On the Grinding Geometry for the Random Spacing of Abrasive Grains on Wheel Surface 利用統計を見る

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On the Grinding Geometry for the Random Spacing

of Abrasive Grains on Wheel Surface

T.Orioka

The g・i頑ng ge・metry・f・・the・and・m．・pacing・f abm・ive g。ain。。n・the Whee1 face

ヨsconstructed・ At first，“Abbott’s Curve，’and 4‘Roughne88 Distrib耐ion Curve・’fer

’：th・g・・und・u・face a・e d・・ived， a・別ming that g・ain tips a・e a・・anged e．xactly。n the

Cylind・ical・u・face・f th・whee1・ln・・d・・t・・btain the exp・e・si。n・， whi・h ag・ee

”with・xpe・im・nta1“ata，・eve・al types・f di・t・ibuti・n in d・pth f・・g・ain晦。 a。e

assumed・and the surface roughness corresponding to them is computed． In the next

．二place， expressions for the chip width and the chip thickness are derived． The size

’di・tibuti・n・f th・m・xplain・the fact that th・・e a・e p・・duced va。i・u。 chip。。f diffe−

’「ent・ize in actua19・inding・perati・ns． It bec・mes clear that mean rati・・f chip

−width to chip thickness is not identical with mean width to depth ratio of individua1

，9・indi・9・c・at・h…Th・・e1・ti。n b・twee血mean 9・ain d・pth。f・ut a。d m。。n。hip

−secti・n・1 a・ea i・・bt．ain・d・At la8ち．th・numb…f 9・inding…at・he8 i』c。mputed

and the m・aning。f th・di・tan・e between・ucce・・iv・・utti・9 P・int・f・・g・indi鳳9

・・盾垂?窒≠狽奄盾獅刀@i8 illustrated．

1．Nemenclature

’Tho following nompnclature is used in the paper：

・D＝wh・el dia甲・t…mm

．D卿＝work 1／diameter， mm

！＋f°「exte「n・19・inding ．、

ノ・・f。r s・rface g・inding ． − fer internal grindiロ9

γ＝wheel speed， m／min

，・V＝：w。rk 8peed， m／min ・d＝＝wheel depth Of cut， mm ・溺＝distance between 8uccess三ve cutt諏9 points on wheel surface， mm ’ 」＝undeformed chip length， mm ・…一・m・11・bgi・・』 ilubt’・nd・d・t ce・t…fwhee1・・d w・・k，・e・p・・ti・e！y， by und・f。・m・d bh ip length（1）．（used in reference（1）by Alden）， ’ 、L ・t．＝「naximum u・d・f・rm・d・hip thi・k・ess・f・・e・age・hip， mm・（used・in・ef・；ence・（4），（5）and （6），by Backer， Reichenbach and Yang， respecti・vely）

C＝鵬an n・mb…f・utting p・i・t・p・・sq mm・f頑eel f・ρ・，1／mm・

〆＝mean ratio of chip width to chip thickne88 ，プ＝mean width to depth rapio of i瓦dividual grinding scτatches 口’＝＝mean width of average chip， mm．（鵬ed by Backe．r ani Reichenbach） h ：mean peak−to〔valley d三stance， mm（used by Yang， Reichenbach and Satg） ka＝・actuahmean peak−to・−valley distance，］mm（used by Yang） 2φ＝grain tip・ngle・f indi・idua19・it 2φ＝mean grain tip ang1e

109

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●

昭和32年7月

_{山梨大学工学部研究報告}

第 8 号鳶＝radius of grain tip fo芳individual grit， mm 矛＝mean radius of grain tip， mm tu2＝mean area occupi6d by single cutting Point on wheel face＝・1／C， mm2 ’ w＝mean distallce between cutting points＝1／C1／2， mm ”12＝mean area occupied by single grain projection on to work gurface， mm2 況2＝mean area occupied by single grain prolection on to section Ol−Oi， mm2 ッ＝height of grain projection on to section O1−02 from ideally finished work surface， mm i ＝infeed per rninute， m／min r ＝height of actually finishe4work surface from ideally finished work surface， mm P（Y）＝probability， in which vertical height of actually finighed work surface from idea11y．’ finished work surface exceeds Y， and so corresponds to‘‘Abbott“s Curve” ∫（r）・＝express ion for‘‘Roughne8s Distribution Curve”， and so derivative of−P（Y） γo＝value of Y， correspOnding to maximum val ue ofノ（Y）， mm ht＝mean height of surface profile from its base， mm

B＝wheel width， mm

プ feed per revolution， mm／rev ∫ numbe；of grinding for Single point on work surface Ho＝maximum depth of distribution in depth for the ， position of grain tips， mm κ，五’，K1， K2， K3＝parameter for surface roughness C1， C2，Cs ：parameter for surface roughness in non−dimensional form P（y，γ）＝expression for 4‘Abbott’s Curve”of surface profile， which is ground jusg to height（y＞ b ＝width of individual chip at height（y）， mm 万＝mean chip width at height（夕）， mm bo ＝＝mean chip width corresponding to height y→0， mm g’＝depth of cut for arbitrary point of grain ctltting edge， mm 冨1 depth of cut for grain tip， mm 9’ depth of cut for arbitrary grain， mm ζepth of cut at height（夕）， mm go depth of cut corresdonding to height y→0， mm mean grain depth of cut， mm st_{@sectional area of indivξdual chip at height（y）， mm2} s mean value of’s’， mm2 so value of s， corresponding to y→0， Mrn2 s mean chip sectional area for total chip length， mm2

M 垣eahrchip。・lu血・， mm3

S mean chip area looked at from above， mm雪・ i’_{@mean numb6r of grinding scratches per mm of surface profile， which is ground just tぴ} height（ツ），1／mm 緬＝mean number of grinding scratches per mm of finally finished work surface，1／mm

2． Introductio皿

It is wel1−known that・the size of grinding chips ig an−important parameter of the grindingl operation． The first expression for maximum chip thickness（の， that is distance RE in Fig．1． wa8 derived by Alden in reference（1）． ’

110

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On the Grinding Geometry for the Random Spacing of Abrasive Grains on Wheel Surface t・e・m・i・（・・＋・・）・・…・………（・） where m is the d｛stance between successive cutting Points on the wheel surface． Guest presented the following expres8ion for t in refe− ・rence（2）． t−2勿÷一（L＋＿laρ Dw）1／2dソ2・………・・…（2）『Howe▽er， a convenient method of finding m is not given．

允蕊「㌶蕊ll愁；；y吻’Sat°de「’ved the“V

別一励・………’一（3） dX

whereωis the mean distance between cutting points，

・which is represented by the mean number of cutting

points per sq mm of wheel face（C）as follows： ω2＝1／C ・・・・・・・・… 一・・（4）・and％o is the mean number of grind ing scratches per mm of the finally finished work surface， obtained from the surface profile given by a roughness tester． H・weve・．， a8 b・1・w・menti・n6d・m dep・nds n・t°nly・up・n Fig．1 ’the kind of grinding whee1， but also tlpon the grinding ・conditions． Furthermore， it is d魯fficult to estimate ne by the profile of grinding scratches． In reference（4）， Backer and others derived the『first reliable not apPear， asguming that the instantaneous c ’℃hipゴthickness，ぢ

〆s

_ぴ D

R

〃んee’ d；ameter（D， R．1 4 賄ぱ d’d・・t．c．・．（D“r） 02 Chip Thickness in External Grinding（by Alden） expression for♂in which物did hip width is a l inear function of the instantaneous and that the mean ratio of chip width to chip thicknes9（〆）i8 identical with the ’mean Width to depth ratio of individual grinding scratches（グ）． They recommended the taper−section ・method described in reference（4）as the best way of estimatingプ・Representing tho mean width lof chips byろ’， and the mean thickness of chips by t／2， th●y Presented the following expression ｛or r andプ’ bl グ＝：グ’＝＝2b’／ま・・・・・・・・・・・・… （5） In reference．（5）， Reichenbach and others also derived the ’expression forオ， which coincided essentially with that of Backβr，、assuming that the shape、， of averageφip is shown in F’ 奄〟D2，，鋤dthatγi8 identical with．〆as Backer． The expression for あd。，i。，d．・by Back。・anユ、R。i。h。nbach，．i．。， f。11。w，、 t Fig．2Mean Shape of Chip

・−2（VCr）1／2（；一＋っ1万）1／4dソ4………（6）（b・R・i・h・nb・・h）

This expression is very useful to find the relation between the chip thickness and the grinding ，conditions． HoweVer， the uSed assumptions by them for． the shap60f chips has not yet been verified． And they expected the mean ratio of chip width to chip thickness（〆）to be identical with the mean width to depth ratio of indjvidual grinding scratches（r）， but the relation between 〆and r has not yet been i撫strated． Furthermore in an actual grinding ogeration are produced vヲrious chips of different size・・but the consideration based upon the average chip can not explain く狽?Edi・tr迦i°n・f・ize f・「a・t・・1・hip・・1・・rder t・in・estig・t・th・・hip・h・p・・the re1・ti・・ between the chip shape and the scratch shape， the chip thickness ahd the size distribution of 1！Z

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昭和32年7月

_{山梨大学工学部研究報告}

_第 ₈ _号 chips・it will b・ugef・1 t・C…id・・th・・a・d・m・paci・g・f・b・a・ive grai・・i・， a g・i・di・g wh・・1・ On the other hand， consider the expression for the roughness of ground surface． In ref・・e・ce（6）・Y・ng・・d Sh・w present・d th・f・11・wi・9 expressi・n t・・th・mean peak・t・．va11。y distance h

万一（可告うη3（十＋貴）・／・㌫．．＿…．…．（7）

H・w・ver・as the a・t・al m・an peak−t・−va11・y di・t・・ce Chのdiffers f・・m l， th・y・・p・・t・d五。 t・ be proportional to h

fia−（−P3｝一）2／3（；＋☆）1／3 “……・・…・…（8）

The expression for h・derived by Reichenbach in reference（5）， ig as follows， if v／IL＜1： h＿（♂／の2 −「− 16 − ’ 1 … 一・・一・・・… （9） In reference（3）， Sato presented the foUowing expression for万

万一1（一；一＋一☆）（e）2彿・＋1，一（÷）2 ………・一（・・）

where no i8 the mean number of grinding scratches per mm of a ground surface， measured transversely to the grooves andたis the mean radius of grain tips． The computed values of surface roughness from their simplified assumptions are very sma11、・bmp・・ed with th・t m・as・・ed i・test・． Why d・・’t experim・・t・l d・ta agree with th・ts c・mput。d by the theory？Tlle reasons will be as follows： 1 Due to the fact that耳brasive grains are not spaced uniformly in a grinding wheel．， 2Due t°built−・p rdg・・n ce・㌻・in ab・a・ive 9r・i・…d・・u…1・etti・g i・d・pth・f・．ccasi・nar gralns．， 3Dq・t・the re1・ti・・di・p1・・ement b・tween theトwh・・1・・d th・w・rk by th・g・i・di・g 、，． resistance， the vibration， and etc．・・DThe「ef。r・・i・・rder t・・bt・in、the exp・e・si…f・u・face r・ugh…s， whi・h・gree・with exp・・im・n− tal． data・it will be useful t・c・nsider the rand。m spacing・f abrasive grains in a grinding whee1・・nd fu・th・・m・・e， t。 t・ke acc・・nt・f th・di・t・ibutiQ・・in d・pth f・r tip・・f・b・a・i。e g・ai・s and their built−up edges on the wheel surface． In this p・per the a・th・r c・mput・d the r・・gh・・s・・f 9・・u・d S・・face and th・g・ai・d・pth・f．9・t by the folIowing assumptions：．〔 1G「i・di・g scrat・h・・are p・。du・ed．・n・W・・kPi・・e l・t th・track。f・b・a・ive・9・ai・・exactly corresponding to their shape and their relative position to the work，、and the piling upi produced beside the s（iratches is ignored． 2・，’The cleavage or the self−dressing and the falling off of abrasive grains during their grinding action and the loading of wheel are ignored． ” 3

4

5

6

There is no relative displacement between the wheel and the work． The distribution of grain tip spaging、 pn the wheel face is at randoln．．；norder to compute the surface roughness， several types of the distribution ］n depth for the position of琴rain tips（inc1Uding tips of built−up edges）on the wheel、 face are assumed．． For the computation of chip width and grain depth of cut， it is assumed that grain tip合 are arranged exactly on the cyl ind ri cal surface of the wheel， because the distribution in． depth fgr grain tips does not play an imPortant role in this case． The shape of grain tips is assume，d principally to be 112

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On the Grinding Geometry for the Random Spacing of Abrasive Grains on Wheel Surface げ（A） acircular cone with a sharp tip， mean vertical angle of which is 25． or forthe comparison with it， to be （B） acircular cone with a spherical tip， mean radius of which is万． The fo110wing equations corresponding to the sharp tip and the spherical t亘p are affixed marks（A）and（B）， respectively． 7 The ratio of work speed to wheel speed＠／τノ）can be ignored forユ， which corresponds to Case l in refer6nce（5）by Reichenbach． The results thus computed seem to agree approximately with the experimental data of many references．、

3．How is a Workpiece Ground by Individual Grits？

o

11

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Whee1

diametev（D）

、、ピ

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ξ3

↓）

u・9f。und w。：い・・f・… （C●vvespending ro Arc BE i見F；9・3）品ケE’・i・‘・dW・・k s・・f・・e （Ce”“P・。di・了鋪 ▼cCF l・烏3） 23

一一’iiiE・；一≡

（a）完σ VVIo r k 、、 F・ d iametev（Dte） Fig．3 Relative Position of Grain Path to Work Surface Nr ieE ＝＝＝＝＝＝＝一◎L・一一・一．．．一． it b‘s≡≡13＝

o一

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口（b） 9

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：：二＝＝＝＝亡＝2＝＝

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111！111E；；1！1；1i：1Ei 3 ＝＝＝＝一： 1

o

the wheel surface， as shown．m Fig．4（b）． single grit， i． e． the depth and the width， surface（which is shown as arc BE in Fig． work（which is sho wn as arc CF in Fig．3）， there． are not so manY grinding scratcheg Fig．4 （a）Cut by Individual Grits in Section OrO2， and （b）Position of Corresponding Grits on Wheel Face ト In order tQ invegtigate the geometry of grinding by individual grits， it is useful to examine h・wthe sp・ti・・0・−0・・f th・w・・k i・Fig・．3i・g・…d． A・sum・th・‘ 唐?Ep・・f・grai・tip t・も・ acircular cone with a vertical angle 2φ． Fig．4（a）shows hoW the section O1−−02 bf the work is cut by 三ndividual grits for externa1， surface and internal grinding． In Fig．4（a）， the number of a grit written in its cut corsesponds to the order of the grit in the direction of grinding on In Fig．4（a）， it is evident that the size of cut by a is larger in the neighborhood of tlle unground work 3），than．that near the finally finished surface of the considering the mean magnitude for many grits． And

per mm in the neighborhood of the unground work

113

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昭和32年7月

_{山梨大学工学部研究報告}

第 8 号の surface， as near the finally finished surface of the work． For any shape of grain tips， there is the same tendency for the size of cut and for the number of grinding scrat ches， as above me泣ioned． The position occupied by a grain tip in Fig・4（a）is defined‘‘grain projeφion on to the section O1−02”of that gr it． The distribution of these projections is computed from that of grain tips on the wheel face． The size and the shape of cut by a single grit in section O1−02 are computed by this distributon， assuming the shape of grain tips． If there are C cutting points per sq mm on the wheel face， the mean area occupied by a single grit（ω2）is 1／C． The root ofω2， that is w， is defined‘‘mean distance between cutting Points” on the wheel surface． For example， Sato’s tests give values of C，ω2 and w for several grinding wheels as follows： Table l Mean Number of Cutting Points per sq mm（C）and Mean Distance between Cutting Points（w）

Kind of Whe，el C1／mm2 〆＝1／Cmm2 wmm

ll 30 H 1．36 0．735 0．85 ＃46L

_{5．00 0．200 0．45}

＃80P l4．6 0．068 0．26 ・ Now， assuming grain tips to be arranged exactly on the cylindrica！surface of the whee1， an individual grit touches the ideally finished work surface at a point corresponding to its position oll the wheel surface for external， surface and internal grinding． This point on the work surface is defined‘‘grain projection on to the work surface”of that grit． Corresponding to the asssumed random distribution of grain tip spacing on the wheel face， the distribution of the grain projections on to the work surface is also at random． The mean area occupied by a single grain projection on to the work surface（w12）is given as follows：

w・2−w2111F“＝“t …・……・…・（・・）

Values of w12 for several grinding wheels are given in Table 3 for the grinding conditions of Table 2． Table 2 Grinding Conditions for the Computation in this Paper Wheel diameter（D）， 180 mm Work diameteτ（Z）w），。。 Wheel speed（の， 1800 m／min Work sgeed（の， 5．5m／min ● Table 3 Mean Area Occupied by a Single Grain Projection on to the ． Work Surface（w12）for the Grinding Conditions of Table 2 Kind of Wheel C1／mm2 紗12 mm2 ＃30H 1．36 2・21×10−3 ＃46］L 5．00 0・62×10−3 ＃80P ’ 14．6 0．21×10−3 See again Fig．3． In usual grinding conditions， d／Dw＜1 andθ2＜1 in a tr三angle O1’02P・If v／V＜1

ツー（B＋☆）x・・…一・…＜・2）

114

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On the Grinding Geometry for the Random Spacing of Abrasive Grains on Wheel Surfac●

一鵬一一2（丁一＋十）1／2y・ノ… 一……（・3）

Substituting y by d and x by l in Equation（13）， the undeformed cllip length（1）for externa1， surface and internal grinding is represented by D，1）w and d， if v／17 is ignored for 1．

’一（；一＋つ1万）−112d・ノ… ◆…………（・4）

From Equation（11）and（13）， the mean area Qccupied by a single grain projection on to the section O1−02（u2）is represented for externa1， surface and internal grinding ag follows：・・−w・2一裂一2fC（ユ＿＋」＿D Dw）1／2夕・ノ・ Variation of％2 with夕are shown in Fig．5for the grinding conditions of Table 2． The above mentioned variations of the size of cut and the number of grinding scratches with the height from the idea皿y finished work surface are owing to the fact that u2 depends upon y． It is worth noting that the shape of cut by a single grit at the height of Y coincides with the shape of undefor血ed chip section perpendicular to the grinding direction at the same height y， and that the pitch on the work 8urface to bq・ ground and the distance between successive cut＿ ting poli nts on the wheel corresponding to it can not been computed from the number of grinding scratches or the scratch width of the finally finished work surface． Now， for plunge grinding of constant infeed， the mean． area occupied by a single gra｛n projec− tion on to the section O1−02（u2）is independent upon the height（ッ）． It iS evident that u？is also in this噛case identical with the mean area of chip section perpendicular to the grinding direction． Representing the infeed per minute by i，。 i u“＝一一 γC 句 § §

8

？

6 s

4

む

、3

× の

32

Fig．・・・・・・・・・・・・…@（15） 0 0．0∫ 0、02 0，03 0，04 008 γ一，閉

5Variation of“Mean Area Occupied by

Single Grain Projection on to Sectio：い Ol−0ピ’（u2）with Height（y）．．乍 E1♂・

c

綿6L）。％繍（

c45・o

b＝∫仔・6 協 2 ‘替80P） the expression for u2 is given as follows：・・・・・・…@；・・… （16）

4．Computing Method of Surface Roughness

■ The shape of cut by a single grain tip in the section O1−02 is generally shown in Fig．6・The width of cut（ξ）depends upon the height（η）．ξis the mean value ofξfor all grits at the height ofη． ξ＝＝ξ（η）・・・・・・… 一一・… （ユ7） Assuming・th・・h・p・。f・g・ain tip t。 b。 a。ircul。。 c。ne with Fig・6 General Shape of Cut by Single asharp tip of vertical angle 2φ，ξ＝： 2ηtanφ． Even if the shape Grain Tip in Section Ol−02 0f a grain tip is not actually a circular cone， all the equations remain unchanged， when the shape of its cut in the section O1−02 is the same as that by a

ZJ5

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昭和32年7月

_{山梨大学工学部研究報告}

第 8 号 circular cone， and、 also wh藺the relation betweenξandηfo ’r a grain tip is equal to・that for a circular cone． As an individual graip tip has a different vertical angle， we define‘‘mean grain tip angle”（2φ）as fol lows：

（A） 2t・㍉＝（2t・・のm，。n ’ ……一・…（18）

therefore

（A） ’ξ＝2ηtanφ， or ξ／η＝2tanφ ………（19） In order to estimateφ， the taper−section method described in reference（4）or an enlarged profile of a ground surface by a roughness tester measured transversely to the grinding scratches will be used， where the mean width to depth ratio of individual grinding scratches（夕）coincides with 2tanφ， i・e．（A） r＝2 tan ￠ ………（20） Now， representing by P（y）the probability， in which the vertical height of the actually finished work surface from the ideally finished work surface at an arbitrary point on a workpiece exceeds Y， it is evident that P（Y）corresponds to the so−called‘‘Abbott’s Curve”， which expresseS the total area of work 8ection parallel to the work surface at an arbitrary height（Y）． At P。esent， ass。m。 th。t g，ai。・i，， are ar・ang・d R exactly on the cylindrical surface of the wheel， and that the shape of grai．n tips is a circul ar cqne with a sharp tip of mean grain tip angle 、 2予． Fig．7（a）， Fig・7（b）and Fig・7（c） show the elevation， the side view and the plan， re・pe・tiVely， illustrating the relative p・s云ti・n of grain paths to the work surface． Ais an arbitrary point on the ideally finished surface of the work， and H is a point at the height of Y above A． Consider the position of the、 grain projection on to the work surface f．or the grit， the cutting edge of which passes the point H． The origin and the axes of co−ordinates are chosen as follows： Origin：the point A x−axis：in the direction AB‘ ‘c｝PI’an y−axis：in the direction AH Fig．7Paths of the Grits， Which Pass Point

z＿axis：in the direction AD Hat Height of Y・（a）Elevation，

C。。，ideri。g。t．fi，st th。9。it，， tip。。f whi。h （b）Side View and（c）Plan pass in thθx−y plane represented by ABCH， it is evident that the points B and C are the grain projections on to the work surface of the grits， tips of which pass the point H．2X is the distance between the points B and C， and X is repregented as follows：

X＝＝（丁一＋▲）一’1／2Y… ………（2・）

Grits corresponding to the points between B and C pass above the point A at the lower height than Y（i．e． the height of the point H）， and grits correspohdi、ng to the points out of B and C

pass above the point A at the higher height than r．・．

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●

(9)

■ On the Grinding Geometry for the Random Spacing of Abrasive Grains on Wheel Surface Consider generally the grits， tips of which pass in the plane z＝z represented by EFPQ． As shown in Fig．7（b）， the point H is passed by the cutting edge PR of the grit， the tip of which passes the point P， which is above the point A at the loweごheight（夕）than（r）for the point H． Eand F corresponding to the point P are the grain projections on to the work surface of the grits， cutting edges of which pasg the point H． Grits corresponding to the poillts between and out of E and F pass above the point A at the lower and higher height than Y， respectively． Therefore， the closed curve BEDFCF’D’E’in Fig・7（c）gives the locus of grain projections on to the work surface of the grits， cutting edges of which pass the point H．． And hence the probability P（Y）， in which the vertical height of the actually finishad work surface from the idea1Jy finished work surface at an arbitrary point A on a workpiece exceedsγ， is identical with the probability， in which there is no grain prolection on to the work surface in this clo8ed curve． The expression for a quarter of the closeユcurve BED in Fig．7（c）is as follows．：

ll；惑蕊司ノ当一・……・（22）

The mean area in the interior of the cl．osed curve（A）is represented as follows：

（A）互一4∫1・飼・33・（÷＋万1訂一1／2r・ノ・ ………・一（23）

whereプ＝2tanφ．

Variations of X，2max（＝γtanφ）， and／1 with Y are given in Fig．8for the grinding conditions of Table 2，の

・…mi・gth・t・25−120・i．・． r−v3．46・ミ

N・w，as・・m・th・t th・・e areN・P・i・tsa・士；・d・min∼P ，

慧：’iltl蕊ご1；b三蒜：1’罵1惣：1”1・

as fo110ws： P＝（1−A／ノ11）M ・・・・・・・・・・・・… （24） Representing the mean area occupiel by a singl e po桓t by／lo（・＝ノ11／2V1）， the probability P is found to be P＝exp（−A／ノlo）・・・・・・・・・・・・… （25） Then， in order to obtain the expression for the above ment ioned probability P（Y）， substitute P，ノ1 and∠40 in this general equation（25）by P（r），ノ1（see Equation（23）） Y−．戸 Fig． 8 へ VariatiOn Of叉，2疏x and／1 withγfor Grinding Conditions of Table 2， As− suming that 2tP ＝120°i．e．プ＝3：46 and the mean area occupiθd by a single grain projection on to the work surface助2（see Equation （11）），respgctively・ Thus P（Y）is found to be

（A） P（Y）一・xp（一一4／w・2）一・xp（−KY3／2） …一……・・（26）

where

（A）K−・・33（B＋オ％）−1ノ】ξエー………・（27）

It may be useful to derive the expression for th6‘‘Roughness Distribution Curve”∫（Y）， where ∫（Y）dr rneans the total area of the surface elements at the height be細een Y and Y十dY from the base of the profile for the ground surface． It is evident that∫（Y）is generally the derivative of−P（Y）， i．e． ∫（Y）＝−d」P（Y）／d】r ’ ……・・・… …（28） Hence∫（Y）is re．presented as follows：＿』tt 、 U7

(10)

昭和32年7月

_{山梨大学工学部研究報告}

第 8 号 ■ 3 ● 2

〉

潔と1 ／ α904ト（Coγ’e‘pon4i鴨g t㎞1元） 0．午81（（。r7e＄P。ndi日rtb Y●） 0 α’ o：2 03 0仔 05 06 0．7 08 十f（Y） Fig． 9 Roughness Distribution Curve in Non−Dimen一副onal Expression for Abrasive Grains with Sharp Tips Arranged Exactly on Cylindrical Surface of Whee1 〃 1．O e．9 e．8 to・7 1トξα6 e．5

T

as _潤D4 ＞三、o．3

巽a2

eu O

Fig．10

●

o

ぷ

遥2

毒

ぶ、

C

0 5 ∫0 1オ c一垢柄苫ら Variation ofκ，1！控ノ3， Yo and ht with Cfor Grinding Conditions of Table 2， When Abrasive Grains are Assumed to「 Have Sharp Tips and夕＝3．46 、一

H

ﾊ 2

子 M’

_ρY

ム M2ro鴻亨（Y両γ）

y

D’ Fig．11 Diagram of Portion ill Section Ol 一’ 02 for the Computation of P（Y） ●

（A）∫（r）一」』緩）一喜κγ・ノ・exp（一五Y・1・）一・・…一・（29）

where K is given in Equation（27）． The curve i孕Fig・gshows the variation of∫（Y）with r， represented as a non−diMensional expression．∫（Y）has the maximum value， when Y＝γo． ht gives the lnean height of the surface profile from its base（y・＝O）．γo and ht are represented as follows：

（A）r・一・・396（丁＋☆）1／3（i，Z，i）213 …・・…一…（3・）

（A）n・一・・744C；一＋☆）113（毒）2／3 ・・一…・…（3・）

ぺ Variations of K，1／K2／3，γo and ht with C are given in Fig．10 for・the grinding conditions of Table 2， assuming thatプ＝3．46． It is note worthy that the expressioR for P（γ）is obtained also from the consideration of the grain projection on to the section O1−02． See Fig．11， showing the section O1−02． It is evident that P（Y）is identical with the probability， in which there is・no grain projection on to the section

118

(11)

On the Grinding Geometry for the Random Spacing of Abrasive Grains on Wheel Surface 01−02in a triangle HDD’． The probability dP（r）， in which there is no grain projection in a trapezoid LMM’L，＠ts area is 2tanφ（Y−y）dy）， is represented as follows：

（A）dP（Y）−exp〔−2tanφ￡拳テタ）dy〕 ……・……・・（32）

where u2／2 is used in place of u2 as the mean area occupied by a single grain projection on to the section O1−02， because arc BH as weU as arc HC in Fig．7（a）， showing the paths of grits， are together to． consider for the computation of the surface roughness． Therefore， P（r）is represented as followS，：

（A）P（Y）−exp卜∫：2（≒考誓an中d夕〕・…・………・（33）

Of course， Equation（33）coincides with Equation（26）・This computing method for deriving Equation（33）will be used for the computation of the grain depth of cut in the later paragraph・ In the next place， assume that the shape of a grain tip is a circular cone with a spherica1晦， radiug of which is k， and that all the cutting edge， actually concerning the grinding action， exists in this spherica1． portion． We define“mean radius of grain tips”ωas follows：（B） le ＝｛（ゐ1／2）mean｝2 …・・…・……（34） The similar consideration and the similar computing method to thats for deriving Equation （26）are used in this case． Assuming that grain tips are arranged exactly on the cylindrical surfac6 0f the wheel，．‘‘Abbott’s Curve”P（Y）and“Roughnesg Distribution Curve”f（Y）are represented as follows：（B） P（Y）＝exp（−K’y） ’ ． ……・……・・（35） x

（B）ノ（Y）一」盤）−Kf・xp（−K’Y）． …・…・……・（36）

where

（B）K’−4・45万・／・（B＋−DIEb−）一’112−！7tt −………・（37）

●

らと

4

｛c・・…r・ndi・t Tt） 0、e．1α2α30．仔050．60708a．el，O k．・fm Fig．12 Roughness Distribution Curve in Non− Dimensional Expression for Abrasive Grains with Spherical Tips Arranged Exactly on Cylindrical Surface of Wheel

119

e．2

叉卯

‘ 1定 τ m、、e O．03 0．05 0， 08

0 5 ∫o一な・

c一掲柄・ Fig．13 Variation Of ht with C for Grinding Conditions of Table 2， When Abrasive Grains are As8umed to Have Spherical Tips andゐ・＝0．C3，0．05 and O．08mm ’

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昭和32年7月

_{山梨大学工学部研究報告噺}

’第 8 号 The curve in Fig・12 shows the variation of∫（｝つwith y，・represented as a non−dimensional expression． The mean height of the surface profile from its base（ht）is represented as follows：

（B）万・一「一・・225“，／、（十＋芸万）1／2☆ ………（38）

Variation of ht with C is given in Fig．13， for the grinding conditions of Table 2， assuming that k＝0．03，0．05and O．08mm． The roughness distribution curve for the spherical tip represented by Equation（36）and shown in Fig・12 does not agree with axperimental data as betow rnentioned， and so it will be岨reasonable to assume a grit to have a spher’奄モ≠戟@tip， for the computation of surface roughness． Therefore， in the next paragraph the computation of surface roughness for a spherical tip is omitted． Now， consider for example a cylindrical grinding， where the wheel width is B mm and・the feed of work is∫mm per revolution． A poillt on the work surface is groundプtimes during its passing in front of the grinding wheel， where ノ＝＝五∠プ』・・・・・・・・・・・・… （39） In this case，（ヲis useヨin place of C in the equations for P（Y）anユf（Y）， because it is assurned that there is Ilo relativ6 displacement between the wheel and the work． In any similar case to this， the silnilar substitution is used． 5．Surfらce Roughness Correspo皿｛1ing to the I）istribution i皿Depth f（）rthe Position of Grain Tips on the Wheel Face In the l ast paragraph the“Roughness Distribution Curve”∫（Y）is represented by Equation（29） for the grain tip shape of a circular cone with a sharp tip・Its shape is asymmetric， and leaves a．trail not towards the inner part but towards the surface of the work， as．shown in Fig．9．且owever， Honda states in reference（7）that the roughness distribution curve of an actually fi・i・h・d w・・k・u・face f・・th・g血di・g・tP・・ati・n h・・th・《・）R，、t・・ y・f・・D・・t・M・“ ．shape of the normal distribution， measuring lnany surface 一工 A ●_{profiles obtained by a reughness tester． The difference of} the shape between the experimental curve and the cQmputed curve will be Qwing to the fact thqt all the actual grain tips are arranged not exactly on the cylindrical surface of the wheel． Therefore， it will be useflll to consider the distri−− bution in depth for the position of grain tips on the wheel face． This di8tribution is due to the dressing of the wheel by diamond， the self−dressing or the cleavage of abrasive grains in the grinding operation， aud built−up edges produced on certain abrasive grains． When built−−up edges are produced on grits， tips of built−up edges are regarded as their grain tiPS． Now， assume the distribution in depth for the position of grain tips on the wheel face as follows：（a） Rectangular Distribution， as shown in Fig．14（a）（b） Triangular Distribution．， as shown in Fig．14（b）（c） Parabolic Distribution， as shown in Fig．14（c） where e．g． the rectangular distribution means that from 120 畳エ

8u

O 工 8酪、1｛《b， Tv；a”t『“1▲■ D輌stv；b“tion Suvfice ef Wheel i−・

苫主 s・・f…

°2㌫・㌔：； °珊“1

《c）脇訪ot輌‘D輌5t7三‘ut；o” 春一合工

a）

0 2111即 Fig． 14 Types of Distribution in Depth for Position of Grain Tips on

Wheel Surface．（a）Rectangular

Distribution，（b）Triangular

Distribution and （c）Parabolic Distribution S“tface 。f Wheel

(13)

On the Grinding Geometry for the Random Spacing of Abrasive Grains on Wheel Surface tlle wlleel face to the plane iII the depth of Ho there are grain tips of the equal number per sq mm in layers of the equal thickness parallel to the wheel face． Even if grain tips are arrallged exactly on the cylindrical surface of the whee1， the following equations remain unchanged， when by certain causes there is produced a distribution in depth for grinding scratches corresponding to the above mentioned assumptions of the distribution in depth for grain tips．． Now， defining the grain projection on to the work surface of a grit as a point， at which the normal distance between its tip and the ideally finished work surface takes the minimum value during its passing for the work surface， there is produced a three−dimensional distribution of grain projections on to the work surface． It is evidenC that the probability P（Y）， in which the verical height of the actually finished work surface from the ideally finished work surface at an arbitrary point on a workpiece exceeds y， is identical with the probability， in which there is no grain _{projection on to the work surface in a certain mean volume W（r， Ho）． In order to comp ute the} mean volume Pγ（Y，Ho）， conside；the layer of thickness dH at the height H fro m・the ideally finished work surface． If there is no grain‘projection in the interior of the closed curve in this layer correspolldillg to the closed curve BEDFCF’D’E’in Fig．7（c）， the height of the actually finished work surface from the lde．ally finished work surface exceedsγ． Refering to Equation（23）， the area of this closed curve F（Y，H）is represented as a function Qf Y and H as follows；

（A）F（Y・H）一・・33・（十＋」㌃一）−1／2（γ一の・… 一・一…（4・）

Then，1η（Y， Ho）is found to be if・Y＜H・W（Y・・Hi・）＝＝∫JF（r・H）dH／・・・・・・・・・・・・・… （41） if・Y≧H・W（Y・H・）一∫ご゜F（Y，H）dH S The mean volume occupied by a single grain projection PVo（H）is a function of the depth from the wheel face（H）． Hence P（Y）is represented as follows：

ifγ＜U・P（Y）一・・p〔一∫滞鵠Σd⇒／

i͡P（Y）−6xp〔一∬綴考）dHU ’’’”◆’命’’’’’”（42）

（a） Rectangular Distribution in Depth for Grain Tips For the rectangular distribution in depth for the position of grain、tips on the wheel face， as shown in Eig・14（a）， Wo（H）and W（Y， Ho）are、represented as follows：

vv・（・・）−H・一完一 ……・…・・…（43）

lif r＜師（Y・昨・・533・（b“rBtl）−1／2Y・／“t ／

（A）撃奄?@．≧H。 fv（卿一・．533・C；．・＋吉）一・ノ・｛Y…一（Y−H。）…｝1”◆”◆°°’’’’’”（44） Therefore，“AbbJtt s Curve”P（Y）and‘‘Roughness Distribution Curve”ノ（Y） are represented as follow8：

（A）1：1ぱ畿ご蕊灘一（Y−U。），ノ2｝〕｝ ………・・一（45）

and 1 ＞

121

●

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昭和32年7月

_{山梨大学工学部研究報告}

第 8 号（A）

where

（A） if Y＜Ho if Y≧Ho ゾ（Y）一÷κ1γ3／2exp〔−K・Y5／2〕 f（Y）一三K・｛γ3／2−一（Y−H・）3／2｝exp〔−K・｛Y・1・一（Y−H・り5・2｝〕 0．533

Kl＝

Ho （」＿＋＿LD _{Dw）−1／2}

_γcr

》

v ‘ 〉＼

l

l， … ■■・… t■・…（46） H．☆2f‘ H、 9 tt・

H5・o

0 α’ 0L2 α3 0．4 α∫ 0‘ 07 f（Y）換

Fig．15 R餌ghness Dist ribution Curve of

Grindihg Whee1善 30 H（C＝1．361／mm2）for Grinding Conditions of Table 2， When Abra−

sive Grains are Assumed to Have Sharp

Tips（r＝3．46）and Distribution in Depth for Grain Tips are Assumed to be Rectangular Type ……… i47） Ho ”’4戸

H●＝2A

tA

o‘o．s l．o ts

f（Y）一〃・

Fi琴・16 Roughness Distribution Curve of

t_{Grinding Wheel1着 SO P（C＝14．61／mm2）for} Grinding Conditions of Table 2， When Abra− sive Grains are Assumed to Have sharp Tips（プ＝3．46）and Distribution in Depth for Grain Tips are Assumed to be Rect．angular Type When the grinding wheels烙 30 H（C＝1．361／mm2）and善 80 P（C＝14．61／mm2）are used， variations ofノ（y）with y are shown in Fig．15 and Fig．16， respectiv61y， for the grinding conditions of Table 2，assuming thatプ＝3・46 and Ho＝1，2and 4μ（1μ＝0．OOlmm）． The curve for Ho＝・O in Fig．15 corresponds to the roughness distribution curv6 for the grain tips arranged exactly on th● cylindrical surface of the whee1．（b） Tria4gular Distribution in Depth for Grain Tips For the triangular distribution in depth for the position of grain tips on the wheel face， as showll in Fig・14（b）， the mean volume occupied by a single grain projection Wo（H）is represented as fo110ws：

W・（H）一一霊…完「一…・・…・（48）

4‘_{`bbott’s Curve”P（Y）and“Roughness Distribution Curve”∫（γ）are represented as follows：} （A）

P隠㌫き㌫（γ＿＋三五。）｝〕｝・…・・………（49；

122

(15)

On the Grinding Geometry for the Random Spacing of Abrasive Grains on Wheel Surface ●

and

Wllere

（A）κ・一゜

ﾝ。14（丁＋−i’bo−）−1／2γξ「 Whep the grinding wheels＃ 30 H（C＝1361／mm2）and S SO of∫（Y）withγare shown in Fig．17 and Fig．18， respectively， Table 2， assuming thatグ＝346 and互o＝1，2and 4μ・ 6 s ＞−2 H。一 Ho ・＝2μ Oo∫ρzρ3．α争・・o．so．6α7 f（Y）一〃

Fig．17 Roughness Distribution Curve of

G；inding Wheel ti 30 H（C＝1．361／mm2）for Grinding Conditions of Table 2， When Abra−

sive Grains are Assumed to Have Sharp

Tips（グ＝3．46）and D三stribution in Depth ・for Grain Tips are Assumed to be Triapgular Type

〉

／／・・・・・・・・・・・・…@（50）・・・・・・・・・・・・…@（51） P（C＝＝14．61／mm2） are used for the

grinding

variations ， conditions of H，＝ eN 凡宕2！・ H。ロ∫！4 O e．5 to ／．T f（Yl一万

Fig．18 Roughness Distribution Curve of

Grinding Whee1拳 SO P（C＝14．61／mm2）・for Grinding ConditiohS of Table 2， When Abra−

sive Grains are Assumed to Have Sharp

Tips’（プ＝3．46）and Distribution in Depth for Grain Tips are Assumed to be Tτiangular．tt Type （c） Parabolic Distribut三〇n in Depth for Grain Tips For the parabolic distribution in depth for the position of grain tips on the wheel face， as shown） in Fig・14（c）， the mean volume occupied by a single grain projection酩）（H）is tepresented as fo110ws：

囎）…霊音一 ……・・…・…（52）

‘‘Abbott’s Curve”P（｝つand‘‘Roughness Distributien Curve”ノ（y）are represented as follows：（ fif γ＜Ho 2P（y）＝exp〔−K3 yg／2〕 `）l

戟@if Y≧H。−P（巧…・〔一五・｛Y・…一（r疏γ＋誓銚一醐〕卜・・……・…一・（53）

123

(16)

昭和32年7月

_{山梨大学工学部研究報告}

第

8 1号

and

if Y＜Ho （A）if｝ア≧Ho ∫①弓κ・Y7・・e・p←K・y・ノ・〕

∫（汗÷κ・櫛≦y−H・）3・・（Y2＋号H・Y＋誓珊｝

×・XP〔−K・｛y・ノ・一（悟暢日・Y＋誓瑚（Y−H・）5”2｝〕｜ L．…＿……（54） ∫

where

（A）K・一゜

T（う一＋芸ω）’1／2γ茅γ ・・……一（55）

When the grinding wheel s襲30 H（C＝1．361／Mm2）and轄80 P（C＝14・61／mm2）are used・variationg of∫（y）withγare shown in Fig．1g and Fig．20， respectively， for the grinding conditions of ’Table 2， assuming that夕＝3．46 and Ho＝1，2and 4μ・ 6

＞

6 ’s

4

3 2 lte e 2A H，F’1k O O．1 ρ」2 0．3 424」 05 06 07 チ（Yい〃

F三9．19 Roughness Distribution Curve of

Grinding Whee1630 H（C＝1．361／mm2）for

Grinding Conditions of Table 2， When Abra−

sive Grains are A6sumed to Have Sharp

Tips（r＝＝3．46）and Distribution in Depth for Grain Tips are Assumed to be Par・abolic Type 1

＞

Ho”4M

H．宙”

01 αnj． t．0 1．s

f‘Y）一匁

Fig．20 Roughness Distribution Curve of

Grinding Whee1襲 80 p（C＝14．61／mm2）for G血ding C皿ditions qf Table 2， When Abra−

sive Grains are Assumed to Have Sharp

Tip9（r＝3．46）and Distribution in Depth for

Grain Tips are Assumed to be Parabolic

Type

Table 4 Variation of．Yo with」Uo for the Grinding Wheel善30 H（C・＝1．361／Mm2） and for the Grindlng Conditions of Table 2 Distribution in Depth for Grain Tips Rectangul ar Distribution Triangular Distribution Parabolic Distribution 4 2．15μ 3．16 3．73 Values of Hoμ 2 ． 1 1．63μ 2．03 2．14 1．09μ 1．2フ 1．30 0 0．52μ 0．52 0．52

124

(17)

On the Grinding Geometry for the Randorn・Spacing of Abrasive Grains on Wheel Surface 、 Table 5 Variation of Yo with Ho for the Grinding Whee1轄 80 P（C＝14．61／mm2） a’ nd for the Grinding Conditions of Table 2 Distribution in Depth ， Values of Hoμ for Grain Tips 4 2 1 0 Rectangular Distribution O．83μ 0．63μ 0．48μ 0．11μ Triangular Distribution 1．61 1．08 0．73 0．11 Parabolic Distribution 2．20 1．39 0．87 0，11 Now， Table 4 and Table 5 show variations of Yo with Ho， for the grihding wheels 630 H（C＝1．36 1／Mm2）and勢SO P（C＝14．61／Mm2）， respectively， whereノ《y）has the maximgm value when Y＝ Yo． Ho＝O corresponds to the grain tips arranged exactlY on the cylindrical 8urface of the wheel． lt is note worthy that the value of Ho is very important for determining the surface roughnegs．

6．Non−Dimensional Expression fDr Roughness Distribution Curve

In order to investigate the general 8hape of‘‘Roughness Distribution・Curve，’for each assumed −type of the distribution in depth for grain tips， it will be useful to derive non−dimensional expres− sions for the roughness distribution curve． Now， non−dimensional quantitieS z， C1，C2 and C3 are defined as follows： z＝Y／Ho， C1 ＝一 Ki Ho5／2t C2 ＝＝ K2Ho712 and C3＝KsElog／2 ・・………・…（56） Then， equations in the last paragraph take simple forms．．（a） Rectangular Listribution in Depth for Grain Tips

（A）｛lll：1㌶こ認蕊｛1：1：1（．一、）、／，｝〕｝ …一・……（57）

and

（A）

o：：：i≡：蕊：ll：：≡：：二pl＿一＿〕｝一・………（53）

where

（A）C・一・・533 H・Fノ・（一；一＋晶一1／2撃 ……・…・・…（59）

〈b） Triangular Distribution in Depth for Grain Tips （A）

B1：：蕊：：㌫一、）・ノ・（．＋号）｝〕｝・…一…・（6・）

and

斗：…㌫ξ1｛∴一一・（6・）

where

（A）C・一一・・3・4H・3・・（十＋☆）−1／2γ…プ・・…・…・一（62）

〈c） Parabol ic Distributign in Depth for Grain Tips 125

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昭和32年7月

_{山梨大学工学部研究報告}

第 8、 _号（A）

o：ll二1：：：：：：1：：：：㌫ぽ＋9．＋乎＿〕｝…・・一・…（63）

and

（A）1：：：：ll：1二：1：：籔：瓢：＋9．＋♀）｝い＿一

、 ×・・p〔一C・｛・g／・一（・・＋；・＋晋）（・一一・）・ノ・｝〕l Wllere

（A）C・一・・2・3・…3・・（−b＋十）−112字 ………・…・・（65）

When g血ding wheels＃30 H（C＝1．361／mm2），＃46 L（C＝5．001／mm‘2） ’ and＃80 P（C＝14．61／mm2＞ are used， variations of C1， C2 and C3 with 」Uo are shown in Fig．21， Fj9．22 and Fig．23夕 respectivel y， for the grinding conditions of Table 2， assllming that r・＝3．46．

心

∫0 o．1

Fig．21

／0

M

91

o∫ ／ 2 、3 4 ’Ho−！4 Variation of Cl with Ho for Grinding Conditions of Table 2， Whenア＝3．46 ／． 2 3

₄

− ぱ80マ、多、、んφ ` ・んム年もレぱ／0 @1、ごα’ 1 5．bO ヱｽ

崇30い

_i

3 1．36漏C多・1

_{2 ．3 4し}

H。∼ノ4 Fig．22 Variation of C2 with Ho for Grind ing Conditions of Table 2， When r＝＝3．46

Fig．23

Variation of C3 with Ho for Grinding Conditions of Table 2， When r＝3．46

H。−A

Z26

(19)

● On the Grinding Gaometry for the Random Spacing of Abras ive Grains on Wheel Surface

＞f

ll

N

2，e c，＝t C，’t／o 0 05 t．0 1．5 20 2．5 H．f｛Y）

Fig．24 Roughness Distribution Curve in

Non。，Dimensional Exp宇ession Corresponding to Rectangular Type of Distribution in Depth for Grain Tip8

25

2 ：、 ’．｛〉→エ● 1、 G・’ 田．￡0 ρgo・8

N

_ら法1o o 0，470．、 0

₀₅

￡o ’5 ₂₀

_2s

H。f（Y）

Fig．25 Roughness Distribution Curve in

Non−Dimensional Expression Corresponding

to Triangular Type of Distribution in Depth for Grain Tips If the shape of the roughness distribution curv6 for actual grinding Operat ions coincides with that of the normal distribution curve as stated in reference（7）， which type of the distribution in depth for the positien of grain tips Qn the wheel face agrees with it？Assumi㎎ that C1， C2 and C3 are l and 10， Fig．24， Fig．25 and

Fig．26 show the shape of the’computed roUghness

distribution curve for each type． Comparing the degree of symmetry for these curves， it will be reaSonable to consider that actual grinding wheels have a type of the distribution in d6pth for the position of grain tips similar to the triangular or parabolic type．

7．Width of Undeformed Chips

The distribution in depth for the position of grain tips on the wheel face does not play an important role in determining the chip thicknes and the chip width， and so we assume that grain tips are arranged exactly en the cyl indrical surface of the whee1， in order to derive the following expression8 for tlle c See the section O1 一一 020f the work in Fig．3．・finished、 surface but just to the height． smallest height of the grain projections on to the section O1−02 till this instant．＞lr“ _（3＝1 G蓄ノρ

O as LO LS 202．SユO

Hof《Y） Fig．26 Roughness D．istribution Curve

in Non−Dimensional Expression

Corresponding to Parabolic Typ● of Distribution in Depth for Grain Tips hip width and the gra’i n depth of cut． The work sutface is ground npt to the finally of y by the whee1， where y is exactly defined as the section O1 一一 02 for the grits， which cut the The profile of the section Ol 一一 020f the work at this instant is

127

(20)

‘

昭和32年7月

_{山梨大学工学部研究報告}

_第

8 号 shown in Fig．27 assuming that the grain tip shape is a circular cone with a l sharp tip． Consider the“Abbott’s Ctlrve”P（y， y）for thig profile， where P（y．｝っcorresponds to the total length of the distances marked with arrows in Fig．27． The computing method deriving Equation ・i・g1・grai・p・・」・・ti・… t。 th。、e、ti。。 O、＿02 Fig・27 P・°fi1・・f Secti・n O1−0・・f W・・k・ Which is Ground Just to Height（y） is to use in place of u2／2 in Equati’on（33）． For the general foζm of the grain tip shape， i．e．ξ＝ξ（η）（refer to Fig．6）， P（y，｝りis repre− sented as follows：

P（y・Y）＝＝exp−∫嵩ξ；箭・4・〕 …一一（66）

whereξ（Y−T）is the value ofξforη＝＝Y−〔and｛u（y，ア）｝2 is the mean area occupied by a single grain projection on to the section・Ol−02 at the height of（γ十T）． Now， assume the grain tip shape to be a circular cone with a sharp tip of mean grain tip angle 2中。r Then， P（夕， Y）i8 found to be

（A）P（元・Y）一・・p〔一∫：2緩∂莞竿4・一・…一・（67）

Fig．28 show the shape of cut in the section Ol−02 by a single grit， of which the grain pro一 lection。n to the section O1−021ies at the hei− 、ght of y from the ideally finished work surfaceo It is note worthy that the shape of cut at the height of y is identical with the shape of the chip 8ection perpendicular to the grinding direction at

the same height（y）・In Fig・28 the distance

marked by b is the chip width， andろ1 and b2 are the chip widthg in the right side and in the left

，id。 f。・m it・tip． 1㎡・・ll7 Fi・i・あ・d．W・・k 5・・f・・e

b＝b1十b2 ………（68） Fig・28 Chip Width at Height（ツ） It is easy to prove that the probability P（bl）， in which the chip width in the right sidθexceeds ろ1＝γtanφ， is eqnal to P（夕，T）・ For the case， in which the mean area occupied by a single grain prolection on to the section Ol−02（u2）doe3 not depend upon its height， as for a plunge grinding with con3tant infeed（refer to Equation〔16〕）， P（y，ア）is found to be

（A）P（y・・）一・・p〔一∫：≡寡tanφd・一・・p−…：： ……・・一・（69）・

where夕＝2tanφ． Therefore，

（A）P（b・）−exp〔一三劉

……… i7・） In order to obtain a non−dimensional expression for Equation〔70〕，zt i8 defined as follows

（A）・F蕊 …・一・…（・・）

：

128

(21)

On the Grinding Ge，o血etry’fσr the Random 5pacing of Abrasive Grains on Wheel Surface ’ Then， the probability P（zl）， in which zl exceeds z1， is repres3nted as follows：

（A）P（・・）一・XP（一芽） …・…一…（72）

Representing by f（Z1）dzl the probability in which the chip width in the right 8ide corresponding to the value between 21 and z1十dzi is produced， f（Zl）is the derivative of−P（zl）， and so （A）f（・・）一一d1t

_{y！：1（z’）一・・exp（一丁1三） …………・・く73）}

Variation of f（Zl）with zl is shown in Fig・29・ The non−dimensional exPression for the distribution of the chip width b＝b1十b2 is represented as fo110ws：（A）∫（・）㏄∫；｛∫；’22・・exp（一一9；三）d・1｝・・exp（−f’2）d・・．……一…（74）

where

・一』1ら，・・一そ，亀・and…多，％ …一・一・（75）

三匡

kL 仔， 6 ∫ 夢言1ミ3

二、

0 α， α2 ￠3 0．皇 05 06 07 f《2∂ ， Fig．29 Size Distribution for Right Half of Chip Width（bl）in Non−Diまnensional

Expres・i・n o

6し∫ α2 0．3 ψ三↓9L O．S’

By the numerical calculation，ゾ（z）is sh・wn f‘2）

kノへ i・Fig・30・whi・h e・p1・i・s th6 fact that the「e Fig．30 size Di・t・ibuti・n・f chip Widthジコ are produced various ch｝ps of different width （b）in Non−Dimensional Expression in actual grinding operations．． Now， the lnean chip width （b）for all chips at the height ofツis generally ropresented as a2 fvmction of y as follows：

了一∫rτω｛−dP砦）｝d・ ………一（76）

Assuming that u2 dees not depend upon y， b is represented for each gτain tip shape as follows：（A） b＝1・253グ1’2u ………（77a）（B） b＝＝2．576ゐ1／eu雪’3 ………（77b） where（A）corregponds to a circular cone with a sharp tip of重nean gTain tip angle・2φ（r＝tan 2 φ）and（B）corresponds to a circular cone wilth a spherical tip（mean radius of grain tipた）． The width of cut by a grit， i． e． the scratch width by this grit in the section O1−02 becomes smaUer by degrees with following grits， which pass the section O1−02 at the lower height， and 129 ．・

On the Grinding Geometry for the Random Spacing of Abrasive Grains on Wheel Surface 利用統計を見る