*Corr. Author’s Address: Xi ‘an Technological University, No. 2 Xuefu Middle Road, Xi’an, China, maqun@xatu.edu.cn 426
Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 426-439 Received for review: 2023-11-28
© 2024 The Authors. CC BY-NC 4.0 Int. Licensee: SV-JME Received revised form: 2024-04-28
DOI:10.5545/sv-jme.2023.884 Original Scientific Paper Accepted for publication: 2024-05-27
Simulation Research on the Control Method  
of Bow-Collapse in Gear Cold Roll-Beating
Ma, Q. – Cha, L. – Zhang, X.
Qun Ma* – Li Cha – Xiangwei Zhang
Xi’an Technological University, School of Ordnance Science and Technology, China
Bow-collapse is a type of geometric defect in the gear cold roll-beating process. In order to effectively control bow-collapse and improve material 
utilization, this paper calculates the cross-section radius of the blank according to the volume invariance principle of metal plastic deformation, 
and then performs FE simulation of the cold roll-beating process by using the cyclic beating of adjacent tooth spaces and intermittent blank 
feeding method. The flow state of metal particles is investigated, revealing that the movement of metal particles along the axial direction of the 
blank is the main reason for the formation of bow-collapse. This paper proposes a method to correct the cross-section radius of the blank and 
thus control for bow-collapse, where the loss coefficient K characterizes the state of metal particle loss in an infinitesimal thickness region on 
the cross-section. The analytical equation for calculating the corrected value of the cross-section radius is derived, and the corrected value is 
calculated. Conducted on the modified blank, cold roll-beating FE simulation results show that bow-collapse is effectively controlled, and the 
loss coefficient K correctly reflects the loss state of metal particles on the cross-section. The simulation results also show that after cold roll-
beating, the gear teeth with standard face width and tooth depth can be obtained after two turning end faces and turning tip circle processes. 
The bow-collapse control method proposed in this paper effectively improves material utilization.
Keywords: cold roll-beating, bow-collapse, FE simulation, loss coefficient, cross-section radius
Highlights
• 	 The reasons behind the formation of bow-collapse are revealed. 
• 	 A blank size modifying method is proposed to control the bow-collapse without limiting material flow. 
• 	 The loss coefficient K is defined to characterize the loss state of metal particles on the blank cross-section.
• 	 The correction equation of the cross-section radius is derived, solved, and calculated.
• 	 The effectiveness of the bow-collapse control method and the characterization of the metal particle loss state on the cross-
section by the loss coefficient K are verified.
0  INTRODUCTION
Cold roll-beating is a near-net forming process 
to produce gears via plastic deformation, and the 
advantages of this process include high productivity, 
short process chain, and high surface quality. The cold 
roll-beating forming motion consists of three parts: 
the rotation of the roller, the continuous indexing of 
the blank, and the feeding of the blank. The blank is 
a cylindrical bar, and the radius of its cross-section 
can be calculated according to the principle of volume 
invariance. However, after cold roll-beating, the blank 
produces geometric defects, such as flashes, concave 
tooth tips, and bow-collapses (Fig. 1).
Flashes can be removed by turning the end face. 
Concave tooth tips have little effect on the strength 
of the gear teeth and can be solved by removing 
materials other than the tip circle (e.g., the spline 
shaft in Fig. 2). Bow-collapse is a defect where the 
gear tooth appears high in the centre and low at both 
ends along the face width, similar to a bow shape 
(Fig. 1c). As the bow-collapse range is large and to 
ensure sufficient face width, the current solution is 
to increase the length of the blank, then all collapsed 
parts at both ends are removed after the gear is formed 
(Fig. 2). However, the above solution results in wasted 
material, and the material utilization rate is especially 
low when forming gear parts with small face width.
Fig. 1.  Geometric defects of cold roll-beating:  
a) concave tooth tips, b) flashes, and c) bow-collapse
Fig. 2.  Spline shaft formed by the cold roll-beating process

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 426-439
427 Simulation Research on the Control Method of Bow-Collapse in Gear Cold Roll-Beating 
To control for geometric defects, such as bow-
collapse, it is necessary to analyze the deformation 
process and material flow states to reveal the 
formation mechanism. While experimental studies are 
most consistent with real situations, measuring during 
the rolling process is challenging, and there is no clear 
physical evidence linking the results and parameters. 
In addition, it is difficult to show the plastic flow 
process of materials in experimental results. Current 
experimental studies on cold roll-beating are mainly 
on process parameters [1] and [2], surface forming 
characteristics [3], microstructure morphology [4] and 
[5], and residual stress analysis [6] and [7]. 
The development of finite element (FE) 
simulation technology has made it one of the most 
effective tools for studying complex metal plastic 
deformation processes and forming mechanisms. 
Byon et al. [8] predicted the crack growth length and 
growth direction in the steel strip rolling process via 
the FE method and verified the results with physical 
experiments. Niţu et al. [9] and [10] simulated a 
groove rolling process of complex profiles with the 
ABAQUS software and investigated the effects of 
radial force and blank diameter on productivity and 
groove quality. Hwang et al. [11] simulated the rolling 
process of aluminium wheels via the FE software 
and investigated the process parameters to improve 
geometric accuracy. Peng et al. [12] studied how 
material flow formed gear teeth during the cold rolling 
process via the FE method, then calculated tooth 
deviation and verified with experiments. Khodaee 
et al. [13] investigated the effects of blank geometry 
on tooth shape deviation via FE simulation and 
concluded that FE simulation can be used to optimize 
blank geometry in similar cold rolling cases. Li et al. 
[14] investigated the plastic deformation mechanism 
and material flow in the three rolling passes process of 
gears via FE analysis. Nistor et al. [15] investigated the 
influence of gear geometry, such as teeth number, and 
the deformation mechanism via FE analysis in terms 
of the evolution in tooth formation and formation 
loads. Ma and Zhang [16] calculated the cold roll-
beating forming force via the FE simulation software 
DEFORM-3D, which was verified with experimental 
measurements. Deng et al. [17] conducted an FE 
simulation of the hot rolling process of face gear and 
verified the feasibility of hot rolling spur face gear. Li 
et al. [18] investigated the effects of initial bite depth, 
friction condition, and teeth number on the slippage in 
gear rolling via FE analysis and verified the result’s 
reliability through experiments.
FE simulation is critical to understanding 
geometric defect control in the gear cold roll 
forming process. Wu et al. [19] proposed a gear-
rolling process using conical gear rollers and used 
FE simulation to analyse the change of tooth shape 
and forming mechanism at different stages, where 
the bow-collapse and the rabbit ear defect (similar 
to the concave tooth tip in cold roll-beating) were in 
agreement with experimental results. Li et al. [20] 
simulated the gear rolling process through the FE 
software, and analysed the formation mechanism 
of rabbit ears, then verified the phenomenon and 
morphology of rabbit ears through experiments. Li et 
al. [21] investigated the formation causes and control 
methods of rabbit ear defects in gear roll forming 
via FE simulation. Ma et al. [22] investigated the 
optimization of rolling tools to reduce the height of 
rabbit ears during the roll-forming process through 
analytical models, FE methods, and experiments. 
Wu et al. [23] investigated the effect of the revised 
rollers on the control of rabbit ear defects through FE 
simulations and experiments, and the experimental 
results agreed with the simulation results. Ma et al. 
[24] investigated the effect of rolling parameters, the 
teeth number and rolling tools on rabbit ear defects. 
Current research on geometric defects mainly focuses 
on controlling rabbit ears, while there is no research 
on controlling bow-collapse.
Therefore, the paper uses FE simulation to study 
the control method of bow-collapse. First, the cross-
section radius of the blank is calculated according 
to the volume invariance principle. Second, FE 
simulation of cold roll-beating is performed to analyse 
the deformation process and flow state of materials to 
reveal the causes for the formation of bow-collapse 
and to propose a method to control for bow-collapse. 
Last, the proposed bow-collapse control method is 
verified via FE simulation.
1  THEORETICAL CALCULATION  
OF THE CROSS-SECTION RADIUS OF THE BLANK
Fig. 3 illustrates the volume invariant principle to 
calculate the cross-section radius of the blank, where 
r
f
 and r
p
 are the root circle radius of the gear and the 
cross-section radius of the blank, respectively; A
1
 and 
A
2
 respectively represent the two diagonal shaded 
areas and A
3
 is the area of the mesh. When the roller 
beats the blank, the metal in the tooth space is expected 
to produce plastic deformation, moving upward along 
the tooth flank to form a gear tooth, while metal on the 
other parts of the blank is not deformed. Therefore, 
the two diagonally shaded areas in Fig. 3 should be 
equal in area, which can be expressed as:

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 426-439
428 Ma, Q. – Cha, L. – Zhang, X.
 A
1
 = A
2
 = A – A
3
, (1)
where A is the area of a single tooth in the blank’s 
cross-section.
The cross-section area of the blank is A
p 
, and  
A
p
 = πr
p
2
. The area below the root circle of the gear is 
A
f
, and A
f
  = πr
f  
2
. The area of all the teeth on the gear is 
Az, and z is the teeth number. According to the volume 
invariance principle, we get A
p
 = A
f
  + Az, and it can be 
expressed as:
πr
p
2
 = πr
f
  
2
 + Az .
Then, the cross-section radius of the blank can be 
determined with Eq. (2):
 r
Az
r
pf


2
. (2)
For an involute spur gear, A can be expressed as 
[25] and [26]:
A
r
r
z
r
z
b
af aa
f


 






 
2
33 2
2
32
2
tant an 




invi nv
invi inv
f






, (3)
where r
b
 is the radius of the base circle, r
b
 is the radius 
of the tip circle, α, α
a
 and α
f
  are the pressure angles on 
the reference circle, the tip circle, and the root circle, 
respectively.
Fig. 3.  Volume invariant principle to calculate  
the cross-section radius of the blank
For an involute spur gear with module m = 2 and 
teeth number z = 20, if α = 20° and the modification 
coefficient is 0, it is known that r
b
 = 18.8 mm, r
f
  = 17.5 
mm, r
a
 = 22 mm and α
a
 = 31.3°. However, since r
f
  < r
b
, 
the part below the base circle is no longer an involute 
but a fillet curve, so the area A of a single tooth cannot 
be calculated directly by Eq. (3).
The fillet curve below the base circle does 
not engage and is only related to the strength of 
the tooth root, so the fillet curve taken by different 
manufacturers is also different. The distance from 
the root circle to the base circle is only 1.3 mm, so 
an approximate calculation is made in this paper, as 
shown in Fig. 4. The fillet curve can be approximated 
as consisting of the straight line BC (B’C’) and the 
circular arc CD (C’D’), where the straight line BC 
(B’C’) is tangent to the involute line. Extending the 
straight lines BC and B’C’, and intersecting with the 
root circle at the points E and E’, then:
 A = A
ab
 + A
bf
 + A
cf
 , (4)
where A
ab
 is the area of the region between the tip 
circle and the base circle, A
bf
 is the area of the sector 
ring BB’E’E between the base circle and the root 
circle, and A
cf
 is the area of the region enclosed by 
the fillet curve, the root circle and the straight line CE 
(C’E’).
Fig. 4.  Calculation of tooth area A with an m = 2 and z = 20 
involute spur gear
Since the pressure angle on the base circle is 0, 
A
ab
 can be expressed as:
 
A
r
r
z
r
z
ab
b
aa a
b
 













2
32
2
32
2
tan
.





invi nv
inv (5)
A
bf
  can be expressed as:
 Ar r
bf
b
bf



2
22
, (6)

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 426-439
429 Simulation Research on the Control Method of Bow-Collapse in Gear Cold Roll-Beating 
where φ
b
 is the centre angle of the circle corresponding 
to points B and B’.
Then, φ
b
 can be expressed as:
 


b
z
 2inv . (7)
For A
cf 
, the arc DE can be approximated as a 
line segment perpendicular to the line CE, and both 
line segments DE and CE are tangent to the circular 
arc CD. If the radius of the circular arc CD is 0.5 
mm, then A
cf
  ≈ 0.054 mm
2
.
With all parameters determined, the results are 
A = 13.3 mm
2
 and r
p
 = 19.8 mm.
2  FE SIMULATION OF THE COLD ROLL-BEATING PROCESS
2.1  Establishment of the FE model
In order to observe the forming process of the gear 
tooth and analyse the causes of bow-collapse, the FE 
simulation software DEFORM-3D is used to simulate 
the cold roll-beating process. DEFORM-3D does not 
have a modelling function, so the 3D solid models are 
modelled with SOLIDWORKS.
2.1.1  Geometric Model of the Roller
The roller′s geometric model of m = 2 and z = 20 
involute spur gear is established. The profile curves 
of the roller coincide with the profile curves of the 
tooth space. The curves between the root circle and 
the base circle are the fillet curves shown in Fig. 4, 
and the curves above the base circle are two involutes. 
The maximum outside diameter of the roller is 40 mm 
and the thickness is 12.5 mm (Fig. 5).
Fig. 5.  Geometric parameters of the roller
2.1.2  Geometric Model of the Blank
The theoretical calculation of r
p
 is only a reference. 
Considering that the formation process of defects 
should be presented completely, so r
p
 cannot be a 
small value, and a slightly larger value will not affect 
the correction. Therefore, this paper increases r
p
 
slightly and integers it to a value of r
p
 = 20 mm. 
The FE simulation of metal plasticity requires 
significant computer resources and time, so it is 
meaningless to define the tooth width too large. If 
the definition of tooth width is very small and the 
distance between the front and back ends is very 
close, the movement law of metal particles along 
the axial direction of the blank cannot be fully 
presented. Therefore, the tooth width is set as 22 
mm. Then the length of a part with a cross-section 
radius of 20 mm is set to 22 mm, and at the same 
time, there is a shaft with a diameter of 28 mm at 
each end face of the blank.
2.1.3  Importing Geometric Models
The geometric models of the roller and the blank are 
converted to STL data format and imported into the 
DEFORM-3D pre-processor (Fig. 6). Establishing 
a space rectangular coordinate system, the origin is 
located in the geometric centre of the blank, the Z-axis 
coincides with the axis of the blank, and the Y-axis 
is the symmetry axis of the profile curve of the tooth 
space formed by the roller.
The following definitions are established: The 
end face of the blank (Z = 11 mm) that is beat first is 
the front end face; the other end face (Z = –11 mm) is 
the back end face. The blank is divided into two parts 
along the XY plane: the part with Z ≥ 0 is the front 
half, and the part with Z < 0 is the back half. 
 
Fig. 6.  FE models in the DEFORM-3D pre-processor:  
a) the frontal view, and b) the left side view
2.2  Pre-processing for Simulation
The simulation conditions in the DEFORM-3D pre-
processor are set as follows:

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 426-439
430 Ma, Q. – Cha, L. – Zhang, X.
(1) The motion parameters of cold roll-beating: The 
rotation speed of the roller is set as ω
g 
= 2000 r/
min; the indexing angular speed of the blank is 
set as ω
p 
= 100 r/min; the feeding speed of the 
blank is set as v = 2 mm/r; and the rotation radius 
of the roller is set as R = 100 mm.
(2) Fully constrained boundary conditions are 
applied to the outer surfaces of the shafts with a 
diameter of 28 mm to simulate the clamping state 
of the blank in the cold roll-beating process.
(3) Disregarding the elastic deformation of the roller 
and the elastic restitution of the blank, the roller 
is defined as a rigid body, and the blank is defined 
as a plastic body. The material of the blank is set 
as Aluminium Alloy AL-1100.
(4) The blank is meshed, and the local mesh 
refinement is conducted in the main deformation 
zone to generate tetrahedral meshes. The element 
number is 217967, and min edge length is 0.1797 
mm.
(5) Cutting fluid is used in the process of cold roll-
beating, so it can be considered as pure rolling 
between the roller and the blank. To simplify the 
calculation, the friction coefficient between the 
roller and the blank is set to 0.
(6) Since cold roll-beating is conducted at room 
temperature, the simulation ambient temperature 
is defined to be constant at 20 °C.
2.3  FE Simulation of Cold Roll-Beating
Since the indexing and feeding speed of the blank is 
exceedingly low relative to the rotation speed of the 
roller, it has little effect on forming; the simulation of 
the beating process ignores the indexing and feeding 
movement of the blank, i.e., the blank is considered 
stationary.
In order to match the simulation process with 
the actual cold roll-beating process, two adjacent 
tooth spaces are cyclically beaten while the blank 
is intermittently fed. When the simulation of the 1
st
 
beating process of a tooth space is finished, the blank 
is rotated clockwise by 18° for the simulation of the 
1
st
 beating process of the adjacent tooth space. Then, 
the blank is rotated counterclockwise by 18° (back to 
the first tooth space), and at the same time, the blank 
is moved by 2 mm in the negative Z-axis direction 
to simulate the 2
nd
 beating process of the first tooth 
space. Follow the above steps, alternately simulating 
the beating processes of two adjacent tooth spaces 
and moving the blank until both tooth spaces are fully 
formed.
The calculation results show that the longest 
action time of one beating process is less than 
1.31×10
–3 
s, thus the time increment of the simulation 
is set as 1×10
–5 
s and the number of simulation steps 
is set as 140.
2.4  Simulation Results
Fig. 7 shows the FE simulation of the cold roll-
beating process, and Fig. 8 shows the concave 
tooth tip, flashes, and bow-collapse formed by the 
FE simulation. As the blank is fed, the grooves 
gradually deepen, and the gear tooth gradually bulges, 
eventually forming a concave tooth tip, flashes, and 
bow-collapse similar to the shapes in Fig. 1.
a)   b) 
c)   d) 
Fig. 7.  FE simulation of the cold roll-beating process:  
a) the 1
st
 beating process of two tooth spaces, b) the 2
nd
 beating 
process of two tooth spaces, c) the 11
th
 beating process of two 
tooth spaces, and d) the final forming of two tooth spaces
Fig. 8.  Geometric defects formed by FE simulation:  
the concave tooth tip and flashes and the bow-collapse
2.5  Velocity Vector Diagrams of Metal Particles
The velocity vector diagrams of the metal particles 
when the roller moves to different positions are shown 
in Fig. 9:
(1) The flow state of metal particles in the front 
and back half of the blank is approximately 
the same. Metal particles mainly present two 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 426-439
431 Simulation Research on the Control Method of Bow-Collapse in Gear Cold Roll-Beating 
motion states: one moving in the XY plane 
and the other moving perpendicular to the 
XY plane, i.e., moving along the Z-axis in the 
positive or negative directions.
(2) On the sides of the tooth space, metal particles 
are more likely to move in the XY plane, 
while on the bottom of the tooth space, metal 
particles are more likely to move perpendicular 
to the XY plane. 
(3) Metal particles located in front of the roller 
move in the same direction as the roller, and 
metal particles located behind the roller move 
in the opposite direction to the roller, which 
is similar to the plastic deformation issue of a 
rigid cylinder rolling a rough surface [27] and 
[28].
(4) The closer to the front and back end faces, the 
more obvious the movement trend of metal 
particles along the Z-axis. At the front end face, 
movement is mainly in the positive direction 
along the Z-axis, i.e., consistent with the roller, 
while at the back end face, movement is mainly 
in the negative direction along the Z-axis, i.e., 
opposite to the roller.
(5) When the roller beats a tooth space, metal 
particles of the adjacent tooth space also 
move, which proves that the cold roll-beating 
simulation of only a single tooth space is not 
consistent with actual working conditions.
2.6  Trajectory Tracking of Metal Particles
Trajectory tracking of metal particles is conducted 
using the point tracking tool in the DEFORM-3D post-
processor to study the movement state of materials. 
Plastic deformation of materials outside the root 
circle is the main reason for the formation of tooth 
spaces and gear teeth. To simplify the calculation 
and consider for symmetry, the region X ∈ [–3.5, 0],  
Y ∈ [17, 20], and Z ∈ [–11, 11] on the blank before 
beating is taken, and the metal particles within are 
subjected to point tracking.
2.6.1  Moving State of Metal Particles in the Back Half of 
the Blank
Trajectory tracking is first performed on metal 
particles in the back half of the blank (Z ∈ [-11, 0]). 
Along the Z-axis direction, sections are taken at 
intervals of 1 mm, and a 17×9 array of particles is 
taken on each section, then trajectory tracking is 
performed for these 153 particles. The metal lattices 
are considered untwisted. Thus, the spatial position 
relationship between metal particles is unchanged, 
which is proven by analysing the trajectories of 
Fig. 9.  V elo city	v ect o r s	o f	me tal	par ticles	when	the	r o ller	mo v es	t o 	dif f erent	po sitio ns:	a)	the	fr o nt	ha lf	Z	≈	5	mm,	b)	the	fr o nt	ha lf	Z	≈	8	mm,	
c)	the	fr o nt	half	Z	≈	1 1	mm,	d )	the	bac k	half	Z	≈	–5	mm,	e)	the	bac k	half	Z	≈	–8	mm,	and	f)	the	b ac k	half	Z	≈	–11 mm

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 426-439
432 Ma, Q. – Cha, L. – Zhang, X.
metal particles at each simulation step. Due to 
the large amount of data, only the metal particle 
positions in the last simulation step are given in this 
paper to compare with the initial positions (Fig. 10).
(1)  Movement of particles in the Z-axis direction
 Metal particles in contact with the side profiles of 
the roller all show negative movement directions 
along the Z-axis, i.e., towards the back end face, 
and the closer to the back end face, the greater the 
displacement.
 Metal particles below the roller (at the bottom 
of the tooth space), on the other hand, show two 
states of motion. In the range Z ∈ (–5, 0], they 
move in the positive direction of the Z-axis, but as 
they approach the back end face (Z ∈ [–11, –5]), 
they move towards the back end face with 
increasing displacement.
(2) Movement of particles in the Y-axis direction
 As shown in Fig. 10, combined with the velocity 
vectors of particles in Fig. 9, when farther away 
from the back end face, metal particles show 
more movement along the positive direction of 
the Y-axis, and when close to the back end face, 
the displacement of metal particles along the 
positive direction of the Y-axis decreases, and 
they gradually change to move along the negative 
direction of the Y-axis. In the range Z ∈ [-11, -8], 
all metal particles transform to move in the 
negative direction of the Y-axis.
 Most metal particles near the root circle move 
in the negative direction of the Y-axis, and metal 
particles within the root circle (Y < 17.5 mm) are 
also forced to move downwards.
2.6.2  Moving State of Metal Particles in the Front Half of 
the Blank
Fig. 11 shows the state of movement of metal particles 
in the front half of the blank. Compared to Fig. 10, 
the section positions of the pickup particles are only 
slightly different near the front end face.
As shown in Fig. 11, combined with the velocity 
vectors of particles in Fig. 9, the moving state of metal 
particles at the front half of the blank is essentially the 
same as that at the back half. Farther away from the 
front end face, metal particles mainly move along the 
positive direction of the Y-axis, while the closer to the 
front end face, the greater the displacement of metal 
particles along the negative direction of the Y-axis and 
the positive direction of the Z-axis. 
2.7  Analysis and Discussion
(1) As shown in Figs. 10 and 11, the closer to the 
front (back) end face, the greater the displacement 
Fig. 10.  Movement of metal particles in the back half of the blank

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 426-439
433 Simulation Research on the Control Method of Bow-Collapse in Gear Cold Roll-Beating 
3  CORRECTION OF THE CROSS-SECTION  
RADIUS OF THE BLANK 
As mentioned above, limiting the movement of metal 
particles along the Z-axis direction is a method to 
control bow-collapse, which can be achieved with two 
approaches. The first is to limit the front and back end 
faces, which can be realized by clamping the front and 
back end faces with special fixtures, but the fixtures 
will affect the movement of the roller, thus failing to 
form a complete tooth space. The second is to increase 
the axial length of the blank so that metal particles 
within the face width move away from the front and 
back end faces, then turning to remove the collapsed 
part, which is consistent with the current production 
process, as shown in Fig. 2. The second approach 
results in a waste of materials and also complicates the 
machining process.
Therefore, this paper proposes a method to control 
the metal plastic forming geometry, which does not 
limit the plastic flow of the material but modifies 
the blank size to supplement (or reduce) the lack (or 
excess) of material in the forming process. For the 
bow-collapse of cold roll-beating, it is necessary to 
increase the cross-section radius of the blank.
As shown in Fig. 3, according to the volume 
invariance principle, A
1
 = A
2
 = A – A
3
. However, due 
to the movement of metal particles along the Z-axis, 
of metal particles along the positive (negative) 
direction of the Z-axis, and a large number of 
metal particles near the front (back) end face 
move out of the range of the face width, forming 
flashes. The reason for this phenomenon is that 
the end faces and the outer surface of the blank 
are all free surfaces; according to the principle 
of least resistance to plastic flow of metals, the 
closer to the end faces, the resistance of metal 
particles to move along the Z-axis direction is 
smaller. Thus, they move towards the end faces. 
Away from the end faces, most metal particles 
have less resistance to move towards the outer 
surface of the blank, and they show more upward 
movement along the tooth flank, forming a raised 
gear tooth. 
(2) The movement along the XY plane does not cause 
the loss of metal particles on the cross-section of 
the blank, whereas the movement along the Z-axis 
direction leads to the loss of metal particles on 
the cross-section, which is the main cause for the 
formation of bow-collapse.
(3) Bow-collapse can only be controlled in two 
ways, either by limiting the movement of metal 
particles along the Z-axis or by increasing the 
cross-section radius of the blank to replace the 
lost metal particles.
Fig. 11.  Movement of metal particles in the front half of the blank

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 426-439
434 Ma, Q. – Cha, L. – Zhang, X.
A
1
 changes to A
1
' and A
2
 remains unchanged so that 
A
1
' ≤ A
2
.
In this paper, we define the loss coefficient K as:
 K
AA
A


21
2
'
. (8)
It is assumed that the loss coefficient K is 
independent of the size of the cross-section and is 
only related to the axial position. In this paper, the loss 
coefficient K is used to characterize the proportion of 
metal particles lost in an infinitesimal thickness region 
on the cross-section. Since A
1
' ≤ A
2
, in general 0 ≤ K ≤1. 
By increasing the cross-section radius r
p
, A
2
 
and d, A
3
 also increase, but the loss coefficient K is 
unchanged. Assuming that A
2
 and A
3
 become A
2
' and 
A
3
', cross-section radius r
p
 is the corrected value when 
Eq. (9) is satisfied.
 A
1
' = (1 – K)A
2
' = A – A
3
 . (9)
Therefore, the correction method for the cross-
section radius r
p 
is:
(1) Calculate A
1
' and A
2
 based on the results of the 
FE simulation, 
(2) Calculate the loss coefficient K via Eq. (8);
(3) Calculate A
2
' on the blank’s cross-section;
(4) Increasing the cross-section radius r
p
 until Eq. (9) 
is satisfied.
3.1  Calculation of A
1
' and A
2
The concave tooth tip should be included in the 
calculations of A
1
'. Fig. 12 shows the shape of 
the concave tooth tip in the Z = 0 cross-section. 
The forces on both sides of the gear tooth are 
essentially identical, and, as can be seen from Figs. 
1 and 12, the concave tooth tips are approximately 
symmetrically distributed along the centre of the 
gear tooth. However, as shown in Fig. 7d, the 
shapes and sizes of the concave tooth tip varies 
across different cross-sections, so it is difficult to 
accurately calculate the area of concave tooth tips.
Here, we calculate the average depth of the 
gear tooth containing concave tooth tips and then 
calculate A
1
' according to the tooth area calculations.
Therefore, the average gear tooth depth H can be 
estimated via Eq. (10), which is expressed as:
 H
HH


maxm in
,
2
 (10)
where H
max
 is the distance from the highest point of 
the concave tooth tip to the root circle; H
min
 is the 
distance from the lowest point of the concave tooth tip 
to the root circle.
Fig. 12.  The concave tooth tip on the cross-section Z=0 (in mm)
In the DEFORM-3D post-processor, the 
coordinates of the highest and lowest points of the 
concave tooth tip are extracted at 0.5 mm intervals 
along the Z-axis direction. The average gear tooth 
depth H is calculated via Eq. (10) and fitted to a smooth 
curve (Fig. 13). The minimum value of H appears at 
the two end faces, even lower than the outer surface 
of the blank, and only in the range of Z ∈ [-2, 0.5] 
which basically reaches the standard full tooth depth 
(4.5 mm).
Fig. 13.  Variations of the average gear tooth depth 
Assuming that the radius of the tip circle after 
cold roll-beating is r
i
, since the gear tooth depth is 
insufficient, we have r
i
 < r
a
.
At this point, A
1
' can be calculated via Eq. (11) 
and expressed as:
A
r
r
z
r
z
b
ip ii
p
1
2
33 2
2
32
2
't an tan 

 











invi nv
inv  






inv
p
, (11)
where a
i
 is the pressure angle at the tip circle, and α
p
  is the pressure angle at the outer circle of the blank.
α
p
 = arccos(r
b
/r
p
), α
i
 = arccos(r
b
/r
i
), and r
i
 can be 
determined by the gear tooth depth curve in Fig. 13.

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 426-439
435 Simulation Research on the Control Method of Bow-Collapse in Gear Cold Roll-Beating 
Since A
2
 does not change, it can still 
be expressed as A
2
 = A – A
3
 = A
1
, then A
2
 can be 
calculated via Eq. (12):
A
r
r
z
r
z
b
ap aa
p
2
2
33 2
2
32
2


 







tant an 




invi nv
inv  






inv
p
, (12)
3.2  Calculation of the Loss Coefficient K
From the results of A
1
' and A
2
, the loss coefficient K 
is calculated for any cross-section of the blank and 
fitted to a smooth curve (Fig. 14).
Fig. 14. Variations of loss coefficient K
It should be noted that the loss coefficient 
K > 1 when close to the front and back end faces. 
In comparison with Fig. 13, near the front and back 
end faces, the radius of the tip circle formed by cold 
roll-beating is smaller than the cross-section radius 
of the blank, thus A
1
' is negative, which shows that 
K > 1. Once a flash is created, the lower surface of the 
flash is also a free surface so that near the front and 
back faces, the metal particles move not only along 
the Z-axis but also downwards. This results in the 
formation of collapse instead of raised gear teeth near 
the front and back end faces.
3.3  Correction Equation for the Cross-Section Radius r
p
The calculation of A
2
' on the cross-section of the blank 
is given in Fig. 15. Increasing r
p
 so that the gear tooth 
depth reaches the standard value, the corrected A
2
 
becomes A
2
', and therefore A
2
' cannot be calculated 
via Eq. (12).
Make the outer circle of the blank intersect 
the tooth profile curves of two adjacent gear teeth 
at points N and P. Make the root circle intersect the 
straight lines NO and PO at points N' and P'.
Since the areas of the two mesh regions in 
Fig. 15 are equal, i.e., A
r1
 = A
r2
, the corrected A
2
' can 
be expressed as:
 A
2
' = A
NN'P'P
 – A
3
', (13)
where A
NN'P'P
 is the area of the sector ring NN'P'P 
with an angular pitch of θ
P
.
A
3
' is the area of a single gear tooth within 
the blank outer circle after the increase of r
p
. 
Although A
3
' in Fig.15 increases due to the increase 
of r
p
, it can still be expressed as:
 A
3
' = A – A
1
'. (13)
Substitute Eq. (14) into Eq. (13), and we get:
 A
2
' = A
NN'P'P
 – A + A
1
'. (14)
Since K is independent of the cross-section size 
and is only related to the cross-section position, 
A
1
' = (1 – K)A
2
' still holds, then the correction equation 
for the cross-section radius can be given as: 
  A
1
' – (1 – K)A
2
' = A
1
' – (1 – K)(A
NN'P'P
 – A + A
1
') = 0. (15)
After collating, we get:
 KA
1
' – (1 – K)(A
NN'P'P
 – A + A
1
') = 0. (16)
A
NN'P'P
 can be calculated by Eq.(17):
 ANNPP
rr
pp f
'' 



22
2
, (17)
where, θ
P
 = 2 π/z.
As shown in Fig. 15, N and P are located on 
the left tooth surface of the adjacent gear teeth, 
respectively, and are also on the same circle with r
p
 as 
the radius. Therefore, whether the teeth number is odd 
or even, and regardless of the value of r
p
, θ
P
  = 2 π/z is 
satisfied.
Increasing r
p
 so that r
i
 = r
a 
, we have:
A
r
r
z
r
z
b
ap aa
p
1
2
33 2
2
32
2
't an tan 

 











invi nv
inv  






inv
p
. (18)
Substitute Eq. (17) and Eq. (18) into Eq. (16):
K
r
r
z
r
z
b
ap aa
p
2
33 2
2
32
2
tant an 





 










invi nv
inv 











 












 inv
p
pf
K
rr
z
A 10
22
.(19)
Since α
P
 can be written as a function of r
p
, i.e., 
α
P
 = arccos(r
b 
/r
p
), Eq. (19) is the correction equation 
for the cross-section radius with r
p
 as the variable.

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 426-439
436 Ma, Q. – Cha, L. – Zhang, X.
Fig. 15.  Method of calculating A
2
'
3.4  Correction of the Cross-Section Radius r
p
It is difficult to directly solve the transcendental 
equation given by Eq. (19). Therefore, in this paper, 
we use MATLAB to prepare a program to iteratively 
calculate the approximate numerical solution of r
p
. 
Considering that the corrected value of r
p
 should be 
larger than the theoretically calculated cross-section 
radius and smaller than the radius of the tip circle, the 
solution interval for r
p
 is set to [19.8, 22]. The solution 
is fitted to a smooth curve (Fig. 16).
Fig. 16.  Values of r
p
 in calculation and in correction
By analysing the variations of r
p
, we get:
(1) r
p
 varies little in the range of Z ∈ [–5, 5], with 
a minimum value of 19.87 mm and a maximum 
value of roughly 20 mm.
(2) The variation curve of r
p
 is roughly symmetrically 
distributed with respect to the Z = 0 cross-section.
(3) r
p
 transitions from Z = ±11 along an approximate 
circular curve to Z = ±5.
Based on the variation curve of r
p
, and also 
considering that the blank is easy to machine, r
p
 is 
corrected as follows:
(1) Taken according to a symmetric distribution with 
respect to the Z = 0 cross-section.
(2) Taken as a constant value in the range of 
Z ∈ [–5, 5].
(3) Taken as a circular curve in the interval from 
Z = ±10 to Z = ±5 to facilitate the machining of the 
blank.
(4) After the gear is formed, the edges of two end 
faces are generally chamfered, so the corrected 
values of r
p
 in the ranges of Z ∈ [–11, –10] and 
Z ∈ [10, 11] are taken as the values at Z = ±10.
According to the above principles, the corrected 
values of r
p
 at Z = ±10 and Z = ±5 are set as 22 mm and 
20 mm, respectively (Fig. 16).
3.5  FE Simulation of Cold Roll-Beating of the Corrected 
Blank
The 3D models of the corrected blank and the roller 
are imported into the DEFOEM-3D pre-processor, 
and chamfers of 1×45° are added to the edges of end 
faces (Fig. 17a). The simulation is still conducted in 
accordance with the method of the cyclic beating of 
adjacent tooth spaces and intermittent feeding of the 
blank; the simulation results are shown in Fig. 17b. 
a)    b) 
Fig. 17.  FE simulation of cold roll-beating of the corrected blank:  
a) the pre-processing models, and b) the simulated gear tooth
Fig. 18 shows the top profile curve of the gear 
tooth. The middle of the curve is slightly higher, and 
there are small collapses on both sides, which indicate 
that the bow-collapse is effectively controlled. 
Compared with Fig. 16, the top profile curve of the 
gear tooth in Fig. 18 is in good agreement with the 
differences between the correction and calculation 
values given in Fig. 16.

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 426-439
437 Simulation Research on the Control Method of Bow-Collapse in Gear Cold Roll-Beating 
Comparing Fig. 18 with Fig. 8b, flashes of the 
corrected blank are obviously larger. According to 
the definition of the loss coefficient K in this paper, 
the loss of metal particles on the cross-section can be 
expressed as KA
2
. Near the end faces, there is A
2
' > A
2
, 
while K is unchanged, so the loss of metal particles 
near the end faces after correction (expressed as 
KA
2
') is larger than that of the blank before correction 
(expressed as KA
2
), i.e., the flashes of the blank after 
correction are larger than those of the blank before 
correction.
Fig. 18.  Top profile curve of the gear tooth and the flash 
co m pared	with	Fig.	8b
In the DEFORM-3D post-processor, the part 
above 22 mm is removed using the Slicing tool with 
a plane (Y = 22) tangent to the top land of the gear 
tooth (Fig. 19). It can be seen that the rest of the part 
reaches the top land, except for a small pit on each of 
the position corresponding to the two collapses in Fig. 
18. The maximum height of the cut-off part is only 
1.05 mm, which indicates that the correction of r
p
 has 
controlled for bow-collapse, and also proves that the 
loss coefficient K correctly reflects the state of loss of 
metal particles on the cross-section.
Fig. 19.  Slicing of the top land of the gear tooth (in mm)
Using the slicing tool, the parts other than the 
gear tooth are removed, and a gear tooth with standard 
face width and tooth depth is obtained (Fig. 20). 
Fig. 20 shows that after cold roll-beating, the 
flashes and concave tooth tips can be removed by two 
processes: turning the tip circle and turning end faces, 
resulting in a standard tooth depth and full face width. 
This bow-collapse control method removes relatively 
less material and improves material utilization.
Fig. 20.  Gear tooth with standard tooth depth and full face width
4  CONCLUSIONS
(1) According to the volume invariant principle of 
metal plastic deformation, the cross-section radius 
of the blank is calculated, the FE simulation of 
the cold roll-beating process is conducted with 
the cyclic beating of adjacent tooth spaces and 
intermittent blank feeding method, and geometric 
defects are obtained in accordance with the actual 
situation. 
(2) Metal particles roughly present two motion states: 
movement perpendicular to the blank axis (in the 
XY plane) and movement along the blank axis (in 
the Z-axis direction), and the latter is the main 
reason for the formation of bow-collapse. 
(3) The corrected value of the cross-section radius is 
calculated by solving the analytical equation with 
the iterative method, and the corrected blank is 
subjected to cold roll-beating FE simulation. The 
results show that bow-collapse is controlled, and 
the loss coefficient K correctly reflects the state 
of loss of metal particles on the cross-section.
(4) The simulation results prove that after cold 
roll-beating of corrected blank, gear teeth with 
standard face width and tooth depth can be 
obtained by turning the tip circle and turning 
end faces. Bow-collapse can be controlled, 
and material utilization can be improved by the 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)9-10, 426-439
438 Ma, Q. – Cha, L. – Zhang, X.
method proposed in this paper for correcting the 
cross-section radius of the blank. 
5  SCIENTIFIC AND INDUSTRIAL CONTRIBUTIONS 
(1) The loss coefficient K is introduced to 
characterize the loss state of metal particles on 
the cross-section of the blank, and the analytical 
equation for calculating the correction value of 
the cross-section radius of the blank is derived. 
This work provides a calculation method for 
determining the size correction of blank in metal 
plastic forming.
(2) The correction method studied in this paper can 
be applied to cold roll-beating production of 
spline, optimize the cold roll-beating process, 
shorten the process chain, and improve material 
utilization rate.
6  PROSPECTS FOR FUTURE RESEARCH
(1) Conducting experimental studies on the control 
method of bow-collapse is a direction for future 
research. Modifications of ordinary machine 
tools to meet the requirements of cold roll-
beating forming motion have been reported in 
experimental studies of cold roll-beating process 
parameters and force-energy parameters [1] to 
[3], and [16]. However, for the study of geometry 
control, equipment is required to have higher 
structural stability and sufficient power, which 
is difficult to accomplish with ordinary machine 
tools. The design of specialized cold roll-beating 
experimental equipment is a prerequisite for the 
experimental study of geometry control.
(2) As mentioned above, of the two methods of 
controlling bow-collapse, the method investigated 
in this paper is the second method, which does 
not control the flow of materials, but supplements 
the loss caused by the flow of the materials. The 
first method, controlling the flow of materials, 
although not easy to achieve, is the essence of 
plastic forming of metal and the method with the 
highest material utilization. Controlling the flow 
of materials by reasonably designing fixtures of 
the blank and the shape of the roller and ensuring 
the cold roll-beating forming movement is also a 
direction for future research.
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