625_1

Download

Proceedings of IDETC/CIE 2005 ASME 2005 International Design Engineering Technical Conferences & Computers and Information in

Engineering Conference September 24-28, 2005, Long Beach, California, USA

DETC2005-84085

COMPUTERIZED MODELING AND SIMULATION OF SPIRAL BEVEL AND HYPOID GEARS MANUFACTURED BY GLEASON FACE HOBBING PROCESS

Qi Fan, Ph.D

The Gleason Works 1000 University Avenue

Rochester, NY 14607, USA [email protected]

Proceedings of IDETC/CIE 2005 ASME 2005 International Design Engineering Technical Conferences

& Computers and Information in Engineering Conference September 24-28, 2005, Long Beach, California USA

DETC2005-84085

ABSTRACT

The Gleason face hobbing process has been widely applied by the gear industry. But so far few papers have been found regarding the mathematic models of the tooth surface generations and tooth contact analysis (TCA). This paper presents the generalized theory of the face hobbing generation method, mathematic models of tooth surface generations and the simulation of meshing of face hobbed spiral bevel and hypoid gears. A generalized description of the cutting blades is introduced by considering four segments of the blade edge geometry. A kinematical model of a bevel gear generator is developed by breaking down the machine tool settings and the relative motions of the machine elemental units and applying coordinate transformations of the elemental motions. A generalized and enhanced TCA algorithm is proposed.

The face hobbing process has two categories of generation methods applied to the gear tooth surface generations, which are non-generated (Formate) and generated methods. In both categories the pinions are always finished with the generated method. The proposed tooth surface generation model covers both categories with left-hand and right-hand members. Based upon the developed theory, an advanced tooth surface generation and TCA program is developed and integrated into Gleason CAGE for Windows System. Two numerical examples are provided to illustrate the implementation of the developed mathematic models. Keywords: Face Hobbing, Tooth Surface Generation, Tooth Contact Analysis (TCA), Spiral Bevel and Hypoid Gears

INTRODUCTION Basically, there are two types of generation processes, face

milling and face hobbing, for manufacturing spiral bevel and

ed From: http://proceedings.asmedigitalcollection.asme.org/ on 05/07/2015 T

hypoid gears. The face hobbing process is implemented on the CNC hypoid gear generating machines by using TRI-AC or PENTAC face hobbing cutter systems [1, 2, 3].

Few papers have been found regarding the mathematic models of the tooth surface generations and tooth contact analysis (TCA) of face hobbed gear drives [4]. Most published papers are related to the face milling process [5, 6, 7, 8]. Generalized modern theory of gearing has been developed and can be applied to specific types of gear drives [9, 10,11, 12, 13].

The major differences between the face milling process and face hobbing process are: (1) in face hobbing, a timed continuous indexing is provided while in face milling, the indexing is intermittently provided after cutting each tooth side or slot, which is also called single indexing. Similar to face milling, in face hobbing the pinion is cut with the generated method and the gear can be cut with either the generated method or the non-generated (Formate) method. The Formate method offers higher productivity than the generated method because the generating roll is not applied in the Formate method. However, the generated method offers more freedoms of controlling tooth surface geometries; (2) the lengthwise tooth curve of face milled bevel gears is a circular arc with a curvature radius equal to the cutter radius while the lengthwise tooth curve of face hobbed gears is an extended epicycloid; and (3) face hobbing gear designs use the uniform tooth depth system while most face milling gear designs use tapered tooth systems.

Theoretically, the face hobbing process is based on the generalized concept of bevel gear generation in which the mating gear and the pinion can be considered respectively generated by the complementary generating crown gears as shown in Fig. 1. The tooth surfaces of the generating crown gears are kinematically formed by the traces of the cutting

1 Copyright 2005 by ASME

erms of Use: http://asme.org/terms

Dow

edges of the tool blades as shown in Fig. 2. The generating crown gear can be considered as a special case of a bevel gear with 90 pitch angle. Therefore, a generic term generating gear is used. The concept of complementary generating crown gear is considered when the generated mating tooth surfaces of the pinion and the gear are conjugate. In practice, in order to introduce mismatch of the mating tooth surfaces, the generating gears for the pinion and the gear may not be complementarily identical. The rotation of the generating gear is represented by the rotation of the cradle on a hypoid gear generator (Fig. 3).

Fig.1: Basic Concept of Bevel Gear Generation.

Fig. 2: Generation of Extended Epicycloids.

The advanced face hobbing system uses inverse optimization design methodology to formulate the parameters of the generating gears and parameters of the generating motions for the generations of a pair of comprehensively crowned gear and pinion. As a result, the face hobbed spiral bevel and hypoid gear sets offer optimized and stabilized bearing contact, bias direction and function of transmission errors, which result in reduced working stresses, vibration and

nloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 05/07/2015 T

noise. The optimized face hobbed gear sets are not sensitive to errors of alignment.

Generating gear

Cutter head Blades Work piece

Fig. 3: The Relationship of the Cutting Tool, Generating Gear and the Work.

RELATIVE MOTIONS OF FACE HOBBING PROCESS As described previously, the face hobbing process has

non-generated and generated methods defined in terms of the generation process for the gear. However, in both methods the pinion is always cut in generated method.

In the generated method, two sets of related motions are defined. The first set of related motion is the rotation of the tool (cutter head) and rotation of the work (work piece), namely,

t

w

w

t

NN

=

(1)

here, t and w denote the angular velocities of the tool and the work; and denote the number of the blade groups and the tooth number of the work respectively. This related motion provides the continuous indexing between the tool and the work. The indexing relationship can also be represented by the rotation of the tool and the generating gear as,

tN wN

t

c

c

t

NN

=

(2)

where c and denote the angular velocity of the generating gear and the tooth number of the generating gear respectively. Meanwhile, the indexing motion between the tool and the generating gear kinematically forms the tooth surface of the generating gear with an extended epicycloid lengthwise tooth curve as shown in Fig. 2. The radii of the rolling circles of the generating gear and the tool are determined respectively by

cN

sNN

NR

ct

cc += (3)

and

sNN

NR

ct

tt += (4)

where s is the machine radial setting.



Dow

The second set of related motion is the rotation of the generating gear and rotation of the work. Such a related motion is called rolling or generating motion and is represented as,

aw

c

c

w RNN

==

(5)

where is called the ratio of roll. aRIn the non-generated (Formate) face hobbing process,

only the first set of motion is provided for the gear tooth surface generation. Therefore, the gear tooth surfaces are actually the complementary copy of the generating tooth surfaces.

FACE HOBBING GENERATION MODEL Fig. 4 shows a kinematical model of monolithic column

design of Gleason CNC hypoid gear generators. The application of sophisticated CNC machines does not complicate the existing understanding on the bevel gear cutting theory and technologies that were established based upon the traditional mechanical cradle-style machines. Bevel gear engineers can still use the conventional terminologies and design tools to design the spiral bevel and hypoid gear drives, and to determine the machine tool settings which are related to the mechanical machines. Special computer codes have been developed to translate the mechanical machine tool settings and kinematical motions into the digitized instructions to control the motions of the axes of a CNC machine. The axes of the CNC machine move together in a numerically controlled relationship with changes in displacements, velocities, and accelerations to implement the prescribed motions and produce the target tooth surface geometry.

Fig. 4: Monolithic Column Model of Gleason CNC Hypoid

Generators.

In this paper, a generalized face hobbing generation model is developed. Based on a physical mechanical spiral bevel and hypoid gear generator, a kinematical model is developed (Fig. 5), which consists of eleven motion elements listed in Table 1. The cradle represents the generating gear, which provides

3

nloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 05/07/2015 T

generating roll motion between the generating gear and the generated work. In the non-generated process, the cradle is held stationary.

Fig. 5: Kinematical Model of a Mechanical Hypoid Gear

Generator.

Table 1: Machine Motion Elements and Axes of Rotation

No. Names of elements and related motion 1 Machine frame, motion reference 2 Cradle, rotation/cradle angle 3 Eccentric, radial setting 4 Swivel, swivel setting 5 Tilt mechanism, tilt setting 6 Tool/cutter head, rotation 7 Work, rotation 8 Work support, offset setting 9 Work head setting 10 Root angle setting 11 Sliding base setting a Cradle axis b Eccentric axis c Cutter head/tool spindle axis d Work spindle axis e Swivel pivot axis for root angle setting

Face hobbing machine settings are: ratio of roll ,

sliding base , radial setting aR

bX s , offset , work head setting , root angle

mE

pX m , swivel , and tool tilt . Although the kinematical model in Fig.5 is based on the mechanical machine, considering the ability of the CNC

j i

Copyright 2005 by ASME


D

machines, we can generally represent the machine setting elements as,

)(0 acaa RRR += (6) )(0 bcbb XXX += (7)

)(0 csss += (8)

)(0 mcmm EEE += (9) )(0 pcpp XXX += (10)

)(0 mcmm += (11) )(0 cjjj += (12)

)(0 ciii += (13) The first terms in Eqs. (6)-(13) represent the basic constant

machine settings and the second terms represent the dynamic changes of machine setting elements which might be kinematically dependent upon the motion parameter of the cradle rotation angle . These motions can be translated and implemented through computer codes on the CNC machines. Eqs. (6)-(13) provide strong flexibility of generating all kinds of comprehensively crowned and corrected gear tooth surfaces.

TOOL GEOMETRY The face hobbing process uses TRI-AC or PENTAC

face hobbing cutters which are different from face milling cutters. The cutter heads accommodate blades in groups. Normally, each group of blades consists of an inside finishing blade and an outside finishing blade with mean point M located at a common reference circle (see Fig. 6). Basically, the tool geometry can be defined by the major parameters of the blades and their installation on the cutter heads. Since the face hobbed tooth surfaces are kinematically generated by the cutting edges of the blades, exact description of the cutting edge geometry in space is important. The blade edge geometry can be described in the coordinate system that is connected to the cutter head with rotation parameter

tS (Fig. 6a). The

major parameters that define the cutting edge geometry in the cutter head are: nominal blade pressure angle , slot offset angle , rake angle , and effective hook angle .

(a) (b)

Fig. 6: Face Hobbing Tool and Blades.

ownloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 05/07/2015 T

Fig. 7 shows a basic geometry of the inside and outside blade edges which are represented in the coordinate system that is fixed to the front face of the blade. The origin coincides with reference point

bS

bOM with reference height .

Generally, the blade geometry consists of four sections: (a) blade tip, (b) Toprem, (c) profile, and (d) Flankrem. Sections (a) and (d) are circular arcs with radii and respectively. Sections (b) and (c) can be straight lines, circular arcs, or other kinds of curves. Section (c) generates the major working part of a tooth surface. Toprem and Flankrem relieve tooth root and tip surfaces in order to avoid root profile interference and tooth tip edge contact. In order to obtain a continuous tooth surface, the four sections of the blade curves should be in tangency at connections. For a current cutting point P on the blade the position vector and the unit tangent can be defined in the coordinate system ,

bh

er fr

bS)(ubb rr = (14) )(ubb tt = (15)

where u is the parameter.

(a) Inside Blade

(b) Inside Blade

Fig. 7: Blade Geometry.



Eqs. (14) and (15) can be represented in the cutter head

coordinate system , as tS)(),,,( uR bbtbt rMr = (16) )(),,,( uR bbtbt tMt = (17)

Here, matrix denotes the coordinate transformation from system to . Coordinate system is used to represent the rotation of the cutter head with an angular displacement

tbM

bS tS iS .

From Fig. 6, one can obtain the transformation matrix and itM ),()()( uu ititi rrMr == (18)

),()()( uu ititi ttMt == (19)

APPLIED COORDINATE SYSTEMS In order to mathematically describe the generation process,

we breakdown the relative motion elements of the kinematical model of a hypoid generator (Fig. 5). Coordinate system (Fig. 8a), called machine coordinate system, is fixed to the machine frame and considered as the reference of the related motions. System defines the machine plane and the machine center.

mS

mS

Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 05/07/2015 T

Fig. 8: Relationships of Coordinate Systems , , , and .

mS cS sS

rS

(a)

(b)

(c)

Fig. 9: Relationships of Coordinate Systems , , ,

and . rS pS oS

wS



Do

System is connected to the sliding base element 11 and represents its translating motion. System (Fig.8b) is connected to the machine element 10 and represents the machine root angle setting. System (Fig.8c) is connected to the cradle and represents the cradle rotation with angle

sS

rS

cS

+= 0qq , where is the initial cradle angle and 0q is the cradle increment parameter.

System is connected to the machine element 9 and represents the work head setting motion (Fig.9a). System is connected to the machine element 8 and represents the work head offset setting motion (Fig.9b). is connected to the work 8 and represents the work rotation with angular parameter

pS

oS

wS

(Fig.9c). System is connected to the eccentric setting element

and represents the radial setting (Fig.10a). System is connected to the tilt wedge base element which performs rotation relatively to the eccentric element to set the swivel angle (Fig.10b). System is connected to the cutter head carrier which performs rotation relatively to the tilt wedge base to set the tool tilt angle i (Fig.10c).

eS

jS

j iS

Through the coordinate transformation from to shown above, the position vector and the unit tangent at the current cutting blade point P can be represented in the coordinate system , namely,

iS wS

wS),( uiwiw rMr = (20) ),( uiwiw tMt = (21)

where is a resultant coordinate transformation matrix and is formulated by the multiplication of the following matrices representing the sequential coordinate transformations from to ,

wiM

iS

wS

jiejcemcsmrspropwowi MMMMMMMMMM = (22)

where matrices )(woM , , , )( mop EM )( ppr XM )( mrs M , , )( bsm XM )(mcM , , and can be

obtained directly from Figs.(8)-(10). By considering the related motions represented by Eqs. (1) and (5). Eqs. (20) and (21) can be generally re-written as,

)(sceM )( jejM )(ijiM

(a) (b)

6

wnloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 05/07/2015 T

(c)

Fig.10: Relationships of Coordinate Systems , , and

. mS eS jS

iS

),,( uww rr = (23) ),,( uww tt = (24)

here subscript w denotes that the vectors are represented in the coordinate system . wS

REPRESENTATION OF TOOTH SURFACE OF A NON-GENERATED (FORMATE) MEMBER

As discussed above, the non-generated (Formate) gear tooth surface is the complementary copy of the generating gear tooth surface which is kinematically formed by sweeping the cutting blade edge along an extended epicycloid lengthwise curve. During the gear cutting process, the cradle is held stationary. Therefore, parameter is assumed to be zero and Eqs. (23) and (24) can be re-written as

),( uww rr = (25) ),( uww tt = (26)

which give the position vector and unit tangent of the gear tooth surface. The unit normal of the gear tooth surface can be derived as,

www tkn = = ),( uwn (27) where the unit vector

=w

w

r

r

k w (28)

which represents the unit vector of hobbing speed. Eqs. (25)-(28) provide position vector, unit tangent, and unit normal of a non-generated gear tooth surface.

REPRESENTATION OF TOOTH SURFACE OF A GENERATED MEMBER

In addition to the relative hobbing motion or the indexing motion, for a generated member, either gear or pinion, the generating roll motion is provided and the generated tooth surface is the envelope of the family of the generating surface. The generated tooth surface can be represented as,



Do

=====

0),,(),,(

),,(),,(

www

ww

ww

ww

ufu

uu

vnnnttrr

(29)

where is obtained from Eq. (27), and the relative generating velocity is determined by

wn

wv

cw

= wr

v (30)

Equation 0),,( == www uf vn is called the equation of meshing [9, 10]. Eq. (29) defines the position vector, the unit tangent, and the unit normal of a generated work tooth surface. All these vectors are considered and represented in the coordinate system that is connected to the work. A unit cradle angular velocity, i.e.,

wS1=c might be considered. The

equation of meshing is applied for determination of generated tooth surfaces.

(a) Pinion Tooth Surfaces

(b) Gear Tooth Surfaces

Fig. 11: Pinion and Gear Tooth Surfaces of a Face Hobbed

Hypoid Gear Drive. The tooth surface generation models described above are

referred to the left hand work members. For the right hand members, the initial cradle angle in Fig. 10 should be replaced by . Fig. 11 shows a pair of mating pinion and gear tooth surfaces of a face hobbed hypoid gear drive (Fig. 12a), which is generated based on the described mathematic models.

0q

0q

7

wnloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 05/07/2015 T

TOOTH CONTACT ANALYSIS (TCA) Tooth Contact Analysis (TCA) is a computational

approach for analyzing the nature and quality of the meshing contact in a pair of gears. The concept of TCA was originally introduced in early 1960s as a research tool and applied to spiral bevel and hypoid gears [14, 15, 16]. Application of TCA technology has resulted in significant improvement in the development of bevel gear pairs under given contact conditions. TCA involves iteration processes and must be implemented on a digital computer. As the development and application of modern high-speed computers, TCA theory has been substantially enhanced. Tooth contact under load has been investigated and the Loaded TCA (LTCA) has been developed [17, 18, 19, 20]. Meanwhile, TCA has been not only used as a tool of analysis, but also integrated as a part of synthesis procedures [4, 5, 6, 9].

(a) Meshing of a Hypoid Gear Drive

(b) TCA Coordinate Systems

Fig. 12: Face Hobbing Tooth Contact Analysis (TCA) Model.



D

TCA is a process of simulation of meshing and contact of a

pair of gears under light load. Therefore, simulation of changes of the assembly errors and misalignments are taken into account. For this purpose, the adjusting parameters E, G, P, and are incorporated into the TCA model (Fig. 12b). The output of a TCA program are the scaled graphs of transmission errors and tooth bearing contact patterns that predict the results from a bevel gear testing machine.

By replacing the subscript w in the previous equations with subscript 1 and subscript 2, respectively, a pair of mating tooth surfaces of the pinion and the gear can be represented in the coordinate systems and that are connected to the pinion and the gear respectively (Fig. 12b), as follows,

1S 2S

for the pinion (31)

====

0),,(),,(),,(

),,(

1111

11111

11111

11111

ufuu

u

ttnnrr

for generated gear (32)

====

0),,(),,(),,(

),,(

2222

22222

22222

22222

ufuu

u

ttnnrr

for non-generated gear (33)

===

),(),(

),(

2222

2222

2222

uu

u

ttnnrr

Further, the mating tooth surfaces are transformed into a

common coordinate system that is fixed to the frame of the

gear drive and with origin located at the theoretical crossing point of the gear axis. The relationship of coordinate systems in Fig. 12b simulates the running meshing of the gear pair shown in Fig.12a and incorporates the adjusting parameters E, G, P, and . In case of spiral bevel gear drives

fS

fO

00 =E in Fig. 12b. The axes and are coincide with the rotation axes of the pinion and the gear respectively. The origins and of systems and would be at the theoretical crossing point if the adjusting parameters G=0 and P=0. Denoting

1Z 2Z

1O

2O 1S 2S

1 and 2 as the rotational motion parameters of the pinion and the gear respectively in Fig. 12b, we can obtain

for the pinion (34)

=

=

=

=

0),,(

),,,(

),,,(

),,,(

1111

1111]1[]1[

1111]1[]1[

1111]1[]1[

uf

u

u

u

ff

ff

ff

tt

nn

rr

8

ownloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 05/07/2015 Te

for generated gear (35)

=

=

=

=

0),,(

),,,(

),,,(

),,,(

2222

2222]2[]2[

2222]2[]2[

2222]2[]2[

uf

u

u

u

ff

ff

ff

tt

nn

rr

for non-generated gear (36)

=

=

=

),,(

),,(

),,(

222]2[]2[

222]2[]2[

222]2[]2[

u

u

u

ff

ff

ff

tt

nn

rr

The basic condition of two tooth surfaces 1 and 2 (Fig.

13) being in contact at a common point P is that the position vectors of the pinion and the gear are equal and their normals are co-linear. Mathematically, following equations are satisfied,

(37)

=

=]2[]1[

]2[]1[

ff

ff

nn

rr

However, it is difficult to computationally implement Eq.

(37) because of changing of the normal directions on the tooth surfaces. Therefore, following equivalent equations are proposed to replace Eq. (37),

=

=

=

0

0)(

0

]1[]2[

]1[]2[]2[

]2[]1[

ff

fff

ff

nt

ntn

rr

(38)

or

=

=

=

0

0)(

0

]2[]1[

]2[]1[]1[

]2[]1[

ff

fff

ff

nt

ntn

rr

(39)

Geometrically, vector and (i=1, 2) are two

orthogonal vectors that lie in the tangent planes of the pinion tooth surface

][][ if

if tn

][ift

1 (i=1) and gear tooth surface 2 (i=2). When the two surfaces contact at point P, the tangent planes coincide and become a common tangent plane (Fig. 13). Considering the equations of meshing, Eq. (38) or (39) generally yields five independent equations. Given a motion parameter, say 1 , which physically means rotation of the driving member, Eq. (38) or (39) can be solved for the rest of the parameters if the related Jacobian differs from zero [9,10]. And, consequently, a series of contact points and the corresponding transmission errors can be obtained, which formulate the TCA output as the bearing contact patterns and the graph of transmission errors. The transmission error (TE) is defined as,


rms of Use: http://asme.org/terms

Fig. 13: Tooth Surface Contact.

2

1101202 )()( N

NTE = (40)

where 10 and 20 are the initial angular displacement of the pinion and the gear when the tooth surfaces are in contact at the initial conjugate contact position where the TE equals zero; and are the tooth numbers of the pinion and the gear respectively.

1N

2N

IMPLEMENTATION OF THE TCA ALGORITHM Based on the developed mathematic models for the face

hobbing generation of spiral bevel and hypoid gears and the TCA algorithm, an advanced TCA program for the face hobbing process has been developed for both non-generated and generated spiral bevel and hypoid gear drives. The program incorporates simulation of meshing of the whole tooth surface contact including those surface parts that are generated by the blade Toprem and Flankrem. Tooth edge contact simulation is also developed and included in the program.

There are two cases of bearing contact patterns that are identified as single meshing pattern and multiple meshing pattern. In the multiple meshing patterns, the adjacent pairs of meshing teeth are considered simultaneously. And, the path of contact reflects about one pitch meshing because the following pairs of teeth take over the meshing process. However, in single meshing pattern, only one pair of tooth meshing is investigated and the adjacent meshing is ignored. In this case, the contact pattern covers larger area of the tooth surface. Meanwhile, tooth tip edge contact or Toprem contact might be observed in the single meshing case because no neighboring teeth interrupt the meshing process. In both cases, three curves of the transmission errors are shown in order to visualize the meshing positions and the corresponding transmission errors.

In practice, when the surface crowning of the mating gears is small or the two mating surfaces are close to conjugate, a large contact pattern might be observed from the testing machine, which may correspond to the single meshing contact pattern of TCA (Fig.14 and 16). When the magnitude of tooth surface crowning is significant, the multiple meshing contact

9

Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 05/07/2015 T

patterns might be obtained from the testing machine (Fig. 15 and 17).

The developed program is able to simulate both the single meshing and the multiple meshing of a pair of gear set. TCA results for the mean, toe, and heel positions are provided, which is the Gleason convention of TCA outputs. The developed TCA program accommodates all kinds of modified and corrected tooth surfaces generated by the face hobbing process. Fig. 14 and 16 show the single meshing TCA results. The tip edge contact and the Toprem contact are observed, which actually can be avoided if the surface crowning is sufficient. Fig. 15 and 17 show the multiple meshing TCA results at mean, toe and heel positions respectively. The adjusting parameters E and P are computed in order to achieve the corresponding bearing contact pattern positions. The program offers the abilities for the users to simulate meshing contact under user defined misalignments or user defined initial conjugate contact positions. The single meshing and multiple meshing can be simulated by selecting different roll pitch factors.

Table 2 shows the dimension data of two numerical examples which are a non-generated (Formate) hypoid gear drive (Design A, left-hand pinion) and a generated spiral bevel gear drive (Design B, right-hand pinion). Fig. 14 and 16 show the single meshing TCA results of Design A and B respectively at the mean initial contact position. Fig. 15 and 17 show the multiple meshing TCA results of Design A and B respectively at mean, toe and heel contact positions.

Table 2: Data of Design A and B

Design A

(Hypoid)

(Formate Gear)

Design B

(Spiral Bevel)

(Generated Gear)

Pinion

(Left

Hand)

Gear

(Right

Hand)

Pinion

(Right

Hand)

Gear

(Left

Hand)

Number of Teeth 11 39 14 29

Diametral Pitch 5.015 3.175

Face Width 1.611 1.325 1.835 1.835

Pinion Offset 1.5 0.0

Shaft Angle 90 90

Outer Cone

Distance

3.821 4.390 5.071

Mean Cone

Distance

2.965 3.681 4.154

Outside

Diameter

3.737 7.825 4.894 9.288

Cutter Radius 88 mm 105 mm



Do

Fig. 14: Single Meshing (Design A) - Mean Contact.

(a) Multiple Meshing (Design A) - Mean Contact

(b) Multiple Meshing (Design A) - Toe Contact

10

wnloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 05/07/2015 Te

(c) Multiple Meshing (Design A) - Heel Contact

Fig. 15: TCA Output of Design A.

Fig. 16: Single Meshing (Design B) - Mean Contact.

(a) Multiple Meshing (Design B) - Mean Contact


rms of Use: http://asme.org/terms

D

(b) Multiple Meshing (Design B) - Toe Contact

(c) Multiple Meshing (Design B) - Heel Contact

Fig. 17: TCA Output of Design B.

CONCLUSIONS This paper presents the theory of the face hobbing process.

The kinematics of the face hobbing process is described in comparison with the face milling process. A generalized spiral bevel and hypoid gear tooth surface generation model for the face hobbing process is developed. The related coordinate systems are directly associated with the physical machine setting and motion elements of a bevel gear generator. The generation model covers both non-generated (Formate) and generated methods of the face hobbing process. A new TCA algorithm is developed for the face hobbed gear drives, which can also be applied to other types of gearings. An advanced TCA program has been developed and integrated into CAGE for Windows System. Two TCA examples of Design A (a hypoid gear drive) and B (a spiral bevel gear drive) are illustrated with TCA outputs.

1

ownloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 05/07/2015 T

ACKNOWLEDGMENTS This paper presents part of the work on the development of

the Gleason Advanced TCA System. The author would like to acknowledge his colleagues, Ronald S. DaFoe, John Swanger, Arthur Pastor, Theodore J. Krenzer, Qiming Lian, Lowell Wilcox, Frank Peppers and Robert Middleton, for their contributions in the development of the advanced TCA technology and software which has been released by The Gleason Works.

REFERENCES 1. Krenzer, T. J., 1990, Face-Milling or Face Hobbing,

AGMA, Technical Paper, 90 FTM 13. 2. Stadtfeld, H. J., 2000, Advanced Bevel Gear Technology,

The Gleason Works, Edition 2000. 3. Pitts, L. S. and Boch, M. J., 1997, Design and

Development of Bevel and Hypoid Gears using the Face Hobbing Method, Cat. #4332, The Gleason Works.

4. Fan, Q., 2001, Computerized Design of New Type Spur, Helical, Spiral Bevel and Hypoid Gear Drives, Ph.D. Thesis, University of Illinois at Chicago, USA.

5. Litvin, F. L., Fan, Q., Fuentes, A. and Handschuh, R. F., 2001, Computerized Design, Generation, Simulation of Meshing and Contact of Face-Milled Formate-Cut Spiral Bevel Gears, NASA Report, /CR-2001-210894, ARL-CR-467.

6. Litvin, F. L. and Zhang, Y., 1991, Local Synthesis and Tooth Contact Analysis of Face-Milled Spiral Bevel Gears, NASA Contractor Report 4342.

7. Zhang, Y., Litvin, F. L., and Handschuh, R. F., 1995, Computerized Design of Low-Noise Face Milled Spiral Bevel Gears, Mechanism and Machine theory, 30 (8), pp. 1171-1178.

8. Lewicki, D. G., Handschuh, R. F., Henry, Z. S. and Litvin, F. L., 1994, Low-Noise, High-Strength Spiral Bevel Gears for Helicopter Transmissions, Journal of Propulsion and Power, Vol. 10, No. 3.

9. Litvin, F. L., 1994, Gear Geometry and Applied Theory, Prentice Hall.

10. Litvin, F. L., 1989, Theory of Gearing, NASA Reference Publication 1212.

11. Dooner, D. B., 2002, On the Three Laws of Gearing, ASME Journal of Mechanical Design, December 2002, Vol. 124, pp.733-744.

12. Dooner, D. B., and Seireg, A. A., 1995, The Kinematic Geometry of Gearing: A Concurrent Engineering Approach, John Wiley and Sons, Inc., New York.

13. Honda, S., 1996, A Pair of Tooth Surfaces without Variation of Bearing Loads, ASME Proceedings of the 7th International Power Transmission and Gearing Conference, San Diego, USA.

14. Tooth Contact Analysis Formulas and Calculation Procedures, The Gleason Works Publication, SD 3115, April 1964.

15. Basic Geometry and Tooth Contact of Hypoid Gears, The Gleason Works Publication, SD 4050A, August 1971.



Do

16. Krenzer, T. J., 1981, Understanding Tooth Contact Analysis, The Gleason Works Publication, SD3139B/381/GMD.

17. Krenzer, T. J., 1981, Tooth Contact Analysis of Spiral Bevel and Hypoid Gears Under Load, The Gleason Works, SD3458 (SAE 810688).

18. Vijayakar, S. M., and Houser, D. R., 1991, Contact Analysis of Gears Using a Combined Finite Element and Surface Integral Method, AGMA Paper 91FMT16.

19. Gosselin, C., Cloutier, L., and Nguyen, Q. D., 1992 The Influence of the Kinematical Motion Error on the Load Transmission Error of Spiral Bevel Gears, AGMA Paper 92FMT10.

20. Wilcox, L. E., Chimner, T. D. and Nowell, G. C., 1997, Improved Finite Element Model for Calculating Stresses in Bevel and Hypoid Gear Teeth, AGMA, 97FTM5.


wnloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 05/07/2015 Terms of Use: http://asme.org/terms

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ]>> setdistillerparams> setpagedevice

Documents

625_1