Archive > 2015 > May 2015 > Hypoid Gears with Involute Teeth

# Hypoid Gears with Involute Teeth

David B. Dooner

This paper presents the geometric design of hypoid gears with involute gear teeth. An overview of face cutting techniques prevalent in hypoid gear fabrication is presented. Next, the specification of a planar involute rack is reviewed. This rack is used to define a variable diameter cutter based upon a system of cylindroidal coordinates; thus, a cursory presentation of cylindroidal coordinates is included. A mapping transforms the planar involute rack into a variable diameter cutter using the cylindroidal coordinates. Hypoid gears are based on the envelope of this cutter. A hypoid gear set is presented based on an automotive rear axle.

**Background**

Cylindrical gearing is the simplest of all gear types and is used more than any other. Bevel and hypoid gear manufacturing analysis entail spatial, geometric relations, whereas spur and helical gear manufacturing analysis entail mostly planar geometric relations. Existing methods of manufacture for different gear forms are type-specific and generally unrelated. For example, the machines used to produce hypoid gears cannot be readily used or for fabricating spur cylindrical gears. The methods of manufacture associated with bevel and hypoid gears do not allow these gears to be treated with the same type of geometric considerations that currently exist for cylindrical gears. To illustrate, spur cylindrical gears are helical gears with a zero helix angle; both gear types can be produced using the same machine. But spur hyperboloidal gears cannot be readily produced using existing fabrication techniques for spiral hyperboloidal gears. The majority of hypoid and bevel gear manufacture today is the focus of The Gleason Corporation and Klingelnberg-Oerlikon. The following three companies provide the machines and machine tools necessary for the production of hypoid gears:

- The Gleason Works (www.gleason.com)
- Klingelnberg-Oerlikon (www.klingelberg.com)
- Yutaka Seimitsu Kogyo LTD (http://www.yutaka. co.jp/Y_hp6/default2.htm)

Depicted in Figure 1 are circular face cutters used today for fabricating spiral bevel and hypoid gear elements. Certain limitations of existing crossed-axis gear technology can be realized by focusing on Figure 2. The theoretical or ideal shape of these crossed-axis gears is the “hour-glass” — or hyperboloidal — shape shown. Current design and manufacturing techniques approximate a portion of the hour-glass shape by a conical segment, as shown. This approximation results in the following restrictions:

- Face width
- Minimum number of teeth
- Spiral angle
- Pressure angle

These restrictions in turn limit candidate gear designs. Face
cutting further places restrictions on the above limitations,
together with the gear ratio. An overview of face cutting methods
for hypoid gear design and manufacture is provided by
Shtipelman (Ref. 1); Stadtfeld (Ref. 2); Wu and Lou (Ref. 3);
Wang and Ghosh (Ref. 4); and Litvin and Fuentes (Ref. 5).
Radzevich (Ref. 6) and Kapelevich (Ref. 7) provide u*P*_{d}ated
approaches for gear design and manufacture. Preliminary investigations
into the “ideal” kinematic geometry of spatial gearing
have been recognized by Xiao and Yang (Ref. 8); Figliolini et
al. (Ref. 9); Hestenes (Ref. 10); and Ito and Takahashi (Ref. 11).
Grill (Ref. 12) uses an “equation of meshing” to establish a relation
between the curvature of one body to that of another body
and applies his results in the context of gearing. Baozhen et al
use Lie Algebra for a coordinate-free approach akin to screw theory (Ref. 13). Phillips (Ref. 14)
proposes a qualitative approach
for point contact of hypoid “involute”
teeth. A hyperboloidal cutter
was proposed as part of a unified
methodology for the analysis,
synthesis, and manufacture of
generalized gear pairs (Ref. 15).

The manufacture of generalized
gear elements is proposed by
introducing a hyperboloidal or
variable diameter cutter to mesh
with a desired gear. An illustration
of a hyperboloidal hob cutter and hypoid gear/work piece is
depicted in Figure 3. The desired gear depends upon the cutter
geometry along with its position and orientation relative to the
gear. Two toothed bodies in mesh where the sum ѱ_{pi} + ѱ_{po} of the
spiral angles is non-zero is established to determine the cutter’s
position and orientation relative to the gear element.

The most common occurrence where ѱ_{pi} + ѱ_{pc} ≠ 0 is for
crossed cylindrical gears. The included angles αpi and αpc for
cylindrical toothed bodies are zero, and the angle between the
two axes *$*_{i} and $c reduces to ѱ_{pi} + ѱ_{pc} = Σ. Meshing conditions
where ѱ_{pi} + ѱ_{pc} ≠ 0 and α_{pi} + α_{pc} ≠ 0 are defined as crossed hyperboloidal
gears. The I/O relationship for the meshing or generating
cylindroid between two crossed hyperboloidal gears in mesh
is identified by an “s” subscript and is uniquely defined as the
swivel I/O relationship gs. Generating conditions are determined
using gs, the *swivel center distance E*_{s}, and the *swivel shaft angle*
Σ_{s}.

**Rack Coordinates**

Introducing the rack as an intermediate step for defining a candidate
cutter is based on its simplicity and usefulness in transforming
rotary motion into linear motion. Rack coordinates
used to parameterize a gear tooth repeat each pitch *P*_{d}, thus, it
is necessary to parameterize candidate rack tooth profiles for
one pitch. The “r” subscript is used to designate that the indicated
variable is in regards to the rack. If the teeth are symmetric
about a line through the center of the tooth, then candidate
tooth profiles need to be specified only for one-half of the
pitch *P*_{d}. The Cartesian coordinates (*x,*_{r}, *y*_{r}) for the rack shown
in Figure 4 are divided into three regions — 1) crest; 2) active
region; and 3) fillet.

This is achieved by specifying the coordinates (*x,*_{r}, *y*_{r}) for the
active region according to a particular application. For example,
if zero errors in the I/O relationship g must be achieved
for small changes in center distance E, then, as anticipated, the
active profile becomes a straight line. Subsequently, the crest is
determined by the “optimal” fillet of the generated gear blank.
This occurs because the crest of the cutter determines the fillet
of the generated blank. The fillet of the cutter is determined
such that the crest of one gear pair does not interfere with the
fillet of the mating gear.

The diametral pitch *P*_{d} is defined as the number of teeth per
inch of pitch diameter for spur circular gears. For two toothed
wheels in mesh, this leads to:

Where

*N*_{i} = Number of teeth on the input

*u*_{pi} = Pitch radius of the input

*N*o_{ }= Number of teeth on the output

upo = Pitch radius of the output

Recognizing that *u*_{pi} + *u*_{po} = E, where E is the distance between
the two axes of rotation, the diametral pitch *P*_{d} is expressed:

The module md is used in the metric system, where:

Such an expression for the tooth size is ingenious and is used
to specify the addendum and dedendum height. The distance
pn between adjacent teeth can also be expressed in terms of
diametral pitch *P*_{d}. Two gears in mesh must have the same pn or
normal pitch. In turn, this normal pitch can be resolved into a
transverse pitch *p*t_{ }and an axial pitch *p*_{a}. At this point it is convenient to temporarily abandon this terminology and introduce
the distance between adjacent teeth as the *circular pitch c*_{p} with
no indication as to whether it is the transverse, axial, or normal
pitch.

A mapping is used to transform the rack coordinates (*x,*_{r}, *y*_{r}) to
polar gear tooth coordinates (*u*_{c}, *v*_{c}). This transformation can be
envisioned as wrapping a rack onto a pitch circle with the desired
pitch radius up. This transformation is the envelope of the rack as
it meshes with a circle of radius up. Depicted in Figure 5 is a rack
being wrapped onto a pitch circle with radius *u*_{p}.

**Cylindroidal Coordinates**

A system of curvilinear coordinates is used to parameterize
the kinematic geometry of motion transmission between skew
axes. These curvilinear coordinates are based upon the cylindroid
determined by the two axes of rotation, *$*_{i} and *$ _{o}*, and are
referred to as cylindroidal coordinates. Cylindroidal coordinates
consist of families of pitch, transverse, and axial surfaces. Pitch
surfaces are specified in terms of the axes of rotation

*$*

_{i}and

*$*.

_{o}*$*

_{i}is the input axis (pinion) of rotation and

*$*is the output (ring) axis of rotation. Pitch surfaces are a family of ruled surfaces, and axodes are the unique pitch surfaces that depend upon a particular I/O relationship. For this reason, the pitch surfaces are referred to as the reference pitch surfaces.

_{o}A system of curvilinear coordinates (*u, v, w*) is used to
describe spiral bevel and hypoid gears. The coordinates (*u, v, w*)
used to parameterize these families of pitch, transverse, and
axial surfaces are formulated using the cylindroid defined by
the input and output axes of rotation. A design methodology
for spatial gearing analogous to cylindrical gearing begins with
the equivalence of friction cylinders. Figure 6 shows two such
wheels along with candidate generators. The I/O relationship g
defines which generator of the cylindroid is used to parameterize
the input and output friction wheels. These generalized friction
surfaces are two ruled surfaces determined by the instantaneous
generator. The transmission of motion between the
two generally disposed axes *$*_{i} and *$ _{o}* via two friction surfaces
requires knowledge of the instantaneous generator. The location
of the instantaneous generator relative to the two axes

*$*

_{i}and

*$*depends upon:

_{o}- Distance
*E*along the common perpendicular to axes of rotation*$*_{i}and*$*_{o} - Angle Σ between axes of rotation $
_{i}and*$*_{o} - Magnitude of the I/O relationship
*g*

Motion transmission between the two skew axes *$*_{i} and *$ _{o}*
results in a combination of an angular displacement about the
instantaneous generator and a linear displacement along the
instantaneous generator. The ratio

*h*of linear displacement to that of the angular displacement is the

*pitch*associated with the instantaneous generator. The pitch

*h*

_{isa}associated with the instantaneous generator is the

*instantaneous screw axis*, or

*ISA*.

A transverse surface is an infinitesimally thin surface used to
parameterize conjugate surfaces for direct contact between two
axes. Candidate generators for the reference pitch surface are
determined by the generators of the cylindroid (*$*_{i}; *$ _{o}*). Given g,
each position angular

*v*

_{i}and axial position wi define a unique point p in space. Allowing

*g*to vary from -∞ to ∞, the point

*p*traces a curve in space. Another value of the input position

*v*

_{i}defines the same cylindroid. There is an angular displacement between these two cylindroids. It is this two-parameter loci of points p that compose the transverse surface. The Cartesian coordinates r for the single point p on the generator

*$*

_{ai}are:

Rotating the above curve *r* about the *z*_{i}-axis an amount *v*_{i}
leads to:

Where

*u* radius of hyperboloidal pitch surface (at throat)

*v* angular position of generator on pitch surface

*w* axial position along generator of pitch surface

*α* angle between generator and central axis of pitch surface

The axial surface provides the relationship between successive
transverse surfaces. For each value of *v*_{i}, the axial surface
is the loci of generators determined by *g*, where -∞ < g < ∞. The
curves defined by holding two of the three parameters *u, v,* and
*w* constant are coordinate curves. Two parameters used to define a surface are the curvilinear coordinates
of that surface: the pitch surface by *v*_{i} and
*w*_{i} (*u _{i} = constant*), the transverse surface by

*u*

_{i}and

*v*

_{i}(

*w*), and the axial surface by

_{i}= constant*u*

_{i}and

*w*

_{i}(

*v*). Depicted in Figure 7 are the pitch, transverse, and axial surfaces determined using cylindroidal coordinates (

_{i}= constant*u*

_{i},

*v*

_{i},

*w*

_{i}). Three surfaces are used to describe the geometry of gear elements.

The curvilinear coordinates (*u _{c}, v_{c}, w_{c}*)
used to parameterize the proposed cutters
are defined by introducing a cutter-cylindroid
(

*$*). This enables cutters to be designed in pairs analogous to the design of gear pairs where two cutters are proposed for the fabrication of spiral toothed bodies. One feature of the cutter cylindroid is that expressions involving the cutters are obtained by simply changing the trailing subscripts in existing expressions involving the input gear from

_{ci}; $_{co}*“i”*to

*“c”*. In order to minimize the notation necessary to distinguish the input cutter from the output cutter, only a

*“c”*subscript is used with no indication as to whether it is the input cutter or the output cutter. Implicit in the cutter designation will be an

*“o”*subscript when describing the input gear. Likewise, when describing the output gear body, it will be assumed that associated with the cutter is an

*“i”*subscript to identify that it designates the input cutter. The above reasoning is that two toothed bodies in mesh involve an input and an output body. The three possibilities being:

- Input gear body and an output gear body
- Input gear body and an output cutter
- Input cutter and an output gear body

The two twist axes *$ _{ci}* and

*$*are the two screws of zero pitch on the cutter cylindroid (

_{co}*$*). The generators $pc are determined by also introducing a cutter I/O relationship gc. Expressions for the radius uac and the angle α

_{ci}; $_{co}_{ac}are identical to those for uai and α

_{ai}, except

*E*, Σ, and

*g*are replaced by

*E*

_{c}, Σ

_{c}, and

*g*, respectively.

_{c}**Hyperboloidal Cutter Coordinates**

General hyperboloidal cutter elements are defined by introducing
a mapping within a system of cylindroidal coordinates.
The purpose of this mapping is to utilize knowledge of conjugate
curves for motion transmission between parallel axes and
apply it to conjugate surfaces for motion transmission between
skew axes. A visual representation of this mapping is shown in
Figure 8. There exists a single generator within a system of curvilinear
coordinates as part of the cylindroid (*$*_{i}; *$ _{o}*) that is coincident
with each point (

*u, v*). For an arbitrary axial position wc along this generator, a transverse surface exists. Each value (

*u, v*) defines a different generator. The distance wc along each of these generators from (

*u, v*) to a single transverse surface is constant. It is the image of these datum points (

*u, v*) upon a given transverse surface that defines the mapping. This mapping is valid for

any type of cutter tooth profile (viz., involute, cyclodial, circulararc, and convuloid).

The planar coordinates (*u, v*) used to define conjugate curves
are polar coordinates where v is an angular position about the
“z-axis” and u is the corresponding radius. Use of coordinates
(*u, v*) to specify conjugate curves in the plane are fashioned
such that conjugate surfaces in space are obtained using the cylindroidal
coordinates (*u _{c}, v_{c, wc}*). This is achieved by assigning
a value to the axial position wc and defining uc ≡ uc and vc ≡ v.
Cutter coordinates must be “scaled” to satisfy the appropriate
transverse pitches. Such scaling is illustrated in Figure 9 and is
obtained by recognizing that the virtual length of the striction
curve spc is the component of its length perpendicular to the
tooth. This scaling is performed prior to the “wrapping” of the
rack onto the circular disk depicted in Figure 3 and depends on
the diametral pitch. The diametral pitch

*P*

_{d}used to parameterize the cutter teeth depends on the size or radius of the input and output cutter. The x-scaling or stretch along the x-axis is shown in Figure 9 and depends on the cone angle αpc; thus, for an arbitrary angle vc, the corresponding parameter xr used to evaluate the expressions for the tooth profile becomes:

Where

The angle *a*_{pc} = *y*_{pc} at the throat (i.e., *w*_{c} = 0). It is the diametral
pitch at the throat that is used to specify the pitch of the cutter
profiles. The cutter is expressed using the Cartesian coordinates
(xc, yc, zc) as follows:

The image of the coordinates
(xc, yc, zc) upon the transverse surfaces
must account for the cutter spiral.
Consequently, a transverse angular
displacement Δ*v*_{ѱc} is superimposed
on the mapping as follows:

The cutter spiral depends on the ratio between the axial displacement
Δ*w*_{ѱc} and the angular displacement Δ*v*_{ѱc}. The displacement
Δ*v*_{ѱc} is based on a constant lead for a given transverse
surface and the spiral angles ѱ_{c} for each radii uc are different.
Note that the displacement Δ*v*_{ѱc} is based on the lead for the reference
pitch surface and the spiral angles ѱ_{c} change for each
radius *u*_{c}.

**Illustrative Example**

This example presents a spiral hypoid gear set for motion transmission between skew axes using Delgear software (Ref. 16). The shaft angle is 90° and the shaft offset is 25 mm. The speed ratio 3.27; 11 teeth on the pinion and 36 teeth on the ring gear. The face width is 35 mm, the axial contact ratio is 3.0 and the nominal spiral angle is 61°. The tooth profile is a standard invo- Figure 11 Input and output gears with involute teeth. Figure 10 Rack, transverse profile, hyperboloidal cutter. Figure 9 Scaling of tooth profile on cutter element. 60 GEAR TECHNOLOGY | May 2015 [www.geartechnology.com] technical lute tooth profile. The normal pressure angle is 20°, the transverse contact ratio is 1.25, the addendum constant is 1.0 and the dedendum constant is 1.2. The variable diameter cutter has three teeth and the nominal lead angle is 10°. Figure 10 shows the rack tooth, a transverse segment of the cutter, and a virtual model of the cutter. The gear pair is depicted in mesh in Figure 11.

**Summary**

Demonstrated is the specification of involute gear teeth on hypoid gears. This process involves the specification of a classical involute rack, a mapping that transforms this rack to a planar circular profile. A system of cylindroidal coordinates is used to define hyperboloidal cutters. Another transformation is used to map the planar circular profile to a hyperboloidal cutter with suitable geometry for specifying general spiral bevel and hypoid gear pairs. An example of an automotive rear differential gear set is presented to illustrate the process.

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This paper was originally presented at the 2014 International Gear Conference, Lyon Villeurbanne, France. It is republished here with the author’s permission.