*Corr. Author’s Address: South China University of Technology, School of Mechanical and Automotive Engineering, China, meyzchen@scut.edu.cn 483
Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 483-493 Received for review: 2024-01-04
© 2024 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2024-04-16
DOI:10.5545/sv-jme.2024.917 Original Scientific Paper Accepted for publication: 2024-06-05
Design and Stress Analysis of Bevel Line Gears  
with Vertical Flank Suitable for Micromachining
Shao, Y . – Chen, Y . – Xiao, X – Zheng, M. – He, W.
Yanjie Shao
1
 – Yangzhi Chen
1,2,*
 – Xiaoping Xiao
2
 – Maoxi Zheng
1
 – Weitao He
1
1 
South China University of Technology, School of Mechanical and Automotive Engineering, China 
2 
Guangdong Ocean University, School of Mechanical and Energy, China
The line gear (LG) exhibits promising applications in micromachinery owing to its simple structure and minimal teeth count. This paper aims to 
advance the meshing theory of LG and introduce a conical LG tooth configuration specifically tailored for micro-machining that is capable of 
driving with high performance. Based on the meshing principle of LG, a conical line gear pair with vertical flank (VFLG) of the driving gear was 
designed. Subsequently, adhering to the curvature non-interference principle that no local interference occurs in the meshing process and the 
specified range of fitting error in one direction, the design parameters for LG were determined. A comparison of the contact stress between 
two sets of different contact line types revealed the distinct advantage of VFLG with parameters identical to those of traditional LG. The 
experimental results conclusively demonstrate a transmission ratio error of 0.004, affirming the feasibility of the design. The curvature non-
interference design formula proposed in this paper refines the LG design theory, and the novel LG design method presented holds significant 
implications for subsequent micromachining. 
Keywords: line gear, bevel gear, meshing theory, stress analysis, micromachining
Highlights
• 	 A VFLG of a 2.5D structure was proposed, which is tailored for micromachining.
• 	 Compared to the typical LG, the contact stress of the VFLG is significantly lower.
• 	 LG’s curvature non-interference theory was advanced, and its overall design theory was refined. 
• 	 The feasibility of the design was verified by kinematic experiments.
0  INTRODUCTION
The growing demand for microsystems and 
miniaturized components has accelerated the need 
to manufacture high-quality microgears efficiently 
and economically [1] and [2]. Miniature gears are 
categorized based on their outer diameter range, 
with 0.1 mm to 10 mm considered the overall range, 
1 mm to 10 mm termed as middle gears, and 0.1 
mm to 1 mm classified as micro gears [3]. Notably, 
microgears have become key elements in a variety 
of microsystems owing to their ultra-lightweight, 
smaller size, and compactness [4]. The performance 
of the microsystem mainly depends on the quality 
of the microgear, and the overall quality of the gear 
depends on the manufacturing process. Among 
the commonly used processing technologies for 
microgears, additive processing stands out for its 
ability to fabricate complex components that are 
challenging to achieve through alternative methods 
[5]. However, a drawback is the tendency to generate 
pores during additive processing [6], leading to 
subpar mechanical properties. Additionally, the 
surface quality is constrained by the size of powder 
particles [7]. Electrical discharge machining (EDM) 
is known for its effectiveness and high precision, 
albeit with the drawback of low material removal 
rate and elevated equipment costs [8]. Furthermore, 
several widely employed micro-machining methods 
like micro-hobbing, micro-milling, and laser ablation 
are primarily tailored for machining spur gears, 
presenting limitations in processing traditional bevel 
gears [1] and [9]. Consequently, it is a viable approach 
to improve the geometric design of bevel gears to 
align with commonly used micro-machining methods 
[10].
In recent years, the emergence of new gear 
meshing theories has garnered significant attention. 
Tan et al. [11] introduced a novel tooth profile design 
method grounded in the conjugate curve theory, 
resulting in the development of the conjugate curve 
bevel gear. However, the normal section of the 
designed gear is convex with a convex arc, posing 
challenges for micro-machining. An et al. [12] 
presented the meshing theory of curve-pair bevel 
gears and designed a multi-point contact curve-
pair bevel gear capable of achieving divisible axis 
intersection angles. Nevertheless, the tooth surface 
remains intricate and difficult to minimize. The line 
gear (LG) represents a novel gear mechanism rooted 
in spatial curve meshing theory [13]. Known for its 
characteristics, including that the minimum number of 
teeth can be 1 [14], flexible design, and controllable 
sliding rate [15], LG offers substantial advantages in 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 483-493
484 Shao, Y. – Chen, Y. – Xiao, X – Zheng, M. – He, W.
terms of miniaturization and lightweight applications, 
such as vertical shaft reducer with transmission 
efficiency exceeding 99 %, lightweight parallel-axis 
LG reducer for sweeping robots, and non-trapped oil 
double-arc LG pumps [16] and [17]. Consequently, LG 
provides a feasible possibility for the miniaturization 
of conical gears. Currently, a conical LG with an 
outer diameter of 2.83 mm has been processed, and 
a transmission experiment has been conducted. 
However, the design principle and processing method 
impose limitations on further miniaturization; 
moreover, it suffers from the drawback of substantial 
contact stress [18].
While LG boasts a notable advantage in design 
flexibility, its design theory is not flawless. Chen et 
al. [19] derived the non-interference condition by 
testing the contact degree and kinematics. Chen et 
al. [13] proposed that it is crucial to ensure that there 
is no point on the driving wheel within the radius of 
the driving wheel tooth to avoid interference with the 
tooth surface during the LG parameter design process. 
However, the preceding discussion relies on kinematic 
geometric constraints in the design parameters, which 
alone cannot guarantee the correct mesh transmission 
of LG.
In this paper, based on the LG meshing theory, 
the meshing principle and tooth surface generation 
method for line gear with vertical flank (VFLG) are 
introduced. Characterized by a 2.5D structure, the 
gear is well-suited for micro-machining and allows 
for the accommodation of installation errors in a 
single direction. Moreover, the non-interference 
theory of LG curvature is proposed, contributing 
to the enhancement of LG’s design theory. Also, a 
comparison of contact strength between VFLG and 
traditional LG is conducted. Ultimately, the accuracy 
of the design theory is validated through transmission 
tests.
1  THE MATHEMATICAL MODEL OF VFLG
1.1  The Meshing Theory of VFLG
The traditional LG meshing principle can be described 
as follows: the driving-driven contact line represents 
a pair of spatial conjugate curves designed by the 
active gear. This line consistently maintains tangential 
contact throughout the transmission process, with 
a shared principal normal at the tangent point. Each 
instantaneous contact point corresponds to a main 
normal plane for both the driving wheel and the driven 
wheel. The continuous surface formed by these curves 
constitutes the tooth surface of the driving and driven 
gears. In contrast, the VFLG meshing theory adopts 
the point contact meshing between the vertical surface 
and the line. The vertical tooth face is 2.5D, making it 
more amenable to processing. Moreover, the “vertical 
surface-line” combination offers superior bearing 
capacity compared to the “line-line” configuration.
1.2  Coordinate System and Coordinate Transformation
As shown in Fig. 1.  Meshing coordinate system of 
LG, in the space curved meshing coordinate system, 
the active contact line is fixed in coordinate system 
S
1
(o–x
1
y
1
z
1
), and coordinate system S
1
 rotates around 
coordinate system S
o
(o–xyz) with angular velocity ω
1
, 
and the rotating angle is ϕ1. The driven contact line is 
fixed in the coordinate system S
2
(o
2
–x
2
y
2
z
2
), and the 
coordinate system S
2
 rotates around the coordinate 
system S
p
(o
p
–x
p
y
p
z
p
) at the angular velocity ω
2
, and 
the rotation angle is ϕ
2
. a represents the distance from 
point o
p 
to the z axis, while b represents the distance 
from point o
p
 to the x-axis; θ represents the angle 
between the x-axis and x
p
.
Fig. 1.  Meshing coordinate system of LG
The coordinate transformation matrix between 
coordinate system S
i
(o
i
–x
i
y
2
z
i
) and coordinate system 
S
j
(o
j
–x
j
y
j
z
j
) is denoted as M
ij
. Here, three coordinate 
transformation matrices, M
o1
, M
po
 and M
2p,
 are given, 
as shown in Eqs. (1) to (3)  respectively.
 M
o1














coss in 00
sinc os
,


11
11
00
00 10
00 01
 (1)

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 483-493
485 Design and Stress Analysis of Bevel Line Gears with Vertical Flank Suitable for Micromachining 
  M
po
ba
ba

 




coss in sinc os
sinc os coss in
 
 
0
0100
0
000 1
 








, (2)
 M
2
22
22
00
00
00 10
00 01
p














coss in
sinc os
.


 (3)
The transformation matrix between coordinate 
system S
1
 and coordinate system S
2
 can be obtained, 
as shown in Eq. (4). 
 MM MM
21 21

pp oo
. (4)
1.3  Meshing Equation
The relative speed of the main and slave driving wheel 
at the engagement point is expressed in Eq. (5).
 v
12
112
11 12
12


 

y
xa x
bz
o
oo
o
()
() ()
()
(c os )
() cos
() si
 
 
 n n
sin
,
()

 




















y
o
12
 (5)
where, 
xyz
ooo
11 1
() () ()
,,
 can be obtained by xyz
1
1
1
1
1
1 () () ()
,, 
through M
o1
.
The meshing equation is denoted in Eq. (6).
 v
12
   0, (6)
where β represents the principal normal vector of the 
driving contact line at the meshing point.
ϕ
1
 = t can be deduced by substituting the relative 
velocity in Eq. (7).
1.4  Contact Curve and Centre Curve
In the coordinate system S
1
, the position vector of the 
active contact line is shown as Eq. (7).
 r      
1
()
(( )cos ,( )sin ,( ), 1)
1
= mg tt mg tt ng t (7)
And r
13
1
− D
()
 is shown in Eq. (8).
 r    
13
1


D
mg tt mg tt ng t
()
(( )cos ,( )sin ,( )). (8)
The equations for the driving contact curve are 
given in Eq. (9).
 M
()
() cos
() sin
()
,
1
1
1
1









xm gt t
ym gt t
zn gt
 (9)
where n=i
12
m, and the driven contact curve can be 
obtained in Eq. (10).
M
()
(c os sin) () cos( /)
(c os sin) ()
2
21 2
1

 
 
xm ng tt i
ym ng t

s sin( /)
(c os sin)()
. ti
zn mg t
12
1
 






 (10)
As depicted in Fig. 2.  Diagram of VFLG’s 
normal vector, in contrast to the conventional LG, 
the normal vector can be directly obtained from M
(1)
 
and represents the normal vector projected onto the 
bottom surface of the active contact line, as illustrated 
in Eq. (11) 
 n
()
[s in ,cos ,].
1
0  tt
T
 (11)
The equations for the driven centring curve are 
expressed in Eq. (12) 
 Cr
2
2
23
2
2
2
2
() () ()
/.  
 D
l n (12)
Fig. 2.  Diagram of VFLG’s normal vector
1.5  Geometrical Modelling
The conventional LG tooth surface construction 
method is illustrated in Fig. 3.  Construction method of 
driven LG tooth surface. Following the determination 
of the cross-section profile, a scanning method is 
employed for modelling [18]. In this approach, the 
contact curve serves as the scan line; the centre line 
acts as the guideline.
The modelling method of VFLG is simpler, as 
depicted in Fig. 4.  Construction method of driving 
VFLG tooth surface. Initially, the plane contour is 
determined, followed by a combination of stretching 
and rotary cutting.

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 483-493
486 Shao, Y. – Chen, Y. – Xiao, X – Zheng, M. – He, W.
Fig. 3.  Construction method of driven LG tooth surface
Fig. 4.  Construction method of driving VFLG tooth surface
2  CORRECT MESHING CONDITIONS CONSIDERING 
CURVATURE INTERFERENCE AND MISALIGNMENT
2.1  Curvature Non-Interference Conditions
Illustrated in Fig. 5.  Diagram of tangent plane and 
common normal, the two surfaces are tangent at 
point M, with n representing the common normal 
line. Assuming that the n direction corresponds to 
the spatial direction of the driving wheel teeth, the 
curvature does not interfere, and the induced curvature 
k
b1
 should satisfy the condition k
b1
≥0 in any direction.
The induced curvature in any direction can be 
obtained by Euler’s formula, as shown in Eq. (13).
 kk k
b11
2
2
2
   cos( )s in () .  (13)
As shown in Fig. 6, where tan2 φ  = 2 τ
a1
/(k
a2
–k
a1
), 
k
1 
and k
2
 are the principal curvature of the surface and 
k
1
 > k
2
, τ
a1
 is the tangential induced torsion.
Fig. 5.  Diagram of tangent plane and common normal
Fig. 6.  The relationship between k
n
 and τ
n
  
in all directions on a surface
Here, k
b1
 can also be expressed as Eq. (14).
   kk k
ba aa 11
2
12
2
2     coss in coss in .  (14)
According to Section 2.1, Eq. (14) must be 
a non-negative definite quadratic function, then  
(2τ
a1
)
2
 – 4 k
a1
k
a2
 ≤ 0. Therefore, the curvature non-
interference condition of LG can be summarized as 
Eq. (15).
 
kk
kk
aa
aa a
12
12 1
2
0 ,
.








 (15)
The pair of tangential and normal induced 
curvature, perpendicular to each other, can be 
incorporated into the calculation. In the above formula, 
k
a1
 is defined as the tangential induced curvature, k
a2
 
is defined as the normal induced curvature, and τ
a1
 
is the tangential induced geodesic torsion, which are 
solved below.
2.2  Requirements for Value of Parameters
2.2.1 Tangential Induced Curvature
The normal curvatures of curves M
(1)
 and M
(2)
 are 
shown in Eqs. (16) (17)  respectively.
 k
1
1
1
11
1
12
1
11
3
 rr r
() () ()
/, (16)
 k
1
2
1
21
1
22
1
21
3
 rr r
() () ()
/. (17)

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 483-493
487 Design and Stress Analysis of Bevel Line Gears with Vertical Flank Suitable for Micromachining 
From Meusnier’s theorem, the induced curvature 
of the tangent can be obtained as Eq. (18).
 kk k
a11
1
11
2
2
 cosc os ,  (18)
where φ
1
(turning angle), φ
2
 are the angles of β
1
 and 
β
2
 respectively with n, as shown in 2.2.2  Normal 
Induced Curvature.
Fig. 7.  Diagram of the turning angle of LG
2.2.2  Normal Induced Curvature
As shown in Fig. 8, the normal induced curvature can 
be indicated in Eq. (19) 
 kR Rll
a21 21 2
11 22  // // , (19)
where, R
1
, R
2
, l
1
, l
2
 are respectively the radius of 
curvature and the diameter of curvature at the meshing 
point.
Fig. 8.  Schematic diagram of normal surface curvature radius
2.2.3  Tangentially Induced Geodesic Torsion
The geodesic torsion of M
(1)
 to ∑
1
 and M
(2)
 to ∑
2
 are 
denoted in Eqs. (20) d (21)  respectively.
   
g
dd s
1111 11
  /, (20)
   
g
dd
2222 22
  /, (21)
where 
11
11
1
12
1
13
1
11
1
12
 () /
() () () () ()
rr rr r and 

22
21
2
22
2
23
2
21
2
22
 () /.
() () () () ()
rr rrr
Finally, the tangential-induced geodesic torsion is 
expressed as Eq. (22).
 
agg 112
. (22)
Hence, the curvature non-interference condition 
of LG can be translated into Eq. (23), allowing for 
the derivation of the value requirements for relevant 
parameters in the modelling of VFLG.
rr
r
rr
r
r
1
11
1
12
1
11
3 1
1
21
1
22
1
21
3 2
1
11
0
() ()
()
() ()
()
(
cosc os



 
) )( )
()
() ()
()
cosc os











r
r
rr
r
1
12
1
11
3 1
1
21
1
22
1
21
3 2
1
2

l
 











2
2
1
2
1
1
11
1
12
1
13
1
11
1
12
l
a
a


() (
() () ()
() ()
rr r
rr
r
2 2
21
2
22
2
23
2
21
2
22
1
1
2
2
() () ()
() ()
) 












rr
rr
d
ds
d
ds
  



(23)
2.3  Misalignment Adaptability for One Direction
If there is a certain installation error in the direction of 
the driving wheel axis. Assume that the installation 
parameter b is in one direction in Fig. 9 has an 
installation error δb. The equations for the driven 
contact curve are donated as M
(2)
 
in Eq. (10), and 
M
δ b
() 1
 can be expressed as Eq. (24) 
 M


b
mg tt
mg tt
ng tb
()
() cos
() sin
()
.
1







 (24)
It can be observed that, in comparison with M
(1)
 
and M
δ b
() 1
, there is only a deviation in the Z-axis 
direction, as illustrated in Fig. 10. The offset contact 
line remains on the tooth surface of the driving wheel. 
Consequently, VFLG exhibits adaptability to 
installation errors in one direction. Theoretically, its 
allowable error range is h
1
 (the height of the vertical 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 483-493
488 Shao, Y. – Chen, Y. – Xiao, X – Zheng, M. – He, W.
plane), and specific values can be determined based 
on the required error range.
Fig. 9.  Assembly error diagram of VFLG
Fig. 10.  Assembly error adaptability of VFLG
3 STRESS ANALYSIS AND COMPARISON  
WITH CONVENTIONAL LG
To compare the load-carrying capacity of VFLG and 
LG ( φ
1
 = π/6) [18], a finite element model was utilized 
to analyse the load stress on both gear types. Since 
many analyses have been done in previous research, 
two sets of different types of contact lines were 
used here for analysis and comparison. The material 
selected was stainless steel, featuring a Young’s 
modulus of 2e5 MPa and a Poisson’s ratio of 0.3. In 
Section 4.1, the maximum outside diameter of the 
driving wheel was 3.2 mm, the maximum size of the 
triangular unit on the tooth surface was 2e–5 mm, the 
applied torque was increased to 5e–3 Nmm from 0 
Nmm, and the rotational speed was kept at 10 r/min. 
In Section 4.2, the maximum outer diameter of the 
driving wheel was 11 mm, the maximum size of the 
unit was 2e–4 mm, the applied torque was increased 
to 5e–2 Nmm from 0 Nmm, and the rotational speed 
was kept at 10 r/min. The speed and torque settings 
are depicted in Fig. 11, and two pairs of rigid surfaces 
were built to apply rotation velocity and torque (Fig. 
11).
Fig. 11.  Schematic of finite element model
3.1  Equidistant Spiral Bevel Gear
The equations for the driving contact curve are given 
as Eq. (25).
 M
xg tt
yg tt
zg t
()
() cos
() sin
()
.
1
1
1
1
2









 (25)
For LG and VFLG, g(t) is t and e
t
, respectively, 
and the results obtained after substituting the curvature 
calculation condition Eq. (23) are deduced as Eqs. 
(26) and (27)  respectively.
                          
22
5 2400 940 124 5
32 08 80 17 12
2
32
24 6
2
24
2
ll
tt tt
tt








/
t tt tt tt
24
2
24 24
22 08 80 17 

 
 
,     (26)
                         
22
81 41 72 06 82
34 4
12
2
32
42
2
ll
tt tt
pt qt
tt pt








/
//
() ()
()
2 2
32
42
42 2
32
2
17 20 68
16 88 153 17 18



 




/
/
() si
tt
tt tt t qt n nc os
.
3
2
34
3
2
2
t
t
t 






























     (27)

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 483-493
489 Design and Stress Analysis of Bevel Line Gears with Vertical Flank Suitable for Micromachining 
Here, while solving for the curvature diameter 
at the meshing point of VFLG, as illustrated in Fig. 
12.  Outline of VFLG tooth surface in normal section, 
the yellow normal phase contour is acquired, and 
subsequently, the curvature at the meshing point is 
determined through approximate fitting.
To facilitate the comparison of contact stress, it is 
imperative to ensure that curvature interference does 
not occur under the same parameters. The parameters 
chosen through the integrated formula Eqs. (26) and 
(27) are defined as Eq. (28), with w
1
 representing the 
width of VFLG gear teeth.
 
  // /
.
.
33 11 40
03
12 1
 
 



t
llw mm
 (28)
Fig. 12.  Outline of VFLG tooth surface in normal section
Table 1.  Modelling parameters of equidistant spiral LG and VFLG
Parameters
m 
[mm]
n 
[mm]
l
1
/w
1
 
[mm]
l
2
 
[mm]
a,b 
[mm]
i
12
θ g(t)
LG 1 2 l
1
=0.3 0.3 0,0 2 π/2 t
VFLG 1 2 w
1
=0.3 0.3 0,0 2 π/2 t
The VFLG and LG gear pairs were established 
using the parameters listed in Table 1.  Modelling 
parameters of equidistant spiral LG and VFLG 
are depicted in Figs. 13a and b, respectively. A 
comparison of maximum stress changes between the 
two is illustrated in Fig. 14.
a)       b) 
Fig. 13.  The model diagram used in finite element simulation:  
a) VFLG; b) LG
Fig. 14.  Comparison diagram of maximum contact stress on tooth 
surfaces of equidistant helix LG and VFLG
3.2  Equiangular Spiral Bevel Gear
Similarly, the calculation results for LG and VFLG 
curvature non-interference are depicted in Eqs. (29)  
and (30), respectively. 
 22
25
12 26 5
12
// , ll
t



e
 (29)
     
22 63
4731 0
3
2
31 0
3
2
12
2
ll
tt
t
 
 

e
(c os sin)
. (30)
The range of parameters selected is expressed in 
Eq. (31).
 
  // /
.,.
.
44 11 40
06 08
11 2
 
 



t
lw l mm mm
 (31)
The comparison of maximum stress changes 
between LG and VFLG is shown in Fig. 15, and 
parameters used for modelling are listed in Table 2.
Fig. 15.  Comparison diagram of maximum contact stress on tooth 
surfaces of equiangular helix LG and VFLG

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 483-493
490 Shao, Y. – Chen, Y. – Xiao, X – Zheng, M. – He, W.
stress is inversely proportional to the product of the 
major semi-axis and the short semi-axis of the contact 
ellipse. Clearly, the contact ellipse of VFLG is larger 
and exhibits better bearing capacity.
The meshing principle of VFLG is improved 
to vertical bearing on the basis of LG. During the 
transmission process, the contact stress is optimized 
by increasing the contact ellipse. The results of 
finite element analysis indicate that VFLG exhibits a 
significant advantage in bearing capacity.
4  TRANSMISSION EXPERIMENT
The designed driving wheel is a microgear with 
a diameter of 0.9 mm. It was manufactured using 
a stereo lithography apparatus (SLA) after being 
enlarged 20 times in equal proportion. The speed 
of the driving and driven wheels was collected by 
torque sensors, and the instantaneous transmission 
ratio can be calculated based on the collected speed 
data. The parameters used are listed in Table 3. In 
the experiment, the power of the driving gear was 
transmitted by a servomotor at a speed of 300 r/min, 
and the load was provided by a magnetic damper at 
20 Nmm. Subsequently, a transmission experiment 
was conducted, and the actual manufactured VFLG 
sample aligns with the theoretical model in Fig. 17. 
The transmission ratio test was performed on the 
VFLG kinematics test facility depicted in Fig. 18, and 
the results are shown in Fig. 19.
As evident from Fig. 19, the instantaneous 
transmission ratio of the gear pairs in the processed 
sample line fluctuates around 2, with an average value 
of 2.004 and a standard deviation of 0.003. In the 
actual transmission process, the elastic deformation 
of the gear and the load sharing of the contact teeth 
will affect the transmission ratio [21]. In this study, the 
influence of elastic deformation on the transmission 
ratio is dominant as the material used is photosensitive 
resin. Specifically, as shown in Fig. 20, when the 
Table 2.  Modelling parameters of equiangular spiral LG and VFLG
Parameters
m 
[mm]
n 
[mm]
l
1
/w
1
 
[mm]
l
2
 
[mm]
a,b 
[mm]
i
12
θ g(t)
LG 1 2 l
1
=0.6 0.8 0,0 2 π/2 e
t
VFLG 1 2 w
1
=0.6 0.8 0,0 2 π/2 e
t
3.3  Contact Stress Analysis
As depicted in Fig. 14, in Section 4.1, the contact 
stress is relatively large when the single-tooth 
meshing is about to end. This phenomenon arises 
from the transition from single-tooth load to two teeth 
at approximately 43º. The contact between the driving 
and driven contact line teeth at the top of the tooth 
causes stress concentration, leading to an increase in 
contact stress.
In Section 4.2 (Fig. 15) and Section 4.1 for VFLG, 
the position of the maximum contact stress is around 
5º. This is attributed to the transformation of the two-
tooth transmission into a single-tooth transmission, 
where the load is not shared, consequently amplifying 
the contact stress.
From Figs. 14 and 15, it is evident that the 
average contact stress of VFLG is nearly half that 
of LG, irrespective of whether the contact line is an 
equidistant helix or an equiangular helix. Without 
parameter optimization, when the active contact line 
is an isometric helix, the maximum contact stress of 
LG, modelled by the optimal contact stress method 
in [18], is 2404.1 MPa, whereas that of VFLG is 
1665.5 MPa. Similarly, when it is an equiangular 
helix, the maximum contact stress is 662.2 MPa for 
LG and 463.9 MPa for VFLG, respectively. During 
the single tooth meshing period in Section 4.2, both 
the finite element analysis results, as shown in Fig. 
16, indicate that the approximate contact ellipse long 
axis and short axis for LG are 34.01 μm and 7.93 μm, 
respectively, while those for VFLG are 39.2 μm and 
9.58 μm. According to [20], the size of the contact 
a)           b) 
Fig. 16.  Finite element analysis results of LG and VFLG; a) LG; b) VFLG

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 483-493
491 Design and Stress Analysis of Bevel Line Gears with Vertical Flank Suitable for Micromachining 
load increases, the elastic deformation causes the 
actual point of engagement to shift from M point 
to M’ point, and the corresponding theoretical and 
actual instantaneous pressure angles are denoted as 
α and α’, respectively. Obviously, the influence of 
the load causes the pressure angle to decrease. It is 
worth noting that the pressure angle has a significant 
effect on the transmission ratio [22] and [23]. As 
the pressure angle decreases, the torque capacity 
of the driving wheel increases [24], thus increasing 
the linear speed of the driven wheel; at the same 
time, manufacturing errors and installation errors 
also have a certain impact, ultimately resulting in a 
slightly higher transmission ratio than the theoretical 
value. Nevertheless, it demonstrates that the meshing 
principle and vertical tooth configuration proposed in 
this paper are feasible.
Table 3.  Parameters of VFLG used for the experiment
Parameters
m 
[mm]
n 
[mm]
l
1
/w
1
 
[mm]
l
2
 
[mm]
a,b 
[mm]
i
12
θ g(t)
VFLG 0.5 1 w
1
=0.2 0.1 0,0 2 π/2 t/5
5  CONCLUSIONS
Previous work on LG primarily focused on contact 
analysis, slide rate, centre distance separability of 
parallel-axis gears, etc. However, there is a lack of 
further research on the feasibility of generating tooth 
surfaces, and a deficiency in simple and feasible LG 
tooth configuration design in the field of micro and 
nano drives.
In this paper, the mathematical model of the 
vertical gear pair is described, emphasizing the 
advantages of the new gear pair in terms of load 
capacity and machining difficulty through analysis. 
Building upon this foundation, the design criteria of 
the gear pair are improved, and the conditions of non-
interference of the curvature of the LG are deduced. 
Finally, a pair of special gears is designed and 
manufactured, validating the feasibility of the design 
of the VFLG. 
The conclusion of this paper is as follows:
1. The VFLG features a 2.5D tooth configuration, 
offering the advantages of low difficulty and easy 
accuracy guarantee in the machining of micro-
gears. In theory, it can adapt to commonly used 
micro-machining methods such as micro-milling 
and laser ablation.
2. The curvature non-interference formula of LG is 
derived, restricting the value of the turning angle 
Fig. 17.  Vertical gear pair used in the experiment
Fig. 18.  Schematic diagram of experimental apparatus
Fig. 19.  Schematic diagram of kinematic experiment results
Fig. 20.  Schematic diagram of instantaneous pressure angle 
variation at the point of meshing

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 483-493
492 Shao, Y. – Chen, Y. – Xiao, X – Zheng, M. – He, W.
and the curvature radius at the meshing point of 
the normal plane.
3. The maximum contact stress of VFLG is lower 
than that of traditional LG, indicating better 
bearing capacity. This feature positions VFLG 
as a promising candidate for applications in the 
micro and nano fields.
These findings have propelled the development 
of LG in the field of micro and nano applications. 
However, this paper solely designed an LG suitable 
for micro-machining and feasible in principle; the 
micromachining has not been carried out, and the 
actual tribological performance, transmission capacity 
and transmission efficiency of VFLG are uncertain. 
Future work will involve utilizing the new gear pair 
for micro-machining objects, such as laser ablation, 
to assess its accuracy level. Additionally, micro-
transmission experiments will be conducted to study 
transmission accuracy and explore its applications in 
the field of MEMS, such as micro pumps and micro 
robots.
6  ACKNOWLEDGEMENTS
The authors gratefully acknowledge the support 
from the National Natural Science Foundation of 
China (No. 52405056), Open Fund of Guangdong 
Provincial Key Laboratory of Precision Gear Flexible 
Manufacturing Equipment Technology Enterprises, 
Zhongshan MLTOR Numerical Control Technology 
Co., LTD & South China University of Technology 
(No. 2021B1212050012-08). It is our honor to thank 
the reviewers and editors for their valuable criticisms 
and comments.
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