Gear Geometry and Applied Theory Episode 2 Part 2 doc

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11.3 Conditions of Nonundercutting 313
and N
c
(N
2
is given):
r
ac
N
c
[
−N
c
sin(φ
c
+ ) cos φ
2
+ N
2
cos(φ
c
+ ) sin φ
2
]
=−r
a2
sin

w

a2
2r
a2

r
ac
N
c
[
N
c
sin(φ
c
+ ) sin φ
2
+ N
2
cos(φ
c
+ ) cos φ
2
]
= r
a2
cos

w
a2
2r
a2


.
(11.3.11)
The first guess for the solution of system (11.3.11) is based on considerations similar
to those previously discussed:
Step 1: Transforming equation system (11.3.11), we obtain
cos
2
(φ
c
+ ) =
r
2
a2
−r
2
ac
r
2
ac

N
2
N
c

2
− 1

. (11.3.12)

We take for the first guess N
c
= 0.8N
2
and obtain (φ
c
+ ) from Eq. (11.3.12). Param-
eter φ
2
is determined from Eq. (11.3.10).
Step 2: Knowing N
c
, φ
c
, and φ
2
for the first guess, and using the subroutine for
the solutions of equation system (11.3.11), we can determine the exact solution for
Axial Generation
Radial Generation
0
0
20
40
60
80
100
120
140
160

180
200
20
40 60
80
100 120 140 160 180 200
No. of Gear Teeth
No. of Shaper Teeth
Figure 11.3.5: Design chart for pressure angle α
c
= 30
◦
.
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314 Internal Involute Gears
Table 11.3.1: Maximal number of shaper teeth
Pressure angle Generation method Gear teeth Shaper teeth
Axial 25 ≤ N
2
≤ 31 N
c
≤ 0.82N
2
− 3.20
α
c
= 20
◦
Axial 32 ≤ N

2
≤ 200 N
c
≤ 1.004N
2
− 9.162
Two-parameter 36 ≤ N
2
≤ 200 N
c
≤ N
2
− 17.6
Axial 17 ≤ N
2
≤ 31 N
c
≤ 0.97N
2
− 5.40
α
c
= 25
◦
Axial 32 ≤ N
2
≤ 200 N
c
≤ N
2

− 6.00
Two-parameter 23 ≤ N
2
≤ 200 N
c
≤ N
2
− 11.86
α
c
= 30
◦
Axial 15 ≤ N
2
≤ 200 N
c
≤ N
2
− 4.42
Two-parameter 17 ≤ N
2
≤ 200 N
c
≤ N
2
− 8.78
N
c
. Computations based on the above algorithms allow us to develop charts for de-
termination of the maximal number of shaper teeth, N

c
, as a function of N
2
and the
pressure angle α
c
. An example of such a chart developed for axial generation and two-
parameter generation is shown in Fig. 11.3.5. Table 11.3.1 (developed by Litvin et al.
[1994]) allows us to determine the maximal number of shaper teeth for various pressure
angles.
11.4 INTERFERENCE BY ASSEMBLY
We consider that the internal gear with the tooth number N
2
was generated by the
shaper with tooth number N
c
and the condition of nonundercutting was observed.
Then, we consider that the internal gear is assembled with the pinion with the tooth
number N
1
> N
c
. The question is what is the limiting tooth number N
1
that allows
us to avoid interference by assembly. Henceforth, we consider two possible cases of
assembly – axial and radial.
Axial Assembly
Axial assembly is performed when the final center distance E
(2)

= (N
2
− N
1
)/2P is
initially installed and the pinion is put into mesh with the internal gear by the axial
displacement of the pinion. Radial assembly means that the pinion is put into mesh
with the internal gear by translational displacement along the center distance. The center
distance E by the radial displacement of the pinion is changed from (N
2
− N
1
− 4)/2P
to (N
2
− N
1
)/2P .
Interference in the axially assembled drive occurs if the tip of the pinion tooth gener-
ates in relative motion a trajectory that intersects the gear involute profile. The trajectory
is an extended hypocycloid. The solution is based on the same approach that was applied
for axial generation. The limiting number N
1
of pinion teeth is a little larger than N
c
due to the lessened dimension of the pinion addendum in comparison with the shaper
addendum.
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11.4 Interference by Assembly 315

Radial Assembly
The investigation of interference in the radial assembly is based on the following con-
siderations:
(i) We represent in fixed coordinate system S
f
equations of profiles of several gear
spaces and pinion teeth by the following vector functions:
r
(2)
f
(θ
2
, j δ
2
, N
2
), r
(1)
f
(θ
1
, j δ
1
, E, N
1
). (11.4.1)
Here, θ
i
is the parameter of the involute profile (i = 1, 2); δ
i

= (2π )/N
i
is the
angular pitch; j is the space (tooth) number; and E is the variable center distance
that is installed by the assembly. The superscripts “1” and “2” indicate the pinion
and the gear, respectively; N
2
is considered as given.
(ii) Interference of pinion and gear involute profiles occurs if
r
(2)
f
(θ
2
, j δ
2
, N
2
) − r
(1)
f
(θ
1
, j δ
1
, E, N
1
) = 0. (11.4.2)
(iii) Equations (11.4.2) provide a system of two scalar equations
f

j
(θ
2
,θ
1
, j δ
2
, j δ
1
, E, N
1
, N
2
) = 0(j = 0, 1, 2, ,m). (11.4.3)
We consider the most unfavorable case when the point of interference lies on the
addendum circles of the pinion and the gear (Fig. 11.4.1), and therefore parameters θ
1
and θ
2
are known. We will determine (N
1
, E) if the solution of equation system (11.4.3)
exists. The solution for N
1
= N
(r )
1
determines the maximal number N
(r )
1

of the pinion
that is allowed by radial assembly.
Figure 11.4.1: Interference by radial assembly.
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316 Internal Involute Gears
Figure 11.4.2: Radial assembly.
After the gears are radially assembled and the final center distance E
(2)
is installed, the
tip of the pinion generates an extended hypocycloid while the pinion and gear perform
rotational motions. Interference of the hypocycloid with the gear involute profile is
avoided by making the number of pinion teeth N
1
≤ N
(a)
1
, where N
(a)
1
is the number
of pinion teeth allowed by axial assembly. The designed number of pinion teeth should
not exceed N
(a)
1
and N
(r )
1
.
Figure 11.4.2 illustrates the computerized simulation of radial assembly of the pinion

and gear. The computations were performed for a gear drive with N
1
= 25, N
2
= 40,
diametral pitch P = 8, and pressure angle α
c
= 20
◦
.
We can avoid the investigation of interference by radial assembly if the pinion tooth
number N
(r )
1
satisfies the inequality N
(r )
1
≤ N
(r )
c
where N
(r )
c
is the shaper tooth number
allowed by radial–axial generation (see Table 11.3.1).
Nomenclature
E
c
distance between gear and cutter axes (Fig. 11.2.1)
N

1
pinion teeth number
N
2
gear teeth number
N
c
shaper teeth number
P diametral pitch
j tooth number
m
ij
transmission ratio of gear i to gear j
r
a1
radius of pinion addendum circle
r
a2
radius of gear addendum circle (Fig. 11.3.2)
r
ac
radius of cutter addendum circle (Fig. 11.2.2)
r
b1
radius of pinion base circle
r
b2
radius of gear base circle (Fig. 11.3.2)
r
bc

radius of cutter base circle (Fig. 11.2.2)
r
p2
radius of gear pitch circle (Fig. 11.3.2)
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11.4 Interference by Assembly 317
r
pc
radius of cutter pitch circle (Fig. 11.2.2)
s
ac
tooth thickness of the cutter on the addendum circle (Fig. 11.2.2)
s
pc
tooth thickness of the cutter on the pitch circle (Fig. 11.2.2)
w
a2
space width on the gear addendum circle (Fig. 11.3.2)
w
p2
space width on the gear pitch circle (Fig. 11.3.2)
2 angle of tooth thickness on the cutter addendum circle (Fig. 11.2.2)
α
c
pressure angle of cutter
θ
i
parameter of gear involute profile (i = 1, 2) (Fig. 11.3.1)
φ

2
angle of gear rotation (Fig. 11.2.1)
φ
c
angle of cutter rotation (Fig. 11.2.1)
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12 Noncircular Gears
12.1 INTRODUCTION
Noncircular gears transform rotation between parallel axes with the prescribed gear
ratio function
m
12
=
ω
(1)
ω
(2)
= f (φ
1
)
where φ
1
is the angle of rotation of the driving gear. The center distance between the
axes of rotation is constant. The most typical examples of application of noncircular
gears are (i) as the driving mechanism for a linkage to modify the displacement function
or the velocity function, and (ii) for the generation of a prescribed function.
Figure 12.1.1 shows the Geneva mechanism that is driven by elliptical gears. The
application of elliptical gears enables it to change the angular velocity of the crank of
the mechanism during the crank revolution. A crank–slider linkage that is driven by

elliptical gears is shown in Fig. 12.1.2. A kinematical sketch of the mechanism is shown
in Fig. 12.1.3(a). Application of elliptical gears enables it to modify the velocity function
v(φ) of the slider [Fig. 12.1.3(b)]. Oval gears (Fig. 12.1.4) are applied in the Bopp and
Reuter meters for the measurement of the discharge of liquid; the oval gears are shown
in the figure in three positions. Figure 12.1.5 shows noncircular gears with unclosed
centrodes that are applied in instruments for the generation of functions. Figure 12.1.6
shows a noncircular gear of a drive that is able to transform rotation between parallel
axes for a cycle that exceeds one gear revolution. During the cycle the gears perform
axial translational motions in addition to rotational motions.
Noncircular gears have not yet found a broad application although modern man-
ufacturing methods enable their makers to provide conjugate profiles using the same
tools as are applied for spur circular gears. The following sections are based on work
by Litvin [1956].
12.2 CENTRODES OF NONCIRCULAR GEARS
We consider two cases, assuming as given either (i) the gear ratio function m
12
(φ
1
), or
(ii) the function y(x) to be generated.
318
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12.2 Centrodes of Noncircular Gears 319
Chain conveyor
Elliptical
gears
Figure 12.1.1: Conveyor driven by the Geneva mechanism and elliptical gears.
Case 1: The gear ratio function
m

12
(φ
1
) ∈ C
1
, 0 ≤ φ
1
≤ φ
∗
1
(12.2.1)
where φ
1
is the angle of rotation of the driving gear 1 is given. Here,
m
12
(φ
1
) =
ω
(1)
ω
(2)
=
dφ
1
dφ
2
where ω
(i )

(i = 1, 2) is the gear angular velocity.
Figure 12.1.2: Conveyor based on application of the crank–slider linkage and elliptical gears.
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Figure 12.1.3: Combination of elliptical gears with a crank–slider linkage.
I
II
III
Figure 12.1.4: Oval gears of a liquid meter.
320
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12.2 Centrodes of Noncircular Gears 321
Figure 12.1.5: Noncircular gears applied in in-
struments.
Figure 12.1.6: Twisted noncircular gear.
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322 Noncircular Gears
The centrode of gear 1 is represented in polar form by the equation
r
1
(φ
1
) = E
1
m
12
(φ
1

) ± 1
(12.2.2)
where E is the center distance. The centrode of the driven gear 2 is determined with the
equations
r
2
(φ
2
) = E
m
12
(φ
1
)
m
12
(φ
1
) ± 1
,φ
2
=

φ
1
0
dφ
1
m
12

(φ
1
)
. (12.2.3)
Function φ
2
(φ
1
) ∈ C
2
is a monotonic increasing function, and the gear ratio function
m
12
(φ
1
) ∈ C
1
must be positive. The difference between m
12 max
and m
12 min
is to be
limited to avoid undesirable pressure angles (see Section 12.12). We have to differentiate
the angle of rotation φ
i
of gear i from the polar angle θ
i
that determines the position
vector of the centrode (i = 1, 2). Angles φ
i

and θ
i
are equal, but they are measured in
opposite directions.
The orientation of the tangent with respect to the current position vector of the
centrode is designated by angle µ, where
tan µ
i
=
r
i
(φ
i
)
dr
i
dφ
i
. (12.2.4)
Equations (12.2.1), (12.2.3), and (12.2.4) yield
tan µ
1
=−
m
12
(φ
1
) ± 1
m


12
(φ
1
)
(12.2.5)
tan µ
2
=±
m
12
(φ
1
) ± 1
m

12
(φ
1
)
. (12.2.6)
Here, m

12
= (∂/∂φ
1
)[m
12
(φ
1
)].

Function µ
i
(φ
1
)(i = 1, 2) is used for determination of variations of the pressure angle
in the process of meshing (see Section 12.12). The upper (lower) sign in the above
expressions with double signs corresponds to the case of external (internal) gears. Angle
µ
i
is measured in the same direction as θ
i
.
The following discussion is limited to the case of external noncircular gears. The
subscripts “1” and “2” in expressions for µ
1
and µ
2
indicate gears 1 and 2, respectively.
Case 2: Function y(x) to be generated is given
y(x) ∈ C
2
, x
2
≥ x ≥ x
1
.
Rotation angles of the gears are determined as
φ
1
= k

1
(x − x
1
),φ
2
= k
2
[y(x) − y(x
1
)] (12.2.7)
where k
1
and k
2
are the scale coefficients of constant values. Equations (12.2.7) represent
in parametric form the displacement function of the gears.
The gear ratio function is
m
12
=
dφ
1
dφ
2
=
k
1
k
2
y

x
(12.2.8)
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12.3 Closed Centrodes 323
where y
x
= dy/dx; y
x
(x) ∈ C
1
, and x
1
≤ x ≤ x
2
. The gear centrodes are represented
by the equations
φ
1
= k
1
(x − x
1
), r
1
= E
k
2
y
x

k
1
+ k
2
y
x
(12.2.9)
φ
2
= k
2
[y(x) − y(x
1
)], r
2
= E
k
1
k
1
+ k
2
y
x
. (12.2.10)
In the case when the derivative y
x
changes its sign in the area x
1
≤ x ≤ x

2
, the di-
rect generation of y(x ) by noncircular gears becomes impossible. This obstacle can be
overcome as follows:
(i) Consider that the noncircular gears generate instead of y(x) the function
F
1
(x) = y(x) + k
3
x (k
3
is constant). (12.2.11)
(ii) A pair of circular gears generates simultaneously the function
F
2
(x) = k
3
x. (12.2.12)
(iii) Functions F
1
(x) and F
2
(x) are transmitted to a differential gear mechanism, and
then the given function y(x) will be executed as the angle of rotation of the driven
shaft of the differential mechanism.
The maximal values of the scale coefficients are determined by the equations
k
1 max
=
φ

1 max
x
2
− x
1
, k
2 max
=
φ
2 max
y(x
2
) − y(x
1
)
(12.2.13)
where φ
i max
= 300
◦
∼ 330
◦
for gears with unclosed centrodes. Knowing function
y
x
(x) and the coefficients k
1
and k
2
, we are able to determine function µ

1
(φ
1
) and
estimate the variation of the pressure angle. In some cases, it becomes necessary to
use a sequence of two pairs of noncircular gears to decrease the maximal value of
the pressure angle (see Section 12.8).
12.3 CLOSED CENTRODES
Noncircular gears designated for continuous transformation of rotational motion must
be provided with cl os ed centrodes. This yields the following requirement for m
12
(φ
1
).
The gear ratio function m
12
(φ
1
) must be a periodic one, and its period T is related with
the periods T
1
and T
2
of the revolutions of gears 1 and 2 as
T =
T
1
n
2
=

T
2
n
1
(12.3.1)
where n
1
and n
2
are whole numbers.
Let us now consider the following design case:
(i) The centrode of gear 1 is already designed as a closed curve.
(ii) Gears 1 and 2 must perform continuous rotations, and n
1
and n
2
are the numbers
of revolutions of the gears.
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324 Noncircular Gears
The question is, what are the requirements to be satisfied to obtain that the gear 2
centrode is a closed curve as well. The solution is based on the following ideas:
(i) We consider that the centrode of gear 1 is represented as a closed curve by the
periodic function r
1
(φ
1
) ∈ C
2

and r
1
(2π) = r
1
(2π/n
1
) = r
1
(0).
(ii) The angle of rotation of gear 2, φ
2
= 2π/n
2
, must be performed while gear 1
performs rotation of the angle φ
1
= 2π/n
1
.
(iii) Taking into account that
2π
n
2
=

2π
n
1
0
dφ

1
m
12
(φ
1
)
(12.3.2)
and
m
12
(φ
1
) =
r
2
(φ
1
)
r
1
(φ
1
)
=
E − r
1
(φ
1
)
r

1
(φ
1
)
. (12.3.3)
we obtain
2π
n
2
=

2π
n
1
0
r
1
(φ
1
)
E − r
1
(φ
1
)
dφ
1
. (12.3.4)
Equation (12.3.4) can be satisfied with a certain value of center distance E, with
which the centrode of gear 2 will be a closed curve.

Problem 12.3.1
Consider that the centrode of gear 1 is an ellipse (Fig. 12.3.1) and the number of
revolutions of the gears are n
1
= 1 and n
2
= n. The center of rotation of gear 1 is focus
O
1
of the ellipse. The centrode of gear 1 is represented in polar form by the equation
r
1
(φ
1
) =
p
1 + e cos φ
. (12.3.5)
Here, p = a(1 −e
2
), e = c/a (Fig. 12.3.1).
Figure 12.3.1: Elliptical centrode.
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12.3 Closed Centrodes 325
Equation (12.3.4) yields
2π
n
=


2π
0
p
E − p + Ee cos φ
1
dφ
1
=
2π p
[(E − p)
2
− E
2
e
2
]
1
2
. (12.3.6)
The derivation of Eq. (12.3.6) is based on the following considerations:
(i)

2π
0
dφ
a +b cos φ
=

π
−π

dφ
a +b cos φ
where a = E − p, b = Ee.
(ii) The substitution
tan
φ
2
= y
yields that

dφ
a +b cos φ
=
2
(a +b)

dy
1 +


a −b
a +b

0.5
y

2
=
2
(a

2
− b
2
)
0.5

dz
1 + z
2
=
2
(a
2
− b
2
)
0.5
tan
−1
z
where
z =

a −b
a +b

0.5
y =

a −b

a +b

0.5
tan

φ
2

.
(iii) Finally, we obtain

π
−π
dφ
a +b cos φ
=
2
(a
2
− b
2
)
0.5

tan
−1


a −b
a +b


0.5
tan

φ
2







π
−π
=
2π
(a
2
− b
2
)
0.5
.
The derivations above confirm Eq. (12.3.6).
Using Eq. (12.3.6), we obtain the following expression for E:
E =
p
1 − e
2


1 + [1 + (n
2
− 1)(1 − e
2
)]
1
2

. (12.3.7)
For the case when n = 1, we obtain
E =
2p
1 − e
2
= 2a,
and the gear centrode is an ellipse as well.
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326 Noncircular Gears
12.4 ELLIPTIC AL AND MODIFIED ELLIPTICAL GEARS
Modification of Elliptical Centrode
The modification of an elliptical centrode is based on the following ideas proposed by
Litvin [1956]:
(i) Consider that a current point M of the elliptical centrode is determined with the
position vector [Fig. 12.4.1(a)]
O
1
M = r
1

(φ
1
), 0 ≤ φ
1
≤ π. (12.4.1)
(ii) We determine the respective point M
∗
of the modified centrode as
O
1
M
∗
= r
∗
1

φ
1
m
I

0 ≤ φ
1
≤ π, |r
∗
1
|=|r
1
|. (12.4.2)
(iii) The same principle of centrode modification is applied for the lower part of the

ellipse [Fig. 12.4.1(b)]; the modification coefficient is m
II
. Generally, m
II
= m
I
.
(iv) The initial elliptical centrode and the modified one are shown in Fig. 12.4.1(c).
Figure 12.4.1: Modified elliptical centrode.
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12.4 Elliptical and Modified Elliptical Gears 327
Figure 12.4.2: Ordinary and modified elliptical centrodes.
Figure 12.4.2 shows the modification of identical elliptical centrodes for the case
where m
I
= 3/2, n
1
= n
2
= 1, and e
1
= 0.5. Figure 12.4.2 illustrates the principle of
centrode modification. Noncircular gears with modified elliptical centrodes transform
rotation with a nonsymmetrical gear ratio function m
12
(φ
1
). This function is symmetrical
for noncircular gears with elliptical centrodes.

Figures 12.4.3 and 12.4.4 illustrate noncircular gear drives whose driving gear is pro-
vided with an elliptical centrode. The driving gear performs two and three revolutions,
respectively, while the driven gear performs one revolution. It was proven by Litvin
[1956] that the centrodes of driven gears are modified ellipses.
Figure 12.4.3: Conjugation of an elliptical centrode
and an oval centrode for two revolutions of the driving
gear.
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328 Noncircular Gears
Figure 12.4.4: Conjugation of an ellip-
tical centrode and a mating centrode for
three revolutions of the driving gear.
Two oval gears with identical centrodes (Fig. 12.4.5) are a particular case of mod-
ified elliptical gears when the coefficients of modification are m
I
= m
II
= m = 2 (see
Fig. 12.4.1). The gear centrode is represented by the equation
r
1
=
a(1 − e
2
)
1 − e cos 2φ
1
=
p

1 − e cos 2φ
1
. (12.4.3)
The center of gear rotation is the center of symmetry of the oval. The oval gears are
used in the meter for liquid discharge (Fig. 12.1.4). A gear drive formed by an oval
centrode and a deformed ellipse is shown in Fig. 12.4.6. The driving gear performs two
revolutions for one revolution of the driven gear.
Figure 12.4.5: Oval centrodes.
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12.5 Conditions of Centrode Convexity 329
Figure 12.4.6: Conjugation of an ellip-
tical centrode with the mating centrode
for four revolutions of the driving link.
12.5 CONDITIONS OF CENTRODE CONVEXITY
Noncircular gears with convex–concave centrodes can be generated by a shaper but not
by a hob. The condition of convexity of a gear centrode means that ρ>0, where ρ is
the centrode curvature radius. In the case of concave–convex centrodes, there is a point
of the gear centrode where ρ =∞.
The curvature radius of a gear centrode is represented by the equation
ρ =

r
2
+

dr
dφ

2


3/2
r
2
+ 2

dr
dφ

2
−r
d
2
r
dφ
2
. (12.5.1)
The condition of centrode convexity (ρ>0) yields
r
2
+ 2

dr
dφ

2
−r
d
2
r

dφ
2
> 0. (12.5.2)
Using Eqs. (12.2.2) and (12.2.3), we can represent the condition of centrode convexity
in terms of function m
12
(φ
1
) and its derivatives:
(i) For the driving gear we have
1 + m
12
(φ
1
) + m

12
(φ
1
) ≥ 0. (12.5.3)
(ii) For the driven gear we obtain
1 + m
12
(φ
1
) + (m

12
(φ
1

))
2
− m
12
(φ
1
)m

12
(φ
1
) ≥ 0. (12.5.4)
Here, m

12
= (d/dφ
1
)(m
12
(φ
1
)), m

12
= (d
2
/d
2
φ
1

)(m
12
(φ
1
)). When the inequalities
(12.5.3) and (12.5.4) turn into equalities, there is a centrode point where ρ =∞.
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330 Noncircular Gears
In the case of generation of given function f (x), we obtain the following conditions
of convexity for the driving and driven gears, respectively:
(i)
k
1
k
2
[ f

(x)]
3
+ k
2
1
[ f

(x)]
2
+ 2[ f

(x)]

2
− f

(x) f

(x) ≥ 0. (12.5.5)
(ii)
k
2
[ f

(x)]
3
[k
1
+ k
2
f

(x)] + f

(x) f

(x) − [ f

(x)]
2
≥ 0. (12.5.6)
Problem 12.5.1
Consider an oval centrode given as [see Eq. (12.4.3)]

r
1
=
a(1 − e
2
)
1 − e cos 2φ
1
.
Determine the condition of centrode convexity.
Solution
The gear ratio function and its derivatives are
m
12
(φ
1
) =
E − r
1
(φ
1
)
r
1
(φ
1
)
=
1 − 2e cos 2φ
1

+ e
2
1 − e
2
(12.5.7)
because E = 2a.
m

12
(φ
1
) =
4e sin 2φ
1
1 − e
2
, m

12
(φ
1
) =
8e cos 2φ
1
1 − e
2
. (12.5.8)
Equations (12.5.3), (12.5.8), and (12.5.7) yield
1 + 3e cos 2φ
1

≥ 0, (12.5.9)
which yields e ≤ 1/3.
12.6 CONJUGATION OF AN ECCENTRIC CIRCULAR GEAR WITH A
NONCIRCULAR GEAR
Figure 12.6.1 shows that the center of rotation O
1
of the eccentric circular gear 1 does
not coincide with the geometric center of the circle of radius a. The centrode of gear 2
must be conjugate with the eccentric circle, the centrode of gear 1. Such drives can be
applied with n = 1, 2, 3, ,n where n is the total number of revolutions of gear 2.
The centrode of the eccentric circular gear is represented by the equation
r
1
(φ
1
) = (a
2
− e
2
sin
2
φ
1
)
1/2
− e cos φ
1
= a[(1 − ε
2
sin

2
φ
1
)
1/2
− ε cos φ
1
] (12.6.1)
where ε = e/a, and e is the eccentricity. The gear ratio function m
21
(φ
1
)is
m
21
(φ
1
) =
r
1
(φ
1
)
E − r
1
(φ
1
)
=
c

c − (1 −ε
2
sin
2
φ
1
)
1/2
+ ε cos φ
1
− 1 (12.6.2)
where c = E/a, m
21 max
= (1 +ε)/(c − (1 + ε)), m
21 min
= (1 −ε)/(c − (1 − ε)).
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12.7 Identical Centrodes 331
Figure 12.6.1: Conjugation of an eccentric
circular centrode with the mating centrode.
The centrode of gear 2 will be a closed curve if the following equation is observed
[see Eq. (12.3.4)]:
2π
n
=

2π
0


c
c − (1 −ε
2
sin
2
φ
1
)
1/2
+ ε cos φ
1
− 1

dφ
1
. (12.6.3)
The solution of Eq. (12.6.3) for c can be accomplished numerically, using an iterative
process of computations. The first guess for c is [Litvin, 1968]
c = (1 +n)

1 −
(n − 12)ε
2
4n

. (12.6.4)
The centrode of gear 2 is determined with the following equations:
r
2
= E −r

1
= a[c − (1 − ε
2
sin
2
φ
1
)
1/2
+ ε cos φ
1
] (12.6.5)
φ
2
=

φ
1
0
(1 − ε
2
sin
2
φ
1
)
1/2
− ε cos φ
1
c − (1 −ε

2
sin
2
φ
1
)
1/2
+ ε cos φ
1
dφ
1
. (12.6.6)
The curvature radius of the gear 2 centrode is
ρ
2
=
ar
1
(φ
1
)[E − r
1
(φ
1
)]
[r
1
(φ
1
)]

2
+ Ee cos φ
1
. (12.6.7)
The condition of convexity of the gear 2 centrode is
[r
1
(φ
1
)]
2
+ Ee cos φ
1
≥ 0. (12.6.8)
12.7 IDENTIC AL CENTRODES
In some rare cases the centrodes of mating gears can be designed to be identical. This
goal can be achieved if the following requirements are satisfied:
(i)
φ
2 max
= F (φ
1 max
) = φ
1 max
(12.7.1)
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332 Noncircular Gears
max
max

max
Figure 12.7.1: Displacement function
for gears with identical centrodes.
where
F (φ
1
) = φ
2
is the displacement function.
(ii)
F (φ
1 max
− F (φ
1
)) = φ
1 max
− φ
1
. (12.7.2)
The displacement function F (φ
1
) that satisfies Eqs. (12.7.1) and (12.7.2) is shown
in Fig. 12.7.1. Points m and m

of the graph are conjugate points of the function.
At these points we have
φ

2
= φ

1
= β, φ

1
= φ
2
= δ, tan α =
dφ
2
dφ
1
=
dφ

1
dφ

2
. (12.7.3)
It is easy to verify that elliptical gears that have identical centrodes satisfy the above
requirements.
Another example of design of noncircular gears with identical centrodes is the
case where the gears generate the function
y =
1
x
, x
2
≥ x ≥ x
1

. (12.7.4)
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12.8 Design of Combined Noncircular Gear Mechanism 333
The design is based on the relations
φ
1
= k
1
(x − x
1
),φ
2
= k
2

1
x
1
−
1
x

(12.7.5)
k
1
=
φ
1 max
x

2
− x
1
, k
2
=
φ
2 max
1
x
1
−
1
x
(12.7.6)
where φ
1 max
= φ
2 max
.
The displacement function of the gears is
φ
2
= F (φ
1
) =
a
2
φ
1

a
3
+ a
4
φ
1
(12.7.7)
where a
2
= k
2
, a
3
= k
1
x
2
1
, a
4
= x
1
. Coefficients a
2
, a
3
, and a
4
are related as
φ

2 max
= φ
1 max
=
a
2
φ
1 max
a
3
+ a
4
φ
1 max
, (12.7.8)
which yields
a
2
− a
3
a
4
= φ
1 max
. (12.7.9)
Using Eq. (12.7.7), we may represent the required functional relations (12.7.2)
as follows:
a
2
[φ

1 max
− F (φ
1
)]
a
3
+ a
4
[φ
1 max
− F (φ
1
)]
= φ
1 max
− φ
1
. (12.7.10)
It is easy to verify that Eq. (12.7.10) is satisfied with expressions (12.7.8) for φ
1 max
and (12.7.7) for F (φ
1
). Thus, the requirement for the design of identical centrodes
is observed, and function y = 1/x can be generated by a noncircular gear with such
centrodes.
12.8 DESIGN OF COMBINED NONCIRCULAR GEAR MECHANISM
A combined mechanism of noncircular gears (Fig. 12.8.1) enables us to generate function
y(x) with substantial variation of the derivative y
x
(x). Application of a mechanism with

only one pair of noncircular gears might cause undesirable pressure angles. There are
important reasons to require that gears 1 and 3, respectively gears 2 and 4, be provided
with identical centrodes. The design of such a combined mechanism of noncircular gears
is considered in this section.
We designate with α and δ the angles of rotation of gears 1 and 4 (Fig. 12.8.1) and
introduce the equations
α = k
1
(x − x
1
),δ= k
4
(y − y
1
) (12.8.1)
where
k
1
=
α
max
x
2
− x
1
, k
4
=
δ
max

y
2
− y
1
. (12.8.2)
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334 Noncircular Gears
Figure 12.8.1: Combined gear mechanism with identical
centrodes for (i) gears 1 and 3, and (ii) gears 2 and 4.
Because the above identity of gear centrodes must be provided, it is required that
α
max
= β
max
= γ
max
= δ
max
. (12.8.3)
It is obvious that β = γ because gears 2 and 3 perform rotation as a rigid body. The
requirement that the centrodes be identical can be represented by the following equation:
ψ(α) = f ( f (α)). (12.8.4)
Here,
δ = ψ(α) (12.8.5)
is the function to be generated. Function
β = f (α),δ= f (β)
that relates the angles of rotation β and α (δ and β) is to be determined. In other words,
considering function ψ(α) as given, we have to determine function f (α).
Generally, Eq. (12.8.4) can be solved only numerically, using an iterative process for

computations. Such a process is based on the idea that if f (α
i
) is known for a fixed
value α
i
, and f (α
i
) = α
i
, we can determine the desired function f (α) numerically, in
discrete form, using the equations
α
i +1
= f (α
i
), f (α
i +1
) = ψ(α
i
). (12.8.6)
The above procedure is illustrated graphically in Fig. 12.8.2; more details are given
in Litvin [1968].
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12.9 Generation Based on Application of Noncircular Master-Gears 335
Figure 12.8.2: Iterative process of the solution of Eq.
ψ(α) = f ( f (α)).
The first guess for the numerical solution of Eq. (12.8.4) is based on the approximate
solution
f (α) ≈ [ψ(α)α

n
]
1/(1+n)
(12.8.7)
where
n =

αψ

(α)
ψ(α)

1/2
.
Another approximate solution for f (α) was proposed by Kislitsin [1955]:
f (α) ≈
ψ(α) + α[ψ

(α)]
1/2
1 + [ψ

(α)]
1/2
. (12.8.8)
12.9 GENERATION BASED ON APPLIC ATION OF
NONCIRCULAR MASTER-GEARS
Initially, the generation of noncircular gears was based on application of devices that
simulated the meshing of a noncircular gear with a tool. Figure 12.9.1 shows the Fellow
device where the noncircular master-gear 1 is in mesh with a master-rack. The rack-

cutter and gear being generated are designated by 3 and 4, respectively. The device
developed by Bopp and Reuter is based on simulation of meshing of a noncircular
master worm gear c with a worm f that is identical to the hob d (Fig. 12.9.2); a is
the spur noncircular gear being generated; the cam b and the follower e form the cam
mechanism designated for simulation of the required variable distance between c and
f . Weight g maintains the continuous contact between the cam and the follower.
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336 Noncircular Gears
1
2
3
4
Figure 12.9.1: Generation of noncircular gears by application of the noncircular master-gear and the
rack-cutter.
12.10 ENVELOPING METHOD FOR GENERATION
The main difficulty in application of the devices discussed above was the necessity of
manufacturing noncircular master-gears. A general method for generation of noncircu-
lar gears that does not require master-gears is based on remodelling existing equipment
designed for manufacture of circular gears or using computer-controlled machines, as
proposed by Litvin [1956, 1968]. Patents based on this idea were claimed by Litvin and
d
a
b
e
f
c
g
Figure 12.9.2: Generation of a noncircular gear by application of a noncircular worm-gear.
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12.10 Enveloping Method for Generation 337
Figure 12.10.1: General principle of generation of noncircular gears.
followers in 1949–1951. The proposed approach is based on the following ideas:
(a) The noncircular gears are generated by the same tools (rack-cutters, hobs, and
shapers) that are used for manufacture of circular gears.
(b) Conjugate tooth profiles for noncircular gears are provided due to the imaginary
rolling of the tool centrode over the given gear centrode.
(c) The imaginary rolling of the tool centrode over the centrode of the gear being
generated is accomplished with proper relations between the motions of the tool
and the gear in the process of cutting.
Figure 12.10.1 illustrates the principle of conjugation of tooth shapes for two mating
noncircular gears that are generated by a rack-cutter. The gear centrodes 1 and 2 and
the rack-cutter centrode 3 are in tangency at the instantaneous center of rotation I . The
rack-cutter centrode is a straight line. Pure rolling of each centrode over the other one
is provided if the instantaneous linear velocities of point I of each centrode are equal
under the magnitude and direction. While the gears perform rotation about O
1
and
O
2
, respectively, the rack-cutter translates along the tangent t–t to the gear centrodes
and along the center distance E and rotates about I. The pure rolling of the rack-
cutter over the gear centrodes is provided if the following vector equations are observed
(Fig. 12.10.1):
v
(3)
= v
(1)
= v

(2)
. (12.10.1)
Here,
v
(i )
= ω
(i )
× O
i
I (i = 1, 2)
is the gear velocity in rotation about O
i
.
v
(3)
= v
(3)
t
+ v
(3)
e