Computer simulation of stone frame sawing process using diamond blades

Computer simulation of stone frame sawing process using diamond blades

International Journal of Machine Tools & Manufacture 43 (2003) 559–572 Computer simulation of stone frame sawing process using diamond blades C.Y. Wa...

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International Journal of Machine Tools & Manufacture 43 (2003) 559–572

Computer simulation of stone frame sawing process using diamond blades C.Y. Wang a,∗, R. Clausen b a

Institute of Manufacturing Technology, Guangdong University of Technology, 510090 Guangzhou, People’s Republic of China b AB5-12, Technical University of Hamburg-Harburg, 21073 Hamburg, Germany Received 18 June 2001; received in revised form 28 October 2002; accepted 3 January 2003

Abstract Based on the results of scratching tests with single point tools and single segments in a previous study, the contact area between stone and blade, the number of effective cutting edges and the cutting forces per diamond grit, per segment and blade in the frame sawing process are simulated by computer. The number of diamond grits per unit area, the distribution of diamond grits per segment and the effective depth of cut of diamond grits are calculated by a new method. The Monte Carlo method is applied to generate the position of diamond grits randomly. The results show that in cutting stroke, only half of the diamond grits on the segment surface cut stone with the depth of cut increasing and a limited moving distance. The interactions of the grooves created by different segments remove the stone and generate the saw kerf. The simulation results of cutting forces are consistent in tendency with the data tested in a frame sawing machine. Cutting feed and the cutting performance of the segments are the major factors which determine the cutting forces and segment wear. The optimized constant contact area between blade and stone depends on the segment spaces, the segment number and the segment size, the ratio of the length of block and the effective cutting length of blade etc.  2003 Elsevier Science Ltd. All rights reserved. Keywords: Frame sawing; Diamond blade; Simulation; Stone

1. Introduction Frame sawing using diamond blades is a widely used machining technique in the stone industry. On the block surface the blades welded with diamond segments move back and forth alternately and feed downward at the same time to cut off the stone. In general, it is nearly impossible to do tests about frame sawing except using it in practice. Traditionally, manufacturers and customers can make or use the blades only according to their experience. The simulation of frame sawing will provide a practical alternative that can be used for the design and the application of sawing blades. By means of scratching tests using single point cutting tools and a single segment, the basic theory of frame sawing has been studied in order to know the action between diamond grit and stone in [1]. The results showed that the



Corresponding author. E-mail address: [email protected] (C.Y. Wang).

cutting mechanism of marble could be described as the plastic deformation and the brittle fracture of stone, which was affected by the cutting conditions such as the depth of cut, coolant, the shape of cutting tool tip and the properties of stone. The surfaces machined in the frame sawing process were formed by the co-actions of the diamond grits distributed on the surface of different segments in different positions randomly. In each cutting stroke, only half of the diamond grits on the surface of the segments cut stone [1]. Different methods for modeling and optimizing of grinding process were provided in [2-4]. By means of kinematic simulation a comprehensive concept for the modeling of grinding processes was presented in [5]. As the topography of the wheel is the key issue to establish the grinding model, many proposals have been published in this field by different test methods [2–6]. The Monte Carlo method was used to simulate the random positions of grits on grinding wheel by some researchers. It has also been used recently to simulate the grinding process and grinding wheel wear [7–9].

0890-6955/03/$ - see front matter  2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0890-6955(03)00019-1

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Based on the results obtained from frame sawing machine and the model machine with 3-segments diamond blades in the Institute of Cutting Technology, Technical University Hamburg-Harburg, we aim at simulating the frame sawing process in this paper. Some models to simulate the dynamic loading acting on the segment are established referring to a single scratching test and kinematic analysis of the sawing movement in [1]. A new method is proposed to calculate the number of diamond grits and their distribution on a segment surface in theory. The simulation is programmed with Microsoft Visual Basic 6.0 and Microsoft Access 97. The cutting forces per blade, per segment and per diamond grit can be calculated under different cutting conditions. The number of effective cutting edges (ECE) and the traces of diamond grits in the cutting process can also be presented. 2. Principle of the frame sawing process As shown in Fig. 1(a), the diamond blade cut the stone by the reciprocating movement from the start point to

Fig. 1.

the endpoint of a cutting stroke in frame sawing. The reciprocating cutting movement is generated by the rotation of a crank connected with a rod, which transfers the rotation to the horizontal movement of a frame. The frame is fixed with more than 25 diamond blades of 3000–4000 mm length. The standard blade thickness is 3.0–3.5 mm with a height of 180 mm. In general 29–39 diamond segments are welded on the bottom of a blade with the segment spaces, see Fig. 1(b). The feed motors drive the frame to move down for the blades cutting into the stone block continuously. Based on the previous study in [1], the frame sawing system is presented in Fig. 2. The major factors affecting the frame sawing process are the dimensions of the blade; segment and stone; the value of blade pre-tension; the cutting performance of diamond (shape, mesh, strength etc.) and diamond segment (bond, diamond concentration etc.); the properties of the stone (minerals, hardness, cleavage etc.); and the cutting parameters (cutting feed, rotation speed of crank etc.) limited by the power of frame sawing machine. The cutting forces of the blade and the segments not only decide the energy

Basic aspect of frame sawing: (a) frame sawing machine; (b) diamond blade.

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Fig. 2.

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Processing system of frame sawing by diamond blades.

consumption but also affect the deformation of the steel blade and the wear of the segments and diamonds. The stability of cutting will decide not only the surface quality of slabs, but also the numbers of the slabs split from a block. In order to simulate the frame sawing process, some models must be introduced. In this section, the models of sawing movement, the models of the depth of cut per diamond grit, the models of cutting forces and the diamond wear per segment will be discussed. 2.1. Model of the movement of frame sawing process In [1], we have discussed the kinematical formula of frame sawing process. The moving paths of blade, diamond segments and diamond grits in the direction of Xaxis (in the direction of cutting forward) and Y-axis (in the direction of cutting feed) can be calculated respectively by Eqs. (1) and (2) and are drawn in Fig. 3. lh x ⫽ (1⫺cos2pnkts) 2 y⫽

Vf t 60 s

(1) (2)

where Vf (mm/min) is the cutting feed that the blade moves down, lh is stroke length, nk is rotation speed of crank (rpm), and ts (s) is cutting time in a cutting stroke. The moving distance of the blade, segments and diamonds in the direction of cutting forward is equal to the stroke length lh = 500 mm. In a cutting stroke the cutting feed distance is about 58 µm while cutting time ts = 0.333 s, nk = 90 min⫺1 and Vf = 10 mm/min, as shown in Fig. 3(a). The effective cutting length of the cutting paths of a diamond grit depends on its position which relates to the side of stone block in cutting, see Fig. 3(b), and its protrusive height on the segment surface [1]. By means of the beginning contact position and the ending contact position of diamond grit the effective depth of cut per diamond grit can be deduced from Eq. (2).

Fig. 3. Movement of blade and segment in frame sawing process: (a) cutting distance of blade versus cutting time; (b) cutting paths of diamond segments.

2.2. Model of diamond wear The diamond wear rate was almost constant in frame sawing. In frame sawing the diamond wear on the segment surface was divided into six categories: fresh diamonds, sharp diamonds, micro-chipping diamonds,

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macro-chipping diamonds (including blunt diamonds), dropping diamonds. The percentage of each category measured by test is marked as k1, k2, k3, k4, and k5 on a segment surface, which presents the performance of segment wear resistance [11–12]. It was indicated that in the testing with a model machine, the ratios of different categories of diamond wear for stone were nearly identical to those obtained through frame sawing machinery with similar cutting parameters, if the possible fluctuation of the values was considered. The percentages of diamond wear classification can also be detected easily by means of a short time wear test with a 3-segment blade in the modelmachine with various combinations of segments and stone [14].

Fig. 4.

Diamond wear classification in frame sawing.

2.3. Model of cutting forces The cutting force in frame sawing can be defined as the feed cutting force Ff which is vertical to the cutting direction, and the cutting force Fc which is along the cutting direction. In [1], we have measured the cutting forces by a sharper carbide tool and the dressing diamonds. We also found that the sharper carbide tool can be used to simulate the action of the sharper diamond grits and the dressing diamonds can be used to simulate the cutting of the blunt diamond grits. We can calculate the cutting forces of diamond grits in frame sawing referring to their wear categories. The formulas of cutting forces by a sharper carbide tool are used for fresh diamonds, sharper diamonds and micro-chipping diamonds. The formulas of cutting forces by dressing diamond tools are used for the macro-damaged diamonds. Because the diamond grits on the surface of segments may cut or nearly cut in the same grooves which will cause overlapping chippings and decrease the cutting forces calculated by formula, we use the coefficients to adjust them. The cutting forces per segment are the sum of the cutting forces of effective cutting edges. The cutting forces of segments will be added to get the total cutting forces of sawing blade.

segment and per diamond; effective cutting edges per segment in sawing process. All input data are managed by a database management system so that we can add, delete and update them easily. The simulating results are displayed as graphs and can be output as text file for further analysis. The flow chart of the simulation is shown in Figs 4 and 5.

3. Simulation of sawing process 3.1. Sawing process of frame sawing by simulation We simulate the frame sawing process by computer based on the above models. The simulator is implemented by using Microsoft Visual Basic 6.0. Microsoft Access 97 is used as database management system. After choosing parameters including stone, diamond grit mesh, diamond concentration, blade size, segment spaces, block size and cutting parameters, we can obtain the data of the contact area between blade and per segment; cutting forces of blade, cutting forces per

Fig. 5.

Flow chart of simulation programming.

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3.2. Basic input data for simulation Basic data for simulation are listed in Tables 1–3. Most blades listed in Table 2 have been tested in the model machine and frame sawing machine [14]. For most simulation the major input data are as follows: Thassos as stone material, wet cutting, Vf = 10 mm/min, nk = 90 min⫺1, lh = 500 mm, lst = 2300 mm, Segment Type 3, Blade BUE1. As shown in Fig. 3, the blade is initially located at the left dead point of a cutting stroke. In the first cutting stroke the blade cuts forward from the start point to the endpoint of the stroke, which is marked as cutting forward-1 then the cutting direction is changed at the turn point and the blade cuts backward in the second cutting stroke which is marked as cutting backward-1. For the next double cutting stroke, the forward cutting is defined as cutting forward-2. 3.3. Generation of diamonds on a segment surface by the Monte Carlo method 3.3.1. Theoretical number of diamond grits on a segment surface The number of effective cutting edges on a segment surface is one of the important parameters for simulating the sawing process. Some equations for calculating the number of effective cutting edges in cylindrical grinding of metal and ceramic have been provided by experiments [2–7]. However, these equations cannot be applied directly to frame sawing because frame sawing is similar to surface grinding. Here we use a simplified method to calculate the number of diamond grits on a segment surface, which has been mentioned in [10]. The volume of single diamond grit Vd (mm3) can be described as: Vd ⫽

200 gNc

(3)

where Nc is the number of diamond grit per carat; g is

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the specific gravity of diamond grit, which equals 3.53 mg/mm3. We assume that the diamond section is a cubic block and the volume of diamond grits per segment is proportional to their section area. The diamond volume per unit volume V (mm3) and the diamond section area per unit area As (mm2) can be written as: V ⫽ 0.25 K

(4)

As ⫽ 0.25 K

(5)

where K (%) is the concentration of diamond grits in a segment. The weight of one-carat diamond is 200 mg. 1 carat/cm3 is equal to (or =) 4.4∗K. The average projecting section area per diamond Ad (mm2) can be derived from Eq. (6). Ad ⫽

Vd 200 ⫽ dg gNcdg

(6)

where dg (mm) is the projecting diameter per diamond grit in tangential direction, which can be decided by the distribution of diamond grit size referring to the diamond grit mesh No. From formula [5] and [6], the number of diamond grits per unit area Ns (mm2) can be calculated as Eq. (7). Ns ⫽

As ⫽ 4.4125 ⫻ 103dgNcK Ad

(7)

3.3.2. Number of effective diamond grits on a segment surface Referring to the discussions in 2.2 the effective number of diamond grits Nsa will decrease, as shown in Eq. (8), because of the dropping of diamonds grits on the surface of segments. Nsa ⫽ (k1 ⫹ k2 ⫹ k3 ⫹ k4)Ns

(8)

Table 1 Properties of segments and segment wear categories Test stone Types of diamond segments Categories of diamond grits wear and Fresh (k1) their percentages (%) on the surface of Sharping (k2) segments [14] Micro chipping (k3) Macro chipping and blunt (k4) Dropping (k5) Number of diamond grits on the segment surface Diamond mesh and concentration Number of diamond grits per carat Bond of diamond segment Segment size (length × width × Height)

Thassos Segment type 3

Jura Gelb Segment type 5 Segment type 6

14 13 13 22 21 26 16 16 19 17 22 10 31 28 32 61 80 86 D302, 22.72% D302 16.5%+D427 5.35% D302: 4560 (Measured) D427: 1850 (from DeBeers, SDA plus) Ni-based Co- based Co-based 20 × 5 × 12 mm3

Dimension of Blade (See also Fig. 1)

28 28 1) Blade BE2: Segment spaces=50.74 mm 2) Blade BUE2: segment spaces: 160-150-150-90-130-90-130-80110-70-110-70-120-80-120-70-110-70-110-80-130-90-130-90-150150-160

30 1) Blade BE1: Segment spaces=84.66 mm 2) Blade BUE1: segment spaces: 120-60-115-60-11560-110-60-110-60-95-55-75-55-95-60-110-60-110-60115-60-115-60-120

3950 3000 475

3950 3000 475

3950 2990 747.5

B3

Length of blade: L1 (mm) Effective cutting length of blade: L4 (mm) Length of segment: No.1 to blade side L5 (mm) Number of segments: n Segment spaces: L 1,2⫺L 2,3⫺L 3,4…⫺L 27,28…L n,n + 1 (mm)

B2

B1

No.

Table 2 Dimension of blade and segment spaces

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Table 3 Stone block size and cutting forces formulas Lst = 2300

Length of stone block (mm) Formula of cutting forces per diamond grit [1]

Sharp and micro chipping diamonds: Jura Gelb: Ff = 0.003ap 2 + 0.525ap + 0.1521; F c = 0.002ap 2 + 0.323ap + 0.0918 (N) Thassos: Ff = 0.0028ap 2 + 0.0931ap; Fc = 0.0026ap 2 + 0.024ap (N) Macro chipping and blunt diamonds: Jura Gelb: Ff = ⫺0.0032ap 2 + 0.4984ap ; Fc = 0.0026ap 2 + 0.1651ap (N) Thassos: Ff = 0.002ap 2 + 0.2429ap; Fc = 0.0023ap 2 + 0.094ap (N) (ap: depth of cut of the diamond grits, µm, wet cutting)

Correction coefficient of cutting forces (referring to [1])

3.3.3. Protrusive height of diamond grit above the bond and effective depth of cut per diamond grit Even if some techniques have been applied to get well-distributed spaces, the diamond grits on a segment surface are distributed randomly because diamond grits are mixed with metal powders and sintered in a graphite mould. In the simulation of the frame sawing process we only need to consider the action of the diamond grits on a segment surface. The random number generator in Visual Basic is used to simulate the positions of diamond grits referring to the principles of the Monte Carlo method. The protrusive heights of diamond grits are also generated randomly. The maximum protrusive height is limited within dg / 3. If the protrusive height is higher than this value, the diamond grit will drop referring to some test results in [13]. The effect of the bond wear to the protrusive height of the diamond grit is neglected. Categories of diamond wear have been discussed in 2.2 and their percentages k1, k2, k3 and k4 on the surface of segments have also been listed in Table 2. The number of the diamond grits per segment for each diamond wear category can be calculated. Eighty-percent of macro-chipping and blunt diamonds are defined as blunt wear. It is marked that (k1 + 0.8 k4)Nsa diamond grits with lower protrusive height act as the fresh diamonds. The rest of Nsa diamond grits are marked as sharp diamonds, micro-chipping diamonds and macro-chipping diamonds referring to respective percentages on a segment surface randomly. As shown in Fig. 6, some of macro-chipping diamonds grits have serious breakages and locate beneath the segment surface, so that the protrusive heights of twenty percentages of them are defined to be zero. Fig. 6 shows a sample of the distribution and protrusive heights of the diamond grits on the surface of one segment, which is generated randomly. After the data of the distribution and protrusion height of diamonds on the segments of a blade are generated randomly, all diamonds are marked with a number together with a segment. By these data we can simulate the sawing process under different conditions with the same blade.

0.25 (Ff), 0.15(Fc)

3.3.4. Effect of diamond mesh and diamond concentration The diamond grit mesh and diamond concentration affect the number of diamond grits on the segment surface. Fig. 7 presents the curves of a theoretical number of diamond grits with the method discussed in the previous section. Comparing the data with the results in [15–16], we find that they are similar, but the measured data of diamond grits on the surface of segments listed in Table 1 are less than those gained from calculation. We think that the discrepancy is due to the wrong data of the diamond concentration of segments provided by the manufacturer.

4. Results and discussions The processes of forward cutting and backward cutting in double stroke present the basic principles of the frame sawing process and are discussed as follows. The contact area between segment and stone, moving traces of diamond grits on a stone surface, effective cutting edges (ECE) of segments and blade; depth of cut and cutting forces of single diamond grit, cutting force of segment and blade will be analyzed step by step as the major output of the simulation. 4.1. Contact area between segment and stone As shown in Fig. 8, in double cutting stroke the total contact area between blade and stone block varies with cutting time. At the half of the stroke length, where the cutting time is respective 0.166 s and 0.499 s, the total contact area reaches the peak value. The dimension of blade and the segment spaces listed in Table 3 decide the positions of the segments. The segment spaces, block length and stroke length will affect the curves of the total contact area versus cutting time. The contact area of segment Nos. 4, 16 and 28 are also shown in Fig. 8. In Fig. 3(b) we can see that the cutting paths of the segment Nos. 4, 16 and 28 also depend on the dimension

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Fig. 6.

Generation of the distribution and protrusion of diamond grits on a segment surface.

of the stone block and its position related to the blade side in a cutting stroke. When a cutting stroke begins, segment No. 4 cuts stone several seconds later and stops inside the block because it is located on the left side of the blade. Segment No. 28 cuts the block from the start of a cutting stroke and stops cutting outside the block at the end of the stroke because it locates on the right side of the blade. Segment No. 16 moves forward and backward only inside the block because it is located in the middle of blade. These segments are taken as the typical segments in the following analysis to indicate the effects of the segment position in frame sawing. 4.2. Effective cutting edges Fig. 7. The number of diamond grits per unit area on segment surface.

Fig. 8.

Contact area of blade and segments.

Because the diamond grits are located on the surface of the segment randomly, their positions determine the effective depth of the cut per diamond grit. The cutting situation per diamond grit can be identified by its protrusive height and the relative height to the highest contacting diamond grit on a segment surface. It also depends on the cutting feed Vf, the position of diamond grit on a segment surface, the moving distance of blade and the length of block lst. As shown in Fig. 9, only a few diamond grits contact with the block in the first forward cutting (cutting forward-1), for all segments the number of ECE increases with the cutting time although the depth of cut per diamond grit is different. The diamond will be loaded with an impacting force as the depth of cut increases about 2–6 µm within 33 microseconds at the turn point of a cutting stroke [1], it is assumed that the diamond grits cut into stone block 2 µm at the endpoint of the cutting stroke, referring to the cutting paths in Fig. 3. This value will be added to the depth of cut of the diamond grits that contact with stone at the turn point. The ECE will be constant for segment No. 16 in the next cutting process. The ECE of

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Fig. 9.

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Effective cutting edges (ECE) of several segments versus cutting time.

segment No. 4 and segment No. 28 change with cutting directions. These results imply that the loads of diamond grits vary with cutting direction. The average ECE of segment Nos. 4 and 28 are fewer than that of segment No. 16, and the segment wear will also be different. The average number of ECE of segment Nos. 1–4 and segment Nos. 26–30 will drop to zero when they cut outside. The average number of ECE of other segments and the average total number of ECE of blade are nearly constant for normal cutting process such as cutting backward-1 and cutting forward-2, so that the cutting process will be stable, see also Fig. 9(a). In a single segment cutting test the number of ECE was 10–13 when the depth of cut was 25 µm [1]. By Eq. (2), we calculate the maximum depth of cut as 58 µm and the number of ECE can be estimated as 30. Only 40% of diamond grits cut stone because the effective number of diamond grits on the segment surface is about 80 in the test. In the simulation, at the endpoint of a cutting stroke the average number of effective cutting edges is about 58; the total number of diamond grits on a segment surface is 118 by the simulation. The percentage of the number of ECE on a segment surface is proportional to feed Vf. Even if we use a faster feed here only half the diamond grits cut stone with a different depth of cut in the cutting process. Although the number of diamond grits on a segment surface varies in the test of Ref. [1] and in the simulation, the percentages of the number of ECE on a segment surface of them are similar. 4.3. Traces of diamond grits on the stone block surface The moving traces of diamond grits simulated are drawn with a line. At the beginning of the first cutting stroke, only six diamond grits contact the stone block. With the increase of the cutting distance more diamonds contact the stone surface and the number of ECE

increases, so that the density of moving traces of diamond grits grows. These traces show the positions of the cutting grooves generated by the diamond grits. These grooves remove the stone materials, form the bottom surface of the saw kerf and finally cut off the slabs. At the starting point of backward cutting in the second cutting stroke, all the diamond grits that contact the stone are displayed in Fig. 10. The ECE of segment No. 26 is shown in detail. All diamond grits move a constant distance from beginning the point of sawing, which equals stroke length lh as described in Fig. 3(b). However, some of diamond grits cut the block in a shorter distance than stroke length before they move outside the block. The density of traces near the outside of the cutting area is lower than that in the middle, which affects the roughness of the sawn surface. The overlapping cutting that generates the intersection of cracks plays an important role in the formation of the cutting surface referring to the study in [1]. Segment spaces can change the density of traces and the practical loading on diamond grits. Even if we consider the size of diamond and the segment, the scratch of diamond grits cannot cover the total surface of the bottom of the saw kerf, which is nearly flat in the middle and uneven outside as shown in Fig. 10. 4.4. Depth of cut and cutting forces per diamond grit As shown in Fig. 11, the depth of cut per grit is calculated by means of the position of grits and feed. The diamond grits on the surfaces of different segments show a different depth of cut because of their different positions. The depth of cut per grit increases with the cutting time at constant feed. As the diamond grit moves inside the stone block, the depth of cut always increases from near zero to maximum. The depth of cut will decrease to zero when the diamond grits locate outside the block, see Fig. 11(a) and Fig. 11(c). The cutting forces depend on the depth of cut per dia-

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Fig. 10.

Fig. 11.

Cutting traces of diamond grits on block surface.

Depth of cut of diamond Grit No. 117 in several segments versus cutting time.

mond grit. Fig. 12 presents the cutting forces of diamond grit No. 117. During the cutting process, the cutting forces per diamond grit grow from near zero to maximum. At the turn point of the cutting direction, the cutting forces keep a small value more than zero under the action of depth of cut 2 µm. The maximum cutting

Fig. 12.

force per diamond grit is about 4 N. In general, the compressive strength of a single diamond grit for stone machining is about 15–18 kg. The cutting forces simulated here indicate that the cutting force on a single diamond is not large enough to break the diamond. Because of the impact loading as well as the direction change of

Cutting forces of diamond Grit No. 117 in several segments versus cutting time.

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Fc at the start point and end point in a cutting stroke, the load on a single diamond grit will be more complex than that in simulation. 4.5. Cutting forces of segments and blade The cutting forces per segment increase with cutting time as shown in Fig. 13. The mean cutting forces per segment for cutting forward and cutting backward in the normal cutting process are similar. The average feed cutting force Ff per segment is about 70 N, the cutting force Fc is about 40 N by simulation. Comparing with the results obtained from [1], it can be found that the mean value of cutting forces is similar to that of single segment cutting, where the maximum feed cutting force Ff per segment is about 80 N with the maximum depth of cut 60 µm. The simulation results show that the cutting forces of blades increase to maximum and drop to near zero as cutting direction is changed because of the smaller initial depth of cut of the diamonds at the turn point (ts = 0 s for cutting backward-1 and cutting forward-2) of the cutting direction, see Figs. 11 and 12. In practice the cutting forces are held at an almost constant level. It means that the cutting forces will keep at the high level at the turn point of cutting direction, even if the cutting feed downward decrease nearly to zero. We have assumed that all contacted diamond grits at the turn point of the stroke have an initial depth of cut 2 µm as a rapid load although the real situation is still not clear. The support of chips and the plastic deformation to the diamond grits and segments etc., may be responsible for keeping the cutting force nearly at a constant level [1].

Fig. 13.

Cutting forces of segments.

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The simulation results of cutting forces indicate the same trend as that tested by a single segment cutting and frame sawing machine as shown in Fig. 13. The mean cutting forces of the blade are similar and the maximum cutting forces of one blade are higher than those of the measured value (about 1200 N) in a frame sawing machine [14]. Although the values of cutting forces are not too exact, we can still get a better understanding of frame sawing by simulation. We consider the following reasons cause the higher simulated cutting forces. 1. As studied in [1], the orientation of dressing diamond resulted in the distribution of the cutting forces of a single diamond tool in a wider range than sharp cutting tools, but the complex shape of a single diamond grit is simplified as a sharp or blunt diamond in simulation. 2. The simplified coefficients are used to express the influence of the intercross cutting of segments on the stone surface to the cutting forces in simulation. However, as we knew from the previous study in [1], when diamonds cut in the same grooves continuously or cut apart from the previous grooves at a shorter distance, the cutting forces per diamond grit should be less than those we simulate now and cannot be described with the cutting forces formulas with correction coefficients simply. When we cut stone with a diamond blade welded with 30 segments under water, the overlapping cuttings of the diamonds are also very complicated that we simulate now. 4.6. Effects of some factors on the frame sawing process 4.6.1. Effect of the segment spaces In the frame sawing process, the segment spaces on the blade affect the moving trace per diamond grit because the segment spaces can change the contacting situation of a diamond with a stone block. The segment spaces will also affect the spaces to contain chips, the deformation of the blade, and the diamond wear. However, they will change a lot as to the segments on the sides of the blades such as segment Nos. 1–4 and segment Nos. 28–31. For example, in Fig. 14, when the unequal segment spaces on the blade, marked as blade BUE1 (see also Table 2), the contact area of segment No. 4 has a higher value in a longer period and the depth of cut of diamond grit No. 117 shows also larger final value, comparing with the blade with uniform segment space (BE1). The total number of average ECE and cutting forces of blade only show a little variation when the segment spaces are changed. In order to optimize segment spaces the relativity of stroke length lh, and the size of stone block lst must be taken into account at the same time. In Fig. 15, the mean contact area of the blade decreases with the increase of

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Fig. 14.

Cutting situations of segment No. 4 with different segment spaces versus cutting time.

stroke length lh. With the same blade, the contact area decreases when the length of block decreases. Blades with different segment spaces contact the stone block with more contact area. Fig. 16 shows clearly that using the blades with different segment spaces the moving traces of diamond grits are changed to achieve a different

final cutting surface. Aiming on maximum material removal and uniform segment wear of the blade, the segment spaces can be optimized by the contact area curves (Fig. 8) and moving trace of diamond (Fig. 16). 4.6.2. Effect of the cutting feed In Fig. 17, the effects of cutting feed on cutting process are presented. The number of ECE and the depth of cut of diamond grits increase with the increasing of the cutting feed so that the cutting forces of the diamond grit and blade are proportional to the cutting feed. The mean cutting forces of blade gained by the measurement in the frame sawing machine show the similar values [14], see Fig. 17(c). The simulation results of cutting forces per segment and per diamond grit can be used to optimize the cutting feed in various conditions. 4.6.3. Effect of stone The difference of the cutting forces between stone Thassos and Jura Gelb wouldn’t be compared in this simulation since we don’t have data of segment wear classification of the same kind of segments. The properties of stone will affect the diamond wear and the cutting forces, which have been discussed in Ref. [1] and Ref. [14].

Fig. 15. Contacted area with different segment spaces, block size and stroke length: (a) cutting time versus contacted area; (b) stroke length versus contacted area.

Fig. 16.

Cutting traces with different segment spaces.

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Fig. 17.

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Cutting force of blades with different cutting feed.

4.6.4. Effect of segment type The selection of segments is very important for cutting process optimization. The classifications of diamond wear show the wear resistance and the cutting performance of the segments, which plays an important role in simulation. Two segment types are applied to simulate the Jura Gelb cutting process. Their classifications of diamond wear are different, as shown in Tables 1 and 3. The number of ECE in the cutting process is also different. Segment Type 5 has more blunt diamond grits than Segment Type 6, so that it has larger cutting forces, as described in Fig. 18.

Fig. 18. Average cutting forces of segments with different segment types.

5. Conclusions 1. A new method is proposed to calculate the number of effective diamond grits on the surface of a segment. The Monte Carlo method is used to generate the three dimensional position and the protrusive height of diamond grits. 2. Based on the results of the segment wearing test, a software package is developed to simulate the frame sawing under different cutting conditions, such as segment type, stone, feed, segment spaces, stroke length, stone block size etc. The cutting forces per diamond grit, per segment and blade can be given. The effective cutting edges of segments and blade, contacting area between stone and blade, as well as moving traces of segments on stone surface, can also be provided. 3. In each cutting stroke, only half of the diamond grits on the segment surface cut stone. The cutting distances of the diamond grits on the surface of stone depend on their positions on the blade. The interactions of the grooves created by the diamond grits on the surface of different segments generate the bottom surfaces of block saw kerf, which is nearly flat in the middle and uneven outside. The intersection of segments depend on the traces of segments, the protrusion of diamond grits above the matrix, effective cutting edges and stone etc. 4. The simulated data of cutting forces of blades are larger than the actually measured value but the tendency is the same and in the same level. The optimized constant contact area between blade and stone is very important for uniform segment wear of total blade, which depends on the segment spaces, segment number and segment size, the ratio of the block length to the effective cutting length of blade. 5. The optimized association of cutting parameters and

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the blades can be decided with the limitation of cutting forces. Cutting feed and the cutting performance of segments are the main factors to determine the cutting forces and segment wear. When finishing a short time test in the model machine we can obtain a segment wear model. By means of inputting some cutting conditions, the optimized size of saw blades and cutting techniques can be suggested by the simulation.

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Acknowledgements This project is funded by the National Natural Science Foundation of China, Chinese Scholarship Committee and Technical University of Hamburg-Harburg. The author thanks Dipl.Ing. J. Stangenberg for helping with the test, and Winter & Ernst Steinbearbeitng GmbH for supplying the diamond segment.

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