Faydor L. Litvin, Alessandro Nava, Qi Fan, and Alfonso Fuentes

University of Illinois at Chicago, Chicago, Illinois

New Geometry of Worm Face Gear Drives With

Conical and Cylindrical Worms: Generation,

Simulation of Meshing, and Stress Analysis

NASA/CR—2002-211895

November 2002

ARL–CR–0511

U.S. ARMY

RESEARCH LABORATORY
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Faydor L. Litvin, Alessandro Nava, Qi Fan, and Alfonso Fuentes

University of Illinois at Chicago, Chicago, Illinois

New Geometry of Worm Face Gear Drives With

Conical and Cylindrical Worms: Generation,

Simulation of Meshing, and Stress Analysis

NASA/CR—2002-211895

November 2002

National Aeronautics and

Space Administration

Glenn Research Center

ARL–CR–0511

U.S. ARMY

RESEARCH LABORATORY

Prepared under Grant NAG3–2450
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Available electronically at http://gltrs.grc.nasa.gov
NASA/CRó2002-211895 1 

New Geometry of Worm Face Gear Drives With Conical and 

Cylindrical Worms: Generation, Simulation of Meshing, 

and Stress Analysis 

Faydor L. Litvin, Alessandro Nava, Qi Fan, and Alfonso Fuentes 

University of Illinois at Chicago 

Chicago, Illinois 60607 

Abstract 

New geometry of face worm gear drives with conical and cylindrical worms is proposed. The generation of 

the face worm-gear is based on application of a tilted head-cutter (grinding tool) instead of application of a 

hob applied at present. The generation of a conjugated worm is based on application of a tilted head-cutter 

(grinding tool) as well. The bearing contact of the gear drive is localized and is oriented longitudinally. A 

predesigned parabolic function of transmission errors for reduction of noise and vibration is provided. The 

stress analysis of the gear drive is performed using a three-dimensional finite element analysis. The 

contacting model is automatically generated. The developed theory is illustrated with numerical examples. 
NASA/CRó2002-211895 2 

1. Introduction 

Invention of face worm gear drives with conical and cylindrical worms by O. Saari [13, 14] was a 

substantial contribution. One of the greatest advantages of such drives is the higher contact ratio as the 

result of simultaneous contact of several pairs of teeth during the cycle of meshing. Initially the design of 

the invented gear drives was based on application of worms provided by axial profiles as straight lines. The 

worms have been designed as helicoids of a constant lead angle. Investigation of the invented gear drives 

has been performed by Saari and his followers [13, 14], Goldfarb [3, 4], Dudas [1], and other researchers. 

Spiroid worm gear drives have been discussed as well in the Gear Handbook [16]. 

A new modification of the face gear drives has been proposed by Litvin, De Donno, Argentieri et al [8, 9, 

10, 11]. The bearing contact of the proposed gear drives is a localized and a favorable shape of 

transmission errors for reduction of noise and vibration is provided. This is achieved by double crowning of 

the worm. 

The generation of face worm gear drives of all types of existing design is based on application of a hob for 

generation of the face-gear. The hob is similar to the worm of the drive. The disadvantage of such method 

of generation is the low precision of a hob used as a generating tool especially in the case of small 

dimensions of hob. 

The new geometry proposed in this paper is based on application of head-cutters or head grinding tools that 

have higher precision and larger dimensions in comparison with a hob and enable to provide a higher 

productivity of the process of generation. The gear drives of new geometry may be generated similar to the 

generation of spiral bevel gears and hypoid gears [7, 15]. The proposed new face worm gear drives (with 

conical and cylindrical worms) are provided with a localized bearing contact and predesigned parabolic 

function of transmission errors of low level. Such a function is able to absorb discontinuous linear functions 

of transmission errors caused by errors of alignment [8, 9]. Therefore vibrations and noise of gear drives 

might be reduced. 

The localization of bearing contact is achieved by the mismatch of surfaces of generating head tools 

applied for generation of the face worm-gear and the worm. Figures 1 and 2 show in 3D space face worm 

gear drives with conical and cylindrical worms that have been investigated in this paper. The proposed new 

design of a face gear drive is based on adaptation of the design parameters of a similar gear drive of 

existing design (based on generation of face-gear by a hob). This permits a common analysis between both 

designs but affects the dimensions of applied head cutters (Section 2). 

The contents of the paper cover the generation and analytical determination of tooth surfaces of the facegear and the worm, simulation of meshing and contact, and stress analysis. 

The theory developed is illustrated with numerical examples. 

The determination of conjugation of tooth surfaces of the proposed gear drive required the application of 

theory of enveloping that is the subject of differential geometry and theory of gearing represented in the 

works by Zalgaller [17], Zalgaller and Litvin [18], Favard [2], Korn [6], Litvin [8, 9], and other works. 
NASA/CRó2002-211895 3 

2. Basic Ideas for Generation of Face-Gear and Conical Worm 

Face Worm-Gear Generation. The generation is performed by a tilted head-cutter with straight line 

profiles of blades (Fig. 3). The tilt of the head-cutter avoids interference of the head-cutter with teeth that 

neighbor to the space being generated. 

Figure 4 illustrates schematically the generation of the face-gear. The tool is mounted on the cradle c of 

the generating machine (Fig. 4) and performs rotation about the tool axis. The face-gear Σ2 and the cradle 

c are fixed and the tooth surface of the face-gear is generated as the copy of the tool surface. 

Indexing of face-gear has to be provided for generation of each space of the gear. 

Blades of the gear head-cutter are shown in Fig. 5(a). The angles α g of blade profile are of different 

magnitude for the convex and concave sides of the space of the face worm-gear. Circular arc profiles of the 

blade fillet are provided for the generation of the fillet of the gear. 

 

Worm generation. The generation of the worm is performed by a tilted head-cutter mounted on the cradle 

d of the generating machine (Fig. 6). The worm and the cradle d perform related rotations determined as 

1

2

( )

( )

N

N

d

w

= ω

ω

 (1) 

where N2 and N1 are the number of teeth of the face-gear and number of threads the worm, respectively. 

The rotation of the head-cutter is provided for obtaining the desired velocity of cutting (grinding) and is not 

related with the meshing of cradle d and the worm. Figure 7 illustrates the installment of the worm with 

respect to the cradle d . 

The process of generation of the worm simulates its meshing with the face-gear. The face-gear is 

represented in Fig. 7 as cradle d that carries the head-cutter. The worm head-cutter and cradle d may be 

considered as rigidly connected to each other in the process of meshing of cradle d with the worm. The 

surface of the head-cutter simulates the tooth surface of the face worm-gear. 

The process of worm generation provides for the gear drive: (i) localization of bearing contact, and (ii) a 

parabolic function of transmission error. Localization of contact is achieved by deviation of profiles of 

blades of worm head-cutter with respect to the blades of gear head-cutter (see Section 3). Parabolic 

function of transmission errors is provided by variation of distance ( ) Em ψ 1 (Fig. 7) during the worm 

generation where (ψ1 ) is the angle of worm rotation performed during the process of worm generation. 

3. Analytical Determination of Face-Gear Tooth Surface Conjugated to Conical 

Worm 

The derivation of face-gear tooth surface of new design is based on the following procedure: 

(i) Derivation of face gear tooth surfaces * Σ2 of existing design generated by a hob. 
NASA/CRó2002-211895 4 

(ii) Determination of pitch point P of existing design. 

(iii) Determination of circle C of surface * Σ2 that passes through point P and two other points of 

* Σ2 . The radius of circle C is the radius of the head-cutter to be used for generation of facegear tooth surface of new design. 

(iv) Determination of profile of blade of head-cutter considering the cross-section of * Σ2 at pitch 

point P .

(v) Derivation of surface of head-cutter applied for generation of surface * Σ2 of new design. 

(vi) Determination of installment of the head-cutter on the cradle c (Fig. 4). 

(vii) Determination of surface Σ2 of the new design as the copy of the gear head-cutter surface. 

The described procedure is applied for derivation of both tooth sides. 

Derivation of Face-Gear Tooth Surface * Σ2 of Existing Design. Surface * Σ2 is determined in coordinate 

system 2 S rigidly connected to the face-gear as follows [10]: 

( ) h h h u ,θ ,ψ *

2

*

2 r = r (2) 

( ) , , 0 * *

2

*

2

*

2 = = ∂

∂ ⋅ 















∂

∂

×

∂

∂

h h h

h h h

f u

u

θ ψ θ ψ

r r r

 (3) 

Here: equations (2) represent the family of hob surfaces in 2 S ; equation (3) is the equation of meshing; 

( , ) h h u θ are the parameters of hob surface; ψ h is the generalized parameter of motion in the process of 

generation of surface * Σ2 of existing design. 

Equations (2) and (3) represent surface * Σ2 by three related parameters. 

We may represent * Σ2 in two parameter form by using the theorem of implicit function system existence as 

follows [6, 8, 9, 17, 18]: 

(i) Consider that point ( ) 0 0 0 , , h h h M u θ ψ satisfies equation 0 * fh = and an inequality 

0

* *

≠

∂

∂

+

∂

∂

h h

f

u

f

θ (4) 
NASA/CRó2002-211895 5 

is observed at M, say because / 0 * ∂f ∂θ h ≠ . Therefore equation 0 * fh = may be solved in the 

neighborhood of M by function 

( ) h h h h θ = θ u ,ψ , 1 θ h ∈C (5) 

Then surface * Σ2 may be represented locally as 

( )( ) ( ) h h h h h h h u ,θ u ,ψ ,ψ u ,ψ *

2

* r2 = R (6) 

Determination of Pitch Point P. Pitch point P is the point of tangency of the cone of the worm and the 

pitch cone of the existing design (Fig. 8). Determination of P requires application of equations (2) and (3) 

and parameters Em and L of installment of the worm. 

Design Parameters and Installment of Generating Tool. The generating tool is installed in a plane 

(designated by Π ) that is determined by three chosen points of surface * Σ2 generated by a hob. These 

points are designated as A1 , P , and A2 (Fig. 8(a)) where P is the pitch point, A1 and A2 are the points 

of * Σ2 that are generated by the middle point of the axial profile of the hob. 

Plane Π is shown in 3D (Fig. 8(b)). Circle C that lies in Π and passes through A1 , P , and A2 is of 

radius ρ determined by using the coordinates of A1 , P , and A2 . The axis g z of the generating tool is 

perpendicular to plane Π and passes through point Og that is the center of circle C . The orientation of 

plane Π is determined with respect to plane T of the drawings (Fig 8(b)). The installment of the 

generating tool with respect to coordinate system 2 S rigidly connected to face worm-gear is determined by 

direction cosines of coordinate system g S of the generating tool and position vector g 0 02 . 

The blade of the gear head-cutter is a straight line (Fig. 5(a)) and profile angle αg is determined from the 

condition that the straight line is a tangent to the cross-section of the face-gear at point P . 

Derivation of Surface of Gear Head Cutter. The surface of the head-cutter is illustrated in Fig. 5(a). It 

may be represented as a combination of a cone in the working part and a torus of the fillet part. The gear 

tooth surface is generated as a copy of the tool surfaces and is represented in S2 by the matrix equation 

 ( ) ( ) g g g g g g r2 u ,θ = M2 r u ,θ (7)

Here: ( , ) g g u θ are the surface coordinates of the head-cutter represented by vector function ( ) g g g r u ,θ . 

Matrix M2g describes the coordinate transformation from the tilted head-cutter to coordinate system 2 S . 

Elements of matrix M2g are determined by position vector g 0 02 and direction cosines of coordinate axes 

of g S (Fig. 8). 

Avoidance of Interference. The investigation of avoidance of interference may be accomplished 

analytically but it requires derivation of equations of tooth surfaces of many teeth and equations of their 
NASA/CRó2002-211895 6 

intersection with the head-cutter. The authors have used computer graphics as shown in Fig. 9. Avoidance 

of interference may require in some case the increase of the tilt of the head-cutter. 

4. Analytical Determination of Thread Surface of the Worm

Coordinate systems applied for derivation are shown in Fig. 10. Fixed coordinate systems n S and mS are 

rigidly connected to the frame of the generating machine. Movable coordinate systems d S and 1 S are 

rigidly connected to the cradle d of the generating machine and the worm, respectively. Schematics of 

worm generation are shown in Figs. 6 and 7. 

The worm is generated by a tilted head-cutter (Figs. 6 and 7). 

The blades of the head-cutter are of parabolic profile and deviate from the straight line profiles of the gear 

head-cutter (Fig. 11). Such mismatch is necessary for the localization of bearing contact of the gear drive. 

The installment of the worm during the generation is shown in Figs. 7 and 10. We emphasize that the 

shortest distance Em between the worm and the cradle is executed in the cycle of meshing as a parabolic 

function 

( ) 2

m ψ1 0 plψ 1 E = E + a (8) 

where E0 is the normal value of the center distance; pl a is the parabola coefficient of the plunge 2

plψ1 a . 

The plunge ( 2

aplψ1 ) provides a predesigned parabolic function of transmission errors of the gear drive (see 

Section 5). 

Remembering that the worm is generated by the head-cutter that is installed on the cradle d (Fig. 6) 

whereas the worm and the cradle d perform related rotations. The worm thread surface is determined as 

the envelope to the family of head-cutter surfaces as follows: 

( ) () w w d dw w w w r1 u ,θ ,ψ1 = M1 M r u ,θ (9) 

 ( ) , , 0 1

1

1 1 1 = = ∂

∂

⋅ 















∂

∂

×

∂

∂ θ ψ θ ψ w w

w w

f u

u

r r r

 (10) 

Here: ( ) w w w r u ,θ is the vector function that determines the surface of the head-cutter in coordinate system 

d S of the cradle d ; ( ) w w u ,θ are the surface parameters of the generating tool (the head-cutter or the 

grinder); ψ1 is the generalized parameter of motions; M1d and Mdw describe the coordinate 

transformation from d S to 1 S and w S to d S . 

Matrix equation (9) determines the family of head-cutter surfaces in 1 S . The cradle d and the head-cutter 

are considered in derivations as one rigid body. Rotations of the head-cutter and cradle d are considered in 

derivations as rotation of one rigid body. Rotation of the head-cutter about its axis provides the desired 

velocity of cutting and does not affect the process of generation. 
NASA/CRó2002-211895 7 

Equation (10) is the equation of meshing. The cross product in equation of meshing is the normal to the 

surface of the head-cutter represented in 1 S . Vector 1 1 ∂r /∂ψ is equivalent to the vector of relative velocity 

in meshing of the head-cutter and the worm. Figure 12 represents in 3D space the worm thread surface Σ1 .

The generated surface Σ1 requires separate generation of both sides of thread surfaces because the 

plunging motion (see equation (8)) may require different parabola coefficients for each side. Multithread 

worm requires separate generation of each surface of the thread. 

5. Simulation of Meshing and Contact 

The purpose of simulation of meshing and contact is for determining the bearing contact and transmission 

errors of a misaligned face worm gear drive. The computerized procedure for simulation of meshing and 

contact is performed by application of computer programs that were developed. 

The meshing of misaligned gear tooth surfaces is based on continuous tangency of contacting tooth 

surfaces (Fig. 13). 

Surfaces Σ1 and Σ2 of the worm and face worm gear and their normals are represented in a fixed 

coordinate system f S . The continuous tangency of Σ1 and Σ2 is represented by equations: 

r ( , , ) − r ( , , ) = 0 2 2 2

(2)

1 1 1

(1)

f u θ φ f u θ φ (11) 

n ( , , ) − n ( , , ) = 0 2 2 2

(2)

1 1 1

(1)

f u θ φ f u θ φ (12) 

For the purpose of simplicity the designations of variables of equations (11) and (12) in comparison with 

previous designations were changed. The solution of equations (11) and (12) is an iterative process and 

permits the determination of the paths of contact on tooth surfaces of the face-gear and the worm and 

transmission errors [8, 9]. Figure 14(a) (the output of TCA (Tooth Contact Analysis)) shows that the 

function of transmission errors is obtained indeed as a parabolic function of low magnitude; Figs. 14(b) and 

14(c) show that the bearing contact is localized indeed and it has a longitudinal direction. 

The bearing contact is formed as a set of instantaneous contact ellipses (Figs. 14(b) and 14(c)). The 

orientation and magnitude of axes of contact ellipses are determined by the principal curvatures and 

directions of the contact tooth surfaces and their elastic approach [8, 9]. The output of TCA (Fig. 14) has 

been obtained for a gear drive with design parameters represented in Table 1 (see Section 6). 

6. Stress Analysis 

Preliminary Information. The goals of finite element analysis developed are the following: 

(i) Investigation of formation of the bearing contact during the cycle of meshing. 

(ii) Determination of the contact and bending stresses. 

(iii) Determination of the contact ratio. 
NASA/CRó2002-211895 8 

A stress analysis has been performed by application of finite element analysis [19] by a general purpose 

computer program [5]. 

The finite element method requires definition of the contacting surfaces and establishment of the boundary 

conditions. 

The approach developed by the authors has the following advantages: 

(a) Using the equations of the tooth surfaces and of the fillets and considering the portion of the 

rim, the finite element models are generated automatically. Loss of accuracy introduced by 

CAD computer programs for the generation of the mesh is avoided. 

(b) The proposed approach does not require an assumption on the load distribution in the contact 

area. To get the contact area and the stresses the contact algorithm of the general purpose 

computer program [5] is used. 

(c) Finite element models are developed numerically at the chosen point of the path of contact 

and at the chosen installment of the gear. Stress convergence is assured because the face-gear 

and the worm are in contact at least in one point. 

(d) Finite element models of five pairs of teeth are applied and therefore the boundary conditions 

are far enough from the loaded areas of the teeth. 

Development of Finite Element Models. The development of finite element analysis using CAD 

computer programs in an interactive way is time expensive, requires skilled users for application of 

computer programs and has to be done for every case of development of gear geometry, position of 

meshing and installment desired. The approach developed is free of all these disadvantages and is 

summarized as follows: 

Step 1: It is possible to represent analytically the volume of the tooth using the equations of the surfaces 

and the portions of the corresponding rim. Fig. 15(a) shows the designed body for one tooth model of the 

face-gear. 

Step 2: Six surfaces (Fig. 15(b)) determined analytically divide the volume in six parts and control the 

discretization of these tooth subvolumes into finite elements. 

Step 3: Analytical determination of node coordinates is performed automatically by choosing the desired 

number of elements in longitudinal and profile directions (Fig. 15(c)). All the nodes of the finite element 

analysis are determined analytically and the points of the intermediate surfaces of the tooth belong to the 

real gear tooth surfaces. 

Step 4: Discretization of the model by finite elements (using the nodes determined previously) is 

accomplished as shown in Fig. 15(d). 

Step 5: The boundary conditions for the face-gear and the worm are set automatically under the following 

conditions: 

(i) Nodes on the bottom of the worm rim are considered as fixed (Fig. 16(a)). 
NASA/CRó2002-211895 9 

(ii) Nodes on the two sides and the bottom part of the face-gear rim form a rigid surface (Fig. 

16(b)). Such rigid surface is a three-dimensional structure that may perform translation and 

rotation but cannot be deformed. 

(iii) Considering the face-gear rim rigid surfaces the variables of motions of the face-gear (its 

translation and rotation) are associated with a single point M located on the axis of the facegear chosen as the reference point. Point M has only one degree-of -freedom (rotation about 

the face-gear axis) and all the other degrees of freedom are fixed. The torque T is applied 

directly to the face-gear at its reference point M . 

Step 6: The definition of the contact surfaces necessary for the finite element analysis is accomplished 

automatically, identifying all the elements of the model required for the formation of such surfaces. 

A symmetric master-slave method has been applied. This method causes the algorithm to treat each surface 

as a master surface. 

Numerical Example: The finite element analysis has been performed for a face-worm gear drive with 

conical worm of common design parameters represented in Table 1. For the analysis the model of the 

whole worm (Fig. 17) and the whole face-gear (Fig. 18) have been substituted by a model of five-pair of 

teeth in meshing (Fig. 19) in order to save computational time. Elements C3D8I of first order have been 

used for the finite element mesh. The total number of elements is 55672 with 68671 nodes. The material 

used is steel with Young's Modulus E=2.068 x 108 mN/mm and Poisson's ratio 0.29. The torque applied to 

the face-gear is 50 N·m. Figures 20, 21, and 22 show how the bearing contact looks at the beginning, at the 

middle, and at the end of a cycle of meshing. 

Table 1: Design Parameters of Face Worm Gear Drive with Conical Worm 

Number of threads of the worm 2 

Number of teeth of the face-gear 60 

Gear ratio 30 

Axial module 1.334 mm 

Convex side pressure angle 10.0 deg

Concave side pressure angle 30.0 deg 

Shaft angle mγ 90.0 deg 

Minimal distance between the axes Em 35.0 mm 

Worm axial displacement L 29.907 mm 

Worm pitch diameter Dp 11.480 mm 

Worm cone angle 5.0 deg 

Inner radius of the face-gear 38.53 mm 

Outer radius of the face-gear 52.50 mm 

Mean radius of the head-cutter 24.398 mm 

Plunging coefficient pl a (convex side) 0.003 mm 

Plunging coefficient pl a (concave side) -0.003 mm 
NASA/CRó2002-211895 10 

The results obtained by finite element analysis confirm the longitudinal path of contact and the avoidance 

of edge contact. Figure 23 shows the variation of bending and contact stresses on the face-gear from the 

beginning of the contact to the end of contact on one tooth of the face-gear. 

 The stresses are represented as unitless parameters in function of the worm rotation 

 

 

max1

1

1

σ

σ

σ m = (13)

 

max 2

2

2

σ

σ

σ m = (14)

where σ1 and σ 2 are the bending and contact stresses of Von Mises and σ max1 and σ max 2 are the 

maximum bending and contact stresses of Von Mises on the face-gear. In the example developed, 

2 σ max1 = 168 N/mm and 2 σ max 2 = 1090 N/mm . The load is always shared by 3 pairs of teeth, therefore 

the contact ratio is 3. 

7. Geometry, TCA, and Stress Analysis of Face Worm Gear Drive with Cylindrical 

Worm

The geometry discussed is considered as a particular case of the geometry of the gear drive with a conical 

worm. The generation of the face-gear and the worm are based on the same principles as described is 

sections 2, 3, and 4. It requires tilted head-cutters as well but generally of a larger value of the tilt of the 

tool. Figure 24 shows the cylindrical worm that is generated by a tilted head-cutter. 

The TCA has been performed for a gear drive of design parameters represented in Table 2. 

Table 2: Design Parameters of Face Worm Gear Drive with Cylindrical Worm 

Number of threads of the worm 2 

Number of teeth of the face-gear 60 

Gear ratio 30 

Axial module 1.334 mm 

Convex side pressure angle 10.0 deg

Concave side pressure angle 30.0 deg 

Shaft angle mγ 90.0 deg 

Minimal distance between the axes Em 35.0 mm 

Worm axial displacement L 29.907 mm 

Worm pitch diameter Dp 11.480 mm 

Worm cone angle 0.0 deg 

Inner radius of the face-gear 39.763 mm 

Outer radius of the face-gear 49.753 mm 

Mean radius of the head-cutter 23.105 mm 

Plunging coefficient pl a (convex side) 0.003 mm 

Plunging coefficient pl a (concave side) -0.003 mm 
NASA/CRó2002-211895 11 

A predesigned parabolic function of transmission errors indeed absorbs the discontinuous functions of 

transmission errors caused by errors of alignments (Fig. 25(a)). Figures 25(b) and 25(c) show that the 

bearing contact of a misaligned gear drive is localized and directed longitudinally as designed. The bearing 

contact is localized, and the path of contact is directed longitudinally. 

Stress analysis has been developed following the principles exposed in Section 6. A numerical example has 

been accomplished for a gear drive of design parameters represented in Table 2. Figure 26 shows the finite 

element model of five pairs of teeth. The torque applied to the face-gear is 50 N·m. The total number of 

elements is 55672 with 68671 nodes. Variation of bending and contact stresses on the face-gear are shown 

in Fig. 27. The maximum bending stress (σ max1 ) is 195 N/mm2

. The maximum contact stress (σ max2 ) is 

960 N/mm2

. The load is shared by three pairs of teeth during all the cycle of meshing, therefore the contact 

ratio is 3. 

8. Conclusions 

Based on the performed research, the following conclusions might be drawn. 

1. A new geometry of face worm gear drives with conical and cylindrical worms is proposed. The 

geometry is based on generation of face worm-gear by a tilted head-cutter or a grinding tool 

instead of a hob. The conjugated thread surface of the worm is generated by a head-cutter or a 

grinding tool as well. 

2. The tooth surface of the face worm-gear and the worm thread surface are determined analytically. 

3. The bearing contact of the face worm gear drive is localized due to the mismatch of generating 

surfaces. The obtained bearing contact is oriented longitudinally. 

4. A predesigned parabolic function of transmission errors of a low magnitude is provided that 

reduces the noise and vibration of the gear drive. 

5. Variation of the surface is obtained by tool plunging during the worm generation. 

6. An enhanced approach for stress analysis has been developed that automates the development of 

the finite element contacting model. 
NASA/CRó2002-211895 12 

References

 

[1] Dudas, I., The Theory and Practice of Worm Gear Drives, Penton Press, 2000. 

[2] Favard, J., Course of Local Differential Geometry, Gauthier-Villars, Paris (in French, translated 

into Russian), 1957. 

[3] Goldfarb, V.I., Airapetov, E.L.e.S., Novosyelov, V.Yu., Analytical and Experimental Assessment 

of Spiroid Gear Tooth Deflection, Proceedings of the 10th World Congress on TMM, vol. 6, Oulu, 

Finland, June 20ñ24, 1999, p. 2257ñ2262. 

[4] Goldfarb, V.I., Kuniver A.S., Koshin D.V., Investigation of spiroid gear tooth tangency under 

action of errors, Proceedings of the International Conference on Gearing, Transmission and 

Mechanical Systems, Notthingam, London, 2000, p. 65ñ73. 

[5] Hibbit, Karlsson & Sirensen, Inc., ABAQUS/Standard 6.1 User's Manual,1800 Main Street, 

Pantucket, RI 20860ñ4847, 1998. 

[6] Korn, G.A. and Korn, T.M., Mathematics Handbook for Scientist and Engineers, McGraw-Hill, 

Inc., 2nd Ed., 1968. 

[7] Krenzer, T.J., Computer Aided inspection of Bevel and Hypoid Gears, S.A.E. paper 831266, 

Milwaukee, WS, 1983. 

[8] Litvin, F.L., Gear Geometry and Applied Theory, Prentice Hall, Inc., Englewood Cliffs, New 

Jersey, 1994. 

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

Publication 1406, ARLñTRñ1500, 1998. 

[10]Litvin, F.L. and De Donno M., Computerized design and generation of modified spiroid wormgear drive with low transmission errors and stabilized bearing contact, Comput. Methods Appl. 

Mech. Engrg. 162(1998) 187ñ201. 

[11]Litvin, F.L., Argentieri G. , De Donno M., Hawkins M., Computerized design, generation and 

simulation of meshing and contact of face worm-gear drives, Comput. Methods Appl. Mech. 

Engrg. 189(2000) 785ñ801. 

[12]Nelson, W.N., Spiroid Gearing, Machine Design February 16, March 2ñ16, 1961. 

[13]Saari, O.E., Speed-Reduction Gearing, Patent No. 2,696,125, United States Patent Office, 1954. 

[14]Saari, O.E., Skew Axis Gearing, Patent No. 2,954,704, United States Patent Office, 1960. 

[15]Stadtfeld, H.J., Handbook of Bevel and Hypoid Gears: Calculation, Manufacturing, and 

Optimization, Rochester Institute of Technology, Rochester, New York, 1993. 

[16]Townsend, D.P. Dudley's Gear Handbook, McGraw-Hill, Inc., New York, 2nd Ed., 1991 

[17]Zalgaller, V.A., Theory of Envelopes, Publishing House Nauka (in Russian), 1975. 

[18]Zalgaller, V.A. and Litvin, F.L., Sufficient Condition of Existence of Envelope to Contact Lines 

and Edge of Regression on the Surface of the Envelope to the Parametric Family of Surfaces 

represented in Parametric Form, Proceedings of Universities: Mathematics (in Russian), Vol. 178, 

No. 3, pp. 20ñ23, 1977. 

[19]Zienkiewicz, O.C. and Taylor, R.L., The Finite Element Method, John Wiley Sons, 5th Ed., 2000. 
NASA/CRó2002-211895 13 

Figure 1: Face-worm gear drive with a conical worm
NASA/CRó2002-211895 14 

 

Figure 2: Face-worm gear drive with a cylindrical worm.
NASA/CRó2002-211895 15 

Figure 3: Generation of face worm-gear.
NASA/CRó2002-211895 16 

Figure 4: Schematic illustration of the head-cutter, the cradle c , and the face worm-gear.
NASA/CRó2002-211895 17 

Figure 5: Blade and generating cones applied for generation of face worm-gear: (a) blade; 

(b) and (c) generating cones for concave and convex sides.
NASA/CRó2002-211895 18 

Figure 6: Schematic of worm generation.
NASA/CRó2002-211895 19 

Figure 7: Installment of the worm with respect to the cradle d.
NASA/CRó2002-211895 20 

Figure 8: (a) Points A1 , P , and A2 that belong to surface * Σ2 generated by the hob; (b) 

Plane Π in 3D space, determined by the condition that Π passes through A1 , P , and A2 . 
NASA/CRó2002-211895 21 

Figure 9: Face-gear and head-cutter in 3D space: avoidance of interference.
NASA/CRó2002-211895 22 

Figure 10: Coordinate systems applied for worm generation.
NASA/CRó2002-211895 23 

Figure 11: Illustration of mismatch of worm head-cutter surface.
NASA/CRó2002-211895 24 

Figure 12: Conical worm thread surfaces in 3D space.
NASA/CRó2002-211895 25 

Figure 13: Tangency of contacting surfaces.
NASA/CRó2002-211895 26 

Figure 14: Output of TCA for a face gear drive with conical worm: (a) Transmissions 

errors with ∆E = 0.1 mm, ∆L = 0.1 mm, ∆γ = 3 arc min; (b) path of contact on the facegear surface with ∆E = 0.1 mm, ∆L = 0.1 mm, ∆γ = 3 arc min; (c) path of contact on the 

worm surface with ∆E = 0.1 mm, ∆L = 0.1 mm, ∆γ = 3 arc min.
NASA/CRó2002-211895 27 

Figure 15: Illustration of: a) the volume of the designed body, b) auxiliary intermediate 

surfaces, c) determination of nodes for the whole volume, and d) discretization of the 

volume by finite elements.
NASA/CRó2002-211895 28 

Figure 16: Schematic illustration of: a) boundary condition for the worm, and b) 

boundary conditions for the face-gear.
NASA/CRó2002-211895 29 

Figure 17: Finite element model of the whole worm.
NASA/CRó2002-211895 30 

Figure 18: Finite element model of the whole face-gear.
NASA/CRó2002-211895 31 

Figure 19: Face-gear drive with conical worm: finite element model of five teeth in mesh.
NASA/CRó2002-211895 32 

Figure 20: Face-gear drive with conical worm: bearing contact at the beginning of the 

cycle of meshing.
NASA/CRó2002-211895 33 

Figure 21: Face-gear drive with conical worm: bearing contact at the middle point of the 

cycle of meshing.
NASA/CRó2002-211895 34 

Figure 22: Face-gear drive with conical worm: bearing contact at the end of the cycle of 

meshing.
NASA/CRó2002-211895 35 

Figure 23: Face-gear drive with conical worm: a) Bending stress on the face-gear tooth in 

function of the pinion angle; b) Bending stress on the face-gear tooth in function of the 

pinion angle.
NASA/CRó2002-211895 36 

Figure 24: Cylindrical worm thread surfaces in 3D space.
NASA/CRó2002-211895 37 

Figure 25: Output of TCA for a face gear drive with a cylindrical worm: (a) Transmission 

errors with ∆E = 0.1 mm, ∆L = 0.1 mm, ∆γ = 3 arc min; (b) path of contact on the facegear surface with ∆E = 0.1 mm, ∆L = 0.1 mm, ∆γ = 3 arc min; (c) path of contact on the 

worm surface with ∆E = 0.1 mm, ∆L = 0.1 mm, ∆γ = 3 arc min.
NASA/CRó2002-211895 38 

Figure 26: Face-gear drive with cylindrical worm: finite element model of five teeth in 

mesh.
NASA/CRó2002-211895 39 

Figure 27: Face-gear drive with cylindrical worm: (a) Bending stress on the face-gear 

tooth in function of the pinion angle; (b) Bending stress on the face-gear tooth in function 

of the pinion angle.
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Available electronically at http://gltrs.grc.nasa.gov

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ARL–CR–0511

E–13577

45

New Geometry of Worm Face Gear Drives With Conical and Cylindrical Worms:

Generation, Simulation of Meshing, and Stress Analysis

Faydor L. Litvin, Alessandro Nava, Qi Fan, and Alfonso Fuentes

Gears; Transmissions

Unclassified -Unlimited

Subject Category: 37 Distribution: Nonstandard

WU–722–90–06–00

NAG3–2450

1L162211A47A

University of Illinois at Chicago

Chiago, Illinois 60612

Project Manager, Robert F. Handschuh, U.S. Army Research Laboratory, Vehicle Technology Directorate, Structures and

Acoustics Division, NASA Glenn Research Center, organization code 5950, 216–433–3969.

New geometry of face worm gear drives with conical and cylindrical worms is proposed. The generation of the face

worm-gear is based on application of a tilted head-cutter (grinding tool) instead of application of a hob applied at present.

The generation of a conjugated worm is based on application of a tilted head-cutter (grinding tool) as well. The bearing

contact of the gear drive is localized and is oriented longitudinally. A predesigned parabolic function of transmission

errors for reduction of noise and vibration is provided. The stress analysis of the gear drive is performed using a threedimensional finite element analysis. The contacting model is automatically generated. The developed theory is illustrated

with numerical examples.