Faydor L. Litvin, Qi Fan, Daniele Vecchiato, and Alberto Demenego

University of Illinois at Chicago, Chicago, Illinois

Computerized Generation and Simulation of

Meshing of Modified Spur and Helical

Gears Manufactured by Shaving

NASA/CR—2001-210893

May 2001

ARL–CR–468

U.S. ARMY

RESEARCH LABORATORY
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NASA/CR—2001-210893

May 2001

National Aeronautics and

Space Administration

Glenn Research Center

Prepared under Grant NAG3–2450

Faydor L. Litvin, Qi Fan, Daniele Vecchiato, and Alberto Demenego

University of Illinois at Chicago, Chicago, Illinois

Computerized Generation and Simulation of

Meshing of Modified Spur and Helical

Gears Manufactured by Shaving

ARL–CR–468

U.S. ARMY

RESEARCH LABORATORY
Available from

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Available electronically at http://gltrs.grc.nasa.gov/GLTRS
NASA/CR—2001-210893 1 

Computerized Generation and Simulation of Meshing of 

Modified Spur and Helical Gears Manufactured by Shaving 

 

 

Faydor L. Litvin, Qi Fan, Daniele Vecchiato, and Alberto Demenego 

University of Illinois at Chicago 

Department of Mechanical Engineering 

Chicago, Illinois 60607 

Abstract 

Modification of geometry of spur and helical gears with parallel axes and helical gears with crossed axes is 

proposed. The finishing process of gear generation is shaving. The purposes of modification of the gear 

geometry are to localize and stabilize the bearing contact, and to reduce noise and vibration. The goals 

mentioned above are achieved by using profile crowning and plunging the shaver by a prescribed motion 

during pinion generation. The pinion becomes double crowned. The gear member is generated as a 

conventional involute gear. A tooth contact analysis (TCA) program for simulation of meshing and contact 

was developed and the analysis is illustrated with TCA results for spur and helical gears.

Nomenclature 

αn Pressure angle in normal section 

β l (l=c, t) Skew angle of rack-cutter applied for generation of pinion c and gear t shaver 

βi (i=1,2,s) Helix angle of pinion 1 and gear 2, and shaver s 

∆λ2 Error of lead angle of gear 2

∆γ Error of crossing angle γ formed between pinion and gear axes 

∆E Error of center distance E between pinion and gear axes 

φ i (i=1,2) Angles of rotation of pinion 1 and gear 2 in the process of meshing 

∆φ2 (φ 1) Function of transmission errors 

∆ψp Additional rotation of the pinion due to the plunging motion 

γs Crossing angle formed between pinion and shaver axes 

Σl (l=c, t) Surface of rack-cutter applied for generation of pinion shaver and gear shaver 

Σd (d=s, g) Surface of shaver applied for generation of pinion and the gear 

Σ i (i=1,2) Surfaces of pinion 1 and gear 2

ψd (d=s, g) Angle of rotation of the shaver in the process for generation of the pinion and the gear 

a Parabola coefficient of predesigned parabolic function of transmission errors 

ac Parabola coefficient of profile crowned rack-cutter 

apl Parabola coefficient of plunging motion of the shaver 

E Shortest distance between the axes of the pinion and gear 

Es1 Shortest distance between the axes of pinion and shaver 

lp Displacement of the pinion in axial direction during the plunging motion 

mn Normal module of gear teeth 

nf

(i) (i=1,2) Surface i unit normal in coordinate system Sf 

Ni (i=1,2,s) Number of teeth of pinion 1 , gear 2, and shaver s

p1 Screw parameter of the pinion 

rf

(i) (i=1,2) Position vector of surface i in coordinate system Sf

ri (i=1,2,s) Radii of pitch cylinders of pinion 1 , gear 2, and shaver s

v(ld)

 (l=c,t) 

 (d=s,g) 

Relative velocity of contact point in coordinate system Sl with respect to coordinate system 

Sd
NASA/CR—2001-210893 2 

 

1. Introduction 

 

Shaving was invented as a finishing process of generation of spur and helical involute gears [1]. The basic 

idea of shaving is based on simulation of meshing of two helical gears or a helical and spur gears that 

perform rotation between crossed axes. One gear of the pair is a shaver that is designed as a helical gear 

provided with gashed teeth to obtain cutting edges (Fig. 1). In the process of meshing of the shaver with the 

workpiece, due to sliding in relative motion, the cutting edges of the shaver act as blades and remove very 

thin pieces of material from the teeth of the workpiece. 

The shaver and the workpiece are in point contact at every instant. Contact pressure is provided wherein the 

shaver is installed at the shaving machine and therefore the surface contact is spread over an elliptical area 

of small dimensions. The simulation of meshing of crossed gears in point contact permits the paths of 

contact on the shaver and the workpiece tooth surfaces (Fig. 2) to be obtained. There is only a narrow strip 

on the tooth surface of the workpiece in the neighborhood of the path of contact that is shaved by the 

shaver. Therefore, to shave the whole tooth surface of the workpiece, it is necessary to provide for the 

shaver or the workpiece translational motion along the axis of the workpiece. Such translational motion 

must be accompanied with additional rotation of the workpiece to provide the resulting motion as a screw 

one with the same parameter of the workpiece that is considered as a helicoid. 

 

Using a conventional shaving process, one can obtain the shaved surface of the workpiece as an involute 

surface. Therefore, the shaved tooth surfaces of the pinion and gear will be in line contact in an aligned 

gear drive. 

 

A new trend in the design of gears is directed at substitution of instantaneous line tangency of contacting 

surfaces by instantaneous point contact. This permits, if a proper design is accomplished, a reduction in 

sensitivity of the mating gears to misalignment and a reduction in the shift of the bearing contact caused by 

errors of alignment. 

 

In addition to observation of point contact of shaved surfaces, it is required as well to reduce the vibration 

of shaved gears. This can be obtained by the reduction of magnitude of transmission errors and obtaining a 

favorable shape of the function of transmission errors (see below). 

 

The shaving process has been described in a gear handbook [1] and it was a subject of research represented 

in [2]. Modification of helical and spur gears was a subject represented in [3]. 

 

The main goals of the paper are modification of the existing shaving process that satisfy the following 

requirements: 

1. The shaved pinion-gear tooth surfaces are in point contact (not line contact). Therefore, the shift of the 

bearing contact caused by misalignment is reduced. 

2. The shape of the function of transmission errors of the gear drive and the low magnitude of transmission 

errors reduces vibration and noise of the gear mesh. 

 

These goals are achieved by application of an approach based on the following ideas: 

1. The shaved pinion tooth surface is generated as a double-crowned surface. The term doublecrowning means crowning (modification) in two directions, profile and longitudinal ones. Profile 

crowning of the pinion tooth surface is achieved by application of a shaver with a tooth surface 

that is deviated from the conventional screw involute surface usually applied for a pinion shaver. 

Crowning of the pinion in the longitudinal direction is accomplished by variation of the plunge of 

the shaver executed as variation of the shortest distance E between the axes of the shaver and the 

pinion. The variation of the plunge is represented by equation 
NASA/CR—2001-210893 3 

2

0 pl p −= laEE (1) 

 

 Here: E and E0 are the current and initial distances between the axes of the pinion and the shaver; lp is the 

axial displacement in the direction of the pinion axis that is measured from the middle of pinion tooth 

length; apl is the parabola coefficient of parabolic function ( 2

pl p la ). 

 

2. The gear tooth surface is not modified: it is generated as a screw involute tooth surface. 

 

3. A parabolic function of transmission errors [4, 5] is provided for the meshing of shaved pinion and gear 

tooth surfaces. Such a function reduces noise and vibration of the gear drive that are caused by errors of 

alignment. 

 

4. The analytical determination of the crowned pinion tooth surface and gear tooth surface is based on 

application of two-parameter enveloping process wherein two independent sets of parameters of motion 

are considered: (i) one set of parameters of motions are the angles of rotation of the shaver and the 

workpiece (that is the pinion or the gear), and (ii) the other set are the parameters of feed motion 

(including the shaver plunge wherein the pinion generation is considered). 

 

5. The meshing and contact of shaved pinion and gear tooth surfaces is simulated by developed Tooth 

Contact Analysis (TCA) computer program. A TCA computer program has been developed for shaved 

spur gears and helical gears with parallel and crossed axes. 

 

The developed theory is illustrated with numerical examples. 

 

2. Derivation of Shaver Tooth Surfaces 

 

Imaginary Rack-Cutter Surfaces. A surface of a helical gear may be considered as generated by an 

imaginary rack-cutter that performs translational motion wherein the helical gear performs related rotation 

[4]. Henceforth, we will consider two cases of derivation of shaver tooth surfaces: 

 

Case 1: In the existing shaving process the pinion and gear are considered as conventional involute gears 

and pinion shaver and gear shaver are designed as screw involute gears. In this case a common rack-cutter 

with straight-line profiles may be applied for derivation of screw involute surfaces of the pinion and gear 

shavers. The disadvantages of such an approach have been mentioned above. 

 

Case 2: The pinion tooth surface is shaved as a double-crowned one and the gear tooth surface is shaved as 

a conventional involute surface. In case 2 we apply two rack-cutters with mismatched profiles as shown in 

Fig. 3(a). The profiles are the normal sections of skew rack-cutters. One of the rack-cutters with straightline profiles is applied for generation of gear screw involute shaver. The other rack-cutter is provided with 

parabolic profiles (Figs. 3(a) and 3(b)) and is applied for generation of profile crowned pinion. Application 

of rack-cutters shown in Fig. 3 improves the existing shaving process (see section 6). 

 

Rack-Cutter Surfaces. We designate by Σc and Σt surfaces of rack cutters applied for generation of pinion 

and gear shavers, respectively. 

Coordinate systems Sa and Sb (Fig. 3(b)) are rigidly connected to the parabolic profile that is the normal 

section of Σc. The origin of coordinate system Sa coincides with point Q of tangency of mismatched rackcutter profiles (Fig. 3(a)). Axes xa and ya are the tangent and the normal to the parabolic profile of the rackcutter. Axis ya passes through point P. 

 
NASA/CR—2001-210893 4 

Surface Σc of the rack-cutter is generated wherein coordinate system Sb with the parabolic profile of the 

rack-cutter performs translation along OO bc (Fig. 4). Axes zc and zb form angle βc that determines the 

orientation of skew teeth of the rack-cutter. Parameters uc (Fig. 3(b)) and θc =| OO bc | (Fig. 4) are the surface 

parameters of the skew rack-cutter. Fig. 4 shows the right-hand skew rack-cutter. 

 

Surface Σc of the rack-cutter is determined by the following matrix equation (Figs. 3(b) and 4) 

[ ] T

10)()(),(

 2 θccc cb θc ba ca MrMMr ca −== uauuu ccc (2) 

where c a is the parabola coefficient of parabolic function ( 2

c

u

c −a ). Equation (2) means that surface Σc is 

represented by vector function (u  ), ccc r which components are 

cccncccncc

cccncccncc

cccncc

luauz

luauy

luaux

α α β θ β

α α β θ β

α α

[ sin )( cos ]( sin ) cos

[ sin )( cos ]cos sin

cos )( sin

+−+−=

++−=

++=

 (3) 

where (Fig. 3(b)) 

4

cos nn

c

m l π α = (4) 

Normal Nc to rack-cutter surface Σc is represented as 

c

c

c

c

c u ∂θ

∂

×

∂

∂ = rr N (5) 

Similarly, we may represent surface Σt of rack-cutter of the gear shaver by vector function (u  ), ttt r taking 

into account drawings of Fig. 3(c) and coordinate transformation from Se to St. Here, 

[ ] T 100)()(),( θttc tk θt ke te == MrMMr te uuu e (6) 

 

Normal Nt to Σt is represented as 

t

t

t

t t u ∂θ

∂

×

∂

∂ = rr N (7) 

 

Derivation of Shaver Tooth Surface. The shaver tooth surface is generated by the rack-cutter that 

performs translational motion ddd = rs ψ wherein the shaver performs rotational motion determined by 

rotation angle ψd (Fig. 5). The axodes of the rack-cutter and the shaver are represented by plane Π and 

cylinder dr , respectively. Axis P-P is the instantaneous axis of rotation. Designation l=c, t and d=s, g

indicate rack-cutter surfaces Σc and Σt and the shaver surfaces Σs and Σg designated for the pinion and the 

gear, respectively. Coordinate system Sm is the fixed one where the motions of the rack-cutters and the 

shavers are represented. 

 
NASA/CR—2001-210893 5 

Surface Σd of the shaver is represented by the equations 

 ) ( , , ) ( ) ( , d ul θl ψd dl ψd l ul θl r = M r (8) 

 0 fdl(ul ,θl ,ψ d ) = (9) 

Here: matrix equation (8) represents the family of rack-cutter Σl (l=c, t) in coordinate system Sd . Equation 

(9) is the equation of meshing that may be determined considering the scalar equation 

 0 ( ) ⋅ = ld

l N v (10) 

or the rule that the normal to rack-cutter surface Σl at the line of tangency of Σl and Σd must pass through 

the instantaneous axis of rotation P-P [4]. 

3. Derivation of Pinion Tooth Surface Σ1

 

Kinematics of Pinion Shaving. We consider two independent sets of parameters of motion that are 

performed during the shaving of the pinion (Fig. 6): 

Set 1: Combination of related rotations ψ s and ψ1 of the shaver and the pinion, where 1 2 1 ψ s = (N / N )ψ . 

Set 2: Combination of translation lp in direction of pinion axis, additional rotation of the pinion ∆ψ p and 

plunge pl ∆s in the direction of shortest distance Es1 . Here, 

 p1

l p ∆ψ p = (11) 

where 1 p is the screw parameter of the pinion and 

 2

1

0

1 1 ( ) s s pl s pl p E = E − ∆s = r + r − a l (12) 

Here 0 Es1 and Es1 are the initial and current magnitudes of the shortest distance; apl is the parabola 

coefficient of the parabolic function ) ( 2

pl p a l ; 1

r and sr are the radii of the pitch cylinders of the pinion and 

the shaver, respectively. 

 

In reality the second set of parameters of motion (it is the set of feed motions) becomes related with the 

first one after the speed of the feed is chosen. Consideration of the second set as an independent one with 

the respect to the first one permits determination of the shaved pinion tooth surface as independent to the 

feed motion. 

 

Applied Coordinate Systems. Coordinate systems Ss and 1 S are rigidly connected to the shaver and the 

pinion (Fig. 7). The meshing of the shaver and the pinion is considered in the basic fixed coordinate system 

0 S . Rotation of 1 S is considered in coordinate system Sw that performs translational motions pl and pl s

with respect to 0 S . Rotation of the shaver is considered in additional fixed coordinate system Su . 

 

Axes s z and 1z form a crossed angle s γ determined as (Fig. 8) 

 β β1 γ s = s ± (13) 
NASA/CR—2001-210893 6 

where the upper (lower) sign correspond to application of the same (opposite) directions of the helices of 

the shaver and the pinion. 

 

Pinion Tooth Surface. The derivation of the pinion tooth surface is based on application of two-parameter 

enveloping process [4,6] and requires the following procedure: 

Step 1: Considering surface Σs of the pinion shaver as given, we represent equations of the family of Σs in 

coordinate system S1 rigidly connected to the pinion as follows 

 ) ( , , , ) ( , ) ( , 1 s s s p 1s s p s us s r u θ ψ l = M ψ l r θ (14) 

where ψ s and pl are the generalized independent parameters of motion; ) ( , us θs are the surface parameters 

of the shaver. 

The normal (s) Ns to the shaver surface Σs is represented in S1 by the following equation 

 ) ( , , , ) ( , ) ( , ( )

1 1 s s

s

s s s p s s p s

s N u θ ψ l = L ψ l N u θ (15) 

where L1s is the (3×3) submatrix of (4×4) matrix M1s . 

Step 2: In the case of two-parameter enveloping, we have to consider two (not one) equations of meshing 

that may be represented as follows 

 0 ( , , , ) (1 , )

1

( )

1

(1)

1 = ⋅ = s s s

s s s s p f u l ψ θ ψ N v (16) 

where (1 , )

1

s s ψ v is the relative velocity obtained for the condition that ψ s is varied and pl is constant, and 

 0 ( , , , ) (1 , )

1

( )

1

(2)

1 = ⋅ = p s s l

s s s s p f u θ ψ l N v (17) 

 

Designation (1 , )

1

p s l v means that the relative velocity is determined wherein parameter ψ s is held at rest and 

pl is varied. 

Step 3: Instead of relative velocities, we may consider in equations (16) and (17) infinitesimal 

displacements in relative motion and represent the equations of meshing as follows 

( , , , ) 0 ( ) 1

1

(1)

1 = ∂

∂ = ⋅

s

s

s s s s p f u l

ψ

θ ψ

r N ) (l const p = (18) 

( , , , ) 0 ( ) 1

1

(2)

1 = ∂

∂ = ⋅

p

s

s s s s p l f u l r θ ψ N ) ( const ψ s = (19) 

The partial derivatives ∂ψ s

∂ 1r and 

p ∂l

∂ 1r can be expressed by the following matrix equations 

 





+

∂

∂ = ∂

∂ ( ) ( ) ( ) ( ) 1 1 1

wo p ou p uj js s

s

w

s

l l ψ

ψ

ψ

ψ

M M M M r M
NASA/CR—2001-210893 7 

 s

s

js s

w wo poup uj ll r

M

MMMM 





∂

∂

ψ

ψ

ψ

)( )()()( 11 (20) 

and 

 









+

∂

∂ = ∂∂ )()()( )( 111

wo poup uj js s

p

w

p

ll

ll ψ ψ MMMM Mr

 +

∂

∂

)()( )( )( 11 pou uj js s

p

wo p

w l

l

l

ψ MMM ψ

M

M

 uj js ss

p

pou

w wo p l

l

l rMM

M

MM









∂

∂

)( )( )()(ψ11 ψ (21) 

Matrix ) ( M w ψ11 depends not only on the parameter of motion ψ1 but also of the plunging motion 

pl because (see equation (11)) variation of the axial position of the pinion must be accomplished by the 

additional rotation ∆ψ p .

Step 4: Equations (14), (18), and (19), considered simultaneously, represent the shaved pinion tooth 

surface. 

 

4. Derivation of Gear Tooth Surface Σ1

 

We use for derivation an approach that is similar to the discussed one in section 3. Surface Σ2 is 

determined as the envelope to the two-parameter family of surfaces of gear shaver Σg. Two 

independent sets of motions are used for determination of the envelope Σ2, but in the second set of 

motions the plunge is taken as zero. The derivations performed yield that gear tooth surface Σ2 is an 

involute one if the gear shaver surface Σg is a screw involute surface. 

 

5. Tooth Contact Analysis (TCA) 

 

The authors have developed a TCA computer program for simulation of meshing and contact of a 

shaved pinion and gear tooth surfaces Σ1 and Σ2. The purposes of the simulation are: 

1. That the point contact is localized and stabilized indeed. 

2. Determination of the path of contact and bearing contact of shaved pinion-gear tooth surfaces 

Σ1 and Σ2. 

3. The parabolic function of transmission errors of low magnitude is provided. 

4. Investigation of influence of errors of alignment: (i) on the shift of the bearing contact, and 

(ii) on the transmission errors. 

 

The developed TCA computer program has been used for simulation of meshing and contact of helical 

and spur gears with parallel axes and helical gears with crossed axes. 

 

In the cases of spur and helical gears with parallel axes, the pinion tooth surface is shaved as a doublecrowned surface. 
NASA/CR—2001-210893 8 

In the case of involute helical gears with crossed axes, the pinion-gear tooth surface are already in 

point contact and errors of alignment such as change of center distance, change of crossing angle and 

lead angle do not cause transmission error. Therefore, the pinion and gear shavers are provided with 

screw involute surfaces and double-crowning of the pinion is not required (see section 8). 

 

Algorithm of TCA for Simulation of Meshing. For the purpose of simplification of discussions, we 

consider that surfaces Σ1 and Σ2 and their normals N1 and N2 are represented in two-parameter form 

in coordinate systems S1 and S2 . 

We apply movable coordinate systems S1 and S2 rigidly connected to the pinion and gear, 

respectively, and fixed coordinate system S f (Fig. 9). An auxiliary coordinate system Sr (Fig. 9) is 

used for simplification of coordinate transformation. Three types of errors are considered: change of 

center distance ∆E , shaft angle ∆γ , and error ∆λ of the lead angle of the gear. 

 

The algorithm of TCA is based on simulation of continuous tangency of pinion-gear tooth surfaces. 

Using coordinate transformation, we represent surfaces Σ1 and Σ2 in fixed coordinate system S f (Fig. 

9). Continuous tangency of shaved surfaces Σ1 and Σ2 is provided by observation of the following 

vector equations (Fig. 10) 

 r ( , , ) − r ( 2 , 2 , 2 ) = 0 (2)

1 1 1

(1)

f u θ φ f u θ φ (22) 

 n ( , , ) − n ( 2 , 2 , 2 ) = 0 (2)

1 1 1

(1)

f u θ φ f u θ φ (23) 

where ) ( 1,2 ( ) i = i n f is the surface unit normal. 

 

Vector equation (23) yields only two independent equations since 1. (1) (2) n f = n f = Instead of equality 

of surface unit normals, we may require collinearity of surface normals using the equation 

 (1) (2) N f = λN f (λ ≠ 0) (24) 

Equation (22) and (23) provide a system of five nonlinear equations in six unknowns represented as 

 

 fi (u1 ,θ 1 ,φ 1 , u 2 ,θ 2 ,φ 2 ) = 0 (i = 1,...,5) (25) 

The Theorem of Implicit Functions System Existence yields that we may consider one of the 

parameters, say φ1 , as the input one and obtain the solution to equation system (25) by functions 

 { } (1)

1 1 1 1 2 1 2 1 2 1 u (φ ),θ (φ ),u (φ ),θ (φ ),φ (φ ) ∈C (26) 

if the following conditions are observed 

 ,

1 fi ∈C 0

( , , , , )

( , , , , )

1 1 2 2 2

1 2 3 4 5 ≠

D u θ u θ φ

D f f f f f (27) 

Then, we may obtain [4,5]: 

(i) the paths of contact on surfaces Σ1 and Σ2 determined as 

 )), ( ( ), ( i ui φ1 θi φ1 r (i = 1,2) (28) 
NASA/CR—2001-210893 9 

and 

(ii) the function of transmission errors 

1

2

1

1212 φ φ φ φ )()( φ N

N −=∆ (29) 

 

Parabolic Function of Transmission Errors. The approach developed enables a predesigned 

parabolic function of transmission errors to be obtained: 

2 φ −=∆ aφ12 (30)

for shaving spur and helical gears. 

In the case of shaving of spur gears, function (30) is provided by application of a profile crowned 

pinion shaver. Longitudinal crowning of the pinion is necessary for localization of the bearing contact 

and is accomplished by the plunging of the shaver during the process of shaving of the pinion (see 

section 3). 

 

In case of shaving of helical gears, parabolic function (30) is provided due to plunging of the shaver 

during the shaving of the pinion. Profile crowning of the pinion shaver provides localization of the 

bearing contact. 

 

It has been proven that a parabolic function of transmission errors is able to absorb a discontinuous 

linear function of transmission errors caused by misalignment [4, 5]. Discontinuous functions of 

transmission errors cause vibration and noise. Therefore, absorption of discontinuous linear functions 

of transmission errors by application of developed parabolic function of transmission errors is a great 

advantage of the developed approach for shaving. 

 

Simulation of Bearing Contact. The shaver and workpiece tooth surfaces, as well the pinion-gear 

tooth surfaces, are in point contact at every instant. The contact is spread over an elliptical area due to 

the elastic deformation of tooth surfaces. The bearing contact during the process of shaving and during 

the meshing of shaved tooth surfaces is formed by a set of instantaneous contact ellipses. The 

determination of dimensions and orientation of the contact ellipse require knowledge of principal 

curvatures and principal directions of contacting surfaces and the approach of contacting surfaces as 

the result of elastic deformation. We refer to [4, 5] for the solution of this problem. 

 

The shaved gear tooth surface is a screw involute surface and determination of principal curvatures and 

directions for such a surface is a simple problem [4, 5]. However, it is a more complex problem for a 

double-crowned pinion tooth surface. Taking into account that the deviations of a double-crowned 

pinion from an involute surface are in the range of ( 0.01-0.05)mm, we may determine the pinion 

principal curvatures and directions as for an involute surface. 

 

The contact of the shaver and the workpiece is also spread over an elliptical area. For the purpose of 

simplification, we will consider the contact of the shaver and the workpiece as the contact of crossed 

involute gears. 

 

Numerical Example 1: Contact Ellipse for Crossed Helical Gears. We will consider two cases of 

crossed helical gears: (i) with the crossing angle $ λ = 20 , and (ii) with the crossing angle $ λ = 90 . The 

first case corresponds to the contact of a shaver with the workpiece. The second case corresponds to a 
NASA/CR—2001-210893 10 

helical gear drive with crossed axes. The numerical example illustrates the influence of the crossing 

angle on the dimensions of the instantaneous contact ellipse. 

 

The design parameters are given in Tables 1 and 2. 

 

 Table 1: Design Parameters ( )20

$ γ =

 

N1 Tooth number of the pinion 25 

N2 Tooth number of the gear 77 

b Gear width (mm) 40 

mn Normal module (mm) 5 

αn Pressure angle (degree) 27.5

β1 Helix angle of the pinion (degree) 10 

β2 Helix angle of the gear (degree) -10 

γ Cross angle (degree) 20 

 Table 2: Design Parameters ( )90

$ γ =

 

N1 Tooth number of the pinion 25 

N2 Tooth number of the gear 77 

b Gear width (mm) 40 

mn Normal module (mm) 5 

αn Pressure angle (degree) 27.5

β1 Helix angle of the pinion (degree) 45 

β2 Helix angle of the gear (degree) -45 

γ Cross angle (degree) 90 

The contact ellipses obtained under very low load are shown at three positions of meshing (Fig. 11(a) 

and (b)) and are represented in Tables 3 and 4. The cross-hatched contact ellipse corresponds to the 

contact at the pitch point. The results obtained show that the crossing angle of helical gears affects 

substantially the dimensions of the contact ellipse. 

Table 3: Parameters of Contact Ellipse ( )20

$ γ =

Major axis 10.80 mm 

Minor axis 1.45 mm 

Area 12.28 mm2

Table 4: Parameters of Contact Ellipse ( )90

$ γ =

Major axis 3.90 mm 

Minor axis 1.91 mm 

Area 5.85 mm2
NASA/CR—2001-210893 11 

 6. Output of TCA for Spur Gears with Parallel Axes 

 

As a reminder that in the case of spur gears, the purpose of profile crowning of the pinion shaver is to 

provide a parabolic function of transmission errors of the gear drive. Such a function is able to absorb 

(see [4]) the discontinuous linear ones that are caused by the errors of alignment of the gear drive. The 

plunging of the shaver during the process of shaving of the pinion permits localization of the bearing 

contact in the middle of the tooth and avoids edge contact. 

 

However, another type of edge contact may occur, if the point of contact reaches the top of the tooth. 

In case of spur gears, such an edge contact may be avoided by the displacement down of point Q of 

tangency of mismatched profiles of rack-cutters (Fig. 3). 

 

The meshing of shaved spur gears has been simulated by the computer program developed. The design 

parameters of the simulated gear drive are given in the following table: 

 

Table 5: Design Parameters of Spur Gears 

N1 Tooth number of the pinion 25 

N1 Tooth number of the gear 77 

b Gear width (mm) 50 

mn Normal module (mm) 5 

αn Pressure angle (degree) 27.5

ac Parabola parameter of profile crowning 4×10-4

apl Parabola parameter of plunging 1×10-4

 

The results of TCA for the paths of contact on the gear tooth surface are shown in Figs. 12(a), 12(b), 

and 12(c). In all three cases the dashed line shows the path of contact for an aligned gear drive. 

 

The influence of change ∆γ of the crossing angle γ =∆ 3( arcmin) on the shift of the bearing contact 

is shown in Fig. 12(a). 

 

Drawings of Figs. 12(b) and 12(c) show that the shift of the bearing contact caused by ∆γ may be 

compensated by the correction of ∆λ , the lead angle of the gear or the pinion. In the case of spur 

gears, the nominal value of the lead angle is $ λ = 90 . Fig. 12(b) shows the shift of bearing contact 

caused by 3 λ2 −=∆ arc min. Fig. 12(c) shows that the shift of bearing contact is zero for 

γ λ2 =∆+∆ 0 . 

 

The results of TCA show: (i) that the shift of contact caused by change ∆E of center distance is zero; 

(ii) profile crowning of the pinion shaver provides a predesigned parabolic function of transmission 

errors. The functions of transmission errors obtained are represented in Figs. 13(a), 13(b), 13(c), and 

13(d). The drawings illustrate that the resulting functions of transmission errors (Figs. 13(b), 13(c), 

and 13(d)) have the same slope as the predesigned parabolic function of transmission errors shown in 

Fig. 13(a). The displacement of the origin of the resulting function of transmission errors with respect 

to the predesigned one indicates the error of angular position of the driven gear caused by 

misalignment. 
NASA/CR—2001-210893 12 

 

7. Output of TCA for Helical Gears with Parallel Axes 

 

In the case of helical gears with parallel axes, the localization of the bearing contact is achieved by 

application of profile crowned shaver for the pinion and a conventional involute shaver for the gear. 

The bearing contact on the shaved gear (or pinion) tooth surface of an aligned gear drive is a helix, 

similar to the case of N-W (Novikov-Wildhaber) gears [4]. 

 

In addition to localization of the bearing contact of the shaved pinion and gear, it is necessary to 

provide a parabolic function of transmission errors designated for the absorption of transmission errors 

caused by misalignment of the gear drive. A predesigned parabolic function of the transmission errors 

of the gear drive is provided due to the parabolic plunge of the pinion shaver during the pinion 

shaving. 

 

The developed TCA (Tooth Contact Analysis) computer program has been applied for simulation of a 

helical gear drive with the following design parameters (Table 6): 

 

Table 6: Design Parameters of Helical Gear Drive 

N1 Tooth number of the pinion 25 

N2 Tooth number of the gear 77 

b Gear width (mm) 40 

mn Normal module (mm) 5 

αn Pressure angle (degree) 27.5

β1 Helix angle of the pinion (degree) 20 

β2 Helix angle of the gear (degree) 20 

γ Cross angle (degree) 0 

ac Parabola parameter of profile crowning 1.4×10-3

apl Parabola parameter of plunging 8×10-5

 

The influence of three types of errors of alignment has been investigated: change of center distance 

∆E , shaft angle ∆γ , and change ∆λ2 of the lead angle of the gear. 

 

The results of the investigation are illustrated with the drawings of Figs. 14(a), 14(b), 14(c), and 14(d) 

that show the paths of contact on the gear tooth surface. 

 

Fig. 14(a) corresponds to the case wherein only the profile crowning of the pinion shaver has been 

accomplished and the parabolic plunge of the pinion shaver has not been applied. The path of contact 

on the tooth surface of an aligned gear drive in such a case is a helix. 

 

Figs. 14(b), 14(c), and 14(d) correspond to the cases wherein the pinion shaver is profile crowned and 

a parabolic plunge during the pinion shaving is executed. Due to the plunge, the path of contact of the 
NASA/CR—2001-210893 13 

gear drive deviates from a helix, even for an aligned gear drive. The magnitude of the deviation 

depends on the assigned magnitude of the predesign parabolic function of transmission errors. 

 

The path of contact indicated in Fig. 14(b) by continuous line corresponds to γ =∆ 3 arcmin. The 

dashed line in Fig. 14(b) illustrates the path of contact for γ =∆ 0 . 

 

Similarly, 14(c) indicates paths of contact obtained for 3 λ2 −=∆ arcmin and 0 λ2 =∆ . 

 

A combination of errors of alignment 0 γ λ2 =∆+∆ enables to avoid the shift of the bearing contact. 

 

Fig. 14(d) shows that error of alignment 05 E =∆ .0 mm does not cause the shift of the bearing contact. 

 

The paths of contact shown in Figs. 14(b), 14(c), and 14(d) deviate from a helix shape of path of 

contact as the result of a parabolic plunge of the pinion shaver. However, such a plunge is required 

because it provides a parabolic function of transmission errors and therefore the noise and vibration of 

the drive should be reduced. The existence of a parabolic function of transmission errors is confirmed 

by application of the developed TCA computer program. 

 

8. Output of TCA for Helical Gears with Crossed Axes 

 

We consider a gear drive formed by helical gears with screw involute tooth surfaces and crossed axes. 

The crossing angle γ is determined as 

 β β21 γ ±= (31) 

where βi is the helix angle. The upper and lower sign in (31) corresponds to the same and opposite 

directions of teeth, respectively. 

 

The investigation of meshing of helical gears with crossed axes accomplished in [4,6] shows: 

 

(i) The gear tooth surfaces are in point contact and therefore their bearing contact is localized. The path 

of contact on gear tooth surface is shown in Fig. 15. 

 

(ii) The advantage of crossed involute helical gears is that the change ∆γ of the crossing angle, change 

∆E of the shortest distance between the axes, and change ∆β of the helix angle do not cause 

transmission errors. This statement has been proven in [6] and confirmed by application of TCA 

computer program developed by the authors of this paper. 

 

(iii) However, due to small dimensions of the instantaneous contact ellipse (see Fig. 11) helical gears 

with crossed axes may be applied to gear drives that operate with light loads. 

 

The specific conditions of meshing of crossed involute helical gears are illustrated with drawings of 

Fig. 16. The line of action of the gears is the line of intersection of two planes that are tangent to the 

respective base cylinders of the helical gears. 

 

There are two lines of action that correspond to meshing of both sides of tooth surfaces. Both lines of 

action in an aligned gear drive intersect each other at a common point P that lies on the shortest center 

distance. In case of a misaligned gear drive, the lines of action are still tangents to the base cylinder but 

do not intersect each other at pitch point P. 

 

Unlike spur and helical gear drives with parallel axes, there is no need for double-crowning of helical 

gears with crossed axes. As it was mentioned above, such gear tooth surfaces are in point contact, the 
NASA/CR—2001-210893 14 

bearing contact is localized, and the gear drive is not sensitive to errors of alignment. A shaved screw 

involute tooth surface can be obtained by application of a conventional involute shaver. 

 

Numerical Example. The computer program developed has been applied for simulation of meshing 

and contact of a helical gear drive with the following design parameters (see Table 7): 

 Table 7: Design Parameters of Crossed Helical Gears 

 

N1 Tooth number of the pinion 25 

N2 Tooth number of the gear 77 

b Gear width (mm) 50 

mn Normal module (mm) 5 

αn Pressure angle (degree) 27.5

β1 Helix angle of the pinion (degree) 45 

β2 Helix angle of the gear (degree) -45 

γ Cross angle (degree) 90 

The results of TCA have confirmed that: (i) errors of alignment ∆γ , ∆λ , and ∆E do not cause 

transmission errors of the drive, and (ii) do not shift substantially the bearing contact. 

 

9. Conclusions 

Based on the results presented in this study the following several conclusions can be made: 

1. A modified geometry of spur and helical gears with parallel axes manufactured by shaving has 

been developed. The bearing contact of shaved gears is localized and the transmission errors 

caused by misalignment are of a favorable shape and low magnitude. 

2. The modification of shaved pinion-gear tooth surfaces of spur and helical gears with parallel axes 

is based on application of a profile crowned shaver for the pinion and execution of a parabolic 

plunge of the shaver during the process of pinion shaving. 

3. Simulation of meshing and contact of pinion-tooth surfaces has been performed by the computer 

programs developed. 

4. An approach for shaving of crossed helical gears has been developed and investigated. 

References 

 

[1] Townsend, D. P.: Dudley's Gear Handbook, Second Edition, McGraw-Hill, Inc. 

[2] Kim, D.: Computerized Simulation of Meshing for Finishing Operations of Production of Spur and 

Helical Gears, Ph. D. Thesis, University of Illinois of Chicago, 1995. 

[3] Patent: Low Noise, Reduced Stresses, High Precision of Helical and Spur Gears Design and 

Manufactured with Double-Crowned Tooth Surfaces, Inventors: Litvin, F. L. and others, US Patent 

Serial No: 09/340.824 (pending). 

[4] Litvin, F. L.: Gear Geometry and Applied Theory, Prentice Hall, 1994. 

[5] Litvin, F. L.: Development of Gear Technology and Theory of Gearing, NASA Reference 

Publication 1406, ARL-TR-1500, 1998. 

[6] Litvin, F. L.: Theory of Gearing, 2nd edition, Nauka, Moscow, 1968, (in Russian). 
NASA/CR—2001-210893 15

Figure 1.—Illustration of a shaver as a helical gear with

 gashed teeth.

Figure 2.—Contact paths on shaver and workpiece tooth surfaces.
NASA/CR—2001-210893 16

Figure 3.—Profiles of rack-cutters in normal section. (a) mismatched normal

 profiles of two rack-cutters. (b) parabolic profiles of rack-cutter. (c) straight-

 lined profiles of rack-cutter.

Figure 4.—For derivation of rack-cutter surfaces Σc and Σt

.
NASA/CR—2001-210893 17

Figure 5.—Generation of a shaver tooth surface Σd.

Figure 6.—Schematic of shaving of the pinion.
NASA/CR—2001-210893 18

Figure 7.—Coordinate systems applied for pinion shaving.

Figure 8.—Determination of crossing angle γs

.
NASA/CR—2001-210893 19

Figure 9.—Coordinate systems applied for TCA.

Figure 10.—Tangency of surfaces in ideal gear train.
NASA/CR—2001-210893 20

Figure 11.—Contact ellipses for helical gears with crossed axes.

 (a) (γ = 20°) and (b) (γ = 90°).

Figure 12.—Spur gears: path of contact on gear tooth surface for

 aligned and misaligned gear drives.
NASA/CR—2001-210893 21

Figure 13.—Functions of transmission errors for spur gears with parallel axes.

 Errors of alignment are ∆γ, ∆λ2 and ∆E.

Figure 14.—Paths of contact on the gear tooth surface for

 aligned and misaligned helical gears with parallel axes.
NASA/CR—2001-210893 22

Figure 15.—Path of contact on the gear tooth surface for helical gears

 with crossed axes.

Figure 16.—Formation of line of action in a helical gear drive with crossed axes.
This publication is available from the NASA Center for AeroSpace Information, 301–621–0390.

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Gears; Transmissions

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Subject Category: 37 Distribution: Nonstandard

University of Illinois at Chicago

Department of Mechancial Engineering

Gear Research Center

Chicago, Illinois 60680

Computerized Generation and Simulation of Meshing of Modified

Spur and Helical Gears Manufactured by Shaving

Faydor L. Litvin, Qi Fan, Daniele Vecchiato, and Alberto Demenego

Project Manager, Robert Handschuh, U.S. Army Research Laboratory, Structures and Acoustics Division, NASA Glenn

Research Center, organization code 5950, 216–433–3969.

Modification of geometry of spur and helical gears with parallel axes and helical gears with crossed axes is proposed.

The finishing process of gear generation is shaving. The purposes of modification of the gear geometry are to localize

and stabilize the bearing contact, and to reduce noise and vibration. The goals mentioned above are achieved by using

profile crowning and plunging the shaver by a prescribed motion during pinion generation. The pinion becomes double

crowned. The gear member is generated as a conventional involute gear. A tooth contact analysis (TCA) program for

simulation of meshing and contact was developed and the analysis is illustrated with TCA results for spur and helical

gears.