GEOMETRIC CONSTRAINTS AND INTERFERENCE-PROOF

GEOMETRIC CONSTRAINTS AND INTERFERENCE-PROOF CONDITIONS OF HELIXCURVE MESHING-WHEEL MECHANISM 1

Yangzhi Chen1, Jiang Ding1, Yueling Lv1 School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou, P.R. China E-mail address: [email protected]; [email protected]; [email protected] Received Month 0000, Accepted Month 0000

No. 00-CSME-00, E.I.C. Accession Number 0000

ABSTRACT In some recent papers, a gear mechanism named Space-Curve Meshing-Wheel (SCMW) has been proposed by the present authors. As a transmission between two conjugate curves through continuous point contact, it has advantages like a large transmission ratio, small size, light weight and high design flexibility. This paper presents the geometry of the Helix-Curve Meshing-Wheel (HCMW), the most common SCMW at present. The driven contact curve equation is optimized by removing a redundant length parameter. Then, both geometric constraints and interference-proof conditions are calculated in the following three aspects: contact curves, HCMW pair and HCMW train. A parameter selection of a novel reducer with HCMW trains is shown as a practical example. The theories in this paper are mainly applied to determine the feasibility range of the geometric parameters within the HCMW, and provide foundation to its industrial standardized production. Keywords: gear geometry; Space-Curve Meshing-Wheel; Helix-Curve Meshing-Wheel; geometric constraint; interference-proof condition; reducer.

CONTRAINTES GÉOMÉTRIQUES ET CONDITIONS NON-INGÉRENCE DU MÉCANISME D'ENGRENAGE COURBE HÉLICOÏDAL

RÉSUMÉ Dans certaines études récentes, un mécanisme d'engrenage nomméL'Engrenage Courbe Spatial a été proposé par les auteurs du présent article. Comme une transmission entre deux courbes conjuguées à travers un point de contact permanent, il présente les avantages d’avoir un taux important de transmission, de petite taille, poids léger et une grande flexibilité de conception. Dans cet article, nos études se concentrent sur la géométrie de L'Engrenage Courbe Hélicoïdal, Une Engrenage Courbe Spatial le plus courant àl'heure actuelle. L'équation de la courbe de contact entraînée est optimisée par la suppression d'un paramètre de longueur superflue. Ensuite, toutes les contraintes géométriques et les conditions noningérence sont calculées dans les trois aspects suivants: courbes de contact, paire de L'ENGRENAGE COURBE SPATIAl et train de L'Engrenage Courbe Spatial. A la fin de l’article, nous illustrerons nos idées par une selection de paramètres d'un nouveau réducteur àtrain de L'Engrenage Courbe Spatial est montrée comme un exemple pratique. Les théories dans cet article sont principalement appliquées pour déterminer la gamme de faisabilité des paramètres géométriques dans L'Engrenage Courbe Spatial, et jeter la base àsa production standardisée industrielle. Mots-clés : géométrie des engrenages; Engrenage Courbe Spatial; Engrenage Courbe Hélicoïdal; contraintes géométriques; conditions non-ingérence; réducteur.

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NOMENCLATURE distances from point OP to axis z and to axis x a ,b taper of driven contact curve C2 height of HCMW pair H max helix heights of driving and driven contact curves h1 , h2 transmission ratio i12 meshing radii of driving and driven contact curves m1 , m2 m2S , m2E initial and termination meshing radii of driven contact curve number of driven wheels N pitch coefficient of driving contact curve n fixed Cartesian coordinate, rotational Cartesian coordinate and o  xyz , o1  x1 y1 z1 , o1  11 z1 rotational cylindrical coordinate of driving wheels o p  x p y p z p , o2  x2 y2 z2 , o2  2 2 z2 fixed Cartesian coordinate, rotational Cartesian coordinate and rotational cylindrical coordinate of driven wheels pitch of driving contact curve p1 P2 , Pz 2 , Pr 2 generatrix pitch, axial pitch and radial pitch of driven contact curve maximum radius of HCMW pair Rmax radii of driving and driven tines r1 , r2 initial and terminal meshing points tS , tE difference from initial meshing point t numbers of driving and driven tines Z1 , Z 2 Greek symbols reference cone angle of driven contact curve 2 field angle of driven wheels S maximum field angle of driven wheel  max included angle between the kth and reference planes 1k included angle between adjacent pair planes  k 1k included angle between angular velocity vectors  helix angles of driving and driven contact curves 1 , 2t contact ratio   1 , 2 included angles of driving and driven contact curves

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INTRODUCTION

Gear geometry has been one of the most difficult subjects within gear research. In 1733, Camus’ Theorem [1] was proposed for the conjugation of tooth profiles. In 1765, Euler [2] expounded the relationship between the normal and principal curvatures of a surface. In 1962, Baxter [3] analyzed the basic geometric and tooth contact of hypoid gears. In 1980s, Litvin et al. [1] systematically proposed the gear geometry theory. In 1990s, computer aided analysis on gear drives got decent progresses [4]. In the past decade, Fong [5] proposed a mathematical model for universal hypoid generator to simulate primary spiral bevel and hypoid cutting methods; similar method for Klingelnberg's bevel gears was proposed by Lelkes et al. [6]; Dooner [7] proposed three laws of gearing to establish a generalized geometric theory

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comparable to the existing theory for planar kinematics; Puccio et al. [8] proposed the research approach without reference systems.

Fig. 1.

Space-Curve Meshing-Wheel Mechanism.

As shown in Fig. 1, an innovative gear mechanism named Space-Curve Meshing-Wheel (a.k.a. SCMW) was proposed by Chen et al. [9]. It was based on the space-curve meshing theory, instead of the classical space-surface meshing theory above. After its invention, the SCMW has been improved in aspects like meshing equations as proved in [9-14], contact ratio in [15], design criterions in [16] and manufacture technology in [17]. With a large transmission ratio, small size, light weight and high design flexibility, it has been used in some micro machines, like micro reducers in [11, 18]. A brief comparison between bevel gear and SCMW is shown in Table 1. Currently, the Helix-Curve Meshing-Wheel (a.k.a. HCMW) is the most common SCMW. Table 1. Brief comparison between bevel gear and SCMW. Gear Theoretical Foundation Working Components

Transmission Ratio Size Bearing Capability

Bevel Gear Space surface meshing Teeth surfaces Can be 0°-180°, Most likely 90° Small, commonly lower than 6 Conventional Conventional

Manufacture Process

Machining

Process Efficiency Main Application

Low Conventional machines

Phase margin

SCMW Space curve meshing Contact curves on the tines Any angle within 0°-180° Large, up to 50 Mini to conventional Relatively lower Rapid Prototyping (Selective Laser Melting) High Mini/Micro machines

The main aim of the paper is to present interference-proof conditions for the geometric parameters of the HCMW in following three aspects: contact curves, HCMW pair and HCMW train. Before that, geometric constraints are proposed in those three aspects for two proposes: descript the geometric parameters in the interference-proof conditions and systematically calculate them for industrial applications. Geometric parameter determination of a reducer with HCMW trains is provided as a practical example.

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GEOMETRIC CONSTRAINTS AND INTERFERENCE-PROOF CONDITIONS OF CONTACT CURVES

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The HCMW mechanism contains a pair of driving and driven wheels, as shown in Fig. 1. The wheels commit a transmission through tines symmetrically arranged around the wheel bodies. As the reacting parts within the HCMW, the contact curves on the tine surface are the core elements, as shown in Fig. 2.

Fig. 2.

Contact curves of driving and driven wheels.

2.1

Geometric Constraints of Driving Contact Curve The driving contact curve in the HCMW is a cylindrical helix. Its equation in o1  x1 y1 z1 is as Eq. (1) [9-12], and in the corresponding cylindrical coordinate, o1  11 z1 , as Eq. (2):  xM1  m1 cos t  1 (1)  yM  m1 sin t ,  1  zM  n  nt   M1  m1  1 , (2)  M  t  1  zM  n  nt where m1 is the meshing radius of the driving contact curve; n is the pitch coefficient, n  0 ; t is the parameter indicating the curve range, tS  t  tE . t S and t E are initial and termination meshing points, which satisfy Eq. (3) [15]: t  t  Z (3)  = E S 1 1, 2 where  is the contact ratio and Z1 is the driving tine number. Assuming the initial meshing point at

the plane xoy , zM1  n  ntS  0 , that is, ts   . Define t  t  tS  t   .

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Fig. 3.

Geometric constraints of driving contact curve.

As shown in Fig. 3, the cylinder enveloped by the helix is defined as the driving pitch cylinder. The geometric constraints of the driving contact curve are as Table 2: Table 2. Geometric constraints of driving contact curve. Geometric constraint Meshing radius

Symbol & Definition

m1   M  1

p1  2 n

Pitch Helix height

h1  zM1

Helix angle

1  arctan

Contact curve included angle

tE tS

 n  tS  t E 

n p1  arctan 2 m1 m1

1 

2 Z1

According to Eq. (2) and Fig. 3, when n  0 , the cylindrical helix is right hand screw; when n  0 , it is left hand screw. n  0 is set as an example in this paper, as the case when n  0 is similar. 2.2

Geometric Constraints of Driven Contact Curve The driven contact curve equation is attained in the space-curve meshing coordinates [12], as shown in Fig. 4. The transmission ratio is denoted as i12 , the distance from o p to z axis as a , the distance from o p to x axis as b , and the included angle between the angular speeds of the driving and driven wheels

as  , 0     .

Fig. 4.

Space-curve meshing coordinates.

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As axis z and axis z p are in the same plane, the center lines of the driving and driven wheels are also in the same plane. Therefore, the HCMWs can be divided into three cases according to the value of  : intersecting axes, vertical axes, and parallel axes. The equation of the driven contact curve in o2  x2 y2 z2 is as Eq. (4) [12], and in the corresponding cylindrical coordinate, o2  2 2 z2 , as Eq. (5): t    2  xM   m1  a  cos   n  nt  b  sin   cos i 12    2 t  ,  yM    m1  a  cos   n  nt  b  sin   sin i12   z  2   m  a sin   n  nt  b cos  1     M     2   m  a  cos   n  nt  b  sin  1  M t    2 .  M   i12   z  2   m  a sin   n  nt  b cos  1     M

(4)

(5)

2.2.1 Optimized Equations of Driven Contact Curves Two length parameters ( a and b ) and one angle parameter (  ) are in Eq. (5) to indicate the relative positions of the driving and driven wheels. However, as their center lines are in the same plane, only one length parameter ( a or b ) and one angle parameter (  ) are needed. The redundant length parameter ( b or a ) creates a distance between the driven contact curve and the coordinate o p  x p y p z p . To remove the redundant length parameter, the initial meshing point of the driven tine is supposed at the plane x p o p y p , that is, zM 2    m1  a  sin    n  ntS  b  cos  0 . (6) Solution of Eq. (6) and derivation of optimized driven contact curves equation are both related to the value of  , and their results are shown in Table 3. Table 3. Optimized equation of driven contact curve. Cases



Intersecting axis   0,  2, 

Vertical axis   2

Parallel axis   0, 

Solution of Eq. (6)

b   m1  a  tan 

a  m1

b0

Driven contact curve equations

  2  m1  a   nt sin   M  cos   t   2  M   i12   z  2  nt cos   M 

   2    nt  b   M t   2  M   i12   z  2  0  M

   2    m  a  1  M t   2  M   i12   z  2  nt  M

Curve shape

General conical helix curve

Planar Archimedes spiral

Cylindrical helixes

The driven contact curve in intersecting axis case is a general conical helix curve, as shown in Fig. 2. The driven contact curve in vertical axis case is a planar Archimedes spiral, which can be consider as a conical helix with zero-height, as shown in Fig. 5.

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Fig. 5.

Contact curves in vertical axis case.

The driven contact curves in the parallel axis cases are cylindrical helixes, which can be considered as conical helixes with invariable radii, as shown in Fig. 6. If all the parameters in Figs. 6(a) and 6(b) are equal, the shapes of driven contact curves are exactly the same, except their coordinate directions. However, their parameters cannot be equal simultaneously, which will be illustrated in Section 2.3.

Fig. 6.

2.2.2

Contact curves in parallel axis case: (a)   0 , (b)    .

Geometric Constraints of Driven Contact Curve

Fig. 7.

Geometric constraints of driven contact curve.

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As the driven contact curves in each case can be considered as conical helixes, the cone enveloped by the conical helix is defined as the driven pitch cone, as shown in Fig. 7. Geometric constraints of the driven contact curve are as Table 4. Table 4. Geometric constraints of driven contact curve. Value

Geometric constraint

Symbol & Definition

Meshing radius

m2   M 

Axial pitch

Pz 2  zM 

2 i12 n cos 

0

Radial pitch

Pr 2  M 

2 i12 n sin 

2 i12 n

Generatrix pitch

P2  Pr22  Pn22

Helical height

h2  zM 

nt cos 

0

Taper

C2  1: cot  2

1: cot  2



Reference cone angle

tan  2  tan 

 2  arctan tan 



Generatrix helix angle

  2

 m1  a 

2

cos 

2

2

Contact curve included angle

2 

P2

 nt sin 

n  nt  b

2 i12 n

2

2t  arctan

  2

arctan

2 M 2

2 Z2

i12 n m  a  1   nt sin  cos  2 Z2

arctan

i12 n n  nt  b

Specially in Table 4, when t  tS , the initial meshing radius is denoted as m2S ; when t  tE , the termination meshing radius is denoted as m2E ; C2 and  2 make sense only when    2 . Moreover, when    2 , the driven contact curve is planar; when 0     2 , right hand screw; when  2     , left hand screw. (With n  0 assumed in Section 2.1) 2.3

Interference-Proof Conditions of Contact Curves

Fig. 8.

Interference of driven contact curves.

To prevent self-interference, the meshing radius of contact curves cannot be zero. As a cylindrical helix, the driving contact curve of HCMW is naturally interference-proof, but not the driven curve. As shown in Fig. 8, all the driven contact curves are concurrent when within interference, forming two opposite tapers with the same center lines, instead of one driven pitch cone. The interference-proof condition of the driven contact curves is： (7) M 2  0 .

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To solve Eq. (7), the screw direction of the driven contact curves should be considered to avoid unavailable results. The following presents the interference-proof conditions in intersecting axis case as an example. The available results in vertical and parallel axis cases are shown in Table 5. When   0,  2,  , the HCMW is in intersecting axis case, Eq. (7) is solved as below: when 0     2 , (8) a  m1 , or a  m1   tE    n sin  cos ;

(9)

when  2     , a  m1 ,

(10)

or a  m1   tE    n sin  cos .

Fig. 9.

(11)

Interference-proof conditions of driven contact curves.

The meshing diagrams of Eqs. (8) - (11) are shown in Figs. 9 (a) - (d), respectively. From Fig. 9, the essence of Eq. (7) can be concluded as below: the center line of the driven wheel cannot intersect with the meshing curve. In geometry, the center line of the driven pitch cone cannot intersect with the common tangent between the driving pitch cylinder and the driven pitch cone. However, situations in Figs. 9(a) and 9(c) are unavailable considering their screw direction. For example, in Fig. 9(a), when 0     2 and a  m1 , the driven contact curve is left hand screw, which is inconsistent with Section 2.2.2. The available results are shown in Table 5. Table 5. Interference-proof conditions between driven contact curves.

 0   2

Interference-proof conditions

Screw direction

Engagement

Intersecting axis

a  m1   tE    n sin  cos 

right hand

internal

 2  

a  m1   tE    n sin  cos

left hand

external

Vertical axis

  2

b  n  tE   

planar

critical

Parallel axis

 0  

a  m1

right hand

Internal

a  m1

left hand

external

Cases

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From Table 5,  M 2  0 in each case, so m2   M 2 = M 2 . Also, from Figs. 9(b) and 9(d), the initial meshing radius is not less than the termination meshing radius, that is, m2 S  m2 E .

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GEOMETRIC CONSTRAINTS AND INTERFERENCE-PROOF CONDITIONS OF HCMW PAIR

3.1

Key Points of HCMW Pair The HCMW pair can be considered as tangency of the driving pitch cylinder and the driven pitch cone, whose center lines are coplanar. The key points of the HCMW pair are shown in Fig. 10, and their values in o  xyz are shown in Table 6.

Fig. 10. Key points of HCMW pair. Table 6. Values of key points in o  xyz . Initial circle

AS

 2S cos  a

BS

 a

CS DS

3.2 1)

 a  a

Termination circle

0 b  2 S sin  

T

2 S

b

T

BE

0 b

T

 2 S

AE

CE

b

T

DE

 2 E cos  a  z2 E sin   z2 E sin   a  z2 E sin   a  z2 E sin   a

0 b  z2 E cos  2 E sin  

T

2 E b  z2 E cos 

T

0 b  z2 E cos  

T

 2 E

b  z2 E cos 

T

Geometric Constraints of HCMW Pair Maximum radius of HCMW pair. From Fig. 10, whatever the value of  is, the maximum radius

point must be among  m1 0 0 , AS and BS  CS  . The maximum radius is attained as Eq. (12): T

Rmax  max{m1 , y AS , xB2S  yB2S } .

(12)

2) Height of HCMW pair. From Eq. (6) and Fig. 10, whatever the value of  is, the height point is AE , the height of the HCMW pair can be attained as Eq. (13): H max  z AE . Transactions of the Canadian Society for Mechanical Engineering, Vol. 00, No. 0, 0000

(13)

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Fig. 11. Geometric constraints of HCMW pair.

3)

Field angle of driven wheel. As shown in Fig. 10 and Table 6, ES has the same z value with BS ,

CS and DS , and its coordinate value in o  xyz is 0 0 (a  m1 ) tan   . As shown in Fig. 11, T

 S =BS ES DS is defined as initial field angle of the driven wheel, and tan

S 2



a

2 S

. Similarly, EE

locates at 0 0 h2 cos   a  m1  tan   , and  E =BE EE DE is defined as the termination field  h sin   a angle of the driven wheel, tan E  2 . According to Section 2.3, the initial field angle is the 2 2 E T

maximum field angle of the driven wheel, and  max   S . The geometric constraints above are attained based on the contact curves. However, tines with radii will be designed in practical to provide the contact curves on their surface. The results attained should be corrected if the value of the tine radii cannot be ignored. Denote the radii of the driving and driven tines as r1 and r2 , respectively. Corrected constraints with tine radii are presented in Table 7. Table 7. Geometric constraints corrected with tine radii. Original constraints

Corrected constraints

m1

m1'  m1  r1

m2

m2'  M   r2

Rmax

' Rmax  Rmax  r2

H max

' H max  H max  r1  r2

S

E

2

 S'  2arctan

 E'  2arctan

a  2r2

2 S

z2 sin   a  2r2

2 E

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3.3

Interference-Proof Conditions of HCMW Pair Considering the tine radii, distances between the corresponding driving and driven contact curves should be concerned to prevent interference within HCMW pairs. As shown in Fig. 12, the driving pitch cylinder is expanded along its circumference. The circumference distances between the driving contact curves should be larger than the sum of the driving and driven tine diameters: (14) m1 1 sin 1  2  r1  r2  .

Fig. 12. First interference-proof condition of HCMW pair.

As shown in Fig. 13, the driven pitch cone is expanded along its circumference. Similarly, the circumference distances between the driven contact curves should be larger than the sum of the driving and driven tine diameters: m2 2 sin 2t  2  r1  r2  . (15)

Fig. 13. Second interference-proof condition of HCMW pair.

Eqs. (14) and (15) are defined as interference-proof condition of HCMW pair. As (seen in Section 2.3), Eq. (15) can be equally rewritten as: m2 E 2 sin 2 E  2  r1  r2  . (16)

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GEOMETRIC CONSTRAINTS AND INTERFERENCE-PROOF CONDITIONS OF HCMW TRAIN

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A HCMW train referred in this paper contains a driving wheel and at least two driven wheels [18]. Within a HCMW, each driven wheel meshes with the driving wheel at different points, and forming a HCMW pair, respectively. 4.1

Geometric Constraints of HCMW Train The plane formed by intersected center lines of the driving and driven wheels is defined as a pair plane. Obviously, all the pair planes are collinear with the center lines of the driving wheels and perpendicular to xoy .

Fig. 14. Geometric constraints of HCMW train.

As shown in Fig. 14, the included angle between the k th ( k  2,3,..., N , N is the number of the driven wheels) pair plane and the reference one is denoted as 1k . Included angle between the adjacent pair planes is denoted as  k 1k , and  k 1k  1k  1k 1 . If the driven wheels are axially symmetrical distributed around the center line of the driving wheel, their pair planes will be in the same distribution. Included angles between the adjacent pair planes are: 2  k 1k   k  2,3,..., N  . N 4.2

Interference-Proof Conditions of HCMW Train As shown in Fig. 14, half of the field angle sum of the two adjacent driven wheels should be less than the included angle of their pair planes, that is: 1 (17)  max k 1   max k    k 1k . 2 Also, all the driven wheels should be distributed within a circle, that is: N

 i 1

max i

 2 .

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(18)

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Eqs. (17) and (18) are defined as interference-proof condition of HCMW train. It is worth noting that they are exactly equivalent if the driven wheels are axially symmetrical distributed around the center line of the driving wheel.

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APPLICATION EXAMPLE: HCMW REDUCER DESIGN

As shown in Fig. 15, a regular tetrahedron HCMW reducer [18] is illustrated as an application example. In the regular tetrahedron HCMW reducer, all the center lines of the driven wheels are the lateral edges of the triangular pyramid, while the center line of the driving wheel is its center line. As following presents the determination process of its parameter a .

(a)

(b)

Fig. 15. Regular tetrahedron HCMW reducer: (a) material objet; (b) perspective drawing

According to the direction of the lateral edges of the triangular pyramid,   0.696 . According to the working stress [19], r1  0.6mm and r2  0.7mm are given. According to the transmission requirement, i12  2 is given. The driving wheel with five tines is adopted, that is, Z1  5 . According to the transmission ratio, Z 2  10 . Supposing   1.25 [15] and tS   , as below can be attained from Eq. (3): tE    2 ,   t    2 . The driving contact curve is determined by m1 and n [12], which depend on the input situation. Here, m1  5 mm and n  6 mm are given. From Eqs. (2) and (5), the equations of the driving contact curve in o1  11 z1 and the driven contact curve in o2  2 2 z2 are derived as Eqs. (19) and (20), with only a indeterminate:   M1  5  1 , (19)  M  t  1  zM  6  6t

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  2  5  a   6t sin 0.696   M  cos  0.696     2 t .  M   2   zM 2  6t cos  0.696   

(20)

Fig. 16. Selection range of a .

Suppose the geometric constraints demanding are Rmax  20.916 mm and H max  30.000 mm . They are solved with Eqs. (12) and (13). Proof-interference conditions are as Eqs. (11), (14) and (16)~(18). Among them, Eq. (14) in this example contains no a , and can be ignored. As shown in Fig. 16, Eqs. (11) ~ (13) and (16) ~ (18) are solved through MATLAB, and the range of a is attained as Eq. (21): (21) 11.990 mm  a  15.180 mm . The maximum radius and height of the HCMW are demanded in this example. However, other geometric constraints mentioned in the previous sections should be considered if desired. The interference-proof conditions can be applied either to determine the ranges of the design parameters or to examine their feasibility.

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CONCLUSION

This paper presents the geometry of the HCMW, the most common SCMW at present. The equations of the driven contact curves are optimized by removing a redundant length parameter. Geometric constraints and interference-proof conditions are proposed in three aspects: contact curves, HCMW pair and HCMW train. Parameter calculation and determination methods in this paper are capable of developing industrial application of the HCMW.

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