journal of engineering for industry volume 92 issue 3 1970 [doi 10.1115%2f1.3427827] fujii, s.;...

7/27/2019 Journal of Engineering for Industry Volume 92 issue 3 1970 [doi 10.1115%2F1.3427827] Fujii, S.; DeVries, M. F.;

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S. F U J I !R e se a r ch A s s i s t a n t

M . F. DeVRIESAss i s t a n t Pr o f e sso r .

M e m . A S M E

S. M . WUPr o f e sso r .

M e m . A S M ED e p a r t m e n t of M e c h a n i c a l E n g i n e e r i n g ,

T he U n i v e r s i t y of W i s c o n s i n ,M a d i s o n , W i s .

I n A n a l y s i s of D r i l l i e o m e t ry for O p t i m u mD r i l l D e s i g n I f C o m p i l e r .P a r t I D r i l l G e o m e t r y A n a l y s i sA comprehensive analysis of the twist drill point geometry is made in order that thehigh-speed digital com puter can be used asan aid in the design ofa drill. This subjectis treated in two parts. In Part I. the drill geometry is analyzed with respect to the drillflute and flank contours by considering cross sections of the drill cut by planes perpendicular to its axis. Since several important drill angles aredefined in planes inclined, to the drill axis, the analysis is extended tocover the general case where the drillis cut by any plane inclined to its axis.

IntroductionT H E O R D I N A R Y twist drill is a c o m m o n c u t t in g to o l ,y e t it is c h a r a c te r iz e d by a c o m p le x g e o m e tr y . T h is c o m p le x i ty

h a s led to difficulties in u n d e r s t a n d i n g the basic dri l l ing operat io n . W i th the a d v e n t of theh ig h - s p e e d d ig i ta l c o m p u te r andI t s a v a i la b i l i ty as a m a n u f a c t u r i n g and design aid, p r e v io u sa n a ly s e s [ 1 , 2 ] 1 of the d r i l l w e r e f o u n d in a d e q u a te ; h e n c e , ac o m p r e h e n s iv e g e o m e tr ic a l a n a ly s is of the twist dri l l is needed.

T h e p u r p o s e of th i s p a p e r is to m a k e an a n a ly s is of the dril lp o in t g e o m e tr y s u c h th a t a c o m p u te r c a n beutil ized to d e s c r ib eand further analyze the d r i l l g e o m e tr y . The in v e s t ig a t io na s s u m e s a dril l with s trai ght cut t ing edges. Th e dril l f lute andf lank shapes are f irs t analyzed in a p la n e p e r p e n d ic u la r to thedril l axis . D ril l poin t cross sections in planes inclined tothe dril l axis were analyzed to describe the dril l point geometry including the face rake and nomina l re lief angles. The dril l marginis not considered in th is analysis .

I n th i s in v e s t ig a t io n , ano r th o g o n a l c u t t in g p la n e isdefined asa p la n e p e r p e n d ic u la r to the dril l axis . A n oblique cutt ing planeis then defined as a p la n e th a t c u ts thedrill at a n y a n g le o th e rth a n p e r p e n d ic u la r to the drill axis.

Geometrical Analysis of the Twist Drill Pointin an Orthogonal Cutting PlaneT h e i m p o r t a n t f e a t u r e s of a dril l point are described by its

flute and f lank shapes. The drill point is a th r e e - d im e n s io n a l

1 N u m b er s inbracke ts designate R eferences at en d of paper.Contributed by the Production E ngineering D ivision and presented at theProduction Engineering Conference, Madison, Wise,March 23-25, 1970, of THE AMERICAN SOCIETY OF MECHANICALE N G I N E E R S . Manuscript received at A S M E H e a d q u ar t er s , December 18, 1969. Pap er N o. 70-Prod-5.

b o d y ; h o w e v e r , the b a s ic g e o m e tr y of the dril l point can bea n a ly z e d in tw o d im e n s io n s b y g e n e r a t in g a series of cross sectionsin o r th o g o n a l or o b l iq u e c u t t in g p la n e s , as i l lu s t r a te d in F ig s .1(a) and 1(6), respectively. The flute and f lank surfaces appearin the two-dimensional cross sections as curves which can beex pressed by m a th e m a t ic a l f u n c t io n s th a t can be e v a l u a t e d by ac o m p u t e r .

The dril l f lu te contours in theo r th o g o n a l r e f e r e n c e p la n e aredeveloped f irs t . The flute location in tin arbi tral '} ' o r th o g o n a lcutt ing plane is th e n r e la te d to its location in the referencep la n e . The in te r s e c t io n s of the dril l grinding cones and ano r th o g o n a l c u t t in g p la n e areell ipsesa portion of w h ic h d e te r mines the dril l f lank contou r. O nce the f lute and f lank contoursin a p a r t ic u la r p la n e are k n o w n , the two can be c o m b in e d toobtain dril l point cross section in th a t p la n e .

T h e a n a ly s is of thec o n v e n t io n a l tw is t d r i l l p o in t g e o m e tr y isb a s e d u p o n th e a p p r o a c h in t r o d u c e d b y G a l lo w a y [ 1 ] a n d a s s u m e sconical grinding (described inA p p e n d ix 1 ). I l lu s t r a te d inF ig . 2is the r ig h t - h a n d c o o r d in a te s y s te m u s e d by G a l l ow a y . Thec o o r d in a te s y s te m isdescribed as:

1 The z-axis is the drill axis with the posit ive directionto w a r d th e d r i l l s h a n k .

2 Th e //-axis is a p e r p e n d ic u la r c o m m o n to the extensions ofthe two cutt ing edges and the z-axis .''> The x-axis isp e r p e n d ic u la r to th e y- and z-axes.

Thro ugho ut th is paper th e ,r


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Fig. 1 O r t hogonal and ob l ique cu t ting p lanes

lat ionship bet w een d r i l l , grinding cone, and coordinate system

in Fig. 2(d), is designed by considering both the dril l s tre ngth andchip ejectio n space. Th e oth er half of the flute, i.e., the cur veB C ( o r B ' C ) i n F i g . 2(d), is designed to produce a s traight cutt ingedge for a specified combination of point angle 2K , helix angle y0,w e b th ic k n e s s 21 , a n d d r i l l d ia m e te r 2R .

The flute contour analysis that , leads to a dri l l having straightcutt ing edges was originally developed in reference [1]. Th eflute contour in the orthogonal reference plane (x, if) can bed e s c r ib e d in te r m s o f th e p a r a m e tr ic e q u a t io n s :

r t cosec ( I )v = -f- t tan 70 cot 4> col. Kwh ere / and v are the polar coordinates of the ( lute contour inthe reference plane. While r is a function of the web thickness 21and the web angle 4> , the angle v is re lated to the web thickn ess21, the helix angle 70, the half po int angle K, and the web angle< p. By var yin g the w eb a ngle r/>, for given value s of 70, I a n d K,values of the polar coordinates r an d v can be calculated whichtrace the locus of the f lute contou r in the reference plane. Th eposition of the drill flute contour in the reference plane was firstestablished in the posit ion as shown in Fig. 2(d) a n d th e u p p e rview in Fig. 3 . Th e posit ion of the f lute conto ur in any or thogo nalcutt ing plane can be determined by rotating the f lute cross section in the reference plane around the dril l axis by an angle f

Nomenclature-R = dril l radius( = l /a dri l l web thickness

7 , = dril l helix angle a t t he per ip h e r yK = V2 drill point, angleQ = grinding cone semiangle d = ^-coordinate of grinding conevertex for left side flanksurface (Fig. 2)

ip = angle between the . r-axisand the l ine of in tersect i on A A ' of t h e t a n g e n tplane of the cone at i tsg e n e r a t r ix p a s s in g th r o u g hthe corresponding outerand chisel edge cornerswith the :r-) plane

X = angle betwe en the cone, axisa n d a c u t t in g p la n e ( F ig s .4 and 11)

p = inverse cosine of cone axisdirection cosine n' ino b l iq u e c u t t in g p la n e a n a l ysis

p , i] rotation angles which defineth e o b l iq u e c o o r d in a te

s y s te m {it, v, w) to theo r th o g o n a l c o o r d in a te s y s te m (x, y, z)

f = d is ta n c e b e tw e e n th e o r th o g onal reference plane anda n y o r th o g o n a l c u t t in gp la n e

ft = d is ta n c e b e tw e e n th eoblique reference plan eand any paralle l obliquec u t t in g p la n e

p,q,h = p a r a m e te r s in th e t r a n s f o r m a t io n e q u a t io n s b e tw e e n c o o r d in a te s y s te m s(.To, J/o) a n d (x u yf) or(o, Wo) a n d {iii, i)

(Continued on next pagp)

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Fig. 3 Flute contour rotat ionwhich is re lated to the dril l helix angle 70 and the distance /Uetween the ortho gon al cutt ing plane and the reference plane.The ro tatio n angle f can be observe d in Fig. 3 as:

f = tan 7 (2 )

In any orthogonal plane cutt ing the dril l , e .g . , the lower plane.- lwn in Fig. 3 , the p rojection s of th e x- and (/-axes from th ereference plan e are d enote d as th e Xo- and j/ 0-axes in th e c u t t in gp la n e . I n th e s a m e a r b i t r a r y p la n e th e x'- and ( / ' -axes representthe original x- and ( /-axes as the} ' are ro tate d by th e angle f .[11 con junc tion wi th th e x- a n d j/-axes, the f lute posit ion in thereference plan e is rot ate d by the sam e angle f . A lso, the poi nt\' in the reference plane can be denote d after r ota tion by th ep o in t A w i th c o o r d in a te s ( x 0, ?/,/).

Th e relat ionsh ip bet ween the co ordinat e syste ms (xo, 1/0) and{x', y') is g iven by the following orthogona l tran sfo rm atio n:

co s tsin f

- s in 1cos f (3 )

Flank Shape. Th e dril l flank surfaces are genera ted by grind ing cones symmetric about the dril l axis as shown in Fig. 14 inA ppendix 1 . Th e two flank surfaces are theoreti cally the sam eb e c a u s e th e y a r e g e n e r a te d b y s y m m e tr ic g r in d in g c o n e s Aand B. Th rou gh out th is pape r only the left-s ide f lank surfaceof the dril l is considered.

The dril l f lank surfaces are determined by the dril l point angle2K, the nominal re lief angle a t the outer corner a 0 , and the characteris t ics of the grindi ng cone. Th e grind ing cone charac teris t ics are given by the cone semiangle 6 a n d th e r e la t iv e p o s i t io n

betw een the grind ing cone and the dril l . Th e grindin g coneposit ion is defined by the cone vertex p osit ion and the directioncosines of the cone axis with respect to the x-, y-, and z-axes.Th e cone verte x l ies on the ex tension of the c utt in g edge an d,as show n in Fig . 2, its co ord ina tes (x, ;/, z) ar e:

xv = -dVv = ~t i.4)z = xv co t K = d co t K

w h e r e d is an inde pen dent ly detern iined var iable that specifiesthe cone vertex posit ion. Th e z-coord inate of the cone vertexis directl y related to the half point angle K and x ,. . Th e coneaxis direction cosines I, m, a n d n are with respect to the x-, ;/-,and z-axes. Th e direction cosines are based on the definit ionof the cone vertex posit ion given by equation (4) and are shownin A p p e n d ix 1 .

O n c e th e g r in d in g c o n e c h a r a c te r is t ic s a r e d e te r m in e d , th edril l f lank surface is a portion of the grinding cone surface.There fore, a portion of the interse ction res ult ing from the cut t ingof the grinding cone by an orthogonal cutt ing plane is the dril lf lank conto ur. To obtain a reasonab le flank shape, the direction cosine n of the cone axis with respect to the drill axis is rela ted to a sup ple me nta ry angle X defined as :

x = T- ( " > )Th e l imits of X are > X > Q to yield an ellipse, a portion ofwhich is the f lank conto ur in an orth ogon al cutt in g piano.Therefore, the f irs t s tep in developing the dril l f lank contour isto generate the proper e ll ipse.

The procedure for generating the e ll ipse can be best explainedthro ugh th e use of a figure. Consid er the cross section of thegrindin g cone A C D in the plane which includes the cone axisA B a n d i s p e r p e n d ic u la r to th e o r th o g o n a l c u t t in g p la n e O D E a sshow n in the lower par t of Fig. 4 . Th e angle betwe en the coneaxis and th e cutt in g plane is X. Th e z-coordina te of the coneverte x A is shown as z and is defined in term s of the originalc o o r d in a te s y s te m . T h e d is ta n c e b e tw e e n th e o r th o g o n a l r e f e re n c e p la n e a n d th e c u t t in g p la n e is d e n o te d b y / . T h u s , th ec u t t in g p la n e p a s s e s th r o u g h th e g r in d in g c o n e a t a d i s ta n c e cf r om t h e c o n e v e r te x w h e r e :

/ = d co t K + / mFor a given grinding cone and value of/ , the e ll ipse that results from the intersection of the grinding cone and the cutt ingplane is shown in the upper pa rt of Fig. 4 . Th e ell ipse in theupper part of Fig. 4 is determined by projecting i ts major axisCD f r om th e lo w e r p a r t a n d b y th e lo c a t io n of P o in t F . T h ecoordinates of Point F are determined from the upper part of

Nomenclature-(x, y, z) = coordinate system defined

f or th e o r th o g o n a l c u t t in gp la n e a n a ly s is d e s c r ib e dby Fig. 2

(it, v, w) = coord inate sy stem definedfor the oblique cutt ingp la n e a n a ly s is d e s c r ib e dby Fig. 8

(xo, i /o) = projections, through the distance / , of the x- a n d y-a x e s o n a n y o r th o g o n a lc u t t in g p la n e

(o, vo ) = p r o je c t io n s , th r o u g h th e d is ta n c e / i , o f th e u- a n d

w-axes on any obliquec u t t in g p la n e

(xi, ?/i) = auxiliary coordinate systemin a n o r th o g o n a l c u t t in gplane used to express ac o n ic in s ta n d a r d iz e d f o rmand related to the coordinate system (xo, y )thro ugh the a ngle co andt r a n s l a t i o n p a r a m e t e r s

('i, VI ) = a u x i l ia r y c o o r d in a te s y s te min an oblique cutt i ngplane used to express ac o n ic in s ta n d a r d iz e d f o r mand relate d to the coord i n a t e s y s t e m (no, %)

thro ugh the angle co andt r a n s la t io n p a r a m e te r s(x', ( / ' ) = auxiliary coordinate systemin an orthogonal cutt ing

plane used to describeflute contour rotation andrelated to the coord inatesy st em (xo, ,Vo) th ro ug hthe angle f

(I, m, n) = cone axis direction cosinesin the orthogonal coordin a te s y s te m (x , (/, z)

(V, m', n') = cone axis direction cosinesin the oblique coordinates y s te m (u, v, w)

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8, 8 2F i g . 4 F l a n k e l l i p s e d e t e r m i n a t i o n

Fig. 4 by noting that Point F is the intersection of a cone generat r ix ( A " F ) a n d a l ine th r o u g h P o in t B p e r p e n d ic u la r to th e c o n ea x is A " B a n d ly in g o n th e c u t t in g p la n e CD E . T h e e l lip s e c a nbe expressed in the auxiliary coordinate system (xi, i/i) in standardized form as:

1 (7 )w h e r e a a n d h a r e th e s e m im a jo r ( CO ' in F ig . 4 ) a n d s e m im in o r( G O ' in Fig. 4) axes, respectively, and

a = - e j co t ( X - 8) - cot (X + 6)}e tan 0\ cot (X - 0) - cot (X + 0)\_

~ 2 sin X[{cot (\~^W^coi Xj {cot, X ^ ^ T ( X ~ + 7 ) j T ^a s d e riv e d in A p p e n d ix 2 .

Complete D rill Point Cross S ections. Th e co mp let e drill poi ntcross section can be drawn by su perim posing the dril l f lu te contour in an orthogonal cutt ing plane and the ell ipse result ingfrom the interse ction of th at cutt ing plane wit h the grindingc o n e . T h e m a t h e m a t ic a l p r o c e d u r e u s e d to ac h ie v e th i s o b jective is to mak e a transfor mat ion of the e ll ipse given in Fig.4 from the coordinate system (x, , i/i) to th e c o o r d in a te s y s te m(x0, j /o) . Pr eviou sly, th e :r0- and j/o-axes were defined as t heprojections of the x- a n d j / -axes of the original coordinate systemon the cutt in g plane. The Xi-axis coincides with the projectionof the cone axis on the cutt ing plane.

Th e tran sform ation can be ma de in two step s: f irst , byfinding th e rotat ion angle co, i .e ., the angle betw een the x vaxisand the Xi-axis; and second, by establis hing the relat ionshi p between the coordinates of the projected cone vertex expressedw i th r e s p e c t to b o th th e c o o r d in a te s y s te m s (x h j/i) and (x0, j/ 0 ) .The equation for the transformation of coordinates is (seeA p p e n di x 3 ) :

x0 \ / cos coVo l \ si " oi

si n coco s co

w h e r ep = - x t co s oi + y, si n co hq = x sin co j/ cos coh = | c { c o t (X - 0 ) + col (X + 6) \

tan |co| = | - if - > 0, co < 0 an d if < 0, co > 0.Fig. 5 portrays the results of the superimposit ion of the f lute

contou r and the flank ell ipse on a cutt ing plane. Th e portionof the ellipse PQ that lies between drill flutes is the drill flankconto ur in th at p lane. O nly one f lute con tour (referred to asFlu te 1 in Fig. 5) is considered in the preceding analysis a nd theintersection of that contour and the ell ipse (referred to as Ell ipse1) is a t Point P .

The complete shape of a dri l l point cross section in an orthogon a l c u t t in g p la n e i s g e n e r a te d b y n o t in g th a t F lu te 2 a n d F la n k2 of Fig. 6 are symm etric to Flute 1 and F lan k 1 of Fig. 5 withrespect to the origin of the coordinate system (x 0, yQ). T h e p o r -

(8 )F i g . 5 S u p e r p o s i t i o n o f d r i l l f l u t e c o n t o u r a n d fl an k e l l ip s e

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Fig. 6 Dr i l l point cross sect ion in orthogo nal plane

c u t t in g p la n e ( F ig . 1(b)) will y ield addi tiona l useful inform ationa b o u t th e d r i ll g e o m e tr y . T h e o b l iq u e c u t t in g p la n e c a n b erelated to the original orthogonal cutt ing plane and coordinatesyste m. The f lute and the f lank conto urs in oblique cutt ingplanes can then be analyzed.

N ew Coordinate System for the O blique Cutting Plane. In Fig . 8p la n e 1\ is inclined to the drill axis on which the cross section ofthe dril l will be obtained . A plane, P0, which is paralle l to thec u t t in g p la n e Pi and passes through the origin of the originalo r th o g o n a l c o o r d in a te s y s te m (x, y, z), is used as a new obliquereference plane: and, a new coord inate sy stem (w, v, w) is established with respect to plane l\ as follows:

1 Th e (/-axis corresponds to the intersect ing l ine A B of theoriginal reference plane ( i .e . , : r i /-plane) and the new referencep la n e P0.

F ig . 7 Inf luence of the f dimension on the f lank el l ipse an d f lute contour

t ions PQ an d R S of the two ell ipses located betw een Flute s 1 and2 are the flanks of the drill point, while the cross section of a drillp o in t in th e o r th o g o n a l c u t t in g p la n e is th e s h a d e d a r e a P Q R Sbounded by the four curves in Fig. 6 .

Th e change in the dril l cross section in different cut t ing p lanesc a n b e in v e s t ig a te d b y r e p e a t in g th e p r o c e d u r e o u t l in e d a b o v ein other orthogonal cutt ing planes, i .e . , by varying the distance/ between the reference plane and the cutt ing plane. A s anexample, the relations hip betw een two cross sections is show nin Fig . 7. Th e ellipse described in Figs . 4 an d 5 is show n inFig. 7 as Ell ipse 1 with dim ensions an d coordi nates given by thesuperscript 1 . Th e flank ell ipse in a second plan e, designa tedas cutt ing plan e 2 , is referred to as Ell ipse 2 with dimen sionsa n d c o o r d in a te s g iv e n b y th e s u p e r s c r ip t 2 . N o te th a t th e d is tance betwe en the reference plane and cutt ing plane 2 is lessthan the dist ance between the reference plane and cutt ingplane 1 . Th us , Ell ipse 2 will be smaller th an Ell ipse 1 and willbe shif ted alon g the zi-axis in the negative d irection as shown inFig. 7 . The f lutes in each cutt ing plan e are located in theirproperly rotat ed posit ions corresp onding to the distan ce / andthe dril l helix angle . Therefo re, the f lank contours obt ainedfor the two cutt ing planes are P 'Q 1 a n d P 2 Q 2 , respectively.

Geometrical Analysis of the Twist Drill Pointin an Oblique Cutting PlaneA n analysis uti l iz ing a dri l l poin t cross section in an obli que F ig . 8 Relat ionship between the orthogona l (x, y, z) and ob l ique (u ,reference systems , w )

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2 Th e u-axis is perpe ndicul ar to the v-axis passing t hro ughthe origin and l ies on the new reference plane Pa .

3 Th e (('-axis is per pen dic ula r to the u- and (.'-axes and passesthrough the origin of the original coordinate system.T h e n e w c o o r d in a te s y s te m (u, v, w) is obtai ned by :

1 l io tat iou of the x- and ;i/-axes clockwise by an angle p aroundthe z-axis as shown in Fig. 8(a) yielding the x'- a n d j / ' -axes .

2 R o ta t io n o f th e x'~ and z-axes counterclockwise by an angleTJ aroun d the ( / ' -axis as shown in Fig. 8(6) yielding th e u- a n d w-ax es .

3 Th e (/'-axis is desi gna ted as the (.'-axis in th e new c oor din ates y s t e m .A ngles p and >; are posit ive when the rota tion s follow the directions defined above.

T h e r e la t io n s h ip b e tw e e n th e c o o r d in a te s y s te m s (x, y, z) a n d(u, v, w) i s g iv e n b y th e t r a n s f o r m a t io n m a t r ix M , as well asthe posit ion of the cone vertex (x y,,, zv) and (., v, w v), a n dthe direction cosines of the cone axis (I, m, n) a n d (V, m', n')( A p p e nd i x 4 ) :

= M

Oblique Cutting Plane P,

a n d = M l m (9 )

w h e r e

\l =/c os p cos >;I sin p\eos p s in r ;

sm p cos T)cos p

sin p sin ij

sm i)0

CO S t]

I n a d d i t io n , fh which is analogous to / , is used to describe thedistance between the reference and cutt ing planes in the obliquecase.

Flute Shape. Th e f lute shape in an oblique cut t ing plan e canbe determined from the intersecting points of the oblique cutt ingplane and the f lutes located in ortho gonal cut t ing p lanes byv a r y in g th e d is ta n c e / . Co n s id e r th e o b l iq u e c u t t in g p la n e Pia n d th e o r th o g o n a l c u t t in g p la n e P 2 in Fig. 9 whe re the f lutecontours in plane P 2 a r e s h o w n a s A B a n d CD . T h e lin e of in te r section of planes P x a n d Pi cuts the orthogonal plane f lute eon-t o u r s A B a n d C D a t P o i n ts G a n d H . T h e s e p o i n t s G a n d I Iare therefore points in common with the f lute contours in bothth e o r th o g o n a l a n d o b l iq u e c u t t in g p la n e s .

A s th e d is ta n c e b e tw e e n th e o r th o g o n a l r e f e r en c e p la n e a n dc u t t in g p la n e Pi varies , the f lute contours A B and CD in P 2changes. I f the oblique cu tt ing pl ane Pi is f ixed, the l ine ofin te r s e c t io n b e tw e e n P, a n d th e v a r y in g p la n e P 2 will trace outthe f lute contours in the oblique cutt ing plane.

The cutt ing plane P, expressed in the original coordinates y s te m (x, y, z) is:

x cos p sin i) y sin p sin T) -\- z co s i) ~ fi = 0 (10)A s s o c ia te d w i th th e o r th o g o n a l c u t t in g p la n e P 2 is the distance/ which, when substi tuted for z in equation (10), g ives the l ineof intersection between the two planes.

Flank Shape. E qua tio n (9) defines the relatio nship betw eenthe grinding cone and the new coordinate system ( , v, w). B ydenoting the projections of the u- and (. '-axes on the cutting planeb y th e v.- and w 0-axes, the f lank shape can be determined withrespect to (he coo rdinate syste m (uo, o) by specifying the d ista n c e / , . T h e d i r e c t io n c o s in e n' of the cone axis with respectto the (-axis determines the cross section of the f lank surfacein the oblique cutt ing plane as one of three types of conies;namely, an ell ipse, a parabola , or a hyperbola as follows:

w h e r e

Orthogonal Culling Plane Pj

F i g , 9 Flute contour analysis in an obl iqu e cut t ing plane

(1) ellipse - > X >( 2 ) p a r a b o la X = 8( 3 ) h y p e r b o la 6 > X > 0

\ =- - 11 if n' > 0

( H )(116)( l i e )

X = , * - if n' < 0an d

(a) Ellipse. If the angle X is betw een TT/2 a n d 0 as shown ine q u a t io n ( H a ) , th e f lan k s h a p e ca n b e d e te r m in e d in th e o b l iq u ecutting plane from the f lank ell ipse analysis in an orthogonal cutt ing plane after modification from the orthogonal toth e o b l iq u e c o o r d in a te s y s te m . W h e n n' = 1, a circle will appea r as a special case of the e ll ipse. I f the projection E of thecone ver tex lies on the po sitiv e side of the id-axis (t he :ri-axis inFig. 4) , the following m odification is necessary in the transform a t io n e q u a t io n g iv e n b y e q u a t io n ( 8 ) :

h = - 7 g\ cot (X - d) + cot (X + 8)}w h e r e

9 /.(b) Parabola. If th e angle X equ als 6, the f lank shape in the

cutt ing plane is pa rt of a para bola . Th e equa tion for thep a r a b o la i s g iv e n iu th e a u x i l ia r y c o o r d in a te s y s te m ( 1; v\ ) ofFig. 10 as:

~ - \I 1

:E

^ 8 _,,.,-Grinding Cone

^ \ C(o) 6 ^ T - Reference Plena

- ^ iC utt ing Plane- | n | ifn'0

F i g . 10 Flank analysis For an obl ique culling plane: par abol ic conf our

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Fig. 11 Flank analysis for an obl iqu e cu l l ing plane: hyperbol ie contour

Mi = aih (12)1

(13)

co t 69 = |w - / i |

T h e r e la t io n s h ip b e tw e e n th e c o o r d in a te s (u h v t) and (u0, Vo) isio \ / cos co si n co \ / ui pVo ) \ sin co cos co/ \ vx q

wherep = M co s co + vv si u co hq = si n cu W co s coh. = j cot 25I TO i TO,ta n jcoj = | if > 0, co < 0 and

if < 0, co > 0id Hyperbola. I f the angle X is smaller th an 0 b u t g r e a te r

than zero, the f lank shape in the cutt ing plane is part of ahyper bola . Th e equ ation for the hyperbola is g iven in the auxil iary coordinate system (u t, \) of Fig. 11 as:

i _-b" - (14)

w h e r e= - f f j c o t (0 - X) + cot (0 + X)j

g tan 0{cot (0 - A ) + co t (0 + X)}~ 2 sin X[{cot (0 - X) + cot X){c ot X - eo tTe + xTjT 1^

0 = i>. /.!I f the rotation of the angle r\ in Fig. 8 is counterclockwise,

the flank shape is repr esen ted by ne gative u values if )j < w a n dby posit ive u values if r\ > w. In Fig . 11 the flank shap e for thecase vvliere -q is less than TT is shown. The f lank shape for theV > TT case is a mirro r image abo ut the vr axis. If X = 0(that, is , the cutt i ng p lane is paralle l to the cone axis im plying ' = 0) , . and 6 in equation (14) reduce t o:

a = g co t 0b = g

In the case where the f lank shape is a part of a hyperbola therelationships between the two coordinates ( id , V\ ) and (o, i'o)are the same as equation (13) with the following modification:

- f / { c o t ( 0 X) - co t (0 + X)}sign, and ifhere if l'n'(w ~- j\) > 0, h t a k e s th e

l'n'(w r ~ .A) < 0, h take s the sign.The notation for a ll three of the f lank surface conies involvest h e p a r a m e t e r s a, b, g, a n d h. N o t e t h a t t h e s e p a r a m e t e r s a r edefined differently for each conic.

Specific O blique Cuffing Planes for A ngles of Importance. Th eanalysis of the dril l point geometry in an oblique cutt ing planecan be of use in describing several of the angles related to drillperfor man ce. The se angles include the nom inal re lief angleand the face rake angle , both of which can be described in termsof basic dri l l parameters in oblique planes.

(a) N ominal Relief A ngle. Th e nom ina l relief angle a is definedin reference [1] . In essence, the nom inal re lief angle is the anglebetw een the orthog onal reference plane and a tan gen t to (he f lankconto ur a t , a poi nt a long the dril l cutt in g edge. Th e nom inalrelief angle is measured in an oblique cutt ing plane which isparalle l to the dril l axis and is perpendic ular to a radius fromthe dril l axis to a point on the cutt in g edge. O nce the dril l f lankcontour in the oblique cutt ing plane is known for a specifiedpoint on the dril l cutt ing edge, the nominal re lief angle can beo b ta in e d .

To be specific , the nom inal re lief angle can be obtain ed bythe comp uter pr ogram in four s te ps: Firs t , specify a poin t onthe dril l cutt ing edge; second, determine the proper transformat io n m a t r ix M in terms of p a n d -q ; th ird , obtain the dril l f lankcontour iu the oblique cutt ing plane; and fourth , construct, thetange nt, to the relief con tou r at, the specified point,. Th e deter mination of the nominal re lief angle as if varies a long the dril lcutt ing edge can be best i l lustrated by an example as shown inFig. 12. S uppo se the relief angle at Poin t Ai on the dril l cutt i ngedge is to be dete rm ined . A s shown in Fig. 12 the coo rdinates(xi, ;(/i, 2 i) of Point A i iu the orth ogon al coord inate syste m are:

Xi = / ' [ C O Syi = / ' i s i n cZl = /

(15)

S u b s e q u e n t ly , th e t r a n s f o r m a t io n m a t r ix . 1 / c a n be d e te r m in e d inte r m s o f th e p a r a m e te r s p a n d -q as given by equa tion (9). H e-c a l l th a t th e t r a n s f o r m a t io n m a t r ix M enables the f lank contoursto be described in the coordinate system (u, v, w). F r o m t h edefinit ion of the nominal re lief angle , the oblique cutt ing planein w h ic h a is defined has the par am eter s p = 4>i a l l c ' V = w/'^-Thus, the coordinates of Point Ai in the coordinate system(u, v, w) c a n b e s h o w n a s :

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Fig . 12 Determination of nominal relief angle

M l = 2 l

Hi = 0 (16)M'I = n

W ith these coordin ates known, the nominal re lief angle a t PointA t c a n b e d e te r m in e d .

N o te a ls o th a t th e c u t t in g p la n e P [,- for Point A 2 is not paralle lto th e c u t t in g p la n e Pu (Fig. 12(a)) . Con sequ ently , a new coordinate system for Point A 2 i.s det erm ine d in ter ms of th eparameter c/>2. Thi s implies that a t every point on the cutt ing:edge a new coordinate system must be defined in order to obtainthe nominal re lief angle .

(b) Face R ake A ngle. Th e face rak e angle ya is also dennedin reference [1]. The face rake angle is described in an obliquecuttin g plane which is perpe ndicul ar to the l ine joining thecorrespond ing outer and chisel edge corners . Th e me thod forobtain ing the face rake angle is analogo us to that for the n omin alrelief angle except tha t i t is the f lute conto ur ra t her th an theflank contour that is described in the oblique cutt ing plane. S uppose the face rake angle a t Point Ai on the cutt ing edge in Fig.13(a) is to be determ ined. Th e coordinates of Point Ai canbe expressed in the coordinate system (x, y, z) a s :

:i'i = / tan K!h = -I (17)*i = /

The pl ane in which face rake angle is defined is g iven byp = 0 and r) K in e q u a t io n ( 9 ) , h e n c e th e t r a n s f o r m a t io n m a t r ixAI can be deter min ed. Th us, the f iute contou r can be describedin th e c o o r d in a te s y s te m (u, v, w) a s p r e v io u s ly s h o w n . T h ec o o r d in a te s of P o in t A i w i th r e s p e c t to th e c o o r d in a te s y s te m(u, v, w) a r e :

, = 0Vi = ~t (18)

wi = / sec Kand the ta nge nt to the f lute contou r as Poi nt A i g ives the facer a k e a n g le a t th a t p o in t . N o te th a t th e c u t t in g p la n e Pa forP o in t A 2 i s p a r a l le l to th e c u t t in g p la n e Pu , as shown in Fig.654 / A U G U S T 1 9 7 0

--T ange nt lo the Flute a! the Point A,(c) Cross Section on the Plane PLI

Fig . 13 Determination of face rake angle

13(6), and thu s i t is not necessary to determ ine a new coordinate system for each oblique cutt ing plane along the cutt ing edge.

In the determination of the face rake angle the f lute contourin an oblique cutt in g pla ne was uti l ized; howev er, the f lankshape was unnecessary s ince the face rake angle is defined withrespect to the f lute contou r. O n the other hand , in the deter mination of the nominal re lief angle , i t was necessary to generatethe f lank shape in an oblique cutt ing plane while the f lute contour is not used.References

1 G alloway, D . F., "S ome Experim ents on the Influence of Various Factors on D ril l Performance," T R A N S . A S ME , Vol. 79, 1957,pp . 191-231.2 O xford, C. J., Jr. "O n the D rilling of Me tals I, Basic Mechanics of the Proc ess," T R A N S . A S ME , Vol. 77, 1955, pp. 103-114.

A P P E N D I X 1Principle of Conical G rinding. Th e con cept of con ical grin din g

was given by G alloway [1] and is i l lust rated in Fig. 14. Th eflank of a twist dri l l is grou nd by the grinding w heel G a s th edril l is rota ted aro und th e axis X X . S uppose the drill is cons id e r e d s ta t io n a r y w h i le th e g r in d in g w h e e l i s r o ta te d a r o u n dthe axis X X . Th e plan e of the grinding wheel face gene rates acone with the f lank surface of the dril l being a portion of thecone surface. This cone is referred to as the grinding cone(thus the name conical grind ing). Th e plane of the grindingw h e e l f a ce a n d th e a x is X X in te r s e c t a t P o in t O . T h e v e r te xof the grinding cone is a t the poin t O , while the cone sem iangle6 is the angle between the axis X X a nd the gener atr ix lying onth e g r in d in g w h e e l s u r f a c e . T h e r e a r e tw o s y m m e tr ic g r in d in gcones, labeled as A and B in Fig. 14, with eac h cone gene ratin gone of the dril l flank surfaces. Th e grindin g motion of mostdril l-grinding machines is usually more complex than thatmen tioned a bove, as ther e is some sliding mot ion betwe en thegrind ing wheel face and the dril l point. H owe ver, in those cases,the grinding motion is reducible to the purely conical grindingmotion at any instant of t ime.

The grinding cone location with respect to the dril l axis isspecified once the four parameters K, \fr, 6, a n d d are f ixed. G allo-

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Grinding Whael

Grinding Cone 'A-)}

a n d X and e are given by e q u a t i o n s (5) and (6).Co n s id e r in g a n o th e r in te r s e c t in g p o in t ( P o in t F) of the ellipse

a n d a p la n e A"BF in the u p p e r p a r t of Fig. 4 which includes thecone axis AB and is p e r p e n d ic u la r to p la n e ACD, the c o o r d in a te s(x F, yF) of Point. F can be obtained from the g e o m e t r y of p la n e sA " B F and ACD.

I n the p la n e ACD,O E = /i = 01) + DE

( 2 2 )= - c f c o t (X - 6) + cot (X + 0)}

O B = OE - BE= h e cot X

I n the p la n e A"BF,B F = A"B tan d

= e cosee X tan 9Tints , considering the signs of c o o r d in a te s (xF, yF) w ith r e s p e c tto the c o o r d in a te s y s te m (xi, ?/i),

X]f = -O ByF = BF

( 2 3 )

S in c e P o in ts C and F are p o in ts on the ellipse, theexpressions fora and 6 in e q u a t i o n (7) can be o b ta in e d by s u b s t i t u t i n g the coo r d in a te s g iv e n by e q u a t i o n s (21) and (23) in e q u a t io n (7) andsolving the s im u l ta n e o u s e q u a t io n s .

A P P E N D I X 3The Derivat ion of the Tran sform at ion Equat ions Def ined by Equat ion

(9). R e f e r r in g to Fig. 15, the a n g le w b e tw e e n the xr and theXi-axes wasd e r iv e d as [1]:t a n i>i = -

w h e r e I and in are the direction cosines of cone axis given bye q u a t i o n (19) inA p p e n di x 1.

T h e r e la t io n s h ip b e tw e e n the original and the a u x i l ia r y coo r d in a te s y s te m s (x0, y0) and (xi, yi), r e s p e c t iv e ly , is s u c h th a tth e XT and !/i-axes are r o t a t e d by the a n g le u> clockwise in thel e f t - h a n d s y s te m a r o u n d its origin and t r a n s l a t e d to the originof the c o o r d in a te s y s te m (x0, J/ 0) as s h o w n in Fig. 15 (refer alsoto Fig. 5).

I f new x'- and j / ' -axes are o b t a i n e d Vjy r o t a t i n g the x\- andj/i-axes by the a n g le co a r o u n d the origin of the c o o r d in a te s y s te m(xi, ?/i), a p o in t w i th the c o o r d in a te s (xi, j/i) is expressed in then e w c o o r d in a te s y s te m (x1, y') by:

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- K M

Projector) of theCone Verle*

F i g . 1 5 T r a n s f o r m a t i o n o f c o o r d i n a t e s y s t e m ( x j , y d t o c o o r d i n a t e s y s t e m(xo, yo)

!/-, and vj-axea with respect to the original x-, i/-, and 2-axes be Xi,Hi , ci; Xj, p 2 , i' 2; X3, ^3, v3, r e s p e c t iv e ly . F r o m F ig . 8 e q u a t io n ( 9 )can be given bj ' :

X ix2X,M ip-1Ms

v\JV / *r*w h e r e

X! = cos p cos >;

Hi = co s

P i = C O S

p I cos ?; = sm p cos r/

-s in JJ

X2 cos | - - p sm p

w h e r eT = S i l l CO

C O S CO

S ince the co ord ina te syste ms (.To, yo ) and (x ' , y') are paralle l ,the following relationship exists between them:

(25)w h e r e r and s are show n in Fig. 15. Th e coordina tes of the projected cone vertex E in the x 0(/ 0-plane a re exp res sed by (x,., //)a n d ( /(, 0) with respect to the c oordi nate sy stem s (xo, i/o) and(xi, yi), respectiv ely. Theref ore, the following relatio nship between these two coordina tes of the point E is obta ined by appl ying equations (24) and (25):

7 -h 0Solving equation (26) with respect to r and s yields:

T T- Vv r

(26)

(271T h u s , s u b s t i tu t in g e q u a t io n s ( 2 4 ) a n d ( 2 7 ) in to e q u a t io n ( 2 5 ) a n dsimplifying, the expressions in equa tion (8) are obtai ned. Th eexpression for h in equa tion (8) is equ atio n (22) in A ppendix 2 .N ote t ha t if the point E l ies on the posit ive s ide of the Xi-axis,the expression for h in equation (8) must be modified as follows:

1 cjcot (X - d) + cot (X + S) \

A P P E N D I X 4The R elationship Between the O rthogonal and the O blique Coordinate

System. R eferring to Fig. 8 , le t the direction cosines of the u- ,

cos pv, = cos - = 02

cos p cos - V cos p sm rj

/J.3 CO S - V -sm p sm i]Vz = C O S 1)

A plane perp endicu lar to th e -uj-axis and passing th roug h ap o in t w i th th e c o o r d in a te s (x p, yp, zp) is expressed by:

/oX3(x - x ) + kfx3(y - yp) + kv3(z - zp) 0 (28)w h e r e k i s a n a r b i t r a r y n u m b e r w h ic h i s n o t z e ro . D iv id in gequation (28) by k y ie ld s :

X 3 ( x - x p ) + f i : s ( y ~~ y) + v3(z - zp) = 0 (29)S ince the (, '-plane i'o (Fig. 8) passes through the origin 0 withthe coo rdinate s (0 , 0 , 0) , the plane P 0 is expressed by:

X3x + mi + vs z 0S in c e th e p la n e Pi p a s s e s th r o u g h th e p o in t 0 ' ( F ig . 8 ) w i th t h ecoordi nates (0 , 0 , / i) expressed with respect to the coo rdina tes y s te m (u, v, iv), the coord inates of the point O ' are expressed inth e c o o r d in a te s y s te m ( x , y,z) u s in g e q u a t io n ( 9 ) a s :

x = /i c o s P sin 7}V = /i sin p sin r\zP = / i c o s V

(30)

S u b s t i tu t io n o f e q u a t io n ( 3 0 ) in e q u a t io n ( 2 9 ) y ie ld s e q u a t io n(10).

656 / A U G U S T 1 9 7 0

journal of engineering for industry volume 92 issue 3 1970 [doi 10.1115%2f1.3427827] fujii, s.;...

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