# ORTHOGONAL CUTTING MECHANICS OF MEDIUM DENSITY FIBERBOARDS.

ABSTRACT

The modeling and experimental analysis of orthogonal cutting mechanics are presented for machining medium density fiberboard (MDF). The forces exerted on the tool during cutting are decomposed into rake face and flank face components. A Coulomb friction model is assumed on both the flank and rake faces, and the friction constants on both faces of the tool are predicted. The experimental results showed that the friction on the rake face is rather small, and the pressure exerted by the uncut chip on the rake face mainly dominates the force on the rake face. The cutting forces are expressed as functions of cutting constants, tool geometry, and uncut chip area. The cutting constants are a function of the MDF density, which changes along the board thickness, and the rake angle of the tool. The proposed cutting constants can be used to predict the cutting forces in machining MDF with tools having complex geometry such as router bits.

The mechanics of cutting materials are useful in estimating the cutting forces, predicting the vibration stability of machining operations, and predicting the surface finish on the machined surface. In addition, the laws of mechanics could be used to relate the tool geometry to machining performance parameters such as temperature and stress loading of the tool and in explaining the chip formation mechanism. While the cutting mechanics for metals have been studied extensively in the literature, wood, and especially medium density fiberboard (MDF), cutting mechanics have not receive as much attention [5-9].

In this paper, the mechanics of cutting MDF were studied based on the experience gained in metal cutting mechanics and dynamics [1,2]. Since MDF is not a homogenous material like metal, the shear laws used in metal cutting are not directly applicable. The authors propose to identify the MDF cutting mechanics by using a mechanistic force model around the cutting-wedge. The parameters in the force models are identified as a function of the depth of cut and the rake angle. The density and hardness distribution of MDF along the thickness are also related to the cutting force parameters. The resulting orthogonal cutting force parameters can be used in predicting the cutting forces in oblique cutting [1,2], such as milling with a complex shape cutter [4].

CUTTING FORCE MODEL

There are two fundamental cutting forces in orthogonal cutting: 1) a tangential force ([F.sub.t]), which is in the direction of the cutting speed (V); and 2) feed force ([F.sub.f]), which is perpendicular to the cutting speed (Fig. 1). In orthogonal cutting, the deformation is assumed to be uniform along the width of the cut. Some of the cuffing forces are contributed by the flank edge and the rubbing of the finished surface against the flank face. In metal cutting, the cutting forces on the rake face can be expressed as a function of the material's shear stress, average rake face-chip contact friction, and shear angle. However, the shear law based model is not directly applicable in MDF cutting due to the non-uniform material properties and fracture of chips. Instead, the mechanistic force model shown in Figure 1 is proposed for machining MDF.

The cutting force ([F.sub.c]) resulting from the contact between the rake face and the chip is decomposed into a tangential component ([F.sub.tc]) and a feed component ([F.sub.fc]). Similarly, force ([F.sub.f]) resulting from the contact between the flank face and the finished MDF surface is decomposed into its ([F.sub.te], [F.sub.fe]) components. Let h denote the uncut chip thickness (or feed rate in orthogonal cutting) and b denote the width of cut. The tangential and feed forces can each be decomposed into a shear force and an edge force. The shear force is proportional to the uncut chip area (bh), whereas the edge force is proportional to the cutting edge contact length or width of cut (b). It follows that:

[F.sub.t] = [F.sub.tc] + [F.sub.te] = [K.sub.tc]bh + [K.sub.te]b

[F.sub.f] = [F.sub.fc] = [F.sub.fe] = [K.sub.fc]bh + [K.sub.fe]b [1]

where:

[K.sub.tc], [K.sub.fc] and [K.sub.te], [K.sub.fe] = the cutting coefficients corresponding to the shearing and edge forces, respectively

The cutting constants have to be identified from measured tangential ([F.sub.t]) and feed ([F.sub.f]) forces. A series of orthogonal cutting tests are conducted at incremental feed rates, and the two orthogonal cutting forces are measured. By extrapolating the measured data at zero uncut chip thickness (i.e., h = 0), the edge force constants ([K.sub.te], [K.sub.fe]) can be evaluated. After subtracting the edge forces from each measurement, the cutting constants ([K.sub.tc], [K.sub.fc]) can be evaluated as well, as presented in the experimental section.

Based on this force model, it is now desired to estimate the friction forces on the rake face and the flank face. The forces on both the rake and flank faces can be decomposed into their normal and friction components, as shown in Figure 1. Let [gamma] denote the clearance angle and [alpha] the rake angle. The measured tangential and feed forces can then be expressed as:

[F.sub.f] = [F.sub.rf]cos [alpha] - [F.sub.rn] sin [alpha] + [F.sub.fn] cos [gamma] + [F.sub.ff] sin [gamma]

[F.sub.t] = [F.sub.rf] sin [alpha] + [F.sub.rn] cos [alpha] - [F.sub.fn] sin [gamma] + [F.sub.ff] cos [gamma] [2]

where:

[F.sub.rf], [F.sub.rn] = rake face friction and normal forces

[F.sub.ff], [F.sub.fn] = flank face friction and normal forces

For simplicity, the dominant sliding friction forces between the chip and the rake face, and between the freshly cut MDF surface and tool flank, are assumed in both cases to satisfy a Coulomb friction model. Consequently, the friction coefficients for the flank face

and rake face are given by [[micro].sub.f] = [F.sub.ff]/[F.sub.fn]

and [[micro].sub.r] = [F.sub.rf]/[F.sub.rn], respectively. The distribution of normal loads on the friction surfaces is not known and is difficult to measure in MDF cutting. For simplicity, a uniform distribution of pressure [[sigma].sub.r] on the rake face and the pressure [[sigma].sub.f] on the flank face is assumed, which yields [F.sub.rn] = [[sigma].sub.r][bl.sub.r], [F.sub.rf] = [[micro].sub.r][[sigma].sub.r][bl.sub.r], [F.sub.fn] = [[sigma].sub.f][bl.sub.f], and [F.sub.ff] = [[micro].sub.f][[sigma].sub.f][bl.sub.f], where [l.sub.r] and [l.sub.f] denote, respectively, the chip contact length on the rake face and the workpiece-flank contact length on the flank face. The chip contact length ([l.sub.r]) is known to be proportional to the uncut chip thinkness, [l.sub.r] = [eta]h, where [eta] is a proportionality constant. The flank contact depends on the sharpness of the cutting edge, wear land, recess, and the elastic springback of the freshly cut MDF surface. The flank contact length ([l.sub.f]) is assumed to be independent of the chip load, i.e., the feed rate h, since it depends on the tool sharpness. These assumptions yield:

[F.sub.f] = [[sigma].sub.ro]bh([[micro].sub.r] cos [alpha] - sin [alpha]) + [[sigma].sub.f][bl.sub.f](cos [gamma] + [[micro].sub.f] sin [gamma])

[F.sub.t] = [[sigma].sub.ro]bh([[micro].sub.r] sin [alpha] + cos [alpha]) + [[sigma].sub.f][bl.sub.f](- sin [gamma] + [[micro].sub.f] cos [gamma]) [3]

where:

[[sigma].sub.ro] = [[eta][sigma].sub.r]

The following definitions are used in subsequent calculations:

[F.sub.fc] = [[sigma].sub.ro]bh([[micro].sub.r] cos [alpha] - sin [alpha])

[F.sub.fe] = [[sigma].sub.f][bl.sub.f](cos [gamma] + [[micro].sub.f] sin [gamma])

[F.sub.tc] = [[sigma].sub.ro]bh([[micro].sub.r] sin [alpha] + cos [alpha])

[F.sub.te] = [[sigma].sub.f][bl.sub.f](- sin [gamma] + [[micro].sub.f] cos [gamma]) [4]

At zero uncut chip thickness (h = 0), we have [F.sub.fe]/[F.sub.te] = [[sigma].sub.f][bl.sub.f](cos [gamma] + [[micro].sub.f] sin [gamma])/[[sigma].sub.f][bl.sub.f](- sin [gamma] + [[micro].sub.f] cos [gamma]),

Which yields the following expression for the flank face friction coefficient:

[[micro].sub.f] = [F.sub.te] + [F.sub.fe] tan [gamma] / [F.sub.fe] - [F.sub.te] tan [gamma] [5]

The force components [F.sub.fe] and [F.sub.te] correspond to edge forces and are given by [F.sub.fe] = [bK.sub.fe] and [F.sub.te] = [bK.sub.te]. Consequently, the friction coefficient on the flank face can be rewritten as [[micro].sub.f] = ([K.sub.te] + [K.sub.fe] tan [gamma])/([K.sub.fe] - [K.sub.te] tan [gamma]). Similarly, the friction coefficient [[micro].sub.r] on the rake face is given by [[micro].sub.r] = ([F.sub.fc] + [F.sub.tc] tan [alpha])/([F.sub.tc] - [F.sub.fc] tan [alpha]). Since the sliding friction may be too small in MDF cutting, care must be taken in evaluating the average friction constant on the rake face. In order to keep the influence of feed rate on the friction, the forces can be normalized with respect to width of cut (b), i.e., [F.sub.uf] = [F.sub.f]/b, [F.sub.ut] = [F.sub.t]/b, and the friction constant can be evaluated for every measurement using:

[[micro].sub.r] = ([F.sub.uf] - [K.sub.fe]) + ([F.sub.ut] - [K.sub.te]) tan [alpha] / ([F.sub.ut] - [K.sub.te]) - ([F.sub.uf] - [K.sub.fe]) tan [alpha] [6]

The general Equations [1] through [6] are helpful in modeling the mechanics of cutting MDF, and can be used in predicting the cutting forces produced by any oblique cutting tool such as router bits and shape cutters used in the furniture industry.

EXPERIMENTAL RESULTS

The cutting tests were conducted on a Cincinnati Falcon 300 CNC lathe. Sandvik carbide inserts with relief angle ([gamma]) of 10 degrees and rake angle ([alpha]) of 0, 10, and 15 degrees were used. A Kistler three component force dynamometer type 9257 B A was used to measure the forces. The measured forces were first recorded on an XR 310 Analog Cassette Data Recorder. The measured forces were filtered using an analog low-pass filter with a cutoff frequency 10 times the spindle speed. The filtered signals are then sampled at 2000 Hz, and used in analyzing the mechanics.

FACE TURNING TESTS

The cutting tests were performed on an MDF disk 500 mm in diameter that was attached with nine screws to an aluminum plate; the plate was attached to the spindle of the lathe (Fig. 2). Orthogonal cutting tests were conducted on MDF ribs that were 6mm wide and 4 mm high. The ribs were obtained by cutting 4-mm-deep grooves on both sides of the desired rib location. Three test series were conducted and the cutting conditions are summarized in Table 1. Due to the large number of cutting tests that had to be performed, each test was conducted only once. However, extreme care was taken to ensure that the collected data are accurate.

A quick examination of the both the tangential and feed forces from the test series 1 revealed that the forces do not greatly change with changes in the cutting speed. An example is shown Figure 3 for the case of h = 0.075 mm/rev and [alpha] = +15 degrees. Consequently, the measured forces corresponding to the same feed rate and rake angle, but different cutting speeds, are averaged. The resulting average forces are used in the modeling process based on Equations [1] through [6]. Figure 3 also shows that the measured forces decrease in magnitude with the increase in the depth at which the MDF is being cut. The decrease in the force magnitude is due to the decrease in the MDF density and hardness from the surface towards the inner MDF core.

Figure 4 illustrates the change in the forces as a function of both the depths of cut (a) and feed rate. While the tangential forces linearly increase with the feed rate, the feed forces do not demonstrate strong dependence on the feed rate. Later, the proposed mechanistic model is used to explain the observed trend. Since the forces are normalized with respect to the width of cut, the edge force coefficients ([K.sub.te], [K.sub.fe]) for each depth of cut can then be obtained from the normalized force curves at h = 0 mm/rev. After subtracting the edge forces from the normalized measured tangential and feed forces, the cutting constants [K.sub.tc], [K.sub.fe]) can be identified as a function of the depth (a) at which the MDF is being cut. High order polynomial curves are used to fit the cutting coefficients, with an average error of less than 3 to 4 percent between the measured and predicted forces. The results are compiled in Table 2. The layer (a = 0.3 mm) corresponds to the hard surface of the MDF. The har dness decreases between 0.3 [less than or equal to] a [less than or equal to] 3.6, and remains approximately constant at the inner soft region 3.6 [less than or equal to] a [less than or equal to] 9. The MDF board is assumed to have symmetric hardness in its second half (9 [less than or equal to] a [less than or equal to] 18). The friction coefficients on both the flank and the rake faces are evaluated and are included in previous report (3), but are not shown here due to space limitations. However, they can be evaluated by applying Equations [1] through [6] to the cutting coefficients given in Table 2. The resulting force models are useful in describing cutting operations such as drilling, where the tool is perpendicular to the MDF surface, and the tool penetrates from the top towards inner layers. However, in operations such as milling and routing, the feed motion is parallel to the board surface, therefore plunge turning tests must be used in order to be able to describe such operations.

PLUNGE TURNING

The setup is similar to that used in the face turning tests except that the tool now cuts the periphery of the disk and moves in a radial direction towards the center of the disk (Fig. 5). Three distinct MDF layers, each 3 mm thick, are considered in the cutting tests. Smaller increments were not possible due to the difficulty in measuring the small magnitude forces accurately. The cutting conditions are summarized in Table 3.

As in face turning, only slight changes in the forces are observed as the cutting speed changes. Consequently, measured forces for different cutting speeds, but otherwise similar cutting conditions, are averaged. The resulting average forces are used in the modeling process. The variation of the normalized tangential and feed forces as a function of the layer being cut, the feed rate, and the rake angle, is shown in Figure 6. The identified cutting constants, shown in Table 4, are a function of the MDF layer and the rake angle. The tangential forces increase linearly with the increase in the uncut chip thickness or feed rate (h). However, Figure 6 also indicates that the feed forces decrease with the increase in the feed rate. Such behavior of the feed force leads to a negative friction coefficient if regular orthogonal cutting mechanics laws are used. The identified friction coefficients as a function of the rake angle, uncut chip thickness, and axial MDF layer being cut are shown in Figure 7. The cutting f orces are extremely small at very small feed rates (h [less than] 0.050 mm/rev). Consequently, the force readings at such feed rates may not reflect the cutting mechanism correctly, and may lead to negative rake friction coefficients as shown in Figure 7. The friction on the flank face is independent of chip thickness, and its variation at different MDF layers and rake angles is also shown in Figure 7.

In plunge-turning tests, the cut chip was usually of the size of a small particle. The MDF chip simply fractures, flows away from the rake face and, consequently, does not move on the rake face as a rigid body. Such chip behavior leads to a very small friction coefficient on the rake face for all rake angles, as indicated in Figure 7. Therefore, the dominant force on the rake face is the normal force exerted by the uncut chip on the rake contact area. The normal force on the rake face can be derived from Equations [1] through [6] and Figure 1 as shown in Equation [7].

The normal forces evaluated using Equation [7] are shown in Figure 8. It can be seen that the normal rake forces are quite insensitive to the rake angle. Moreover, the normal forces are approximately linearly dependent on the feed rate (h). Since the hardness at each MDF layer is different, the normal force becomes dependent only on the MDF hardness and the uncut chip area (bh).

With negligible rake friction coefficients, the cutting force on the rake face can be approximated with the normal rake force. Consequently, for a given rake angle, as the feed rate increases, the normal force increases in magnitude (Fig. 8). The latter leads to an increase in the magnitude of the normal rake force projection in the feed direction, which would be opposing the feed force. Assuming a constant flank force for all feed rates, it follows that the projections of the flank forces in the feed direction are of constant magnitude. As shown in Figure 1, the feed force is mainly represented by the sum of the projections, in the feed direction, of the normal rake force and the flank forces. Consequently, the feed force decreases in magnitude as the feed rate increases and could even become negative if the normal rake force is large enough.

If flank friction and pressure are identified, and rake face friction force is neglected, the normal cutting force could be identified from the bending strength, which changes as a function of density along the MDF thickness. The hardness of MDF at each layer was measured using a Rex Model D Durometer, and correlated to the normal rake force as:

[F.sub.rn] [approximate] [delta] H bh

0.267 [less than or equal to] [delta] [less than or equal to] 0.280 [8]

where:

H[N] = the MDF hardness at the layer being cut

Equation [8] can be used as an approximate relationship for average calculations in tool design. For more accurate calculations, the proposed procedure and cutting constants given in Table 4 must be used. The authors measured the density of the MDF, and used a relationship between the density and bending strength of MDF given in a report by Suchsland and Woodson [10]. A similar relationship was found between the normal force and bending strength of the MDF.

CONCLUSION

The mechanics of orthogonal cutting of MDF were modeled. Orthogonal cutting tests, both perpendicular and parallel to the MDF surface, have been conducted. These tests are useful in describing the mechanics of drilling and routing when machining MDF parts. The variation of cutting constants with the rake angle, MDF density changes along the thickness, and feed rate are identified. With the aid of the proposed mechanics model, it is shown that the sliding friction on the rake face of the tool in MDF machining is rather small, and the pressure on the rake face mainly dominates the cutting force. The pressure on the rake face is found to be linearly proportional to the uncut chip area and bending stress/hardness of MDF boards. The cutting constants and the mechanics model presented in this paper can be used in predicting the cutting forces in routing using orthogonal to oblique cutting transformation as presented in a separate article [4]. More information can be found in a previous article [3].

The authors are, respectively, Student, Univ. of British Columbia (UBC), Dept. of Mechanical Engineering, 2324 Main Mall, Vancouver, BC, Canada V6T 1Z4; Technical Officer and Research Officer, National Res. Council, Innovation Centre, 3250 East Mall, Vancouver, BC, Canada, V6T 1W5; and Professor, UBC. This paper was received for publication in July 1999. Reprint No. 9010.

LITERATURE CITED

(1.) Altintas, Y. 2000. Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design. Cambridge Univ. Press, New York. ISBN 0521650291.

(2.) Budak, E., Y. Altintas, and E.J.A. Armarego, 1996. Prediction of milling force coefficients from orthogonal cutting data. Transactions of the American Soc. of Mechanical Engineers (ASME). J. of Manufacturing Sci. and Engineering (former name: J. Engineering for Industry) 118(2):216-224.

(3.) Dippon, J., H. Ren, S. Engin, F. Ben Amara, and Y. Altintas. 1999. Cutting mechanics of medium density fiberboard machining. Tech. Rept. The National Res. Council Canada - Innovation Centre, Vancouver, B.C., Canada.

(4.) Engin, S., Y. Altintas, and F. Ben Amara. 200_. Mechanics of routing medium density fiberboards. Forest Prod. J. (in press).

(5.) Fischer, R. 1997. A way to observe and calculate edge wearing in cutting wood materials. In: Proc. 13th Inter. Wood Machining Seminar. Univ. of British Columbia, Vancouver, B.C., Canada. Vol. II. pp. 631-640.

(6.) Koch, P. 1964. Wood Machining Process. Ronald Press Co., New York.

(7.) McKenzie, W.M. 1960. Fundamental aspects of the wood cutting process. Forest Prod. J. 10(9):447-456.

(8.) Sitkei, G. 1997. On the mechanics of oblique cutting of wood. In: Proc. 13th Inter. Wood Machining Seminar. Univ. of British Columbia, Vancouver, B.C., Canada. Vol. II. pp. 469-476.

(9.) Stewart, H.A. 1988. Analysis of tool forces and edge recession after cutting medium density fiberboard, In: Proc. 9th Inter. Wood Machining Seminar. 320-341. Univ. of California, Forest Prod. Lab., Richmond, Calif.

(10.) Suchsland, O. and G.E. Woodson. 1987. Fiberboard manufacturing practices in the United States. USDA, Gov. Printing Office, Washington, D.C.

The modeling and experimental analysis of orthogonal cutting mechanics are presented for machining medium density fiberboard (MDF). The forces exerted on the tool during cutting are decomposed into rake face and flank face components. A Coulomb friction model is assumed on both the flank and rake faces, and the friction constants on both faces of the tool are predicted. The experimental results showed that the friction on the rake face is rather small, and the pressure exerted by the uncut chip on the rake face mainly dominates the force on the rake face. The cutting forces are expressed as functions of cutting constants, tool geometry, and uncut chip area. The cutting constants are a function of the MDF density, which changes along the board thickness, and the rake angle of the tool. The proposed cutting constants can be used to predict the cutting forces in machining MDF with tools having complex geometry such as router bits.

The mechanics of cutting materials are useful in estimating the cutting forces, predicting the vibration stability of machining operations, and predicting the surface finish on the machined surface. In addition, the laws of mechanics could be used to relate the tool geometry to machining performance parameters such as temperature and stress loading of the tool and in explaining the chip formation mechanism. While the cutting mechanics for metals have been studied extensively in the literature, wood, and especially medium density fiberboard (MDF), cutting mechanics have not receive as much attention [5-9].

In this paper, the mechanics of cutting MDF were studied based on the experience gained in metal cutting mechanics and dynamics [1,2]. Since MDF is not a homogenous material like metal, the shear laws used in metal cutting are not directly applicable. The authors propose to identify the MDF cutting mechanics by using a mechanistic force model around the cutting-wedge. The parameters in the force models are identified as a function of the depth of cut and the rake angle. The density and hardness distribution of MDF along the thickness are also related to the cutting force parameters. The resulting orthogonal cutting force parameters can be used in predicting the cutting forces in oblique cutting [1,2], such as milling with a complex shape cutter [4].

CUTTING FORCE MODEL

There are two fundamental cutting forces in orthogonal cutting: 1) a tangential force ([F.sub.t]), which is in the direction of the cutting speed (V); and 2) feed force ([F.sub.f]), which is perpendicular to the cutting speed (Fig. 1). In orthogonal cutting, the deformation is assumed to be uniform along the width of the cut. Some of the cuffing forces are contributed by the flank edge and the rubbing of the finished surface against the flank face. In metal cutting, the cutting forces on the rake face can be expressed as a function of the material's shear stress, average rake face-chip contact friction, and shear angle. However, the shear law based model is not directly applicable in MDF cutting due to the non-uniform material properties and fracture of chips. Instead, the mechanistic force model shown in Figure 1 is proposed for machining MDF.

The cutting force ([F.sub.c]) resulting from the contact between the rake face and the chip is decomposed into a tangential component ([F.sub.tc]) and a feed component ([F.sub.fc]). Similarly, force ([F.sub.f]) resulting from the contact between the flank face and the finished MDF surface is decomposed into its ([F.sub.te], [F.sub.fe]) components. Let h denote the uncut chip thickness (or feed rate in orthogonal cutting) and b denote the width of cut. The tangential and feed forces can each be decomposed into a shear force and an edge force. The shear force is proportional to the uncut chip area (bh), whereas the edge force is proportional to the cutting edge contact length or width of cut (b). It follows that:

[F.sub.t] = [F.sub.tc] + [F.sub.te] = [K.sub.tc]bh + [K.sub.te]b

[F.sub.f] = [F.sub.fc] = [F.sub.fe] = [K.sub.fc]bh + [K.sub.fe]b [1]

where:

[K.sub.tc], [K.sub.fc] and [K.sub.te], [K.sub.fe] = the cutting coefficients corresponding to the shearing and edge forces, respectively

The cutting constants have to be identified from measured tangential ([F.sub.t]) and feed ([F.sub.f]) forces. A series of orthogonal cutting tests are conducted at incremental feed rates, and the two orthogonal cutting forces are measured. By extrapolating the measured data at zero uncut chip thickness (i.e., h = 0), the edge force constants ([K.sub.te], [K.sub.fe]) can be evaluated. After subtracting the edge forces from each measurement, the cutting constants ([K.sub.tc], [K.sub.fc]) can be evaluated as well, as presented in the experimental section.

Based on this force model, it is now desired to estimate the friction forces on the rake face and the flank face. The forces on both the rake and flank faces can be decomposed into their normal and friction components, as shown in Figure 1. Let [gamma] denote the clearance angle and [alpha] the rake angle. The measured tangential and feed forces can then be expressed as:

[F.sub.f] = [F.sub.rf]cos [alpha] - [F.sub.rn] sin [alpha] + [F.sub.fn] cos [gamma] + [F.sub.ff] sin [gamma]

[F.sub.t] = [F.sub.rf] sin [alpha] + [F.sub.rn] cos [alpha] - [F.sub.fn] sin [gamma] + [F.sub.ff] cos [gamma] [2]

where:

[F.sub.rf], [F.sub.rn] = rake face friction and normal forces

[F.sub.ff], [F.sub.fn] = flank face friction and normal forces

For simplicity, the dominant sliding friction forces between the chip and the rake face, and between the freshly cut MDF surface and tool flank, are assumed in both cases to satisfy a Coulomb friction model. Consequently, the friction coefficients for the flank face

and rake face are given by [[micro].sub.f] = [F.sub.ff]/[F.sub.fn]

and [[micro].sub.r] = [F.sub.rf]/[F.sub.rn], respectively. The distribution of normal loads on the friction surfaces is not known and is difficult to measure in MDF cutting. For simplicity, a uniform distribution of pressure [[sigma].sub.r] on the rake face and the pressure [[sigma].sub.f] on the flank face is assumed, which yields [F.sub.rn] = [[sigma].sub.r][bl.sub.r], [F.sub.rf] = [[micro].sub.r][[sigma].sub.r][bl.sub.r], [F.sub.fn] = [[sigma].sub.f][bl.sub.f], and [F.sub.ff] = [[micro].sub.f][[sigma].sub.f][bl.sub.f], where [l.sub.r] and [l.sub.f] denote, respectively, the chip contact length on the rake face and the workpiece-flank contact length on the flank face. The chip contact length ([l.sub.r]) is known to be proportional to the uncut chip thinkness, [l.sub.r] = [eta]h, where [eta] is a proportionality constant. The flank contact depends on the sharpness of the cutting edge, wear land, recess, and the elastic springback of the freshly cut MDF surface. The flank contact length ([l.sub.f]) is assumed to be independent of the chip load, i.e., the feed rate h, since it depends on the tool sharpness. These assumptions yield:

[F.sub.f] = [[sigma].sub.ro]bh([[micro].sub.r] cos [alpha] - sin [alpha]) + [[sigma].sub.f][bl.sub.f](cos [gamma] + [[micro].sub.f] sin [gamma])

[F.sub.t] = [[sigma].sub.ro]bh([[micro].sub.r] sin [alpha] + cos [alpha]) + [[sigma].sub.f][bl.sub.f](- sin [gamma] + [[micro].sub.f] cos [gamma]) [3]

where:

[[sigma].sub.ro] = [[eta][sigma].sub.r]

The following definitions are used in subsequent calculations:

[F.sub.fc] = [[sigma].sub.ro]bh([[micro].sub.r] cos [alpha] - sin [alpha])

[F.sub.fe] = [[sigma].sub.f][bl.sub.f](cos [gamma] + [[micro].sub.f] sin [gamma])

[F.sub.tc] = [[sigma].sub.ro]bh([[micro].sub.r] sin [alpha] + cos [alpha])

[F.sub.te] = [[sigma].sub.f][bl.sub.f](- sin [gamma] + [[micro].sub.f] cos [gamma]) [4]

At zero uncut chip thickness (h = 0), we have [F.sub.fe]/[F.sub.te] = [[sigma].sub.f][bl.sub.f](cos [gamma] + [[micro].sub.f] sin [gamma])/[[sigma].sub.f][bl.sub.f](- sin [gamma] + [[micro].sub.f] cos [gamma]),

Which yields the following expression for the flank face friction coefficient:

[[micro].sub.f] = [F.sub.te] + [F.sub.fe] tan [gamma] / [F.sub.fe] - [F.sub.te] tan [gamma] [5]

The force components [F.sub.fe] and [F.sub.te] correspond to edge forces and are given by [F.sub.fe] = [bK.sub.fe] and [F.sub.te] = [bK.sub.te]. Consequently, the friction coefficient on the flank face can be rewritten as [[micro].sub.f] = ([K.sub.te] + [K.sub.fe] tan [gamma])/([K.sub.fe] - [K.sub.te] tan [gamma]). Similarly, the friction coefficient [[micro].sub.r] on the rake face is given by [[micro].sub.r] = ([F.sub.fc] + [F.sub.tc] tan [alpha])/([F.sub.tc] - [F.sub.fc] tan [alpha]). Since the sliding friction may be too small in MDF cutting, care must be taken in evaluating the average friction constant on the rake face. In order to keep the influence of feed rate on the friction, the forces can be normalized with respect to width of cut (b), i.e., [F.sub.uf] = [F.sub.f]/b, [F.sub.ut] = [F.sub.t]/b, and the friction constant can be evaluated for every measurement using:

[[micro].sub.r] = ([F.sub.uf] - [K.sub.fe]) + ([F.sub.ut] - [K.sub.te]) tan [alpha] / ([F.sub.ut] - [K.sub.te]) - ([F.sub.uf] - [K.sub.fe]) tan [alpha] [6]

The general Equations [1] through [6] are helpful in modeling the mechanics of cutting MDF, and can be used in predicting the cutting forces produced by any oblique cutting tool such as router bits and shape cutters used in the furniture industry.

EXPERIMENTAL RESULTS

The cutting tests were conducted on a Cincinnati Falcon 300 CNC lathe. Sandvik carbide inserts with relief angle ([gamma]) of 10 degrees and rake angle ([alpha]) of 0, 10, and 15 degrees were used. A Kistler three component force dynamometer type 9257 B A was used to measure the forces. The measured forces were first recorded on an XR 310 Analog Cassette Data Recorder. The measured forces were filtered using an analog low-pass filter with a cutoff frequency 10 times the spindle speed. The filtered signals are then sampled at 2000 Hz, and used in analyzing the mechanics.

FACE TURNING TESTS

The cutting tests were performed on an MDF disk 500 mm in diameter that was attached with nine screws to an aluminum plate; the plate was attached to the spindle of the lathe (Fig. 2). Orthogonal cutting tests were conducted on MDF ribs that were 6mm wide and 4 mm high. The ribs were obtained by cutting 4-mm-deep grooves on both sides of the desired rib location. Three test series were conducted and the cutting conditions are summarized in Table 1. Due to the large number of cutting tests that had to be performed, each test was conducted only once. However, extreme care was taken to ensure that the collected data are accurate.

A quick examination of the both the tangential and feed forces from the test series 1 revealed that the forces do not greatly change with changes in the cutting speed. An example is shown Figure 3 for the case of h = 0.075 mm/rev and [alpha] = +15 degrees. Consequently, the measured forces corresponding to the same feed rate and rake angle, but different cutting speeds, are averaged. The resulting average forces are used in the modeling process based on Equations [1] through [6]. Figure 3 also shows that the measured forces decrease in magnitude with the increase in the depth at which the MDF is being cut. The decrease in the force magnitude is due to the decrease in the MDF density and hardness from the surface towards the inner MDF core.

Figure 4 illustrates the change in the forces as a function of both the depths of cut (a) and feed rate. While the tangential forces linearly increase with the feed rate, the feed forces do not demonstrate strong dependence on the feed rate. Later, the proposed mechanistic model is used to explain the observed trend. Since the forces are normalized with respect to the width of cut, the edge force coefficients ([K.sub.te], [K.sub.fe]) for each depth of cut can then be obtained from the normalized force curves at h = 0 mm/rev. After subtracting the edge forces from the normalized measured tangential and feed forces, the cutting constants [K.sub.tc], [K.sub.fe]) can be identified as a function of the depth (a) at which the MDF is being cut. High order polynomial curves are used to fit the cutting coefficients, with an average error of less than 3 to 4 percent between the measured and predicted forces. The results are compiled in Table 2. The layer (a = 0.3 mm) corresponds to the hard surface of the MDF. The har dness decreases between 0.3 [less than or equal to] a [less than or equal to] 3.6, and remains approximately constant at the inner soft region 3.6 [less than or equal to] a [less than or equal to] 9. The MDF board is assumed to have symmetric hardness in its second half (9 [less than or equal to] a [less than or equal to] 18). The friction coefficients on both the flank and the rake faces are evaluated and are included in previous report (3), but are not shown here due to space limitations. However, they can be evaluated by applying Equations [1] through [6] to the cutting coefficients given in Table 2. The resulting force models are useful in describing cutting operations such as drilling, where the tool is perpendicular to the MDF surface, and the tool penetrates from the top towards inner layers. However, in operations such as milling and routing, the feed motion is parallel to the board surface, therefore plunge turning tests must be used in order to be able to describe such operations.

PLUNGE TURNING

The setup is similar to that used in the face turning tests except that the tool now cuts the periphery of the disk and moves in a radial direction towards the center of the disk (Fig. 5). Three distinct MDF layers, each 3 mm thick, are considered in the cutting tests. Smaller increments were not possible due to the difficulty in measuring the small magnitude forces accurately. The cutting conditions are summarized in Table 3.

As in face turning, only slight changes in the forces are observed as the cutting speed changes. Consequently, measured forces for different cutting speeds, but otherwise similar cutting conditions, are averaged. The resulting average forces are used in the modeling process. The variation of the normalized tangential and feed forces as a function of the layer being cut, the feed rate, and the rake angle, is shown in Figure 6. The identified cutting constants, shown in Table 4, are a function of the MDF layer and the rake angle. The tangential forces increase linearly with the increase in the uncut chip thickness or feed rate (h). However, Figure 6 also indicates that the feed forces decrease with the increase in the feed rate. Such behavior of the feed force leads to a negative friction coefficient if regular orthogonal cutting mechanics laws are used. The identified friction coefficients as a function of the rake angle, uncut chip thickness, and axial MDF layer being cut are shown in Figure 7. The cutting f orces are extremely small at very small feed rates (h [less than] 0.050 mm/rev). Consequently, the force readings at such feed rates may not reflect the cutting mechanism correctly, and may lead to negative rake friction coefficients as shown in Figure 7. The friction on the flank face is independent of chip thickness, and its variation at different MDF layers and rake angles is also shown in Figure 7.

In plunge-turning tests, the cut chip was usually of the size of a small particle. The MDF chip simply fractures, flows away from the rake face and, consequently, does not move on the rake face as a rigid body. Such chip behavior leads to a very small friction coefficient on the rake face for all rake angles, as indicated in Figure 7. Therefore, the dominant force on the rake face is the normal force exerted by the uncut chip on the rake contact area. The normal force on the rake face can be derived from Equations [1] through [6] and Figure 1 as shown in Equation [7].

The normal forces evaluated using Equation [7] are shown in Figure 8. It can be seen that the normal rake forces are quite insensitive to the rake angle. Moreover, the normal forces are approximately linearly dependent on the feed rate (h). Since the hardness at each MDF layer is different, the normal force becomes dependent only on the MDF hardness and the uncut chip area (bh).

With negligible rake friction coefficients, the cutting force on the rake face can be approximated with the normal rake force. Consequently, for a given rake angle, as the feed rate increases, the normal force increases in magnitude (Fig. 8). The latter leads to an increase in the magnitude of the normal rake force projection in the feed direction, which would be opposing the feed force. Assuming a constant flank force for all feed rates, it follows that the projections of the flank forces in the feed direction are of constant magnitude. As shown in Figure 1, the feed force is mainly represented by the sum of the projections, in the feed direction, of the normal rake force and the flank forces. Consequently, the feed force decreases in magnitude as the feed rate increases and could even become negative if the normal rake force is large enough.

If flank friction and pressure are identified, and rake face friction force is neglected, the normal cutting force could be identified from the bending strength, which changes as a function of density along the MDF thickness. The hardness of MDF at each layer was measured using a Rex Model D Durometer, and correlated to the normal rake force as:

[F.sub.rn] [approximate] [delta] H bh

0.267 [less than or equal to] [delta] [less than or equal to] 0.280 [8]

where:

H[N] = the MDF hardness at the layer being cut

Equation [8] can be used as an approximate relationship for average calculations in tool design. For more accurate calculations, the proposed procedure and cutting constants given in Table 4 must be used. The authors measured the density of the MDF, and used a relationship between the density and bending strength of MDF given in a report by Suchsland and Woodson [10]. A similar relationship was found between the normal force and bending strength of the MDF.

CONCLUSION

The mechanics of orthogonal cutting of MDF were modeled. Orthogonal cutting tests, both perpendicular and parallel to the MDF surface, have been conducted. These tests are useful in describing the mechanics of drilling and routing when machining MDF parts. The variation of cutting constants with the rake angle, MDF density changes along the thickness, and feed rate are identified. With the aid of the proposed mechanics model, it is shown that the sliding friction on the rake face of the tool in MDF machining is rather small, and the pressure on the rake face mainly dominates the cutting force. The pressure on the rake face is found to be linearly proportional to the uncut chip area and bending stress/hardness of MDF boards. The cutting constants and the mechanics model presented in this paper can be used in predicting the cutting forces in routing using orthogonal to oblique cutting transformation as presented in a separate article [4]. More information can be found in a previous article [3].

The authors are, respectively, Student, Univ. of British Columbia (UBC), Dept. of Mechanical Engineering, 2324 Main Mall, Vancouver, BC, Canada V6T 1Z4; Technical Officer and Research Officer, National Res. Council, Innovation Centre, 3250 East Mall, Vancouver, BC, Canada, V6T 1W5; and Professor, UBC. This paper was received for publication in July 1999. Reprint No. 9010.

LITERATURE CITED

(1.) Altintas, Y. 2000. Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design. Cambridge Univ. Press, New York. ISBN 0521650291.

(2.) Budak, E., Y. Altintas, and E.J.A. Armarego, 1996. Prediction of milling force coefficients from orthogonal cutting data. Transactions of the American Soc. of Mechanical Engineers (ASME). J. of Manufacturing Sci. and Engineering (former name: J. Engineering for Industry) 118(2):216-224.

(3.) Dippon, J., H. Ren, S. Engin, F. Ben Amara, and Y. Altintas. 1999. Cutting mechanics of medium density fiberboard machining. Tech. Rept. The National Res. Council Canada - Innovation Centre, Vancouver, B.C., Canada.

(4.) Engin, S., Y. Altintas, and F. Ben Amara. 200_. Mechanics of routing medium density fiberboards. Forest Prod. J. (in press).

(5.) Fischer, R. 1997. A way to observe and calculate edge wearing in cutting wood materials. In: Proc. 13th Inter. Wood Machining Seminar. Univ. of British Columbia, Vancouver, B.C., Canada. Vol. II. pp. 631-640.

(6.) Koch, P. 1964. Wood Machining Process. Ronald Press Co., New York.

(7.) McKenzie, W.M. 1960. Fundamental aspects of the wood cutting process. Forest Prod. J. 10(9):447-456.

(8.) Sitkei, G. 1997. On the mechanics of oblique cutting of wood. In: Proc. 13th Inter. Wood Machining Seminar. Univ. of British Columbia, Vancouver, B.C., Canada. Vol. II. pp. 469-476.

(9.) Stewart, H.A. 1988. Analysis of tool forces and edge recession after cutting medium density fiberboard, In: Proc. 9th Inter. Wood Machining Seminar. 320-341. Univ. of California, Forest Prod. Lab., Richmond, Calif.

(10.) Suchsland, O. and G.E. Woodson. 1987. Fiberboard manufacturing practices in the United States. USDA, Gov. Printing Office, Washington, D.C.

Face turning conditions. Test series Rake angle [alpha] Cutting speed (V) Feed rate (h) (degrees) (m/min.) (mm/rev.) 1 0,15,30 100,506,1048 0.075,0.25,0.4 2 15 506 0.075,0.125,0.2,0.25,0.3,0.4 3 15 506 0.075,0.125,0.2,0.25,0.3,0.4 Test series Width of cut (b) Cutting depth (a) (mm) 1 6 0 [less than or equal to] a [less than or equal to] 4 2 6 0 [less than or equal to] a [less than or equal to] 4 3 6 4 [less than or equal to] a [less than or equal to] 8 Cutting force coefficients in face turning. Rake angle ([alpha]) Cutting coefficients (degrees) (N/mm) Edge force coefficients 0 [K.sub.te] [K.sub.fe] 15 [K.sub.te] [K.sub.fe] 30 [K.sub.te] [K.sub.fe] Cutting force coefficients 0 [K.sub.tc] [K.sub.fc] 15 [K.sub.tc] [K.sub.fc] 30 [K.sub.tc] [K.sub.fc] Cutting depth ([alpha]) a [less than or equal to] 0.3 (mm) Edge force coefficients 4.7473 8.642818 5.051898 10.3381 4.4224 10.46288 Cutting force coefficients 26.2621 8.02475 15.94095 3.14762 4.45866 -2.8575 0.3 [less than or equal to] a [less than or equal to] 3.6 Edge force coefficients 0.3479[[alpha].sup.2] - 2.5868[alpha] + 5.514 0.5892[[alpha].sup.2] - 4.3737[alpha] + 9.9019 0.3422[[alpha].sup.2] - 2.583[alpha] + 5.796 0.7249[[alpha].sup.2] - 5.2971[alpha] + 11.862 0.2978[[alpha].sup.2] - 2.2172a + 5.0608 0.8363[[alpha].sup.2] - 5.8346[alpha] + 12.138 Cutting force coefficients -4.143[alpha] + 27.505 -1.3435[alpha] + 8.4278 -2.6235[alpha] + 16.728 -0.9366[alpha] + 3.4286 -0.5838[alpha] + 4.6338 +0.805[alpha] - 3.099 3.6 [less than or equal to] a [less than or equal to] 9 Edge force coefficients 0.7103 1.7926 0.93211 2.1871 0.93836 1.971888 Cutting force coefficients 10.1044 2.7851 5.4573 -0.50512 2.18184 0.282 Plunge turning cutting test conditions. Rake angle ([alpha]) Cutting speed (V) Feed rate (c) (degrees) (m/min.) (mm/rev.) 0, 15, 30 100, 506, 1048 0.05, 0.1, 0.2, 0.3, 0.5 Rake angle ([alpha]) Width of cut (b) Layer (degrees) (mm) 1 0, 15, 30 3 2 3 Rake angle ([alpha]) (degrees) 0 [less than or equal to] a [less than or equal to] 3 0, 15, 30 3 [less than or equal to] a [less than or equal to] 6 6 [less than or equal to] a [less than or equal to] 9 Edge force and cutting force coefficients in plunge turning. Rake angle (a) Edge force coefficients (N/mm) 0 [K.sub.te] 0.0395[a.sup.2] - 0.75a + 8.0439 [K.sub.fe] 0.0558[a.sup.2] - 1.0601a + 10.1827 15 [K.sub.te] 0.0372[a.sup.2] - 0.7074a + 7.4048 [K.sub.fe] 0.0644[a.sup.2] - 1.2243a + 10.9367 30 [K.sub.te] 0.048[a.sup.2] - 0.9126a + 8.6457 [K.sub.fe] 0.0858[a.sup.2] - 1.6295a + 13.6966 Rake angle (a) Cutting force coeff- icients (N/mm) 0 [K.sub.tc] 0.4055[a.sup.2] - 7.7054a + 55.8045 [K.sub.fc] -0.0249[a.sup.2] + 0.4734a - 2.6394 15 [K.sub.tc] 0.3376[a.sup.2] - 6.4143a + 53.4345 [K.sub.fc] -0.0447[a.sup.2] + 0.8496a - 7.9731 30 [K.sub.tc] 0.2134[a.sup.2] - 4.0537a + 39.3695 [K.sub.fc] -0.0956[a.sup.2] + 1.8157a - 21.2389

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Author: | DIPPON, JOERG; REN, HAIKUN; AMARA, FOUED BEN; ALTINTAS, YUSUF |
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Publication: | Forest Products Journal |

Date: | Jul 1, 2000 |

Words: | 4335 |

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