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Computerized End-Milling Force Predictions with Cutting Models Allowing for Eccentricity and Cutter Deflections
Computerized End-Milling Force Predictions with Cutting Models Allowing for Eccentricity and Cutter Deflections E. J. A. Armarego ( l ) , N . P. Deshpande, The University of Melbourne, Victoria/Australia Received on January 14, 1991 In this paper further developments of computer based mechanics of cutting models and software modules for the prediction of the average and fluctbating fsrce components and torqhe in End mL11lng are considered and assessed by numerical simulation and experimental testing. Three models are considered namely, the ‘ideal’ model for rigid cutters with no eccentricity, the rigid cutter ‘eccentricity’ model and a more comprehensive ‘deflection’ model ailowing for both eccentricity and cutter deflections. It is shown that the ‘ideal’ model provides useflu1 qualitative trends for the average forces and torques which are similar for all three models but yields poor qualitative force fluctuation predictions. The ‘eccentricity’ model yields good predictions of the average forces and torque while the fluctuation predictions aro also good but deteriorate far unfavourable cutter deflection while the ‘deflection‘ model provides the best predictions and correlation with the 360 experimental test results but at the expense of excessive computer processing time. The practical implications of these models and the importance of both eccentricity and cutter defleczion on the force fluctuations are discussed. KEYWORDS : End-milling, Eccentricity, Cutting forces, Milling, CAD/CAM LNTRODUCTION The need for more readily available and reliable quantitative machining performance information has been recognized for decades and re-emphasized in a recent CIRP survey [l]. In view of the wide variety of machining operations and the numerous 1nflUenCing variables for each operation the development o f models for the quantitative prediction of machining performance characteristics represents a formidable task. Nevertheless such models should be achieved to satisfy the above need for the effective and efficient use of machining as well as to establish machining on a sound scientific and quantitative basis. In this paper further developments of Computer based mechanics of cutting models and software modules for the prediction of the average and fluctuating force components and torque in End-milling [21 are presented and assessed by numerical simulation and experimental testing. Three comprehensive models are considered, viz. the ‘ideal‘ model for rigid cutters with no eccentricity which allows for the many process variables, the rigid cutter ’eccentricity’ model which includes the addition71 inevitable eccentricity and the most sophisticated deflection’ model which incorporates both eccentricity and cutter deflections.
from the plane normal to the feed motion. At this inis engaged in cutting over a stant one tooth (i.2) number of elements M (=&/db) with in the range ui and y u = e x . It has been shown [6,71 that, irrespective of the milling model used, the effect of the feed speed on the ‘dynamic’ tool geometry and cutter ?ath during tooth engagement can be ignored. Thus the static’ and specified normal rake angle y n and inclination angle X., which are constant for all points and elements, may be used in the mechanics of cutting analysis for force prediction. the tooth path may be considered circular and the resultant cutting speed V. found from the cutter w, i.e. y n = tan-l(tanrrcos6); A. = 6 a V. = v = wR (1,2,3) Furthermore, the elemental oblique force components dFp, dFq which vary from element to element, will be tangential and normal to the circular path so that in the fixed these will need to be resolved ‘practical’ force directions x and y to simplify the summation o f the instantaneous forces on the whole cutter. The elemental practical force component, torque and power are given by (4) dFx = dFpcosu+dFqsinu dFy = dfpsinv-dfqcosu (5) dFz = dFr. dT = R.dFp, dP = w-R.dFp ( 6 , 7 . 8 )
MECHANICS OF CUTTING APPROACH FOR FORCE MODELS The basic strategy for the three milling models is to develop computer based analyses to predict the instantaneous force components, torque and power for the cutter as a whole in any selected orientation and to vary the angular orientation over one cutter revolution o r cycle to determine the average and fluctuating forces, torque and power charactecistics f o r a given set of cutting conditions and cutter specification. Thin shear zone mechanics of cutting analyses and orthogonal cutting data bases [ 3 , 4 , 5 ] are used to predict the instantaneous milling forces by partitioning the cutter teeth into a series of axial elemental oblique cutting tools, identifying the tooth elements actively engaged in cutting for a given cutter orientation, determining the fundamental tool geometry, cut thickness and width of cut f.or each element to obtain the elemental force components from the ‘classical’ oblique cutting analysis, resolving the elemental components in the practical milling force directions and summing the force component contributions o f all the elements on each tooth. The elemental force components may also be used to evaluate the elemental torque and power which can be summed to yield the instantaneous values for the cutter as a whole. The cutter orientation is identified in terms of the angular orientation B r of the leading point on a ‘reference’ tooth with respect to a plane normal to the feed motion (Fig.1). Thus if tooth 1 is chosen as the ‘reference’ tooth er=@ti=etr and the angular orientation of the i t ” tooth Bti can be found for the known tooth angular pitch e b , i.e. EIti=Br-(i-l)eb. By varying Br over 360’ the predicted variations and average values can be obtained, althoygh, where possible, explicit expressions for the ‘peak and average values will be sought. Since the three milling models considered adopt the same strategy but involve sophisticated extensions of the basic ‘ideal’ case model, the common aspects of these analyses will be outlined first followed by the specific equations and solution procedures for each model. In this way a better appreciation of’the differences i n the predictive models may be achieved. Fig.1 shows the instantaneous geometry of an Endmillin? Operation at orientation & (based on the ‘ideal case) together with the relevant geometry and force components for a typical ‘active’ oblique cutting element of width db (=Rdv/tanh) at an angle u
Annals of the CIRP, Vol. 40/1/1991
FIG 1 : Rebted oblique cut Geometry for an End mill tooth element. The elemental oblique force components are found from the ‘classical’ oblique cutting analysis
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tan(0.+Rn ):tan(h )~os7~/[tan(ne )-sinrn*tan(Xs ) I (16) Kep = Cep*cosh, KBQ-Ceq and Ker:O.2S*Kep*sinX. (17) The Kek and Kck factors in eq.(9) can be found from from eqs.(lO to 17) given the basic cutting data (K,ri,O,Cep,Ceq) in the data bank [3,4,5,71 and the elemental tool geometry from eqs. ( 1 6 2 ) . However, to predict dFp, dFq and dFr in eq.(9) and the corresponding practical force components and torque f r o m eqs.(4 to 8) it is necessary to know the elemental cut thickness t(x) for the given width of cut db. The required t(u) is h i g h l y dependent on the milling model considered i . e . whether the 'ideal', 'eccentricity' o r 'deflection' model i s used. Given that t(u) can be found for each element, as will be shown below, the practical f o r c e components and torque acting on each engaged tooth at at orientation 8 t t is found b y summing the elemental forces and torque within the relevant u1 and u u limits given in Table ( 1 ) below so that
.-
u=uu
J-Yu
>=XU
Fjt(eti)
= ZdFj
; Tt(8ti)
= tdT
u=u1
; Pt(8ti)-
-
idP >=>I
u3111
j=x,y,z (18,19,20) Finally, the instantaneous Torce components, torque and power on each active tooth from eqs.(i8 to 20) can be added to yield the instantaneous forces components, torque and power on the cutter as a whole at the corresponding given orientation er i . e . Fj(B,)
=
i=Nt rdFjt(8ti) i =1
; T(%)
=
i =Nt ITt(Bt0;
i :Nt P(Br)=XPt(Bti)
i=l
i-1
j=x,y,z For
o s YU
ett
= eti
For ( e t i - e w ) ill
=
0
<
(21,22,231
et i z ex uu 5
o
= ex
o < ( e t l - e w )( ex = (eti-8.)
(eti-ew) 2 u1
= ex
ex
By repeating the above procedure for different Br values over a revolution or cycle, the fluctuations in the forces, torque and power can be obtained from which the maximum, minimum and average forces can be found or evaluated. As noted earlier, the cut thickness t(u) at each element is the only remaining quantity required to predict the forces, torque and power from the above analysis. This fundam'ental variable, which results from the.interference o f the element under consideration with the transient surface generated by an element on a preceding tooth, is in fact the main distinguishing feature of the three proposed milling models. The overall analysis and solution procedure for force predictions can be substantially affected by the milling model adopted.
IDEAL CASE MODEL Under ideal cutting conditions of no eccentricity and cutter deflections, every tooth element interferes with the path of the immediate preceding tooth element and the cut thickness can be found from two consecutive circular tooth paths and approximated to 124) t(u) = ftsin(u) Thus for the 'ideal' case model the cut thickness can be found from the simple explicit expression in eq.(24) given the feed per tooth ft and the orientation angle v of the required element. It follows that in this model the instantaneous force components, torque and power at the given cutter orientation 0r can be predicted from the above analysis provided the cutter specification (D,Nt or B b , yn or . y r . A. or a). the cutting conditions (ar.a.,ft or Nt,w) and the basic cutting quantities (r,rll!3,Cep,Ceq)in the orthogonal cutting data base are known. In addition no iterations are required for predicting the cut thickness t(u). It has also been shown [2,71 that because of the relatively simple expression for t(u) in eq.(24) explicit expressions for the average force components, torque and power could be established in terms of many o f the major process variables such as ft,aa,Nt, while the cyclic force fluctuations were on a per tooth basis with zero fluctuations occurring when the tool and cut variables satisfied the equation 2nR-m = Nt&tan6; m 1,2,..m (25) However it has not been possible to establish the maximum forces and torque b y explicit expressions and recourse to numerical studies using the computer based predictive model was necessary. It is apparent that the 'ideal' case model is comprehensive since it allows for the vast majority of the process variables. ECCENTRICITY ECCENTRICITY ~~0000~~44 I n practice D r a c t i c e it i t has has been been found found that t h a t an an ececIn c e n t r i c i t y i n c u t t e r axis a x i s of o f rotation r o t a t i o n occurs occurs primarily primarily centricity in cutter
26
from t h e chuck o r t o o l h o l d i n g d e v i c e r a t h e r t h a n due t o e r r o r s i n c u t t e r manufacture [2,71. Furthermore, e c c e n t r i c i t i e s a r e u s u a l l y i n e v i t a b l e and these can v a r y each t i m e t h e same c u t t e r i s mounted i n a h o l d e r . The e c c e n t r i c i t y can be s p e c i f i e d by t h e d i s t a n c e e between t h e c u t t e r g e o m e t r i c a l a x i s and i t s a x i s of r o t a t i o n t o g e t h e r w i t h t h e minimum p o s i t i v e a n g l e er w i t h r e s p e c t t o t h e r a d i a l l i n e from i t s a x i s t o t h e l e a d i n g p o i n t o f t h e r e f e r e n c e t o o t h as r e p o r t e d earl i e r [ a ] . I t has a l s o been shown t h a t when eccent r i c i t y i s p r e s e n t t h e c u t t e r may be represented by a v a r i a b l e r a d i u s c u t t e r about t h e a x i s o f r o t a t i o n such t h a t t h e r a d i i o f a l l p o i n t s on t h e c u t t i n g edges w i l l v a r y w i t h i n and between t e e t h a l t h o u g h t h e a n g u l a r pitch Bb, normal r a k e angle .yn and h e l i x angle d a r e e s s e n t i a l l y constant a t t he s p e c i f i e d values [2,7]. For a c u t t e r o r i e n t a t i o n & , t h e r a d i u s of the p o i n t ( e l e m e n t ) on t h e c u t t i n g edge o f t h e i t h t o o t h ( R ’ u ) i o r i e n t e d a t x i t o t h e p l a n e normal t o t h e feed m o t i o n i s g i v e n by e.cos(Br-Er-Yi (26) (R’u)i = R Due t o t h e d i f f e r e n c e s i n r a d i i between elements on d i f f e r e n t t e e t h , t h e c u t t h i c k n e s s no l o n g e r r e s u l t s f r o m t h e i n t e r f e r e n c e o f two c i r c u l a r p a t h s o f c o n s e c u t i v e t e e t h w i t h t h e same r a d i i and d i s p l a c e d by ft as i n t h e ‘ i d e a l ’ case model, b u t may r e s u l t from the int er f er enc e w i t h t he nth preceding t o o t h ( i .e . (i-n)Ch t o o t h ) p a t h d i s p l a c e d by n . f t w i t h r e s p e c t t o t h e ith F i g . Z ( a ) shows t h e r e l e v a n t geometry f o r t h e e c c e n t r i c i t y ’ model where t h e d i f f e r e n t r a d i i (R’u)i and ( R ’ u l i - n a r e d e p i c t e d t o g e t h e r w i t h t ( u ) i and t h e displacement i n t h e f e e d d i r e c t i o n between t h e two t e e t h . While i n F i g . Z ( a ) t h e t o o t h paths a r e n o t circular i t has been shown t h a t a c i r c u l a r t o o t h p a t h i s a v e r y good a p p r o x i m a t i o n as i s u i 2 u ’ t - n 1 7 1 so t h a t t h e c u t t h i c k n e s s i s g i v e n by (27) t ( u i ) = n - f t s i n ( u i )+(R’u)i -(R’il)i-n, n=l t o Nt Further the radius (R’illi-n can be found from eq.(26) f o r t h e corresponding ili-”v a l u e s which equals ( u i +neb so t h a t t ( u i ) = nftsin(ili )+e[cos(er-Er-ili-n8b)-cos(e~-€~-ui ) I (28) n = i t o Nt The r e l e v a n t n is t h a t t o o t h which g i v e s t h e minimum v a l u e o f t ( v i ) ) 0 . I t i s w o r t h n o t i n g t h a t due t o t h e d i f f e r e n c e s i n t h e r a d i i and t h e s m a l l v a l u e o f ft if. i s p o s s i b l e t h a t an element does n o t t a k e a ‘ b i t e even though i t s o r i e n t a t i o n ili i s w i t h i n the l i m i t s when c u t t i n g i s l i k e l y t o occur ( i . e . u 1 c u i L ~ ~ ) as i n F i g . 1 . Such a c o n d i t i o n occurs when any o f t h e t ( u i ) from eq.(28) a r e l e s s t h a n zero.
!oath.
I t i s apparent t h a t when e c c e n t r i c i t y i s a l l o w e d for, an e x p l i c i t e q u a t i o n f o r t(ut) i s no l o n g e r a v a i l a b l e a l t h o u g h p r o v i d e d e and E V a r e known ( o r measured) t ( v , i ) can be e v a l u a t e d by ‘ s e a r c h i n g ’ f o r t h e r e q u i r e d ‘ n ’ v a l u e by s o l v i n g e q . ( 2 8 ) f o r minimum t f x i ) > 0 , o r i d e n t i f y i n g elements which do n o t engage t h e workpiece i . e . t ( i l i ) s 0.
Ci-n)‘h Tooth Dath
(i-nlth
;/T ,;
d
T o o t h path
2: +aqJpJDXAFJ?
%
P r q ~ “
nrBb (a) Eccentricity
1
‘ Y3- n
ru1
Q
nreb
( b l E c c e n t r i c i t y & D ef l e ct i o n
Fig.2: E l e m e n t a l c u t t h i c k n e s s
in
milling
For t h i s model, t h e instantaneous f o r c e compon e n t s , t o r q u e and power a t a g i v e n c u t t e r o r i e n t a t i o n Br can be p r e d i c t e d i f e and Er a r e g i v e n i n a d d i t i o n t o the t o o l geometrical variables, c u t t i n g conditions and b a s i c c u t t i n g d a t a as f o r t h e ‘ i d e a l ’ case model. However f o r t h e ‘ e c c e n t r i c i t y ’ model t ( v i ) has t o be s o l v e d n u m e r i c a l l y f o r each element on t h e c u t t e r t e e t h a t each c u t t e r O r i e n t a t i o n so t h a t computer a s s i s t a n c e i s e s s e n t i a l and w i l l i n v o l v e more computat i o n a l t i m e t h a n f o r t h e ‘ i d e a l ’ case model. Furthermore, because t ( u i ) has t o be s o l v e d n u m e r i c a l l y i t has n o t been p o s s i b l e t o e s t a b l i s h e x p l i c i t express i o n s between t h e average force components, t o r q u e and power and t h e p r a c t i c a l process v a r i a b l e s such as f t , aa, N t as was p o s s i b l e i n t h e ‘ i d e a l ’ model. Nevertheless t h e ‘ e c c e n t r i c i t y ’ model i s a more s o p h i s t i c a t e d model a l l o w i n g f o r the inevitable e c c e n t r i c i t y noted i n p r a c t i c e and o f t e n i g n o r e d i n machining f o r c e p r e d i c t i o n s .
DEFLECTION MODEL It is well recognized that because End-milling cutters are only supported at one (built-in) end, they are prone'to deflections under the action o f the cutting fortes [2,8,9]. It has also been noted [ 2 , 7 1 that under some cutting conditions the rigid cutter 'eccentricity' model above becomes inadequate for force fluctuation predictions and the cutter deflections, which will affect t(>i) and the resulting cutting forces, need t o be incorporated in addition to eccentricity in predictive cutting models. The 'deflection' model can be represented by a cutter with variable radii (Rv')i and constant a a . yn and A. for all elements (to account for eccentricity) together with the instantaneous deflections in the x and y directions of the axis of rotation due to the instantaneous forces Fx(Br) and Fy(& ) . It should be pointed out that the cutter axis deflecTions vary along the axis since the cutter is 'built-in at one end so that for a given cutter orientation B r . the deflection of the axis of the cutter (rotation) W i l l be different for all elements on a tooth i . e . depentooth. Similarly, tne cutter dent on vi for the h'i axis deflection for any given element on a tooth will be different for each cutter orientation Br since both Fx(e,) and Fy(Br) are expected to vary or fluctuata in End-milling. Thus the deflections depend on B r and Y i . The relevant interference geometry for establishing the cut thickness t(v1) n: i t h tooth ?t v , is shown in Fig.2cb). As for the eccentricity model, because the radii ( R v ' i ) vary within and between teeth while the cutter axis of rotation also deflects, the cut thickness t(uc) of the element on the i t n tooth rotating about point D can be due to the interference with the nth preceding tooth ((i-n)th tooth) path of different radius rotating about its instantaneous axis of rotation F. The deflections of the axes o f rotation of the it? tooth (Dxso, Dybo) and the (i-n)th tooth (DXLF, Dyrf) about their corresponding 'rigid' cutter or undeflected axes o f rotation B and A (Fig.Z(b)) will be different because the cutter orientations are different (with a phase angle of fk - ( u c - u ' i - n + n e b ) 2. er - nOa),From an analysis of the geometry in Fig.P(b) and noting that y i 9 ~ ' 1 - n it has been shown [73 that t ( w ) = nftsinvi + (R'v)i - (R'v)i-" + (DXnF-DXBo )sinui+(Dyso-DxrF )cosvi , n=l , N t (29) or,t(vi )=nftsinui + e[cos(&-er-ui-neb )-cos(Br-sr-Yi ) I + ( D X A F - D X B D)sinui+(Dyeo-Dxa~ ) c o s ~ i, n=l .Nt ( 3 0 ) This expression for t(vi) shows that in addition to e , er for a given Br and III the deflections o f the i t h and (i-n)th teeth axes of rotation with respect to their undeflected positions need t o be known before the required ' r i ' f o r least positive t(ui) is found b y solving eq.(:O) for n = 1 to N t . As for the 'eccentricity model it is possible that some elements can be shown not to engage the workpiece if any of the 'n' values searched yields t(vi) < 0. Unfortunately, for this model the deflections Dxso, D Y B O for it" tooth at cutter orientation er and DXAF and D Y A F for the (i-nIth tooth at ( 8 r - n O b ) required to solve eq.(30) are not independent process variables but are related to the instantaneous forces Fx(B7) and Fy(&) and distributions acting on the cutter as a whole at corresponding cutter orientations as well as the cutter diameter D and second moment of area I overhang length 1 and Young's modulus E. Provided the instantaneous force component distributions a;e known ( o r assumed) it is possible to use Mohr's Bending Moment Area' method [ l o ] to obtain the deflections in the x and y directions. This method is shown in Fig.3 for the deflection in the x direction (DX(€k,vi)) for an element on the i t h tooth at angle Ui and cutter orientation er so that its distance z from the free end and I are given b y z = R(er-(i-l)BD-~i)/tanx. and I = D4/48 (31),(32) It should be noted that eq.(32) reported in the earlier literature [81 has recently been confirmed using finite element analysis techniques [ 9 1 .
Moment p) Deflection at A' = Fig.3:Steps in obtaining
of
basis over the complete cycle. To 'speed up' the computational process, the assumed or input forces for the first cycle for use in the 'deflection' model are those predicted b y the simpler 'eccentricity' model. Details of the computer flow chart and software may be found in 1 7 1 and will form part of a separate paper. ASSESSMENT AND VERIFICATION OF FORCE MODELS The three predictive models have been assessed and compared by means of numerical simulation and experimental testing of the forces over a wide range of conditions encountered in End-milling. An S1214 free machining steel work material whose basic cutting data were known [41 has been used for the numerical and experimental studies while a computer based data acquisition system coupled t o a Kistler dynamometer was used to measure and evaluate the instantaneous force component fluctuations, peak forces and averages for each cutting condition tested. The qualitative and quantitative effects o f the major variables on the average, peak and fluctuating forces have been c m s;dered for the three models assessed. NUMERICAL SIMULATION RESULTS In this study in excess of 10,000 predictions have been made covering the three models and process variable values within the following ranges: D,10-40mm aa,0.5R-2R mm: Nt,2-6 teeth: 6(X~).10'-30'; yn,5'-20': ft,0.03-0.08mm/tooth; ap,0.5R-R mm; V..15-30m/rnin; e.0.02-0.04mm: e r , O ' - e b ' . The qualitative effects of the individual major Process variables Nt , % ,ft ,V,yn , A 8 ,ar and R(=D/Z) on the average force components and torque for the three models are illustrated in Fig.4(a) to (h). Since all three models have been found to yield the same trends t h e results for eccentricity model are depicted in Fig.4 for illustration purposes. It is noted that the average forces and torque increase linearly with increases in number of teeth N t , axial depth o f cut a and feed per tooth ft (Fig.4(a)-(c)) due to the proportional increases in the total instantaneous interference areas of cut. These trends f o r all three models are consistent with those found b y the explicit expressions for the average forces and torque established previously for the 'ideal' case model 121. By contrast, the cutting speed V has little influence 06 these characteristics (Fig.4(d)) while increasing the normal rake angle y n reduces all forces (Fig.4(e)) and torque as expected from machining theory [31. 2%
T
'
-0 I
717 *
0
01 t
Yn
,
,
'
:
/
0
;
I
t
Fx
area N about dZ~(e,*) El
the deflection From the above equations it is evident that the introduction of cutter deflections complicates the t ( v 0 equation and the solution procedure since the force component distributions along the cutter axis as well a s their variation for one complete cutter revolution or cycle must be known or assumed to predict the t(vi) and elemental forces on all active elements at each and every cutter orientation Br over a 360' range. In effect the forces the model is intended to predict, have to be assumed as inputs so that intricate iterative techniques are required in the use of this predictive model such that the assumed (input) and predicted forces match on an instant by instant
Fig.4:Average forces and torque trends with eccentricity Similarly varying the helix or inclination angle has little effect on Fx,Fy and torque for A. 5 40' while Fz increases noticeably (Fig.4f) as noted for Fr in oblique cutting studies C31 since Fz=Fr. Increasing the radial cut thickness ar also increases the forces and torque but in a non-linear manner (Fig.4(g)) due to the non-linear variations in t(u) and the complex relationships between dFx, dFy and the oblique cutting force components dFp and dFq in (eqs.(4 8 511, which A.
27
also cause the thrust force Fy to change direction as a7 increases. Surprisingly, increases in cutter radius does not significantly affect the torque but decrease
69\Nt-2 (FlIxt
--?--
-5 sc4
[Fllxg- t I
--- _ _---
10 a a m 30 (dl 0 10 a, ,mm 30 Fig.5:Fiuctuation indices in End m k g wlth eccentmity The Qualitative effects of the process variables on the force fluctuations have also been studied. For this purpose the force and torque Fluctuation Index [FI) defined as FI=(Max(Fj ,T)-Min(Fj ,T))/Average(FJ ,T) j:x,y,z has been used. Again it has been found that the three models give similar trends for all force components and torque. Fig.(5) shows an example of the effect of the major process variables on the feed force Fluctuation Index (FI). (based on the ‘eccentricity model’) when plotted again the axial depth of cut & . The dotted lines in each figure represent the higher level of the process variable considered. It is seen that the Fluctuation Index decreases with decreases in cutter radius R (Fig.5(a)) and increases in helix angle 6 (Fig.S(b)) and number of teeth N t (Fig.5(c)). It is also noted that the Fluctuation Indices become minimum at certain axial depths of cut & values. These h values have been found to correspond to those of eq.(25) for the ‘ideal’ model where the Fluctuation Index is zero 1.e. no force fluctuations. Hence R , N t and 6 will affect the a. for minimum Fluctuation Index as in Figs.S(a,b,&c), where FI is zero for the ‘ideal’ model but non-zero for the ‘eccentricity‘ and ‘deflection’ models. By contrast, increases in radial depth of cut ar (Fig.5(d)) decreases the Fluctuation Index but does not affect the value of a which give minimum (or zero) Fluctuation Index. Similarly, yn,fc and V do not affect the aa value for minimum fluctuations (since these variables,like a r , are not included in eq.(25)), but also have no influence on the Fluctuation Index and are therefore not illustrated in Fig.5. While the qualitative effects of the major process variables on the averages and Fluctuation Indices are similar for all three models there are basic differences in force pulse shapes as well as the quantitative values of the predictions. Fig.6 shows an example of the feed force pulse shapes f o r the three models over one cutter revolution Fx(0,). It is clear that the force cycle for the ‘ideal’ model is on a pe‘; tooth basis while the ‘eccentricity’ and ‘deflection models are on a per cutter revolution basis.
(cFO
c u t t e r orientation, deg 3M1 Fig.6: Predicted feed f o r c e pulses Introducing eccentricity, the force pulses per tooth become uneven and the peak force is higher while the minimum force is lower than for the ‘ideal model. The ‘deflection‘ model force pulses are similar to those of the ‘eccentricity’ model except that allowing for deflection decreases the peak force and increases the minimum force per revolution, tending to ‘smooth’ the pulse and counteract the effect of eccentricity. From a quantitative point of view, the percentage differences in the forces and torque between the ‘ideal’ and either of the other models e.g. (Fa..vid..i )*10O/FI,d..i have shown that both the eccentricity and ‘deflection’ models yield noticeably lower average and significantly higher peak values than the ‘ideal’ model. Table ( 2 ) illusFrates these differences between the ‘eccentricity and ‘ideal’ model for all the conditions simulated. Fx FY Fz T Avg Max Avg Avg Avg Max Max Max -32 -42 8 3 42 -16 Min.Dif. 8 -41 57.5 -11 3 9 . 5 -16.7 AVg.Dif. 5 8 . 2 -13.1 6 1 . 7 -0.1 =-3 95. 190 8 Max.Dif. 142 s-3 =-3 1 4 : Jable ( 2 ) : Percentase d ifferences between ideal and ‘eccentricity’ model Dredictions.
Generally,the ‘deflection’ model gives slightly higher averages and noticeably lower peak forces than the ‘eccentricity’ model (as in Fig.6) at least for cutting conditions when deflections become significant. In summary, the ‘ideal’ model may be very useful for qualitative predictions of average forces but for realistic and quantitative predictions the other more sophisticated models must be considered after experimental verification. EXPERIMENTAL RESULTS AYD QUAYTITATIVE VFRIFICATION The ‘eccentricity and deflection models have been quantitatively tested by an extensive experimental plan involving over 3 5 0 different cutting conditions spanning the recommended practical ranges for most variables. The ranges of the variables used in the test plan were: normal rake angle yn 8’-19‘; cutter radius R , 3/8” (9.52mm)helix angle 6, 3 0 ’ ; li”(38.10mm), number of teeth N t , 2-6; axial depth of cut a = , lmm-2Rmm; radial depth of cut a r , lmm-Rmm: feed per tooth ft, 0.015mm-0.254mm: V.15m/min-3lm/min. The eccentricity values could not be independently set but the magnitudes of e and € 7 varied between 0 . 0 1 5 m m to 0.05mm and about 0’-90 , respectively. Since the End-milling cutter is supported as a cantilever, it is prone to significant deflections that can affect the force pulses, especially for heavier cutting conditions. It has been reported in the authors’ earlier work C21 that any lack of ‘rigidity’ during cutting or significant cutter deflection can be detected by comparing the ‘static’ (measured) and the ‘best fit’ e , ~r values. The ‘best fit’ values are the combination of e. ~r values that generate the predicted pulse ‘best fitting’ the experimental pulse. When significant cutter deflections occur the ‘static’ and ‘best fit’ e , E~ values differ noticeably it is inappropriate to predict the forces based on the ‘eccentricity’ model which assumes a rigid cutter which is likely to produce poor force pulse predictions. Hence for the verifrcation of predictive ‘eccentricity’ model only the cutting conditions not showing the effects of cutter deflections have been chosen i.e. when the ‘static’ and ‘best fit’ e , ~r values matched. The ‘deflection‘ model has been initially verified, based on the cutting conditions showing significant cutter ‘deflection’ effects. Since the ‘deflection‘ model i s the most sophisticated model it has also been assessed for the cutting conditions not showing cutter 'deflection effects’. The histograms of percentage deviations(P0) between the experimental and predicted average and peak )/Fpr.d. force components defined as ( F p r m d . - F . x , , t . based on the ‘eccentricity’ model are shown in Fig.7. The average percentage deviations for feed (Fx) and thrust (Fy) force components are less than 5% whereas for the axial force component (Fz) they are less than 10%. Thus the predictions of the ‘eccentricity’ model when the significant cutter deflection effects are not present are very good. In addition to very good average and peak force predictions the overall force pulse shapes are also very well predicted. Fig.8(a) shows a typical example of good agreement between predicted and experimental feed force pulses for one of the cutting conditions not exhibiting the effect of cutter deflection. Similarly, good correlation has been obtained for the other comoonents.
0
28
F i g .7: Histograms of percentage deviations wlth eccentricity m o d e I,[Only the tests not showh dehctbn,affects) However, when the cutter ieflection effects are significant, the match between experimental pulse and predicted pulse for the ‘eccentricity’ model deteriorates as illustrated in Fig.B(b) for thrust force pulse. As can be expected, for such cutting conditions, there is a significant improvement in the agreement between pred:cted and experimental pulses based on the ‘deflection model, as shown in Fig.e(c). Similar improvements have been noted for the other cutting conditions for which significant cutter ‘deflection effects’ were noted. For all such cutting conditions the percentage deviations between experimental and predicted average as well as peak
[a) Cutting conditions: R - 19.0 5mm Nt -6, ?A- 19’
tested (270/360275%) satisfied this criterion and that the computational time for force prediction using the ‘eccentricity’ model can be less than a tenth of that for the ‘deflection’ model, this ‘simpler’ model can also be useful in practice.
6-30’ a,-1 5 . 2 m m a, - 6 . 3 5 m m ft
-0.084m,m
V-3lm/min
e-0.042mm 6-250
CONCLUS’IONS Three comprehensive computer based mechanics of cutting mode;s and software for End-milling force, torque and power predictions have been developed and assessed qualitatively and quantitatively by numerical simulation and experimental testing. The simplest ‘ideal’ model for r i g i d cutters and no eccentricity provides useful qualitacive trends for the average forces and torque for a majority of the process variables, which are similar for all the models, as well as guidelines for process variable selection for minimum force fluctuations. The r i g i d cutter ‘eccentricity’ model, yields good Quantitative predictions for the average forces and torque while the fluctuation predictions are also good but deteriorate for unfavourable cuttfr deflectiys. The third and most sophisticated deflection model provides the best overall quantitative predictions and has scope for predicting machined component surface profiles. Each of the three models can provide Useful predictions of use in practice. A significant feature of this work is that the mechanics of cutting approach has been shown to be a sound scientific and quantitative basis for machining performance prediction. NOMENCLATURE.
2Cd
I
1
0 ’cuttei orien’tation: deg ’ 3 60 Fig.8: Predicted and experimental force pYlSeS
forces have been obtained based on the deflection’ model. The resulting histograms are shown in Fig.9. It may be noted that the average percentage deviations for different force components are less than 10% there by proving the good predictive capability of the ‘deflection’ model when significant cutter deflections are present.
a , ar dFp,dFq,dFr dFx,dFy,dFz dT, dP Cep,Ceu db, D D x , DY e, ft Fjt(),j:x,y,z Fj(),j=x,y,z
Axial depth of cut, radial depth of cut Oblique force components on one element Force components on one element Torque and power due to one element Orthogonal ‘edge’ force intensities Width of elemental cut, cutter diameter Deflections in x and y directions Eccentricity magnitude, feed per tooth Instantaneous force function for a tooth Instantaneous force function for Cutter i Tooth number Kck, k=p.q,r Oblique 'cutting' force intensities Kek, k=p,q,r Oblique ‘edge force intensities N t , R(=D/2) Number of teeth, Cutter radius Pt Assumed working plane ri Chip-length ratio (R’Y)l Actual radius with eccentricity for an element on i t h tooth at orientation Y t( 1 Cut thickness function Instantaneous torque for tooth 8 cutter Tt ( ) , T( ) V. Resultant cutting speed B,Bn Friction and normal friction angles 0,0n ‘Shear plane h normal shear plane angles T, n, Shear stress, chip-flow angle As. d Inclination angle, Helix angle -fr I -fn Radial and normal rake angle Er, w Eccentricity orientation, angular speed I) Angular orientation of an element Yi Orientation o f an element on i t h tooth Yi -” Orientation of an element on nth preceding tooth w.r.t. i t h tooth Y’i -” Orientation of line joining point of intersection of shear plane with free surface and the path of axis Upper 8 lower limits of tooth engagement YU , Y 1 Tooth orientation, cutter orientation e t t , er Tooth pitch, phase angle, exit angle ex eb, ew, REFERENCES
Fig.9:Histograms of percentage deviations with deflection mode I (Only the tests showing deflection effects) As noted above, the ‘deflection’ model is a more general model and is expected to yield good force pre; dictions, even when the cutter ‘deflection effects are not significant.A summary of the percentage deviation for the cutting conditions not showing deflection effects, based on the ‘deflection’ model is given in Table ( 3 ) . where very good correlation i s shown. (Fx)m.x (Fx).vg (Fy)max (Fy)rrs (Fz1.a~ (FZ)avg Min. -15.8 -8.01 -15.0 -15.0 -16.5 -23.1 4.1 0.83 2.7 8.0 9.7 Avg. 3.8 Max. 28.9 22.5 24.5 15.1 27.9 32.9 Table (31: Percentage deviations with deflection model Thus, the ‘deflection’ model is a more comprehensive and general model for force prediction purposes and has very good force, torque and power prediction capability with potential for component surface profile prediction. By contrast, the ‘eccentricity’ model also yields very good force predictions but only within the limits defined b y the ‘best fit’ 8, EP criterion. Bearing in mind that a h i g h percentage of the cutting conditions
1. KAHLES,J.F.,CIRP Tech. Report,Annals of CIRP, = / 2 , ( 1 9 8 7 ) . D 523. 2. ARMAREGO’, E.J.A. and DESHPANDE. N.P.,Annals Of CIRP 38/1, (1989). p 45. UNESCO/CIRP Seminar on Manuf. 3. ZMAREGO,E.J.A., Tech., Singapore , (1982) , p p 167-182. 4. ARMAREGO, E.J.A. and WHITFIELD. R.C., Annals.CIRP. %/1, (19861, pp 65-69. 5. WHITFIELD, R.C., Ph.D. Thesis., The University of Melbourne, (1986). 6. ARMAREG0,E.J.A. and EPP.,C.J., 1nt.J. of Mach. Tool Des. and Res., lo, (19701, pp. 273-291. 7. DESHPANDE.N.P., Ph.D. Thesis., The University Of Me1 bourne, ( 1990). 8. SUTHERLAND, J . W . and DeVOR, R.E., J.of.Engg. for Industry, (1986). pp.269-279. 9. K0PS.L. and VO,D.T.,Annals of CIRP,B/1,(1990),p 93 lO.RYDER,G.H..”Strength of Materials , Cleaver-Hume press ltd., London, (t953).
a,
ACKNOWLEDGMENTS The authors wish to acknowledge the financial support given b y the Australian Research Council and the assistance of their colleague Dr. A.J.R.Smith in the later stages of this research work.
29
from the plane normal to the feed motion. At this inis engaged in cutting over a stant one tooth (i.2) number of elements M (=&/db) with in the range ui and y u = e x . It has been shown [6,71 that, irrespective of the milling model used, the effect of the feed speed on the ‘dynamic’ tool geometry and cutter ?ath during tooth engagement can be ignored. Thus the static’ and specified normal rake angle y n and inclination angle X., which are constant for all points and elements, may be used in the mechanics of cutting analysis for force prediction. the tooth path may be considered circular and the resultant cutting speed V. found from the cutter w, i.e. y n = tan-l(tanrrcos6); A. = 6 a V. = v = wR (1,2,3) Furthermore, the elemental oblique force components dFp, dFq which vary from element to element, will be tangential and normal to the circular path so that in the fixed these will need to be resolved ‘practical’ force directions x and y to simplify the summation o f the instantaneous forces on the whole cutter. The elemental practical force component, torque and power are given by (4) dFx = dFpcosu+dFqsinu dFy = dfpsinv-dfqcosu (5) dFz = dFr. dT = R.dFp, dP = w-R.dFp ( 6 , 7 . 8 )
MECHANICS OF CUTTING APPROACH FOR FORCE MODELS The basic strategy for the three milling models is to develop computer based analyses to predict the instantaneous force components, torque and power for the cutter as a whole in any selected orientation and to vary the angular orientation over one cutter revolution o r cycle to determine the average and fluctuating forces, torque and power charactecistics f o r a given set of cutting conditions and cutter specification. Thin shear zone mechanics of cutting analyses and orthogonal cutting data bases [ 3 , 4 , 5 ] are used to predict the instantaneous milling forces by partitioning the cutter teeth into a series of axial elemental oblique cutting tools, identifying the tooth elements actively engaged in cutting for a given cutter orientation, determining the fundamental tool geometry, cut thickness and width of cut f.or each element to obtain the elemental force components from the ‘classical’ oblique cutting analysis, resolving the elemental components in the practical milling force directions and summing the force component contributions o f all the elements on each tooth. The elemental force components may also be used to evaluate the elemental torque and power which can be summed to yield the instantaneous values for the cutter as a whole. The cutter orientation is identified in terms of the angular orientation B r of the leading point on a ‘reference’ tooth with respect to a plane normal to the feed motion (Fig.1). Thus if tooth 1 is chosen as the ‘reference’ tooth er=@ti=etr and the angular orientation of the i t ” tooth Bti can be found for the known tooth angular pitch e b , i.e. EIti=Br-(i-l)eb. By varying Br over 360’ the predicted variations and average values can be obtained, althoygh, where possible, explicit expressions for the ‘peak and average values will be sought. Since the three milling models considered adopt the same strategy but involve sophisticated extensions of the basic ‘ideal’ case model, the common aspects of these analyses will be outlined first followed by the specific equations and solution procedures for each model. In this way a better appreciation of’the differences i n the predictive models may be achieved. Fig.1 shows the instantaneous geometry of an Endmillin? Operation at orientation & (based on the ‘ideal case) together with the relevant geometry and force components for a typical ‘active’ oblique cutting element of width db (=Rdv/tanh) at an angle u
Annals of the CIRP, Vol. 40/1/1991
FIG 1 : Rebted oblique cut Geometry for an End mill tooth element. The elemental oblique force components are found from the ‘classical’ oblique cutting analysis
25
tan(0.+Rn ):tan(h )~os7~/[tan(ne )-sinrn*tan(Xs ) I (16) Kep = Cep*cosh, KBQ-Ceq and Ker:O.2S*Kep*sinX. (17) The Kek and Kck factors in eq.(9) can be found from from eqs.(lO to 17) given the basic cutting data (K,ri,O,Cep,Ceq) in the data bank [3,4,5,71 and the elemental tool geometry from eqs. ( 1 6 2 ) . However, to predict dFp, dFq and dFr in eq.(9) and the corresponding practical force components and torque f r o m eqs.(4 to 8) it is necessary to know the elemental cut thickness t(x) for the given width of cut db. The required t(u) is h i g h l y dependent on the milling model considered i . e . whether the 'ideal', 'eccentricity' o r 'deflection' model i s used. Given that t(u) can be found for each element, as will be shown below, the practical f o r c e components and torque acting on each engaged tooth at at orientation 8 t t is found b y summing the elemental forces and torque within the relevant u1 and u u limits given in Table ( 1 ) below so that
.-
u=uu
J-Yu
>=XU
Fjt(eti)
= ZdFj
; Tt(8ti)
= tdT
u=u1
; Pt(8ti)-
-
idP >=>I
u3111
j=x,y,z (18,19,20) Finally, the instantaneous Torce components, torque and power on each active tooth from eqs.(i8 to 20) can be added to yield the instantaneous forces components, torque and power on the cutter as a whole at the corresponding given orientation er i . e . Fj(B,)
=
i=Nt rdFjt(8ti) i =1
; T(%)
=
i =Nt ITt(Bt0;
i :Nt P(Br)=XPt(Bti)
i=l
i-1
j=x,y,z For
o s YU
ett
= eti
For ( e t i - e w ) ill
=
0
<
(21,22,231
et i z ex uu 5
o
= ex
o < ( e t l - e w )( ex = (eti-8.)
(eti-ew) 2 u1
= ex
ex
By repeating the above procedure for different Br values over a revolution or cycle, the fluctuations in the forces, torque and power can be obtained from which the maximum, minimum and average forces can be found or evaluated. As noted earlier, the cut thickness t(u) at each element is the only remaining quantity required to predict the forces, torque and power from the above analysis. This fundam'ental variable, which results from the.interference o f the element under consideration with the transient surface generated by an element on a preceding tooth, is in fact the main distinguishing feature of the three proposed milling models. The overall analysis and solution procedure for force predictions can be substantially affected by the milling model adopted.
IDEAL CASE MODEL Under ideal cutting conditions of no eccentricity and cutter deflections, every tooth element interferes with the path of the immediate preceding tooth element and the cut thickness can be found from two consecutive circular tooth paths and approximated to 124) t(u) = ftsin(u) Thus for the 'ideal' case model the cut thickness can be found from the simple explicit expression in eq.(24) given the feed per tooth ft and the orientation angle v of the required element. It follows that in this model the instantaneous force components, torque and power at the given cutter orientation 0r can be predicted from the above analysis provided the cutter specification (D,Nt or B b , yn or . y r . A. or a). the cutting conditions (ar.a.,ft or Nt,w) and the basic cutting quantities (r,rll!3,Cep,Ceq)in the orthogonal cutting data base are known. In addition no iterations are required for predicting the cut thickness t(u). It has also been shown [2,71 that because of the relatively simple expression for t(u) in eq.(24) explicit expressions for the average force components, torque and power could be established in terms of many o f the major process variables such as ft,aa,Nt, while the cyclic force fluctuations were on a per tooth basis with zero fluctuations occurring when the tool and cut variables satisfied the equation 2nR-m = Nt&tan6; m 1,2,..m (25) However it has not been possible to establish the maximum forces and torque b y explicit expressions and recourse to numerical studies using the computer based predictive model was necessary. It is apparent that the 'ideal' case model is comprehensive since it allows for the vast majority of the process variables. ECCENTRICITY ECCENTRICITY ~~0000~~44 I n practice D r a c t i c e it i t has has been been found found that t h a t an an ececIn c e n t r i c i t y i n c u t t e r axis a x i s of o f rotation r o t a t i o n occurs occurs primarily primarily centricity in cutter
26
from t h e chuck o r t o o l h o l d i n g d e v i c e r a t h e r t h a n due t o e r r o r s i n c u t t e r manufacture [2,71. Furthermore, e c c e n t r i c i t i e s a r e u s u a l l y i n e v i t a b l e and these can v a r y each t i m e t h e same c u t t e r i s mounted i n a h o l d e r . The e c c e n t r i c i t y can be s p e c i f i e d by t h e d i s t a n c e e between t h e c u t t e r g e o m e t r i c a l a x i s and i t s a x i s of r o t a t i o n t o g e t h e r w i t h t h e minimum p o s i t i v e a n g l e er w i t h r e s p e c t t o t h e r a d i a l l i n e from i t s a x i s t o t h e l e a d i n g p o i n t o f t h e r e f e r e n c e t o o t h as r e p o r t e d earl i e r [ a ] . I t has a l s o been shown t h a t when eccent r i c i t y i s p r e s e n t t h e c u t t e r may be represented by a v a r i a b l e r a d i u s c u t t e r about t h e a x i s o f r o t a t i o n such t h a t t h e r a d i i o f a l l p o i n t s on t h e c u t t i n g edges w i l l v a r y w i t h i n and between t e e t h a l t h o u g h t h e a n g u l a r pitch Bb, normal r a k e angle .yn and h e l i x angle d a r e e s s e n t i a l l y constant a t t he s p e c i f i e d values [2,7]. For a c u t t e r o r i e n t a t i o n & , t h e r a d i u s of the p o i n t ( e l e m e n t ) on t h e c u t t i n g edge o f t h e i t h t o o t h ( R ’ u ) i o r i e n t e d a t x i t o t h e p l a n e normal t o t h e feed m o t i o n i s g i v e n by e.cos(Br-Er-Yi (26) (R’u)i = R Due t o t h e d i f f e r e n c e s i n r a d i i between elements on d i f f e r e n t t e e t h , t h e c u t t h i c k n e s s no l o n g e r r e s u l t s f r o m t h e i n t e r f e r e n c e o f two c i r c u l a r p a t h s o f c o n s e c u t i v e t e e t h w i t h t h e same r a d i i and d i s p l a c e d by ft as i n t h e ‘ i d e a l ’ case model, b u t may r e s u l t from the int er f er enc e w i t h t he nth preceding t o o t h ( i .e . (i-n)Ch t o o t h ) p a t h d i s p l a c e d by n . f t w i t h r e s p e c t t o t h e ith F i g . Z ( a ) shows t h e r e l e v a n t geometry f o r t h e e c c e n t r i c i t y ’ model where t h e d i f f e r e n t r a d i i (R’u)i and ( R ’ u l i - n a r e d e p i c t e d t o g e t h e r w i t h t ( u ) i and t h e displacement i n t h e f e e d d i r e c t i o n between t h e two t e e t h . While i n F i g . Z ( a ) t h e t o o t h paths a r e n o t circular i t has been shown t h a t a c i r c u l a r t o o t h p a t h i s a v e r y good a p p r o x i m a t i o n as i s u i 2 u ’ t - n 1 7 1 so t h a t t h e c u t t h i c k n e s s i s g i v e n by (27) t ( u i ) = n - f t s i n ( u i )+(R’u)i -(R’il)i-n, n=l t o Nt Further the radius (R’illi-n can be found from eq.(26) f o r t h e corresponding ili-”v a l u e s which equals ( u i +neb so t h a t t ( u i ) = nftsin(ili )+e[cos(er-Er-ili-n8b)-cos(e~-€~-ui ) I (28) n = i t o Nt The r e l e v a n t n is t h a t t o o t h which g i v e s t h e minimum v a l u e o f t ( v i ) ) 0 . I t i s w o r t h n o t i n g t h a t due t o t h e d i f f e r e n c e s i n t h e r a d i i and t h e s m a l l v a l u e o f ft if. i s p o s s i b l e t h a t an element does n o t t a k e a ‘ b i t e even though i t s o r i e n t a t i o n ili i s w i t h i n the l i m i t s when c u t t i n g i s l i k e l y t o occur ( i . e . u 1 c u i L ~ ~ ) as i n F i g . 1 . Such a c o n d i t i o n occurs when any o f t h e t ( u i ) from eq.(28) a r e l e s s t h a n zero.
!oath.
I t i s apparent t h a t when e c c e n t r i c i t y i s a l l o w e d for, an e x p l i c i t e q u a t i o n f o r t(ut) i s no l o n g e r a v a i l a b l e a l t h o u g h p r o v i d e d e and E V a r e known ( o r measured) t ( v , i ) can be e v a l u a t e d by ‘ s e a r c h i n g ’ f o r t h e r e q u i r e d ‘ n ’ v a l u e by s o l v i n g e q . ( 2 8 ) f o r minimum t f x i ) > 0 , o r i d e n t i f y i n g elements which do n o t engage t h e workpiece i . e . t ( i l i ) s 0.
Ci-n)‘h Tooth Dath
(i-nlth
;/T ,;
d
T o o t h path
2: +aqJpJDXAFJ?
%
P r q ~ “
nrBb (a) Eccentricity
1
‘ Y3- n
ru1
Q
nreb
( b l E c c e n t r i c i t y & D ef l e ct i o n
Fig.2: E l e m e n t a l c u t t h i c k n e s s
in
milling
For t h i s model, t h e instantaneous f o r c e compon e n t s , t o r q u e and power a t a g i v e n c u t t e r o r i e n t a t i o n Br can be p r e d i c t e d i f e and Er a r e g i v e n i n a d d i t i o n t o the t o o l geometrical variables, c u t t i n g conditions and b a s i c c u t t i n g d a t a as f o r t h e ‘ i d e a l ’ case model. However f o r t h e ‘ e c c e n t r i c i t y ’ model t ( v i ) has t o be s o l v e d n u m e r i c a l l y f o r each element on t h e c u t t e r t e e t h a t each c u t t e r O r i e n t a t i o n so t h a t computer a s s i s t a n c e i s e s s e n t i a l and w i l l i n v o l v e more computat i o n a l t i m e t h a n f o r t h e ‘ i d e a l ’ case model. Furthermore, because t ( u i ) has t o be s o l v e d n u m e r i c a l l y i t has n o t been p o s s i b l e t o e s t a b l i s h e x p l i c i t express i o n s between t h e average force components, t o r q u e and power and t h e p r a c t i c a l process v a r i a b l e s such as f t , aa, N t as was p o s s i b l e i n t h e ‘ i d e a l ’ model. Nevertheless t h e ‘ e c c e n t r i c i t y ’ model i s a more s o p h i s t i c a t e d model a l l o w i n g f o r the inevitable e c c e n t r i c i t y noted i n p r a c t i c e and o f t e n i g n o r e d i n machining f o r c e p r e d i c t i o n s .
DEFLECTION MODEL It is well recognized that because End-milling cutters are only supported at one (built-in) end, they are prone'to deflections under the action o f the cutting fortes [2,8,9]. It has also been noted [ 2 , 7 1 that under some cutting conditions the rigid cutter 'eccentricity' model above becomes inadequate for force fluctuation predictions and the cutter deflections, which will affect t(>i) and the resulting cutting forces, need t o be incorporated in addition to eccentricity in predictive cutting models. The 'deflection' model can be represented by a cutter with variable radii (Rv')i and constant a a . yn and A. for all elements (to account for eccentricity) together with the instantaneous deflections in the x and y directions of the axis of rotation due to the instantaneous forces Fx(Br) and Fy(& ) . It should be pointed out that the cutter axis deflecTions vary along the axis since the cutter is 'built-in at one end so that for a given cutter orientation B r . the deflection of the axis of the cutter (rotation) W i l l be different for all elements on a tooth i . e . depentooth. Similarly, tne cutter dent on vi for the h'i axis deflection for any given element on a tooth will be different for each cutter orientation Br since both Fx(e,) and Fy(Br) are expected to vary or fluctuata in End-milling. Thus the deflections depend on B r and Y i . The relevant interference geometry for establishing the cut thickness t(v1) n: i t h tooth ?t v , is shown in Fig.2cb). As for the eccentricity model, because the radii ( R v ' i ) vary within and between teeth while the cutter axis of rotation also deflects, the cut thickness t(uc) of the element on the i t n tooth rotating about point D can be due to the interference with the nth preceding tooth ((i-n)th tooth) path of different radius rotating about its instantaneous axis of rotation F. The deflections of the axes o f rotation of the it? tooth (Dxso, Dybo) and the (i-n)th tooth (DXLF, Dyrf) about their corresponding 'rigid' cutter or undeflected axes o f rotation B and A (Fig.Z(b)) will be different because the cutter orientations are different (with a phase angle of fk - ( u c - u ' i - n + n e b ) 2. er - nOa),From an analysis of the geometry in Fig.P(b) and noting that y i 9 ~ ' 1 - n it has been shown [73 that t ( w ) = nftsinvi + (R'v)i - (R'v)i-" + (DXnF-DXBo )sinui+(Dyso-DxrF )cosvi , n=l , N t (29) or,t(vi )=nftsinui + e[cos(&-er-ui-neb )-cos(Br-sr-Yi ) I + ( D X A F - D X B D)sinui+(Dyeo-Dxa~ ) c o s ~ i, n=l .Nt ( 3 0 ) This expression for t(vi) shows that in addition to e , er for a given Br and III the deflections o f the i t h and (i-n)th teeth axes of rotation with respect to their undeflected positions need t o be known before the required ' r i ' f o r least positive t(ui) is found b y solving eq.(:O) for n = 1 to N t . As for the 'eccentricity model it is possible that some elements can be shown not to engage the workpiece if any of the 'n' values searched yields t(vi) < 0. Unfortunately, for this model the deflections Dxso, D Y B O for it" tooth at cutter orientation er and DXAF and D Y A F for the (i-nIth tooth at ( 8 r - n O b ) required to solve eq.(30) are not independent process variables but are related to the instantaneous forces Fx(B7) and Fy(&) and distributions acting on the cutter as a whole at corresponding cutter orientations as well as the cutter diameter D and second moment of area I overhang length 1 and Young's modulus E. Provided the instantaneous force component distributions a;e known ( o r assumed) it is possible to use Mohr's Bending Moment Area' method [ l o ] to obtain the deflections in the x and y directions. This method is shown in Fig.3 for the deflection in the x direction (DX(€k,vi)) for an element on the i t h tooth at angle Ui and cutter orientation er so that its distance z from the free end and I are given b y z = R(er-(i-l)BD-~i)/tanx. and I = D4/48 (31),(32) It should be noted that eq.(32) reported in the earlier literature [81 has recently been confirmed using finite element analysis techniques [ 9 1 .
Moment p) Deflection at A' = Fig.3:Steps in obtaining
of
basis over the complete cycle. To 'speed up' the computational process, the assumed or input forces for the first cycle for use in the 'deflection' model are those predicted b y the simpler 'eccentricity' model. Details of the computer flow chart and software may be found in 1 7 1 and will form part of a separate paper. ASSESSMENT AND VERIFICATION OF FORCE MODELS The three predictive models have been assessed and compared by means of numerical simulation and experimental testing of the forces over a wide range of conditions encountered in End-milling. An S1214 free machining steel work material whose basic cutting data were known [41 has been used for the numerical and experimental studies while a computer based data acquisition system coupled t o a Kistler dynamometer was used to measure and evaluate the instantaneous force component fluctuations, peak forces and averages for each cutting condition tested. The qualitative and quantitative effects o f the major variables on the average, peak and fluctuating forces have been c m s;dered for the three models assessed. NUMERICAL SIMULATION RESULTS In this study in excess of 10,000 predictions have been made covering the three models and process variable values within the following ranges: D,10-40mm aa,0.5R-2R mm: Nt,2-6 teeth: 6(X~).10'-30'; yn,5'-20': ft,0.03-0.08mm/tooth; ap,0.5R-R mm; V..15-30m/rnin; e.0.02-0.04mm: e r , O ' - e b ' . The qualitative effects of the individual major Process variables Nt , % ,ft ,V,yn , A 8 ,ar and R(=D/Z) on the average force components and torque for the three models are illustrated in Fig.4(a) to (h). Since all three models have been found to yield the same trends t h e results for eccentricity model are depicted in Fig.4 for illustration purposes. It is noted that the average forces and torque increase linearly with increases in number of teeth N t , axial depth o f cut a and feed per tooth ft (Fig.4(a)-(c)) due to the proportional increases in the total instantaneous interference areas of cut. These trends f o r all three models are consistent with those found b y the explicit expressions for the average forces and torque established previously for the 'ideal' case model 121. By contrast, the cutting speed V has little influence 06 these characteristics (Fig.4(d)) while increasing the normal rake angle y n reduces all forces (Fig.4(e)) and torque as expected from machining theory [31. 2%
T
'
-0 I
717 *
0
01 t
Yn
,
,
'
:
/
0
;
I
t
Fx
area N about dZ~(e,*) El
the deflection From the above equations it is evident that the introduction of cutter deflections complicates the t ( v 0 equation and the solution procedure since the force component distributions along the cutter axis as well a s their variation for one complete cutter revolution or cycle must be known or assumed to predict the t(vi) and elemental forces on all active elements at each and every cutter orientation Br over a 360' range. In effect the forces the model is intended to predict, have to be assumed as inputs so that intricate iterative techniques are required in the use of this predictive model such that the assumed (input) and predicted forces match on an instant by instant
Fig.4:Average forces and torque trends with eccentricity Similarly varying the helix or inclination angle has little effect on Fx,Fy and torque for A. 5 40' while Fz increases noticeably (Fig.4f) as noted for Fr in oblique cutting studies C31 since Fz=Fr. Increasing the radial cut thickness ar also increases the forces and torque but in a non-linear manner (Fig.4(g)) due to the non-linear variations in t(u) and the complex relationships between dFx, dFy and the oblique cutting force components dFp and dFq in (eqs.(4 8 511, which A.
27
also cause the thrust force Fy to change direction as a7 increases. Surprisingly, increases in cutter radius does not significantly affect the torque but decrease
69\Nt-2 (FlIxt
--?--
-5 sc4
[Fllxg- t I
--- _ _---
10 a a m 30 (dl 0 10 a, ,mm 30 Fig.5:Fiuctuation indices in End m k g wlth eccentmity The Qualitative effects of the process variables on the force fluctuations have also been studied. For this purpose the force and torque Fluctuation Index [FI) defined as FI=(Max(Fj ,T)-Min(Fj ,T))/Average(FJ ,T) j:x,y,z has been used. Again it has been found that the three models give similar trends for all force components and torque. Fig.(5) shows an example of the effect of the major process variables on the feed force Fluctuation Index (FI). (based on the ‘eccentricity model’) when plotted again the axial depth of cut & . The dotted lines in each figure represent the higher level of the process variable considered. It is seen that the Fluctuation Index decreases with decreases in cutter radius R (Fig.5(a)) and increases in helix angle 6 (Fig.S(b)) and number of teeth N t (Fig.5(c)). It is also noted that the Fluctuation Indices become minimum at certain axial depths of cut & values. These h values have been found to correspond to those of eq.(25) for the ‘ideal’ model where the Fluctuation Index is zero 1.e. no force fluctuations. Hence R , N t and 6 will affect the a. for minimum Fluctuation Index as in Figs.S(a,b,&c), where FI is zero for the ‘ideal’ model but non-zero for the ‘eccentricity‘ and ‘deflection’ models. By contrast, increases in radial depth of cut ar (Fig.5(d)) decreases the Fluctuation Index but does not affect the value of a which give minimum (or zero) Fluctuation Index. Similarly, yn,fc and V do not affect the aa value for minimum fluctuations (since these variables,like a r , are not included in eq.(25)), but also have no influence on the Fluctuation Index and are therefore not illustrated in Fig.5. While the qualitative effects of the major process variables on the averages and Fluctuation Indices are similar for all three models there are basic differences in force pulse shapes as well as the quantitative values of the predictions. Fig.6 shows an example of the feed force pulse shapes f o r the three models over one cutter revolution Fx(0,). It is clear that the force cycle for the ‘ideal’ model is on a pe‘; tooth basis while the ‘eccentricity’ and ‘deflection models are on a per cutter revolution basis.
(cFO
c u t t e r orientation, deg 3M1 Fig.6: Predicted feed f o r c e pulses Introducing eccentricity, the force pulses per tooth become uneven and the peak force is higher while the minimum force is lower than for the ‘ideal model. The ‘deflection‘ model force pulses are similar to those of the ‘eccentricity’ model except that allowing for deflection decreases the peak force and increases the minimum force per revolution, tending to ‘smooth’ the pulse and counteract the effect of eccentricity. From a quantitative point of view, the percentage differences in the forces and torque between the ‘ideal’ and either of the other models e.g. (Fa..vid..i )*10O/FI,d..i have shown that both the eccentricity and ‘deflection’ models yield noticeably lower average and significantly higher peak values than the ‘ideal’ model. Table ( 2 ) illusFrates these differences between the ‘eccentricity and ‘ideal’ model for all the conditions simulated. Fx FY Fz T Avg Max Avg Avg Avg Max Max Max -32 -42 8 3 42 -16 Min.Dif. 8 -41 57.5 -11 3 9 . 5 -16.7 AVg.Dif. 5 8 . 2 -13.1 6 1 . 7 -0.1 =-3 95. 190 8 Max.Dif. 142 s-3 =-3 1 4 : Jable ( 2 ) : Percentase d ifferences between ideal and ‘eccentricity’ model Dredictions.
Generally,the ‘deflection’ model gives slightly higher averages and noticeably lower peak forces than the ‘eccentricity’ model (as in Fig.6) at least for cutting conditions when deflections become significant. In summary, the ‘ideal’ model may be very useful for qualitative predictions of average forces but for realistic and quantitative predictions the other more sophisticated models must be considered after experimental verification. EXPERIMENTAL RESULTS AYD QUAYTITATIVE VFRIFICATION The ‘eccentricity and deflection models have been quantitatively tested by an extensive experimental plan involving over 3 5 0 different cutting conditions spanning the recommended practical ranges for most variables. The ranges of the variables used in the test plan were: normal rake angle yn 8’-19‘; cutter radius R , 3/8” (9.52mm)helix angle 6, 3 0 ’ ; li”(38.10mm), number of teeth N t , 2-6; axial depth of cut a = , lmm-2Rmm; radial depth of cut a r , lmm-Rmm: feed per tooth ft, 0.015mm-0.254mm: V.15m/min-3lm/min. The eccentricity values could not be independently set but the magnitudes of e and € 7 varied between 0 . 0 1 5 m m to 0.05mm and about 0’-90 , respectively. Since the End-milling cutter is supported as a cantilever, it is prone to significant deflections that can affect the force pulses, especially for heavier cutting conditions. It has been reported in the authors’ earlier work C21 that any lack of ‘rigidity’ during cutting or significant cutter deflection can be detected by comparing the ‘static’ (measured) and the ‘best fit’ e , ~r values. The ‘best fit’ values are the combination of e. ~r values that generate the predicted pulse ‘best fitting’ the experimental pulse. When significant cutter deflections occur the ‘static’ and ‘best fit’ e , E~ values differ noticeably it is inappropriate to predict the forces based on the ‘eccentricity’ model which assumes a rigid cutter which is likely to produce poor force pulse predictions. Hence for the verifrcation of predictive ‘eccentricity’ model only the cutting conditions not showing the effects of cutter deflections have been chosen i.e. when the ‘static’ and ‘best fit’ e , ~r values matched. The ‘deflection‘ model has been initially verified, based on the cutting conditions showing significant cutter ‘deflection’ effects. Since the ‘deflection‘ model i s the most sophisticated model it has also been assessed for the cutting conditions not showing cutter 'deflection effects’. The histograms of percentage deviations(P0) between the experimental and predicted average and peak )/Fpr.d. force components defined as ( F p r m d . - F . x , , t . based on the ‘eccentricity’ model are shown in Fig.7. The average percentage deviations for feed (Fx) and thrust (Fy) force components are less than 5% whereas for the axial force component (Fz) they are less than 10%. Thus the predictions of the ‘eccentricity’ model when the significant cutter deflection effects are not present are very good. In addition to very good average and peak force predictions the overall force pulse shapes are also very well predicted. Fig.8(a) shows a typical example of good agreement between predicted and experimental feed force pulses for one of the cutting conditions not exhibiting the effect of cutter deflection. Similarly, good correlation has been obtained for the other comoonents.
0
28
F i g .7: Histograms of percentage deviations wlth eccentricity m o d e I,[Only the tests not showh dehctbn,affects) However, when the cutter ieflection effects are significant, the match between experimental pulse and predicted pulse for the ‘eccentricity’ model deteriorates as illustrated in Fig.B(b) for thrust force pulse. As can be expected, for such cutting conditions, there is a significant improvement in the agreement between pred:cted and experimental pulses based on the ‘deflection model, as shown in Fig.e(c). Similar improvements have been noted for the other cutting conditions for which significant cutter ‘deflection effects’ were noted. For all such cutting conditions the percentage deviations between experimental and predicted average as well as peak
[a) Cutting conditions: R - 19.0 5mm Nt -6, ?A- 19’
tested (270/360275%) satisfied this criterion and that the computational time for force prediction using the ‘eccentricity’ model can be less than a tenth of that for the ‘deflection’ model, this ‘simpler’ model can also be useful in practice.
6-30’ a,-1 5 . 2 m m a, - 6 . 3 5 m m ft
-0.084m,m
V-3lm/min
e-0.042mm 6-250
CONCLUS’IONS Three comprehensive computer based mechanics of cutting mode;s and software for End-milling force, torque and power predictions have been developed and assessed qualitatively and quantitatively by numerical simulation and experimental testing. The simplest ‘ideal’ model for r i g i d cutters and no eccentricity provides useful qualitacive trends for the average forces and torque for a majority of the process variables, which are similar for all the models, as well as guidelines for process variable selection for minimum force fluctuations. The r i g i d cutter ‘eccentricity’ model, yields good Quantitative predictions for the average forces and torque while the fluctuation predictions are also good but deteriorate for unfavourable cuttfr deflectiys. The third and most sophisticated deflection model provides the best overall quantitative predictions and has scope for predicting machined component surface profiles. Each of the three models can provide Useful predictions of use in practice. A significant feature of this work is that the mechanics of cutting approach has been shown to be a sound scientific and quantitative basis for machining performance prediction. NOMENCLATURE.
2Cd
I
1
0 ’cuttei orien’tation: deg ’ 3 60 Fig.8: Predicted and experimental force pYlSeS
forces have been obtained based on the deflection’ model. The resulting histograms are shown in Fig.9. It may be noted that the average percentage deviations for different force components are less than 10% there by proving the good predictive capability of the ‘deflection’ model when significant cutter deflections are present.
a , ar dFp,dFq,dFr dFx,dFy,dFz dT, dP Cep,Ceu db, D D x , DY e, ft Fjt(),j:x,y,z Fj(),j=x,y,z
Axial depth of cut, radial depth of cut Oblique force components on one element Force components on one element Torque and power due to one element Orthogonal ‘edge’ force intensities Width of elemental cut, cutter diameter Deflections in x and y directions Eccentricity magnitude, feed per tooth Instantaneous force function for a tooth Instantaneous force function for Cutter i Tooth number Kck, k=p.q,r Oblique 'cutting' force intensities Kek, k=p,q,r Oblique ‘edge force intensities N t , R(=D/2) Number of teeth, Cutter radius Pt Assumed working plane ri Chip-length ratio (R’Y)l Actual radius with eccentricity for an element on i t h tooth at orientation Y t( 1 Cut thickness function Instantaneous torque for tooth 8 cutter Tt ( ) , T( ) V. Resultant cutting speed B,Bn Friction and normal friction angles 0,0n ‘Shear plane h normal shear plane angles T, n, Shear stress, chip-flow angle As. d Inclination angle, Helix angle -fr I -fn Radial and normal rake angle Er, w Eccentricity orientation, angular speed I) Angular orientation of an element Yi Orientation o f an element on i t h tooth Yi -” Orientation of an element on nth preceding tooth w.r.t. i t h tooth Y’i -” Orientation of line joining point of intersection of shear plane with free surface and the path of axis Upper 8 lower limits of tooth engagement YU , Y 1 Tooth orientation, cutter orientation e t t , er Tooth pitch, phase angle, exit angle ex eb, ew, REFERENCES
Fig.9:Histograms of percentage deviations with deflection mode I (Only the tests showing deflection effects) As noted above, the ‘deflection’ model is a more general model and is expected to yield good force pre; dictions, even when the cutter ‘deflection effects are not significant.A summary of the percentage deviation for the cutting conditions not showing deflection effects, based on the ‘deflection’ model is given in Table ( 3 ) . where very good correlation i s shown. (Fx)m.x (Fx).vg (Fy)max (Fy)rrs (Fz1.a~ (FZ)avg Min. -15.8 -8.01 -15.0 -15.0 -16.5 -23.1 4.1 0.83 2.7 8.0 9.7 Avg. 3.8 Max. 28.9 22.5 24.5 15.1 27.9 32.9 Table (31: Percentage deviations with deflection model Thus, the ‘deflection’ model is a more comprehensive and general model for force prediction purposes and has very good force, torque and power prediction capability with potential for component surface profile prediction. By contrast, the ‘eccentricity’ model also yields very good force predictions but only within the limits defined b y the ‘best fit’ 8, EP criterion. Bearing in mind that a h i g h percentage of the cutting conditions
1. KAHLES,J.F.,CIRP Tech. Report,Annals of CIRP, = / 2 , ( 1 9 8 7 ) . D 523. 2. ARMAREGO’, E.J.A. and DESHPANDE. N.P.,Annals Of CIRP 38/1, (1989). p 45. UNESCO/CIRP Seminar on Manuf. 3. ZMAREGO,E.J.A., Tech., Singapore , (1982) , p p 167-182. 4. ARMAREGO, E.J.A. and WHITFIELD. R.C., Annals.CIRP. %/1, (19861, pp 65-69. 5. WHITFIELD, R.C., Ph.D. Thesis., The University of Melbourne, (1986). 6. ARMAREG0,E.J.A. and EPP.,C.J., 1nt.J. of Mach. Tool Des. and Res., lo, (19701, pp. 273-291. 7. DESHPANDE.N.P., Ph.D. Thesis., The University Of Me1 bourne, ( 1990). 8. SUTHERLAND, J . W . and DeVOR, R.E., J.of.Engg. for Industry, (1986). pp.269-279. 9. K0PS.L. and VO,D.T.,Annals of CIRP,B/1,(1990),p 93 lO.RYDER,G.H..”Strength of Materials , Cleaver-Hume press ltd., London, (t953).
a,
ACKNOWLEDGMENTS The authors wish to acknowledge the financial support given b y the Australian Research Council and the assistance of their colleague Dr. A.J.R.Smith in the later stages of this research work.
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