34th INTERNATIONAL CONFERENCE ON
PRODUCTION ENGINEERING
28. - 30. September 2011, Niš, Serbia
University of Niš, Faculty of Mechanical Engineering
EXPERIMENTAL ANALYSIS AND MATHEMATICAL MODELLING OF THE
ROLLING FORCE
1
Milan JURKOVIĆ1, Zoran JURKOVIĆ2, Asim JUŠIĆ1, Vesna MANDIĆ3
Technical faculty, University of Bihać, Irfana Ljubijankića bb., Bihać, Bosnia and Herzegovina
2
Faculty of Engineering, University of Rijeka, Vukovarska 58, Rijeka, Croatia
3
Mechanical Engineering Faculty, University of Kragujevac, S. Janjić 6, Kragujevac, Serbia
mi.jurkovic@gmail.com, zoran.jurkovic@riteh.hr, asimjusic@hotmail.com, mandic@kg.ac.rs
Abstract: This paper deals with experimental determination of rolling force, mathematical modelling,
analytical calculation and verification of rolling force. The results of performed research indicate that
the mathematical-experimental modelling can be successfully used to define the rolling force and the
technological parameters of the rolling process. In this paper the mathematical model of rolling force F
= f (Δh, ε, δ, k) has been defined. Main achievements are: original experimental identification of rolling
force, original of measuring sensor for measurement of rolling force and mathematical model for
defining the rolling force.
Key words: experimental analysis, mathematical modelling, rolling force, process, model
1. INTRODUCTION
2.1. Measuring system
The objectives of this paper are to define mathematical
model of rolling force in dependence of the basic
influential parameters. This paper is based on
experimental results for the rolling force on the line for
rolling with three machining modules, where the
measurement equipment and measurement system are
used for measuring the rolling force. The initial data are:
cold rolling strip material (DIN St 14), initial thickness h0
= 2,5 mm and width b0 = 350 mm. Diameters of rolls on
machinig modules are: D1 =207,9 mm; D2 = 175,7 mm
and D3 = 157,9 mm. Parameters of the rolling process are:
absolute deformation Δh, relative degree of deformation
ε, coefficient δ and yielding of materials k.
In this paper has been presented plan of the experiments
and its results for the rolling force, procedure of
modelling, testing adequancy of the model, analysis result
of the modelling and experiment, comparison of
experimental force and modelling force, and at the end
simulation of the rolling force [1,2,3,4,5,6,7,8,9].
2. EXPERIMENTAL
ROLLING FORCE
RESEARCH
By means of special measuring converter with strain
gauge (Fig. 2) was realized experimental test mechanical
load of rolling machine in process of cold rolling with
measuring the rolling forces (Fig. 3).
Fig. 1. Frame of rolling machine (1,2), sensor force (3),
rolled section-sheet metal (4), thread spindle (5), roller
cradle (6)
OF
The experimental analyses of rolling process are made in
the aim of measuring the rolling forces which are used for
modelling and simulation of the rolling process (Fig. 1).
Original experiments have carried out in the industrial
conditions (plant “Krajinametal“ Bihać) under macro
project titled: Research and development of flexible
rolling machines (Bosnia and Herzegovina) and project
titled: Modelling and Simulation Processes by Using
Genetic and Stochastic Algorithm (Croatia).
Fig. 2. Converter with strain gauge (measuring element)
5. MATHEMATICAL
ROLLING FORCE
Fig. 3. Measuring system
3. ANALYSIS OF THE INPUT AND OUTPUT
VARIABLES (The choice of the process
parameters and block scheme)
The force modelling has been performed for the four
variables of rolling process (Fig. 4.). The parameters of
the rolling process are defined in the following way:
- input variables: absolute deformation Δh (mm),
relative degree of deformation ε, coefficient δ
=2µ1/Δh, where l= hR , R=D/2 and yield stress
of material k (N/mm2).
- output value – rolling force Fv (kN).
Function of the process state: Fv = f (Δh, ε, δ, k). A
graphic presentation of the input – output values is given
in Fig. 4.
MODELLING
OF
For chosing the type of the mathematical model, there is
no generally applicable rule, that means, that for each
investigated process or system have to be chosen a model
and examine its accuracy and adequacy in relation to the
real process. On the basis of performed experiment and
regression analysis the statistical model is determined by
means of real regression coefficients bi , bii , bim , bimk so
that mathematical model obtains the form:
k
k
i 0
1i m
Yˆ   bi X i   bim X i X m 
N
b0 
1
N
bi 
1
N  n0
X
j 1
oj
N
1
bim 
N  n0
j 1
ij
j 1
i
m
Xk
Y j , za i  1,2,..., k
N
X
i
(1)
X oj  1
Yj ,
X
b X X
1i mk
ij
(2)
X mj Y j , za 1  i m  k
Taking in attention only significant coefficients of
regression, the mathematical model of force has the form:
Y = F= 5113,75 + 394,43 X1 + 406,81 X2 +368,56 X3
(3)
+ 600,43 X4 – 118,06 X1X4 -193,81 X2X3X4 + 91,43
X1X2X3X4
Fig. 4. Scheme for rolling force modelling (input – output
values)
The coded and physical values of influential parameters
are presented in Table 1.
Table 1. Physical xji and coded Xji values
Coded and physical
input values
0,5
1,0
1,5
Influental factors
x1= Δh (mm)
Physical
input
values
x2 = ε
0,20
0,40
0,60
2,0
3,5
5,0
x4 = k (N/mm )
250
375
500
Xi
-1
0
1
x 3= δ
2
Coded
input
values
4. EXPERIMENTAL DESIGN AND RESULTS
The experiments were perform by using plan of
experiments with N = 2k + n0 = 24 + 4 = 20 tests (Table 2).
The design matrix meets the criteria of orthogonality,
symmetry and normativity [1].
The coefficient of multiple regression R=0,982 shows
very good correlation between varying Xi and rolling
force. Encoding the mathematical model (3) is obtained
the physical mathematical model of the rolling force in
the form of:
Y  Fv  992,35  2736,62h  2890,73  264,70
(4)
 3,15k  1353,85h  199,38h  3,40h k
 2075,41  15,46k  1,74  k  6,25 k  1,08h k .
6. ANALYSIS
OF
MODELLING
EXPERIMENT RESULTS
AND
6.1. Comparison of the experimental force and
modelling force
Rolling force obtained results by the experimental
investigation and mathematical modelling were presented
on Figure 5. Results comparisons were shown adequacy
of the model (4) and its possible using for rolling force
prediction. The obtained experimental results (Table 2)
and results obtained by mathematical model (4) show
very good correlation between each other (Figure 5), also
presented by multiple regression coefficient R = 0,982.
According to that, rolling force model (4) describes
accurately enough experimental results within domain of
experiment.
6.2. Simulation of rolling force
3D simulation model (4) was shown, on Figure 6,
dependence of the rolling force in relation to relative
degree of deformation ε and absolute deformation Δh. In
the same time other input parameters coefficient δ and
yield stress of material k were constant on coded level -1
shown in Table 1.
Table 2. Experimental results
trial run Nj
Experimental forces Fj (kN)
trial run Nj
Experimental forces Fj (kN)
1
3370
11
5795
2
4460
12
6140
3
3880
13
5880
4
4900
14
5924
Fv experimental
5
3780
15
5828
6
4825
16
6730
7
5245
17
4790
8
6190
18
4868
9
4520
19
4890
10
5440
20
4820
Fv model (4)
7000
Rolling force Fv (kN)
6500
6000
5500
5245
5000
4825
4821,58
4460
5204,92
4678,65
3780
4000
3000
4900
4508,82
4500
3500
6730
6636,64
6140
5880 5987,76 5828
5795 6135,82
5924
5440
5785,48
5680,37
5663,30
5113,75
4622,10 5202,03
4842
4520
6190
6094,21
3880
3822,55
3370
3185,31
1
2
3
3790,48
4
5
6
7
8
9
10
11
12
13
14
15
16
17
number of experimental trial Nj
Fig. 5. Comparison of the experimental and modelling force results
4800
4800
4600
4600
rolling force F(kN)
4400
4400
4200
4000
4200
3800
4000
3600
3400
3800
3200
3000
0.6
3600
0.55
0.5
0.45
0.4
0.35

0.3
0.25
e
0.2
0.5
0.6
0.7
0.8
0.9
1
1.1
h (mm)
1.2
1.3
1.4
1.5
3400
3200
h (mm)
Fig. 6. Response surface for rolling force model (4) depends on relative degree of deformation and absolute
deformation with constant values of the coefficient δ = 2 and yield stress of material k = 250 N/mm2
7. CONCLUSION
The obtained model of the rolling force enough correctly
and reliable describes forming force, that are confirmed
with confidence level P=0.95 and the coefficient of
multiple regression R=0.982. On this presented way
obtained model is useful to reduce the production cost and
achieve desired product quality.
[4]
[5]
[6]
[7]
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[3]
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