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Mechanistic cutting force model in band sawing
International Journal of Machine Tools & Manufacture 39 (1999) 1185–1197
Mechanistic cutting force model in band sawing Tae Jo Ko*, Hee Sool Kim School of Mechanical Engineering, Yeungnam University, Gyoungsan, Kyoungbuk 712-749, South Korea Received 4 March 1998; received in revised form 27 November 1998
Abstract In order to establish a mechanistic model of cutting force, specific cutting pressure was first obtained through cutting experiments. The band sawing process is similar to milling in that it involves multi-point cutting, so it is not an easy matter to evaluate specific cutting pressure. This was achieved by making the thickness of workpiece smaller than one pitch of the saw tooth, analogous to fly cutting in the face milling process. Then the cutting force was predicted by analysing the geometric shape of a saw tooth. The tooth shape used was the raker set style that is generally used in band sawing. A set of teeth comprises three teeth, ranked as left, straight, and right. The mechanistic model developed in the research considered the shape of each tooth in a set. The predicted cutting forces coincided well with those measured in the validation experiment. Therefore, the predicted cutting forces in band sawing can be used for the adaptive control of saw-engaging feed rate in band sawing. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Band sawing machine; Mechanistic cutting force model; Multi-point cutting; Saw blade; Single-point cutting; Tooth form
1. Introduction Sawing machines are of primary value for the preparation of raw materials to be machined, and constitute some of the most important machine tools found in a machine shop. Common types of cutoff machines include reciprocating saws, horizontal endless band saws, universal tilt frame band saws, abrasive saws, and cold saws [1]. Band saw machines use a steel band blade in the form of a band with the teeth on the edge. The band saw machine is widely used for the following reasons: it has a high cutting efficiency because the band cuts continuously with no
* Corresponding author. Tel: ⫹ 82-53-810-2576; fax: ⫹ 82-53-813-3703; e-mail: [email protected] 0890-6955/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 9 8 ) 0 0 0 8 7 - X
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wasted motion; material loss is small due to the small kerf of the saw cut; the feed rate through the material may be varied; the machine can handle workpieces of large dimensions. In general, such as when the workpiece is a cylindrical rod, the length of the cut changes progressively during in-feed of the band saw. In the case of constant in-feed rate, the load on the toothed edge increases along with the radial engagement of the cut, and this variation of the load induces vibration in the machine. The vibration affects the kerf of the saw cut and leads to severe tooth wear and breakage, causing deterioration of surface roughness and tolerance of the cut dimensions. With regard to this problem, Ulsoy and Morte [2] developed the equation of motion, based on Hamilton’s principle, to the vibration of wide band saw blades, and obtained approximate solutions using both the classical Ritz and finite element-Ritz methods. Carlin et al. [3] analysed the buckling and vibration of a circular saw blade subjected to a combination of loading conditions approximating those encountered in operation. Chandrasekaran and colleagues [4,5] researched tooth chipping during the band sawing of steel, and Sarwar and colleagues [6,7] researched the relationship between cutting forces and friction characteristics and the parameters affecting the performance of a tooth blade. To deal with these problems at their origin, it is necessary to be able to predict the magnitude and variation of cutting force. However, it is not an easy matter to predict cutting force in band sawing since it is not a single-point cutting but multi-point cutting as in the milling process, and because the geometric shape of the cutting edge varies due to the offset on each side to provide clearance for the back of the blade. In previous studies, Sarwar et al. [8] analysed cutting forces with a finite element method and a single-point cutting technique using a turning lathe, and Henderer et al. [9] suggested an analytical method for predicting the cutting force of a saw blade in two-dimensional cutting. However, these models are based on two-dimensional cutting, and require prior knowledge of the shear angle. Nor can they provide the pulsation cutting forces needed in the vibration analysis, but give only the static mean cutting forces. In this work, we developed a mechanistic model for predicting cutting forces in multi-point cutting by a band saw. The cutting force system in face milling has been extensively studied both analytically and empirically. In analytical modelling approaches, theories of single-point cutting such as energy method, flow stress method, matrix method, and single-shear plane method may be applied. While such a model may be sound in principle, it requires knowledge of the shear angle, dynamic stress, friction angle, etc., parameters which are usually not easy to determine in practice. As a result, a more empirical approach to modelling a face milling has been popular. The mathematical model for predicting cutting forces has been widely used after Martellotti [10,11], who developed mathematical equations for the milling process, calculating analytically with the tooth path, instantaneous undeformed chip thickness, etc. He also introduced the notion that the average undeformed chip thickness could be used in establishing a relationship between the conditions of the cut and specific pressure required. Later, Koenigsberger and Sabberwal [12] also observed that there is a strong relationship between the instantaneous chip thickness and the tangential cutting force. Tlusty and MacNeil [13] examined the variation of cutting forces in flatend milling at steady state and transient cutting conditions. Most of the research to date has dealt with the development of force equations and the modelling of specific cutting pressure under the simplest of conditions. However, Kline et al. [14] have established a mechanistic force model for end-milling under various cutting environments. More recently, Fussell and Srinivasan [15] investigated the capability of the model developed by Kline et al. [14] under varying machining
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conditions. Fu et al. [16] used the general approach of Martellotti’s method [10,11] to develop a mechanistic force model for face milling. Armarego and Deshpande [17] studied the effects of cutter runout and developed a computerized cutting force prediction model for flat end-milling. Feng and Menq [18,19] reported a rigid system cutting force prediction model for the ball-end milling process. The mechanistic model in this work for predicting cutting forces in band sawing was developed by introducing Martellotti’s model [10,11] that uses instantaneous undeformed chip thickness and specific cutting pressure. To this end, the specific cutting pressure was obtained by a single-point cutting technique, analogous to fly cutting in face milling. Single-point cutting can be performed by using the workpieces with a thickness smaller than the interval between adjacent teeth on the saw. Then, the cutting forces were predicted by applying the specific cutting pressure to a geometric model, which considers the geometric profile of a band saw tooth such as left-bent, straight, right-bent tooth.
2. Materials and terminology 2.1. Geometry of saw blade The shape of a saw blade is described in terms of tooth form, set and pitch. Saw tooth forms are referred to as ‘standard’, ‘skip’, or ‘hook’. The standard form gives accurate cuts with a smooth finish [1], and is used in this research. The teeth of a saw blade must be offset on each side to provide clearance for the back of the blade. Set forms include ‘raker’, ‘straight’, and ‘wave’. The raker set is used in general sawing and selected in the research. The pitch of a saw blade is the number of teeth per inch; in this case, the pitch is three. Fig. 1 shows the geometry of the saw blade.
Fig. 1. Geometry of saw blade.
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2.2. Undeformed chip thickness Fig. 2 represents the cutting mechanism during band sawing. In band sawing, the feed per tooth is equal to the depth of cut of each tooth, since they are measured in the same direction. This depends on both the feed rate and cutting velocity, and can be determined by the following expression ti ⫽
p·f
(1)
where ti is the depth of cut, p is the distance between successive teeth, f is the in-feed rate, and is the cutting velocity. As shown in Fig. 2(a), the saw blade is tilted with a slope angle of a for efficient cutting. However, the slope angle is very small; the actual depth of cut d is approximated as ti. Therefore, the undeformed chip area, as shown in Fig. 2(b), is given in Eq. (2): A ⫽ b·ti
(2)
where b is the width of the saw blade.
Fig. 2. Cutting mechanism of band sawing (a) cutting mechanism of band saw; (b) actual cutting area of saw tooth.
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3. Development of the force system model 3.1. Specific cutting pressure A rectangular Cartesian coordinate is set up with the origin at the center of edge end and with the X-axis in the cutting direction. The Z-axis is perpendicular to the machined surface and directed downward. The Y-direction is then determined by the right-hand rule, as shown in Fig. 3. The normal cutting force is defined as the X-directional force, that is, normal to the tooth face. The Z-direction is radial to the tool face and directed downward. The Y-direction, lateral to the tool face, is then determined by the right-hand rule. Martellotti [10,11] has proposed that the normal cutting force acting on the chip cross-section is the product of the undeformed chip area and the specific cutting pressure, ks. The lateral and the radial force acting along the cutting edge are obtained by multiplying the normal force by the empirical constants, ky, kz, respectively. By ignoring the effects of tooth geometry, a specific cutting pressure ks can be obtained by dividing the X-directional mean cutting force per tooth by the undeformed chip area A of Eq. (2). Specific cutting coefficients ky, kz of the Y-, Z-directions are obtained by dividing Y-, Z-directional cutting forces by X-normal force, respectively. Accordingly, specific cutting pressure and specific cutting coefficients are written as follows: ks ⫽
Fx A
ky ⫽
Fy Fx
kz ⫽
Fz Fx
(3)
where Fx, Fy and Fz are the mean cutting forces per tooth of X, Y, Z direction, respectively.
Fig. 3. Forces in cutting edge.
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Fig. 4.
Set patterns.
3.2. Cutting force model from geometry of saw tooth In the raker set, as shown in Fig. 4, the saw blade has left-bent, straight, and right-bent tooth, iteratively. Therefore, the cutting force model considers these three kinds of geometric shape. First, as shown in Fig. 5, in the case of a left-bent tooth, the instantaneous cutting force per tooth can be modelled by decomposing normal, lateral and radial forces as
冦
FXl(i ⫺ 1,d)
冧冤
cos␥z
FYl(i ⫺ 1,d) ⫽ sin␥z FZl(i ⫺ 1,d)
0
⫺ sin␥z
冥冦
冧
⫺ sin␥xsin␥z Fxl(i ⫺ 1,d)
cos␥xcos␥z
sin␥xcos␥z
⫺ sin␥xcos␥z
cos␥x
Fig. 5. Left bent saw tooth.
Fyl(i ⫺ 1,d) Fzl(i ⫺ 1,d)
(4)
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where ␥x and ␥z are the tooth rotational angles with respect to the X, Z axes, respectively, i is the order of tooth, d is the cutting distance. The symbol l means left. Fx(i,d), Fy(i,d), and Fz(i,) is the normal, lateral and radial component of the tooth face as shown in Fig. 5, respectively. FX(i,d), FY(i,d), and FZ(i,d) is the X-, Y-, and Z-directional instantaneous cutting force, respectively. Substituting a specific cutting pressure, specific cutting coefficient and undeformed chip area into Eq. (4), we can obtain instantaneous cutting forces. Instantaneous cutting forces of the straight tooth following the left-bent tooth are similar to Eq. (4), except that the normal direction of the undeformed chip area is coincident with the X axis, as shown in Fig. 6. Therefore, cutting forces can be modelled as follows:
冦 冧 冤 冥冦 冧 FXs(i,d)
1 0 0 Fxs(i,d)
FYs(i,d) ⫽ 0 1 0
FZs(i,d)
(5)
0
0 0 1 Fzs(i,d)
where s represents straight. On the other hand, similarly to the left-bent tooth, instantaneous cutting forces of the rightbent tooth, as shown in Fig. 7 are modelled as follows:
冦
FXr(i ⫹ 1,d)
冧冤
cos␥z
⫺ sin␥z
冥冦
冧
⫺ sin␥xsin␥z Fxr(i ⫹ 1,d)
FYr(i ⫹ 1,d) ⫽ sin␥z ⫺ cos␥xcos␥z ⫺ sin␥xcos␥z Fyr(i ⫹ 1,d) FZr(i ⫹ 1,d)
0
⫺ sin␥xcos␥z
cos␥x
Fzr(i ⫹ 1,d)
(6)
where r represents right. Eqs. (3)–(6) are used to describe forces on a single tooth. Therefore, in the case of multi-point
Fig. 6. Straight saw tooth.
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Fig. 7. Right bent saw tooth.
cutting, where more than one tooth is engaged simultaneously in cutting, the cutting forces are predicted by summing up Eqs. (4)–(6) as follows:
冦 冧 冘冦 冦 FX
n
FXs(i,d)
FXr(i ⫹ 1,d)
␦l FYl(i ⫺ 1,d) ⫹ ␦s FYs(i,d) ⫹ ␦r FYr(i ⫹ 1,d)
FY ⫽ FZ
冧 冦 冧 冦
FXl(i ⫺ 1,d)
i⫽1
FZl(i ⫺ 1,d)
FZs(i,d)
FZr(i ⫹ 1,d)
冧冧
(7)
where n is the total number of teeth, and ␦ is a Kronecker delta. ␦ is 1 when the tooth is engaged in the cutting, and zero when the tooth is out of the workpiece. FX, FY, and FZ is X-, Y-, and Z-directional cutting force, respectively. 4. Cutting experiments Fig. 8 shows the horizontal band saw machine (KDBS 450A: Kyoung-Dong Co.) used in the experiment. The tooth blade (Bearcat M42: STARRETT) is a standard tooth form, three pitch, and raker set. In order to measure three directional cutting forces, a tool dynamometer (9257A: KISTLER) is mounted on the table of the saw machine. A fixture, to hold the workpiece, is bolted on to the dynamometer. The cutting force signal from the tool dynamometer is amplified by the charge amplifier, and is collected by the computer through the A/D converter. The workpiece is a rectangular rod of mild steel (AISI 1010) 8 ⫻ 25 mm. The distance between each tooth is 8.3 mm, so that single-point cutting is possible if we cut parallel to the 8 mm edge. Hence, mean cutting force per tooth can be obtained, and specific cutting pressure can be calculated from the mean cutting force. Whereas Sarwar et al. [8] obtained specific cutting pressure using one tooth cut from a saw
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Fig. 8. Experimental set-up.
blade by turning a cylindrical part with the tooth fixed in the tool holder of a turning lathe, we could measure mean cutting force per tooth simply by adjusting the width of the workpiece. Thus, rotating the workpiece to cut a 25 mm thickness engages three teeth in the sawing. The corresponding force was measured in the same way and used to check the cutting force model. Experimental conditions used in the test are summarized in Table 1. 5. Results and discussion 5.1. Modelling of specific cutting pressure In order to predict cutting forces in sawing, first, a model of specific cutting pressure is built. To this end, a total of 20 kinds of cutting force signals were measured, as shown in Fig. 9: four cutting speeds (30–77 m/min), and five feed rates (15–332 mm/min). It is possible to use observed average forces to estimate the values of average ks. Therefore, we obtain various ks with respect Table 1 Experimental conditions Cutting speed (m/min) Feed rate (mm/min) Tilt angle about X axis (rx) Tilt angle about Y axis (rz) Saw blade dimension (mm) Sampling frequency (Hz) Number of sampled data Workpiece material
30, 48 15, 40, 84, 189 5° 6° 4670 ⫻ 38 ⫻ 1.3 1000 1024 SS41
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Fig. 9. Measured cutting forces with fly cutting.
to different cutting conditions by applying average cutting forces to Eq. (4). To model a specific cutting pressure, in general, an equation of ks ⫽ ␣·A is used. This is a relationship between specific cutting pressure and undeformed chip area, where ␣ and  are constants and A is the undeformed chip area. From this equation, the following model is fitted; Fig. 10 shows a graph of its value. ks ⫽ 1427.6A−0.295
(8)
The other specific cutting coefficients of ky, kz are calculated using Eq. (3). 5.2. Model verification in fly cutting Fig. 11 shows the simulated and measured cutting forces with respect to the cutting conditions of feed rate 84 mm/min, and cutting speed 30 m/min. The simulation cutting force signal is trapezoidal, whereas the measured cutting force is not. This is due to external factors such as band vibration. However, the magnitude of the cutting force is coincident. Cutting force in the X direction, that is, cutting direction, is two times larger than cutting force in the Z direction. In
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Fig. 10. Estimation of specific cutting force.
Fig. 11. Measured and simulated cutting forces with fly cutting (cutting speed: 30 m/min, feed rate: 84 mm/min).
general, the ratio of thrust force to main cutting force is 0.4, while the ratio is 0.5 in the saw cutting. The reason for the difference is that the feed direction coincides with the thrust direction. The force in the Y direction is very small and it is difficult to measure the cutting force. This verifies that cutting force in the Y direction does not affect the cutting process. As shown above, in the case of single-point cutting, cutting forces can be predicted well by a specific cutting
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pressure. Whereas the external component has a dominant effect on the variation of total cutting force, the geometric shape of the tooth blade (left, right, or straight) has none. 5.3. Model verification in multi-point cutting In multi-point cutting, using a workpiece of 25 mm thickness, three teeth engage in the sawing, simultaneously. Fig. 12 shows the measured and simulated cutting forces under the conditions of cutting speed 48 m/min, and feed rate of 189 mm/min. The pattern of the cutting force signal is trapezoid, similar to that for single-point cutting. The simulated cutting force is good enough to predict the real cutting force. Furthermore, the Y-directional cutting force is negligible and the ratio of the X- and Z-directional cutting force is the same as in single-point cutting. Thus, in multipoint cutting, the cutting force model is satisfactory. However, to predict cutting force even more precisely, the runout of tooth-in-tooth generation and band vibration should also be considered. In the experiments, the cutting forces of each direction are three times those in single-point cutting. Therefore, in multi-point cutting, the cutting forces can be predicted simply by multiplying the total engaged tooth number by the cutting forces for single-point cutting. 6. Conclusion In order to predict cutting forces in band sawing, a mechanistic model was developed by considering the geometry of the saw tooth. A specific cutting pressure is necessary for using a mech-
Fig. 12. Measured and simulated cutting forces with multipoint cutting (cutting speed: 48 m/min, feed rate: 189 mm/min).
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anistic model. To this end, a single-point cutting method, like fly cutting in milling, was introduced to obtain specific cutting pressure. Using the cutting forces from single-point cutting, a specific cutting pressure model was built. From the verification experiments for both single-point and multi-point cutting, we conclude that the mechanistic model developed could predict cutting forces in band sawing very well. However, further work should also consider non-uniform tooth shapes, transverse vibration of the saw blade, and imperfect balance of the driving wheel, for even more precise predictions of the cutting force. Acknowledgement This work was partly supported by a KOSEF grant, no. 95-2-09-04-01-1. References [1] P.F. Ostwald, J. Munoz, Manufacturing processes and systems, John Wiley and Sons, Inc., 1997. [2] A.G. Ulsoy, C.D. Morte, Vibration of wide band saw blades, ASME Journal of Engineering for Industry 104 (1982) 71–78. [3] J.F. Carlin, F.C. Appl, H.C. Bridwell, R.P. Dubois, Effects of tensioning on buckling and vibration of circular saw blades, ASME Journal of Engineering for Industry February (1975) 37–48. [4] H. Chandrasekaran, S. Svensson, M. Nissle, Tooth chipping during power hack sawing and the role of saw material characteristics, Annals of the CIRP 36 (1) (1987) 27–31. [5] H. Chandrasekaran, H. Thoors, H. Hellbergh, L. Johansson, Tooth chipping during band sawing of steel, Annals of the CIRP 41 (1) (1992) 107–111. [6] M. Sarwar, D. Gillibrand, S.R. Bradbury, Forces, surface finish and friction characteristics in surface engineered single- and multi-point cutting edges, Surface and Coating Technology 49 (1991) 443–450. [7] W.M.M. Hales, M. Sarwar, Geometrical parameters affecting hacksaw blade performance, Ninth International Conference on Production Engineering Research, Cincinatti, 1987, pp. 1802–1810. [8] M. Sawar, S.R. Bradbury, M. Dinsdale, An approach to computer aided band saw teeth testing and design, Proceedings of the Fourth National Conference on Production Research, 1988, pp. 494–501. [9] W.E. Henderer, J.D. Boor, J.R. Holston, Estimation of cutting forces in band sawing metals, Transactions of NAMRC 24 (1996) 33–38. [10] M.E. Martellotti, An analysis of the milling process, Transactions of ASME 63 (1941) 667. [11] M.E. Martellotti, An analysis of the milling process. Part II: Down milling, Transaction of ASME 67 (1945) 233. [12] F. Koenigsberger, A.J.P. Sabberwal, An investigation into the cutting force pulsation during milling operation, International Journal of Machine Tool Design and Research 1 (1961) 15. [13] J. Tlusty, P. MacNeil, Dynamics of cutting forces in end-milling, Annals of CIRP 24 (1) (1975) 21–25. [14] W.A. Kline, R.E. DeVor, J.R. Lindberger, The prediction of cutting forces in end milling with application to cornering cut, International Journal of Machine Tool Design and Research 22 (1) (1982) 7–22. [15] B.K. Fussell, K. Srinivasan, An investigation of the end-milling process under varying machining conditions, ASME Journal of Engineering for Industry 111 (1989) 27–36. [16] H.J. Fu, R.E. DeVor, S.G. Kapoor, A mechanistic model for the prediction of the force system in face milling operation, ASME Journal of Engineering for Industry 106 (1984) 81–88. [17] E.J.A. Armarego, N.P. Deshpande, Computerized predictive cutting models for forces in end-milling including eccentricity effects, Annals of CIRP 38 (1) (1989) 45–49. [18] H.Y. Feng, C.H. Menq, The prediction of cutting forces in the ball-end milling processes. Part 1: Model formation and model building procedure, International Journal of Machine Tools Manufacture 34 (5) (1994) 697–710. [19] H.Y. Feng, C.H. Menq, The prediction of cutting forces in the ball-end milling processes. Part 2: Cut geometry analysis and model verification, International Journal of Machine Tools Manufacture 34 (5) (1994) 711–720.
Mechanistic cutting force model in band sawing Tae Jo Ko*, Hee Sool Kim School of Mechanical Engineering, Yeungnam University, Gyoungsan, Kyoungbuk 712-749, South Korea Received 4 March 1998; received in revised form 27 November 1998
Abstract In order to establish a mechanistic model of cutting force, specific cutting pressure was first obtained through cutting experiments. The band sawing process is similar to milling in that it involves multi-point cutting, so it is not an easy matter to evaluate specific cutting pressure. This was achieved by making the thickness of workpiece smaller than one pitch of the saw tooth, analogous to fly cutting in the face milling process. Then the cutting force was predicted by analysing the geometric shape of a saw tooth. The tooth shape used was the raker set style that is generally used in band sawing. A set of teeth comprises three teeth, ranked as left, straight, and right. The mechanistic model developed in the research considered the shape of each tooth in a set. The predicted cutting forces coincided well with those measured in the validation experiment. Therefore, the predicted cutting forces in band sawing can be used for the adaptive control of saw-engaging feed rate in band sawing. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Band sawing machine; Mechanistic cutting force model; Multi-point cutting; Saw blade; Single-point cutting; Tooth form
1. Introduction Sawing machines are of primary value for the preparation of raw materials to be machined, and constitute some of the most important machine tools found in a machine shop. Common types of cutoff machines include reciprocating saws, horizontal endless band saws, universal tilt frame band saws, abrasive saws, and cold saws [1]. Band saw machines use a steel band blade in the form of a band with the teeth on the edge. The band saw machine is widely used for the following reasons: it has a high cutting efficiency because the band cuts continuously with no
* Corresponding author. Tel: ⫹ 82-53-810-2576; fax: ⫹ 82-53-813-3703; e-mail: [email protected] 0890-6955/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 9 8 ) 0 0 0 8 7 - X
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wasted motion; material loss is small due to the small kerf of the saw cut; the feed rate through the material may be varied; the machine can handle workpieces of large dimensions. In general, such as when the workpiece is a cylindrical rod, the length of the cut changes progressively during in-feed of the band saw. In the case of constant in-feed rate, the load on the toothed edge increases along with the radial engagement of the cut, and this variation of the load induces vibration in the machine. The vibration affects the kerf of the saw cut and leads to severe tooth wear and breakage, causing deterioration of surface roughness and tolerance of the cut dimensions. With regard to this problem, Ulsoy and Morte [2] developed the equation of motion, based on Hamilton’s principle, to the vibration of wide band saw blades, and obtained approximate solutions using both the classical Ritz and finite element-Ritz methods. Carlin et al. [3] analysed the buckling and vibration of a circular saw blade subjected to a combination of loading conditions approximating those encountered in operation. Chandrasekaran and colleagues [4,5] researched tooth chipping during the band sawing of steel, and Sarwar and colleagues [6,7] researched the relationship between cutting forces and friction characteristics and the parameters affecting the performance of a tooth blade. To deal with these problems at their origin, it is necessary to be able to predict the magnitude and variation of cutting force. However, it is not an easy matter to predict cutting force in band sawing since it is not a single-point cutting but multi-point cutting as in the milling process, and because the geometric shape of the cutting edge varies due to the offset on each side to provide clearance for the back of the blade. In previous studies, Sarwar et al. [8] analysed cutting forces with a finite element method and a single-point cutting technique using a turning lathe, and Henderer et al. [9] suggested an analytical method for predicting the cutting force of a saw blade in two-dimensional cutting. However, these models are based on two-dimensional cutting, and require prior knowledge of the shear angle. Nor can they provide the pulsation cutting forces needed in the vibration analysis, but give only the static mean cutting forces. In this work, we developed a mechanistic model for predicting cutting forces in multi-point cutting by a band saw. The cutting force system in face milling has been extensively studied both analytically and empirically. In analytical modelling approaches, theories of single-point cutting such as energy method, flow stress method, matrix method, and single-shear plane method may be applied. While such a model may be sound in principle, it requires knowledge of the shear angle, dynamic stress, friction angle, etc., parameters which are usually not easy to determine in practice. As a result, a more empirical approach to modelling a face milling has been popular. The mathematical model for predicting cutting forces has been widely used after Martellotti [10,11], who developed mathematical equations for the milling process, calculating analytically with the tooth path, instantaneous undeformed chip thickness, etc. He also introduced the notion that the average undeformed chip thickness could be used in establishing a relationship between the conditions of the cut and specific pressure required. Later, Koenigsberger and Sabberwal [12] also observed that there is a strong relationship between the instantaneous chip thickness and the tangential cutting force. Tlusty and MacNeil [13] examined the variation of cutting forces in flatend milling at steady state and transient cutting conditions. Most of the research to date has dealt with the development of force equations and the modelling of specific cutting pressure under the simplest of conditions. However, Kline et al. [14] have established a mechanistic force model for end-milling under various cutting environments. More recently, Fussell and Srinivasan [15] investigated the capability of the model developed by Kline et al. [14] under varying machining
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conditions. Fu et al. [16] used the general approach of Martellotti’s method [10,11] to develop a mechanistic force model for face milling. Armarego and Deshpande [17] studied the effects of cutter runout and developed a computerized cutting force prediction model for flat end-milling. Feng and Menq [18,19] reported a rigid system cutting force prediction model for the ball-end milling process. The mechanistic model in this work for predicting cutting forces in band sawing was developed by introducing Martellotti’s model [10,11] that uses instantaneous undeformed chip thickness and specific cutting pressure. To this end, the specific cutting pressure was obtained by a single-point cutting technique, analogous to fly cutting in face milling. Single-point cutting can be performed by using the workpieces with a thickness smaller than the interval between adjacent teeth on the saw. Then, the cutting forces were predicted by applying the specific cutting pressure to a geometric model, which considers the geometric profile of a band saw tooth such as left-bent, straight, right-bent tooth.
2. Materials and terminology 2.1. Geometry of saw blade The shape of a saw blade is described in terms of tooth form, set and pitch. Saw tooth forms are referred to as ‘standard’, ‘skip’, or ‘hook’. The standard form gives accurate cuts with a smooth finish [1], and is used in this research. The teeth of a saw blade must be offset on each side to provide clearance for the back of the blade. Set forms include ‘raker’, ‘straight’, and ‘wave’. The raker set is used in general sawing and selected in the research. The pitch of a saw blade is the number of teeth per inch; in this case, the pitch is three. Fig. 1 shows the geometry of the saw blade.
Fig. 1. Geometry of saw blade.
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2.2. Undeformed chip thickness Fig. 2 represents the cutting mechanism during band sawing. In band sawing, the feed per tooth is equal to the depth of cut of each tooth, since they are measured in the same direction. This depends on both the feed rate and cutting velocity, and can be determined by the following expression ti ⫽
p·f
(1)
where ti is the depth of cut, p is the distance between successive teeth, f is the in-feed rate, and is the cutting velocity. As shown in Fig. 2(a), the saw blade is tilted with a slope angle of a for efficient cutting. However, the slope angle is very small; the actual depth of cut d is approximated as ti. Therefore, the undeformed chip area, as shown in Fig. 2(b), is given in Eq. (2): A ⫽ b·ti
(2)
where b is the width of the saw blade.
Fig. 2. Cutting mechanism of band sawing (a) cutting mechanism of band saw; (b) actual cutting area of saw tooth.
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3. Development of the force system model 3.1. Specific cutting pressure A rectangular Cartesian coordinate is set up with the origin at the center of edge end and with the X-axis in the cutting direction. The Z-axis is perpendicular to the machined surface and directed downward. The Y-direction is then determined by the right-hand rule, as shown in Fig. 3. The normal cutting force is defined as the X-directional force, that is, normal to the tooth face. The Z-direction is radial to the tool face and directed downward. The Y-direction, lateral to the tool face, is then determined by the right-hand rule. Martellotti [10,11] has proposed that the normal cutting force acting on the chip cross-section is the product of the undeformed chip area and the specific cutting pressure, ks. The lateral and the radial force acting along the cutting edge are obtained by multiplying the normal force by the empirical constants, ky, kz, respectively. By ignoring the effects of tooth geometry, a specific cutting pressure ks can be obtained by dividing the X-directional mean cutting force per tooth by the undeformed chip area A of Eq. (2). Specific cutting coefficients ky, kz of the Y-, Z-directions are obtained by dividing Y-, Z-directional cutting forces by X-normal force, respectively. Accordingly, specific cutting pressure and specific cutting coefficients are written as follows: ks ⫽
Fx A
ky ⫽
Fy Fx
kz ⫽
Fz Fx
(3)
where Fx, Fy and Fz are the mean cutting forces per tooth of X, Y, Z direction, respectively.
Fig. 3. Forces in cutting edge.
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Fig. 4.
Set patterns.
3.2. Cutting force model from geometry of saw tooth In the raker set, as shown in Fig. 4, the saw blade has left-bent, straight, and right-bent tooth, iteratively. Therefore, the cutting force model considers these three kinds of geometric shape. First, as shown in Fig. 5, in the case of a left-bent tooth, the instantaneous cutting force per tooth can be modelled by decomposing normal, lateral and radial forces as
冦
FXl(i ⫺ 1,d)
冧冤
cos␥z
FYl(i ⫺ 1,d) ⫽ sin␥z FZl(i ⫺ 1,d)
0
⫺ sin␥z
冥冦
冧
⫺ sin␥xsin␥z Fxl(i ⫺ 1,d)
cos␥xcos␥z
sin␥xcos␥z
⫺ sin␥xcos␥z
cos␥x
Fig. 5. Left bent saw tooth.
Fyl(i ⫺ 1,d) Fzl(i ⫺ 1,d)
(4)
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where ␥x and ␥z are the tooth rotational angles with respect to the X, Z axes, respectively, i is the order of tooth, d is the cutting distance. The symbol l means left. Fx(i,d), Fy(i,d), and Fz(i,) is the normal, lateral and radial component of the tooth face as shown in Fig. 5, respectively. FX(i,d), FY(i,d), and FZ(i,d) is the X-, Y-, and Z-directional instantaneous cutting force, respectively. Substituting a specific cutting pressure, specific cutting coefficient and undeformed chip area into Eq. (4), we can obtain instantaneous cutting forces. Instantaneous cutting forces of the straight tooth following the left-bent tooth are similar to Eq. (4), except that the normal direction of the undeformed chip area is coincident with the X axis, as shown in Fig. 6. Therefore, cutting forces can be modelled as follows:
冦 冧 冤 冥冦 冧 FXs(i,d)
1 0 0 Fxs(i,d)
FYs(i,d) ⫽ 0 1 0
FZs(i,d)
(5)
0
0 0 1 Fzs(i,d)
where s represents straight. On the other hand, similarly to the left-bent tooth, instantaneous cutting forces of the rightbent tooth, as shown in Fig. 7 are modelled as follows:
冦
FXr(i ⫹ 1,d)
冧冤
cos␥z
⫺ sin␥z
冥冦
冧
⫺ sin␥xsin␥z Fxr(i ⫹ 1,d)
FYr(i ⫹ 1,d) ⫽ sin␥z ⫺ cos␥xcos␥z ⫺ sin␥xcos␥z Fyr(i ⫹ 1,d) FZr(i ⫹ 1,d)
0
⫺ sin␥xcos␥z
cos␥x
Fzr(i ⫹ 1,d)
(6)
where r represents right. Eqs. (3)–(6) are used to describe forces on a single tooth. Therefore, in the case of multi-point
Fig. 6. Straight saw tooth.
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Fig. 7. Right bent saw tooth.
cutting, where more than one tooth is engaged simultaneously in cutting, the cutting forces are predicted by summing up Eqs. (4)–(6) as follows:
冦 冧 冘冦 冦 FX
n
FXs(i,d)
FXr(i ⫹ 1,d)
␦l FYl(i ⫺ 1,d) ⫹ ␦s FYs(i,d) ⫹ ␦r FYr(i ⫹ 1,d)
FY ⫽ FZ
冧 冦 冧 冦
FXl(i ⫺ 1,d)
i⫽1
FZl(i ⫺ 1,d)
FZs(i,d)
FZr(i ⫹ 1,d)
冧冧
(7)
where n is the total number of teeth, and ␦ is a Kronecker delta. ␦ is 1 when the tooth is engaged in the cutting, and zero when the tooth is out of the workpiece. FX, FY, and FZ is X-, Y-, and Z-directional cutting force, respectively. 4. Cutting experiments Fig. 8 shows the horizontal band saw machine (KDBS 450A: Kyoung-Dong Co.) used in the experiment. The tooth blade (Bearcat M42: STARRETT) is a standard tooth form, three pitch, and raker set. In order to measure three directional cutting forces, a tool dynamometer (9257A: KISTLER) is mounted on the table of the saw machine. A fixture, to hold the workpiece, is bolted on to the dynamometer. The cutting force signal from the tool dynamometer is amplified by the charge amplifier, and is collected by the computer through the A/D converter. The workpiece is a rectangular rod of mild steel (AISI 1010) 8 ⫻ 25 mm. The distance between each tooth is 8.3 mm, so that single-point cutting is possible if we cut parallel to the 8 mm edge. Hence, mean cutting force per tooth can be obtained, and specific cutting pressure can be calculated from the mean cutting force. Whereas Sarwar et al. [8] obtained specific cutting pressure using one tooth cut from a saw
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Fig. 8. Experimental set-up.
blade by turning a cylindrical part with the tooth fixed in the tool holder of a turning lathe, we could measure mean cutting force per tooth simply by adjusting the width of the workpiece. Thus, rotating the workpiece to cut a 25 mm thickness engages three teeth in the sawing. The corresponding force was measured in the same way and used to check the cutting force model. Experimental conditions used in the test are summarized in Table 1. 5. Results and discussion 5.1. Modelling of specific cutting pressure In order to predict cutting forces in sawing, first, a model of specific cutting pressure is built. To this end, a total of 20 kinds of cutting force signals were measured, as shown in Fig. 9: four cutting speeds (30–77 m/min), and five feed rates (15–332 mm/min). It is possible to use observed average forces to estimate the values of average ks. Therefore, we obtain various ks with respect Table 1 Experimental conditions Cutting speed (m/min) Feed rate (mm/min) Tilt angle about X axis (rx) Tilt angle about Y axis (rz) Saw blade dimension (mm) Sampling frequency (Hz) Number of sampled data Workpiece material
30, 48 15, 40, 84, 189 5° 6° 4670 ⫻ 38 ⫻ 1.3 1000 1024 SS41
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Fig. 9. Measured cutting forces with fly cutting.
to different cutting conditions by applying average cutting forces to Eq. (4). To model a specific cutting pressure, in general, an equation of ks ⫽ ␣·A is used. This is a relationship between specific cutting pressure and undeformed chip area, where ␣ and  are constants and A is the undeformed chip area. From this equation, the following model is fitted; Fig. 10 shows a graph of its value. ks ⫽ 1427.6A−0.295
(8)
The other specific cutting coefficients of ky, kz are calculated using Eq. (3). 5.2. Model verification in fly cutting Fig. 11 shows the simulated and measured cutting forces with respect to the cutting conditions of feed rate 84 mm/min, and cutting speed 30 m/min. The simulation cutting force signal is trapezoidal, whereas the measured cutting force is not. This is due to external factors such as band vibration. However, the magnitude of the cutting force is coincident. Cutting force in the X direction, that is, cutting direction, is two times larger than cutting force in the Z direction. In
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Fig. 10. Estimation of specific cutting force.
Fig. 11. Measured and simulated cutting forces with fly cutting (cutting speed: 30 m/min, feed rate: 84 mm/min).
general, the ratio of thrust force to main cutting force is 0.4, while the ratio is 0.5 in the saw cutting. The reason for the difference is that the feed direction coincides with the thrust direction. The force in the Y direction is very small and it is difficult to measure the cutting force. This verifies that cutting force in the Y direction does not affect the cutting process. As shown above, in the case of single-point cutting, cutting forces can be predicted well by a specific cutting
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pressure. Whereas the external component has a dominant effect on the variation of total cutting force, the geometric shape of the tooth blade (left, right, or straight) has none. 5.3. Model verification in multi-point cutting In multi-point cutting, using a workpiece of 25 mm thickness, three teeth engage in the sawing, simultaneously. Fig. 12 shows the measured and simulated cutting forces under the conditions of cutting speed 48 m/min, and feed rate of 189 mm/min. The pattern of the cutting force signal is trapezoid, similar to that for single-point cutting. The simulated cutting force is good enough to predict the real cutting force. Furthermore, the Y-directional cutting force is negligible and the ratio of the X- and Z-directional cutting force is the same as in single-point cutting. Thus, in multipoint cutting, the cutting force model is satisfactory. However, to predict cutting force even more precisely, the runout of tooth-in-tooth generation and band vibration should also be considered. In the experiments, the cutting forces of each direction are three times those in single-point cutting. Therefore, in multi-point cutting, the cutting forces can be predicted simply by multiplying the total engaged tooth number by the cutting forces for single-point cutting. 6. Conclusion In order to predict cutting forces in band sawing, a mechanistic model was developed by considering the geometry of the saw tooth. A specific cutting pressure is necessary for using a mech-
Fig. 12. Measured and simulated cutting forces with multipoint cutting (cutting speed: 48 m/min, feed rate: 189 mm/min).
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anistic model. To this end, a single-point cutting method, like fly cutting in milling, was introduced to obtain specific cutting pressure. Using the cutting forces from single-point cutting, a specific cutting pressure model was built. From the verification experiments for both single-point and multi-point cutting, we conclude that the mechanistic model developed could predict cutting forces in band sawing very well. However, further work should also consider non-uniform tooth shapes, transverse vibration of the saw blade, and imperfect balance of the driving wheel, for even more precise predictions of the cutting force. Acknowledgement This work was partly supported by a KOSEF grant, no. 95-2-09-04-01-1. References [1] P.F. Ostwald, J. Munoz, Manufacturing processes and systems, John Wiley and Sons, Inc., 1997. [2] A.G. Ulsoy, C.D. Morte, Vibration of wide band saw blades, ASME Journal of Engineering for Industry 104 (1982) 71–78. [3] J.F. Carlin, F.C. Appl, H.C. Bridwell, R.P. Dubois, Effects of tensioning on buckling and vibration of circular saw blades, ASME Journal of Engineering for Industry February (1975) 37–48. [4] H. Chandrasekaran, S. Svensson, M. Nissle, Tooth chipping during power hack sawing and the role of saw material characteristics, Annals of the CIRP 36 (1) (1987) 27–31. [5] H. Chandrasekaran, H. Thoors, H. Hellbergh, L. Johansson, Tooth chipping during band sawing of steel, Annals of the CIRP 41 (1) (1992) 107–111. [6] M. Sarwar, D. Gillibrand, S.R. Bradbury, Forces, surface finish and friction characteristics in surface engineered single- and multi-point cutting edges, Surface and Coating Technology 49 (1991) 443–450. [7] W.M.M. Hales, M. Sarwar, Geometrical parameters affecting hacksaw blade performance, Ninth International Conference on Production Engineering Research, Cincinatti, 1987, pp. 1802–1810. [8] M. Sawar, S.R. Bradbury, M. Dinsdale, An approach to computer aided band saw teeth testing and design, Proceedings of the Fourth National Conference on Production Research, 1988, pp. 494–501. [9] W.E. Henderer, J.D. Boor, J.R. Holston, Estimation of cutting forces in band sawing metals, Transactions of NAMRC 24 (1996) 33–38. [10] M.E. Martellotti, An analysis of the milling process, Transactions of ASME 63 (1941) 667. [11] M.E. Martellotti, An analysis of the milling process. Part II: Down milling, Transaction of ASME 67 (1945) 233. [12] F. Koenigsberger, A.J.P. Sabberwal, An investigation into the cutting force pulsation during milling operation, International Journal of Machine Tool Design and Research 1 (1961) 15. [13] J. Tlusty, P. MacNeil, Dynamics of cutting forces in end-milling, Annals of CIRP 24 (1) (1975) 21–25. [14] W.A. Kline, R.E. DeVor, J.R. Lindberger, The prediction of cutting forces in end milling with application to cornering cut, International Journal of Machine Tool Design and Research 22 (1) (1982) 7–22. [15] B.K. Fussell, K. Srinivasan, An investigation of the end-milling process under varying machining conditions, ASME Journal of Engineering for Industry 111 (1989) 27–36. [16] H.J. Fu, R.E. DeVor, S.G. Kapoor, A mechanistic model for the prediction of the force system in face milling operation, ASME Journal of Engineering for Industry 106 (1984) 81–88. [17] E.J.A. Armarego, N.P. Deshpande, Computerized predictive cutting models for forces in end-milling including eccentricity effects, Annals of CIRP 38 (1) (1989) 45–49. [18] H.Y. Feng, C.H. Menq, The prediction of cutting forces in the ball-end milling processes. Part 1: Model formation and model building procedure, International Journal of Machine Tools Manufacture 34 (5) (1994) 697–710. [19] H.Y. Feng, C.H. Menq, The prediction of cutting forces in the ball-end milling processes. Part 2: Cut geometry analysis and model verification, International Journal of Machine Tools Manufacture 34 (5) (1994) 711–720.