Precise modeling of arc tooth face-gear with transition curve

Chinese Journal of Aeronautics, (2013),26(5): 1346–1351

Chinese Society of Aeronautics and Astronautics & Beihang University

Chinese Journal of Aeronautics [email protected] www.sciencedirect.com

Precise modeling of arc tooth face-gear with transition curve Cui Yanmei a, Fang Zongde a b c

a,*

, Su Jinzhan b, Feng Xianzhang c, Peng Xianlong

a

School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072, China School of Construction Machinery, Chang’an University, Xi’an 710064, China School of Mechatronics Engineering, Zhengzhou Institute of Aeronautical Industry Management, Zhengzhou 450015, China

Received 8 October 2012; revised 27 November 2012; accepted 16 December 2012 Available online 28 April 2013

KEYWORDS Face-gear; Geometry; Kinematics; Mechanical drive; NC machine; Transmissions

Abstract A fabrication method is adopted for which an imaginary gear simultaneously realizes conjugated meshing with an arc tooth cylindrical gear and an arc tooth face-gear. The cutter fillet and tooth crest edge form the tooth root fillet of the gear, and the linear tooth surface equation of the imaginary gear and the position vector of the curvature center of the cutter fillet are constructed with certain cutter inclination to deduce a working arc tooth surface equation. The tooth root fillet equation of the arc tooth face-gear is derived from the meshing geometry and kinematics. A numerically controlled machining model of the arc tooth face-gear is established through the transformation of adjustment parameters from the cutter-tilt milling machine to a common multi-axis NC machine. Motion parameters of each movement axis of the NC machine are acquired. A processing example is presented to verify the precision of the fabrication method in processing the arc tooth face-gear. The method provides a theoretical and tentative basis for the analysis of tooth surface contact stress, tooth root bending stress and dynamics. A hobbing test is conducted to demonstrate the good meshing condition of the arc tooth face-gear pair. ª 2013 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA.

1. Introduction Arc tooth face-gear transmission is a new mechanical drive that achieves spatially intersecting or cross-axis gear drive and is characterized by stable transmission, high load carrying capacity, low meshing noise and simple structure. The transmission system is widely used in aviation, ocean vessels and * Corresponding author. Tel.: +86 29 88493958. E-mail addresses: [email protected] (Y. Cui), fauto@nwpu. edu.cn (Z. Fang). Peer review under responsibility of Editorial Committee of CJA.

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automobiles. As an arc tooth face-gear is a point-contact local-conjugate gear pair that adopts paired processing and application, the geometry of the tooth surface is complex and it is difficult to adjust the machining tool and construct an accurate geometric model of the tooth surface.1–8 Khoshnaw and Ahmed9 presented an equation for the tooth root fillet of a spiral bevel gear, constructed a three-dimensional model of the spiral bevel gear in Pro/E software, and generated a finite element mesh for the gear in ABAQUS software. Litvin and his research institution made a great contribution to research on a linear tooth-surface gear10–14 and presented new computerized developments in the design, generation, simulation of the meshing and stress analysis of gear drives. Tsay and Fong15 proposed a novel profile modification methodology for the molded face-gear drive to enhance the controllability of the contact pattern and transmission characteristics.

1000-9361 ª 2013 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. http://dx.doi.org/10.1016/j.cja.2013.04.006

Precise modeling of arc tooth face-gear with transition curve Guingand et al.16 proposed a simple formula for predicting the width of the wheel as a function of the main design parameters. Employing the formula, they were able to simulate the geometry and the quasistatic loaded behavior of a face gear. A genetic algorithm17 was used to determine the robust areas suitable for tooth modification. Studies have investigated face gears in terms of modeling and meshing, computer simulation of the contact face, integrated localized tooth contact, and manufacture.18–25 The manufacturing method described in this paper ensures the precise transmission of arc tooth face-gear pairs. The cutter fillet and tooth crest edge form the tooth root fillet of the gear, and the tooth root fillet equation of the arc tooth face-gear is deduced from the meshing geometry and kinematics to achieve smooth transition between the working and fillet tooth surfaces. The adjustment parameter for a cutter-tilt milling machine is converted to that for a common multi-axis numerically controlled (NC) machine according to spatial kinematics. A matrix is established for the transformation from the cutter coordinates to the machined workpiece coordinates. The matrix ensures the same relative position and motion for the cutter-tilt milling machine and the common multi-axis NC machine. The paper also establishes an NC machining model of the arc tooth face-gear. The model provides motion parameters for each movement axis of the NC machine, which offers precise processing parameters for NC processing. The arc tooth face-gear pair meshing condition is verified in a hobbing test.

2. Design of an arc tooth face-gear pair 2.1. Analysis of the manufacturing principle of an arc tooth facegear pair Fig. 1 is the tooth profile of a machining cutter. The working tooth surface is processed by a linear cutter edge. Because a fillet is machined with the top tooth edge of a gear shaper cutter, the tooth root fillet can be machined with a blade fillet and top tooth edge to decrease the tooth root bending stress. The fabricated gear is assumed to be a conjugate arc tooth cylindrical gear and arc a tooth face-gear to ensure the processed arc tooth face-gear can correctly mesh with an arc tooth cylindrical gear. The coordinate system Sa is fixed on the pitch line of the cutter and midpoint of the tooth width. In the figure, af is the tooth addendum of the cutter, a is the tooth profile angle, point C is the center of the top edge curvature of the cutter, M is a random point on the arc of the tooth root fillet, and q is the radius of the blade fillet. The equation for the tooth surface used for the linear cutter is: 3 2 lf cos a  af 7 6 ra1 ¼ 4 ðlf sin a þ bf  af tan aÞ 5 ð1Þ 0 where ‘+’ indicates the outer blade cutter, generating gear concave, and ‘’ corresponds the inner blade cutter, generating gear convex. lf is the modulus |M0M|, a parameter of the gear tooth rack and surface, bf ¼ pmn =4 is half gullet thickness. The coordinates of point C are deduced geometrically ðaf þ q; pmn =4  q sin a  ha mn cot aÞ, which can be denoted

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Fig. 1

Tooth profile of the cutter fillet.

as rc1 ¼ ½c d, where ha is the addendum coefficient and mn is normal modulus . Fig. 2 shows the process of manufacturing the gear. The cutter tooth is equivalent to one tooth of an imaginary gear. The movement of the cutter consists of two parts. One is a cutting movement, which rotates the cutter about its own shaft; this movement is the main movement of the cutting gear blank with the cutter tooth, and has nothing to do with the imaginary gear and generation of the gear blank. The other is the motion that fabricates the gear, and it involves the rotation of the imaginary gear, which includes the tooth surface of the imaginary gear formed by the linear cutter edge and the cutting edge of the blade fillet. When the imaginary gear and gear blank mesh with each other and rotate about their own axes according to the specified gear ratio, the fabricated gear surface covers a series of positions in the coordinate system of the gear blank, and the envelope of the surface of the generated gear on these positions is the tooth surface of the machined gear. To prevent the finished gear tooth surface from being cut during machining, the cutter has a certain tilt c, which results in a very small difference in the curvature radius between convexity and concavity of the tooth surface of a machined gear while improving the curvature properties of the tooth surface, and rp0 is the radius of the cutter gear. Sb (xb, yb, zb) is fixed at the center Of of the cutter, and rotates an angle of hf about this center and also about the center of the fabricated gear. Sc (xc, yc, zc) is achieved by rotating Sb an angle of c about the z axis; Sd (xd, yd, zd) is the auxiliary coordinate system of the cutter, which rotates about the center

Fig. 2

Coordinate systems of the fabricated gear.

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Od of the fabricated gear; Se (xe, ye, ze) is fixed at the midpoint of the tooth width of the pitch circle of the fabricated gear, and rotates about the center Oe of the fabricated gear; and S0 (x0, y0, z0) is fixed on the fabricated gear. By transforming the coordinates of the tooth-surface equation of the linear cutter to coordinates of the imaginary-gear coordinate system S0, the position vector and normal vector become ( rc0 ¼ M0a ra1 ð2Þ nc0 ¼ orohc0f  orolc0f where hf is the rotating angle of the cutter af sinðB=2rf Þ 6 hf 6 af sinðB=2rf Þ, rf is the nominal radius of the cutter, B is the gear tooth width, and M0a is a 4 · 4 matrix representing the transformation of the radius vector from the tooth surface of the cutter to the fabricated flank. M0a ¼ M0e Med Mdc Mcb Mba

ð3Þ

The matrix Mba describes the translation from Sa to Sb; The matrix Mcb describes rotation about z-axis from Sa to Sb, forming cutting angle c; The matrix Mdc describes rotation about x-axis from Sc to Sd, forming cutting major movement; The matrix Med describes translation from Sd to Se, which is the middle section of the tooth width on the pitch circle; The matrix M0e describes translation from Se to S0, which is the center of the cutter-gear rotation. Under the transformation of the coordinates of the center C of the top edge curvature of the cutter into the imaginary gear coordinate system S0, the position vector is r00 ¼ M0a rc1

ð4Þ

with the arc cylinder-gear, the center of the cutter is placed at rp2, on the pitch circle of the arc tooth face-gear. By directly transforming the position vector and normal vector of the tooth surface of the fabricated gear into the coordinate system of the arc tooth face-gear, the position vector and normal vector of the tooth surface of the arc tooth facegear are obtained as rc2 ¼ M20 rc0 nc2 ¼ L20 nc0

ð5Þ

where M20 is a 4 · 4 matrix representing the transformation of the radius vector from the fabricated gear surface to the tooth surface of the arc tooth face-gear. L20 is the matrix obtained by removing the last column and last row of the matrix. M20 ¼ M2i Mih Mh0

ð6Þ

In Fig. 3, Matrix M2i describes rotation /2 about x-axis from Si to S2, matrix Mih describes along the x-axis translation rp0 and z-axis translation rp2 from Sa to Sb, and rp2 is the radius of the arc tooth face-gear; matrix Mh0 describes rotation /0 about z-axis from S0 to Sh.A meshing equation can be written for the processing of the arc tooth face-gear: nc2 v20 2 ¼ f20 ðlf ; hf; /2 Þ ¼ 0

ð7Þ

v20 2

where is the relative velocity of the contact point between the cutting face of the cutter and the processed arc tooth face-gear in coordinate system S2. Solving simultaneous Eqs. (5) and (7), we get the equation for the tooth surface of an arc tooth face-gear: r2 ¼ r2 ð/2 ; hf Þ

ð8Þ

2.2. Working tooth surface model of the arc tooth face-gear

2.3. Tooth root fillet model of the arc tooth face-gear

As shown in Fig. 3, the fabrication method is adopted to process an arc tooth face-gear. S0 is fixed to the fabricated gear; Sh is the auxiliary coordinate system, which is fixed to the frame and provides the installation position of the fabricated gear; S2 is fixed to the arc tooth face-gear to be processed; and Si is another auxiliary coordinate system which is fixed to the frame and represents the installation position of the arc tooth facegear to be machined. To obtain good meshing transmission

The tooth root fillet equation of the arc tooth face-gear can be deduced from the meshing geometry and kinematics. We assume that the angular velocity of the imaginary gear cutter in the fixed coordinate system Sh is x1 = [0 0 x1]T. The angular velocity of the machined arc tooth face-gear in the fixed coordinate system Sh is x2 = [x2 0 0]T. The distance between the centers of the gear cutter and processed arc tooth face-gear in the fixed coordinate system Sh is assumed to be E = [rp0 0 rp2]T. The sliding velocity of the center C of the top edge curvature of the cutter in the fixed coordinate system Sh is v12 h ¼ ðx1  x2 Þ  rc1 ¼ E  x2

ð9Þ

The equation of the normal vector to the position vector rc1 of the fillet curvature center C in the fixed coordinate system Sh is N¼

orc1  v12 h ohf

ð10Þ

The unit normal vector is n¼

N jNj

ð11Þ

The position vector of a random point M on the arc of the transition curve in the fixed coordinate system Sh is Fig. 3 Coordinate systems for the processing of the arc tooth face-gear.

rmh ¼ r00 þ jnjq

ð12Þ

Precise modeling of arc tooth face-gear with transition curve

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By rewriting Eq. (12) for the coordinate system S2 fixed on the machined arc tooth face-gear, the tooth surface equation of the transition curve on the arc tooth face-gear is r2c ðhf ; /2 Þ ¼ M2h rmh

ð13Þ

where M2h is a 4 · 4 matrix representing the transformation of the radius vector from the fabricated gear surface to the tooth surface of the arc tooth face-gear. M2h ¼ M0a M20

ð14Þ

3. Processing of the arc tooth face-gear 3.1. NC machining model of the arc tooth face-gear Because of the complex structure of gear cutting machines with a tool-inclining mechanism that is difficult to adjust, the arc tooth face-gear gear will be processed instead with multi-axis NC machining tools. The machine named Free-form can simulate any tool-inclining mechanism or transgender mechanism. The machine concept is shown in Fig. 4. All the required movements are implemented along or about the six NC axes of gear cutting. The six axes, namely, three translational axes and three rotational axes, provide six degrees of freedom with which to flexibly control the position and movement of the workpiece and the tool in space. The machining is performed by the instantaneous control of the relative position and movement of the tool and the workpiece, which is fully realized by an NC machine, with three translational axes X, Y, and Z and three rotational axes A, B, and C. The rotation about the C axis is the main movement. The relative movement of the tool and the workpiece is decomposed into linear X, Y, and Z motions and rotary A, B, and C motions. The workpiece and the cutter are driven along or about the six NC axes through composite spatial movement. The motions are described as follows. Z axis: the negative direction is to the left and the positive direction to the right. Y axis: the negative direction is upward and the positive direction is downward. The Y axis has an inclination of 15. Taking the cutter axis as vertical, the axis concentric is 0 for the workpiece and tool plate. X axis: the negative direction is inward and the positive direction is outward. Taking the cutter axis as horizontal, the axis concentric is 0 for the workpiece and tool plate. C axis: the positive direction is counterclockwise. The cutter disc used to process arc tooth face-gear pairs is a Gleason cutter, as shown in Fig. 5. The cutter shape is based

Fig. 5

Gleason cutter.

on the modulus selection for straight teeth with standard pressure angle a. The profile angle c of the inner and outer cutting edges is adjusted according to the calculation of the cutter angle; the outer edge has an angle of (a  c) and the inner edge an angle of (a + c). The cutter teeth have the simplest linear shape, and are distributed on the rotating conical surface, which makes it easy to manufacture, sharpen, and test. The double-sided tool plate with an inner blade and outer blade can sweep convex and concave side surfaces of the cutting gear (cutting rack) using an appropriate tool at an inclined angle, which ensures a cylindrical arc and the appropriate meshing of the arc tooth face-gear. Thus, using the same tool to process the intermeshing gear pair reduces the number of tools required and greatly simplifies the processing. The fabrication relationships are determined by the relative position and movement of the tool and workpiece axes. To ensure their relative position and relative motion during coordinate conversion, matrices are created for the transformation from the moving coordinate system of the cutter plate to the movable coordinate system of the workpiece. The matrix guarantees the same relative position and movement of the workpiece and the tool in the tool-tilted machine and general multi-axis NC machine. MC2h ¼ M2h

ð15Þ

where M2h is a 4 · 4 matrix representing the transformation of the radius vector from the fabricated gear surface to the tooth surface of the arc tooth face-gear. ‘‘C’’ denotes the fabrication of the face gear in a multi-axis NC machine. The position vector follows the relationship 2 3 2 3 0 0 607 607 6 7 6 7 Oh O2 ¼ M2h 6 7 ¼ Oh OC2 ¼ MC2h 6 7 ð16Þ 405 405 1

1

From Eqs. (15) and (16) and the NC machining geometric relationship, it follows that 8 l ¼ f1 ð/2 Þ > > > > > ht ¼ f2 ðhf Þ > > > < w ¼ f3 ð/2 Þ ð17Þ > X ¼ f4 ð/2 Þ > > > > > Z ¼ rp2 > > : Y ¼ f5 ð/2 Þ

Fig. 4

Machine concept of free-form.

The movement is only along the Z-axis and Y-axis, and the cutter and workpiece have no relative movement along the X-axis. Rotary movements w and l are used to describe the

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rotation of the workpiece and the rotating angle of machine changing between the workpiece axis and the tool axis, and ht, which denotes the rotation of the tool plate around its axis, is not involved in forming the arc tooth face-gear. To conveniently program the NC machine and improve machining accuracy, all axis movements can be expressed in a high-order polynomial. The fourth-order Taylor expansion for motion along the Z-axis is z ¼ z0 þ z1 t þ z2 t2 þ z3 t3 þ z4 t4

ð18Þ

The coefficient of each order can be obtained from the position at the reference point t = 0. Coefficients of Eq. (18) are expressed as 8 z0 ¼ zjt¼0 > > > > > z1 ¼ oz j > ot t¼0 > < 1 o2 z z2 ¼ 2 ot2 jt¼0 ð19Þ > 3 > 1 o z > > z3 ¼ 6 ot3 jt¼0 > > > 4 : z4 ¼ 241 oot4z jt¼0

Fig. 6

An arc tooth face-gear fabricated by NC machining.

3.2. Machining example Table 1 presents the basic design parameters for an example of fabricating an arc tooth face-gear pair. Table 2 lists the movement parameters for each axis of a common multi-axis NC machine fabricating an arc tooth face-gear. The movement parameters of the multiple axes of an NC machine in the complete machining of an arc tooth face-gear are obtained by multi-numerical superposition. The manufactured arc tooth face-gear is shown in Fig. 6. The hobbing test is an important method with which to check the meshing condition. The arc tooth face-gear pair obtained by NC machining is inspected in a 90 hobbing machine, as shown in Fig. 7. The hobbing test inspection of the arc tooth face-gear pairs reveals good meshing, which verifies the precision of the meth-

Table 1

Basic design parameters.

Design parameters

Value

Cutter-gear tooth number z0 Arc face-gear tooth number z2 Pressure angle a() Normal modulus mn (mm) Shaft angle C() Radius of the cutter rf (mm) Initial machining angle c() Normal addendum coefficient Normal dedendum coefficient

42 180 22 3.0 90 120 6 1.0 1.25

Table 2

Fig. 7

90 hobbing machine.

od of fabricating the arc tooth face-gear and demonstrates the smooth transition between the working tooth surface and the curved tooth surface of the arc tooth face-gear. 4. Conclusions

(1) An equation for the linear tooth surface is established, which ensured the accurate meshing of the arc tooth face-gear pair. (2) The tooth root fillet equation of the arc tooth face-gear was derived, which ensures the smooth transition between the working tooth surface and the curved tooth surface of the arc tooth face-gear, and reduces tooth root bending stress of the face-gear. (3) An NC model of the arc tooth face-gear is established. The model provides a theoretical and tentative basis for the analysis of the tooth surface contact stress, tooth root bending stress and dynamics. The parameters of the movement axes in fabrication are obtained. A processing

Motion parameters in fabricating a face gear.

Movement axis

Constant term

1st-order

2nd-order

3rd-order

4th-order

Z X Y l w

17.0125 321.0000 119.0322 0.1402 1.2998

119.0342 0 17.0128 1.0000 0

22.9863 0 60.1021 0 0

20.0012 0 17.3564 0 0

6.9861 0 5.0153 0 0

Precise modeling of arc tooth face-gear with transition curve example verifies that the fabrication method applied in processing an arc tooth face-gear is suitable and it realizes smooth transition between the working tooth surface and the curved tooth surface of the arc tooth face-gear. The hobbing test of the arc tooth face-gear pairs shows a good meshing condition.

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14.

Acknowledgements This study was co-supported by the National Natural Science Foundation of China (No. 51175423), the Aeronautical Science Foundation of China (No. 2011ZB55002), Science & Technology Innovation Talents in Universities of Henan Province, ‘‘HASTIT’’ (No. 2012HASTIT023), and Assistance Scheme of Young Backbone Teachers of Henan Province Colleges and Universities (No. 2010GGJS-147).

15.

16.

17.

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Cui Yanmei received the M.S. degree in mechanical engineering from Tsinghua University in 2006, is a Ph.D. student at the School of Mechanical Engineering, Northwestern Polytechnical University. Her main research interests are mechanical transmission, piezoelectric ceramic drive. Fang Zongde is a professor and Ph.D. supervisor at the School of Mechanical Engineering, Northwestern Polytechnical University. His area of research includes mechanical transmission, vehicle design, and bionic machinery.