A PLANETARY GEAR TRAIN WITH RING-INVOLUTE TOOTH

A PLANETARY GEAR TRAIN WITH RING-INVOLUTE TOOTH

*Yang,Shyue-Cheng

*Professor

** Tsang-Lang Liang

** Associate Professor

Department of Industrial Education and TechnologyNational Changhua University of Education,

Bao-Shan Campus: Number 2, Shi-Da Road, Changhua, 500, Taiwan, R.O.C.* Corresponding author. E-rnail:[email protected]

Received October 2007, Accepted June 2008No. 07-CSME-45, E.Le. Accession 3014

ABSTRACT:This paper proposes a planetary gear train with ring-involute tooth profile. Inherent in a

planetary gear train is the conjugate problem among the sun, the planet gears and the ring gear.The sun gear and the planet gear can be obtained by applying the envelope method to aone-parameter family of a conical tooth surface. The conical tooth rack cutter was presentedin a previous paper [5]. The obtained planet gear then becomes the generating surface. Thedouble envelope method can be used to obtain the envelope to the family of generatingsurfaces. Subsequently the profile of a ring gear of the planetary gear trains can be easilyobtained, and using the generated planet gear and applying the gear theory, the ring gear isgenerated. To illustrate, the planetary gear train with a gear ratio of 24:10:7 is presented.Using rapid prototyping and manufacturing technology, a sun gear, four planet gears, and aring gear are designed. The RP primitives provide an actual full-size physical model that canbe analyzed and used for further development. Results from these mathematical models areapplicable to the design of a planetary gear train.

UN TRAIN D'ENGRENAGE PLANETAIRE AVEC LA DENT ANNEAU-SPIRALE

RESUME:Ce document propose un train d'engrenage planetaire avec Ie profil anneau-spiral de dent.

Inherent dans un train d'engrenage planetaire est Ie probleme conjugue parmi Ie soleil, lesengrenages planetaires et la vitesse d'anneau. La vitesse de soleil et l'engrenage planetairepeuvent etre obtenus en s'appliquant la methode d'enve1oppe a une famille d'un-parametred'une surface conique de dent. Le coupeur de support conique de dent a ete presente dans unpapier precedent [5]. L'engrenage planetaire obtenu devient alors la surface produisante. Ladouble methode d'enveloppe peut etre utilisee pour obtenir l'enveloppe a la famille degeneration des surfaces. Plus tard Ie profil d'un engrenage d'anneau des trains d'engrenageplanetaire peut etre facilement obtenu, et a l'aide de l'engrenage planetaire produit etappliquant la theorie de vitesse, l'engrenage d'anneau est produite. Pour illustrer, Ie traind'engrenage planetaire avec un rapport de vitesse de 24: 10:7 est presentes. Utilisant Ieprototypage et la technologie manufacturiere rapides, un engrenage de soleil, quatreengrenages planetaires, et un engrenage d'anneau sont cons:us. Les primitifs de RP fournissentun modele physique normal reel qui peut etre analyse et employe Ie developpement ulterieur.Les resultats de ces modeles matMmatiques s'appliquent a la conception d'un traind'engrenage planetaire.

Transactions ofthe CSME Ide la SCGM Vol. 32, No.2, 2008 251

1 INTRODUCTION

Planetary gear trains are very important mechanisms in power transmission. Yan and Lai [1,

2] proposed a geometric design of an elementary planetary gear train with cylindrical meshing

elements. Chen et al. [3] reported an innovative design of planetary cam trains based on

pure-rolling contact intended to overcome the drawbacks of gear trains, such as Coulomb friction

and backlash. Regardless, most planetary gear trains have involute teeth. For instance, spur gears

with an involute tooth has some advantages. Such as allowing the use of linear rack cutters which

leads to high production efficiency, and provideing line-contact meshing, constant pressure angle,

insensitivity to center distance variation and simplicity. Nowadays, only two types of planetary

gear trains with involute teeth are used in industry for power transmission between parallel axes.

One is the helical gear type, and the other is spur gear type. In this paper, another kind of

planetary gear train with ring-involute teeth is presented, as shown in Fig. 1. This paper also

presents a method for determining the mathematical model of a planetary gear train with

ring-involute teeth.

The shape of the involute teeth also significantly impacts functioning. For example, factors

influencing errors in alignment for a planetary gear train with a set of planets was presented by

Daniele [4]. This mechanism is capable of self-compensating for various alignment errors. Here,

a ring-involute tooth generated by a rack cutter with conical teeth was presented by Yang [5].

This rack cutter was used to generate the pinion and the gear. The teeth of the obtained gears are

called ring-involute teeth. The condition of undercutting for the proposed gear is also presented.

It is found that kinematic errors of an external gear with ring-involute teeth are independent of the

central distance between the gear centers [5]. Thus based on this paper, a double envelope method

for determining a ring gear of planetary gear train is presented.

In the present work, a conical tooth rack cutter is first used to generate the sun gear and the

planet gear of a planetary gear train. The teeth of the obtained gears are ring-involute teeth. The

obtained planet gear then becomes the generating surface. Based on the double envelope method

which can be reviewed in [6, 7, 8], the mathematical model of a ring gear is an envelope to the

family of generating surfaces when the given generating surface rotates a complete cycle.

To demonstrate the profile of a planetary gear train, a computer program and

computer-aided software were used. Using rapid prototyping and manufacturing technology, a


sun gear, four planet gears and a ring gear with ring-involute teeth are manufactured. Stress

analysis is then conducted according to the obtained geometric modeling of the proposed gear

train. One advantage of this study's method is the ability to provide a rapid and simple geometric

model of a planetary gear train. Based on the above results, mathematical models of the proposed

planetary gear train can be developed using a rapid prototype technology. A numerical example is

presented in Table 1 to demonstrate the geometric model of a planetary gear train with a gear

ratio of24:10:7.

Fig. 1 The proposed planetary gear train with ring-involute teeth.

2 DESIGN OF A CONICAL TOOTH RACK CUTTER

A conical rack cutter is shown in Fig. 2. Region ab of the rack-cutter surface is assumed

to generate the bottom land of the sun gear. The region ef of the rack-cutter surface, the

addendum surface, is designated as the shape of the sun gear blank. Thus, regions ab, bc, and- - -cd are only considered in the generating process of the sun gear. Similarly, regions cd, de ,and

ef are only considered in the generating process of the planet gear. The detailed description is

given in [5]. In this paper, only the mathematical model of the rack cutter is described in terms of

the Sc(Oc' xc' Yc' zJ coordinate system is described:


Fig. 2 Design of a rack cutter with conical teeth for generating the sun gear and the planet

gear.

2-1 REGION ab

(mn 12 - f b ) sin fJ

(mn12 -fb)COSfJ +cymn

-ae+rsin¢e- r, 0 < f b < W, 0 < fJ < 2n , (1)

2-2 REGION bc

(be +aetan¢e +rcos¢e -rsinfJsinfJ

(be +aetan¢e + rcos¢e - rsinf JcosfJ+cymn , 0 < fe

< 7T: 12-rPe' 0 < fJ < 2n, (2)-ae+rsin¢e -rcosfe

1

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2-3 REGION cd

1

(bc -J!h sin¢J sin p(bc - J! hsin ¢c) cos p + Cy7m'l

=J! hcos¢c

1

(3)

2-4 REGION de

xdec

deRde = Yc

e Zdee

(be -at tan¢e -rcos¢e +rsinfd)sinp

(be -at tan¢e -rcos¢e +rsinfd)cosP+cy7l7n , O<J!d <;r/2-rPe,O<P<2Jr, (4)at +rsin¢e -rcosfd

1

In Eqs. (1)-(4), the position vector of the rack cutter is R; and the superscript g indicates

regions ab, bc, cd, and de ofthe rack-cutter surface.

3 GENERATION OF A SUN GEAR, PLANET GEARS, AND A RING GEAR

As shown in Fig. 1, the proposed planetary gear train with ring-involute tooth profile has a

sun gear, four planet gears, and a ring gear. The correspondent coordinate systems for deriving

the mathematical model of the proposed planetary gear are shown in Fig. 3. A conical tooth rack

cutter is used to generate the sun gear and the planet gear, as shown in Fig. 3. The coordinate

system Sc (Oc , xc' Yc' zJ is rigidly attached to the rack cutter. The coordinate system

SI(O" X" Y" zJ is attached to the sun gear. The coordinate system Sz(Oz, xz' Yz' zz)

is attached to the planet gear. The coordinate system S3(03' X 3 ' Y3' zJ is attached to the

ring gear. The angles ¢pi' ¢pZ and ¢p3 , respectively, represent the rotary angle of the sun gear,

the planet gear, and the ring gear about their centers °1 , 0z' and °3 , During the generation

process, the ring-involute rack translates a distance S while the sun gear blank and the planet gear

blank rotate through an angle ¢pi and ¢pZ' respectively. The relationship between the angles


N2 denotes the number of teeth of the planet gear. The relationship between the distance S and

the angles rPpl and rPp 2 is given by S = rp\rPp \ and S = r p2 rPp 2 respectively, where r p1

represents the radius of the pitch circle of the sun gear, and r p2 ' the planet gear. Based on the

gear theory [9], an envelope to the family of the rack-cutter surfaces can be found. The obtained

envelope is called the mathematical models of the sun gear and the planet gear.

Fig. 3 Coordinate systems of the planetary gear train

3-1 OBTAIN THE MATHEMATICAL MODELS OF A SUN GEAR AND A PLANET

GEAR

The method of homogeneous coordinate transformation is a powerful tool for studying the

motion of a rigid body among the different coordinate systems. Based on the 4 x 4

homogeneous coordinate transformation matrices, the matrices M 1c ' M 2c ' and M 32 can be

obtained as follows:


1 0 0 0

0 COS¢pI Sin¢Pl -SCOS¢PI +rpl Sin¢PIM 1C =

0 -sin¢PI COS¢pI Ssin¢PI +rp1 COS¢pI

0 0 0 1

1 0 0 0

0 cos¢pz -sin¢pz - Scos¢pz + rpz sin¢pzM zc =

0 sin¢pz cos¢pz - Ssin¢pz - rpz cos¢pz

0 0 0 1

and

1 0 0 0

0 COS(¢p3 -¢pz) - sin(¢p3 - ¢pz) - (rp1 + rpz ) sin ¢p3M 32 =

0 sin(¢p3 -¢pz) COS(¢p3 -¢pz) (rp1 + rpz )cos ¢p3

0 0 0 1

(5)

(6)

(7)

The mathematical models of the sun gear and the planet gear can be determined by

obtaining the one-parameter family of conical rack cutters and representing them in coordinate

system SI and Sz as:

(8)

where R; is shown in Section 2. The upper sign of vector R; indicates regions ab, be, cd,

and de of the rack-cutter surface.

Based on the envelope theory, the envelope to the family of rack-cutter surfaces should

simultaneously satisfy Eq. (8) and

(9)

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where f~fk' fJ, ¢Pj) is called the equation of meshing. Substituting Eqs. (1)~(4) into Eqs. (8)

and (9), the equation of meshing can now be represented as:

(10)

The values of Eq. (10) xJ, yJ and zJ are shown in Section 2. The superscript g represents

ab, be, cd, and de of the rack-cutter surface; xJp, yJp and zJp are partial derivatives of

g g d g • h fJ S' '1 1 g g d g . 1 d" fx j ' Yj an Zj WIt respect to . lml ar y, Xjik' Yj£k an zPk are partla envatlves 0

XJ, yJ and zJ with respect to f k' By simultaneously solving Eqs. (8) and (10), the

mathematical models of the sun gear and the planet gear can be obtained. The profile of the sun

gear and the planet gear is ring-involute teeth, and the dimensional parameters of their geometric

surface are listed in Table 1. To demonstrate the contours of the sun and the planet gears, the

Turbo C++ programming language and the SolidWorks software package are employed to draw

the complete profile of the sun and the planet gears. The geometric models of the sun and the

planet gears with ring-involute can be evaluated and plotted, as shown in Fig. 4.

Table 1 Major design parameters of the gear with ring-involute teeth

Parameters Sun (Convex) Planet (Concave) Ring(Convex)

Number of teeth N] =40 N2=28 N3=96

Normal module m 5mm 5mm 5mm

Pitch radius mN! m N 2 m (N! +2N2 )

rp ! =-2- rp2 =-2- rp3 =2

fJ o~ 27l" o~ 27l" o~ 27l"

pressure angle ¢c 20° 20° 20°

ac and at 1m 1m 1m

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rr =0.2m r = 0.2m r =0.2m

be mn/4 mn/4 mn/4

Backslash f!.p-0.08 0 -0.08

Fig.4 The geometric models of the sun and the planet gears.

3-2 OBTAIN A RING GEAR

The double envelope concept is used to obtain the mathematical model of a ring gear. The

concept is also called the inverse problem or two-parameter enveloping concept. The concept of

the inverse problem can be reviewed in [6, 7, 8]. The basic idea for determining a ring gear can

be regarded as an envelope to the family of the planet gear surfaces, which are determined in

Section 3-1 where the mathematical model of the planet gear is called the first envelope. The

determined mathematical model of the planet gear is designated as a generating surface in Section

3-1. The ring gear is then obtained as an envelope to the family of the planet gear surfaces when

the ring gear completes a cycle. This obtained envelope is the ring gear of the proposed gear

mechanism and is defined as the second envelope. Subsequently, through Sections 3-1 and 3-2,

the profiles of the proposed gear mechanism can be easily obtained. Therefore, based on the

double envelope concept, the profile of the ring gear can be obtained and its mathematical model

can be represented by

(11)


(R fJ ,f,)- (aRt x aRtJ. aR; - 0g k' ,'f'p3 - afJ aRk a¢P3- (12)

where Eq. (12) is the equation of meshing. Vector R;.g is the position vector of the planet gear

and determined by Eqs. (8) and (10) in Section 3-1. The superscript sign of vector R;.g indicates

regions ab, be, cd, and de of the rack-cutter surface. Matrix M 32 describes the coordinate

transformation from 82 to 83 , Rearranging Eqs. (8), (10) (11) and (12), the equation of

meshing can be obtained:

(13)

where ¢p2 is the rotary angle of the planet gear. Substituting the vector Eqs. (2)~(4) into Eqs. (8),

(10), (11) and (13), the mathematical model of the ring gear with ring-involute teeth is obtained.

Based on the proposed mathematical models, a computer design program has been developed to

assemble the geometric model of the proposed planetary gear train.

4. STRESS ANALYSIS

The most powerful method for determining accurate stress and deflection information is the

finite element (FE) method. The stress analysis presented in this section determines the stress for

the proposed planetary gear train while the computer-aided stress analysis is used to determine

the contact stress for the proposed planetary gear train. This study assumes that there are no

assembly errors. The performed stress analysis is based on a finite method [10] and a general

proposed computer program. A PC version of this program running on Windows XP is used to

obtain the numerical solution for the contact stress analysis.

Two geometric models based on the proposed method have been developed. One is the sun

and the planet gears created by a rack cutter; the other is the assembly of the proposed planetary


gear train. Based on Sections 2-3 and CAD/CAM software packages, the geometric models of the

sun gear, the planet gear and the ring gear are obtained. These models are developed by the CAD

computer program and then transferred to the FE package ANSYS Workbench [11] for stress

analysis. The stress analysis is applied to the mechanism of the sun-planet gear pairs and the

proposed planetary gear train. The material is assumed to be isotropic and homogeneous. The

module material is AfSf CI020 steel with a Young's modulus of E=207Gpa and a Poisson ratio of

v= 0.3. The total number of elements is 58259 with 88611 nodes for the sun gear, where the total

number of elements is 34317 with 52524 nodes for the planet gear. The total number of elements

is 157168 with 241209 nodes for the ring gear. A finer mesh is used in the tooth contact area and

is shown Figs. 5 and 7 .The mass of the sun gear is 3.74 kg. The mass of the planet gear is 1.96

kg. Similarly, the mass of the ring gear is 6.051 kg. In this analysis, only the sun gear has a

constant moment load. A counterclockwise torque is applied to the sun gear, causing contact

between the gear teeth. A torque of 10Nm is applied to the sun gear. The contour plot of

von-Mises stress of the sun-planet gear is shown in Fig. 6. Three-dimensional stress distributions

of the planetary gear train are plotted based on von-Mises stress contours and are shown in Fig. 8.

The maximum stress of von-Mises in Fig. 6 is 7.565 x 106 N/ m 2• The maximum stress of

von-Mises in Fig. 8 is 5.628 x 105 N/ m 2• Figure 9 is greatly exaggerated in magnitude by Fig. 8 .

Finer Mesh

Fig. 5 Finite element model of the sun and the planet gears.


~'It!enl '(Von.MOOs).Slies•.x 106 PaMu:7;~.MiD: L257IitOOJ.?fXJSI4m 2IJ:38

7.51556.7255.81l4SJ)434:2033,362Z.52Z

. ~681Q,S41L~i006

Fig. 6 von-Mises stress in Pa at the contact region.

Finer Mesh

Fig. 7 Finite element model of the proposed planetary gear train with ring-involute teeth.


,Equiyalent (von~~).Str~x le5'Pa ... ,Max:5,Q?8e+W,~

Mi.n: 3:364e~3,~4127,17:46 '

5J~28

5JJ03'4:m'3.7523JZ7~,501

1.8761.~10,625l~e'998,

Fig. 8 von-Mises stress distribution for the proposed planetary gear train.

Fig. 9 A greatly exaggerated version of Fig. 8 in magnitude

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5. RAPID PROTOTYPE FOR PLANETARY GEAR TRAIN MANUFACTURING

Conventional steel modulus is generally CNC machined from a solid block. However, the

fabrication of complex shapes through traditional machining methods requires long lead times

and high costs. The advantage of RP technology simplifies, and shortens the time for making

complex shapes. In this section, the proposed planetary gear train with ring-involute teeth can be

manufactured by a computer-aided manufacturing program, and the selected geometric

parameters are shown in Table 1. However, in actual cases, the selection of transmission ratio will

depend on the application of the planetary gear train.

For industrial application, the presented mathematical models and geometrical models have

been successfully converted to STL, DYNA, and GM codes for SLA (stereo-lithography

apparatus) and SLS (selective laser sintering). Using the proposed mathematical model and rapid

prototyping technology, the sun gear and the planet gear created by a conical rack cutter, and the

ring gear created by the planet gear were obtained. The proposed gear mechanism including ring

gear, sun gear and planet gears is manufactured by rapid prototyping technology in the current

structure. Their materials are ABS plastic. The sun gear is put on the center position. The planet

gears are arranged around the sun gear. Finally, the ring gear is set in the external of the planet

gears. A skill in the assembly of the ring gear is slowly pulling the ring gear, so that the ring gear

is at its position, as shown in Fig. 10. For clarification, one sun gear, four planet gears, and one

ring gear are shown in Fig. 10. If their materials are carbon steel (AISI C1020), the assembly of

the mechanism will be based on the method of thermal expansion and cold contraction. Here, the

ring gear expands with heat and the sun gear contracts with cold. The planet gears are arranged

around the sun gear. Finally, the ring gear is set in the external of the planet gears.

In this paper, the process from development to end product is methodical, consisting of

design, homogeneous coordinate transformation matrix, space engagement theory, stress analysis

and rapid prototype. One advantage of this study's proposed method is its ability to provide a

rapid and simple geometric model of a planetary gear train. On the other hand, the RP primitives

provide an actual full-size physical model that can be analyzed and used for further development.

Transactions ofthe CSME Ide la SCGM Va!. 32, No.2, 2008 264

Fig. 10 The rapid prototype for the proposed planetary gear train

6. CONCLUSION

Based on space engagement theory and differential geometry, this study has developed a

tooth mathematical model of a planetary gear train. The proposed rack cutter with conical tooth is

used to generate sun and planet gears. The teeth of the obtained gear are ring-involute teeth.

Using the double envelope method, the generated planet gear is used to establish the

mathematical model of the ring gear. In this work, the tooth profile and equation of meshing for

the planetary gear train are established. An assembly model of the sun gear, the planet gear and

the ring gear with ring-involute teeth is developed. According to the developed mathematical

model, a three-dimensional stress analysis of the proposed gear concept is conducted.

Finally, the Turbo C++ programming language, SolidWorks, and Mathematica software are

utilized to generate the proposed planetary gear train. To demonstrate the contours of one sun

gear, four planet gears and one ring gear, rapid prototyping and manufacturing technology are

used to make these gears. The proposed mathematical model analysis for the planetary gear train

should be helpful in the design and production of a planetary gear train.

ACKNOWLEDGEMENT

The authors are grateful to the National Science Council of the Republic of China for supporting

this research under Grant NSC95-2212-E-018-003


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A PLANETARY GEAR TRAIN WITH RING-INVOLUTE TOOTH

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