1. Introduction
Grinding is an operation applied in almost every type of manufacturing process. The grinding process is extensively used during finishing operations for discrete components [
1,
2]. When requiring precise tolerances and smooth surfaces for the final machining of components, grinding is expected to be an effective processing method. However, the grinding process is a costly procedure [
3]. In industries, it can account for about 20–25% of the expenditure on machining operations [
3]. Thus, this process should be used at optimal conditions.
Previous research works have studied the problem of optimizing technology parameters in order to enhance quality and productivity in grinding processes. For instance, Peters and Aerens [
4] analyzed process parameters to minimize the total grinding time in an external cylindrical grinding process without intermediate dressing. Since then, this field has been continuously advanced by more constraints [
5]. The relevance to the optimization of grinding and dressing parameters has been proposed for maximizing the material removal rate [
3], minimizing the grinding time [
6], as well as minimizing the dressing and grinding costs [
7]. These works have implemented optimization problems with different grinding methods such as external cylindrical grinding [
4,
8,
9], surface grinding [
10,
11,
12,
13,
14,
15], and internal grinding [
16]. These optimization problems have addressed not only traditional grinding machines [
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16] but also CNC milling machines [
17]. They have established different objective functions. For instance, in a recent study [
9], Pi et al. presented a cost optimization method for internal cylindrical grinding operations. In this work, based on the calculation of the
De.op value, the grinding cost is significantly reduced. Regarding surface grinding in recent years, several research works have focused on optimizing surface grinding process parameters [
10] in order to obtain a high accuracy of the surface finishing process through the analysis of grinding parameters [
11,
12,
13]. A proposed methodology was presented for computing optimal machining parameters in order to achieve a high production rate and good surface quality of machine components [
14]. Recently, an experimental study for surface grinding was proposed in [
15]. In this study, the
De.op value is determined from the experiment.
This paper is concerned with the optimization of the replaced grinding wheel diameter in a surface grinding operation for 9CrSi steel material based on the formulation of the manufacturing cost per piece. This work was continuously developed by the proposed formula [
9], which has not been extended to surface grinding and has not yet been employed to carefully evaluate grinding process parameters and cost components. Therefore, in this paper, the
De.op value is determined by minimizing the cost function. Moreover, the grinding technology parameters, such as the initial grinding wheel diameter, the total dressing depth, the radial grinding wheel wear per dress, and the wheel life are given to evaluate the effects of these parameters on the optimal exchange grinding wheel diameter. Additionally, the influence of the cost components on the
De.op value is also considered. To identify the effect of these parameters, an experiment is built, and a computational program is also established to carry out the experiment. The research results will help manufacturers solve the problem of selecting conditions for initial technological parameters since they allow manufacturers to determine and set up optimal parameters of pre-machining grinding conditions to increase the economic and technical effectiveness of the grinding process.
2. Methodology
In this section, the cost analysis of the surface grinding process is investigated. Based on the findings, the relationship between the cost of the surface grinding process and the replaced grinding wheel diameter is studied. In addition, the relationship between the replaced grinding wheel diameter and the technological parameters in the grinding process is analyzed. This is the theoretical basis on which we determine the
De.op value to minimize the cost of the surface grinding process. For the surface grinding process, the grinding cost per part can be computed as follows:
where:
ts is the manufacturing time (h) which will be discussed in more detail following;
Cmh is the machine tool hourly rate (USD/h) including wages, cost of maintenance etc.;
Cg is the grinding wheel cost per workpiece (USD/workpiece). Cg can be expressed as follows:
where
Cg,p is the grinding wheel cost per piece (USD/piece) and
np,w is the total number of workpieces ground by a grinding wheel.
np,w can be determined as follows [
18]:
where
is the initial grinding wheel diameter (mm);
is the replaced grinding wheel diameter (mm);
Wpd is the radial grinding wheel wear per dress (mm/dress);
is the total depth of dressing cut (mm);
np,d is the number of workpieces per dress.
np,d can be determined as follows:
where
is the longevity of the grinding wheel (h) and
is the grinding time (h). In surface grinding, the grinding time can be determined as follows:
In Equation (5),
is the calculated grinding length (mm);
where
is the length of the workpieces (mm);
is the calculated grinding width (mm);
, where
is the width of the workpieces (mm) (see
Figure 1) and
is the grinding wheel width (mm);
is the total depth of cut (mm);
is the speed of the workbench (m/s);
is the work feed rate (mm/min);
is the downfeed (mm/pass) (see
Figure 1);
is the number of workpieces per grinding time.
To determine
tc by Equation (5), several parameters,
, and
, are determined as follows. With grinding carbon steel, alloy steel, and brass, the work speed
can be computed from the data in [
15], which depends on the Rockwell hardness of workpiece
HRC. Thus,
can be written as the following regression equation:
Based on the required roughness grade number
and the grinding wheel width
, the work feed rate
can be computed from the following regression equation [
18]:
The downfeed
can be calculated as follows [
18]:
where
is the tabulated downfeed (mm/pass). When grinding tool steel, the tabulated downfeed
can be determined as follows [
18]:
where
is the total depth of cut and
is the work feed rate.
In Equation (8),
, and
are coefficients;
depends on the workpiece material and required tolerance grade
. When grinding tool steel,
can be calculated as follows [
15]:
The coefficient
is determined as follows [
15]:
where
is the grinding wheel diameter and
is the density of the workpiece loaded on the machine table. The value of the coefficient
depends on the grinding machine age;
if the age is less than 10 years,
if the age ranges from 10 to 20 years, and
if the age is more than 20 years [
15].
In Equation (1),
can be identified as follows:
Here,
, and
are the grinding time (as shown in Equation (5)), the time for loading and unloading workpiece, the spark-out time, the dressing time per piece, and the time for changing a grinding wheel per workpiece, respectively.
, and
can be expressed as follows:
Substituting Equation (3) into Equation (15),
can be written as follows:
Based on the formulation of the above manufacturing cost per piece, it is indicated that the replaced grinding wheel diameter (
De) affects the cost of the surface grinding process. For a certain technological condition where
;
;
;
;
;
;
, the relationship between the grinding cost and the
De value (calculated by Equation (1)) is built as shown in
Figure 2. It is observed that the
De value strongly affects the cost of grinding operations. When the
De value increases from 200 to 475 mm, the grinding cost decreases from 0.0014 to 0.001 USD/part. However, as the
De value increases from 475 to 500 mm, the grinding cost grows rapidly from 0.001 to 0.0016 USD/part. In particular, the grinding cost is minimum when the
De value equals an optimum value of
, which is much larger than the conventional
De value (in this case about 200 to 250 mm).
From the above analyses, the
value can be determined by minimizing the grinding cost per piece
. Thus, the cost function of the surface grinding process can be expressed as follows:
with the constraint as:
Besides, the
De,op value depends on various technology factors. From the cost analysis of the surface grinding process, it is revealed that there are eight main factors affecting the
De,op value. The eight main factors include the initial grinding wheel diameter
D0, the grinding wheel width
Wgw, the total depth of dressing cut
aed, the Rockwell hardness of the workpiece
HRC, the wheel life
Tw, the radial grinding wheel wear per dress
Wpd, the machine tool hourly rate
Cmh, and the grinding wheel cost
Cgw. Therefore, the function of the optimum replaced grinding wheel diameter can be presented as follows:
4. Results and Discussions
Based on the data reported in
Table 2, the influence of the factors on the
De,op value was determined as shown in
Figure 4. It can be clearly seen that
D0 value has the largest effect on the
De,op value (the left side in
Figure 3). For the other parameters such as
aed,
Tw,
Wed,
Cmh, and
Cgw, the effect of these parameters on the
De,op value is much smaller than that of the
D0 factor. In addition, the
De,op value is not affected by the
Wgw and
HRC parameters.
The influence of the factors can be seen more clearly in
Figure 5, which shows the Pareto chart of the standardized effects for determining the magnitude and the importance of the influence on the
De,op value. For the response model, the parameters are statistically significant at the 0.05 level. As presented in this figure, the magnitude of the influence on the grinding process parameters is arranged from the lowest value to the highest value. The largest influence on the optimum diameter belongs to the initial grinding wheel
D0 (factor A in
Figure 5). The influence is gradually reduced in the sequence of several grinding process parameters, such as the grinding wheel cost
Cg (factor H in
Figure 5), the machine tool hourly rate
Cmh (factor G in
Figure 5), the wheel life
Tw (factor E in
Figure 5), and the total depth of dressing cut
aed (factor C in
Figure 5). The factor with the smallest effect is the radial grinding wheel wear per dress
Wpd (factor F in
Figure 5). Significantly, as noted above, the optimum exchanged diameter is not affected by the grinding wheel width
Wgw (factor B in
Figure 5) and the Rockwell hardness of the workpiece
HRC (factor D in
Figure 5).
However, the Pareto chart (as shown in
Figure 5) merely displays the magnitude of the effects. Thus, the normal plot of the standardized effects was established as shown in
Figure 6 to evaluate the effects of increasing or decreasing the response. The distribution of the standardized effects to most of the factors is close to the reference line (red line in
Figure 6). The positive effects of the 10 factors including
D0,
Tw,
Cmh and the interactions AE, AG, GH, CG, CE, CF, EH increase the
De,op value when these factors change from a low value to a high value. Meanwhile, the other parameters such as
aed,
Cg,
Wpd and the interactions AH, EG, AC, CH have negative effects. When they alter from a low value to a high value, the
De,op value declines. In addition, the initial grinding wheel diameter
D0 has the largest magnitude compared to other factors. Therefore, the optimum replaced grinding wheel diameter is tremendously influenced by the initial grinding wheel diameter
D0.
Figure 7 depicts the estimated effects and coefficients for the
De,op value. It can be recognized that those parameters including
D0,
aed,
Tw,
Wpd,
Cmh,
Cg and the interactions between
D0 and
aed, D0 and
Tw, D0 and
Cmh, D0 and
Cgw, aed and
Tw, aed and
Wpd, aed and
Cmh, aed and
Cgw, Tw and
Cgw, Tw and
Cmh, Wpd and
Cgw, Cmh and
Cgw have significant effects on a response because their
P-values are lower than 0.05. Remarkably, the
D0 factor has the largest effect on the
De,op value in comparison with the other factors.
Consequently, after carrying out experiments and collecting results, the data were analyzed and processed by using Minitab 18 software. Based on that, the mathematical model, which shows the relationship between the
De,op value and the significant effect parameters, is expressed as follows:
In order to evaluate the appropriateness of the formula (20), the
De,op value computed by Equation (20) was compared with the experimental result found in [
15]. The values of the factors used for the optimum diameter calculation in Equation (20) and for the experimental design were the same, i.e.,
D0 = 300 mm;
aed = 0.115 mm;
Tw = 22.5 min;
Wpd = 0.02 mm/dress;
Cmh = 5 USD/h;
Cg = 25 USD/piece. Accordingly, the
De,op value was 266.86 mm according to Equation (20) and was 265 mm as determined by the experiment. The difference between the formulation and the experiment can be expressed as follows:
The results from Equation (21) show that the De,op value is calculated by the Equation (20) in accordance with the optimum value obtained from the experiment. Thus, the proposed method can be used to determine the De,op value in surface grinding operations for 9CrSi steel material.