Optimization of Replaced Grinding Wheel Diameter for Surface Grinding Based on a Cost Analysis

Based on a cost analysis, a method of identifying and predicting optimum replaced grinding wheel diameter (De.op) in a surface grinding operation for 9CrSi steel material was developed in this study. The De.op value was determined by minimizing the cost function. An experimental design was set up, and a computational program was developed to perform the experiment in order to calculate the De.op value. Furthermore, the impact of the grinding process parameters such as the initial grinding wheel diameter, the grinding wheel width, the total dressing depth, the Rockwell hardness of the workpiece, the radial grinding wheel wear per dress, and the wheel life on the De.op value were investigated. Moreover, the impacts of the cost components such as the machine tool hourly rate and the grinding wheel cost on the De.op value were given. Based on that, a mathematical model was proposed to determine the De.op value. The predicted De.op value was also verified by an experiment. The obtained result shows that the difference between the experimental De.op value and the predicted De.op value is within 1.7%, indicating that the mathematical model proposed in the study is reliable.


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.

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 D e.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: t s is the manufacturing time (h) which will be discussed in more detail following; C mh is the machine tool hourly rate (USD/h) including wages, cost of maintenance etc.; C g is the grinding wheel cost per workpiece (USD/workpiece). C g can be expressed as follows: C g = C g,p /n p,w where C g,p is the grinding wheel cost per piece (USD/piece) and n p,w is the total number of workpieces ground by a grinding wheel. n p,w can be determined as follows [18]: n p,w = (D 0 − D e )n p,d / 2(W pd + a ed ) (3) where D 0 is the initial grinding wheel diameter (mm); D e is the replaced grinding wheel diameter (mm); W pd is the radial grinding wheel wear per dress (mm/dress); a ed is the total depth of dressing cut (mm); n p,d is the number of workpieces per dress. n p,d can be determined as follows: where T w is the longevity of the grinding wheel (h) and t c is the grinding time (h). In surface grinding, the grinding time can be determined as follows: In Equation (5), l c is the calculated grinding length (mm); l c = l w + (20 . . . 30) where l w is the length of the workpieces (mm); w c is the calculated grinding width (mm); w c = w w + w gw + 5, where w w is the width of the workpieces (mm) (see Figure 1) and w gw is the grinding wheel width (mm); a e,tot is the total depth of cut (mm); v w is the speed of the workbench (m/s); v f is the work feed rate (mm/min); f d is the downfeed (mm/pass) (see Figure 1); N t is the number of workpieces per grinding time.
where is the longevity of the grinding wheel (h) and is the grinding time (h). In surface 93 grinding, the grinding time can be determined as follows: In Equation (5), is the calculated grinding length (mm); = + (20 … 30) where is the length of the workpieces (mm); is the calculated grinding width (mm);

96
where is the width of the workpieces (mm) (see Figure 1) and is the grinding wheel width 97 (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  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 104 in [15], which depends on the Rockwell hardness of workpiece HRC.
The downfeed can be calculated as follows [18]: f c c c (8) where , is the tabulated downfeed (mm/pass). When grinding tool steel, the tabulated downfeed 110 , can be determined as follows [18]: where is the total depth of cut and is the work feed rate. To determine t c by Equation (5), several parameters, v w , v f , and f d , are determined as follows. With grinding carbon steel, alloy steel, and brass, the work speed v w can be computed from the data in [15], which depends on the Rockwell hardness of workpiece HRC. Thus, v w can be written as the following regression equation Based on the required roughness grade number N Ra and the grinding wheel width w gw , the work feed rate v f can be computed from the following regression equation [18]: The downfeed f d can be calculated as follows [18]: where f d,t is the tabulated downfeed (mm/pass). When grinding tool steel, the tabulated downfeed f d,t can be determined as follows [18]: where a e,tot is the total depth of cut and v f is the work feed rate. In Equation (8), c 1 , c 2 , and c 3 are coefficients; c 1 depends on the workpiece material and required tolerance grade tg. When grinding tool steel, c 1 can be calculated as follows [15]: The coefficient c 2 is determined as follows [15]: where d s is the grinding wheel diameter and D w is the density of the workpiece loaded on the machine table. The value of the coefficient c 3 depends on the grinding machine age; c 3 = 1 if the age is less than 10 years, c 3 = 0.85 if the age ranges from 10 to 20 years, and c 3 = 0.7 if the age is more than 20 years [15]. In Equation (1), t s (h) can be identified as follows: Here, t c , t lu , t sp , t d,p , and t cw,p 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. t sp , t d,p , and t cw,p can be expressed as follows: t cw,p = t cw /n p,w .
Substituting Equation (3) into Equation (15), t cw,p 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 (D e ) affects the cost of the surface grinding process. For a certain technological condition where D 0 = 500 mm; W gw = 40 mm; z = 0.1 mm; C mh = 10 USD/h; C gw = 30 USD/piece; n s = 1450 rpm; n pd = 35; a ed = 0.1 mm; T w = 20 min; W pd = 0.02 mm/dress, the relationship between the grinding cost and the D e value (calculated by Equation (1)) is built as shown in Figure 2. It is observed that the D e value strongly affects the cost of grinding operations. When the D e value increases from 200 to 475 mm, the grinding cost decreases from 0.0014 to 0.001 USD/part. However, as the D e 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 D e value equals an optimum value of D e,op = 475 mm, which is much larger than the conventional D e value (in this case about 200 to 250 mm).
From the above analyses, the D e,op value can be determined by minimizing the grinding cost per piece C part . Thus, the cost function of the surface grinding process can be expressed as follows: with the constraint as: to 0.0016 USD/part. In particular, the grinding cost is minimum when the De value equals an optimum  Besides, the D e,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 D e,op value. The eight main factors include the initial grinding wheel diameter D 0 , the grinding wheel width W gw , the total depth of dressing cut a ed , the Rockwell hardness of the workpiece HRC, the wheel life T w , the radial grinding wheel wear per dress W pd , the machine tool hourly rate C mh , and the grinding wheel cost C gw . Therefore, the function of the optimum replaced grinding wheel diameter can be presented as follows:

Experimental Work
In this section, the eight abovementioned factors are selected to evaluate their effects on the D e,op value. Simultaneously, an experimental design with a two-level factorial design with a half fraction was set up. The values of these input factors in experimental tests are presented in Table 1. Accordingly, 128 = 2 (8−1) experiments were conducted. In addition, based on the equations in Section 2, a computational program was established to determine the D e,op value in each experiment. Various levels of the input parameters and the output response are shown in Table 2. The obtained results of the conducted experiments would be the basis to analyze the influences of these parameters on the D e,op value. Grinding experiments were performed on a surface grinder (MGK7120 × 6, Moto-Yokohama, made in Japan) by using a Vietnamese grinding wheel with a JIS code of Cn80MV1 300 × 127 × 30, as shown in Figure 3. The 9CrSi steel plates with dimensions of 100 mm × 80 mm × 30 mm were used as the work material. The specifications of the grinding conditions and grinding wheels are shown in Table 3.  Table   163 3.

168
Based on the data reported in Table 2

Results and Discussions
Based on the data reported in Table 2, the influence of the factors on the D e,op value was determined as shown in Figure 4. It can be clearly seen that D 0 value has the largest effect on the D e,op value (the left side in Figure 3). For the other parameters such as a ed , T w , W ed , C mh , and C gw , the effect of these parameters on the D e,op value is much smaller than that of the D 0 factor. In addition, the D e,op value is not affected by the W gw and HRC parameters. The influence of the factors can be seen more clearly in Figure 5, which shows the Pareto chart 176 of the standardized effects for determining the magnitude and the importance of the influence on the 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 D e,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 D 0 (factor A in Figure 5). The influence is gradually reduced in the sequence of several grinding process parameters, such as the grinding wheel cost C g (factor H in Figure 5), the machine tool hourly rate C mh (factor G in Figure 5), the wheel life T w (factor E in Figure 5), and the total depth of dressing cut a ed (factor C in Figure 5). The factor with the smallest effect is the radial grinding wheel wear per dress W pd (factor F in Figure 5). Significantly, as noted above, the optimum exchanged diameter is not affected by the grinding wheel width W gw (factor B in Figure 5) and the Rockwell hardness of the workpiece HRC (factor D in Figure 5). belongs to the initial grinding wheel D0 (factor A in Figure 5). The influence is gradually reduced in 181 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   183 5), and the total depth of dressing cut aed (factor C in Figure 5). The factor with the smallest effect is 184 the radial grinding wheel wear per dress Wpd (factor F in Figure 5). Significantly, as noted above, the 185 optimum exchanged diameter is not affected by the grinding wheel width Wgw (factor B in Figure 5) 186 and the Rockwell hardness of the workpiece HRC (factor D in Figure 5).

189
However, the Pareto chart (as shown in Figure 5) merely displays the magnitude of the effects.

190
Thus, the normal plot of the standardized effects was established as shown in Figure 6 to evaluate  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 D 0 , T w , C mh and the interactions AE, AG, GH, CG, CE, CF, EH increase the D e,op value when these factors change from a low value to a high value. Meanwhile, the other parameters such as a ed , Cg, W pd and the interactions AH, EG, AC, CH have negative effects. When they alter from a low value to a high value, the D e,op value declines. In addition, the initial grinding wheel diameter D 0 has the largest magnitude compared to other factors. Therefore, the optimum replaced grinding wheel diameter is tremendously influenced by the initial grinding wheel diameter D 0 .    Figure 7 depicts the estimated effects and coefficients for the D e,op value. It can be recognized that those parameters including D 0 , a ed , T w , W pd , C mh , C g and the interactions between D 0 and a ed , D 0 and T w , D 0 and C mh , D 0 and C gw , a ed and T w , a ed and W pd , a ed and C mh , a ed and C gw , T w and C gw , T w and C mh , W pd and C gw , C mh and C gw have significant effects on a response because their P-values are lower than 0.05. Remarkably, the D 0 factor has the largest effect on the D e,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 D e,op value and the significant effect parameters, is expressed as follows: The results from Equation (21)   In order to evaluate the appropriateness of the formula (20), the D e,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., D 0 = 300 mm; a ed = 0.115 mm; T w = 22.5 min; W pd = 0.02 mm/dress; C mh = 5 USD/h; C g = 25 USD/piece. Accordingly, the D e,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 D e,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 D e,op value in surface grinding operations for 9CrSi steel material.

Conclusions
This study developed a method to minimize the grinding cost based on determining the D e,op value in surface grinding operations for 9CrSi steel material. A mathematical model for determining the D e,op value was proposed. In addition, the influence of eight factors including D 0 , W gw , a ed , HRC, T w , W pd , C mh , and C g on the D e,op value in the surface grinding process was determined by designing and conducting a simulation experiment. From the experimental results of the study, some findings can be presented: (1) The initial grinding wheel diameter D 0 has the largest effect on the D e,op value. Meanwhile, the grinding wheel width (W gw ) and the Rockwell hardness of workpiece (HRC) do not affect the D e,op value. (2) Three factors including D 0 , T w , C mh have positive influences on the D e,op value. Meanwhile, the other parameters such as a ed , C gw , W pd have negative influences on the D e,op value. (3) For a certain set of technological parameters, i.e., D 0 = 300 mm; a ed = 0.115 mm; T w = 22.5 min; W pd = 0.02 mm/dress; C mh = 5 USD/h; C g = 25 USD/piece, the D e,op value is 266.86 mm. After carrying out an experiment with those values of the input parameters, the obtained result shows that the difference between the experimental D e.op value and the prediction D e.op value is 1.7%, indicating that the model proposed in this study is reliable.