A PLANETARY GEAR TRAIN WITH RING-INVOLUTE TOOTH *Yang,Shyue-Cheng *Professor ** Tsang-Lang Liang ** Associate Professor Department of Industrial Education and Technology National 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 2008 No. 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 a one-parameter family of a conical tooth surface. The conical tooth rack cutter was presented in a previous paper [5]. The obtained planet gear then becomes the generating surface. The double envelope method can be used to obtain the envelope to the family of generating surfaces. Subsequently the profile of a ring gear of the planetary gear trains can be easily obtained, and using the generated planet gear and applying the gear theory, the ring gear is generated. 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 a ring gear are designed. The RP primitives provide an actual full-size physical model that can be analyzed and used for further development. Results from these mathematical models are applicable 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, les engrenages planetaires et la vitesse d'anneau. La vitesse de soleil et l'engrenage planetaire peuvent etre obtenus en s'appliquant la methode d'enve1oppe a une famille d'un-parametre d'une surface conique de dent. Le coupeur de support conique de dent a ete presente dans un papier precedent [5]. L'engrenage planetaire obtenu devient alors la surface produisante. La double methode d'enveloppe peut etre utilisee pour obtenir l'enveloppe a la famille de generation des surfaces. Plus tard Ie profil d'un engrenage d'anneau des trains d'engrenage planetaire peut etre facilement obtenu, et a l'aide de l'engrenage planetaire produit et appliquant la theorie de vitesse, l' engrenage d'anneau est produite. Pour illustrer, Ie train d'engrenage planetaire avec un rapport de vitesse de 24: 10:7 est presentes. Utilisant Ie prototypage et la technologie manufacturiere rapides, un engrenage de soleil, quatre engrenages planetaires, et un engrenage d'anneau sont cons:us. Les primitifs de RP fournissent un 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 train d'engrenage planetaire. Transactions ofthe CSME Ide la SCGM Vol. 32, No.2, 2008 251
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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,
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
Transactions ofthe CSME Ide la SCGM Vol. 32, No.2, 2008 252
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:
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Fig. 2 Design of a rack cutter with conical teeth for generating the sun gear and the planet