as shown in Fig. 3. From Fig. 2, the unit vectors sðiÞ (i ¼ 1; 2) are represented by the equations 2 3 ð1Þ sin c1 cos h1 or =ou O1 P 1 f sð1Þ ¼ ¼ ¼ 4 sin c1 sin h1 5 jO1 P j jorð1Þ =ou j 1 f cos c1 2 3 ð2Þ sin c cos h 2 2 orf =ou2 O2 P ¼ 4 sin c2 sin h2 cos c cos c2 sin c 5 ¼ ð2Þ sð2Þ ¼ jO2 P j jorf =ou2 j sin c2 sin h2 sin c þ cos c2 cos c
ð21Þ
ð22Þ
From Eqs. (20)–(22) and taking into account Fig. 3, we obtain for g ¼ b1 b2 cos g ¼
cos c þ sin c1 sin c2 cos c1 cos c2
sin g ¼
nxf sin c cos h1 cos h2 sin c ¼ sin c ¼ cos c2 cos c1 cos c1 cos c2
ð23Þ ðP Þ
ð24Þ
3. Synthesis of the basic surfaces. Application to hypoid gears Choosing several positions of pitch point P in the meshing zone, we can obtain solutions for any given value of the spiral angle and the mean tooth longitudinal curvature; however, this can not be calculated directly and requires the use of numerical techniques with iterative processes. The design of the basic surfaces is based on two equations which relate the angles ci of the pitch cones with the longitudinal disposition of the teeth basic surfaces, bi . The first equation is based on the condition of tangency of primitive basic surfaces at pitch point P and the relation between
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the longitudinal disposition of the teeth basic surfaces on the pitch plane. This equation can be expressed as f1 ðc1 ; c2 ; b1 Þ ¼ 0
ð25Þ
and it is applicable to all the types of hypoid gears. Details of derivation are given in Section 3.1. The second equation is described by f2 ðc1 ; c2 ; b1 ; b2 Þ ¼ 0
ð26Þ
and its expression is different depending on the conditions of synthesis of the hypoid gears, with face-milled tapered teeth or face-hobbed teeth of uniform depth. Details of derivation are given in Section 3.2. 3.1. Derivation of function f1 From Eq. (9) and the relations N1 mn ; 2 cos b1
r1 ¼
r2 ¼
N2 mn ; 2 cos b2
b¼
cos b1 N1 r2 ¼ cos b2 N2 r1
ð27Þ
we obtain 2E cos b1 ¼ mn
N1 N2 ðP Þ b nxf ¼ yA þ cos c1 cos c2
ð28Þ
Taking into account Eqs. (8) and (24), yA can be expressed as yA ¼ N1 cos h1 N2 cos h2 b ¼ ðN1 cos c2 þ N2 cos c1 bÞ
sin g sin c
ð29Þ
Now, for g ¼ b1 b2 we have cos b2 ¼ cos g cos b1 þ sin g sin b1
ð30Þ
where b¼
1 cos g þ sin g tan b1
ð31Þ
Finally, from Eq. (28) and the expression for r2 in Eq. (27) is derived mn r2 cos b2 E cos b1 ¼ ¼ 2 N2 ya The expression for f1 is given by f1 ðc1 ; c2 ; b1 Þ ¼ r2
N2 Eb ¼0 yA
ð32Þ
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Function f1 may be also expressed as a function of the normal module of teeth basic surfaces in the pitch mean cone or of the pinion pitch radius. From Eq. (28) f1 ðc1 ; c2 ; b1 Þ ¼ mn
2E cos b1 ¼0 yA
ð33Þ
From Eq. (28) and the expression for r1 in Eq. (27) f1 ðc1 ; c2 ; b1 Þ ¼ r1
N1 E ¼0 yA
ð34Þ
3.2. Derivation of function f2 The derivation of function f2 is different according to the method of generation used for the hypoid gears. Descriptions may be found in Refs. [6–11]. 3.2.1. Method 1: Hypoid gear with face-milled teeth To avoid singular points in the contact an orientation of the teeth basic surfaces should be chosen in such a way that the unit normal nðiÞ and the WildhaberÕs limit normal [6,10] are not collinear at any point of the contact zone. From this approach, the limit normal to the gear tooth surface at point P forms with the pitch plane the limit pressure angle U0 , which is given by [11]: tan c1 tan c2 ðr2 = sin c2 Þ sin b2 ðr1 = sin c1 Þ sin b1 tan U0 ¼ ð35Þ cos g ðr2 = cos c2 Þ þ ðr1 = cos c1 Þ The intersection of the gear tooth surface with the pitch plane represents a circle of radius rc , the mean radius of the head cutter, which is represented by [11] rc ¼
sec U0 ðtan b1 tan b2 Þ tan U0 ððtan b1 cos c1 =r1 Þ þ ðtan b2 cos c2 =r2 ÞÞ þ ðsin c1 =r1 cos b1 Þ ðsin c2 =r2 cos b2 Þ ð36Þ
Eqs. (35) and (36) considered together represent function f2 for gears with face-milled teeth. 3.2.2. Method 2: Hypoid gear with face-hobbed teeth of uniform depth The derivation of f2 in this case is based on the specific location of the head cutter, which, according to Fig. 4, yields the following expression [10]: rx cosðb2 dx Þ N1 cos b2 sinðb1 b2 Þ r2 rx sin c2 sinðb2 dx Þ N2 cos b1 sin c1 N1 cos b2 sin c2 cosðb1 b2 Þ ¼0 ð37Þ
f2 ðb1 ; b2 ; c1 ; c2 Þ ¼
where, Nx is the number of blades of the head cutter, dx the setting angle and rx the gear cutter radius, which are related by sin dx ¼
Nx r2 cos b2 N2 r x
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Fig. 4. Orientation of the head cutter axis.
3.3. Computational procedure The computational procedure for the solution of the nonlinear equation system require an iterative process that may be structured in two forms: (a) In echelon form, similar to those described in [9,10], with f1 ðc1 ; c2 ; b1 Þ ¼ 0 f2 ðc1 ; c2 ; b1 ; b2 Þ ¼ 0 (b) With a single function f2 ðc1 c2 ; b1 ; b2 Þ ¼ 0, and the relations (31), (32) and (34). The above equations relate the unknowns c1 , c2 and b2 . The input data are E, c, b1 , N1 , N2 , rc (and Nx for method 2: face-hobbed teeth) and r2 or mn or r1 , according to the expression (32), (33)or (34) used for f1 ðc1 ; c2 ; b1 Þ ¼ 0. In the case in which the pitch cone outer radius, ri , and the face width, Fi , were given instead of the pitch cone mean radius, the following relation must be used at each iteration: ri ¼ ri
Fi sin ci ; 2
ði ¼ 1; 2Þ
4. Results The computational procedure has been implemented in Matlab 5.0 and has been used to carry out the following examples and analysis: (1) Application of methods 1 and 2 with Eq. (32) for f1 using both echelon and single-function structures. (2) Study of the influence of the teeth normal module on the limit pressure angle and study of valid range of mean radii of the head cutter and their influence on the gears size, according to the recommendation of Wildhaber.
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4.1. Application of methods 1 and 2 For the application of method 1, the same input data as those in example of [11] have been considered. Table 1 shows those input data and the results obtained from the application of method 1 with both echelon and single-function structures. Similarly, method 2 has been applied to the same input data as those in example of [9], which are presented in Table 2, with the results from the above two structures. Table 1 Results of synthesis by method 1, computational procedures: echelon form and single function Input data
Output data
Hypoid offset, E: 38.1 mm Shaft angle, c: 90° Pinion mean radius, r1 (mm) Pinion number of teeth, N1 : 11 Pinion pitch cone angle, c1 (deg) Pinion spiral angle, b1 : 48° Gear spiral angle, b2 (deg) Gear number of teeth, N2 : 45 Gear mean radius, r2 (mm) Gear face width, F2 : 40.64 mm Gear pitch cone angle, c2 (deg) Gear pitch cone out radius, r2 : 136.79 mm Limit presure angle, U0 (deg) Cutter radius, rc : 114.3 mm Mean normal module, mn (mm)
Echelon form
Single function
37.1313 16.9481 30.0573 117.438 72.2385 )4.44 4.5174
37.0741 16.8793 30.017 117.208 72.3072 )4.492 4.5174
Table 2 Results of synthesis by method 2, computational procedures: echelon form and single function Input data Hypoid offset, E: 38.1 mm Shaft angle, c: 90° Pinion number of teeth, N1 : 13 Pinion spiral angle, b1 : 47° Gear number of teeth, N2 : 46 Gear face width, F2 :31.75 mm Gear pitch cone out radius, r2 : 108.03 mm Number of blades, Nw : 11 Cutter radius, rc : 74 mm
Output data
Pinion mean radius, r1 (mm) Pinion pitch cone angle, c1 (deg) Gear spiral angle, b2 (deg) Gear mean radius, r2 (mm) Gear pitch cone angle, c2 (deg) Mean normal module, mn (mm)
Echelon form
Single function
35.4468 27.9317 24.8090 94.2377 60.2064 3.7191
35.4359 27.794 24.849 94.2377 60.3553 3.7488
Table 3 Influence of the normal module on the limit pressure angle Mean normal module, mn (mm) Pinion mean radius, r1 (mm) Gear mean radius, r2 (mm) Gear spiral angle, b2 (deg) Limit presure angle, U0 (deg)
3.1 25.48 73.68 18.8 )11.149
3.5 28.768 85.648 23.153 )8.777
4.5 36.988 116.873 29.9657 )4.589
6 49.317 165.32 35.254 )0.87
6.33 52.03 176.1 36.027 )0.2979
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Table 4 Influence on the size of gears of mean radii of the head cutter Cutter radius, rc (mm) Pinion mean radius, r1 (mm) Gear mean radius, r2 (mm) Gear spiral angle, b2 (deg) mean normal module, mn (mm) Limit presure angle, U0 (deg)
79.375 18.94 52 4.86 2.3 )17.6
114.3 25.48 73.68 18.8 3.1 )11.149
177.8 37.51 118.81 26.95 4.564 )6.5
228.6 49.24 165 35.22 5.991 )4.383
320 67.84 239.63 39.2 8.2531 )2.9
4.2. Limit pressure angle and range of head cutter radii For these studies, the same input data as those for the application of method 1 in Section 4.1 (see Table 1) have been considered. Table 3 shows the results of the study of the influence of the normal teeth module on the limit pressure angle. Table 4 presents the valid range of mean radius of the head cutter, according to the recommendation of Wildhaber, and the influence of this radius on the size of the gears.
5. Conclusions A new approach for the dimensional synthesis of gear basic surfaces has been developed, whose solution can be structured in two ways: with two functions in echelon form or with a single function. The new method has high versatility thanks to several options for function f1 i.e., for condition f1 ðc1 ; c2 ; b1 Þ ¼ 0, which allow to find solutions for several different problems, as several different parameters of the basic surfaces are considered as known: the gear pitch radius, the teeth normal module in the mean cone or the pinion pitch radius. The limit pressure angle, whose value should be smaller than 16° according to the Wildhaber recommendation, establishes the range of valid sizes of the gears, for a pre-established value of the mean radius of the head cutter. If none of the previous parameters were given, the method yields the gears of minimum size, for the mean radius of the head cutter considered. Finally, the flexibility and generality of the new approach provide a structured solution of the problem, which improves significantly the convergence and accuracy of the results. Additionally, the possibility of considering several different forms for the function f1 ðc1 ; c2 ; b1 Þ ¼ 0 completes traditional approaches, as those presented in [10,11].
Acknowledgements The results discussed in this paper have been obtained in the course of the project ‘‘Tooth contact analysis of gears: influence of profile modifications’’, supported by the Spanish Council for Scientific and Technological Research (Ministry of Education and Culture), Ref. DPI20000427-C02.
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References [1] D.W. Dudley, Manual de Engranajes, Dise~ no, Manufactura y Aplicaci on de Engranajes, CECSA, Mexico, 1973, pp. 41–42. [2] F.L. Litvin, K.M. Petrov, V.A. Ganshin, The effect of geometric parameters of hypoid and spiroid gears on its quality characteristics, ASME Journal of Engineering for Industry (1974) 330–334. [3] K. Minkof, A new approach to basic geometry and classification of non-orthogonal gearing, in: Proceedings International Power Transmission and Gearing Conference (2), Chicago, 1989, pp. 593–598. [4] V.I. Goldfarb, Variants of spiroid gearing from pitch realization point of view, Gearing and Transmissions (1) (1995) 25–34. [5] V. Abadjiev, Aspects of the mathematic modeling of skew-axes gears whose tooth surfaces have a linear contact, Gearing and Transmissions (2) (1994) 24–31. [6] E. Wildhaber, Basic relationships of hypoid gears, American Machinist 90 (4) (1946) 108–111. [7] E. Wildhaber, Basic relationships of hypoid gears––II, American Machinist 90 (5) (1946) 131–134. [8] E. Wildhaber, Basic relationships of hypoid gears––III, American Machinist 90 (6) (1946) 132–135. [9] F.L. Litvin, W.S. Chaing, M. Lundy, W.J. Tsung, Design of pitch cones for face-hobbed hypoid gears, Journal of Mechanical Design 112 (1990) 413–418. [10] F.L. Litvin, Gear Geometry and Applied Theory, Prentice-Hall, Englewood Cliffs, 1994. [11] ANSI/AGMA 2005-C96, Design Manual for Bevel Gears, American Gear Manufacturers Association, Alexandria VA, 1996.