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Inter-Ing 2005„INTERDISCIPLINARITY IN ENGINEERING”

SCIENTIFIC CONFERENCE WITH INTERNATIONAL PARTICIPATION,TG. MUREŞ – ROMÂNIA, 10 -11 November 2005.

NUMERICAL SIMULATION OFWORM-FACE GEAR DRIVE MANUFACTURING

MĂTIEŞ VISTRIAN, TECHNICAL UNIVERSITY OF CLUJ-NAPOCA, NAPĂU ILEANA, „I.D.LAZĂRESCU” TECHNICAL COLLEGE CUGIR, NAPĂU RADU, S.C. TEAM TECHNOLOGY &SERVICES S.R.L., ROMANIA, NAPĂU MIRCEA, NAPĂU IOAN, CRH-NORTH AMERICA INC.,

USA

Key words: Worm-face gear, Computer Manufacturing Simulation

Abstract: The paper presents the manufacturing process a new face worm-gear drive, known asworm-face gear, through a novel computer-based simulation method, developed by authors for thisgear drive set but suitable for virtual manufacturing of any kind of gearing. The whole manufacturingprocess is illustrated with various numerical examples.

1. Introduction

The aim of this paper is to present an approach regarding to simulation ofmanufacturing tooth contact localization of the face worm-gear drive with a cylindrical ZK-type worm, known as worm-plane wheel drive or, as worm-face gear drive as well, with toothcontact localization, related with the patent and invention performed by Sudrijan et. al. [4].The entire presentation is based on 3D CAD simulation using a novel method known as CMS(Computer Manufacturing Simulation), developed by authors in two different CAD softwarepackages, CATIA and Pro/ENGINEER. The method, suitable for manufacturing processsimulation of any type of gear, provides a virtual 3D visual output and the ability to validate(within the precision limits of the CAD software), the accuracy of the calculation results.

2. Face gear teeth generation

As well as all orthogonal skew-axis gear drives, the worm-face gear drives transmituniform rotary motion between two fixed axes, through a combined action of rolling andsliding of two meshing tooth surfaces. The gear drive consists of a cylindrical ZK-type wormand a worm-face gear with teeth curved in a lengthwise direction (Fig. 1).

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Fig. 1. Meshing zone inworm-face gear drive.

Fig. 2. Generation of face gearteeth curves in its pitch plane.

Fig. 3. Face gear teeth generation by a cylindrical hob.

Fig. 4. The kinematics of meshing inworm-face gear drive.

The threads of cylindrical worm engage with transversal spiral teeth of a flat wormwheel, known as plane wheel or face gear. Because of the offset position of worm axis,relative to the face worm-gear axis, the meshing zone is shifted relative to the line O1O2,along the axes of both components worm and face worm-gear, respectively (Fig. 1). Thus, themultiple contact between worm threads and face worm-gear spiral teeth increases the contactratio and has a positive influence concerning noise reduction and load capacity enhancement,respectively. By definition, an extended involute is a product of the translation of a pointalong the line ģ and a uniform rotation about the center O (Fig. 2). Similarly, a worm helix isthe product of a uniform translation along the same line ģ and a uniform rotation about theaxis of the worm. Considering the two rotations such that the common rack generator ģ moveswith the same linear velocity in both elements, then the two curves will mesh correctly, thelocus of mutual contact being limited to a plane section along a straight line ģ. In thisapproach, if the cylindrical hob has a considerable length it is possible to generate teeth on aface gear blank, by simply feeding the hob into the face gear lengthwise, along the line ģ. Toinsure a proper continuity of engagement for any number of revolutions of the hob, the facegear spiral teeth must be equally spaced, with the arcs measured of the face gear base circlewith respect to the center O. This spacing must be exactly equal to the pitch of the generatingworm of the cylindrical hob. Thus, the generated extended involutes will have a constantpitch along a straight line g’. Because the lead angle γ01, of the cylindrical hob thread isconstant, for all its convolutions along the straight line g’, the face gear teeth flanks will beslightly mutilated along a certain portion of its pitch plane. However, this interference can bepractically avoided by making the offending portion of the worm threads slightly thinner onits pitch cylinder from rest of the thread flank, that is, by modifying the worm lead pE. Theresulted face gear teeth are of equal height and width, but of unsymmetrical profile. Theposition of the generating worm of the cylindrical hob, relative to the face gear blank axis, isset orthogonally, so that, its pitch cylinder is tangent to a plane of the face gear, perpendicularto its axis, plane further on called face gear pitch plane (Fig. 3).

From a kinematic prospective, the process of meshing in a worm-face gear drive isconsidered The worm is arranged such that, the projection of its axis on the face gear pitchplane is set orthogonally and offset relative to the face gear axis by the shortest centerdistance a = Rb - p, where Rb and p, are the radius of the face gear base circle, and the degreeof involute modification, respectively. If the worm and the face gear are rotated with theconstant angular velocities n1 and n2, respectively, about their own axes of rotation, theperipheral movement of the worm will be in the direction of the velocity vector v1, tangent toits pitch cylinder and directed perpendicular to its axis, whereas the peripheral movement of

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the face gear will be in the direction of the velocity vector v2, tangent to a circle called thecalculation pitch circle of RC radius.

Fig. 5. Simulation of spatial generationof face gear teeth curve in its pitch plane.

(z1 = 1, z2 = 45, a = 57.15 mm)

Fig. 6. Simulation of planar generationof face gear teeth curve in its pitch plane.

(z1 = 1, z2 = 60, a = 16 mm)

In the calculation point C, the point in which the two curves, the extended involutecurve of the face gear tooth and the worm pitch helix respectively, are tangent and have acommon normal n, which passes through the point I, the well- known equation of meshing istrue, where v12 is the relative velocity vector of the contact point C, situated in the face gearpitch plane, and directed tangent to the two curves in contact (Fig. 4). It can be shown, thatthe face gear base radius Rb, is the same as the face gear pitch radius [3]. Simulation ofspaţial generation of face gear teeth curve is illustrated in figure 5, and the planar one isillustrated in figure 6.

3. Mathematical model of worm-face gear manufacturing simulation

In order to reduce the shift of bearing contact caused by misalignment, and vibrationand noise in worm-face gear drives work, respectively, a mathematical model of a worm-facegear with localized tooth contact is developed. The method principle, adopted from reviewedliterature [1, 2], is adapted and developed to the specifics of the worm-face gear drives, inwhich the worm is of ZK-type.

Mathematical model of contact pattern localization is based on a double crowning ofthe worm thread surfaces with respect to the hob thread surfaces. Thus, through acombination of profile (Fig. 7) and longitudinal crowning (Fig. 8), the proposed method,achieves the tooth contact localization. It may be emphasized that in both cases parabolicconvex curves are used for worm crowning.

Application and development of this method implies synthesis and analysis of followingthree generating processes:

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Fig. 7. Profile crowning of the worm. Fig. 8. Longitudinally crowning of the worm .

Fig. 9. Synthesis of unmodified wormmanufacturing process and theoretical

face gear generation.

Fig.10. Simulation of worm threadgeneration by CMS method.(z1 = 1, z2 = 60, a = 16 mm)

a) unmodified worm thread surface and theoretical face gear teeth surface generation(Fig. 9). The worm surface of the gear set is generated by grinding with a disk shaped tool,simulation of worm thread generation being ilustrated in figure 10. Theoretical face gear teethsurfaces are generated kinematic by an enveloping process, with unmodified generating wormthreads, according to an initial set-up illustrated in figure 11. For some particular angles ofrotation, of generating worm, the contact lines on theoretical face gear teeth convexe flanksare illustrated in figure 12.

Fig. 11. Coordinate systems applied fortheoretical face gear teeth generation.

(z1 = 1, z2 = 45, a = 57.15 mm)

Fig . 12.Contact lines on convexe flanksof theoretical face gear teeth(z1 = 1, z2 = 60, a = 16 mm)

b) hob and face gear teeth manufacturing processes (Fig. 13). The hob active surfacesare generated by grinding also, being determined as the envelope to the family of another diskshaped tool surfaces. The side relief process of the hob teeth is simulated in figure 14. The

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face gear teeth surfaces are generated by hobbing process with a tangential feed, as envelopesto the family of hob teeth surfaces, according to an initial set-up illustrated in figure 16.

Fig. 13. Synthesis of hob and face gearmanufacturing processes.

Fig . 14. Simulation of hob side relief. (z1 = 1, z2 = 45, a = 57.15 mm)

Fig. 15. Coordinate systems applied for face gear manufacturing.

(z1 = 1, z2 = 45, a = 57.15 mm)

Fig . 16. Simulation of face gear teethgeneration by CMS method. (z1 = 1, z2 = 60, a = 16 mm)

c) worm double crowning process (Fig. 17). The correction of ZK-type functionalworm, consists of a combination of profile and longitudinal crowning of the active flanks ofunmodified worm thread surfaces already manufactured, using parabolic convexe function forcrowning. The 3D CAD model of a worm-face gear drive with double-crowned worm isillustrated in figure 18.

Fig. 17. Synthesis worm doublecrowning : in profile and longitudinally

manufacturing process.

Fig. 18. The 3D CAD model of a worm-face gear drive with localized contact.

(i12 = 45:1, a = 57.15 mm)

4. Simulation of contact pattern and contact stress

A procedure has been developed in order to perform the tooth contact analysis andcontact stress for worm-face gear drives. To simulate the idler tooth contact for a gear drive

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with localized contact a combinative algoritm , using CMS simulation method and power ofMathCAD software has been developed.

Fig. 19 Simulation of idler contact pattern on face gear flanks.(i12 = 45 : 1, a = 57.15 mm)

Fig. 20 Von Mises stress distributionpattern on face gear teeth.

(i12 = 60:1, a = 16 mm)Table 1. Geometric parameters of worm-face gear drives

with a ZK-type double-crowned worm.(z1 = 1, z2 = 60, a = 12 mm)

Pitch diameter of the worm (hob) d01 11.035 mmWorm axial module mx 0.554996278 mmWorm medium lead angle γ01 20 47’ 57,98”Direction of worm helicoid rightOuter diameter of the worm grinding disk das 125 mmFace gear outer diameter da2 48 mmFace gear inner diameter di2 35.2 mmPressure angles in worm normale section : - coasting flank - driving flank

αn1αn2

300 00’ 0”100 00’ 0”

Coefficients of the parabolic function for profile crowning : - coasting flank - driving flank

apr1apr2

0.002 [mm-1]0.002 [mm-1]

Coefficients of the parabolic function for longitudinalcrowning: - coasting flank - driving flank

al1al2

0.001 [mm-1]0.002 [mm-1]

Simulation of idler contact pattern on the face gear teeth flanks, is illustrated in figure19. It should be mentioned that, in addition to double crowning of worm, corrections are alsoapplied to the worm lead pE, in order to compensate the shift of contact pattern on the twoflanks of the face gear tooth. For a worm-face gear with localized contact having the designparameters from Table 1. loaded contact analysis has been carried out toghether with contactstress analysis by FEA method using the ANSYS V8 software. The FEM models used weredevelop by CMS method. The equivalent Von Misses stress on the face gear teeth for aninstantaneous contact position of the assembly worm-face gear, is presented in figure 20.

4. Conclusion

Through a novel method of computerized simulation of gear drives (CMS), the virtualmanufacturing process of worm-face gear drives with localized contact is presented. Themethod of simulation, suitable for any type of gear is based on the kinematics of generation,combined with the kinematic capabilities of most popular CAD software.

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6. References

[1] Litvin, F.L., Argentieri, G., Donno, M., and Hawkins, M., 2000, “Computerized design,generation and simulation of meshing and contact of face worm-gear drives,” ComputerMethods in Applied Mechanics and Engineering, 189, pp. 785-801.[2] Litvin, F. L., Nava, A., Fan, Q., and Fuentes, A., 2002, “New Geometry of Worm FaceGear Drives With Conical and Cylindrical Worms: Generation, Simulation of Meshing, andStress Analysis,” NASA Contractor Report No. 211895.[3] Napau, I.,1999,"Contributions to the Worm-Face Gear Drive Study," Ph.D. thesis,Technical University of Cluj-Napoca, (in Romanian).[4] Sudrijan, M., Cismas, P.T., and Napau, I., 1993, "Worm-Plane Wheel Drive”, RO PatentNo. 107302 B1 (in Romanian).