International Journal of Precision Engineering and Manufacturing (2022) 23:429–443
https://doi.org/10.1007/s12541-022-00625-2
Online ISSN 2005-4602
Print ISSN 2234-7593
REGULAR PAPER
Tooth Thickness Error Analysis of Straight Beveloid Gear by Inclined
Gear Shaping
Feihong Zhu1 · Chaosheng Song1
· Caichao Zhu1 · Xuesong Du1
Received: 29 October 2021 / Revised: 6 January 2022 / Accepted: 3 February 2022 / Published online: 28 February 2022
© The Author(s), under exclusive licence to Korean Society for Precision Engineering 2022
Abstract
In this paper, two approaches to calculate the tooth thickness error (TTE) of straight beveloid gear by tilt-type gear shaper
were proposed. The first calculation approach of TTE was established by considering the change of pitch circle radius during
the gear slotting process. The analytical tooth surface model of straight beveloid gear was derived by inclined gear shaping,
and the tooth surface point set was obtained. Then, another calculation methodology of TTE was established based on the
analytical straight beveloid gear model. Two approaches were employed to calculate the TTE of internal and external straight
beveloid gear, respectively. And they were employed to validate each other, and the results show a good consistency. The
influences of design parameters on TTE was analyzed. Results show that the internal/external straight beveloid gears have
a convex/concave TTE along the tooth width direction while cutting by tilt-type gear shaper, which makes the tooth thickness of the heel and toe sides of beveloid gear thinned/thickened, respectively. The TTE of internal and external beveloid
gear both increase with the increase of design cone angle, and decrease with the increase of modulus and number of teeth.
For the same macro gear design parameters, the TTE of external beveloid gear is smaller than that of internal beveloid gear.
The results of the two approaches show good consistency at the middle of tooth surface. The maximum difference between
the results of two approaches gradually increases away from the middle of tooth surface, and it is positively correlated with
the value of TTE.
Keywords Straight beveloid gear · Tooth thickness error · Slotting · Zero backlash · Inclined gear shaping
List of Symbols
𝛿Cone angle of beveloid gear
𝜔cAngular velocity of slotting cutter
𝜔Angular velocity of gear blank
ZcTooth numbers of slotting cutter
ZTooth numbers of gear blank
mnModulus of normal section
mtModulus of transverse section
𝛼nPressure angle of normal section
𝛼tPressure angle of transverse section
𝛼t′Pressure angle at any circle of the gear blank
′ Pressure angle at pitch circle of the gear blank in
𝛼tp
the processing section
h∗anAddendum coefficient of normal section
h∗atAddendum coefficient of transverse section
rpWorking pitch circle radius of gear blank
* Chaosheng Song
chaoshengsong@hotmail.com
1
State Key Laboratory of Mechanical Transmissions,
Chongqing University, Chongqing 400030, China
rpcWorking pitch circle radius of slotting cutter
rReference circle radius of gear blank
rtcReference circle radius of slotting cutter
rbtBase circle radius of gear blank
c
Base circle radius of slotting cutter
rbt
ymRadial displacement of rack cutter
xM Modification coefficient of beveloid gear in the
middle section
xtModification coefficient of beveloid gear in the
section of interest
BDistance from the section of interest to the middle
section
xtcModification coefficient of the slotting cutter cutting section
sTooth thickness at reference circle
ecReference circle space width of the slotting cutter
ecpPitch circle space width of the slotting cutter
rK Radius of any circle between tip circle and root
circle of the machined gear
hkTooth depth coefficient
Vol.:(0123456789)
13
430
International Journal of Precision Engineering and Manufacturing (2022) 23:429–443
sK Tooth thickness at any circle of the gears processed by the rack cutter
scK Tooth thickness at any circle of the gears processed by the slotting cutter
ser
Tooth thickness error at any circle of the gears
K
𝜑cRotated angles of the slotting cutter
𝜑gRotated angles of the gear blank
nkiCommon normal vector of the contact point of the
unit k in coordinate Si
vkiVelocity vector of the contact point of the unit k
in coordinate Si
x, y, zk
X/Y/Z-Axis value of the unit k tooth surface
X/Y/Z-Axis components of the common normal
ncx,y,z
vector of the cutting edge
1 Introduction
Beveloid gears have become an important gear transmission device for transmitting motion and power among certain conditions, due to its advantages of compensating for
axial movement, eliminating backlash and being easy to
manufacture. The internal beveloid gear pair is widely used
in the reducer with parallel shafts transmission, because of
the high transmission ratio and the compact space. For the
gear shaping processing of internal beveloid gear, there is
an inclination angle between the axis of the gear shaper and
the gear blank, which will unavoidably cause the deviation
of tooth surface [1]. Therefore, it is of great significance to
propose a method to quickly and accurately calculate the
tooth thickness error (TTE) of beveloid gear by inclined
gear shaping, which is essential to improve the machining
accuracy of beveloid gear, and realize precise transmission.
Since Mettitt proposed the beveloid gear, many researchers have conducted extensive and in-depth research on beveloid gears. To overcome the weaknesses of beveloid gears
in point contact and low durability, Mitome [2] proposed
a concave conical gear, and pointed out that it is particularly necessary to develop practical gear cutting machines
and gear grinding machines. Sun [3] proposed an efficient
accurate parametric mesh stiffness model of helical beveloid
gear pair based on the potential energy theory. In the aspect
of external beveloid gear manufacturing, Yang et al. [4]
deduced the mathematical model of the beveloid gear tooth
surface considering machine tool adjustment errors. Song
et al. [5–8] studied the hobbing method of marine involute
beveloid gears based on the principle of gear hobbing, established the meshing model of intersecting shaft and cross
shaft beveloid gear transmission, and investigated their
geometric design and contact characteristics. Liu [9] proposed a grinding method for machining beveloid gears. The
proposed method improves the shortcomings of Mitome’s
grinding method by eliminating the transmission error of
13
the helical concave beveloid gear pair. Simulation results
indicate that the gear processed by the proposed grinding
method has a larger contact ellipse than the conventional
beveloid gear pair. Brecher [10] analyzed the influence of
two machine kinematics (tilt and linked feeds) and two tool
concepts (grinding disk and grinding worm) on the microgeometry of the grinded beveloid gears. The simulation
results show that the worm grinding process causes a tooth
surface twist of helical beveloid gears, but this phenomenon
does not exist in grinding disc grinding. Zhao [11] investigated the processing of beveloid gears starting from the traditional machine tool, and provided the relationship between
gear and processing parameters, which facilitated the serial
production of beveloid gear. Ni et al. [12] proposed a parabolic modification tool to enhance the contact characteristics
of crossed beveloid gear transmission, and investigated the
sensitivity of misalignments. Liu et al. [13, 14] investigated
the influences of general tooth surface modifications (helix
crowing, flank line slope, profile crowning, profile slope,
concave and convex modifications) on the gear contact
characteristics.
In the aspect of internal beveloid gear manufacturing,
Wu [15] proposed the method to slot a pair of parallel-shaft
internal meshing gear pairs using the same gear shaper. He
deduced the formulas for the optimization of slotting inclination angle, and then applied parallel shaft beveloid gear to
RV reducer [16]. Li et al. [17, 18] established the tooth surface model of non-involute beveloid gears with intersecting
axes that can achieve line contact using the space engagement theory, and proposed methods to calculate the tooth
profile errors and axial errors. Hu [19] used a ball-end milling cutter to mill internal beveloid gears using a 4-axis horizontal CNC machining center. Song [1] proposed a slotting
method with parallel axes between the slotting cutter and
the gear blank. Although this method can process beveloid
gears without theoretical errors, it cannot be machined with
the current gear shaping machine tool, and the machine tool
must be modified. Liu [20, 21] developed the mathematical
model of internal beveloid gear according to the gear slotting mechanism, and investigated the tooth surface deviations due to the tooth numbers of the slotting cutter. Hu [22]
proposed a design method of special gear shaper cutters for
precision manufacturing internal beveloid gears. The above
papers mostly focus on the machining method, optimization of machining parameters and tooth contact analysis of
beveloid gears. Little efforts have been invested in the generation principle of TTE during the beveloid gear slotting.
Therefore, it is of great significance to propose a method to
calculate the TTE of beveloid gear cutting by tilt-type gear
shaper. This method can guide the selection of parameters
for the design and manufacture of beveloid gears, and does
not require complex calculations. It is easier to apply to engineering practice than traditional methods.
International Journal of Precision Engineering and Manufacturing (2022) 23:429–443
This study mainly focused on the generation principle
of TTE in beveloid gear slotting process. Two approaches
to calculate TTE were proposed for the straight beveloid
gear by tilt-type gear shaper. The first calculation approach
of TTE was established by considering the change of pitch
circle radius during the gear slotting process and the zerobacklash meshing condition. To validate the proposed
approach, a manufacturing mathematical model of straight
beveloid gear was derived. The tooth surface point set was
obtained and another calculation methodology of TTE for
beveloid gears using the slotting cutter with gear shape was
established. Then, two kinds of TTE solving approaches
were used to calculate the TTE of the internal and external straight beveloid gears. Finally, the influences of design
parameters on the TTE were investigated.
431
2 Calculation Method of TTE for Beveloid
Gears Using an Imaginary Rack Cutter
The schematic view of the straight beveloid gear manufacturing method by tilt-type gear shaper is shown in Fig. 1.
The rotating axis of slotting cutter and gear blank form an
angle 𝛿 that is equal to the cone angle of the beveloid gear.
The cutter and the gear blank rotate between the intersecting axes with angular velocities 𝜔c and 𝜔, respectively. Two
angular velocities are related as follows
𝜔c
Z
=
𝜔
Zc
(1)
where Zc and Z are the teeth numbers of the cutter and the
gear blank, respectively.
The slotting process of straight beveloid gears can be
regarded as the meshing between the slotting cutter, the
beveloid gear and the imaginary rack. As shown in Fig. 2a,
taking the machining process of external beveloid gear as an
example, when the slotting cutter meshes with the imaginary
Fig. 1 Schematic view of the
slotting mechanism of beveloid
gear
Fig. 2 Relationship between slotting cutter, beveloid gear and imaginary rack
13
432
International Journal of Precision Engineering and Manufacturing (2022) 23:429–443
rack, the symmetry plane of the imaginary rack is parallel
to the cutter axis I, and the included angle with the beveloid
gear axis II Sis 𝛿. It is obvious that the imaginary rack can
mesh with the beveloid gear, and drive the beveloid gear
to rotate with the other side of its tooth surface, when the
slotting cutter pushes the imaginary rack to move. When
the imaginary rack meshes with the slotting cutter and the
beveloid gear respectively, a fixed transmission ratio can be
achieved. Therefore, the slotting cutter and the beveloid gear
are also transmitted with a fixed transmission ratio. Likewise, the slotting process of internal beveloid gears can also
be regarded as the process of two gears meshing with the
same imaginary rack, as shown in Fig. 2b.
Figure 3 shows the normal section and transverse section
of the imaginary rack. The tooth profile on the normal plane
of the imaginary rack can be considered as a standard tooth
profile. The pressure angle and the addendum coefficient are
𝛼n and h∗an , respectively. The pressure angle of transverse section is 𝛼t = arctan(tan𝛼n cos𝛿) and the addendum coefficient
is h∗at = h∗an ∕cos𝛿.
Since the imaginary rack meshes with the slotting cutter
and the beveloid gear at the same time, the slotting cutter can
be replaced by an imaginary slotting cutter with the tooth
profile of the imaginary rack transverse plane. The imaginary slotting cutter and the beveloid gear rotate between
parallel axes with angular velocities 𝜔c and 𝜔, respectively.
The cutter machining movement includes axial and radial
movement, and the mentioned two movements form an angle
𝛿, as shown in Fig. 4. The imaginary slotting cutter and the
actual slotting cutter have the same tooth number to ensure
that they have the same linear velocity at the pitch cone
generatrix.
The imaginary rack can mesh with the slotting cutter
and the beveloid gear, respectively. Therefore, if the imaginary rack is regarded as a processing tool, it will be able to
Fig. 3 Normal section and
transverse section of the imaginary rack
Fig. 4 Schematic view of the
cutting mechanism of beveloid
gear by imaginary slotting cutter
13
separately process the beveloid gear and the slotting cutter.
Take the machining process of internal beveloid gear as an
example. If the cogging part of the internal beveloid gear is
regarded as the gear teeth of the external gear, both the internal beveloid gear and the slotting cutter can be enveloped by
the imaginary rack. However, it is necessary to change the
center distance between the slotting cutter and the gear blank
when manufacturing the modified gear, which will lead to
the change of pitch circle radius, while that does not show in
the process of rack manufactured modified gear. Therefore,
there is a TTE on the modified gear, which is processed by
the slotting cutter and the imaginary rack.
Figure 5 shows the machining process of gears with different modification coefficients. The rack tool processing the
modified gears with different modification coefficients only
needs to move the displacement ym in the radial direction,
and the working pitch circle rp always coincides with the reference circle r. Therefore, a rack cutter can accurately process
a beveloid gear which satisfies Eq. 2. Machining a straight
beveloid gear with a slotting cutter is equivalent to making the
imaginary slotting cutter move with the same distance ym, like
that of the rack cutter to machine the beveloid gear. As shown
in Fig. 5, this method will cause the pitch circle radius rp to
change, and rp will no longer coincide with the reference circle r, so there will be the TTE in the processed beveloid gear.
Assuming that the modification coefficient of the beveloid gear
middle section is xM , the modification coefficient of a certain
section of the machined gear can be represented as follows
xt = xM + B
tan𝛿
mt
(2)
where B is the distance from the section of interest to the
middle section, which is positive towards the heel side and
negative towards the toe side.
International Journal of Precision Engineering and Manufacturing (2022) 23:429–443
433
Fig. 5 Meshing conditions of imaginary slotting cutter, beveloid gear and imaginary rack
When the modification coefficient of the slotting cutter cutting section is not zero, the center distance between the slotting
cutter and the gear blank will change. The amount of change in
the center distance can be represented as follows
(
)
ym = xt ± xtc mt
(3)
where xtc is the modification coefficient of the slotting cutter
cutting section.
Since the slotting cutter and the gear blank are driven at
a constant angular velocity, the pitch circles of the slotting
cutter and the processed gear are rpc and rp , respectively.
rpc = rtc + ym
rp = r + ym
rtc
r ± rtc
r
r ± rtc
(4)
(5)
where and r are the reference circle radius of the slotting
cutter and the beveloid gear, respectively.
According to the involute function, the pressure angle at
the pitch circle of the beveloid gear processing section can
be calculated by
�
𝛼tp
= arccos
c
rbt
rpc
= arccos
rbt
rp
(6)
The reference circle tooth thickness at a selected section
of the machined gear can be calculated by
s=
mt 𝜋
± 2xt mt tan𝛼t
2
(7)
In Eqs. (3), (4), (5) and (7), the positive sign is used for
the external beveloid gear, and the negative sign is used for
the internal beveloid gear. The reference circle space width
of the slotting cutter can be calculated by
ec =
mt 𝜋
− 2xtc mt tan𝛼t
2
(8)
rtc
13
434
International Journal of Precision Engineering and Manufacturing (2022) 23:429–443
The pitch circle space width of the slotting cutter can be
calculated by
( c
(
))
�
c
c e
ep = rp c + 2 inv𝛼tp − inv𝛼t
(9)
rt
In the gear shaping process, the slotting cutter and the
machined gear should meet the no-backlash meshing condition, so the pitch circle space width of the slotting cutter
should be equal to the pitch circle tooth thickness of the
machined gear.
The pressure angle at any circle of the machined gear can
be calculated by
𝛼t� = arccos
rbt
rK
(10)
where rK is the radius of any circle between the tip circle
and the root circle of the machined gear, which can be calculated by
(
)
rK = r + xt + hk h∗at mt
(11)
where hk is the tooth depth coefficient, and its value range is
[− 1 to 1]. For internal gears, hk at the tip circle, root circle
and tooth depth middle are − 1, 1, 0, respectively, as shown
in Fig. 6.
The tooth thickness at any circle of the gears processed
by the rack cutter can be represented by
(
sK = rK
(12)
The tooth thickness at any circle of the gears processed
by the slotting cutter can be represented by
)
( c
)
(
ep
�
�
c
∓ 2 inv𝛼t − inv𝛼tp
sK = rK
(13)
rp
In Eqs. (12) and (13), the positive sign is used for the
internal beveloid gear, and the negative sign is used for the
external beveloid gear. The TTE at any circle of the involute
straight beveloid gear cutting by tilt-type gear shaper can be
represented as follows
ser
K
=
scK − sK
2
(14)
3 Calculation Method of TTE for Beveloid
Gears Using the Slotting Cutter with Gear
Shape
3.1 Mathematical Model of Gear Slotting
for Straight Beveloid Gears
As shown in Fig. 1, the cutting mechanism of manufacturing
process between the slotting cutter and the machined gear
Fig. 6 Tooth thickness at any circle of the gears processed by the slotting cutter and rack cutter
13
)
)
(
s
∓ 2 inv𝛼t� − inv𝛼t
r
International Journal of Precision Engineering and Manufacturing (2022) 23:429–443
can be depicted in Fig. 7a for internal beveloid gear and
Fig. 7b for external beveloid gear. The coordinate system
Sc(Xc, Yc, Zc) and Sg(Xg, Yg, Zg) are fixed on the slotting cutter and the beveloid gear blank, respectively. The coordinate
systems Ss(Xs, Ys, Zs) and Sf(Xf, Yf, Zf) are fixed to the frame,
and they are the reference coordinate systems of Sc(Xc, Yc,
Zc) and Sg(Xg, Yg, Zg), respectively. In addition, 𝜑c and 𝜑g are
the rotated angles of the slotting cutter and the gear blank
during the cutting process, and rpc and rp are their working
pitch circle radius.
The tooth profile of the slotting cutter in the coordinate
system Sg(Xg, Yg, Zg) can be obtained by the following coordinate transformation
]
]
[
][
][
[
Rg = Mgf Mf s Msc Rc = Mgc Rc
(15)
435
where Mgf , Mfs and Msc are the transfer matrixes from
coordinate system Sc to Sg. Rc and Rg represent the position
vector of the slotting cutter tooth profile in the coordinate
system Sc and Sg, respectively.
For the cutting process, the cutting edge of the slotting
cutter and the tooth surface of the machined gear should be
continuously tangent [23]. The equation of meshing can be
determined in coordinate system Sf as
(
)
cg
g
ncf ∙ vf = ncf ∙ vcf − vf = 0
(17)
where the subscript f indicates that the vector is expressed
cg
in coordinate Sf. ncf and vf represent the common normal
vector and the relative velocity vector of the contact point,
respectively.
The transformation matrix Mgc can be written as follows.
Mgc
⎡ ±cos𝛿cos𝜑c cos𝜑g ±cos𝛿cos𝜑g sin𝜑c cos𝜑 sin𝛿 cos𝜑 (r ∓ rc cos𝛿) ⎤
g
g p
p
⎥
⎢ +sin𝜑c sin𝜑g
−cos𝜑c sin𝜑g
⎥
⎢ cos𝛿cos𝜑 sin𝜑
cos𝛿sin𝜑c sin𝜑g
c
g
±sin𝛿sin𝜑g ±sin𝜑g (rp ∓ rpc cos𝛿) ⎥
=⎢
±cos𝜑c cos𝜑g
⎢ ∓cos𝜑g sin𝜑c
⎥
⎢
⎥
∓sin𝛿sin𝜑c
cos𝛿
±rpc sin𝛿
∓cos𝜑c sin𝛿
⎢
⎥
⎣
⎦
0
0
0
1
(16)
Fig. 7 Coordination relationship between the shaper cutter and gear blank
13
436
International Journal of Precision Engineering and Manufacturing (2022) 23:429–443
Combining Eqs. (15), (16), and (17), the mathematical
model of the beveloid gear tooth surface is obtained in coordinate system Sg as follows
where xc , yc and zc represent the components of the tooth
profile position vector of the slotting cutter cutting edge in
coordinate Sc, respectively. ncx , ncy and ncz represent the com-
�
�
�
�
⎧ xg = xc sin𝜑c sin𝜑g ± cos𝛿cos𝜑c cos𝜑g − yc cos𝜑c sin𝜑g ∓ cos𝛿cos𝜑g sin𝜑c
�
�
⎪
+cos𝜑g rp ∓ rpc cos𝛿 + zc cos𝜑g sin𝛿
⎪
�
�
�
�
⎪
c sin𝜑g �± yc cos𝜑c cos𝜑g ± cos𝛿sin𝜑c sin𝜑g
⎪ yg = ∓xc cos𝜑g sin𝜑c ∓ cos𝛿cos𝜑
�
⎨
±sin𝜑g rp ∓ rpc cos𝛿 ± zc sin𝜑g sin𝛿
⎪
⎪ zg = zc cos𝛿 ± rpc sin𝛿 ∓ xc cos𝜑c sin𝛿 ∓ yc sin𝛿sin𝜑c
�
�
�
�
⎪
B
A
⎪ 𝜑c = arcsin √ 2 2 + arcsin √ 2 2
B +C
B +C
⎩
here
A = ncx rp yc − ncy rp xc ± ncy rpc xc cos𝛿 ∓ ncx rpc yc cos𝛿
(19)
B = ∓ncy rpc 2 cos𝛿 + ncy rpc rp − ncz rpc yc sin𝛿 + ncy rpc zc sin𝛿
(20)
C = ±ncx rpc 2 cos𝛿 − ncx rpc rp + ncz rpc xc sin𝛿 − ncx rpc zc sin𝛿
(21)
Table 1 Geometric design
parameters of beveloid gear
13
ponents of the common normal vector of the slotting cutter
cutting edge, respectively. The mathematical description of
the slotting cutter can refer to the literature [1].
3.2 Tooth Thickness Error Calculation
According to the mathematical model of gear slotting for
straight beveloid gears, the commercial software Matlab was
used to compile the tooth surface generation program, and
Parameters
Value
Parameters
Value
Tooth numbers Z
Normal module mn (mm)
Normal pressure angle 𝛼n (°)
65
1
20
Addendum coefficient h∗an
Bottom clearance coefficient c∗n
Modification coefficient of middle section xM
Facewidth B (mm)
1
0.25
0
Cone angle 𝛿 (°)
Fig. 8 Tooth surface point sets
of straight beveloid gear
(18)
1
12
International Journal of Precision Engineering and Manufacturing (2022) 23:429–443
437
Fig. 9 Definition of TTE in any
section
then the coordinates of each point of the involute beveloid
gear tooth surface were obtained. When the design cone
angle 𝛿 = 3◦ , according to the geometric design parameters
given in Table 1, the tooth surface point sets of the internal and external beveloid gears were calculated as shown in
Fig. 8a and b respectively.
The beveloid gear theoretical tooth surface is constructed based on the geometric characteristics of the involute function. Then the TTE between the machined tooth
surface coordinate point set and the theoretical tooth surface is calculated. The definition of TTE in any section is
shown in Fig. 9. Since the tooth surface of involute straight
beveloid gear is symmetrical, only the left side is shown.
In the beveloid gear blank coordinate system Sg(Xg, Yg,
Z g), the position coordinate of a certain point M of the
machined tooth surface can be expressed as
𝛼M = arccos
rbt
rM
(25)
According to the involute function, the position coordinates of the point M ′ on the theoretical tooth profile can be
obtained as
�
�
M�
⎡ xg ⎤ ⎡ rM cos �𝜂� − inv𝛼M � ⎤
�
⎥ = ⎢ −rM sin 𝜂 � − inv𝛼M ⎥
Mg� = ⎢ yM
(26)
g
⎢ M
⎢
⎥
� ⎥
M
z
z
⎣ g ⎦ ⎣
⎦
g
Then the TTE at point M of the machined tooth surface
can be calculated by
(
| xM |
| xM� |)
| g |
|
|
er
̂� = rM arctan| | − arctan| g � |
sM = MM
(27)
| xM |
| yM |
| g |
| g |
| |
|
|
M
⎡ xg ⎤
⎥
Mg = ⎢ yM
g
⎢ M
⎥
z
⎣ g ⎦
(22)
into Eq. (2), the modifiSubstituting the coordinate zM
g
cation coefficient xt of the cross-section where point M is
located can be obtained. In this section, according to the
geometric characteristics of the modified gear, for theoretical tooth surface, the increase of the tooth thickness at the
reference circle can be obtained as xt mt tan𝛼t , so 𝜂 ′ can be
calculated by
𝜂� =
x m tan𝛼t
𝜋
+ inv𝛼t + t t
2Z
r
(23)
The radius rM and pressure angle 𝛼M at point M can be
calculated by
√
2
rM = xgM 2 + yM
(24)
g
4 Numerical Examples
To validate the feasibility of the two approaches proposed
in this paper, the TTE calculation methods of Sects. 2 and
3 are used to calculate the TTE of the internal and external
beveloid gears. Then the influences of design parameters on
the TTE are studied. Both the external and internal beveloid
gear adopt the same geometric design parameters, which are
listed in Table 1.
4.1 Comparison of the TTE Results of the Two
Calculation Methods
For the internal gear with the design parameters in Table 1,
the two TTE calculation methods for beveloid gears using an
imaginary rack cutter and the slotting cutter are used to calculate the TTE. Both methods show that the internal straight
13
438
International Journal of Precision Engineering and Manufacturing (2022) 23:429–443
Fig. 10 TTE of internal gear
calculated by the method based
on no-backlash meshing condition
Fig. 11 TTE of internal gear
calculated by the method based
on mathematical model of
beveloid gear
Fig. 12 Difference between the
results of internal gear calculated by two methods
beveloid gear cutting by tilt-type gear shaper has a convex
TTE along the tooth width direction, which makes the tooth
thickness of the heel and toe sides of beveloid gear thinned.
There is almost no error in the middle of the tooth width,
and the toe side has a larger TTE compared to the heel side.
The TTE of internal gear calculated by the two methods are
shown in Figs. 10 and 11.
13
From the results, the maximum absolute value of the TTE
calculated by the calculation method of Sect. 2 is 1.4578 μm,
and that of the method of Sect. 3 is 1.4455 μm. The extreme
values of TTE calculated by the two methods are relatively close, with a difference of only 0.0123 μm. Then the
detailed distribution of the difference between the results
of two methods on the tooth surface is obtained, which are
shown in Fig. 12. The two methods are almost completely
International Journal of Precision Engineering and Manufacturing (2022) 23:429–443
439
Fig. 13 TTE of external gear
calculated by the method based
on no-backlash meshing condition
Fig. 14 TTE of external gear
calculated by the method based
on mathematical model of
beveloid gear
Fig. 15 Difference between the
results of external gear calculated by two methods
coincident in the middle of the tooth surface. The difference
between the two methods gradually increases away from the
middle of the tooth surface, and the maximum difference is
0.01344 μm.
For the external gears, both methods show that the external straight beveloid gear by tilt-type gear shaper has a concave TTE along the tooth width direction. There is almost no
error in the middle of the tooth width, and the heel and toe
sides of beveloid gear have very similar errors, which makes
the tooth thickness of the heel and toe sides of beveloid gear
thickened. In Figs. 13 and 14, the maximum absolute value
of the TTE calculated by the calculation method of Sect. 2 is
0.2887 μm, and that of the method of Sect. 3 is 0.2931 μm.
The extreme values of TTE calculated by the two methods
are very close, with a difference of only 0.0044 μm.
The detailed distribution of the difference between the
results of two methods on the tooth surface is shown in
Fig. 15. Similar to the internal beveloid gear, the results of
the two methods are almost completely coincident in the
middle of the tooth surface. The difference between the two
methods gradually increases away from the middle of the
tooth surface, and the maximum difference is 0.02347 μm.
13
440
International Journal of Precision Engineering and Manufacturing (2022) 23:429–443
Fig. 16 Effects of design cone
angle on the external gear tooth
thickness error
Fig. 17 Effects of design cone
angle on the internal gear tooth
thickness error
Fig. 18 Effects of normal module on the external gear tooth
thickness error
5 Effects of Design Parameters on Tooth
Thickness Error
The internal beveloid gear with too large design cone angle
cannot be accurately processed, so the selection range of
design cone angle parameters in this paper is 𝛿=1–5°. Figure 16a shows the changing trend of the maximum TTEs of
13
the external beveloid gear, which are calculated by the above
two methods when the design cone angle 𝛿 is 1–5°. The
results show that the TTE increases with the design cone
angle, and the maximum value is 7.4795 μm. The maximum difference between the results of two methods also
increases with the design cone angle, and the extreme value
is 0.7718 μm, as shown in Fig. 16b. For the internal beveloid
gear, Fig. 17a shows the influence of the design cone angle
International Journal of Precision Engineering and Manufacturing (2022) 23:429–443
on the maximum value of the TTE. The TTE also increases
with the increase of design cone angle, and the maximum
value is 43.98 μm. The maximum difference between two
methods also increases with the increase of design cone
angle, and the extreme value is 2.901 μm, which is shown
in Fig. 17b. In summary, whether the internal or the external
beveloid gear, the TTE caused by gear slotting all increases
with the increase of the design cone angle. When the gear
441
design parameters remain unchanged, the TTE of the external beveloid gear is smaller than that of the internal beveloid
gear. The maximum difference between the results of the two
methods also increases with the design cone angle. However,
the maximum difference is in the micron level, when the
design cone angle does not exceed 5°.
For the external beveloid gear, Fig. 18a shows the changing trend of the maximum TTE calculated by the two
Fig. 19 Effects of normal module on the internal gear tooth
thickness error
Fig. 20 Effects of tooth numbers on the external gear tooth
thickness error
Fig. 21 Effects of tooth numbers on the internal gear tooth
thickness error
13
442
International Journal of Precision Engineering and Manufacturing (2022) 23:429–443
methods, when the design cone angle 𝛿=3° and the normal
module mn is 1–5 mm, respectively. The TTE decreases with
the increase of modulus, the maximum value is 2.6609 μm.
The maximum difference between the results of two methods also decreases with the increase of modulus, and the
extreme value is 0.2442 μm, as shown in Fig. 18b. For the
internal beveloid gear, Fig. 19a shows the influence trend
of the modulus on the maximum TTE. The TTE decreases
with the increase of modulus, and the maximum value is
14.1852 μm. The maximum difference between the results
of two methods decreases as the modulus increases, and the
extreme value is 0.4088 μm, which is shown in Fig. 19b.
For the external beveloid gear, Fig. 20a shows the changing trend of the maximum TTE calculated by the two methods, when the design cone angle 𝛿 = 3◦ and the tooth numbers are 65, 73, 81, 89, and 97, respectively. The results
show that the TTE decreases with the increase of the tooth
numbers, and the maximum value is 2.6609 μm. The maximum difference between the results of two methods also
decreases as the tooth numbers increases, and the extreme
value is 0.2442 μm, as shown in Fig. 20b. For the internal beveloid gear, Fig. 21a shows the influence of the tooth
numbers on the maximum TTE. The TTE decreases with
the increase of tooth numbers, and the maximum value is
14.1852 μm. The maximum difference between the results of
two methods also decreases as the tooth numbers increases,
and the extreme value is 0.4088 μm, as shown in Fig. 21b.
In summary, whether the internal or the external beveloid
gear, the TTE caused by gear slotting all decrease with the
increase of the modulus and the tooth numbers. The maximum difference between two methods also decreases with
the increase of the modulus and the tooth numbers, and all
the maximum differences belong to the micron level.
6 Conclusion
(1) For straight beveloid gear, the definition of the TTE
and two approaches to calculate TTE of straight beveloid gear by tilt-type gear shaper were proposed. The
first calculation approach of TTE using an imaginary
rack cutter was established, which considers the change
of pitch circle radius during the gear slotting process.
Another calculation approach of TTE using the slotting
cutter with gear shape was established, and the mathematical model of straight beveloid gear manufacturing
was derived.
(2) The internal/external straight beveloid gear have a
convex/concave TTE along the tooth width direction
while cut by tilt-type gear shaper, which makes the
tooth thickness of the heel and toe ends of beveloid
gear thinned/thickened, respectively. The results of
two approaches are almost completely consistent in the
13
middle of the tooth surface, and the difference between
the results of the two approaches gradually increases
away from the middle of the tooth surface.
(3) The TTE of internal and external beveloid gear increase
with the increase of design cone angle, and decrease
with the increase of modulus and tooth numbers. For
the same macro gear design parameters, the TTE of
external beveloid gear is smaller than that of the internal beveloid gear. The maximum difference between
the results of two methods is positively correlated with
the value of TTE, and the calculation results of two
methods are in good agreement.
Acknowledgements The authors would like to thank the National Key
R&D Program of China (Grant No. 2019YFB2004700)
References
1. Chen, Q., Song, C., Zhu, C., Du, X., & Ni, G. (2017). Manufacturing
and contact characteristics analysis of internal straight beveloid gear
pair. Mechanism and Machine Theory, 114, 60–73.
2. Mitome, K. (2004). Today and tomorrow of conical gear. JSME
Symposium on Motion and Power Transmissions, 11, 270–273.
3. Sun, R., Song, C., Zhu, C., Wang, Y., & Yang, X. (2021). Computational studies on mesh stiffness of paralleled helical beveloid gear
pair. International Journal of Precision Engineering and Manufacturing, 22, 123–137.
4. Yang, X., Song, C., Zhu, C., & Liu, S. (2018). Tooth surface deviation and mesh analysis of beveloid gears with parallel axis considering machine tool adjustment errors. Journal of Advanced Mechanical Design, Systems, and Manufacturing, 12(4), JAMDSM0082.
5. Song, C., Zhu, C., Lim, T. C., & Peng, T. (2012). Parametric analysis of gear mesh and dynamic response of loaded helical beveloid
transmission with small shaft angle. Journal of Mechanical Design,
134(8), 84501.
6. Zhu, C., Song, C., Lim, T. C., & Peng, T. (2013). Pitch cone design
and influence of misalignments on tooth contact behaviors of
crossed beveloid gears. Mechanism and Machine Theory, 59, 48–64.
7. Song, C., Zhou, Y., Zhu, C., Ni, G., & Liu, S. (2018). Loaded tooth
contact analysis of intersected beveloid and cylindrical involute
gear pair with small shaft angle. Journal of Advanced Mechanical
Design, Systems, and Manufacturing, 12(1), 1881–3054.
8. Zhu, C., Liu, L., Song, C., Xiang, Y., & Liu, H. (2014). Pitch cone
design and tooth contact analysis of intersected beveloid gears for
marine transmission. Mechanism and Machine Theory, 82, 141–153.
9. Liu, C. C., & Tsay, C. B. (2002). Mathematical models and contact simulations of concave beveloid gears. Journal of Mechanical
Design, 124(4), 753–760.
10. Brecher, C., Brumm, M., Hübner, F., & Henser, J. (2013). Influence
of the manufacturing method on the running behavior of beveloid
gears. Production Engineering (Berlin, Germany), 7(2), 265–274.
11. Zhao, J., Huang, J., Wen, J., Bian, Y., & Sun, J. (2014). Gear hobbing
of space beveloid gears. Coal Mine Machinery, 35(10), 135–137.
12. Ni, G., Zhu, C., Song, C., Du, X., & Zhou, Y. (2017). Tooth contact
analysis of crossed beveloid gear transmission with parabolic modification. Mechanism and Machine Theory, 113, 40–52.
13. Liu, S., Song, C., Zhu, C., & Ni, G. (2018). Effects of tooth modifications on mesh characteristics of crossed beveloid gear pair with
small shaft angle. Mechanism and Machine Theory, 119, 142–160.
14. Liu, S., Song, C., Zhu, C., Ni, G., & Ullah, N. (2019). Concave and
convex modifications analysis for skewed beveloid gears considering
misalignments. Mechanism and Machine Theory, 133, 127–149.
International Journal of Precision Engineering and Manufacturing (2022) 23:429–443
15. Wu, J., Li, G., & Li, H. (2000). A new slotting technology for a
pair of inside engaged gears. Journal of Xi’an Petroleum Institute
(Natural Science Edition), 15(03), 45–48.
16. Wu, J. (1999). Development of backlash control beveloid gear RV
reducer for robots. PhD Thesis, Harbin Institute of Technology.
17. Li, G., Wen, J., Liu, F., & Li, X. (2004). Profile errors and axial
errors noninnvolute beveloid gears with crossed axes. Journal of
Nanjing University of Science and Technology, 028(6), 585–589.
18. Li, G., Li, X., Wen, J., Xin, Z., & Yu, L. (2004). Meshing theory and
simulation of noninvolute beveloid gears. Mechanism and Machine
Theory, 39(8), 883–892.
19. Hu, D. (2017). Study on theory and processing technology of beveloid internal gear enveloping external-rotor crown worm drive.
Master Thesis, Xihua University.
20. Liu, C. C. (2005). The mathematical model and tooth surface deviations of internal conical gears. In Proceedings of the ASME international design engineering technical conferences and computers and
information in engineering conference (DETC2005).
21. Liu, C. C., & Wang, S. F. (2007). Tooth contact analysis and contact
ellipse simulation of internal conical gear pairs. In 12th IFToMM
world congress, Besancon (France), 18–21 June 2007.
22. Hu, R., Du, X., & Zhu, C. (2020). Design of shaper cutter for precision matching internal beveloid gears. China Mechanical Engineering, 31(549), 19–25.
23. Litvin, F. L. (2004). Gear geometry and applied theory. Cambridge
University Press.
443
Caichao Zhu is currently a professor in State Key Laboratory of
Mechanical Transmissions,
Chongqing University, China.
His research felds include the
dynamics of gear systems, the
tribology of mechanical transmissions, and the design of accurate transmission.
Xuesong Du is currently an associate professor in State Key
Laboratory of Mechanical Transmissions, Chongqing University,
China. His main research interests include mechanical design
and mechanical system optimization analysis.
Publisher's Note Springer Nature remains neutral with regard to
jurisdictional claims in published maps and institutional affiliations.
Feihong Zhu is currently pursuing the Ph.D. degree in mechanical design and theory in Chongqing University. His research
interests include the precision
gear transmission, design of the
complex gear tooth surface and
gear efficiency analysis.
Chaosheng Song is currently a
professor in State Key Laboratory of Mechanical Transmissions, Chongqing University,
China. His research interests
include gear geometry design
and dynamics of geared rotor
system.
13