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Specific machining forces and resultant force vectors for machining of reinforced plastics
CIRP Annals - Manufacturing Technology 60 (2011) 69–72
Contents lists available at ScienceDirect
CIRP Annals - Manufacturing Technology jou rnal homep age : ht t p: // ees .e lse vi er. com/ci rp/ def a ult . asp
Specific machining forces and resultant force vectors for machining of reinforced plastics V. Schulze *, C. Becke, R. Pabst KIT – Karlsruhe Institute of Technology, wbk Institute of Production Science, D76131 Karlsruhe, Germany Submitted by H. Weule (1), Karlsruhe, Germany
A R T I C L E I N F O
A B S T R A C T
Keywords: Machining Composite Cutting edge Process forces
When machining fiber reinforced plastics, the machining forces may induce workpiece damage if they exceed the workpiece’s anisotropic strength values. Knowledge of the resultant force vectors is therefore important to allow optimization of tool geometry and machining strategy. This article deals with experimentally obtained machining forces on short glass fiber reinforced polyester. Specific cutting, passive and axial forces have been determined for varied parameters of cutting velocity, cutting depth, cutting edge rounding and tool inclination. Generic multivariate regression models have been calculated, which, implemented in a kinematic simulation, allow calculation of machining forces (and direction) for arbitrary milling operations. ß 2011 CIRP.
1. Introduction Fiber reinforced plastics (FRP) are being increasingly used because of their superior specific properties, which makes them a good choice for lightweight construction. Even though the overall goal should be to manufacture FRP-parts without any mechanical machining, there is still a need for many subsequent machining operations. Examples are drill holes for screws or rivets, pocket milling operations or contour machining. However, due to the fact that composite materials are designed anisotropically, there are particular machining challenges. Machining forces are in many cases (e.g. drilling) acting in directions of lower material strength, thus causing permanent damages. Resulting damages such as delamination, chipping or spalling are defined as permanent and thus critical [1]. Many researchers later correlated the extent of these damages to process parameters like feed and ultimately to the process forces [2,3]. Optimizations usually aim at reducing the forces by means of special tool geometries for drilling and milling [4,5] or adapted machining strategies such as circular milling [6]. Process strategies which are designed to direct resulting process forces toward the center of the workpiece when machining composites showed significant reduction in workpiece damages [7]. Besides special tool (macro-) geometries, the influence of the cutting edge’s micro-geometry is an important influencing factor affecting process forces. When machining fiber reinforced plastics, significant changes of the cutting edge occur due to the fiber’s highly abrasive nature during the cut. Rounding of the cutting edge is even proposed to be used as wear criterion for composite machining [8]. In consequence of geometry change, machining forces usually increase with increasing tool wear, which favors
* Corresponding author. 0007-8506/$ – see front matter ß 2011 CIRP. doi:10.1016/j.cirp.2011.03.085
damages further. The special influence of cutting edge rounding (radius rb) has lately been investigated by [9] for the machining of a titanium alloy. Cutting forces and feed forces both increased significantly within the investigated range of rb. Summarizing previous work, the most important influencing parameters on process forces are shown and compiled in Fig. 1. From this list the parameters of cutting velocity vc, cutting edge radius rb, cutting thickness h and tool spiral angle l have been selected to be varied experimentally in order to evaluate their significance on process forces. Of course absolute cutting force models are limited in their significance to the investigated and similar workpiece material. However, the direction of the resulting force (Fc/Fp) is an important parameter for the resulting workpiece loading. It also is expected to offer transferability to some extent, as it is highly dependent on the geometric cutting edge engagement conditions, which are investigated in this work. 2. Experimental setup and scope An unconventional experimental setup has been chosen to obtain the data necessary to derive process force models. A machining center (Heller MC16) has been used as planer by using one of the linear axes to provide a linear cutting motion, thus providing constant cutting conditions for a set of process parameters. Plates of short glass fiber reinforced polyester obtained through a sheet molding compounding process (SMC) have been cut to specimen with a length of 100 mm (thickness 2.5 mm). The chopped glass fibers had a length of approximately 25 mm and a diameter of 15 mm. Fiber content was 22.3 vol%; fiber orientation was random. Since the SMC material does not have a particular fiber orientation which could have been varied experimentally, the material has been considered quasi-homogeneous.
[()TD$FIG]
[()TD$FIG]
V. Schulze et al. / CIRP Annals - Manufacturing Technology 60 (2011) 69–72
70
Fig. 1. Schematic presentation of the cutting edge during the cut. Fig. 2. Experimental setup: clamped SMC specimen, cutting insert during the cut, direction of the measured process forces.
Uncoated cemented carbide cutting inserts (WC/Co, WK08) have been mounted to a stationary spindle-held tool holder (rake angle g = 08, clearance angle a = 78). The setup allowed setting vc, h and l (by inclining the tool) directly by the machine center’s controls. Additionally, rb has been varied in five steps. The different radii have been prepared by rotatory drag finishing using an OTEC DF-3. A NanoFocus mSurf has been used to measure the radii following Wyen and Wengener’s approach [9] at three different positions along the cutting edge. Cutting has been done without any coolant or lubricant. An overview of the experimentally varied parameters and their range is given in Table 1. A complete combination of parameters has been used for l = 08 in order to obtain a good data basis. For l = 208 and 408 some less important combinations of parameters have been left out. However, the parameter combination of rb/h = 1 has been additionally investigated for each cutting edge radius, as this particular cutting condition was expected to be specifically important. For each investigated parameter combination forces have been measured in the three spatial axes by using a Kistler 9255B three component dynamometer. As constant cutting conditions have been used, the forces could be immediately related to cutting, passive and transverse force (l > 08). Average values and standard deviations of the forces have been calculated and used for the subsequent derivation of multivariate regression models. The forces have then further been calculated to specific forces (N/mm2) by dividing them by cutting thickness and specimen thickness. The experimental setup and the measured forces at the cutting edge are shown in Fig. 2.
3. Regression models The experimentally obtained force measurement data showed a trend, which is commonly known for many years in metal cutting and led to the formulation of the Victor–Kienzle-formula [10]: disproportionate increase of specific cutting forces (kc) at low cutting thickness. The important role of the cutting edge radius has been added later to the formula by [11] for micro-machining. Both of the models are based on power function approaches, which show the relation of the respective parameters as straight lines in double logarithmic plots. Fig. 3 shows plots of the specific cutting forces kc vs. cutting thicknesses h for the investigated range of cutting edge radii. Power function regressions for one varying parameter (h) are plotted into the diagram. Their parameters and coefficients of determination are given as well. Besides the good fit
the data show a strong interdependence of kc to the cutting edge radius rb. As univariate models are not well suited to describe the interdependencies between multiple parameters, the approach of multiplying power functions of the separate process variables has been chosen. For the example of kc this may be written as: mh
vc kc ¼ vm h c
m
rb rb lml C
(1)
This approach can then be simplified to a linear problem by logarithmizing both sides of the equation, which enables multivariate linear regression to determine the model constants. Combining this equation with the experimentally obtained data a linear system of equations can be derived, which is then used to determine the model constants by the method of least squares. Comparing the process models with the experimental data, a nonlinear deviation is evident (see Fig. 4a). This deviation exists for all the calculated models of kc, kp and kc/kp. Within the investigated range the compliance of regression and experimental data is only as good as 50%. This relatively large error has to be addressed further and compensated as good as possible. Taking into account the assumption that the cutting conditions at the cutting edge are very different for small cutting thicknesses (h < rb) as opposed to larger cutting thicknesses (h > rb), it is proposed to divide the model for the respective two cases. Following this proposal two separate multivariate regressions are performed using only the particular set of experimental data (h < rb and h > rb). Data obtained at the ratio of h = rb are being used for both models, thus reducing the inevitable discontinuities at the intersection. The combination shows a much better fit to the experimental data (20%) and is thus preferable (Fig. 4b). The same is true for the models of specific passive force kp and force direction, represented by the ratio of kc/kp. A list of the model parameters and respective coefficient of determination R2 for the separate regression models is given in Table 2, completing the general formula: mh
vc yðvc ; h; r b ; lÞ ¼ vm h c [()TD$FIG] y ¼ fkc ; k p ; kc =k p g
m
ml
rb r b l
C
(2)
Table 1 Range of the experimentally varied parameters. Parameter
Units
Range
Cutting velocity vc Cutting thickness h
m/min mm
Cutting edge radius rb Tool inclination l
mm
5, 10, 20, 30, 40 0.005, 0.01, 0.05, 0.1, 0.2, 0.3 (add.: 0.015, 0.035, 0.055, 0.075, 0.095) 15, 35, 55, 75, 95 0, 20, 40
8
Fig. 3. Specific cutting forces (kc) vs. cutting thickness (h): experimental data and power function regressions.
V. Schulze et al. / CIRP Annals - Manufacturing Technology 60 (2011) 69–72
71
Table 2 Multivariate regression models for specific cutting force kc and specific passive forces kp.
mvc mh mrb ml C R2
kc
kc
kp
kp
kc/kp
kc/kp
(rb/h < 1)
(rb/h > 1)
(rb/h < 1)
(rb/h > 1)
(rb/h < 1)
(rb/h > 1)
0.025 0.281 0.173 0.001 277.91 0.82
0.0001 0.463 0.508 0.003 354.58 0.94
0.011 0.757 0.665 0.003 244.01 0.95
0.080 0.648 0.758 0.006 237.39 0.97
0.035 0.476 0.493 0.002 1.139 0.96
0.080 0.184 0.250 0.003 1.494 0.92
Units: vc (mm/min), h (mm), rb (mm), l (8).
4. Results and discussion Comparing the influence of the different parameters, it first can be noticed, that vc and l have a much lower significance in contrast to h and rb. As for tool inclination l: an additional force component perpendicular to cutting and passive force exists naturally, which increases with l. Evaluating the presented models, some trends can be validated which have been expected: specific forces increase disproportionately with decreasing cutting thickness and increasing cutting edge radius. Fig. 5 shows a plot of kc vs. h and rb. Also, a sharp bend can be recognized in the surface plot at the intersecting line of the
[()TD$FIG]
two separate models (h = rb). However, the separate models show only a minor discontinuity in this region. It is noteworthy mentioning, that there is no strong correlation between the varied process parameters and the specific cutting force for cutting thicknesses above the cutting edge radius (flat surface). The model for kp is similar, except that the correlation between kc and h is even stronger for rb/h > 1 and there still is a strong influence of rb and h in the region of rb/h < 1. The slightly different behavior of kc and kp toward changing geometric cutting conditions is visualized in the plot of kc/kp vs. h and rb in Fig. 6. For small cutting thicknesses h and larger cutting edge radii rb the passive force becomes increasingly more dominant: up to the factor 2 (within the investigated range). On the other hand small cutting edge radii and large cutting thicknesses lead to a dominance of the cutting force: up to the factor 5. As h and rb are the most significant influencing parameters and the regression models are divided at the geometric ratio of rb/h = 1, a further separate regression has been calculated using rb/h as a combined variable. Comparing this model with the experimental data (Fig. 7), a good agreement between the model and the experimental data is evident. This implies that the direction of the resulting force (kc/kp) is basically only a function of the geometric engagement conditions. The function of this regression model is derived in the same way as described before with separate models for rb/h > 1 and rb/h < 1. It can be written as:
kc kp
kc kp
Fig. 4. Comparison of experiment and regression models ((a) combined, (b) separate) of specific cutting force (kc) and force direction (kc/kp).
[()TD$FIG]
Fig. 5. Regression model of specific cutting force kc plotted vs. cutting edge radius (rb) and cutting thickness (h).
¼ vc0:026
0:486 rb 0:001 l 1:11 h
(3)
¼ vc0:085
0:194 rb l0:002 1:99 h
(4)
ðr b =h < 1Þ
ðr b=h > 1Þ
The plot also shows again the minor significance of vc (influence even decreasing at higher vc), which allows extrapolating the model above the investigated range of cutting velocity to some extent. However, there might be different phenomenological effects coming into play at very high velocities, which are not present at low velocities and thus not being considered in the regression models. [()TD$FIG]
Fig. 6. Regression model of force direction (kc/kp) plotted vs. cutting edge radius (rb) and cutting thickness (h).
[()TD$FIG]
72
V. Schulze et al. / CIRP Annals - Manufacturing Technology 60 (2011) 69–72
5. Conclusions and outlook Multivariate regression models for specific cutting forces, specific passive forces and the direction of the resultant force have been calculated for the investigated short glass fiber reinforced plastic. Two separate models for the parameter ranges of rb/h < 1 and rb/h > 1 have been calculated respectively. As the cutting edge engagement conditions are very different for these two cases, a much better quality of the models could be obtained this way. Evaluating the models the following points can be concluded:
Fig. 7. Regression model and experimental data of force direction (kc/kp) plotted vs. cutting velocity (vc) and engagement condition (rb/h).
[()TD$FIG]
There is a disproportional increase in specific forces when machining with rb/h > 1. This has to be taken into account especially for milling operations with their process-related varying cutting thicknesses from zero to feed per tooth. The cutting edge rounding has a significant influence on the resultant forces and force direction, generally leading to higher forces and a relatively increasing influence of passive forces. A general prediction of force direction (kc/kp) is possible on basis of geometric cutting conditions alone. Using the ratio rb/h the effective force angle h can be calculated as the influence of cutting velocity vc and tool inclination l is considerably less significant. The overall goal for the calculation of the regression models is to finally implement them in a kinematic simulation tool, which then allows calculating resulting machining forces and their direction for arbitrary milling operations.
References Fig. 8. Effective force angle h (b) between kc and the resultant force of kc and kp plotted vs. geometric engagement condition rb/h (a).
Finally the direct influence of the geometric cutting edge engagement conditions on the direction of the resultant force is visualized in Fig. 8a. An effective force angle h is introduced (Fig. 8b), describing the angle between the cutting force (ideal cutting) and the resultant force. The angle is calculated directly from the ratio of kc/kp and is plotted vs. the geometric engagement condition of rb/h. h is calculated from the general regression model with the model constants given in Table 2. The plot clearly shows a sharp bend again at rb/h = 1. For the region rb/h < 1 the ratio of kc/ kp is decreasing much more pronounced than for rb/h > 1, showing the decreasing influence of specific passive forces at large cutting thickness. However, it can be recognized, that the effective force angle is slightly different to 458 at rb/h = 1. In fact it is shifted toward a larger angle, meaning that the passive force is usually higher at the theoretical geometric equilibrium for the two force components.
[1] Ko¨nig W, Grab P (1985) Quality definition and assessment in drilling of fibre reinforced thermosets. Annals of the CIRP 38:119–124. [2] Hocheng H, Dharan C-K-H (1990) Delamination during drilling in composite laminates. Journal of Engineering for Industry 112:236–239. [3] Jain S, Yang D-C-H (1991) Effects of feedrate and chisel edge on delamination in composites drilling. Processing and Manufacturing of Composite Materials, Proceedings, 37–51. [4] Hocheng S, Tsao C-C (2003) Comprehensive analysis of delamination in drilling of composite materials with various drill bits. Journal of Materials Processing Technology 140:335–339. [5] Dennis P (1991) Machining fibre-reinforced plastics. Industrial Diamond Review 51:288–291. [6] Jansen R (2003) Bohren und Zirkularfra¨sen von Schichtverbunden aus Aluminium, CFK und Titanlegierungen. Dissertation, Universita¨t Bremen. [7] Schulze V, Becke C (2011) Machining strategies for hole making in composites with minimal workpiece damage by directing the process forces inwards. Journal of Materials Processing Technology 211:329–338. [8] Faraz A, Biermann D, Weinert K (2009) Cutting edge rounding: an innovative tool wear criterion in drilling CFRP composite laminates. International Journal of Machine Tools and Manufacture 49:1185–1196. [9] Wyen C-F, Wengener K (2010) Influence of cutting edge radius on cutting forces in machining titanium. Annals of the CIRP 59:93–96. [10] Victor H (1956) Beitrag zur Kenntnis der Schnittkra¨fte beim Drehen, Hobeln und Bohren. Dissertation, Technische Hochschule Hannover. [11] Fleischer J, Schulze V, Kotschenreuther J (2009) Extension of cutting force formulae for microcutting. CIRP Journal of Manufacturing Science and Technology 2:75–80.
Contents lists available at ScienceDirect
CIRP Annals - Manufacturing Technology jou rnal homep age : ht t p: // ees .e lse vi er. com/ci rp/ def a ult . asp
Specific machining forces and resultant force vectors for machining of reinforced plastics V. Schulze *, C. Becke, R. Pabst KIT – Karlsruhe Institute of Technology, wbk Institute of Production Science, D76131 Karlsruhe, Germany Submitted by H. Weule (1), Karlsruhe, Germany
A R T I C L E I N F O
A B S T R A C T
Keywords: Machining Composite Cutting edge Process forces
When machining fiber reinforced plastics, the machining forces may induce workpiece damage if they exceed the workpiece’s anisotropic strength values. Knowledge of the resultant force vectors is therefore important to allow optimization of tool geometry and machining strategy. This article deals with experimentally obtained machining forces on short glass fiber reinforced polyester. Specific cutting, passive and axial forces have been determined for varied parameters of cutting velocity, cutting depth, cutting edge rounding and tool inclination. Generic multivariate regression models have been calculated, which, implemented in a kinematic simulation, allow calculation of machining forces (and direction) for arbitrary milling operations. ß 2011 CIRP.
1. Introduction Fiber reinforced plastics (FRP) are being increasingly used because of their superior specific properties, which makes them a good choice for lightweight construction. Even though the overall goal should be to manufacture FRP-parts without any mechanical machining, there is still a need for many subsequent machining operations. Examples are drill holes for screws or rivets, pocket milling operations or contour machining. However, due to the fact that composite materials are designed anisotropically, there are particular machining challenges. Machining forces are in many cases (e.g. drilling) acting in directions of lower material strength, thus causing permanent damages. Resulting damages such as delamination, chipping or spalling are defined as permanent and thus critical [1]. Many researchers later correlated the extent of these damages to process parameters like feed and ultimately to the process forces [2,3]. Optimizations usually aim at reducing the forces by means of special tool geometries for drilling and milling [4,5] or adapted machining strategies such as circular milling [6]. Process strategies which are designed to direct resulting process forces toward the center of the workpiece when machining composites showed significant reduction in workpiece damages [7]. Besides special tool (macro-) geometries, the influence of the cutting edge’s micro-geometry is an important influencing factor affecting process forces. When machining fiber reinforced plastics, significant changes of the cutting edge occur due to the fiber’s highly abrasive nature during the cut. Rounding of the cutting edge is even proposed to be used as wear criterion for composite machining [8]. In consequence of geometry change, machining forces usually increase with increasing tool wear, which favors
* Corresponding author. 0007-8506/$ – see front matter ß 2011 CIRP. doi:10.1016/j.cirp.2011.03.085
damages further. The special influence of cutting edge rounding (radius rb) has lately been investigated by [9] for the machining of a titanium alloy. Cutting forces and feed forces both increased significantly within the investigated range of rb. Summarizing previous work, the most important influencing parameters on process forces are shown and compiled in Fig. 1. From this list the parameters of cutting velocity vc, cutting edge radius rb, cutting thickness h and tool spiral angle l have been selected to be varied experimentally in order to evaluate their significance on process forces. Of course absolute cutting force models are limited in their significance to the investigated and similar workpiece material. However, the direction of the resulting force (Fc/Fp) is an important parameter for the resulting workpiece loading. It also is expected to offer transferability to some extent, as it is highly dependent on the geometric cutting edge engagement conditions, which are investigated in this work. 2. Experimental setup and scope An unconventional experimental setup has been chosen to obtain the data necessary to derive process force models. A machining center (Heller MC16) has been used as planer by using one of the linear axes to provide a linear cutting motion, thus providing constant cutting conditions for a set of process parameters. Plates of short glass fiber reinforced polyester obtained through a sheet molding compounding process (SMC) have been cut to specimen with a length of 100 mm (thickness 2.5 mm). The chopped glass fibers had a length of approximately 25 mm and a diameter of 15 mm. Fiber content was 22.3 vol%; fiber orientation was random. Since the SMC material does not have a particular fiber orientation which could have been varied experimentally, the material has been considered quasi-homogeneous.
[()TD$FIG]
[()TD$FIG]
V. Schulze et al. / CIRP Annals - Manufacturing Technology 60 (2011) 69–72
70
Fig. 1. Schematic presentation of the cutting edge during the cut. Fig. 2. Experimental setup: clamped SMC specimen, cutting insert during the cut, direction of the measured process forces.
Uncoated cemented carbide cutting inserts (WC/Co, WK08) have been mounted to a stationary spindle-held tool holder (rake angle g = 08, clearance angle a = 78). The setup allowed setting vc, h and l (by inclining the tool) directly by the machine center’s controls. Additionally, rb has been varied in five steps. The different radii have been prepared by rotatory drag finishing using an OTEC DF-3. A NanoFocus mSurf has been used to measure the radii following Wyen and Wengener’s approach [9] at three different positions along the cutting edge. Cutting has been done without any coolant or lubricant. An overview of the experimentally varied parameters and their range is given in Table 1. A complete combination of parameters has been used for l = 08 in order to obtain a good data basis. For l = 208 and 408 some less important combinations of parameters have been left out. However, the parameter combination of rb/h = 1 has been additionally investigated for each cutting edge radius, as this particular cutting condition was expected to be specifically important. For each investigated parameter combination forces have been measured in the three spatial axes by using a Kistler 9255B three component dynamometer. As constant cutting conditions have been used, the forces could be immediately related to cutting, passive and transverse force (l > 08). Average values and standard deviations of the forces have been calculated and used for the subsequent derivation of multivariate regression models. The forces have then further been calculated to specific forces (N/mm2) by dividing them by cutting thickness and specimen thickness. The experimental setup and the measured forces at the cutting edge are shown in Fig. 2.
3. Regression models The experimentally obtained force measurement data showed a trend, which is commonly known for many years in metal cutting and led to the formulation of the Victor–Kienzle-formula [10]: disproportionate increase of specific cutting forces (kc) at low cutting thickness. The important role of the cutting edge radius has been added later to the formula by [11] for micro-machining. Both of the models are based on power function approaches, which show the relation of the respective parameters as straight lines in double logarithmic plots. Fig. 3 shows plots of the specific cutting forces kc vs. cutting thicknesses h for the investigated range of cutting edge radii. Power function regressions for one varying parameter (h) are plotted into the diagram. Their parameters and coefficients of determination are given as well. Besides the good fit
the data show a strong interdependence of kc to the cutting edge radius rb. As univariate models are not well suited to describe the interdependencies between multiple parameters, the approach of multiplying power functions of the separate process variables has been chosen. For the example of kc this may be written as: mh
vc kc ¼ vm h c
m
rb rb lml C
(1)
This approach can then be simplified to a linear problem by logarithmizing both sides of the equation, which enables multivariate linear regression to determine the model constants. Combining this equation with the experimentally obtained data a linear system of equations can be derived, which is then used to determine the model constants by the method of least squares. Comparing the process models with the experimental data, a nonlinear deviation is evident (see Fig. 4a). This deviation exists for all the calculated models of kc, kp and kc/kp. Within the investigated range the compliance of regression and experimental data is only as good as 50%. This relatively large error has to be addressed further and compensated as good as possible. Taking into account the assumption that the cutting conditions at the cutting edge are very different for small cutting thicknesses (h < rb) as opposed to larger cutting thicknesses (h > rb), it is proposed to divide the model for the respective two cases. Following this proposal two separate multivariate regressions are performed using only the particular set of experimental data (h < rb and h > rb). Data obtained at the ratio of h = rb are being used for both models, thus reducing the inevitable discontinuities at the intersection. The combination shows a much better fit to the experimental data (20%) and is thus preferable (Fig. 4b). The same is true for the models of specific passive force kp and force direction, represented by the ratio of kc/kp. A list of the model parameters and respective coefficient of determination R2 for the separate regression models is given in Table 2, completing the general formula: mh
vc yðvc ; h; r b ; lÞ ¼ vm h c [()TD$FIG] y ¼ fkc ; k p ; kc =k p g
m
ml
rb r b l
C
(2)
Table 1 Range of the experimentally varied parameters. Parameter
Units
Range
Cutting velocity vc Cutting thickness h
m/min mm
Cutting edge radius rb Tool inclination l
mm
5, 10, 20, 30, 40 0.005, 0.01, 0.05, 0.1, 0.2, 0.3 (add.: 0.015, 0.035, 0.055, 0.075, 0.095) 15, 35, 55, 75, 95 0, 20, 40
8
Fig. 3. Specific cutting forces (kc) vs. cutting thickness (h): experimental data and power function regressions.
V. Schulze et al. / CIRP Annals - Manufacturing Technology 60 (2011) 69–72
71
Table 2 Multivariate regression models for specific cutting force kc and specific passive forces kp.
mvc mh mrb ml C R2
kc
kc
kp
kp
kc/kp
kc/kp
(rb/h < 1)
(rb/h > 1)
(rb/h < 1)
(rb/h > 1)
(rb/h < 1)
(rb/h > 1)
0.025 0.281 0.173 0.001 277.91 0.82
0.0001 0.463 0.508 0.003 354.58 0.94
0.011 0.757 0.665 0.003 244.01 0.95
0.080 0.648 0.758 0.006 237.39 0.97
0.035 0.476 0.493 0.002 1.139 0.96
0.080 0.184 0.250 0.003 1.494 0.92
Units: vc (mm/min), h (mm), rb (mm), l (8).
4. Results and discussion Comparing the influence of the different parameters, it first can be noticed, that vc and l have a much lower significance in contrast to h and rb. As for tool inclination l: an additional force component perpendicular to cutting and passive force exists naturally, which increases with l. Evaluating the presented models, some trends can be validated which have been expected: specific forces increase disproportionately with decreasing cutting thickness and increasing cutting edge radius. Fig. 5 shows a plot of kc vs. h and rb. Also, a sharp bend can be recognized in the surface plot at the intersecting line of the
[()TD$FIG]
two separate models (h = rb). However, the separate models show only a minor discontinuity in this region. It is noteworthy mentioning, that there is no strong correlation between the varied process parameters and the specific cutting force for cutting thicknesses above the cutting edge radius (flat surface). The model for kp is similar, except that the correlation between kc and h is even stronger for rb/h > 1 and there still is a strong influence of rb and h in the region of rb/h < 1. The slightly different behavior of kc and kp toward changing geometric cutting conditions is visualized in the plot of kc/kp vs. h and rb in Fig. 6. For small cutting thicknesses h and larger cutting edge radii rb the passive force becomes increasingly more dominant: up to the factor 2 (within the investigated range). On the other hand small cutting edge radii and large cutting thicknesses lead to a dominance of the cutting force: up to the factor 5. As h and rb are the most significant influencing parameters and the regression models are divided at the geometric ratio of rb/h = 1, a further separate regression has been calculated using rb/h as a combined variable. Comparing this model with the experimental data (Fig. 7), a good agreement between the model and the experimental data is evident. This implies that the direction of the resulting force (kc/kp) is basically only a function of the geometric engagement conditions. The function of this regression model is derived in the same way as described before with separate models for rb/h > 1 and rb/h < 1. It can be written as:
kc kp
kc kp
Fig. 4. Comparison of experiment and regression models ((a) combined, (b) separate) of specific cutting force (kc) and force direction (kc/kp).
[()TD$FIG]
Fig. 5. Regression model of specific cutting force kc plotted vs. cutting edge radius (rb) and cutting thickness (h).
¼ vc0:026
0:486 rb 0:001 l 1:11 h
(3)
¼ vc0:085
0:194 rb l0:002 1:99 h
(4)
ðr b =h < 1Þ
ðr b=h > 1Þ
The plot also shows again the minor significance of vc (influence even decreasing at higher vc), which allows extrapolating the model above the investigated range of cutting velocity to some extent. However, there might be different phenomenological effects coming into play at very high velocities, which are not present at low velocities and thus not being considered in the regression models. [()TD$FIG]
Fig. 6. Regression model of force direction (kc/kp) plotted vs. cutting edge radius (rb) and cutting thickness (h).
[()TD$FIG]
72
V. Schulze et al. / CIRP Annals - Manufacturing Technology 60 (2011) 69–72
5. Conclusions and outlook Multivariate regression models for specific cutting forces, specific passive forces and the direction of the resultant force have been calculated for the investigated short glass fiber reinforced plastic. Two separate models for the parameter ranges of rb/h < 1 and rb/h > 1 have been calculated respectively. As the cutting edge engagement conditions are very different for these two cases, a much better quality of the models could be obtained this way. Evaluating the models the following points can be concluded:
Fig. 7. Regression model and experimental data of force direction (kc/kp) plotted vs. cutting velocity (vc) and engagement condition (rb/h).
[()TD$FIG]
There is a disproportional increase in specific forces when machining with rb/h > 1. This has to be taken into account especially for milling operations with their process-related varying cutting thicknesses from zero to feed per tooth. The cutting edge rounding has a significant influence on the resultant forces and force direction, generally leading to higher forces and a relatively increasing influence of passive forces. A general prediction of force direction (kc/kp) is possible on basis of geometric cutting conditions alone. Using the ratio rb/h the effective force angle h can be calculated as the influence of cutting velocity vc and tool inclination l is considerably less significant. The overall goal for the calculation of the regression models is to finally implement them in a kinematic simulation tool, which then allows calculating resulting machining forces and their direction for arbitrary milling operations.
References Fig. 8. Effective force angle h (b) between kc and the resultant force of kc and kp plotted vs. geometric engagement condition rb/h (a).
Finally the direct influence of the geometric cutting edge engagement conditions on the direction of the resultant force is visualized in Fig. 8a. An effective force angle h is introduced (Fig. 8b), describing the angle between the cutting force (ideal cutting) and the resultant force. The angle is calculated directly from the ratio of kc/kp and is plotted vs. the geometric engagement condition of rb/h. h is calculated from the general regression model with the model constants given in Table 2. The plot clearly shows a sharp bend again at rb/h = 1. For the region rb/h < 1 the ratio of kc/ kp is decreasing much more pronounced than for rb/h > 1, showing the decreasing influence of specific passive forces at large cutting thickness. However, it can be recognized, that the effective force angle is slightly different to 458 at rb/h = 1. In fact it is shifted toward a larger angle, meaning that the passive force is usually higher at the theoretical geometric equilibrium for the two force components.
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