Kinematical Simulation of Face Indexing and Tooth Surface Generation of Spiral Bevel and Hypoid Dr. Qi Fan

Management Summary In addition to the face milling system, the face hobbing process has been developed and widely employed by the industry for manufacturing spiral bevel and hypoid gears. However, the mechanism of the face hobbing process is Dr. Qi Fan is a theoretician not well known by gear researchers and engineers. This paper presents the general- at The Gleason Works of Rochester, NY. ized theory of the face hobbing process, mathematical models of tooth lengthwise Prior to that, he worked in the Gear curve generation and tooth surface generation. The face hobbing indexing motion Research Center at the Mechanical and is simulated and visualized. A face hobbing tooth surface generation model is Industrial Engineering Department at the developed, which is directly related to the kinematical mechanisms of a physi- University of Illinois at Chicago. Fan has cal bevel gear generator. The developed mathematical models accommodate two authored more than 20 papers on bevel categories of the face hobbing system: non-generating (Formate) and generating. gear research.

30 JANUARY/FEBRUARY 2006 • GEAR TECHNOLOGY • www.geartechnology.com • www.powertransmission.com Introduction Gear There are two types of face hobbing processes used to generate the tooth surfaces of spiral bevel and hypoid gears, a non- generated (Formate®) process and a gen- erated process (Refs. 1, 2, 3). However, the is always manufactured using a generated process. The major differences between the face milling process and the face hobbing pro- cess are: (1) in the face hobbing process, a timed continuous indexing is provided, Pinion while in the face milling process, the index- ing is intermittently provided after cut- Generating crown gears ting each tooth side or slot. Similar to the face milling process, in the face hobbing Figure 1—Basic concept of bevel gear generation. process, the pinion is cut with a generated method and the gear can be cut with either a non-generated (Formate®) or a generated method. The Formate method offers higher productivity than the generating method because generating roll is not applied in the Formate method. However, the generating method offers more freedom for controlling tooth surface geometries; (2) the lengthwise tooth curve of face milled bevel gears is a circular arc, while that of face hobbed gears is an extended epicycloid; and (3) face hob- bing gear designs use a uniform tooth depth system, while face milling gear designs use a tapered tooth depth system. 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 respective- ly generated by the complementary generat- ing crown gears, as shown in Figure 1. The tooth surfaces of the generating are kinematically formed by the traces of the cutting edges of the tool blades, as shown in Figure 2. The generating crown gear can be considered as a special case of a bevel gear Figure 2—Relationship of the cutting tool, generating gear and the work. with 90˚ pitch angle. Therefore, a generic result, face hobbed spiral bevel and hypoid term “generating gear” is usually used. gear sets can offer optimized bearing con- The concept of a complementary generating tact, bias direction and function of transmis- crown gear is considered when the gener- sion errors, resulting in reduced working ated mating tooth surfaces of the pinion and stresses, vibration and noise. Optimized the gear are conjugate. In practice, in order face hobbed gear sets are less sensitive to to introduce mismatch and crowning of errors of alignment. the mating tooth surfaces, generating gears Related Motions of the Face for the pinion and the gear may not be Hobbing Process complementarily identical. The rota- In the generated face hobbing method, tion of the generating gear is represented two sets of related motions are defined. The by the rotation of the cradle on a hypoid first set of related motions is the rotation of gear generator. the tool (cutter head) and the rotation of the The face hobbing system can use work (workpiece), namely, advanced design methodology to formulate the parameters of the generating gears. As a www.powertransmission.com • www.geartechnology.com • GEAR TECHNOLOGY • JANUARY/FEBRUARY 2006 31 where ωc and Nc denote the angular veloc- ity of the generating gear and the tooth number of the generating gear, respectively. Meanwhile, the indexing motion between the tool and theFan generating Equations gear kinematical- Draft 1 ly forms the tooth12/9/2006 surface of the generating gear with an extended epicycloid lengthwise tooth curve, asEquation shown in1: Figure 3. The radii of the rolling circlesω w ofN t the generating gear = = Rtw and the tool areω determinedt N w respectively by

Equation N2: R = c ⋅ s (3) ωc N c N t +t N c = = Rtc and ω t N c N t R t = ⋅ s (4) EquationN t 3:+ N c

N c here s is the machineRc = radial setting.⋅ s The second set N oft + relatedN c motions is the rotation of the generating gear and rota- Equation 4: tion of the work. Such a related motion is Figure 3—Generation of extended epicycloids. N called rolling R or = generatingt motion⋅ s and is t + represented as N t N c Equation 5:

ω w N c = = R a (5) ωc N w Equation 6: where Ra is called the ratio of roll. Basically, the second setr b of= relatedrb (u) motions provides generated toothEquation geometry 7: in the profile direction. t b = t b (u) In the non-generatingEquation 8: (Formate®) face hobbing process,rt =onlyM tbthe(δ ,firstλ,κ set, R bof)r motionb (u) (a) (b) or indexing motionEquation is provided9: for the gear = δ λ κ Figure 4—Face hobbing cutter head and blades. tooth surface generation.t t M tb ( Therefore,, , , Rb ) thet b ( ugear) tooth surfacesEquation are actually 10: the complemen- r = M (θ )r (u) = r (u,θ ) tary copy of thei generatingit tootht surfaces.i Equation 11: Tool Geometry t = M (θ )t (u) = t (u,θ ) The Gleasoni face ithobbingt processi uses Equation 12: TRI-AC® or PENTAC® face hobbing x  cutters. Face hobbingg  cos[( cutters1+ R aretc ) θ differ-] − sin[(1+ Rtc )θ ]xt  S H cos(Rtcθ ) + SV sin(Rtcθ )   =    +   ent from face y millingg  sin[( cutters.1+ R tc The)θ ] cut-cos[(1+ Rtc )θ ] yt   S H sin(Rtcθ ) − SV sin(Rtcθ ) ter heads accommodateEquation 13: blades in groups.

Normally, eachR agroup= Ra 0of+ bladesRac (ϕ )consists of an inside finishingEquation blade 14: and an outside fin- ω N w t (1) ishing blade withX = referenceX + X point(ϕ ) M locat- = = Rtw b b0 bc ωt N w ed at a commonEquation reference 15: circle, and the Here, ω and ω denote the angular t w blades are evenlys = spaceds0 + s con(ϕ the) cutter head

velocity of the tool and the work; Nt and (see Fig. 4). InEquation some specially 16: designed cut- ter heads, the inside and outside blades are Nw denote the number of blade groups and the number of teeth in the work, respective- not evenly spaced and therefore the blade ly. This related motion provides indexing radial position is correspondingly adjusted. between the tool and the work. The index- Basically, the tool geometry can be defined ing relationship can also be represented by by the major parameters of the blades and the rotation of the tool and the generating their installation on the cutter heads. Since gear as, the face hobbing tooth surfaces are kinemat- ically generated by the cutting edges of the ω N c t (2) blades, an exact description of the cutting = = Rtc ωt N c edge geometry in space is very important.

32 JANUARY/FEBRUARY 2006 • GEAR TECHNOLOGY • www.geartechnology.com • www.powertransmission.com The blade edge geometry can be described in the coordinate system St that is connected to the cutter head with rotation parameter θ. The major parameters that affect the cutting edge geometry are: nominal blade pressure angle α, blade offset angle δ, rake angle λ, and effective hook angle κ. Figure 5 shows a basic geometry of the inside and outside blade edges, which are represented in the coordinate system Sb that is fixed to the front face of the blade. The origin Ob coincides with reference point M at reference height hb. Generally, the blade geometry consists of four sections: (a) tip, (b) Toprem®, (c) profile, and (d) Flankrem. Sections (a) and (d) are circular arcs with radii re andFan rf Equations respectively. Sections (b) and (c) can beDraft straight 1 lines, circular arcs, or other kinds12/9/2006 of curves. Section (c) generates the majorEquation working 1: part of a tooth surface. Toprem andω FlankremN relieve tooth root and w = t = R tw (a) tip surfacesω t in orderN w to avoid root profile interference and tooth tip edge contact. In order to obtainEquation a 2:continuous tooth surface, ω N the four sectionsc oft the blade curves should = = Rtc be in tangencyωt atN connections.c For a current cutting pointEquation P on3: the blade, the position vector and the unit tangent can be defined N c Rc = ⋅ s in the coordinate system Sb, N t + N c

Equation 4: rb = rb (u) (6) N t Rt = ⋅ s N t + N c Equation 5: t b = t b ( u ) (7) ω w N c = = Ra where ωuc is theN wparameter. EquationsEquation 6 and 6: 7 can be represented in r = r (u) the cutter headb coordinateb system S , as Equation 7: t

t b = t b (u) Equation 8: (8) rt = M tb (δ ,λ,κ , Rb )rb (u) Equation 9: (b) t t = M tb ( δ , λ , κ , R b ) t b ( u ) Equation 10: (9) Figure 5—Blade geometry. t t = M tb (δ ,λ,κ , Rb )t b (u) ri = M it (θ )rt (u) = ri (u,θ ) Equation 11: here matrix Mtb denotes the coordinate Kinematical Simulation of t i = M it (θ )t t (u) = t i (u,θ ) transformation from Sb to St. Coordinate Face Hobbing Indexing Motion Equation 12: system Si is used to represent the rotation As described above, the indexing xg  cos[(1+ )θ ] − sin[(1+ )θ ] x cos( θ ) + sin( θ ) of the cutter head with an Rangulartc displace- Rtc  t  S H Rtc SV Rtc    =  motion  generates+  the cutting path or the ment θ. Fromy g  Figuresin[( 4,1 +oneRtc )canθ ] obtaincos[( 1the+ R tc )θtooth] y t lengthwise  S H sin( R curvestcθ ) − SV thatsin( R aretcθ ) called Equation 13: transformation matrix Mit and represent the the extended epicycloids (see Fig. 3). The Ra = Ra0 + Rac (ϕ) blade geometry in system Si as indexing motion can be simulated and visu- Equation 14: alized in terms of the motion relationship X = X + X (ϕ) r =b M ( bθ 0 ) r ( u bc ) = r ( u , θ ) (10) between the generating gear and the cut- Equationi it 15: t i s = s + s (ϕ) ter. Furthermore, such a motion can be 0 c Equation 16: represented in the cross-section plane that (11) intersects the blade edges at the reference t i = M it (θ )t t (u) = t i (u,θ )

www.powertransmission.com • www.geartechnology.com • GEAR TECHNOLOGY • JANUARY/FEBRUARY 2006 33 point M and is perpendicular to the axis of the generating gear so that the motion can be visualized on a plane, as shown in Figure 6. The simulation is based on the following matrix transformation, which represents the relative motion of the cutter head with respect to the generating gear.

(See Figure 7 for Equation 12)

The coordinate system Sg is connected

to the generating gear (Fig. 6), and xg and

yg are the coordinates of the tooth length- wise curves, i.e., the extended epicycloids.

Coordinates xt and yt specify points on the

blades in system St. SH and SV are the initial horizontal and vertical settings of the cutter head. Giving a value to parameter θ corre- sponds to a specific cutting position. Figures 6 (a) and (b) respectively show a single cut- (a) ting position and multiple cutting positions, from which the kinematical generation of tooth lengthwise curves and the tooth slots can be visualized. Modeling of Face Hobbing Tooth Surface Generation The face hobbing process can be imple- mented on the Gleason Phoenix® series CNC hypoid machines (see Fig. 8). The machine tool settings and the tooth surface generation model can be derived based on traditional cradle-style mechanical machines. Computer codes have been devel- oped to “translate” the machine settings, and CNC hypoid gear generators 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. In this paper, a generalized face hobbing kinematical model is developed. The pro- (b) posed model, shown in Figure 9, is based on the mechanical spiral bevel and hypoid gear Figure 6—Simulation of face hobbing indexing motion. generators. The model consists of eleven motion elements, which are listed in Table xg  cos[(1+ R )θ ] − sin[(1+ R )θ ]x  S cos(R θ ) + S sin(R θ ) = tc tc t + H 1. The cradle-styletc V representstc the generating       gear, which provides generating roll motion y g  sin[(1+ Rtc )θ ] cos[(1+ Rtc )θ ] yt   S H sin(Rtcθ ) − SV sin(Rtcθ )   between the generating gear and the gener-

xg  cos[(1+ Rtc )θ ] − sin[(1+ Rtc )θ ]xt  S H cos(Rtcθ ) + SV sin(Rtcθ ) ated work. In the non-generated (Formate®)   = + process, the cradle is held stationary.  + θ + θ    θ − θ  y g  sin[(1 Rtc ) ] cos[(1 Rtc ) ] yt   S H sin(Rtc ) SV sin(Rtc ) Face hobbing machine settings are: (1)

Figure 7—Equation 12. ratio of roll Ra; (2) sliding base Xb; (3) radial

setting s; (4) offset Em; (5) work head setting

Xp; (6) root angle γm; (7) swivel j; and (8) tool tilt i. Although the kinematical model in Figure 9 is based on the cradle-style genera-

34 JANUARY/FEBRUARY 2006 • GEAR TECHNOLOGY • www.geartechnology.com • www.powertransmission.com Fan Equations Draft 1 12/9/2006

Equation 1:

ω w N t = = Rtw ωt N w Fan Equations Draft 1 12/9/2006Equation 2: ω N Equationc = 1: t = R ω N tc ωwt Nt c = = Rtw ωt N w Equation 3: Equation 2: ω N N c Rc = t ⋅ s c = = Rtc ωt NNct + N c

EquationEquation 3: 4: N c Rc = N ⋅ s N + Nt Rt = t c ⋅ s N t + N c EquationEquation 4: 5: N t Rωt = N ⋅ s w +c =N t N=c Ra Equationωc N5: w ω N Equationw = c6:= R ω N a rbc = rb (wu) Equation 6: Equation 7: rb = rb (u) Equationt b = t b 7:(u )

tEquationb = t b (u 8:) Equation 8: rt = M tb (δ ,λ,κ , Rb )rb (u) r = M (δ ,λ,κ , R )r (u) Equationt tb 9: b b Equation 9: t = M (δ ,λ,κ , R )t (u) t tt = M tbtb(δ ,λ,κ , Rb )tbb (ub) EquationEquation 10: 10: r = M (θ )r (u) = r (u,θ ) rii = Mit it (θ )t rt (u) =i ri (u,θ ) EquationEquation 11: 11: t i = M it (θ )t t (u) = t i (u,θ ) t i = M it (θ )t t (u) = t i (u,θ ) tors, consideringEquation the 12: universal motion abil- Equationx  12: ity of CNC machines,g cos[( the 1machine+ Rtc )θ settings] − sin[( 1+ Rtc )θ ]xt  S H cos(Rtcθ ) + SV sin(Rtcθ )  x  =    +   can be generallyy gg representedsin[(cos[(1 +as1R+tcR)θtc])θ ]cos[(− sin[(1+ R1tc )+θ R] tc)yθt ]xtSH sin(SRHtcθcos() − RSVtcθsin() +RStcθV )sin( Rtcθ ) Equation   = 13:    +   y g  sin[(1+ Rtc )θ ] cos[(1+ Rtc )θ ] yt   S H sin(Rtcθ ) − SV sin(Rtcθ ) Ra = Ra0 + Rac (ϕ) (13) Equation Equation 14: 13:

X R ba == X R ba 0 0 + + X R bc ac ( ϕ( ϕ ) ) Equation 15: EquationX b = X b 014:+ X bc (ϕ) (14) s = s + s (ϕ) X =0 X c + X (ϕ) Equation b 16: b 0 bc Equation 15:

s = s 0 + s c ( ϕ ) (15) Equation 16:

E m = E m 0 + E mc ( ϕ ) (16)

Figure 8—Gleason Phoenix® II 275HC hypoid gear generator. X p = X p 0 + X pc ( ϕ ) (17)

γ m = γ m 0 + γ mc ( ϕ ) (18)

j = j 0 + j c ( ϕ ) (19)

i = i 0 + i c ( ϕ ) (20)

The first terms in Equations 13–20 rep- resent 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 parameters of the cradle rotation angle ϕ. Incurrent practice, higher order polynomial are used (Ref.2). These motions Figure 9—A kinematic model of mechanical hypoid gear generators. can be realized through computer codes on the CNC machine. Equations 13–20 provide strong flexibility for generating all kinds of comprehensively crowned and corrected bevel gear tooth surfaces. Matrix Representation of Motion Elements In order to mathematically describe the generation process, the relative motions of the machine elements and their relation- ship have to be represented by a series of coordinate transformations established and rigidly connected to each motion element listed in Table 1. In current pratice, higher polynominals are used (Ref. 2). System Sm is fixed to the machine frame 1 and is con-

www.powertransmission.com • www.geartechnology.com • GEAR TECHNOLOGY • JANUARY/FEBRUARY 2006 35 Table 1—Machine Motion Elements and Axes of Rotation. sidered as the reference of the related motions. System Ss is connected to the Number of Names of elements and related motion sliding base element 11 and represents motion elements its translating motion. System Sr is con- 1 Machine frame, motion reference nected to the root angle setting element 2 Cradle, rotation/cradle angle 10 and represents the machine root

3 Eccentric, radial setting angle setting. System Sc is connected to the cradle and represents the cradle rota- 4 Swivel, swivel setting tion with parameter ϕ. System S is con- 5 Tilt mechanism, tilt setting p nected to the work head sliding setting 6 Tool cutter/head, rotation element 9 and represents the work head

7 Work, rotation setting motion. System So is connected 8 Work support, offset setting to the work head offset setting sliding element 8 and represents the work head 9 Work head setting offset setting motion. System Sw is con- 10 Root angle setting nected to the work 7 and represents the 11 Sliding base setting work rotation with angular parameter ψ.

a Cradle axis System Se is connected to the eccentric setting element and represents the radial b Eccentric axis setting of the cutter head (see Fig. 11 c Cutter head/tool spindle axis (a)). System Sj is connected to the tilt d Work spindle axis wedge base element 5, which performs e Swivel pivot axis for root angle setting rotation relative to the eccentric element to set the swivel angle j (see Fig. 11 (b)). System S is connected to the cutter head M wi = M wo (ψ )M op (Em )M pr (X p )M rs (γ m )M sm (X b )Mi mc (ϕ)M ce (s)M ej ( j)M ji (i) carrier, which performs rotation relative M wi = M wo (ψ )M op (Em )M pr (X p )M rs (γ m )M sm (X b )M mc (ϕ)M ce (s)M ej ( j)M ji (i) to the tilt wedge base element to set the tool tilt angle i (see Fig. 11 (c)). Figure 10—Equation 23. During tooth surface generation, at a b each cutting instant, a point on the blade edge generates a corresponding point on the work tooth surface. Through the

coordinate transformation from Si to

Sw, the current cutting point P can be represented in the coordinate system

Sw, namely,

r w = M wi ( θ , ϕ , ψ ) r i ( u , θ ) (21)

c t w = M wi ( θ , ϕ , ψ ) t i ( u , θ ) (22)

where Mwi (θ, ϕ, ψ) is a resul- tant coordinate transformation matrix with three parameters and is formulated by the multiplication of the follow- ing matrices representing the sequential

coordinate transformations from Si to

Sw,

(see Fig. 10 for equation 23)

where matrices Mwo, Mop, Mpr,

Mrs, Msm, Mmc, Mce, Mej and Mji can be obtained directly from Figures 12 and 13 and Equations 21 and 22 can

Figure 11—Relationships of coordinate systems Sm, Se, Sj and Si. be simply re-written as

36 JANUARY/FEBRUARY 2006 • GEAR TECHNOLOGY • www.geartechnology.com • www.powertransmission.com r w = r w ( u , θ , ϕ ) (24)

t w = t w ( u , θ , ϕ ) (25) here subscript “w” denotes that the vectors are represented in the coordinate system Sw. Tooth Surface of a Non-Generated (Formate) Member As discussed above, the non-generated (Formate) gear tooth surface is the comple- mentary copy of the generating gear tooth surface, which is kinematically formed as the cutting path of the blade edge along an extended epicycloid lengthwise curve. In Formate gear generation, the cradle is held stationary and parameter ϕ is assumed to be (a) Pinion tooth surfaces zero. Therefore, Equations 24 and 25 can be re-written as

r w = r w ( u , θ ) (26)

t w = t w ( u , θ ) (27) which give the position vector and a unit tangent at the current cutting point P on the gear tooth surface. The unit normal of the gear tooth surface can be derived as

n = k × t = n ( u , θ ) (28) w w w w (b) Gear tooth surfaces where Figure 12—Pinion and gear tooth surfaces of a face-hobbed hypoid gear drive.

∂rw ∂ θ (29) k w = ∂rw ∂θ which represents the unit vector of hobbing speed. Equations 26–29 provide equations of position vector, unit tangent, and unit normal of a non-generated gear tooth sur- face. Tooth Surface of a Generated Member In addition to the relative hobbing motion or the indexing motion for a generat- ed member, either gear or pinion, generating roll motion is provided and the generated tooth surface is the envelope of the family of the generating surface. The generated tooth Figure 13—A TCA output interface. surface can be represented as

rw = rw (u,θ ,ϕ)   t w = t w ( u , θ , ϕ ) (30)  n w = n w (u,θ ,ϕ)   f w (u,θ ,ϕ) = n w ⋅ v w = 0

where nw is obtained from Equation 28, www.powertransmission.com • www.geartechnology.com • GEAR TECHNOLOGY • JANUARY/FEBRUARY 2006 37 Conclusion and the relative generating velocity vw is determined by This paper presents the tooth surface generation theory of the face hobbing pro- ∂rw v w = ω c (31) cess. The kinematics of face hobbing index- ∂ϕ ing are described and simulated. A gener- alized spiral bevel and hypoid gear tooth Equation fw(u,θ,ϕ)=nwx vw=0 is called the equation of meshing (Ref. 4). Equation surface generation model for the face hob- 30 defines the position vector, the unit bing process is developed with the related normal, and the unit tangent at the current coordinate systems that are directly and cutting point P on the generated work tooth visually associated with the physical motion surface. All these vectors are considered elements of a bevel gear generator. The generation model covers both Gleason non- and represented in the coordinate system Sw that is connected to the work. A unit cradle generating and generated methods of the face hobbing process. And, it can also be angular velocity, i.e., ωc = 1 can be consid- ered. The equation of meshing is applied for adapted for the face milling process. Based determination of generated tooth surfaces. on the developed mathematical model, an Implementation of the advanced TCA is developed and integrated Mathematical Models into Gleason CAGE™ for the Windows sys- Figures 12 (a) and (b) show geometric tem, which has been released worldwide. models of a pinion and a gear generated References based on Equations 26–30. A very important 1. Krenzer, T.J. “Face-Milling or Face application of the developed mathematical Hobbing,” AGMA Technical Paper, 90 models is the development of tooth contact FTM 13, 1990. analysis for the face hobbed spiral bevel and 2. Stadtfeld, H.J. Advanced Bevel Gear hypoid gears. Tooth contact analysis (TCA) Technology, The Gleason Works, Edition is a computational approach for analyz- 2000. ing the nature and quality of the meshing 3. Pitts, L.S. and M.J. Boch. “Design and contact in a pair of gears. The concept of Development of Bevel and Hypoid Gears TCA was originally introduced by Gleason using the Face Hobbing Method,” Cat. Works in the early 1960s as a research #4332, The Gleason Works, 1997. tool and applied to spiral bevel and hypoid 4. Litvin, F.L. Gear Geometry and Applied gears (Ref. 7). Today, TCA theory has been Theory, Prentice Hall, 1994. substantially enhanced and generalized and 5. Baxter, M.L. “Effect of Misalignment on applied in advanced synthesis of face milled Tooth Action of Bevel and Hypoid Gears,” spiral bevel gears (Refs. 8, 9). ASME paper 61-MD-20, 1961. Based on the developed mathemati- 6. Krenzer, T.J. “The Effect of the Cutter cal models for face hobbing generation of Radius on Spiral Bevel and Hypoid Tooth spiral bevel and hypoid gears, an advanced Contact Behavior,” AGMA Paper No. TCA program for the face hobbing process 129.21, Washington, D. C., October 1976. has been developed for both non-generated 7. “Tooth Contact Analysis Formulas and and generated spiral bevel and hypoid gear Calculation Procedures,” The Gleason drives. The program incorporates simula- Works Publication, SD 3115, April 1964. tion of meshing of the whole tooth surface 8. Litvin, F.L., Q. Fan, A. Fuentes. and contact including the Toprem and Flankrem R.F. Handschuh. “Computerized Design, contact. Tooth edge contact simulation is Generation, Simulation of Meshing and also developed and included in the program. Contact of Face-Milled Formate-Cut Spiral Figure 13 shows an example of the output Bevel Gears,” NASA Report CR-2001- interface of the advanced TCA. Typically, 210894, ARL-CR-467, 2001. a TCA output illustrates bearing contact 9. Litvin, F.L. and Y. Zhang. “Local patterns on both driving and coast sides Synthesis and Tooth Contact Analysis of the tooth surfaces and the correspond- of Face-Milled Spiral Bevel Gears,” ing errors. Information on NASA Contractor Report 4342, 1991. the gear drive assembling adjustments is also provided.

38 JANUARY/FEBRUARY 2006 • GEAR TECHNOLOGY • www.geartechnology.com • www.powertransmission.com