USING DOUBLE ENVELOPE METHOD ON A · PDF fileUSING DOUBLE ENVELOPE METHOD ON A PLANETARY GEARMECHANISM WITH DOUBLE CIRCULAR-ARCTOOTH ... archelical gear by computerized ... and gear

** Chun-Fang Tsai** Student

USING DOUBLE ENVELOPE METHOD ON A PLANETARY GEAR MECHANISMWITH DOUBLE CIRCULAR-ARC TOOTH

***Tsang-Lang Liang *Shyue-Cheng Yang*** Associate Professor *Professor

Department of Industrial Education and TechnologyNational Changhua University ofEducation,

Bao-Shan Campus: Number 2, Shi-Da Road, Changhua, 500, Taiwan, [email protected]

Received October 2007, Accepted June 2008No. 07-CSME-48, E.LC. Accession 3017

ABSTRACTA geometric model and mathematical model of planetary gear mechanism with double circular-arc

teeth is determined using the imaginary rack cutters and double envelope concept. The mathematicalmodel of a ring gear with double circular-arc teeth in a second envelope has been developed. In this paper,the conditions of the gear meshing and contact load of the gears are simulated by assembly errors. Thegoal of the stress analysis is to determine the contact stress on the planet gear and ring gear, and planetgear and sun gear. Given the assembly errors of the planetary gear mechanism including center distanceerror and mis-aligmment error, the maximum von-Mises stress in the planetary gear mechanism withdouble circular-arc teeth is analyzed using visualNastran desktop package. It is found that the centerdistance error is more critical to mis-alignment error.

SUIVRE LA DOUBLE METHODE D'ENVELOPPE SUR UN MECANISMED'ENGRENAGE PLANETAIRE AVEC LA DOUBLE DENT D'ARC CIRCULAIRE

RESUMEUn modele geometrique et Ie modele mathematique du mecanisme d'engrenage planetaire avec de

doubles dents d'arc circulaire est determine utilisant les coupeurs de support imaginaires et Ie doubleconcept d'enveloppe. Le modele mathematique d'un engrenage d'anneau avec de doubles dents d'arccirculaire sous deuxieme enveloppe a ete developpe. En ce document, les conditions du maillage del'engrenage et de la charge de contact des engrenages sont simules par des erreurs d'assemblee. Le but del'analyse des contraintes est de determiner l'effort de contact sur l'engrenage planetaire et la vitessed'anneau, et l'engrenage planetaire et l'engrenage de soleil. Etant donne les erreurs d'assemblee dumecanisme d'engrenage planetaire comprenant l'erreur de distance centrale et l'erreur de mis-aligmment,l'effort maximum de von-Mises dans Ie mecanisme d'engrenage planetaire avec de doubles dents d'arccirculaire est analyse utilisant Ie paquet d'ordinateur de bureau de visualNastran. On Ie constate queI'erreur de distance centrale est plus critique al'erreur de deviation d'alignement.

Transactions ofthe CSME Ide la SCGM Vol. 32, No.2, 2008 267

INTRODUCTION

Nowadays, three fundamental types of tooth profile are used in industry for powertransmission between parallel axes. They are involute, cyc1odial, and circular arc (or Noviko)[1-3]. However, the involute tooth has relatively poor lubrication and lower load capacity [4].The problem of explicit determination of the corresponding conjugate profile of two meshingcylindrical gears was solved in [5] by solving the ordinary differential equations. In this paper,an imaginary rack cutter with double circular arc tooth is used to develop a mathematical modelof planetary gear mechanism. Herein the profile of double circular arc tooth is shown in Fig. 1.

Chen and Wang [6] presented a compensating analysis of a double circular-arc helical gearby computerized simulation of the meshing. The equations for calculating the contactdeformation of double circular-arc gear tooth are derived using the theory of elasticity of generalcontact problem between arbitrary elastic bodies in three dimensions [7]. Wu [8] investigated thecontact bearings of double circular arc helical gears based on the theory of W-N gears. Fahmyand Jonckheere [9] embodied the results of an experimental investigation to determinedeflections and bending moment on a gear tooth model.

The concept of double circular-arc tooth is based on the circular-arc tooth. Bair [10]developed a gear profile with circular arc teeth which is used on an oil pump with a largerdischarge volume and without the appearance of tooth undercutting. Chen and Tsay[ll] studiedthe undercutting conditions and the characteristics of elliptical gear with circular-arc teeth. Lu etal. [12] computerized the simulation of the meshing and transmission errors. In their paper, thecontact stress analysis of the tooth surface with double circular-arc helical gear was presented.Litvin and Lu [3] investigated the condition of load and the real contact ratio using the finiteelement method. The contact pressure spreads in an elliptical area. El-Bahloul [13] discussed thesurface capacity of gears with circular-arc tooth. Based on the results and analysis of anexperiment to study the effect of depth, hardness and tooth load on the surface capacity of arelatively new type of gearing with circular arc teeth were obtained. It was found that the wearrate for case hardened gears decreases with an increase in case depth, and approaches aminimum at the 1.1mm case in the experiment. Yang [14] applied a method of deviation functionto a gear pump with circular-arc tooth. In his work, the visualNastran for desktop 2004 softwarewas used to analyze contact stress between gear pairs.

In general, researches on double circular-arc gear or circular-arc gear are focused on themathematical model of external spur gear. A type of internal gear for the planetary gearmechanism using imaginary rack cutters and the method of double envelope to determine themathematical model ofplanetary gear mechanism is rarely reported.

The double envelope concept comes from the gear theory and inverse envelope concept [15,16]. Yang [17] used the double envelope method to determine the geometric model of internalgear with asymmetric teeth. Yang [18] also used the inverse envelope concept to determine the


geometric model of a cutter for machining a single screw rotor.

Based on the double envelope concept, the mathematical model of the planet gear isregarded as an envelope to the family of rack-cutter curves. The obtained planet gear becomesthe generating curve and is defined as the first envelope. Through homogeneous coordinatetransformation and gear theory, the planet gear is used to generate the ring gear. The obtainedring gear is defined as the second envelope. Therefore, the geometric model of a planetary gearmechanism with planet, sun and ring gears can be easily obtained.

In this work, the complete profile of a planetary gear mechanism including the planet gear,sun gear and ring gear with double circular-arc teeth are illustrated based on a computer programand computer-aided software. Using Turbo C++ and SolidWorks software, the planetary gearmechanism is designed. One advantage of the present method is the ability to provide a rapid andsimple geometric model of planetary gear mechanism. The developed computer program can beused to determine the geometric properties of the tooth contact stress analysis under assemblyerrors. Based on the above results, the developed planetary gear mechanism can be obtained forgear machining. Finally, a numerical example is presented in Table 1 to demonstrate thegeometric model of a planetary gear mechanism with a gear ratio of 1:3:9.

h.

ef

e.h,

Yn

hf

Fig. 1 The normal section of rack cutter with the double circular-arc tooth.

Transactions ofthe CSME Ide fa SCGM Vol. 32, No.2, 2008 269

GEOMETRY OF RACK CUTTER WITH DOUBLE CIRCULAR-ARC TOOTH

As shown in Fig. 1, an imaginary rack cutter with double circular-arc tooth is discussed inthis section, and developing its mathematical model is useful. In general, the profile of the gearis generated by the position vector of the rack cutter through homogeneous coordinatetransformation and equation of meshing. Such a gear profile can reduce transmission errors ingear manufacturing. Therefore, developing a geometry mathematical modeling of the gearsbased on analytical method is an important method. A rack cutter of double circular-arc teethwith a normal section is show in Fig. I. The shape of the rack cutter consists of eight circular-arc

-------- -forms denoted by ab ~ be ~ ed ~ de ~ ef ~ fg ~ gh ~ hi ~ and ai with respect to the xn -axis.

The normal section of the rack-cutter curve L ~ represented in the 8\ (01' XI' yJcoordinate system can be written as follow:

REGIONS be AND ai OF THE RACK CUTTER

To generate both sides of the working tooth surface of the gear, the two regions be and

ai of the rack cutter are used to generate the planet gear. The parameter Pais the radius of the- -

circular arc on regions be and ai. Equations for regions be and ai, represented in thecoordinate system 8\, can be respectively written as

(1)

and

(2)

where Pa is the radius of convex tooth on the imaginary rack cutter. Symbols ea, la and ef

are the design parameters of imaginary rack cutters. These values are given in Table 1. Parameter- -

8\ is a curvilinear parameter of regions be and ai.

REGIONS ed AND hi OF THE RACK CUTTER

As indicated in Fig.1,regions ed and hi are used to generate the fillet curve of the blank- -

material of the planet gear. Both regions ed and hi connect the convex curve and the concave


curve. The parameter rj is the radius of circular arc on regions cd and hi. The position- -

vectors of regions cd and hi are represented in the coordinate system 81 as:

(3)

and

(4)

where hja represents the vertical distance from point c to Yn -axIS, as given in Table 1.

Parameter ef is the shifting coefficient of the imaginary rack cutter.

REGIONS de AND gh OF THE RACK CUTTER

As show in Fig. 1, the two curves de and gh of the rack cutter are used to generate the- -

working tooth surface of the concave tooth of the planet gear. Both curves de and gh tangent- - - -ef and fg, respectively. Position vectors for regions de and gh, represented in coordinate

system 81 , can be respectively written as

(5)

and

(6)

where parameter Pf denotes the radius of circular arc of the rack cutter in regions de and gh.- -

The variable, 83 , is the curvilinear coordinate of regions de and gh. m is the module of the


proposed planetary gear mechanism.

REGIONS ef AND fg OF THE RACK CUTTER

Regions ef and fg on the normal section of the rack cutter generate the root fillet curve- -

of the blank material of planet gear. The position vectors of regions ef and fg are

represented in the coordinate system 81 as:

(7)

and

(8)

where ()4 represents the curvilinear coordinate that determines the coordinates of any position

on the fillet. Parameter hjf is the vertical distance from point d to Yn -axis. The parameter rgf

- -is the radius of circular arc inn regions ef and fg. Based on Eqs. (1) to (8), Table 1 and the

Mathematica software package, the part profile of the imaginary rack cutter can be obtained.

Table1. Major design parameters of the planetary gear mechanism with doublecircular-arc teeth

Parameters Sun gear Planet gear Ring gear

Number of teeth Nt =30 N 2 =10 N3 =90

Modulem 5m.m

Pressure angle ¢c 20°

Depth of tooth hI 2m

Tooth addendum ha 0.9m

Tooth dedendum hf hI -ha

ea 0.0163m

ef 0.0285m

Pa 1.3m

Pf 1.41m


hja O.16m

hjf O.2m

la O.6289m

If O.6994m

ao 7r /7.5

MATHEMATICAL MODEL OF PLANETARY GEAR MECHANISM WITH DOUBLECIRCULAR-ARC TEETH

As shown in Fig. 2, the imaginary rack cutter can be used to generate a planet gear which isthen the obtained is used to determine the mathematical model of the ring gear. Fig. 2 describesthe relationship between the imaginary rack cutters and the planet gear, the planet gear and thering gear. As shown in Fig. 2, the coordinate system 81(°1, XI' 1';) is attached to the rack

cutter; The coordinate system 82(°2 , X 2 , Y2 ), to the planet gear L: 2 ; and the coordinate

system 83(°3, X 3 , Y3), rigidly attached to the ring gear L: 3. The coordinate system 8 2 is

allowed to rotate at the angle ¢p2 about the Z2 axis; and the coordinate system 83 , at the angle

¢p3 about the Z3 axis. The coordinate system 8f (Of' Xf' Yf ) represents the fixed

coordinate system. Based on the gear theory [19], an envelope to the family of the rack-cuttercurves can be found. The obtained envelope is called the planet gear and is defined as the firstenvelope. Assuming the planet gear becomes a generating surface, when it (first envelope)rotates along the Z2 axis with a rotary angle ¢p2' the ring gear blank rotates along the Z3 axis

with a rotary angle ¢p3' The relationship between the angles ¢p2 and ¢p3 is ¢p3 = N~P2 ,

3

where N2 is the number of teeth of the planet gear and N 3 denotes the number of teeth of the

ring gear. According to the double envelope concept, the form of the ring gear is regarded as anenvelope to the family of the planet gears when the given planet gear rotates for a complete cycle.Here, the obtained envelope is called a second envelope.

Based on gear theory, the imaginary rack cutter translates the linear displacement 8along the Y f -axis. The coordinate transformation matrix from 81 to 82 can be represented as

(9)

where 8 = rp2¢p2' Using Eq.(1)-(8) and homogeneous coordinate transformation matrix

M 21 , the mathematical model of the planet gear can then be obtained and represented in the

coordinate system 8 2 as


Sun CIla.""'.---' ....

" ,\.

\\

I'-=~-+-...~__~B---...Xj I,I

~-....-Xl "..-',---".,..

Fig. 2 Coordinate transformation system for generating planet, sun and ring gears.

(10)

and

(11)

where R: is shown in Section 2. The parameter OJ is a curvilinear coordinate, with subscript

j=I,2, 3, and 4. The superscript of the vector R: indicates regions ab, be, cd, de, ef,

fg, gh, hi, and ai of the rack cutter curve in Section 2. Eq. (10) is the family of imaginary

rack-cutter curves represented in coordinate system 8 2 , Eq. (11) is the equation of meshing for

planet gear. Substituting Eq. (1)-(8) into Eq. (10) and Eq. (11), the equation of meshing can bedetermined as

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(12)

274

Following Eqs. (1) ~(12), the planet gear with a double circular-arc tooth is obtained and definedas the first envelope. In the second envelope, the planet gear rotates along the Z2 axis with a

rotary angle ¢p2' The ring gear blank rotates along the Z3 axis with a rotary angle ¢p3' The

coordinate transfonnation matrix from 82 to 83 can be represented as

[

COS(¢P3 - ¢p2) - sin(¢p3 - ¢p2) (rp3 - rp2)COS¢P3J

M 32 = sin(¢p3 - ¢p2) COS(¢p3 - ¢p2) (rp3 - rp2 )sin¢p3

001

(13)

Using Eqs. (1)-(12) and coordinate transfonnation matrixM32 , the mathematical model of

the ring gear can then be obtained and represented in the coordinate system 83 as

(14)

and

(15)

Eq. (15) is the equation of meshing for the ring gear. Substituting Eqs. (1)~(12) into Eqs.(14) and (15), the equation ofmeshing can be obtained by the following equation.

(16)

Where

Following Eqs. (1) ~(16), the ring gear with a double circular-arc tooth is obtained anddefined as the second envelope. Using Table 1, the geometric model of the planet gear, sun gearand ring gear can be drawn. The first envelope to the family of imaginary rack-cutter surface can


be obtained by Eqs. (1)-(12). This envelope is called the planet gear. The ring gear is a secondenvelope to the family of planet gear obtained by Eqs. (1)-(16). As shown in Fig. 2, the sun gearis represented by dashed lines. Using a method similar to that for generating the plant gear, thesun gear can be obtained. Herein the detailed derivation is omitted. Using Turbo C++programming language and 8olidWorks software package, the complete profile of the planetarygear mechanism with double circular-arc teeth can be obtained, as shown in Fig. 3.

Fig. 3 The planetary gear mechanism with planet, sun and ring gears.

COMPUTER-AIDED STRESS ANALYSIS

In the above sections, the geometric models of the planetary gear mechanism are developedusing the Turbo C++ programming language and a computer aided design (CAD) computerprogram, and then transferred to the visualNastran desktop package for contact stress analysis,where the backlash between the gears is found to be O.04m.

The goal of the stress analysis in this section is to compare the center distance andmisaligned error on the contact stress of planet gears with ring gear and sun gear. In the stressanalysis, a torque of 3Nm was applied at the rotational axis of the planet gear. Given that therestitution coefficient and the friction coefficient is zero, the von-Mises stress distribution can beevaluated by visualNastran desktop. First, the planetary gear mechanism is investigated underideal assembly condition. The result of the von-Mises stress analysis is shown in Fig. 4. Second,the planetary gear is investigated under center distance errors. By changing the center distance,the origin O2 of the co-ordinate system 82 is displaced by LJd with respect to Of' as

shown in Fig. 5. The center distance is defined as d' = d + M , where distance d equals to thesubtraction of rp2 from rp3 ' The center distance error LJd is represented as

L1dJ + L1dyj + L1dzk. In this evaluation, the center distance LJd is decomposed into three


components, !1dx = O.lmm, !1dy = O.lmm and !1dz = O.lmm. The result of the von-Mises

stress analysis under center distance error is shown in Fig. 6. To obtain the misalignment anglesLlx ' Lly and 4, the co-ordinate system Sn may be respectively rotated about axes X n, Yn

and Zn through the respective angles L1x = O.C, !1y = O.C and !1z = O.C with respect to the

fixed co-ordinate system Sf' The von-Mises stress distribution under misalignment errors is

shown in Fig. 7. Comparing Fig. 4, Fig. 6 and Fig. 7, center distance errors effected a greatimportance on von-Mises stress is than the misalignment errors.

Fig. 4 The von-Mises stress distribution under ideal assembly condition

1-"'------1;

X 1(llrx I

Xl(t,)

r;xFig. 5 Coordinate systems for a simulation of assembly errors.


Max. Value = 6.81 e+4 Pa

6.15.e+45.14 e+45.33 e+44.92 e+44.51 e+44.1 e+43.69 e+43.28 e+42.81 e+42.46 e+42.05 e+41.64 e+41.23 e+48.2 e+34.1 e+3o

von Mises Stress (Pa )

Fig. 6 The von-Mises stress distribution under center distance errors I:1dx = O.lmm ,

I:1dy = O.lmm and I:1dz = O.lmm.

Max. Value =1.06e+4 Pa

9.6 e+38.96 e+38.32 e+31.68 e+31.04 e+36.4 e+35.16 e+35.12 e+34.48e+33.84 e+33.2 e+32.56 e+31;92 e+31.28 e+3640o

von Mises Stress (Pa )

Fig. 7 The von-Mises stress distribution under misaligned angles.


CONCLUSIONS

In this paper, a complete mathematical model of the planetary gear mechanism with doublecircular-arc teeth is developed. A mathematical model for the imaginary rack cutter curve isproposed. Based on the double envelope concept, the geometric model of the planet gearbecomes the first envelope to the family of rack-cutter curves. The geometric model of the ringgear becomes the second envelope to the family ofplanet gears. In the high production efficiency,obtained planetary gear mechanism with double circular-arc teeth is found able to shoulder moreload.

The computerized simulation of the stress analysis in the planetary gear mechanism withdouble circular-arc teeth is investigated under two assembly errors, namely center distance error;and misalignment angles. In this study, the stress analysis of assembly errors is performed byvisualNastran for desktop 2004. It is found that a center distance assembly error is more seriousthan a misalignment errors. In other word, center distance assembly errors can quickly damagepower transmission and reduce its lifespan.

ACKNOWLEDGEMENT

The authors are grateful to the National Science Council of the Republic of China forsupporting this research under grant NSC 94-2212-E-018-003

References[1] F. L. Litvin, 1994, Gear Geometry and Applied Theory, Prentices-Hall, Englewood Cliffs,

NJ.[2] F. L. Litvin, 1989, Theory ofGearing, NASA Publication, Washington DC.[3] F. L. Litvin and J. Lu, 1995, "Computerized design and generation of double circular-arc

helical gears with low transmission errors," Comput. Methods Appl. Mech. Engrg. 127, pp.57-86.

[4] D. Door, A. Seireg, 1985, The Kinematic Geometry ofGearing, Wiley, New York.[5] BAR, G., 2003, "Explicit Calculation Methods for Conjugate Profiles," Journal for

Geometry and Graphics, Vol. 7, No.2, pp. 201-210.[6] C. K. Chen and C. Y. Wang, 2001, "Compensating analysis of a double circular-arc helical

gear by computerized simulation of meshing," Proceedings of the Institution ofMechanicalEngineers, Part C: Journal ofMechanical Engineering Science, v 215, n 7, pp. 759-771.

[7] Z. Tingjian, 1992, "Calculation of contact deformation for double circular-arc gears,"American Society of Mechanical Engineers, Design Engineering Division (Publication) DE,v 43 pt 1, Advancing Power Transmission Into the 21st Century, pp. 147-151.

[8] B. Wu, 2000, "Theoretical investigation on state of load distribution among contact bearingsof double circular arc helical gears", Chinese Journal ofMechanical Engineering (EnglishEdition), vB, n 2, Jun, pp. 108-113.

[9] M. A. K. Fahmy and R. Jonckheere, 1980, "Deflection and bending moment distribution ofgear teeth of double circular arc profile using three-dimensional holographic geometry,"American Society ofMechanical Engineers (Paper), n 80-C2/DET-109, 14p.

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[10] B. W. Bair, 2004, "Computer aided design of elliptical gears with circular-arc teeth" ,Mechanism and Machine Theory, 39, pp. 153-168.

[11] C. F. Chen and C. B. Tsay, 2004, "Computerized tooth profile generation and analysis ofcharacteristics of elliptical gears with circular-arc teeth" , Journal of Materials ProcessingTechnology. 148,pp.226-234.

[12] J. Lu, F. L. Litvin and 1. S. Chen, 1995, "Load Share and Finite Element Stress Analysis forDouble Circular-Arc Helical Gears," Journal ofMath I. Comput. Modelling, Vol. 21, No. 10,pp.13-30.

[13] A. M.M. EI-Bahloul, 1996, "Surface capacity if gears of circular-arc tooth-profile.", Wear193, pp. 146-154.

[14] S. C. Yang, 2007, "Application of Envelope theory and deviation function to circular gearpump", Mechanism and Machine Theory. 42, pp. 262-274.

[15] S. K. Kang, K. F. Ehmann and C. Lin, 1996, "A CAD Approach to Helical GrooveMachining-I. Mathematical Model and Model Solution," Int. J Mach. Tools Manufact, Vol.36, No.1, pp.141-153.

[16] H. C. Chyan and K. F. Ehmann, "Tapered-web helical groove machining," Journal ofMechanical Engineering Science Proceedings ofthe Institution ofMechanical Engineers, Vol.213, Part B, pp. 779-785.

[17] S. C. Yang, 2007, "Study on an internal gear with asymmetric involute teeth," Mechanismand Machine Theory. 42, pp. 977-994.

[18] S. C. Yang, 2007, "Modeling and Manufacturing of PP-Type Single Screw Compressor,"Canadian Society for Mechanical Engineering Transactions, Vol. 31, No.2, pp. 219-234.

[19] F. L. Litvin and A. Fuentes, 2004, Gear Geometry and Applied Theory, CambridgeUniversity Press, 2nd Ed, New York.

NOTATION:ef

hjf

hja

la' If

m

Pa

shifting coefficient of a rack cutter

vertical distance from point d to Yn -axis

vertical distance from point c to Yn -axis

design parameters of the rack cutter

module ofthe proposed planetary gear mechanism.radius ofthe convex curve of the rack cutter.

radius of the concave curve of the rack cutter.

radius ofcircular arc on regions cd and hi

position vector of imaginary rack cutter where the upper sign indicates

regions ab, bc, cd, de, ef, fg, gh, hi, and ai of the imagery

rack-cutter curve.the family of imaginary rack- cutter curves

the family ofplanet gears

Transactions ofthe CSME Ide fa SCGM Vol. 32, lVo. 2, 2008 280

S

Si(Oi' Xi' .r;)

()j

¢P2¢P3II

I 2 and I 3

linear displacement of rack cuttercoordinated systems where subscript i = 1, 2, 3, 1 denotes rack cutter

I I , 2 denotes planet gear I 2 , 3 denotes ring gear I 3

curvilinear coordinates of the rack cutter (j=1,2,3 and 4)

orientation angel of the planet gear about z 2 -axis

orientation angel of the ring gear about Z3 -axis

imaginary rack cutter (generating curve)

planet gear and ring gear (generated curves), respectively



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USING DOUBLE ENVELOPE METHOD ON A · PDF fileUSING DOUBLE ENVELOPE METHOD ON A PLANETARY GEARMECHANISM WITH DOUBLE CIRCULAR-ARCTOOTH ... archelical gear by computerized ... and gear