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Observations on the mean values of forces, torque and specific power in the peripheral milling process
Int. J. Mach. Tool Des. Res, Printcd in Great Britain
Vol. 25, No. 4. pp. 337-346. 1985
0020-735718553.00+.00 Pergamon Press Ltd.
OBSERVATIONS ON THE MEAN VALUES OF FORCES, TORQUE AND SPECIFIC POWER IN THE PERIPHERAL MILLING PROCESS I. YELLOWLEY*
(Received 24 July 1984: in final form 9 January 1985) Abstract--The forces arising from the milling process are important practical parameters. In this paper, the author uses a force model which considers both rake and flank face contributions to derive expressions for average values of force, torque and specific power. It is shown that specific power is a unique function of mean chip thickness. The major practical importance of the work is the application of the concepts derived to adaptive control. It is demonstrated that both radial width and axial depth of cut may be estimated from a knowledge of the quasi mean resultant force and the mean value of torque. The possibility of identifying the dimensions of cut, in real time, provides a significant step towards the development of adaptive optimisation strategies for the milling process.
INTRODUCTION
SEVERAL authors have examined the problem of estimating average forces and torque in peripheral milling. The first rigorous analysis of the geometry of the process was carried out by Martellotti [1], who also made experimental measurements of power and speculated that the specific power was primarily a function of average chip thickness. Later work by Koenigsberger and Sabberwal [2, 3], examined tangential forces and specific cutting pressure, again a strong influence of mean chip thickness on these variables was observed, regardless of the geometry of the cutter. In recent years the author [4] has shown how the concept of an analogue tool may be used to estimate average forces and several authors, e.g. [5, 6], have analysed the variation in forces numerically, making various assumptions regarding the instantaneous cutting pressure and ratio of tangential and radial cutting force components. The purpose of this paper is two fold; (a) to demonstrate analytically that the modelling of the cutting forces as the sum of active cutting and edge components leads to the conclusion that specific power is a unique function of average chip thickness in cut. (b) To demonstrate that the average values of force and torque may be useful in practical adaptive control strategies where, because of variations in blank geometry, the instantaneous cutting conditions are not known with any certainty. In this regard one is particularly interested in being able to ascertain current values of radial width and axial depth of cut. Such considerations are particularly important when one wishes to optimise cutting conditions, the two variables having very different influences on cutter wear rate [7, 8]. 2. INSTANTANEOUSVALUES OF FORCES AND TORQUE IN PERIPHERAL MILLING The forces acting upon a single straight tooth in up and down milling are shown in Figs 1 and 2 respectively. As shown in Figs 1 and 2. it is customary to resolve the radial and tangential force components parallel to the feed direction (X) and perpendicular to the feed (Y). In general it has been assumed that the tangential component may be calculated by multiplying the instantaneous area of cut by the specific cutting pressure (K) and that the ratio of radial to tangential component (r) is known. In the simplest case both K and r are assumed constant, however, a recent paper [9] has assumed these * Department of Industrial Engineering, Technical University of Nova Scotia, Halifax. Nova Scotia, Canada B3J 2X4. 337
338
I. YELLOWLEY
F, //~
I I
~
Fx
FIG. 1. Forces in up milling.
0o
I
Fx
FIG, 2. Forces in down milling.
parameters to be known functions of mean chip thickness. Since the chip thickness in peripheral milling is generally very low then it would seem logical, as a first improvement on a linear model of cutting forces, to consider the edge forces, (nose and flank), which are known to constitute a significant proportion of total forces at such conditions. (Several authors have reported experiments aimed at measuring these parasitic forces e.g. [10].) The model developed then considers that the tangential force is made up of two components, one of which is directly proportional to undeformed area of cut, the other directly proportional to length of cutting edge engaged. It will, for the present time, be assumed that the cutting edge is straight with no nose radius or chamfer at the point. The influence of a helix angle is discussed in the following section. Thus the tangential force on a single straight edge may be expressed as FT
= KaSt
sin~ + xa
(1)
where k is the specific cutting pressure, a is the axial depth of cut, St is the feed per tooth, is the instantaneous angle of rotation and x is the edge force constant. Assuming now that one defines a critical chip thickness (h*) as that which yields a total cutting force which is comprised of equal components of cutting and parasitic force, then
or
kah* = xa x = Kh*
and equation (1) may be rewritten in the form. Fv=KaSt(sind~+~).
(2)
Observations In Peripheral MillingProcess
339
It is to be expected that the ratio of tangential to radial components on the rake face and flank of the tool will be considerably different. The radial force is thus expressed in the form FR = KaSt rlsindp + r 2
(3)
where r~ is the ratio of radial to tangential cutting forces and r2 is the ratio of radial to tangential flank forces. In general, typical values for these ratios are the following, 0 . 2 < r ~ < 0.5 1
T = FT.R
where T is torque and R is cutter radius. (a) Up milling F x = F r c o s + + FRsind0 F y = FR cos do -- F r sin +.
(5) (6)
(b) D o w n milling F x = FR sin + - F r c o s c b Fy = FR cos + + F r s i n + .
(7) (8)
3. AVERAGE VALUES OF FORCE. TORQUE AND SPECIFIC POWER 3.1.
Introduction
From the previous section it is evident that the instantaneous values of force and torque are linearly dependent on axial depth of cut. Thus any one of the forces may be expressed in the form, F = a f(eb, h*, rl. 1"2, S,) the corresponding average value of force for a cutter with a single straight tooth is given by ¢b, F = ~
f (~b. h*, rl, r2. St)
(9)
where +~ is swept angle of cut. A consideration of equation (9) suggests that an anoiogy may be constructed by considering the constant force produced by an analogue tool with tooth spacing 8+ each tooth having an axial projection given by
8( =
aa_ 8+.
2v
0o)
340
I. YELLOWLEY
Such a tool may, evidently be regarded as an approximation to a helical tooth with a lag angle of 360 ° between leading and trailing edges. Should a cutter have a number (N) of straight cutting edges which are evenly spaced, then the corresponding axial projection of each tooth in the equivalent analogue cutter is given by Na
817 =
2~
8~b
(ll)
such a representation does not have a particularly relevant physical significance however making the substitution = Nd~
(12)
then from equations (1) and (12) 8e = ~
a
1502
(13)
evidently then the equivalent analogue tool represents a helical cutter with N teeth, the lag angle between leading and trailing edges being equal to the tooth spacing angle, see Fig. 3.
2TR
r
"-I
r
r.i. i._r.l.,ir.i...i.i.ri"'1-"" r.r ot-
r.rI/"
I
IziT FIG. 3. Equivalent "analogue tool".
Observations In Peripheral MillingProcess
341
Having found that the average forces on a straight toothed tool may be estimated by considering an analogue tool which approximates helical teeth, it would be reasonable to assume that the same average force relationship holds true for a cutter with helical teeth. The proof of this is rather straight forward if one assumes that the helix may be represented by a series of straight edges, (the influence of obliquity on the force constants being taken into account). The average value for each of the resulting straight teeth is given by the previous arguments and since the force is linearly dependent on depth it will be found that the average forces are the same for both helical and straight teeth, provided that the variations in chip thickness and the remaining parameters (i.e. K, h*, q, and r2) are identical. Whilst the concept of an analogue tool does not help greatly in the analysis which follows it is certainly a useful visualisation of the process. The author will now continue to derive the expressions for average values of force and torque.
Average torque and specific power From the previous arguments the average torque may be written
3.2.
T(AVE)
-
-
2"rr
Qsind~ + r 2
d(b
(14)
i.e.
T(AVE) --
KaNStR h* 2w [(1--cos+s) + ~
+s].
Now equation (14) may be rewritten in several different forms, i.e.
T(a vm -
KaNS, (d + h* +~R) 2w
(15)
where d is radial width of cut. The form of equation (15) demonstrates that, for h* = 0 the torque is related directly to the width of cut as well as axial depth and feed per tooth. The average torque may also be expressed in terms of the average chip thickness as follows,
T(AVE) --
KaNSt gPsS,n (1 + h* 2v -S-~--)
where S,, is the mean chip thickness in cut. Now the average power is given by
PAVE = TArE to where to is the rotational speed. Thus rewriting equation (16)
P(A VE) --
KaNRto ¢bsSm (1 + h* 2"rr S-m--~ )"
(16)
342
I. YELLOWLEY
Substituting for Sm and collecting terms, to
P(AVE) = KaNR (1-cosdps) St ~
h*
(1 + - - ~ )
i.e. h*
P A W = K a d v (1 + - - ~ )
where v is the feeding velocity and the specific power is given by h* P* = K( 1 + ---d---)am
(17)
Thus the specific power (P*) is only a function of the mean chip thickness, (h* and K assumed constant). 3.3. Average values of forces The equations developed in the previous sections will now be used to obtain expressions for the average values of the forces Fx and F,' in both up and down milling. (a) Up milling.
Fx(A VE) --
aN --K2'rr S, [l
.~b5
(sin¢cosdp + rl sin2dP+
(cos¢ + r2sind~))dd~]
i.e.
Fx(A VE) --
KaNS, [(1-cos2~bs) + rl (2qbs - sin2~b,) + 8"n" •
4h* S, (sind~, + r2 (1-cos+~))]
(18)
similarly KaN F y ( A VE) --
~0S h* 2"tr St[ (rlsin~cos~ - sin2d~ + - ~ - t (r2cos~ - sin~))d~]
i.e. 4h* Fy(A WE) -- KaNS, 8"It [r,(l_cos2~bs)_(2d~s_sin2~bs ) + --S,- (r2sin~b~ - (1-cos~bs))]
(b) D o w n milling. Since the procedure is identical only the final results are given,
(19)
Observations In Peripheral Milling Process
FX~AVE) --
343
KaNSt 4h* 8rr [rl(2d~.~-sin2~bs) -(1-cos2~bs) + ~ (r2(1-cos+,.) -sin+s)]
(20)
4h* KaNS, [rl(l_cos2d~s)+[2+_sin2qb~) + ~ - t ((1-cos+,.) - r2sin+~)]. 8"ff " "
(21)
and
Fy(A
VE)
--
The equations derived have doubtful value in the examination of constraints, since they only give average values of force. Fairly obviously however at extreme conditions of axial depth and width of cut with multiple tooth end mills one would expect that their use may be justified. In writing the paper however the author is not primarily concerned with the identication of constraint violation, but rather with the identification of current cutting conditions. If one knows the current direction and magnitude of cutter traverse rate then the equations may be used to estimate the current values of width and depth of cut. Moreover it would appear that by defining a modified resultant force the need to evaluate direction of motion is obviated, this topic is the subject of section 5.
4.
EXPERIMENTAL RESULTS
In order to demonstrate the validity of the equations developed force measurements obtained from the square shoulder face milling of a medium carbon steel will be utilised. The force measurements were obtained from a two component strain gauge table dynomometer, [11], mounted on a vertical milling machine. Actual force measurements were made at intervals of two milliseconds, the resulting forces were then plotted and representative samples analysed to find average values in the X and Y directions. Since in this case the main purpose of considering experimental data is to verify the form of equations (2) and (3), then the first series of results considered utilise a width of cut equal to cutter diameter, (i.e. full immersion), with a wide variation in cutter feedrate. For the particular case of full immersion, the equations for average force in the X and Y directions reduce to the following,
FX(AVE) = K'S,2"rrrl + 8h*K'r2 and
Fy(AVE) = -(K'St21r + 8h*K') where Ka
K I _
8~ Thus, since both average forces are linear functions of S,, it is possible to evaluate all four of the experimental constants from the slope and intercepts of these linear functions. The resulting values are the following. K rl r2 h*
= = = =
1999.7 N/mm 2 0.5467 0.750 0.02062 mm.
It may be noted that the only surprising feature of the data is the rather low value of r~. It
344
I. YELLOWLEY
is felt that this is attributable to the fact that, in this case, with no wear on the cutter, the nose forces form a considerable proportion of the total parasitic force. Having obtained the necessary information on the constants for the particular work material - - tool pair, the equations derived may be used to predict the forces in both half and quarter immersion, at varying feedrate. The measured data together with predictions for all widths and feedrates are shown in Table 1. It is seen that the errors incurred, even for widely differing machining conditions are reasonably small, thus leading to the conclusion that the m e t h o d employed in characterising the force components is a reasonable approximation. From the data presented and from previous work [4], it is evident that the influence of axial depth of cut and radial width may be separated fairly easily by considering the magnitude of one average force and the ratio of the two mutually orthogonal average forces. On the other hand such an approach, whilst viable, from a physical point of view may be computationally time consuming, since the current direction of table velocity must be measured and measured forces resolved in that direction. It would appear that by defining a modified resultant force, the need to evaluate direction of motion is obviated, albeit with a considerable loss in resolution. This topic is the subject of the following section. 5.
THE RELATIONSHIP BETWEEN MEAN TORQUE AND THE QUASI MEAN RESULTANT FORCE
Suppose one defines a quasi-mean resultant force as follows
F j(AVE)
= ( F x ( A VE) 2 + F y ( A VE)2) ½
where FX(AVE)and FV(AVe) are now the average forces in any two mutually perpendicular directions. Clearly F~(AVE)is independent of the direction of travel and is linearly dependent on axial depth of cut. It should be noted however that F~(AVE)is not equal to the mean resultant force unless the fluctuation in force about the average is negligible. Returning
TABLE 1. COMPARISON OF MEASURED ,AND PREDICTED VALUES OF AVERAGE FORCE Width of cut
Feed per tooth
Measured forces
FX(A VE)
Fy(A rE)
(mm)
(mm)
(N)
(N)
101.6 101.6 101.6 101.6 101.6
0.0235 0.0469 0.0938 0.1876 0.2345
234 376 579 920 1100
- 351 - 569 - 961 -1660 -1953
50.8 50.8 50.8 50.8 50.8
0.0938 0.1876 0.2345 0.3049 0.3753
650 1015 1273 1443 1838
25.4 25.4 25.4
0.1876 0.2814 0.3753
631 857 1138
Calculated forces
FX(AVE)
FytA VE)
(N)
(N)
-0.667 -0.661 -0.602 -0.554 -0.563
248 345 541 931 1127
- 379 - 557 - 915 -1629 -1986
-0.654 -I).619 -0.591 -0.572 -0.567
- 264 - 459 - 572 - 715 -1165
-2.46 -2.21 -2.23 -2.02 -1.58
598 1021 1232 1550 1867
-
257 491 607 782 957
-2.33 -2.08 -2.03 -1.98 -1.95
-
-6.93 -6.30 -5.87
618 865 1112
-
78 124 171
-7.92 -6.98 -6.50
91 136 194
Ratio
Ratio
W o r k material. AISI 1040, Tool, 6 Tooth Carbide Face Mill. D i a m e t e r . 101.6mm, R o t a t i o n a l Speed, 361 r,p.m.. Axial Depth, 2.54 mm.
Observations In Peripheral Milling Process
345
to the concept of the analogue tool, (where FI(AvE) is equivalent to the mean resultant force), it is evident that F~(AVE)is a distinctly nonlinear function of width of cut, as a result of force cancellation. The significance of this fact is most easily shown first if one considers the case where the edge forces are zero, in this case, from the equations in the previous section, the following results are obtained.
T(AVE) =
KadNSt 2~
(22)
and
F I ( A vE) --
KaNS, ( l + r l 2 ) t ( ( l _ c o s 2~bs)2 + (2~bs_sin2d~,)2)½. 8"tr
(23)
As expected equation (22) shows torque to be a linear function of radial width, (it may be noted in passing that the vertical force (FZ(ArE)) will show the same characteristics as torque should an end mill with no nose radius of chamfer be used). On the other hand the quasi mean resultant force is not a simple linear function of width. The author would then suggest that the ratio of mean torque to quasi mean resultant force may be used to infer radial width, any one of the two equations may then be used to calculate axial depth of cut. In order to be specific the relationship between the ratio (T/RFI(AvE)) and width of immersion is shown for two practical values of r~ in Fig. 4. As may be seen from Fig. 4 the ratio varies considerably from half immersion to slotting (as would be expected), the changes below half immersion are relatively slight and experimentation will have to be resorted to before it may be claimed that the strategies based on this approach will be successful. It appears that consideration of the edge forces will in general lead to an increase in the change in the ratio (T/RF~AvE). Figure 5 shows the following ratio.
I..50
R~ =03
1.25
1,00
R~ = 0 5 0.75
O. 50 h*=O
Up milling
0.25
I 0.5
I I0
I 1.5
I 2.0
O/R
FIG. 4. The influence of radial width on the ratio of torque to quasi mean resultant force.
346
I. YELLOWLEY 0.8
r2=l 0.6
r2=2 +
+ r2=3.
0.4
r~ =OL3 Up miLLing o.Z
I 0.2
I 04
I 0,6
h* FI6. 5. The influence of flank forces on the relationship between torque-force ratio and radial width.
Q = (T/RFIAvE) (T/RFIAvE)
(d = 0.1R) (d = 2R)
as a f u n c t i o n o f h * a n d r2. 6.
CONCLUSIONS
The author believes that the findings presented here have practical application to adaptive control strategies, particularly in cases where the geometry of the blank varies. Evidently the same approach may be used for square shoulder face milling, extensions to face milling are however likely to be limited unless the cutter has a considerable offset to the center line of the work. It is the author's intention to conduct an experimental evaluation of the equations presented to more fully ascertain the applicability of the various approaches to the adaptive control of milling operations. ACKNOWLEDGEMENTS
The author gratefully acknowledges financial support from the Natural Science and Engineering Research Council of Canada under operating Grant No. A5812. The author is also grateful to Mr. Y. Altintas, a Graduate Student at McMaster University, who provided the basic force data used to demonstrate the validity of the equations derived in this work.
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
M. E. MAgTEt.LOTn, Trans. ASME, Nov,, 667 (1941). F. KOENIGSBERGEgand A. J. P. SABBERWAL,Ann. CIRP (1960). F. KOENIGSBEXGERand A. J. P. SABBEgWAL,Int. J. Mach. Tool Des. Res. 1, 15 (1961). I. YELLOWLEY,Fertigung, 2, 31 (1978). J. TLusrY and P. MAcNEIL, Ann. CIRP, 24, 21 (1975). W. A. KUNE, R. E. DEVoR and J. R. LINDBERG,Int. J. Mach. Tool Des. Res. 22, 7 (1982). I. YELLOWLEVand G. BARROW,Int. J. Mach. Tool Des. Res. 16, 1 (1976). I. YELLOWLEYand G. BAaROW,Proc. 19th Int. M T D R Conf., p. 443. MacMillan Press, Manchester (1978). H. J. Fu, R. D. DEVORand S. G. KAPOORTrans. A S M E J. Engng. Ind. 106, 81 (1984). N. N. ZOREV, Metal Cutting Mechanics, Pergamon Press (1966). P. YOUNG, McMaster Table Dynomometer, MRDG Report #183, McMaster University (1983).
Vol. 25, No. 4. pp. 337-346. 1985
0020-735718553.00+.00 Pergamon Press Ltd.
OBSERVATIONS ON THE MEAN VALUES OF FORCES, TORQUE AND SPECIFIC POWER IN THE PERIPHERAL MILLING PROCESS I. YELLOWLEY*
(Received 24 July 1984: in final form 9 January 1985) Abstract--The forces arising from the milling process are important practical parameters. In this paper, the author uses a force model which considers both rake and flank face contributions to derive expressions for average values of force, torque and specific power. It is shown that specific power is a unique function of mean chip thickness. The major practical importance of the work is the application of the concepts derived to adaptive control. It is demonstrated that both radial width and axial depth of cut may be estimated from a knowledge of the quasi mean resultant force and the mean value of torque. The possibility of identifying the dimensions of cut, in real time, provides a significant step towards the development of adaptive optimisation strategies for the milling process.
INTRODUCTION
SEVERAL authors have examined the problem of estimating average forces and torque in peripheral milling. The first rigorous analysis of the geometry of the process was carried out by Martellotti [1], who also made experimental measurements of power and speculated that the specific power was primarily a function of average chip thickness. Later work by Koenigsberger and Sabberwal [2, 3], examined tangential forces and specific cutting pressure, again a strong influence of mean chip thickness on these variables was observed, regardless of the geometry of the cutter. In recent years the author [4] has shown how the concept of an analogue tool may be used to estimate average forces and several authors, e.g. [5, 6], have analysed the variation in forces numerically, making various assumptions regarding the instantaneous cutting pressure and ratio of tangential and radial cutting force components. The purpose of this paper is two fold; (a) to demonstrate analytically that the modelling of the cutting forces as the sum of active cutting and edge components leads to the conclusion that specific power is a unique function of average chip thickness in cut. (b) To demonstrate that the average values of force and torque may be useful in practical adaptive control strategies where, because of variations in blank geometry, the instantaneous cutting conditions are not known with any certainty. In this regard one is particularly interested in being able to ascertain current values of radial width and axial depth of cut. Such considerations are particularly important when one wishes to optimise cutting conditions, the two variables having very different influences on cutter wear rate [7, 8]. 2. INSTANTANEOUSVALUES OF FORCES AND TORQUE IN PERIPHERAL MILLING The forces acting upon a single straight tooth in up and down milling are shown in Figs 1 and 2 respectively. As shown in Figs 1 and 2. it is customary to resolve the radial and tangential force components parallel to the feed direction (X) and perpendicular to the feed (Y). In general it has been assumed that the tangential component may be calculated by multiplying the instantaneous area of cut by the specific cutting pressure (K) and that the ratio of radial to tangential component (r) is known. In the simplest case both K and r are assumed constant, however, a recent paper [9] has assumed these * Department of Industrial Engineering, Technical University of Nova Scotia, Halifax. Nova Scotia, Canada B3J 2X4. 337
338
I. YELLOWLEY
F, //~
I I
~
Fx
FIG. 1. Forces in up milling.
0o
I
Fx
FIG, 2. Forces in down milling.
parameters to be known functions of mean chip thickness. Since the chip thickness in peripheral milling is generally very low then it would seem logical, as a first improvement on a linear model of cutting forces, to consider the edge forces, (nose and flank), which are known to constitute a significant proportion of total forces at such conditions. (Several authors have reported experiments aimed at measuring these parasitic forces e.g. [10].) The model developed then considers that the tangential force is made up of two components, one of which is directly proportional to undeformed area of cut, the other directly proportional to length of cutting edge engaged. It will, for the present time, be assumed that the cutting edge is straight with no nose radius or chamfer at the point. The influence of a helix angle is discussed in the following section. Thus the tangential force on a single straight edge may be expressed as FT
= KaSt
sin~ + xa
(1)
where k is the specific cutting pressure, a is the axial depth of cut, St is the feed per tooth, is the instantaneous angle of rotation and x is the edge force constant. Assuming now that one defines a critical chip thickness (h*) as that which yields a total cutting force which is comprised of equal components of cutting and parasitic force, then
or
kah* = xa x = Kh*
and equation (1) may be rewritten in the form. Fv=KaSt(sind~+~).
(2)
Observations In Peripheral MillingProcess
339
It is to be expected that the ratio of tangential to radial components on the rake face and flank of the tool will be considerably different. The radial force is thus expressed in the form FR = KaSt rlsindp + r 2
(3)
where r~ is the ratio of radial to tangential cutting forces and r2 is the ratio of radial to tangential flank forces. In general, typical values for these ratios are the following, 0 . 2 < r ~ < 0.5 1
T = FT.R
where T is torque and R is cutter radius. (a) Up milling F x = F r c o s + + FRsind0 F y = FR cos do -- F r sin +.
(5) (6)
(b) D o w n milling F x = FR sin + - F r c o s c b Fy = FR cos + + F r s i n + .
(7) (8)
3. AVERAGE VALUES OF FORCE. TORQUE AND SPECIFIC POWER 3.1.
Introduction
From the previous section it is evident that the instantaneous values of force and torque are linearly dependent on axial depth of cut. Thus any one of the forces may be expressed in the form, F = a f(eb, h*, rl. 1"2, S,) the corresponding average value of force for a cutter with a single straight tooth is given by ¢b, F = ~
f (~b. h*, rl, r2. St)
(9)
where +~ is swept angle of cut. A consideration of equation (9) suggests that an anoiogy may be constructed by considering the constant force produced by an analogue tool with tooth spacing 8+ each tooth having an axial projection given by
8( =
aa_ 8+.
2v
0o)
340
I. YELLOWLEY
Such a tool may, evidently be regarded as an approximation to a helical tooth with a lag angle of 360 ° between leading and trailing edges. Should a cutter have a number (N) of straight cutting edges which are evenly spaced, then the corresponding axial projection of each tooth in the equivalent analogue cutter is given by Na
817 =
2~
8~b
(ll)
such a representation does not have a particularly relevant physical significance however making the substitution = Nd~
(12)
then from equations (1) and (12) 8e = ~
a
1502
(13)
evidently then the equivalent analogue tool represents a helical cutter with N teeth, the lag angle between leading and trailing edges being equal to the tooth spacing angle, see Fig. 3.
2TR
r
"-I
r
r.i. i._r.l.,ir.i...i.i.ri"'1-"" r.r ot-
r.rI/"
I
IziT FIG. 3. Equivalent "analogue tool".
Observations In Peripheral MillingProcess
341
Having found that the average forces on a straight toothed tool may be estimated by considering an analogue tool which approximates helical teeth, it would be reasonable to assume that the same average force relationship holds true for a cutter with helical teeth. The proof of this is rather straight forward if one assumes that the helix may be represented by a series of straight edges, (the influence of obliquity on the force constants being taken into account). The average value for each of the resulting straight teeth is given by the previous arguments and since the force is linearly dependent on depth it will be found that the average forces are the same for both helical and straight teeth, provided that the variations in chip thickness and the remaining parameters (i.e. K, h*, q, and r2) are identical. Whilst the concept of an analogue tool does not help greatly in the analysis which follows it is certainly a useful visualisation of the process. The author will now continue to derive the expressions for average values of force and torque.
Average torque and specific power From the previous arguments the average torque may be written
3.2.
T(AVE)
-
-
2"rr
Qsind~ + r 2
d(b
(14)
i.e.
T(AVE) --
KaNStR h* 2w [(1--cos+s) + ~
+s].
Now equation (14) may be rewritten in several different forms, i.e.
T(a vm -
KaNS, (d + h* +~R) 2w
(15)
where d is radial width of cut. The form of equation (15) demonstrates that, for h* = 0 the torque is related directly to the width of cut as well as axial depth and feed per tooth. The average torque may also be expressed in terms of the average chip thickness as follows,
T(AVE) --
KaNSt gPsS,n (1 + h* 2v -S-~--)
where S,, is the mean chip thickness in cut. Now the average power is given by
PAVE = TArE to where to is the rotational speed. Thus rewriting equation (16)
P(A VE) --
KaNRto ¢bsSm (1 + h* 2"rr S-m--~ )"
(16)
342
I. YELLOWLEY
Substituting for Sm and collecting terms, to
P(AVE) = KaNR (1-cosdps) St ~
h*
(1 + - - ~ )
i.e. h*
P A W = K a d v (1 + - - ~ )
where v is the feeding velocity and the specific power is given by h* P* = K( 1 + ---d---)am
(17)
Thus the specific power (P*) is only a function of the mean chip thickness, (h* and K assumed constant). 3.3. Average values of forces The equations developed in the previous sections will now be used to obtain expressions for the average values of the forces Fx and F,' in both up and down milling. (a) Up milling.
Fx(A VE) --
aN --K2'rr S, [l
.~b5
(sin¢cosdp + rl sin2dP+
(cos¢ + r2sind~))dd~]
i.e.
Fx(A VE) --
KaNS, [(1-cos2~bs) + rl (2qbs - sin2~b,) + 8"n" •
4h* S, (sind~, + r2 (1-cos+~))]
(18)
similarly KaN F y ( A VE) --
~0S h* 2"tr St[ (rlsin~cos~ - sin2d~ + - ~ - t (r2cos~ - sin~))d~]
i.e. 4h* Fy(A WE) -- KaNS, 8"It [r,(l_cos2~bs)_(2d~s_sin2~bs ) + --S,- (r2sin~b~ - (1-cos~bs))]
(b) D o w n milling. Since the procedure is identical only the final results are given,
(19)
Observations In Peripheral Milling Process
FX~AVE) --
343
KaNSt 4h* 8rr [rl(2d~.~-sin2~bs) -(1-cos2~bs) + ~ (r2(1-cos+,.) -sin+s)]
(20)
4h* KaNS, [rl(l_cos2d~s)+[2+_sin2qb~) + ~ - t ((1-cos+,.) - r2sin+~)]. 8"ff " "
(21)
and
Fy(A
VE)
--
The equations derived have doubtful value in the examination of constraints, since they only give average values of force. Fairly obviously however at extreme conditions of axial depth and width of cut with multiple tooth end mills one would expect that their use may be justified. In writing the paper however the author is not primarily concerned with the identication of constraint violation, but rather with the identification of current cutting conditions. If one knows the current direction and magnitude of cutter traverse rate then the equations may be used to estimate the current values of width and depth of cut. Moreover it would appear that by defining a modified resultant force the need to evaluate direction of motion is obviated, this topic is the subject of section 5.
4.
EXPERIMENTAL RESULTS
In order to demonstrate the validity of the equations developed force measurements obtained from the square shoulder face milling of a medium carbon steel will be utilised. The force measurements were obtained from a two component strain gauge table dynomometer, [11], mounted on a vertical milling machine. Actual force measurements were made at intervals of two milliseconds, the resulting forces were then plotted and representative samples analysed to find average values in the X and Y directions. Since in this case the main purpose of considering experimental data is to verify the form of equations (2) and (3), then the first series of results considered utilise a width of cut equal to cutter diameter, (i.e. full immersion), with a wide variation in cutter feedrate. For the particular case of full immersion, the equations for average force in the X and Y directions reduce to the following,
FX(AVE) = K'S,2"rrrl + 8h*K'r2 and
Fy(AVE) = -(K'St21r + 8h*K') where Ka
K I _
8~ Thus, since both average forces are linear functions of S,, it is possible to evaluate all four of the experimental constants from the slope and intercepts of these linear functions. The resulting values are the following. K rl r2 h*
= = = =
1999.7 N/mm 2 0.5467 0.750 0.02062 mm.
It may be noted that the only surprising feature of the data is the rather low value of r~. It
344
I. YELLOWLEY
is felt that this is attributable to the fact that, in this case, with no wear on the cutter, the nose forces form a considerable proportion of the total parasitic force. Having obtained the necessary information on the constants for the particular work material - - tool pair, the equations derived may be used to predict the forces in both half and quarter immersion, at varying feedrate. The measured data together with predictions for all widths and feedrates are shown in Table 1. It is seen that the errors incurred, even for widely differing machining conditions are reasonably small, thus leading to the conclusion that the m e t h o d employed in characterising the force components is a reasonable approximation. From the data presented and from previous work [4], it is evident that the influence of axial depth of cut and radial width may be separated fairly easily by considering the magnitude of one average force and the ratio of the two mutually orthogonal average forces. On the other hand such an approach, whilst viable, from a physical point of view may be computationally time consuming, since the current direction of table velocity must be measured and measured forces resolved in that direction. It would appear that by defining a modified resultant force, the need to evaluate direction of motion is obviated, albeit with a considerable loss in resolution. This topic is the subject of the following section. 5.
THE RELATIONSHIP BETWEEN MEAN TORQUE AND THE QUASI MEAN RESULTANT FORCE
Suppose one defines a quasi-mean resultant force as follows
F j(AVE)
= ( F x ( A VE) 2 + F y ( A VE)2) ½
where FX(AVE)and FV(AVe) are now the average forces in any two mutually perpendicular directions. Clearly F~(AVE)is independent of the direction of travel and is linearly dependent on axial depth of cut. It should be noted however that F~(AVE)is not equal to the mean resultant force unless the fluctuation in force about the average is negligible. Returning
TABLE 1. COMPARISON OF MEASURED ,AND PREDICTED VALUES OF AVERAGE FORCE Width of cut
Feed per tooth
Measured forces
FX(A VE)
Fy(A rE)
(mm)
(mm)
(N)
(N)
101.6 101.6 101.6 101.6 101.6
0.0235 0.0469 0.0938 0.1876 0.2345
234 376 579 920 1100
- 351 - 569 - 961 -1660 -1953
50.8 50.8 50.8 50.8 50.8
0.0938 0.1876 0.2345 0.3049 0.3753
650 1015 1273 1443 1838
25.4 25.4 25.4
0.1876 0.2814 0.3753
631 857 1138
Calculated forces
FX(AVE)
FytA VE)
(N)
(N)
-0.667 -0.661 -0.602 -0.554 -0.563
248 345 541 931 1127
- 379 - 557 - 915 -1629 -1986
-0.654 -I).619 -0.591 -0.572 -0.567
- 264 - 459 - 572 - 715 -1165
-2.46 -2.21 -2.23 -2.02 -1.58
598 1021 1232 1550 1867
-
257 491 607 782 957
-2.33 -2.08 -2.03 -1.98 -1.95
-
-6.93 -6.30 -5.87
618 865 1112
-
78 124 171
-7.92 -6.98 -6.50
91 136 194
Ratio
Ratio
W o r k material. AISI 1040, Tool, 6 Tooth Carbide Face Mill. D i a m e t e r . 101.6mm, R o t a t i o n a l Speed, 361 r,p.m.. Axial Depth, 2.54 mm.
Observations In Peripheral Milling Process
345
to the concept of the analogue tool, (where FI(AvE) is equivalent to the mean resultant force), it is evident that F~(AVE)is a distinctly nonlinear function of width of cut, as a result of force cancellation. The significance of this fact is most easily shown first if one considers the case where the edge forces are zero, in this case, from the equations in the previous section, the following results are obtained.
T(AVE) =
KadNSt 2~
(22)
and
F I ( A vE) --
KaNS, ( l + r l 2 ) t ( ( l _ c o s 2~bs)2 + (2~bs_sin2d~,)2)½. 8"tr
(23)
As expected equation (22) shows torque to be a linear function of radial width, (it may be noted in passing that the vertical force (FZ(ArE)) will show the same characteristics as torque should an end mill with no nose radius of chamfer be used). On the other hand the quasi mean resultant force is not a simple linear function of width. The author would then suggest that the ratio of mean torque to quasi mean resultant force may be used to infer radial width, any one of the two equations may then be used to calculate axial depth of cut. In order to be specific the relationship between the ratio (T/RFI(AvE)) and width of immersion is shown for two practical values of r~ in Fig. 4. As may be seen from Fig. 4 the ratio varies considerably from half immersion to slotting (as would be expected), the changes below half immersion are relatively slight and experimentation will have to be resorted to before it may be claimed that the strategies based on this approach will be successful. It appears that consideration of the edge forces will in general lead to an increase in the change in the ratio (T/RF~AvE). Figure 5 shows the following ratio.
I..50
R~ =03
1.25
1,00
R~ = 0 5 0.75
O. 50 h*=O
Up milling
0.25
I 0.5
I I0
I 1.5
I 2.0
O/R
FIG. 4. The influence of radial width on the ratio of torque to quasi mean resultant force.
346
I. YELLOWLEY 0.8
r2=l 0.6
r2=2 +
+ r2=3.
0.4
r~ =OL3 Up miLLing o.Z
I 0.2
I 04
I 0,6
h* FI6. 5. The influence of flank forces on the relationship between torque-force ratio and radial width.
Q = (T/RFIAvE) (T/RFIAvE)
(d = 0.1R) (d = 2R)
as a f u n c t i o n o f h * a n d r2. 6.
CONCLUSIONS
The author believes that the findings presented here have practical application to adaptive control strategies, particularly in cases where the geometry of the blank varies. Evidently the same approach may be used for square shoulder face milling, extensions to face milling are however likely to be limited unless the cutter has a considerable offset to the center line of the work. It is the author's intention to conduct an experimental evaluation of the equations presented to more fully ascertain the applicability of the various approaches to the adaptive control of milling operations. ACKNOWLEDGEMENTS
The author gratefully acknowledges financial support from the Natural Science and Engineering Research Council of Canada under operating Grant No. A5812. The author is also grateful to Mr. Y. Altintas, a Graduate Student at McMaster University, who provided the basic force data used to demonstrate the validity of the equations derived in this work.
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
M. E. MAgTEt.LOTn, Trans. ASME, Nov,, 667 (1941). F. KOENIGSBERGEgand A. J. P. SABBERWAL,Ann. CIRP (1960). F. KOENIGSBEXGERand A. J. P. SABBEgWAL,Int. J. Mach. Tool Des. Res. 1, 15 (1961). I. YELLOWLEY,Fertigung, 2, 31 (1978). J. TLusrY and P. MAcNEIL, Ann. CIRP, 24, 21 (1975). W. A. KUNE, R. E. DEVoR and J. R. LINDBERG,Int. J. Mach. Tool Des. Res. 22, 7 (1982). I. YELLOWLEVand G. BARROW,Int. J. Mach. Tool Des. Res. 16, 1 (1976). I. YELLOWLEYand G. BAaROW,Proc. 19th Int. M T D R Conf., p. 443. MacMillan Press, Manchester (1978). H. J. Fu, R. D. DEVORand S. G. KAPOORTrans. A S M E J. Engng. Ind. 106, 81 (1984). N. N. ZOREV, Metal Cutting Mechanics, Pergamon Press (1966). P. YOUNG, McMaster Table Dynomometer, MRDG Report #183, McMaster University (1983).