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FACTA UNIVERSITATIS Series: Mechanical Engineering Vol.1, No 9, 2002, pp. 1193 - 1198

DESIGN AND SIMULATION OF MESHING OF INTERNAL INVOLUTE SPUR GEARS WITH PINION CUTTERS UDC 621.833.16

Bo?idar Rosi?

Faculty of Mechanical Engineering, University of Belgrade 27. marta 80, 11000 Belgrade, Serbia and Montenegro E-mail: boza.r@eunet.yu

Abstract. This paper presents the kinematics and computer aided procedure for the design of internal involute spur gears. Geometric and operating constraints for internal gears are defined. For the defined kinematics model, a computer program is developed. The developed kinematics model is most helpful in designing, contact and stress analyzing, manufacturing, measuring and optimizing internal gear sets. The numerical results are tested by computerized simulation of mesh of internal involute spur gears with pinion cutters to demonstrate the developed model. Key words: internal gear, pinion, interference, design.

1. INTRODUCTION Internal gears have initially found application in design of planetary gear trains and mechanisms for their high gear ratio and small space requirement. In comparison with an external spur gear, the center distance of an internal spur gear is much shorter. Planetary gear trains have a number advantages as compared to the transmission with fixed shafts. Under similar operating conditions the planetary transmissions serve longer and produce less noise compared to the fixed shaft transmission. In recent years, many researchers have successfully investigated the surface geometry of spatial conjugate gear pairs [1,2]. The kinematics and geometric relation between the gear and the cutting tool during the profile generation process is the same relation as that developed when a gear meshes with a rack [3]. These analyses have greatly extended understanding of surface geometry and contact kinematics. In this paper, a kinematics model for the internal gear set is developed based on the cutting design parameters of the pinion cutter form used in generating internal spur gears.

Received January 17, 2003

1194

B. ROSI?

In order to ensure the mounting as well as the correct meshing of the gears, it is necessary to fulfil the requirements regarding their alignment and the clearance between the gears. It is necessary to express the above requirements by the corresponding functional constraints, and based upon them, to identify all relevant values together with the areas of their practical applications. Computer graphics of the gear set are presented to demonstrate the developed kinematics model. 2. TIP INTERFERENCE Interference may occur between internal and external tooth tips. Figure 1. Illustrates the condition when tip interference not exists. The critical point is when the pinion tip intersects the gear tip circle.

Fig. 1. Checking for tip interference. Condition for the actual clearance angles between the profiles of a tooth pair measured on the addendum circle of the tooth pair, must be satisfied:

δ = ?` 2 ? ?x > 0

From Fig. 2. the following equation can be derived:

(1)

ra21 = ra22 + a 2 ? 2 ? a ? ra 2 ? cos ? x

(2)

Design and Simulation of Meshing of Internal Involute Spur Gears with Pinion Cutters

1195

where ra1, ra2 – the tip radii of the pinion and gear respectively, The gear ratio gives the relation between the twist angles where the corresponding rotation of the internal gear is given by: z ? 2 = 1 ? ?1 (3) z2 The angles ?1 and ?2 are then calculated as follows:

` ?1 = arccos 2 ra22 ? ra2 1 ?a 2 ? ra1 ? a

(4) (5)

?` 2 = ? 2 ? ? 20

where ?10 = invα A1 ? invα w ? 20 = invα w ? invα A 2 αA1 and αΑ2 - the pressure angles at pinion and gear tooth tips respectively.

In connection with that, Fig. 2 represents the change of the functional constraints in the function of the rotation of angle ?1, with a different tooth number difference ?z.

2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 0 5 10 15 20

?z = 5 ?z = 10 ?z =15

Fig. 2. The effect of the rotation of angle ?1 upon the space requirements. Based upon the graphic representation of the results obtained, it follows that the given functional constraint is exceptionally sensitive to the change rotation angle of the external gear – pinion cutter. The tooth number difference is the major influential factor on the clearance angle. 2. SIMULATION OF MESHING The kinematics and geometric relation between the gear and the cutting tool during the profile generation process is the same relation as that developed when a gear meshes with

1196

B. ROSI?

a rack. Based on the basic law of conjugate action, the common normal of the surfaces at the contact point must pass through the instantaneous contact points on the surface of the cutting tool. In order to simulate the conditions of meshing, coordinate system O1x1y1 and O2x2y2 that are rigidly connected to pinion 1 and gear 2, respectively as shown in Figure 3. The meshing of the gear tooth surfaces is considered in the fixed coordinate system Cxy that is rigidly connected to the housing.

y1

x

y2

y

1

1

2

1

x1

O1

2

x2

O2

Fig. 3. Coordinates systems for generation of pinion tooth space. Transforming coordinate system O1x1y1 which is attached to the pinion cutter to fixed coordinate system Cxy, we can describe by:

G G ?i1 ? ?cos ?1 sin ?1 ??iG ? = ?G ? ? ? ? sin ?1 cos ?1? ?? j ? ? j1? ?

(6)

At the point of contact, due to the tangency of two contacting gear tooth surfaces, the position vectors and their unit normals of both gear tooth surfaces should be the same. Therefore, the following equations must be observed [5]:

G G ρ (f p ) = ρ (fg ) G G n (f p ) = n (f g )

(7) (8)

Design and Simulation of Meshing of Internal Involute Spur Gears with Pinion Cutters

1197

The shape of the pinion tooth is represented in coordinate system Cxy by the vector equation. Thus G G G G JJJG G ρ = CO1 + ρ1 = ?rw1 j + x1i1 + y1 j1 (9) where rw1 – cinematic circle of pinion cutter. We transfer the pinion tooth shape to fixed coordinate system Cxy, using the matrix equation: G G G ρ = ( x cos ?1 + y1 ? sin ?1 ) ? i + (? x1 ? sin ?1 + y ? cos ?1 ? rw1 ) ? j (10) where 2 ? x1 ? ?r ? ?1 ? cos ?1 ? cos α ? r ? sin ? ? (1 ? 0,5 ? ?1 ? sin 2α) ? ? ? ? ? 2 ? y1 ? = ?r ? ?1 ? sin ? ? cos α + r ? cos ?1 ? (1 ? 0,5 ? ?1 ? sin 2α )? ?t ? ? ? 1 ? 1? ? ? The position vector from O2 to the cutting point, expressed in fixed coordinate system is given by: G G G G G (11) ρ 2 = O2C + ρ = rw 2 j + xi + yj By applying the coordinate transformations, the equation of gear tooth surface can be represented in coordinate system O2x2y2 as follows: G G G ρ 2 = ( x cos ?2 ? y sin ?2 ? rw 2 sin ?2 ) i2 + ( x sin ? 2 + y cos ?2 + rw2 cos ?2 ) j2 (12) where

G ? ?i ? ? ?cos ? 2 ? G? = ? ? ? j? ? ? sin ? 2

G sin ? 2 ? ? i2 ? ?G ? cos ? 2 ? ? ? j2 ?

It is practically impossible to present in this paper all the relation between the gear and the cutting tool during the profile generation process. 5. THE RESULTS Computer graphs of pinion and internal gear can be plotted and represented in fixed coordinate system as shown in Fig 4.

y

x

Fig. 4. Computer graph of internal involute spur gear.

1198

B. ROSI?

6. CONCLUSION In this paper method for computer generation of involute internal spur gears has been developed. The developed kinematics model is most helpful in designing, contact and stress analyzing, manufacturing, measuring and optimizing internal gear sets. The numerical results are tested by computerized simulation of mesh of internal involute spur gears with pinion cutters to demonstrate the developed model. REFERENCES

1. Z. Ou, A. Seireg: Analysis and Synthesis of Circular Arc Gears by Interactive Graphics, ASME Journal of Mechanisms, Transmissions, and Automation in Design, MARCH 1986, Vol. 108, pp 65-71. 2. Colbrune, J. R.: The geometry of Involute gears, Springer–Verlag, 1990. 3. F. L. Litvin, J. Zhang, R. F. Handschuh: Crowned Spur Gears: Methods for Generation and Tooth Contact Analysis – Part I: Basic Concepts, Generation of the Pinion Tooth Surface by a Plane,Journal of Mechanisms, Transmissions, and Automation in Design, September 1988, Vol. 110, pp 337-342. 4. F. L. Litvin, J. Zhang, R. F. Handschuh: Crowned Spur Gears: Methods for Generation and Tooth Contact Analysis – Part 2: Generation of the Pinion Tooth Surface of Revolution, Journal of Mechanisms, Transmissions, and Automation in Design, September 1988, Vol. 110, pp 343-347. 5. F.L. Litvin, R. N. Goldrich, J.J. Coy, Z. B. Zaretsky: Kinematic Precision of Gear Trains, ASME Journal of Mechanisms,Transmissions, and Automation in Design, SEPTEMBER 1983, Vol. 105, No.pp 317-326. 6. Bai Hefeng, Michael Savage, Raymond James Knorr: Computer Modeling of Rack–Generating spur gears, Mechanism and Machine Teory, Volume 20 no 4, pp 351-360, 1985.

KONSTRUKCIJA I SIMULACIJA SPREZANJA ZUP?ANIKA SA UNUTRA?NJIM OZUBLJENJEM SA ALATOM U OBLIKU ZUP?ANIKA Bo?idar Rosi?

U radu je prikazan kinematski postupak kontruisanja zup?anika sa unutra?njim ozubljenjem podr?an odgovaraju?im kompjuterskim programom. Geometrijska i radna ograni?enja za zup?anik sa unutra?njim ozubljenjem su definisana. Za definisani kinematski model razvijen je kompjuterski program. Razvijeni kinematski model je od izuzetne pomo?i u kontruisanju, analizi kontaktnih napona, proizvodnji, kontroli i optimizaciji zup?astih parova sa unutra?njim ozubljenjem. Numeri?ki rezultati su testirani, kompjuterskom simulacijom sprezanja zup?anika sa unutra?njim ozubljenjem sa alatom u obliku zup?anika, da potvrde razvijeni model.

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