*Corr. Author’s Address: Xi’an Technological University, No. 2 Xuefu Middle Road, Xi’an, China, 250239614@qq.com 409
Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 409-421 Received for review: 2023-04-23
© 2023 The Authors. CC BY-NC 4.0 Int. Licensee: SV-JME Received revised form: 2023-06-12
DOI:10.5545/sv-jme.2023.616 Original Scientific Paper Accepted for publication: 2023-06-16
Research on an Analytical Method for the Forming Force  
of External Spline Cold Roll-Beating
Ma, Q. – Zhang, X.
Qun Ma
*
 – Xiangwei Zhang
Xi’an Technological University, School of Ordnance Science and Technology, China
To determine the force and energy parameters of cold roll-beating of external splines, the characteristics of the deformation zone in cold roll-
beating are analysed. The geometric dimensions and position of the deformation zone change with the movement of the roller, the contact 
arc, and the reduction are very small, and there is an incomplete deformation zone in the initial stage of cold roll-beating. A discrete analytical 
method for calculating the unit pressure and deformation force is proposed, and the cold roll-beating process is discretized into an infinite 
number of cold-rolling processes with complex sections. The discrete analytical model of unit pressure and deformation force is established, 
and the unit pressure value and its distribution characteristics are determined. To verify the discrete analytical model, a finite element model 
of cold roll-beating is established, and the forming force is calculated. A horizontal milling machine is modified to carry out the cold roll-beating 
experiment, and the forming force is measured. The predicted results of the discrete analytical model are compared with the simulation and 
experiment results. The results show that the maximum error of radial force compared with the simulation and experiment results is about 7 % 
and 4 % respectively, and the variation curve of radial force is basically consistent, but the time of a cold roll-beating process is slightly shorter. 
The discrete analytical model correctly predicts the magnitude and change process of cold roll-beating forming force.
Keywords: discrete analytical method, cold roll-beating, forming force, deformation zone, radial force
Highlights
• 	 The changing characteristics of the metal deformation zone of external spline cold roll-beating are defined. 
• 	 A discrete analytical model for calculating the forming force of cold roll-beating is established by using the principal stress 
method. 
• 	 The finite element simulation of the cold roll-beating is carried out, the deformation of the workpiece is presented, and the 
forming force of the cold roll-beating is calculated.
• 	 The cold roll-beating forming force is measured via experiment, and the discrete analytical model is verified. The results show 
that the discrete analytical model correctly predicts the magnitude and the change process of cold roll-beating forming force.
0  INTRODUCTION
The cold roll-beating of external splines is a non-
cutting manufacturing technology in which a pair of 
rollers with the same profile and the tooth space shape 
of the workpiece continuously strike the workpiece to 
produce plastic deformation to form a defined tooth 
shape. This kind of processing technology aiming at 
net or near net forming has been rapidly developed 
and plays an important role in industry. The processing 
efficiency of cold-rolling splines is very high, and 
it is widely used in the automobile and agricultural 
machinery industries.
The cold roll-beating process of splines is a 
discontinuous and progressive local deformation 
process, so the deformation mechanism is complex, 
and the theoretical analysis and parameter calculation 
are difficult. The research on this kind of problem is 
mainly carried out by experiments or finite element 
simulations.
Wu et al. [1] and Liu et al. [2] analysed the gear 
cold-rolling process with the finite element method 
and carried out experimental verification. Ma et al. [3] 
analysed the gear roll-forming process with the finite 
element method and calculated the forming force. Fu 
et al. [4] used DEFORM-3D software to study the 
local induction heating rolling method, which is used 
to manufacture large gears. Zhang et al. [5] Quan et 
al. [6] and Wang et al. [7] used ABAQUS and ANSYS 
to conduct numerical simulations of the spline cold 
roll-beating process and calculate the deformation 
force of cold rolling. Wang et al. [8] established the 
finite element model for deformation prediction of 
the split straight bevel gear, and the results of finite 
element calculation were in good agreement with the 
experimental results. Yuan et al. [9] used ABAQUS 
to conduct finite element simulations of racks’ cold 
roll-beating deformation force and then analysed 
the influence of materials, cold roll-beating speed, 
and cold roll-beating mode on the deformation force 
through experiments. However, the finite element 
analyses (FEA) of the complex profile rolling required 
a long central processing unit (CPU) time and large 
storage space. For example, a typical CPU time for 
the 3D FEA of gear rolling with a flat die was about 
one week, using DEFORM code in 2007 [10].

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 409-421
410 Ma, Q. – Zhang, X.
The analytical model (e.g., modelled by the 
principal stress method or slip-line field method 
(SLFM)) has a clear physical interpretation and with 
shorter computation time. Zhang et al. [11] developed 
and improved the slip line field (SLF) and the 
analytical models of rolling force. Li et al. [12] and 
Li et al. [13] established the slip line field of external 
spline rolling forming and calculated the unit pressure 
and rolling pressure. Zhang et al. [14] developed an 
SLFM model with different friction models to analyse 
the spline cold-rolling process. Quagliato and Berti 
[15] established a slip line model for force estimation 
in the radial-axial ring rolling process. Then, the 
SLFM model was perfected to include axial force, by 
which experimental and numerical verifications were 
carried out [16]. Ma et al. [17] proposed an analytical 
model for predicting the pitch error of gear rolling 
with consideration of geometric relations and process 
parameters. 
Although the above research has greatly improved 
the efficiency, the analytical model for the special 
forming process of external spline cold roll-beating 
has not been studied. At present, there is no analytical 
model, empirical formula, or chart for calculating the 
unit pressure and deformation force of external spline 
cold roll-beating.
In this paper, the characteristics of the cold 
roll-beating deformation zone are analysed, and 
the discrete analytical method for calculating the 
forming force is given. A cold roll-beating process 
is discretized into an infinite number of cold-rolling 
processes, and the unit pressure and deformation 
force of each cold-rolling process are calculated to 
determine the deformation force variation curve of a 
cold roll-beating process. Finally, the correctness of 
the discrete analytical method is verified via finite 
element simulation and cold roll-beating experiment.
1  FORMING PRINCIPLE OF EXTERNAL  
SPLINE COLD ROLL-BEATING
The tooth shape of an external spline is distinct, as is 
the cold roll-beating process. The involute spline is 
generally formed by continuous indexing, while the 
rectangular spline and triangle spline are formed by 
intermittent indexing.
Fig. 1 shows the principle of involute spline 
cold roll-beating. The cold roll-beating of an involute 
spline is carried out by continuous indexing at normal 
temperature. Two rollers rotate synchronously and 
reversely at high speed, and the angular velocity is ω
g
. 
The rollers beat the workpiece once every revolution. 
The workpiece rotates continuously with the angular 
velocity ω
p
 and feeds along the axis with the velocity 
v. During the contact time with the workpiece, two 
rollers rotate along their own rotation axis under 
the action of friction, and the angular velocity is ω
r
. 
The transmission system of the machine tool strictly 
ensures that the ratio of ω
g
 to ω
p
 is the number of 
spline teeth so as to process the involute keyway, 
which is consistent with the profile of the roller 
section.
Fig. 1.  Schematic diagram of cold roll-beating of involute spline
2  CHARACTERISTICS OF THE METAL  
DEFORMATION ZONE OF COLD ROLL-BEATING
In this paper, a cold roll-beating process is defined as 
the process of the roller from contacting the workpiece 
to leaving the workpiece in a rotation cycle. In a cold 
roll-beating process, the rotation speed and feed speed 
of the workpiece are far less than the rotation speed 
of the rollers, and the contact time between the rollers 
and the workpiece is very short (several milliseconds); 
it can be approximately considered that the workpiece 
is stationary when the rollers beat it.
The cold roll-beating is similar to a metal rolling 
process, and the research on metal rolling theory 
is more developed, which can provide theoretical 
guidance for the calculation of the forming force of 
cold roll-beating. However, the cold roll-beating is 
different from the general metal rolling process, as the 
location and parameters of the metal deformation zone 
are changing constantly.
Fig. 2 shows the variation of the metal 
deformation zone. O is the rotation centre of the roller. 
R is the radius of rotation of the roller axis. r is the 
radius of the roller. h
f
 is the rolling depth. Arc DE 
and D’E’ are the profile curves of the section formed 
by two cold roll-beating processes on the same tooth 
space. P is the axial feed rate of the workpiece per 
revolution.

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 409-421
411 Research on an Analytical Method for the Forming Force of External Spline Cold Roll-Beating 
It can be seen from Fig. 2a that the position of the 
deformation zone is not fixed but moves continuously 
with the roller. When the roller is at B
1
, the contact 
arc, bite angle and reduction are arc A
1
B
1
, α
1
and Δh
1
. 
When the roller moves to B
i
, the contact arc, bite angle 
and reduction change to arc A
i
B
i
, α
i
 and Δh
i
. When the 
roller moves to the other position on the arc D’E’, 
all of the above deformation zone parameters are 
different. Considering the profile shape of the roller, 
the average width of the deformation zone at different 
positions on the arc D’E’ is also different. According 
to the geometric relationship in Fig. 2a, when the 
spline modulus is small, the contact arc length and 
the reduction are also small relative to the workpiece 
thickness.
It can be seen from Fig. 2b that D’ is not the 
starting point of the cold-rolling process. When the 
roller is at B on the extension line of arc D’E’, the 
cold-rolling process begins, but the deformation zone 
is not complete at this time. The theoretical contact 
arc, bite angle and reduction are arc AB, α and Δh. 
Due to the missing part, the actual deformation zone 
parameters are arc AB’, α’ and Δh’. However, the 
duration of the incomplete deformation zone is very 
short. The length of the contact arc, the bite angle and 
the reduction increase rapidly and reach the maximum 
when the roller moves to D’, then decrease slowly and 
decrease to 0 at E’.
Therefore, there are three main characteristics of 
a cold roll-beating metal deformation zone. First, the 
location and parameters of the deformation zone are 
constantly changing; Second, there is an incomplete 
deformation zone in the initial stage of cold rolling; 
Third, the contact arc and the reduction are both small.
In the process of metal deformation, it is 
impossible to calculate the forming force directly by 
the principal stress method with the change in the 
geometric size of the deformation zone.
3  ESTABLISHMENT OF THE DISCRETE ANALYTICAL MODEL 
A cold roll-beating process is different from a metal 
rolling process, but it is similar when the rollers 
only rotate along their own rotation axis ( ω
g
 = 0). 
The periodic rotation of the roller only changes the 
position and geometric parameters of the deformation 
zone.
As shown in Fig. 2a, when the roller is located 
at B
1
, assuming that ω
g
 = 0 and ω
r
 > 0, the cold roll-
beating process can be approximately regarded as 
the cold-rolling process of the workpiece along the 
direction of C’C in the following time, in which the 
workpiece is rolled in at A
1
 and rolled out at B
1
, and 
the rolling speed is expressed as
 V
1
 = r ω
r1
, (1)
When the roller moves to point B
i
, the cold roll-
beating process can also be approximately regarded 
as a cold-rolling process from A
i 
to B
i
, and the rolling 
speed is expressed as
 V
i
 = r ω
ri
, (2)
Because the rotation angular velocity of the roller 
is constant at any time, V
1
 = V
i
.
      
Fig. 2.  The change of metal deformation zone in cold rolling; a) complete deformation zone; and b) incomplete deformation zone

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 409-421
412 Ma, Q. – Zhang, X.
Since there is no other external force in the cold-
rolling process, a cold roll-beating process can be 
discretized into an infinite number of tension-free 
cold-rolling processes of a complex cross-section with 
the same rolling speed.
Fig. 3 shows the force condition of any differential 
unit in the metal deformation zone. Firstly, it is very 
important to determine the direction of friction t
z
, 
which determines the type of deformation zone. 
The 3D solid model of the workpiece and the roller 
was established by using the Solidworks 3D design 
software, and it was imported into the pre-processing 
program of the finite element software DEFORM-3D. 
The rotation speed of the roller is set to 209 rad/s, 
and the radius of rotation is 90 mm. To simulate 
the forming force in a cold roll-beating process, the 
contour formed by the last cold roll-beating is directly 
established on the workpiece, as shown in Fig. 4.
Fig. 3.  Force condition of any differential unit in deformation zone 
of workpiece
Fig. 4.  The finite element model of cold rolling 
Observe the velocity nephogram of metal 
particles, as shown in Fig. 5. The finite element 
analysis shows that at any time of cold roll-beating, 
the metal flow velocity in the deformation area is 
less than the linear velocity of the roller, so the whole 
deformation area belongs to the backward slip zone, 
and the direction of friction on the contact arc is 
opposite to the linear velocity of the roller. 
As shown in Fig. 3, if the elastic deformation of 
the roller and the workpiece is neglected, the sum of 
the components of the force acting on the differential 
unit along the Z direction is expressed as
 
p
dz
yt
dz
dy dy
zz
z
zz
z
zz
sin
cos
cos
cos
,






    (3)
Fig. 5.  Velocity nephogram of metal particles at a) step 40, and b) steep 120; in [mm/s]

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 409-421
413 Research on an Analytical Method for the Forming Force of External Spline Cold Roll-Beating 
where p
z
 is the unit pressure of the roller on the 
workpiece, and T
z
 is the unit friction force between 
the roller and workpiece. If z and y are the coordinates 
of the contact arc, then tan( φ
z
) = dy / dz, which can be 
substituted into Eq. (3), and the Kalman differential 
equation for calculating the average unit pressure is 
obtained, expressed as
 
d
d
d
d
p
z
K
y
y
z
t
y
zz
  0, (4)
where K is determined by the plastic equation under 
plane deformation, expressed as
 K   
13
2
3
, (5)
where σ
φ
 is the actual deformation resistance of the 
metal.
Assuming that the contact friction condition in 
the deformation zone obeys the dry friction law, i.e.,  
t
z
 = fp
z
, where f is the friction coefficient on the contact 
arc, the solution of the differential Eq. (4) is as follows
 pe C
K
y
ey
z
f
y
z
f
y
z





dd
d () , (6)
where C is integral constant determined by boundary 
conditions. 
Since the difference between the contact arc 
and the chord connecting the roll-in point and the 
roll-out point is very small, the contact arc can be 
approximated to a chord as
 dd z
l
h
y
a


, (7)
where l
a
 is the length of chord AB.
Substitute Eq. (7) into Eq. (6) and integrate 
them, and determine the integral constant according 
to the tension-free boundary condition. Finally, the 
distribution formula of unit pressure is obtained as
 p
Kh h
y
z
 
 

















11


. (8)
Here,  
lf
h
a

, and h is half of the thickness of 
the workpiece at the rolling point.
If the workpiece is an involute spline with Z = 20 
and m = 2 mm, the cold roll-beating parameters are 
p = 1 mm, r = 20 mm, R = 90 mm, and the friction 
coefficient on the contact arc is f = 0.17. After the 
deformation zone parameters such as h and l
a
 are 
determined from Fig. 2, the unit pressure on the 
contact arc and its distribution along the Y direction 
can be determined by Eq. (8).
Fig. 6 shows the distribution of unit pressure 
along Y direction on the contact arc in a single cold-
rolling process. It can be seen that the distribution 
of unit pressure is not uniform. As the roller moves 
towards point E’ (Δh becomes smaller), the unit 
pressure on the contact arc decreases, which is due 
to the decrease of the reduction and the length of the 
contact arc. The unit pressure on the contact arc is 
small in the whole cold roll-beating process because 
the length and reduction of the contact arc are very 
small.
Fig. 6.  Distribution of unit pressure on contact arc: 
1 	Δh	=	0.3 1	mm;	2	Δh	=	0.25	mm;	3	Δh	=	0. 1 9	mm;	
4Δh	=	0. 12	mm;	5	Δh	=	0.06	mm
The unit pressure in the incomplete 
deformation zone should be calculated according to 
the parameters of the theoretical deformation zone. 
Although incomplete, the force condition of any 
differential unit in the deformation area is still as 
shown in Fig. 3.
According to the unit pressure obtained with Eq. 
(8), the deformation force of each cold-rolling process 
is calculated, so as to determine the deformation force 
and its changes during one cold roll-beating process.
The deformation force in each cold-rolling 
process can be decomposed into radial force 
and tangential force (i.e., the component of the 
deformation force along Y and Z directions).
It can be seen from Fig. 3 that the radial force can 
be determined by the following equation as
 
Fb np
dz
bn t
dz
yz
Z
Z
z
z
z
Z
Z
z
z
B
A
B
A




cos
cos
cos
sin
cos
sin








,, (9)

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 409-421
414 Ma, Q. – Zhang, X.
where, b is the average width of the deformation 
zone, n
σ
 is the stress state coefficient, Z
A
 and Z
B
 are 
the Z-axis coordinates of points A and B in Fig. 3.
Since tan( φ
z
) = dy/dz and t
z
 = fp
z
, then
 Fb np dz fp dy
yz z
Z
Z
B
A
 


 (c os ). sin (10)
Substitute Eqs. (5), (7) and (8) into Eq. (10), and 
the radial force can be expressed as
 
Fb n
l
h
f
hh
y
y
a
h
hh
 











 
 











2
3
1






coss in
 



	














1 dy . (11)
After the integration, we can obtain the following 
equation as
 
Fb n
l
h
f
h
hh
y
h
y
a
 







 






















2
3 

coss in
. (12)
The analytical formula of tangential force can 
also be deduced as
 
Fb n
l
h
f
h
hh
y
h
z
a
 







 






















2
3
coss in
.


 (13)
Therefore, the parameters of b , n
σ
 and σ
φ
 must 
be determined when calculating the deformation 
forces.
4  DETERMINATION OF THE PARAMETERS 
4.1  Determination of Average Width of the Deformation 
Zone
Considering the symmetry of the profile of the roller, 
the total pressure on the roller should point to the axis 
of the roller. Therefore, in the calculation, the width of 
the deformation zone b is taken as the width of the 
projection area along the total pressure direction of the 
actual deformation zone, so it is determined by the 
geometric relationship shown in Fig. 2a and the profile 
curve of the roller.
4.2 Determination of Stress State Coefficient n
σ
The stress state coefficient is affected by the size of 
the roller, external friction, external zone metal and 
tension, and can be expressed as
 nn nn n
  
 '' '' '' , (14)
where n
β
 is the stress state coefficient considering 
the influence of workpiece width, n'
σ
 is the influence 
coefficient of external friction, n'''
σ
 is the external 
zone influence coefficient, n'''
σ
 is the tension influence 
coefficient.
A cold roll-beating process is regarded as an 
infinite number of cold-rolling processes, while for 
small module spline, if the friction coefficient on 
contact arc is constant, the influence of n
β
, n'
σ
 and n'''
σ
 
can be ignored.
Since the length of contact arc and the reduction 
are very small, which is similar to thick workpiece 
rolling, considering the influence of the external 
zone on the average unit pressure, the external zone 
influence coefficient can be expressed as
 n
l
h
a
''








04 .
. (15)
4.3  Determination of Actual Deformation Resistance of 
Metal
The actual deformation resistance is affected by the 
nature of metal, temperature, deformation degree 
and deformation speed. Several scholars [18] to [21] 
have studied the mathematical models of deformation 
resistance of various alloys in accordance with the 
characteristics of Chinese materials, which can be 
used to calculate the deformation force of cold roll-
beating.
Studies show that the effect of strain rate on 
deformation resistance is relatively small at low 
temperature [18] and [19], so the effect of deformation 
rate is not considered in the calculation.
5  VERIFICATION OF THE DISCRETE ANALYTICAL MODEL
The accuracy of the discrete analytical method 
is verified by the finite element method and the 
experimental results. Because the effect of workpiece 
indexing is very small, the cold roll-beating forming 
force of rack is measured in the experiment.

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 409-421
415 Research on an Analytical Method for the Forming Force of External Spline Cold Roll-Beating 
5.1  Design of the Roller
It is more complicated to make the roller strictly 
according to the involute tooth profile, and the tooth 
shape does not affect the verification. To reduce the 
experimental cost, the roller with the tooth profile size 
as shown in Fig. 7 is made, and the maximum outer 
circle radius is R = 24 mm. To conform to the cold 
roll-beating process of rack, the roller is designed 
with three tooth profiles. After cold roll-beating, two 
complete teeth and three tooth spaces are formed on 
the workpiece. 
Fig. 7.  a)	3D	mo del	o f	the	r o ller ,	and	b)	its	pr o f ile	size
5.2  Establishment of Finite Element Model
In this paper, the discrete analytical model is used to 
calculate the forming force of a single tooth roller. 
Although the roller shown in Fig. 7 has three teeth, the 
parameters of the three deformation regions are the 
same, so the results of the discrete analytical model 
can be expanded three times to compare with the 
experimental results. 
Consistent with the discrete analytical model, 
the roller is also modelled according to the shape of 
a single tooth, but the tooth shape is modified to the 
size shown in Fig. 7, and the tooth groove size on the 
workpiece is also modified accordingly. Import the 
3D models into DEFORM-3D preprocessor, and the 
symmetry plane of the tooth groove is X = 0, as shown 
in Fig. 8.
Fig. 8.  The finite element models of the roller and workpiece
5.3  Reformation of Experimental Equipment
In this paper, a horizontal milling machine is reformed, 
and the roller is installed on the spindle, as shown in 
Fig. 9. The spindle motor power is 7.5 kW, the spindle 
speed is 30 r/min to 1500 r/min, and the feed motor 
power is 1.5 kW. The radial and tangential forces of 
cold roll-beating are measured with a PCB261A03 
tri-axial forces sensor, which is produced by PCB 
Piezotronics INC in the United States. 
Fig. 9.  Cold roll-beating experimental equipment;  
1 machine tool spindle, 2 workpiece,  
3	triaxial	f o rces	senso r ,	and	4	the	r o ller
5.4  Selection of Workpiece Material
The difference of materials only affects the actual 
deformation resistance of metals but does not affect 
the verification of the discrete analytical model. 
Because the horizontal milling machine is not 
specifically defined as cold roll-beating equipment, 
the stability and the power are insufficient, so the 
workpiece material is selected as copper with good 
ductility and small deformation resistance.
6  RESULTS AND DISCUSSION
6.1  Measurement of Cold Roll-Beating Forming Force
If the cold roll-beating process with the feed of 
2 mm/r is tested, the nearest speed should be selected 
according to the speed ratio of the horizontal milling 
machine gearbox, and it should also meet the 
load requirements of the motor. The experimental 
parameters are as follows: the cold roll-beating 
depth is 2.5 mm, the roller rotation speed is 25.13 
rad/s (240 r/min), and the workpiece feed speed is  

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 409-421
416 Ma, Q. – Zhang, X.
6.2 Finite Element Simulation of Cold Roll-Beating  
6.2.1 Preprocessing of Finite Element Models
The parameters in the DEFORM-3D preprocessor are 
set as follows:
1. Without considering the influence of deformation 
of the roller, the roller is considered as a rigid 
body, the workpiece is considered as a plastic 
body. The material of workpiece is set to BRASS-
CDA-110 in the DEFORM-3D material library, 
and is considered with a constant property in 
room temperature (20 °C);
2. A ring region with thickness of 5 mm of 
workpiece is refined with density ratio of 2, as 
shown in Fig. 12;
Fig. 12.  Preprocessing of finite element models
3. The element number is 39192, and minimum 
element size is about 0.6574 mm;
4. Without considering the rotation of the roller, the 
roller rotates around the spindle with an angular 
velocity of 25.13 rad/s, and the feed speed of the 
workpiece is 476 mm/min (feed rate 1.983 mm/r); 
5. In the process of cold roll-beating, it can be 
considered as pure rolling between the roller and 
the workpiece. To simplify the calculation, the 
friction coefficient between the roller and the 
workpiece is set to 0;
6. The target volume option is set to active in FEM 
+ meshing.
6.2.2 Finite Element Simulation Results
In the DEFORM-3D postprocessor, the distribution 
of the effective stress and the contact state are 
476 mm/min (feed rate 1.983 mm/r). Samples formed 
by cold roll-beating are shown in Fig. 10.
Change curves of radial and tangential forces 
measured in the experiment are shown in Fig. 11. 
As shown in Fig. 11a, the radial force increases 
continuously in the beat-in stage and remains constant 
in the stable stage (about 31.8 kN) and decreases 
continuously in the beat-out stage. As shown in Fig. 
11b, the maximum tangential force is only about 2 kN, 
which is about 6.3 % of the maximum radial force.
Fig. 10.  Samples formed by cold roll-beating
Fig. 11.  Cold roll-beating forming forces measured in the 
experiment; a) the radial force; b) the tangential force

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 409-421
417 Research on an Analytical Method for the Forming Force of External Spline Cold Roll-Beating 
given, as shown in Fig. 13, where the green nodes 
represent the contact nodes. The distribution area of 
the effective stress and contact nodes moves with the 
roller, indicating that the position of the deformation 
zone is also moving constantly. The distribution 
area of contact nodes and the variation of stress are 
increasing and then decreasing, which indicates that 
the parameters of metal deformation zone are also 
changing. The results in Fig. 13 validate the analysis 
of the characteristics of the deformation zone of cold 
roll-beating in this paper.
Fig. 14a shows the deformation of the tooth 
groove. Compared with previous deformation, the 
width of the tooth groove significantly increases, and 
the increments along the workpiece feed direction (Z 
direction) also show a trend of increasing and then 
decreasing, which indicates the trend of variation in 
the width of the deformation zone. Fig. 14b shows 
the variation curves of forming force; it can be seen 
that the maximum radial force is about 10.9 kN, 
while the maximum tangential force is about 0.8 kN, 
which is about 7.3 % of the maximum radial force, 
which is basically consistent with the experimental 
measurement results. The reason that the tangential 
force is very small is that the deformation on both 
sides of the tooth groove is distributed symmetrically 
along the centre of the tooth groove, as shown in Fig. 
14a. Therefore, the tangential force on both sides of 
the tooth groove is equal in magnitude and opposite 
in direction.
6.3  Comparison of Cold Roll-Beating Forming Force
Since the radial force of cold roll-beating is much 
larger than the tangential force, only the radial force 
is considered when calculating the forming force 
with the discrete analytical model. At the stable stage 
in Fig. 11, the radial force variation curve of a cold 
roll-beating process is obtained and compared with 
the discrete analytical model calculation results and 
the finite element simulation results, as shown in Fig. 
15. In the experiment, the roller contains three teeth, 
so the calculation results and simulation results are 
expanded by three times for comparison.
Fig. 13.  Effective stress distribution and contact state on the workpiece

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 409-421
418 Ma, Q. – Zhang, X.
It can be seen from Fig. 15 that the maximum 
value of radial force predicted by the discrete 
analytical model is 30.5 kN, which is about 2.2 kN 
different from the simulation results (about 32.7 kN) 
with an error of 7 %, and only about 1.3 kN different 
from the experimental measurement results at the 
stable stage (about 31.8 kN) with an error of 4 %.
The change trend of radial force predicted by the 
discrete analytical model is approximately the same as 
that of simulation and experimental results, but there 
is an error when the maximum radial force is reached.
The duration of cold roll-beating process 
predicted by the discrete analytical model is in good 
agreement with the simulation results but significantly 
less than the experimental results. 
Fig. 14.  Deformation of the tooth groove and the forming force
Fig. 15.  Comparison of cold roll-beating radial force calculated by different methods

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 409-421
419 Research on an Analytical Method for the Forming Force of External Spline Cold Roll-Beating 
6.4  Reasons for the Longer Duration of Cold Roll-Beating 
Measured in Experiment
The action process of the roller on the workpiece is 
similar to the plastic problem of the wedge or cylinder 
sliding over the smooth surface. Research on this kind 
of problem has shown that the slip line field in the 
metal deformation zone should be as shown in Fig. 16 
(V
r
 is the instantaneous linear velocity of the roller) 
[22] to [24]. As the cold roll-beating progresses, the 
friction coefficient increases [25], and the metal bulge 
in front of the roller, so the time of cold roll-beating 
is longer.
Similar results can be seen in the finite element 
simulation. The workpiece is cut by X = 0 plane, and 
the deformation of the metal at the bottom of the 
groove is obtained, as shown in Fig. 17. The metal 
bulge in front of the roller is obvious, and a protrusion 
is formed on the beat-out face, and a collapse is 
formed on the surface. Since the tooth groove formed 
by the last cold roll-beating has been established on 
the 3D model of the workpiece, the protrusion and 
the collapse are not obvious after simulation, and the 
duration of cold roll-beating process is not increased. 
However, in the experiment, the tooth groove is 
formed after multiple cold roll-beating, so the 
protrusion is more obvious, and the duration is longer.
At the same time, the elastic recovery of metal, 
the insufficient stiffness and the lack of power of 
machine tools will also increase the time of cold roll-
beating. 
Although there is some error in the time of 
cold roll-beating process calculated by the discrete 
analytical method, it does not affect the determination 
of cold roll-beating force and energy parameters. 
Cutting fluid is used in cold roll-beating, and the 
adhesion of metal can be controlled, and the stability 
and power of the special cold roll-beating machine 
tools are more appropriate, so the theoretical value 
will be closer to the true value.
6.5  Prospects for Future Research
Although the discrete analytical model can predict the 
cold roll-beating forming force quickly and accurately, 
it cannot do so with some factors, such as the elastic 
deformation of the roller, the bulge and collapse of 
the metal material. The finite element simulation 
Fig. 16.  Slip line field in the metal deformation zone of cold roll-beating
Fig. 17.  The bulge, collapse, and protrusion of metal

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 409-421
420 Ma, Q. – Zhang, X.
consumes significant simulation time, but it provides 
the movement process of the metal deformation zone 
more intuitively, and shows the shape of the metal 
during and after deformation more completely.
In the future, the research work on the forming 
force of cold roll-beating should be combined with 
the finite element simulation method to simulate the 
whole cold roll-beating process to form a complete 
tooth groove, study the influence of the metal shape 
after deformation on the forming force, study the 
influence of elastic deformation of roller, and further 
improve the discrete analytical model.
7  CONCLUSIONS
1. The characteristics of deformation zone in cold 
roll-beating are analysed. The position and 
parameters of the deformation zone are constantly 
changing. The contact arc length and reduction 
are very small, and there is an incomplete 
deformation zone in the initial stage of cold roll-
beating.
2. The discrete analytical model of unit pressure 
and deformation force in cold roll-beating 
is established. A cold roll-beating process is 
discretized into an infinite number of cold-rolling 
processes. The unit pressure of each cold-rolling 
process is calculated. The deformation force of 
cold roll-beating is calculated.
3. The change curves of cold roll-beating forming 
force are obtained by finite element simulation 
and experimental measurement. The results show 
that the change trend of cold roll-beating forming 
force predicted by the discrete analytical model 
is basically consistent with that of finite element 
simulation and experimental measurement. The 
maximum calculation error compared with the 
simulation and experiment results is about 7 
% and 4 % respectively. The discrete analytical 
method is accurate in predicting the cold roll-
beating forming force.
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