192 DOI: 10.5545/sv-jme.2024.1254
Strojniški vestnik - Journal of Mechanical Engineering   ▪   VOL 71   ▪   NO 5-6   ▪  Y 2025 © The Authors. CC BY 4.0 Int. Licencee: SV-JME
Tooth Contact Analysis of Involute Beveloid Gear Based 
on Higher-Order Curve Axial Modification
Yongping Liu
1,2
  – Qi Chen
1
 – Changbin Dong
1
1 Lanzhou University of Technology, School of Mechanical and Electrical Engineering, China
2 Lanzhou University of Technology, State Wenzhou Engineering Institute of Pump & Valve, China
  cameliu@163.com
Abstract This study investigates the tooth flank contact characteristics of a beveloid gear pair through the lens of higher-order curve tooth modification of 
the involute beveloid gear. The machining coordinate system of the modified gear pair is established, and its tooth surface equations are derived based on 
the principle of gear meshing and coordinate transformation. In this context, a contact analysis of the modified gear is conducted, examining the impact of 
varying parameters on the contact trace and contact ellipses, as well as the implications for meshing characteristics in the presence of assembly errors. 
The findings indicate that the contact form of the high-order curve axial modification of the beveloid gear pair is point contact. Furthermore, the maximum 
modification magnitude and the order of the modification curve influence the meshing performance of the beveloid gear pair. Additionally, the beveloid gear pair 
demonstrates enhanced tolerance to the center distance and the axis crossed error, while exhibiting reduced tolerance to the axis intersected error.
Keywords involute beveloid gear, higher order curve axial modification, tooth contact analysis, transmission error, assembly error
Highlights
 ▪ The model of involute beveloid gear pair with higher-order curve axial modification is constructed.
 ▪ The contact paths and contact ellipses of the modified beveloid gear pair are derived.
 ▪ The transmission error of the modified gear pair is calculated.
 ▪ The transmission error difference of the modified gear pair with different assembly errors is analyzed.
1 INTRODUCTION
Involute beveloid gears offer a number of advantages, including 
ease of manufacture, low cost and high accuracy. As a result, 
they are suitable for a range of applications, including fast ferries, 
all-drive automobiles, aerospace precision machinery and other 
instances of power transmission. However, the beveloid gear pair 
exhibits transmission error, uneven contact stress distribution, and 
other factors during operation, which result in vibration impact and 
a reduction in load-bearing capacity and transmission performance. 
Consequently, there is a need to enhance the durability of the gear 
pair's surface and improve its load-bearing transmission performance.
The load-bearing capacity represents a crucial criterion for 
evaluating the performance of gear transmissions. The accuracy of 
the model of tooth contact stress and inter-tooth load distribution 
coefficient is a fundamental prerequisite for ensuring the precision 
of the calculated results of gear bearing contact mechanics [1]. 
Modification can effectively enhance the gear dynamic load change 
gradient, mitigate shock, vibration and noise, and thus improve the 
quality of gear transmission. Axial modification is one of the methods 
of tooth modification, which can also be termed micro-geometric 
design. It is an effective approach to augment such performance.
The modelling of involute beveloid gears provides the foundation 
for related research. Chen et al. [2] proposed a method for 
machining a straight-toothed beveloid internal gear pair with tooth 
blanks parallel to the gear shaping tool. Additionally, they derived 
theoretical models for the beveloid internal gear and the beveloid 
external gear, and conducted a load-bearing contact analysis. Sun et 
al. [3] put forth a novel approach to modelling the tooth surface of 
involute beveloid gears, employing the theory of minimum potential 
energy in conjunction with the slicing method. They also investigated 
the impact of design parameters on the contact characteristics of 
parallel axis beveloid gears. Şentürk and Fetvacyi [4] developed a 
mathematical method for the prevention of root cuts on the model 
of beveloid gears. They also developed a mathematical method for 
the prevention of root cuts on the model of parallel shaft variable 
thickness gears. Furthermore, they developed a mathematical method 
for the prevention of root cuts generated on beveloid gear. 
The meshing stiffness is also a frequently studied topic in the digital 
modelling of gears. Wen et al. [5] derived the contact line equations 
of a parallelled beveloid gear pair, proposed an analytical algorithm 
for calculating the meshing stiffness of a beveloid gear based on the 
slicing method, and analyzed the effect of changing parameters on 
the meshing stiffness of a beveloid gear. Song et al. [6] put forth a 
methodology for calculating the meshing stiffness of parallelled 
beveloid gears based on the potential method. They also investigated 
the impact of parameters such as pressure angle, pitch cone angle, 
gear displacement coefficient, and others. Zhou et al. [7] developed an 
alternative meshing stiffness model that considered the influence of 
parameters like the direction of inter-tooth friction. The influence of 
specific parameters on the meshing stiffness was examined, including 
pressure angle, pitch cone angle, and gear displacement coefficient. 
Mao et al. [8] enhanced the existing Weber energy method, which is 
based on the gear slicing method, constructed a time-varying mesh 
stiffness solution model for involute beveloid gears, and calculated 
and verified the time-varying meshing stiffness of beveloid gears. Liu 
et al. [9] constructed a three-dimensional model of a beveloid gear 
transmission and a gear dynamics model of a parallelled beveloid 
gear, based on the processing principle of a beveloid gear. They then 
proceeded to analyse the influence factors of the transmission error of 
a beveloid gear.
The microgeometric design of the beveloid gear, also referred 
to as the modification of the beveloid gear tooth, has constituted 

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Structural Design
a significant field of investigation in recent years. Wang et al. [10] 
put forth a methodology for calculating modifications to tooth 
profiles for the purpose of analysing tooth contact. Fuentes et al. 
[11] proposed an enhanced solution for intersecting beveloid gears 
with two distinct types of tooth profile bulge modification, with the 
objective of enhancing load-carrying capacity and reducing noise 
and vibration response. Şentürk et al. [12] developed a computer 
programme to obtain generating and generated surfaces of beveloid 
gears with modification. Morikawa et al. [13] examined the impact 
of tooth profile modifications on tooth surface damage through the 
utilization of a tooth fatigue test. In a further contribution to the 
field, Brecher et al. [14] put forward a method for the design of the 
tooth surface of intersected beveloid gears, taking into account the 
tolerance field for functional tolerance. Brimmers et al. [15] also 
made a significant contribution to the design of tooth flanks of 
variable thickness gears, investigating the possibility of free flank 
modification by means of a weighted objective function of beveloid 
gears with respect to the operating behavior. Ni et al. [16] designed 
a rack cutter with a parabolic modification to enhance the contact 
characteristics of helical beveloid gears. They then proceeded to 
investigate the impact of the parabolic modification on the contact 
path and contact ratio. Finally, they conducted an in-depth analysis 
to determine the sensitivity of beveloid gears to mounting errors. 
Liu et al. [17] put forth a numerical design methodology to augment 
the meshing characteristics of alternated beveloid gears through 
modification on the contact ellipse, contact path, transmission error, 
and relative curvature. Cao et al. [18] introduced the rack cutter with 
parabolic modification into the design of intersected beveloid gear 
pairs and investigated the effects of the modification coefficients on 
the mesh characteristics of the gear pair, including contact mode, 
transmission error, mesh stiffness, contact force, and tooth root stress. 
Zhang et al. [19] proposed a differentiated modification method based 
on a sine function, analyzed the dynamic characteristics and noise 
of gear pairs, comparing unmodified, normally, and differentially 
modified gears.
Presently, the majority of modification designs for involute 
beveloid gears are based on previous design experience. 
Consequently, the innovative modification method for beveloid gears 
is of great significance, as it offers a promising avenue for enhancing 
the meshing characteristics of beveloid gear pair. The involute 
parallelled beveloid gear is distinguished by a distinctive tooth 
surface configuration, which exhibits the phenomenon of automatic 
backlash adjustment. However, modifications to this configuration 
are not recommended for significant alterations in the central portion 
of the tooth surface. Instead, high-order curves are employed to the 
axial modification, resulting in a notable discrepancy between the 
two ends and a relatively minor discrepancy in the central region. 
This paper derives the mathematical equations of the tooth surface 
of the beveloid gear under the corresponding design parameters, 
including the maximum amount of thickened gears. It clearly outlines 
the design steps of the modelling method of the high-order curve axial 
modification model and carries out a tooth contact analysis (TCA) 
for the modification of the parallelled beveloid gear. This aims to 
establishing a perfect modification theory of the involute parallelled 
beveloid gear transmission and promoting the development and 
application of the beveloid gear.
2 METHODS & MATERIALS
The tooth profiles of the involute beveloid gear, as they exist in 
their unmodified state, can be considered to be generated under the 
tooth profile envelope of a usual rack cutter. In accordance with this 
principle, the tooth surface equations of the beveloid gear can be 
derived by undertaking a series of coordinate transformations through 
the rack cutter surface equations, as illustrated in Fig. 1. In the figure, 
the angle formed by the normal coordinate system S
n
(x
n
, y
n
, z
n
) and the 
end coordinate system S
o
(x
o
, y
o
, z
o
) of the rack cutter is the helix angle 
β, while the angle formed by the end coordinate system S
o
 and the 
pitch coordinate system S
c
(x
c
, y
c
, z
c
) is the pitch angle γ. Coordinate 
system S
j
(x
j
, y
j
, z
j
) is a movable coordinate system attached to the 
blank of the beveloid gear, while coordinate system S
b
(x
b
, y
b
, z
b
) is 
a stationary coordinate system with its origin located at the center of 
the end face of the blank of the beveloid gear. The coordinate origin 
position O
n
 of S
n
 is jointly determined by the variable u = O
o
 O
n
 and 
the helix angle β. The angular velocity during the rotation of the 
beveloid gear blank is ω, the radius of the dividing cylinder of the 
beveloid gear is r, the translation speed of the rack cutter is v = ω·r, 
and the real-time rotation angle of the beveloid gear during the 
machining process is φ
j
.
Fig. 1.  Coordinate system for the generation of the beveloid gear
In the normal coordinate system S
n
 of the rack cutter, the normal 
tooth profile vector of the rack cutter is given by
r
nn nn
xyz =[] 1
T
 (1)
The rack cutter surface in the pitch coordinate system, designated 
S
c
, can be obtained by combining the normal tooth profile vectors 
with the following coordinate transformations:
rM r
cc nn ccc
xyz = =[] , 1 (2)
where r
c
 is the position vector of the rack cutter surface in the 
coordinate system S
n
; The transformation matrix, designated as M
cn
, 
is employed for the purpose of effecting a change from the coordinate 
system S
n
 to the coordinate system S
c
, and the transformation matrix 
is given by:
M
cn
u
u



coss in sinc os sinc os sin
coss in sin
sinc os
   
 

0
c cosc os cosc os
.
  u
00 01












 (3)
 
The position vector r
c
 of the tooth face of the rack cutter in the 
section coordinate system S
c
 is combined with the relative motion 
relationship between the rack cutter and the gear blank during the 
machining process. Consequently, the equation representing the tooth 
face of the beveloid gear in coordinate system S
j
 is derived as:
r
j
jc jc jj jj
jc jc j
xx yr
yx yr
:
coss in (cos sin)
sinc os


 
 ( (sin cos)
,
 

jj j
jc
jx cc yc cx c
zz
yy r










nnn /
 (4)

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194   ▪   SV-JME   ▪   VOL 71   ▪   NO 5-6 ▪   Y 2025
where x
j
, y
j
, z
j
 represent the position vectors of the tooth surface of 
the beveloid gear, and n
xc
, n
yc
 and n
zc
 denote the unit normal vectors 
of the rack cutter surface.
In accordance with the tooth equation, a computer program has 
been developed to generate the point set of the tooth profile of an 
involute beveloid gear. Figure 2 illustrates the point cloud model of 
the tooth profile of the beveloid gear with the parameters outlined in 
Table 1.
Table 1. Basic parameters of involute beveloid gears
Parameters Pinion Gear
Normal modules [mm] 2.5
Center distance [mm] 80
Pressure angle [°] 20
Tooth number 27 37
Tooth width [mm] 26 24
Pitch angle [°]) 6 6
Fig. 2. Point cloud model of beveloid gear tooth surface
In the event of a high-order curve axial modification being applied 
on the involute beveloid gear, the amount of modification at each 
location in the axial direction can be expressed as follows:

um m
aB ua 







0
2
1
1
2
, (5)
where a
mi
 represents the modification factor of gear thickness 
modification (i = 0, 1), B denotes the length of the tooth width of the 
beveloid gear, and u signifies the distance from the end face in the 
axial direction at that specific point.
Fig. 3. Beveloid gear with higher order curve axial modification
Figure 3 illustrates the modification of the involute beveloid gear 
under conditions of high-order curve axial modification. The dotted 
and solid lines indicate the tooth surface of the beveloid gear before 
and after modification, respectively. The maximum modification 
magnitude of the beveloid gear is represented by Δ
max
.
3 RESULTS AND DISCUSSION
3.1 Tooth Contact Analysis of Involute Beveloid Gear Based  
on High-Order Curve Axial Modification
Following the axial modification of the involute beveloid gear, the 
tooth face of the beveloid gear assumes a drum-shaped configuration, 
and the equations governing this tooth face exhibit greater complexity 
compared to those of the ordinary involute gear. In order to study the 
meshing characteristics, a meshing coordinate system is established 
for the beveloid gear. The TCA mathematical model of the involute 
beveloid gear pair is then obtained by transforming the coordinate 
system so that the modification of the pinion and the gear can achieve 
the correct meshing under this coordinate system. 
In accordance with the tooth surface equation of the high-
order curve axial modification of the beveloid gear outlined in the 
preceding section, the tooth position vectors r
p
 and r
g
 and normal 
vectors n
p
 and n
g
 of the pinion and gear can be derived using the 
following expressions:
ri jk
ri jk
ni
pppppp
gggggg
pppp x
uv xyz
uv xyz
uv nn
(,)
(,)
(,)



p py pz
gggg xg yg z
n
uv nnn
jk
ni jk









(,)
, (6)
where the functions x
p
, y
p
, z
p
, n
px
, n
py
, and n
pz
 are defined with respect 
to two independent variables, u
p
 and v
p
. Similarly, the functions x
g
, 
y
g
, z
g
, n
gx
, n
gy
, and n
gz
 are defined with respect to two independent 
variables, u
g
 and v
g
.
The beveloid gear pair transmission introduces two new variables, 
designated as ψ
p
 and ψ
g
, which represent the position angles of the 
pinion and the follower, respectively. These variables satisfy the 
conditions at a ratio of i:

gp
i /.  (7)
Following the implementation of the high-order curve axial 
modification scheme, the configuration of tooth contact is 
characterized by point contact. Furthermore, the two position vectors 
and the two normal vectors of the two conjugate tooth surfaces at the 
contact point are equal [20]. In consequence, the mathematical model 
for contact analysis of a beveloid gear pair is given by:
rr
nn
pppp gggg
pppp gggg
uv uv
uv uv
(,,) (,,)
(,,) (,,)
.









 (8)
The partial vectors in the x, y, and z directions corresponding to 
the position and normal vectors in Eq. (8) are equal. Furthermore, 
by coupling Eq. (7), four relatively independent sets of equations 
can be obtained. In the contact analysis calculation of the beveloid 
gear pair, the position angle ψ
p
 of the pinion can be regarded as a 
known quantity and solved separately by substituting a number of 
values within the meshing range. This results in the original set of 
equations becoming composed of four independent equations with 
four independent variables. Consequently, the solution to the set 
of contact equations can be achieved. The initial meshing point is 
designated as the reference point for calculation, and the resulting 
tooth contact trajectory points are shown as discrete red points in Fig. 
4.
In order to analyze the contact area of the tooth surface of the 
beveloid gear pair, it is necessary to calculate the curvature of the 
tooth surface. The first-order and second-order partial derivatives 
of the position vector r can be derived from the first and second 
fundamental quantities of the surface, as follows:

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r
r
r
r
r
r
r
r
r
u
v
uu
uv
vv
uv
u
uv
u
uv
u
uv
uv












(,)
(,)
(,)
(,)
2
2
2
 

















2
2
r(,)
,
uv
v
 (9)
EF G
L
uu vv
uu uv uu
 
 
rr rr
rn rn rn
22
,, ,
,, M= ,N= (10)
n
rr
rr
=
tv
tv
×
×
, (11)
where the r
u
 and r
v
 represent the first-order partial derivatives of the 
position vector r with respect to u and v, respectively. Meanwhile, 
r
uu
, r
uv
 and r
vv
 denote the second-order partial derivatives of the 
position vector r. E, F, and G are the first fundamental quantities of 
the surface, and L, M, and N are the second fundamental quantities of 
the surface. The unit normal vector n at point (u, v) is given by Eq. 
11.
Fig. 4. Contact path for the tooth surface with high-order curve axial modification
The variation of curvature at the tooth contact point can be 
obtained from the principal curvature, which in turn allows for the 
calculation of the size and location of the contact area. The tooth 
profile direction principal curvature (K
1
) and the tooth width direction 
principal curvature (K
2
) are solved by the Gaussian curvature (K) and 
the mean curvature (H). This process is described by Eqs. (12) to 
(15):
K
LN M
EG F



2
2
, (12)
H
LG MF NE
EG F



2
2
2
()
, (13)
KH HK
1
2
  , (14)
KHHK
2
2
  . (15)
The tooth contact form of the beveloid gear with high-order curve 
axial modification is characterized by point contact. However, in 
actual conditions, this contact will extend into an elliptical contact 
area due to elastic deformation, as illustrated in Fig. 5.
Fig. 5. Tooth contact area
In Figure 5, a and b represent the lengths of the semi major axis 
and semi minor axis of the contact ellipses, respectively. The angle 
α denotes the angle between the normal vector z at the contact point 
and the principal direction e of the tangent plane at that point. The 
dimensions and orientation of both the semi major axis and semi 
minor axis of the contact ellipses can be inferred from the curvature 
characteristics of the tooth surface:
a=

 KK gg gg

 
12
1
2
12 2
2
22 cos
, (16)
b=

 KK gg gg

 
12
1
2
12 2
2
22 cos
, (17)



=
1
2
2
22
12
1
2
12 2
2
arccos
gg
gg gg


cos
cos
 (18)
KKgg

 
12
12
2 cos.  (19)
Given that tooth contact analyses are typically conducted under 
light load conditions, the elastic deformation δ in Eq. (16) to Eq. (17) 
is typically assumed to be 0.00025 inch, or 0.00632 mm. K
∑
1
 and 
K
∑
2
 are obtained by adding K
1
 and K
2
 for the pinion and the gear, g
1
 
and g
2
 are obtained by subtracting K
2
 from K
1
 for the pinion and the 
gear, respectively. Figure 6 illustrates the variation in contact ellipses 
of the modified beveloid gear pair throughout the meshing process. 
Fig. 6. Tooth contact ellipses
Figure 7 illustrates the variation in the area of the contact ellipses 
over time for different modification magnitudes. It can be observed 
that an increase in the magnitude of the modification results in a 
reduction in the contact ellipse area, which consequently gives rise 

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In conjunction with the tooth contact trajectory, the theoretical 
transmission error of the modified beveloid gear pair in the 
transmission process can be derived [21], as illustrated in Fig. 8. 
Figure 9 illustrates the cumulative transmission error of the beveloid 
gear pair following the superimposition of the transmission error of 
a single pair of teeth under varying modification curves. Figure 10 
depicts the corresponding peak-to-peak transmission error. It can be 
observed that an increase in the order of the modification curve is 
associated with a reduction in the transmission error of the beveloid 
gear pair. This is accompanied by a decrease in the peak-to-peak 
value of the transmission error, which is conducive to the smooth 
transmission of the gear pair.
3.2 Influence of Assembly Error on Contact Characteristics of 
Involute Beveloid Gear with High-Order Curve Axial Modification
In involute beveloid gear pair, three principal forms of assembly 
errors may occur: center distance error, shaft staggering error, and 
shaft intersection angle error. These are illustrated in Fig. 11.
In the event of a solitary assembly error, the positive and negative 
assembly errors exert an opposing influence on the meshing path, 
while the impact of center distance errors is comparatively negligible. 
In the event of equality between the error values, the shaft intersection 
angle error exerts a more pronounced influence on the meshing path 
than the shaft staggering error.
Fig. 10. Peak-to-peak value of the transmission error of beveloid gear pair
In order to investigate the impact of assembly errors on the 
meshing characteristics of the beveloid gear pair, a series of 
assembly errors have been introduced into the calculation of tooth 
contact analysis, with the resulting contact paths and contact ellipses 
illustrated in Fig. 12.
Figure 13 illustrates the variation in transmission error for the 
beveloid gear pair in the presence of a center distance error of 0.5 
mm, a shaft staggering error of 0.3°, and a shaft intersection angle 
error of 0.3°, respectively. Figure 14 depicts the peak-to-peak value 
of transmission error for the aforementioned three cases. It can 
          
Fig. 11. The forms of assembly errors of beveloid gear pair; a) center distance error; b) shaft staggering error; and c) shaft intersection angle error
to an augmentation in the contact stress experienced by the tooth 
surface.
Fig. 7. Changes in the contact ellipse area
Fig. 8. Theoretical transmission error
Fig. 9. Transmission error of beveloid gear pair

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be observed that any assembly error results in an increase in the 
transmission error of the beveloid gear pair, with the peak-to-peak 
value also being affected. The largest increase in transmission error 
is caused by the presence of shaft intersection angle error, while the 
largest increase in the peak-to-peak value of transmission error is 
caused by the presence of shaft staggering error.
Fig. 13. Transmission error of beveloid gear pair with assembly errors
The combination of the results of the aforementioned analyses 
indicates that the beveloid gear pair with high-order curve axial 
modification exhibits superior tolerance performance with regard 
to center distance error and shaft staggering error. Conversely, the 
tolerance performance with respect to shaft intersection angle error 
is relatively poor.
Based on the aforementioned analysis, the transmission 
characteristics of the variable thickness gear pair were simulated 
using simulation software. The resulting transmission error, as 
depicted in Fig. 15, exhibits a trend that is largely consistent with the 
theoretical transmission error. This finding corroborates the accuracy 
and reliability of the analytical method presented in this study.
The beveloid gear pair with high-order curve axial modification 
shows strong tolerance to center distance error and shaft staggering 
error, but it is relatively poorly tolerated for shaft intersection 
error. The shaft intersection error can cause the contact point of 
gear meshing to shift, leading to significant transmission errors 
and uneven load distribution. This not only affects the stability 
and accuracy of the system but also reduces the service life of the 
gears. Consequently, while beveloid gears with high-order curve 
axial modified can tolerate other forms of assembly errors to some 
extent, their sensitivity to shaft intersection errors necessitates more 
precise control and compensation strategies in practical engineering 
applications.
Fig. 14. Peak-to-peak value of transmission error of the beveloid gear pair with assembly errors
4 CONCLUSIONS
This paper presents a modelling and analysis of a beveloid gear pair 
with high-order curve axial modification. The conclusions drawn 
from this analysis are as follows:
                    
Fig. 12. Contact paths and contact ellipses of the beveloid gear pair with assembly errors; a) center distance error; b) shaft staggering error; and c) shaft intersection angle error
              
Fig. 15. Theoretical transmission error and simulated transmission error of beveloid gear pair with assembly errors;  
a) center distance error; b) shaft staggering error; and c) shaft intersection angle error

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1. Following the axial modification, the tooth surface of the beveloid 
gear pair exhibits point contact. The contact trace is concentrated 
in the middle of the tooth surface, and the contact ellipse gradually 
increases from the root to the top of the tooth, before decreasing 
with the increase of the maximum amount of modification. This 
affects the smoothness of transmission of the gear pair.
2. It can be observed that an increase in the order of the modification 
curve is associated with a reduction in the transmission error of 
the beveloid gear pair. Consequently, the peak-to-peak value of 
the transmission error is also diminished.
3. The transmission error of the beveloid gear pair and its peak-
to-peak values are observed to increase in the presence of 
assembly error. Conversely, the effect is observed to be smaller 
in the presence of center distance error and shaft staggering error. 
The largest effect is observed to occur in the presence of shaft 
intersection angle error.
References
[1] Wang, C., Cui, H., Zhang, Q., Zhang, B. Theoretical research progress of gear 
meshing efficiency. J Univ Jinan (Sci. & Tech.) 29 229-235 (2015) DOI:10.13349/j.
cnki.jdxbn.2015.03.014.
[2] Chen, Q., Song, C., Zhu, C., Du, X., Ni, G. Manufacturing and contact characteristics 
analysis of internal straight beveloid gear pair. Mech Mach Theory 114 60-73 
(2016) DOI:10.1016/j.mechmachtheory.2017.04.002.
[3] Sun, R., Song, C., Zhu, C., Wang, Y., Liu, K. Numerical study on contact force of 
paralleled beveloid gears using minimum potential energy theory. J Strain Anal 
Eng Des 56 249-264 (2020) DOI:10.1177/0309324720936894.
[4] Şentürk, B., Fetvaci, M. Modelling and undercutting analysis of beveloid gears. J 
Fac Eng Archit Gaz 35, 901-916 (2020) DOI:10.17341/gazimmfd.544038.
[5] Wen, J., Yao, H., Yan, Q., You, B. Research on time-varying meshing stiffness of 
marine beveloid gear system. Mathematics 11, 47-74 (2023) DOI:10.3390/
math11234774.
[6] Song, C., Zhou, S., Zhu, C., Yang, X., Li, Z., Sun, R. Modeling and analysis of 
mesh stiffness for straight beveloid gear with parallel axes based on potential 
energy method. J Adv Mech Des Syst Manuf 12, 1-14 (2018) DOI:10.1299/
jamdsm.2018jamdsm0122.
[7] Zhou, C., Dong, X., Wang, H., Liu, Z. Time-varying mesh stiffness model of a 
modified gear-rack drive with tooth friction and wear. J Braz Soc Mech Sci 44 213 
(2022) DOI:10.1007/S40430-022-03517-8.
[8] Mao, H., Fu, L., Yu, G., Tupolev, V., Liu, W. Numerical calculation of meshing 
stiffness for beveloid gear with profile modification. Compute Integra Manuf Sys 
28 526-535 (2022) DOI:10.13196/j.cims.2022.02.017.
[9] Liu, Y., Ren, Z., Wei, Y., Ma, D., Wei, S. Error study of involute beveloid spur gear 
transmission. Mach Tool Hydraulic 51, 1-6 (2023) DOI:10.3969/j.issn.1001-
3881.2023.04.001.
[10] Wang, C., Wang, S., Wang, G. A calculation method of tooth profile modification 
for tooth contact analysis technology. J Braz Soc Mech Sci 40, 1-9 (2018) 
DOI:10.1007/s40430-018-1268-4.
[11] Fuentes, A., Gonzalez-Perez, I., Hayasaka, K. Computerized design of conical 
involute gears with improved bearing contact and reduced noise and vibration. 
VDI Bericht 2108, 635-646 (2010).
[12] Şentürk, B., Fetvaci, M. Modelling and undercutting analysis of beveloid gears. J 
Fac Eng Archit Gaz, 35, 901-916 (2020) DOI:10.17341/gazimmfd.544038.
[13] Morikawa, K., Kumagai, K., Nagahara, M. lnfuence of tooth flank modfication 
on tooth surface damage of conical involute gears. Trans JSME 81 1-8 (2015) 
DOI:10.1299/transjsme.15-00311.
[14] Brecher, C., Löpenhaus, C., Brimmers, J. Function-oriented torlerancing of 
tooth flank modifications of beveloid gears. Procedia CIRP 43 124-129 (2016) 
DOI:10.1016/j.procir.2016.02.155.
[15] Brimmers, J., Brecher, C., Löpenhaus, C. Potenzial von freien Flanken 
modifikationen für Beveloidverzahnungen. Forschung im Ingenieurwesen 81 83-
94 (2017) DOI:10.1007/s10010-017-0232-2.
[16] Ni, G., Zhu, C., Song, C., Du, X., Zhou, Y. Tooth contact analysis of crossed beveloid 
gear transmission with parabolic modification. Mech Mach Theory 113 40-52 
(2017) DOI:10.1016/j.mechmachtheory.2017.03.004.
[17] Liu, S., Song, C., Zhu, C., Ni, G. Effects of tooth modifications on mesh 
characteristics of crossed beveloid gear pair with small shaft angle. Mech Mach 
Theory 119 142-160 (2018) DOI:10.1016/j.mechmachtheory.2017.09.007.
[18] Cao, Y., Ni, G., Fang, Z., Liu, Z. Effects of cutter modification on the meshing 
characteristics of intersected beveloid gear and involute cylindrical gear. J Phys 
Conf Ser 2691 012-022 (2024) DOI:10.1088/1742-6596/2691/1/012022.
[19] Zhang, Y., Zhou, H., Duan, C., Wang, Z., Luo, H. Gear differential flank modification 
design method for low noise. Stroj Vestn-J Mech E 70 569-581 (2024) 
DOI:10.5545/sv-jme.2024.1072.
[20] Wang, X., Lu, J., Yang, S. Sensitivity analysis and optimization design of hypoid 
gears’ contact pattern to misalignments. J Zhejiang Univ-Sc A 20 411-430 (2019) 
DOI:10.1631/jzus.A1900021.
[21] Dong, C., Yang, X., Li, D., Zhao, G., Liu, Y. Service Performance optimization and 
experimental study of a new W-W type non-circular planetary gear train. Stroj 
Vestn-J Mech E 70 128-140 (2024) DOI:10.5545/sv-jme.2023.673.
Acknowledgements This research was funded by the National Natural 
Science Foundation of China No. 52265008; the Youth Science Foundation 
of Gansu Province, No. 23JRRA751; the Education Department of Gansu 
Province: University Teachers Innovation Fund No. 2023A-021; the Wenzhou 
Municipal Basic Scientific Research Project of China No. 2022G0145.
Received: 2024-12-23, revised: 2025-03-14, accepted: 2025-04-09 as 
Original Scientific Paper.
Data Availability The data that support the findings of this study are available 
from the corresponding author upon reasonable request.
Author Contribution Yongping Liu: Data curation, Qi Chen: Writing—original 
draft preparation, Changbin Dong: Writing—review & editing.
Analiza zobnega kontakta evolventnega stožčastega zobnika na 
osnovi aksialne modifikacije s krivuljo višjega reda
Povzetek V tej raziskavi so preučene kontaktne značilnosti bokov zob 
evolventnega stožčastega zobniškega para z vidika aksialne modifikacije 
zobne površine s krivuljo višjega reda. Postavljen je koordinatni sistem 
obdelave modificiranega zobniškega para, izpeljane pa so tudi enačbe 
zobnih površin, ki temeljijo na principu zobniškega ubiranja in koordinatnih 
transformacijah. Izvedena je analiza kontakta modificiranega zobnika, pri 
čemer so preučeni vplivi spreminjajočih se parametrov na kontaktno sled in 
kontaktne elipse ter posledice za lastnosti ubiranja ob prisotnosti montažnih 
napak. Ugotovitve kažejo, da je oblika kontakta pri aksialni modifikaciji s 
krivuljo višjega reda pri stožčastem zobniškem paru v obliki točke. Poleg 
tega maksimalna velikost modifikacije in red krivulje modifikacije vplivata 
na lastnosti ubiranja stožčastega zobniškega para. Stožčasti zobniški par 
izkazuje izboljšano toleranco glede napake medosne razdalje in napake 
križanja osi, obenem pa kaže manjšo toleranco do napake presečišča osi.
Ključne besede evolventni stožčasti zobnik, aksialna modifikacija s krivuljo 
višjega reda, analiza zobnega kontakta, napaka prenosa, napaka pri montaži