Dividing (Workshop Practice)

' W 02K5H0P PRACIICESERIESfrom SpeticlInjeres,ModelBooks 1 Hgrdenfng,Fempering & 13.Workshop Draw/ng TubolCoin )4.A/l ...

Author: Harold Hall

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W 02K5H0P PRACIICESERIESfrom SpeticlInjeres,ModelBooks 1 Hgrdenfng,Fempering &

13.Workshop Draw/ng TubolCoin )4.A/l a/tfng smo//workshop Too/s gonBroy 15workholyfng infhe l-afhe

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unitand the equivalentofa commercialsem i-universolheod. AuthorHorol d Hallhasestablished his repotation as a mentorto tyro engineers throughthe pogesofModelEngineers'Workshopofwhich hewastheeditorforo berofyearsond through hisothertwo books in fhisseries,tathework- A Conw lete nu Comurseandpllliog- A Complete Course.

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utotayIeevxepleth atwillbeunderstoodeasilybyamodelenqineer.Dividingeqoipmentis relativel nsive,sotwofully-detaileddesignsareincludediordividingheads:obasic

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dividing headanda rotarytabl e.The mathematicalaspectsofdi viding arealso covered b

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ItcoversthesubiectofDividing,dealing withthe manymethodsthatcanbe adopted: from sim ple opplicationswithoutspecialised equipmentto the use ofo sem i-universal

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engineerswillbe di scouraged ond willturn el sewhere fortheirnextmodel.This need not be so,forthe principles underlying georcutting ond manyotheraspectsofengineering Where on accurate division ofcirclesis required ore exploined in depth in thisbook.

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cbapter 1

Engl and

An introducti onto oivi ding

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chapter2 12 The Machinery.Discussesthe majoritemsofequipment,Di viding heads,Rotarytabl es,

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Frstpublishedby SpeciallnterestModelBooksLtd.2005

lndexersetc. ,

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C hapter3 TheMethods

The rightofHarpld Hallto be identi fi ed astheAuthorofthis work has been

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as A cs te or f te 1d 98 b8 vhimi naccordancewi ththeCopyri ght,Desi gnsandPatentsRi ghts

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AlIrights reserved.NO partOfthis book may be reproduced in anyform by print,

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tten permission from the ?hotography,microfilm orany othermeansw ithoutwri publisher.

18

Explainshow to use the equipmentin chapterTwoto achievethe required

r esuj txsoexpl ai nsthoughhowevencompl exdi vi di ngcanbeachi evedusi ngnoother .

equipmentthanthatnormallyfoundinthe rkshop,typicall ytheIatheandmilingmachine. - ro

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32

The Mathematics.W hiIstthetablesprovi ded wi th adi viding head,orthose published in thisbook,willprovide the i nformation formostappli cations,occasionally recourse to cajculation may be necessary. This chapter expl ains the process,fortunately,the mathematics is notcomplex.

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Chapter5

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Holeson aPitchCircl e Diameter(PCD)Expl ainsthe mathematicsforplacing holesona

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1SBN 1.85486-238-3

PCD using a m illing m achine,s X and Y table movements and theircalibrated Ieadscrew dials.Al so shows how using a computerspread sheetprogram can be used to Iimi t considerably the work i nvolved.

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Ch apter6 e Dividing Devices.Some sim ple devices thatw illenable dividi 5n0g Shop M ade Sim pl

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activi tieswithoutthe needforadi viding head.

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Shop Made BasicDivi ding Head.A simple di vidingheadthatiseasyto construct,andfor most,willbe adequate foralIthe divi ding tasks required in the workshop.

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6hOP Vade FullFunction Dividinge ead. A dividing head thatis more adaptable than a

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nOtovercomplexto make andwould bea verysatisfyi ng projectto complete.

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ts differentIength lines is m ade easy wi th shop Made uining Tool.Engravi ng a dialwi tb i

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thisIining tool.ltis also aninteresting projecttoconstruct. 1c

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whilstin the homeworkshop,di viding is ThebookcanthereforebeIoosel ydi vided requentoperation, there are though n otaf many ways ofcarrying out the task. This is

d th rime Num bers.Prime num bers an eiruses explained and listed.

chapter 11 98 dingheadratiosanddi visi onpla teholenumbers Tabl es.Tablesforawide range ofdivi rrence.

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into three sections. 1.The machinery

qui t eties diff erent o theitems major kshop i vi wher ert ound wilwor linvar iably 2. 3'The methods a ct hema thetma icsexplanations ofthe angul ar InT addi tion o tt he be the dom ain ofthe lathe anj-oct .

areincl uded.Theseshouldmaketheneedforcalcul ationarareoccu

items,thatofthe m illing machine,with the

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equipmentand how i tisused,the bookalso

s haper c h i p i n n n f e c . n c ued s s ot m e d es i g f oiatr te s t h auin j a i d t j i ) l gp ,, m hg j Oj wi n e gi v m ea r a 'c l aw j ne ba a es ne ls ae rr v i e sd nt oj te ti b t j e j ol s m ae j d u e st i n g j v ho et w jj 0 r o k sn o hs o os g s s ri o y lm r l a o wt j b Ja gt sc m 4 1 R t on thlV

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drillingmachine. l tisthough,thewi derange with some manufacturing tips where ofaccessoriesandmethodsthatm akethe task a daunting One fOr many Workshop owners. ln additi onto choi ce ofm ethod,setting

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considered desirable. Di vidingwillalmostcertainlybe a sm all proportion ofworkshop activi ty,butwi thout the required provisions,andunderstanding,

i tup correctlywilloftenrequirerecourseto someprojectswilleitherbeimpossible,very some m athematics.These,mathematically, di #i cul torofa very inferi orquali ty. are fortunately quite sim ple, but som e understanding oftheirpurpose isessential. Harold HallJanuary 2005

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simplemethodsarenotjustlimi tedtolower numbers,insomeapplications,dialstypi call y,evenhighernumberscanbeachi eved wi thoutany expensebei ng incurred.

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f orthelathespindl etocontrolangleofro- thewayto proceed.Because ofthis,detation Today,greater availability and reduced price ofm odern accessories have made thi s less necessary.There are still instances where i t is wortb consi dering. Carrying outsome work on a component setting up the dividing head on the milling machine and transferring the com ponent to this. lfdi viding is to become an im portant activi ty then acquiring a semi-universaldivi ding head,com merci ally m ade Photo 2 ormade in the workshop,should be con-

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si gnsfortwo shop m ade itemsare included in Chapters 7 and 8. Using a rotary table Photo 3 is al so a possibil ityand whilstthis i tem ofequipment is prim aril y intended forthe machining of

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W hen placing holes on a PCD,working outthe X and Y co-ordinates of each hol e and using these to place the holes using the milling m achine table dialsasshown in Photo 4 can give a very accurate result. This can be padicularly usefuli frequiring a di viding plate foruse on the dividing head

sidered.Asa dividing head willlikely get especi allyiffora one offapplicati on.

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Using the lathe spindl e. Using a divi ding head. Using a rotarytable. U sing an indexer. Di viding wi th no speci alequipment. The I aqerw i llbe dealtwithwhendi scussing

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be in the chuck s backplate.The most convenient point, probably the C'Omm on isto use the gearinthe back gear changewheelsquadrant,willovercome the assem blythatis rigidlyfi tted to the spi ndl e, Iim i tation.Even where the lathe has a

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' /.Indexing using thelathe'sBull

Ifyourlathe does nothave a suitable movementduringthemachini ngoperation. btlllwheelthenmountingagearonthe rear Itisthoughqui teadequateformanytasks

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q T ' E( ... E ( k( . y.; 7 . ' .

.

---

-

. ..

yE

.

'

?y r yyy , . , yj yy; , ;y, y . , y,r ,, , . , ' q l t ,,yyy; chain is setup fordi ffering ratios, rather Iike the changewheels used on a lathe for screw cuding.

vedical ly,Photo 4,orany anglein between. even wi th a wide range ofdi viding pfates

much widerrange ofdivi sions,typically all

The gear chain also has another

The firstpointto note is thatthe rotating

therewiltbemanydivisionsthatcannotbe

Valuesbetween2and380andmanyhi gher purpose and whilsttoo complex to be

achieved, particularly at the higher numbers.Atone time dividing plates were very expensive but m odern com puter

Values.This being achieved bythe addition discussed in detailit i s worth noting for ofa gearchain between the dividing head com pl eteness. Ratherthan being between spindle and the dividing plate i tself, but spindle and dividing pl ate,the gears are

' tt j , l ' l j t ' c , t i ' t y ? ' k ) . ) t t ontrolledproductionmethodsenablethe # . ) .t ,i ; , t . ) tl t )l j , ty t y ) / j y ) t . c j punched sheetsteelvariety to be produced : , r p . q jjjiryyjgj, 4 j .q . x 1) verycjjeapl y.

whilststillusing tue Iimited numberof setupbetweenspindleandoneendofthe

? ); )''

,E ' ))y ,), ) . y ; ) . # ,yl . ' â '. ( ; l q t .t , / j ), g ' . ji . ;

' fittedandwi thatailstock,ei thersupplied

1

, . ,

2a.Universaldividing head Universaldi vidingheads,(theabove being knownassemi-universal)arecapable ofa

i . l '

1 4

t yp (: )7)l

y r j . ' ; l ( k r ( ! d q : . r ' ) 6 . 7 r ) i k è ' .''. . t ; . , r y j! , ? t . sf jp ., 7 ? ' r : , y j j l j q t t i y j p l ----, , ---,. . , , , -,. . -. ,,--.,, -,-. -, y .j ,j , --j .j,--, , .- --, --. . y y . s . a j y j y j y y ; . ( y j y . y . Sj ty' . j j j gt y j ty t j' tk,y yty yy . y yk( )) j ( ) ' . t ' , : ( ; . '. fr'' ' : . l t' . k' . '' . 1 6 , 6 l ' 1j '

as woul d be expected the mostversatile' , wi ththe40:1ratiotherequireddi vi sioncan Photo 3 showsa typicalexampl e.Aswell be obtained by bypassing the appropriate as being used horizontally itcan be setup num berofholesforeachdivi si on However ,

faceplate orchuck,Also a centre can be

'

i

.j t , j . , $ , . g y ,.

g ( j yL-'-':. ,. -.-, ,, . i f ) ! 1 , ' i 7 .. ,., ' -.':r, . .' ) ; ' k , t ., .. : . 'j ' -.-. y , C t -) y -. ..' , ,' .' -l .) .F -t ..à t . r -.. ., . ,'

Mounting the workpiece onto the di viding head follows very closel y thatof : the Iathe having the facili ty for fitting a

' : I l

,

o ummo tput. Aburt aott io heo rs fm4a 0y :1beap av pa ei a larb sl emo s Co n ----- -- - ---. - -. .-. - -. , -, . - --

i ' 0 t '

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,..

fOrOr1O ttlrllZtthc di vidiR0 hozd1 t )

.

IE

.

j

. Di viding plates Thi si s the mostcom plex and costly ofthe are available w i th a wide range of hole dividingtechni quescovered inthisbookbut numbers. By choosing the correctone and

.

.. .

,,. y. jy j . y .

h',-

.. A -uI V1uIRQ IIIZX IS lIttct. zto t1IC uOuY 01 the device and the inputspindle fitted with an arm and detent for locating in tbe dividing plate holes so thatrepeti tive input

:

2.Using a dividing head

1

,

.

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. . .;F ,

j k l , s j s j s' s y ik ,. j , k t z , j y , .t y j . , y, y . -y , jo . a yj i t1 / e j,; .. . . t

m nytur()SZrO FOQUIrwwluw<.xv...v...v.-ina put j

j

'

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using aworm andwor wh c h a -m --, . , , -e .el , ..--s , -u -, -. , -,-t . -h -. --t -

.

j

: r . y l j l . z i z i t l l # ' t i y . : t ' q i $ ) ' i y q j ; t ; @ S è t t ë ' . j : p y t q k j l j y p h t ) 1 j j ) j f k è j . . t l , j ' p ) ' p h r y q r ) ! ? j ) ! t j r ' i l ir t ' b k p ) ; yy , ? ,)t j ,i -) ,; . j t y . j , jj. s r ) . ; ? jy .j . jy k y-;,rty,jj p?y!jyj-tlrj

, ') oravailable as an accessory,between

' '

<

'

q i f ' l jt p l p qr .,#r?-' . :r y : . F p t . , . .. .

14

:

'

.

.

-: *W

.

, . ; L ' yjt) t yl t l t y , ) j centresworkcanbecarriedout. ,..y# -t ,s ty,y . t t 1 V,'' lj; # A semiuniversalhead can also be kk .

.

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ry t , r f) -r: ' p ) C .' ::ètt .. --, . ' . ..

..

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1 : . , 1:t ih t : lrl'e8, 7 4 7 / r d : ;, ii l df i z ' k? '

dividing plate values normally supplied.

milling machine Ieadscrew resul ting in the

'rhe drive from the manualcrank outputfrom the dividing headrotating as

handle to the divi ding head spi ndle i s still the machine table is traversed. By this Via thej worm and worm wheel. However,a method heli calflutes can be cut. Shoul d gearc )ain from the rearend ofthe head thisbe ofinterest,search outotherreading

spindle is then arranged to rotate the onthesubject.

di vidingplatei tsel fbyasmallamount.Thi s

has the effectofmarginally altering the 3.Using a Rotary table

rotationofthe crank handle fora given Thedemarcationbetweenadivi di nghead numberofholes.The outcome ofthis is andtherotarytableisratherblurred both

thatdi visi ons notobtainable with the sem i- having com mon features. The majorone tjnj versajhead can be SetU;. Yhegoar bei ng60thhaveaWorm andWorm Wheel

15

'T '

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f

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ç . ,1 !: t

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': ê ' ï $) ( ) j '. ) y j) L ( ' jy ,, yj y t : yy ' ), .' .-? #, ( j '., tr t t 'k. ( t ktyy,nyt)j.j,t:(,y , $42 . , :.. : t;...., ''', ., # , ; ) .

.

,

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,

.

,

.

jj

,.

.

su/acesand curved slots.Placing holes

Holne workshop m ade items

on a pcD is also practicable forlower num bers w here high precision is not

The com ments above regarding dividing heads and Indexers are made

f ro er qa ui l r aeru g. eln tu wmb oue ldro thfohuogle hsboenqauP eC stD iow nahbeln e

i p nri mmin ad rilyThwey ithc omm egh rcia l y avail abl e te msos do thou inl many cas esi al

1 .

.t. j ' making a dividing plate orforproducing a .è. jj y s

.

7/. 74/ ti ,t moun ng.Also available as an extra in ?' ' V some cases is a dividing psate mounting

'ùt) arrangementasseeninphoto6w ith this

' . . avai l abl etherei sl i tl edi ff erencebetween ' %.

i

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. ) % t , ? # '$., 7 .=:6;:. jt j, ; ,, -: ë ) , ) r) . , j, ),,!, .., , .' . k. zk .. , g,.. .-s,,.-

.

.

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.

.

.

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6.Commercialrotarylab/e, sonenhale a divi ding plate assemblyavailable a. gan

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BCCBSSOW

so Iikely to be usable,as these are aj

.

adjustabl eforheight,see Photo2Chapter 1

'

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1. I $ : ' ' l

d :) l j l r1 l :1 , . 14 E ! ,l k / q k l r 4 :) 11 , . ! <j :) ki1 E 5 l ,4 :4 2 5 ) ,,1 t :

.

This isa very simpl e form ofdi viding head havi ng a rotating work hol ding device that

1

Rhoto 5.lnputto the worm i s determined divisi ons.

l b l i'

byreferencetoacalibrateddialratherlike

7i ' t ' . t '; ) ' ' l sg . j .. l ' '''

4 7' lE .t

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5 ) 1 ' 't' y ,. # ( l p ( j y q ) Divi di ngwi thno,orveryIi ttle, equipmentcan oft a d d i t i o n a l t . . ' , )j y enbecarriedoutandis C i 'V1J? t ; l i : k p V4 t ' t r i , ) ; k

) ( ' E

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ù y , y ,'. 4: ,

-7y

k--

t j eveniftheworkshopis ' )1 /t '1) / l 4 / .: i ,1,.jt y y t j t y ( k j , y , j . , j y j t y t j y $ y-.y yy .t.yg. t y j; g y .#. j j y y . y j j . y .

,

t-. ë)-.., .

. . .. ., .. .. . ,, s,s . .

,,.,,, . .

di ngfacilities.Itmay tt). v t stti l -t y/ i l z ' 1 s' ' , . 1 ' (j''.! . ?.év je è t' . l j s ? r 4, t' ' ttl y, yz . x j t , r y . beexpectedth vi

common.The majordifferoncesare inthe Worm andWormWheel.Angl e ofrotationfor

l

1

i j , n b k i? i , q . s ' .. ; . 1 '

. 7

.. .. . .

worth consi dering fuql yequippedwi thdi atsuch inethodsare I imited to the simplernumbers, 3,4,6,etc.butas willbeseeninthenextChaptermuch more compl exnumbers e possi ble,even those notavail able wi thar a com merci aldi viding head.

available,40!60and90:1alIbeing isrotateddirectlybyhand,therebeingno 13rovision fOrWork holding and in most casesthe method Ofdeterminingthe input vajue. Forworkholding,ratherthanhaving facili tiesforeitherafaceplateorchuck,the rotary tabl e has a T slotted round table,

)r ' t) 7 ' @ C''t j

.

5.Dividing w ith no special equjpm ent

betweenthe manuali nputandthe rotation 4.Using an indexer hough there w ould appearto be more variation in the ratio' s

.

designsandki tsofpadsformakingdividing headsandrotarytables

.

'.

.

tabi e from the same m anufacturer,itis l ikely

J'ê ) , y h 4,. )' j tll i ïjr a t ) è . l , yy,,tjy ty :y thatthedi vidingplateassemblyonthehead )f:t t ' . ), $ pt f y ) t t4 : ; . ' ! )y tTj. k)? , è t)y). : tj) i t ,, #r ,-. t y-. , ' . r j y j ) q r ; j y t , t , . t j y wj j j a j s o f ittherotarytable.Itstail stockis ) y j .. j.) b. . j ( y j, j y , ; k r y . , j y . y t r y , y t l q t q j y : y , y p j . ( y-j y, y .r .. r ) . ) y . .

.

.

'j @ l i ) ' # ë 0 : i ' 1 'ë ) j 1 , 'ù / * . g , l .) ..,,:,

in chapters 7 and 8. In addition, any supp. liers to the hom e workshop mhave

.

.

.

Desi gnsforthatshowninPhoto 7anda much moreadaptable headare included

.

y,.

. 'jyjtjjy, Yjjjjt (l l t lj t y (: y j j y j . t ; yy j y g y. jj , k é y; ' /ty,.

are invariably known as dividing heads.

' t -,' t. è t, x,t ng lj 1 14 y thantheprovisionforworkpiecemounti p) () t@ ,/ . ), ' i j y . r jfthe budgetwillstretch to itand you : . t y . y , ) #,,):!),l)i-t (. ; , ) , . purchaseb0thadividingheadandarotary

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,

'%Yl / :J )' ' 1l , ' ) y

,

a dividing head and a rotary tabie,other .

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'

woul dbemor eapp r opri ate ' ' 'such devices

.

us

.

.

.

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.

applyto itemsm ade inthehom eworkshop. However,many home workshop ownersdo find m uch simplerdesigns adequate for their needs, n pshto totit7 ingan' example. w hit l sh ta ft ori thi :e le b 'e Indexer

tjy ty. gearMostwillalso be suitable forvertical

t

-' J'' --' h '''''K ': ' --tv## ) 'tpi j.k j yt j y. y, ; yj ,) ., jjr yj . yt, k (k ( ., ,, ?$ ,t ) j ) y ? l t ' j 'r.ï hj ; ' ( jy: jyqjjt ' , j

eachdi visi onisei thersetbyreferencetoa 360deg.calibration round the edge Ofthe table orby the use ofa num berofholes round the table,maybe 24,into whi ch a detentIocates giving a range ofcom mon

..

yyyyyy-,,j y z yyyj j y.y -y -q j y a , y y :# y ' <-r '! . ' *-î' l* r .,, ' a1' . : -7) # , 0 :, p ' . /l) ' J ?l. ll 7 1 ,.i . .# .,,f :s'. ,c' vù'k'rpy,)y sy,.,,,,yj, , z. m basic hom e m ade dividing head z

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a

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.

.

lndexersarenormalsyfi ttedwi thonl y

thatof the Ieadscrew dials on a Iathe or one method ofwork holding,being ei thera

millingmachine.Indivi dingapplicati onsthis three-jaw chuck orcollets.This severely .

only works wellifthe required inputfalls exactl y on a calibration,otherwise errors willresul t.

! ' i ' 1

Rotary tables have therefore only substantialangleplate.Commercialitems limi tedusewheredividingisrequired,their are notreally appropriate forthe average

$

m ain use being for machining curved

'

limits the tasks they can perform .Some are also suitableforhori zontalmounting though of course they could be used with a

'

home workshop.

1 i j '

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16 17

t' b'1

&''j.'' iI

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p.a e duaystub m andre/for m otlnting ba h gear d dja/.

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using M inim um

Equipment.

becauseei theradi vi di ngheadoa ortoe t' ae

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three jaw chuck.In this case,placing a third spacewould give therequired resul t. accurate enough formostapplications and

longersecond bar,a 50 t00th gearcoul d

cedainlymuch quickerthansetting up a di vidinghead.Donotfallintothetrapof t h inkaw ingit ahe tjurs tpl aci ng the pla ck n undsi exr oej nht ear posi tion wi lgi vi eg you dj vjsi ons,thi swillonlyworki fthehei ghtof

beusedtocalibratea100di visiondial. Anothersimpleideaand onlyIimited i n t h e n umb e r o f d i v i s i o n s b y apractical maximum , s opwer r o ba a e, orface plai tet ra ipp ha er yu ,n adt sh te rip oc fk pp al pt er

the packing is exactly centre heightminus

sui tablym arked wi ththe divi sionsrequired.

hal fthejaw thickness.

Thismayseem asimpl eideabutthespace

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' r .. r,. , . .-, s ; y , ' L ' ' ( % ' . 1 , F 1 1 3 î L ( ' 2 ' % ' . ' ) 4 f)' tt) # t q j) j ) j . ( é , F ( . i . .ks-.) 7 ' i tv' , ; , i s , t t a t j y y , l 1' t, ' / . ?C ? , )t, ti . t , r ' ( , ' , ) . ' , ? . ' , y , t(.' ,!.F.:. ' i ' / :, . ' 7)'I ; 7 k ., ) ,,,:*1-* z ' . , .v . ,4,x:43-. ' .,

.

; y ? :: ; . t. . , i , ( ; j ,y .,, . ,

pi ece of packing between the bed ofthe

.

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.

d be when needing ften used.The reasons for Typical0fthiswoul equipmentis o tem in the f t en be t o make t hr ee di vi s i ons on an i this can be vari ed,butwillm osto .

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H avingdeal tw iththeequipmentreq uired divisionr iredei sand nota vai lab leolr seq su impl mor e easi y setup inchapter2,i talso mentioned thatdi viding projecti can often be achieved with little orno usingsomeotherm ethod. specialequipment.W hilstusing a dividing head willin mostcases be the i deal,other Using the lathe spindle

1

j

,

...... .,

Also,by carefulchoice ofa thin piece of

I atheand eachchuckjaw inturnwould be packingtoplacebelow thebar,ora slightly

,

An extensi on to this idea having the between each marking,when in the flat,is advantage of provi ding a wide range of Iikely to be a complex value.Say with a

l j

d i visi ons,istoincludeagearwheelintothe 100mm diameter backplate the setup and use this in a similarway,Photo circumference would be 314.1593m m .

' '

'

' I

1 Using a weighton the end ofa pi ece of .

t

'

keep the shortIencth ofbarin place whilst fortunately,there isa sim ple way outofthe

,

.theoperationofcJttingadi aliscarriedout. probl em.

'

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e

80th earanddi albeing madearemounted 1.using a gearon a stub mandrelto cut

adial.

.

g a I i t t l e lo at o dla lllt s lby moa ;d)e' nl' tn loe l't i nl o t ln egcli u dc lC aId S1Y ar e lUI Ry dc lt ot IO r o l i P glo en l Jt ' oh le t f

itaP p . of c use every t00th space, ' i fyou required 15

with a sharp knife atthe overlap,you will

divi sions,a4st00thgearstoppedatevery now haveyour314.1593.m stripofpaper, 19

18

1 2

' y t,

'

w rapyourstripofpaper,

O

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Divi ding this into any num berwould result

string attached tothe chuck,as seen,will i navaluethatwouldbedimculttoworkwith,

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II 1h III I

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s'rqAI GHTEDGEl SSETS0 THAT TsEEDGEoF -IHESQ t JA NE UA F' 'R CEOL NI TS HE t J ? A I I H * I A H N E i ) F 1 I N H A E L E N 9 ' ï ' O FIBEII' EM LINE y

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(Ii kIUCFISMAN'SFQtJARE

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IRANSFELIkikING THE: To THE IIE: SEING DIVIOED.

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Fix this down on a piece ofpaperand on Iong enough, cuttwowiderstrips,peelback this draw a Ii ne atan angle to the strip of a shod Iength ofthe backing paperon one

awprR SHEET OF PLAIx j

px e aa pm er pl a endsayst ayou r ti ng from nedievnisdions ofi tmar .Fokr satn o nstt o gght etheitr. Toe nen sur e hte e qui re d' want edo 19 rd ipj i si r ai can th bet rimr med to ,

the Iine at285mm (19 x 15)and markthis the wi dth required Thoroughly clean the offat15mm increments Using a straight faceplate orchuck's backplate edge to .

.

edge and a draughtsman,s square as

T HAT !To SPA NE SN lH BE S uu àL1 TH UE kEN EU RQo FRn0 i vF l sl oNs

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LINEOFALENCIH IHAICAN

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. ?.A d'vided papfr . Stfl p dCO&r?d chuck' s back plate

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ensure adhesion.The process in SK1 is a

illustrated insK1A,transferthe divisions rareoccurrenceintheworkshopfordivi ding

tel ydi vi dedIfyouiaavealargesheet

REQCIFEZ

to the strip of paper that w ill now be

accura

,

length.

A11themethodsabove haveonemajor

1I j I

ofIinedorgraphpaper,thenanalternati ve, problem,there is no method ofsecurel y

II1

and probably easiermethod is illustrated

Iocking the spindle.W i th onl y a Ii tti e care

1 I t $ :

inSKIB. Fi xthe striptothechuckanduse

this should notpresenta problem forthe

.

11. 11 I

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thi stoachi evetherequiredresultphoto 3

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V ETH-Q D- 2.

Iines. Ani dealpaperto use isanA4 sheetof adhesive paperasused in photocopi ers or com puter printers when making Iarge

SHEET oq - LINED PAPER

sK1 Dividing length

21

20 .

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the system can easily cope with any num ber itwoul d notbe suitable forcutting ' that gear for which you do not have a suitable di viding plate, there is though a

adhesive I abel, sandthe like.Ifthi sisnot simplewayout.

i 1

tasks shown being undedaken.Form ore

shows a di albeing cutin thi s way. As an arduoustasksthough,suchas machining afterthoughtIrealised Icouldhavemarked a hexagon on a turned item, itwould be a tbe strip wi th the posi tions forthe Ionger non starter.Because ofthis,even though

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Y. ? . aceplate markingsare setusesthe same ? ''. ' . f two fixings as thatforthe bullwheeldetent ) . T ' ), . t,. seen in Photo 1 in the Chapter2. r, )) shouId vou be m aking a sm all

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o jjje m jjjjng m achine. The smal11:)ost seen in Photo 7 has a SRZRRPChOXa9Ol 3 thatWaS m achined aS m X ouown in Photo 8. A temporary hexagon

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workpiece would enable twelve divisions and ifonly everythird position was used a square can be machined using a hex head screw.

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headed screw in the bottom is used to set itup foreach facetas shown in Photo 9. W hi1stthisis notthatprecise,i tis,ifcarried outwith care,m ore than adequate forthe purpose.Aiternativei y,itcould be setusing a square offthe table surface setting one face ofthe screw head upright. Using a rui e and a square onthe sam e

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anyone arrivi ng atthis pointin workshop acti vity willalready have gained the skiîls of m ountIng and m achining, there is traversed foreach division (this to be therefore Iittle thatneeds to be stated. Iam described inChapter4)there is notmuch though including a few photographs as l eftto the exerci se otherthan m ounting the exam ples. workpiece and carrying outthe required M ounting the workpiece willnormaily machining operations.Anticipating that be straightforward following ciosely sim ilar Havingdeterm inedwhich di vidingplate to use and the number of holes to be

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24

11.Calibrating a dJ ' a/whl ' /slheld on a stub mandrel.

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operations on the Iathe.Frequentl y itwill to m ove it to the dividing head without beto usethethree orfourjaw chuckeither removing the workpiece from the chuck.lf holding the parti tsel f,Photo 10 orusi ng a a parthas an irregularshape oristoobarge stub mandrelas i n Photo 11.lnm any cases forthe chuck the faceplate willhave to be the com ponent being machined wiflhave broughtinto use as itwould on the Iathe. M ounting the workpiece between previously been turned on the lathe and to m aintain concentricity itwillbe preferable centres is anothermethod ofwork holding * '

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and isobviousl y used where the partto be machined is too l ong forei therthe chuck orthe faceplate.Photo 12 shows a Iong shafthaving splinesmachined into oneend. w hilst appearing to follow exactly the situation when mounting between centres on the Iathe there is one important difference. ln the caseofthe lathe,the Iathe carri er

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on the componentwist be driven by the driving peg mounted on the catchpl ate.The carrierand peg willoften be hel d together wi tha Iooportwoofstring orwire butreiying I argel y onthe machining force to keepthem i n contact.This willf nOt3ufflce When uSil )9 the set-up with a dividing head as the 34 c/ose up ofdividing plate fingers. machining will notnecessarily keep the ' driverand driven together.Because ofthis traversing 14O holes,orinotherterms,two the dri verdi #ers from the Iathe catchplate turns plus 42 hoies per division. The wi th driving peg and enabl es the driverand mathemati csofthisbeing40x49/14 = 140. driven to be rigidly coupied, Photo 13 Itwil be obvious thatwhil sttwo turns wi li shows the method. See also Photo 2 ir be easy, counting the additional42 holes Chapter1 foran iilustrati on ofthe dividing ateach divisi on willbe prone to error. head'sdri ver.Lathe carrierswith bentarms Because ofthi s,dividing heads are

are available butthe position ofthe arm

fitted with adjustable fingers,Photo 14,

would depend on the diam eter of the workpi ece.Unfortunately,Ihave notfound a commercialcarrierthatovercomes thi s problem ,though they may be around. I have therefore developed my own design Seen in the photographandwhichfeatures in Chapter6. Having arrived atthe pointwhere the Component i s ready to be machined and armed wi th the correctdi viding plate and the num ber of holes to be traversed, machining can com mence. Taking as an example the need for14 divi sions using a di viding head having a 40:1 worm/worm Wheelrati o. Typically,thiswilluse adividing Pl ate having 49 hol es and be achieved by

which are set to the num ber of holes between one di visi on and the next.These holdtheirangularsetting whii ststillenabling them to be rotated as a pairbetween each di vision. The construction has some inbuilt friction that enables the fingers to be rotated as a pairw hi lst being suffici entto ensure they rem ain in posi tion during each machining operati on.Mostly,fingers and inputrotate in the same direction butdue to an upper limitto the number of holes thatthe fingercan span,typically around 75% ofthe num berofholes in the ring,for Iarger num bers of holes they rotate in opposite directi ons. SK3 should make the si tuati on clear.

27

15 M achining w#/?a verticaldividlhg head willresultin the cutterallempl/lpg to unscrew the chuck.Very Iightcutsare thereforessential.

usinga 45t00thgearfor15 di visions.W here tbis i s the case the gear can be marked appropriately prior to fitting it onto the di viding head.A coloured chinagraphpencil isexcell entforthi s.The bullwheelin Photo 6 has been marked in thi s way and may be vi sibl e ifthe finalprinting permits i t. H avîng said that in term s of the m achining operation there is no di#erence between normal m illing operations and those using a divi ding head,there is one potentialdisasterarea.Photo 15 shows a handw heeI being m ade by m achining notches around its edge. C arefuI observation ofthe set-up reveal s thatthe cutting actionw illbe attem pting to unscrew the chuck from its mcunting.ltis essentiaf therefore that the chuck is very firm ly screwed on andthatonly very Ii ghtcutsare taken.Shouldthe chuck becom efree,then there wilfcertainl y be a spoi led com ponent and probably a broken cutteralso.Do be aware of this and take great care when working in the verticaimode.Itcan also occur in the horizontal m ode js some specialised si tuati ons.

Using the rotary table.

If you have a fully equi pped rotary table, Ifusing ashopm ade dividing headwith that is with dividing plates and tailstock, ons willlargely a gearto replace the di viding plate,then then the above explanati thefacili tyofadjustablefingerstodetermine appIy w ith on1y som e sm aIlchanges each di visi on is notIikeiy to be included. regarding workpiece mounting. Ifhowever,as i s m ostîikely,you have Thisstatementbasedonthe designs 1have seen,including those i n thi s book.However, justthebasi crotarytableyouwillbefaced Ican see no reason why such a facility with setting the divisions by using the cannotbe builtinto such designs.W ithout m icrom eter dials at the hand wheel. this i twi llbe a case ofcounting the num ber Consi der using a table with a 60:1 ratio ofteeth traversed with considerable care. worm/worm wheeland 60 divisions on the lt is frequently the case though thatthe dialgi ving O.ldeg.rotati on perdivision.If di vision is achieved with one revotuti on of we now consider,asabove,a requirement the gear,such as every third t00th when for14 divisions thiswillrequire and output

28

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16.M achining a hexagon fora Iarge nulusing a m laly lab/e.

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of 360/14 degrees perdivision, that is a PCD forbol ting two flangestogetherthen 25.7143 degrees.This equates to 257.143 a little clearance in the bolt holes should divisi ons,or4 turns plus 17 and a Iittle bi t m ake the approach possibse. Do notthough (0.143)divisions,itisthelittlebitthatisthe simplify the figure to 25.7 degrees as the problem ,lfthe task is to pl ace 14 holes on errorof0.0143 willaccum ul ate overthe 14 1s1

SK3 USING THE DIVID -

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/1i s norm alfordividi ng fingers and the inputarm to m ove in the same direction . Hok d vev' el: && t o a u ppe r I i mi t thatthe dividing fingers can span, around 75% ofthe maximum,for / oN erspans rotati on ofthe fingers l ' sopposite to thatofthe inputarrn as shown aöok'e.

29

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17.Attem pting the task in photo 16 but using a dividl hg head in the verticalm ode requires a Iotofhead room .Itis though a non slarlec see comm entfOrPhoto 15. 18.A/l ' gr// ' r?g centres in preparation for positioni ng workpiece.

divisions m aking the 1ast have an appreciable error.Do though work outa figure for each division,say 257.143, 514.286,771. 429,etc. From this,work out the turns and di visi ons,1stdiv,4 turns + 17.143,2nd di v, 8 turns+ 34.286,3rd div,12turns + 51.429, etc.W hi1st this is possible,it has to be asked is i t practicable having to carefully set tlp each position? For the exam ple

gi ven,probablyjust,butwhetheryouwish to countand set up each positi on in this way willbe a personalchoi ce.Ifgears are to be cutora greaternumberofdivisions are to be made Idoubtifitis practi cable and itwil!be a case ofequipping you rotary table with dividing plates or obtaining a di viding head ofsome form . Some divi ding applicationsare though equall y,ormore appropriate,forthe rotary table.Photo 16 shows a hexagon being made for a Iarge nut.The m aterialwould have been too (arge to be held securely in

the three-jaw chuck mounted on the dividing head and so the rotary tabl e with Keatsangle plate wasbroughtinto use.The rotary table therefore coped with the situati on adm irabl y and requiring only six divisions the table's dials were adequate forsetting each position. Using a dividing head in its vertical m ode andfittedwi th afacepiate forthe task, Photo 17,would seem possible butithas some severe lim i tati ons.Available height in the m achine and rigidi ty ofthe assembly,

30

especially the lightnature ofthe faceplate are both Iikely problem areas.However, as mentioned earlier in the chapter, in this mode rotation ofthe cutteri s endeavouring to unscrew thefaceplate.Itis nottherefore a practicalproposition and should notbe attempted,even with a lightcutl. Mounting a workpiece onto the rotary table,typi cally as in Photo 16, is not as Sim ple as first envisaged as it m ust be concentric with the table.To do this mark the workpiece with a centre punch mark, Orcentre dri ll,Then,pl ace a centre in both the rotary table and the machine spindie and align them as in Photo 1B using the machi ne' s X and Y movements. Place the Workpiece onto the tabl e and tine i tupusing the machine' scentre and clam p inposition, Photo 19. Thi s should be accurateenough fOra!lbutthe mostdemanding si tuation, in

thi s case finalpositioning can be setai ded by a dialtest indicator and rotating the rotary tabl e.

Dealing w ith backlash Depending on the equi pm entbeing used there m ay be appreciable backlash in the assem bly,butevenwiththe more precision item s i t will notbe totally absent. ln a11 cases,iti s advi sable to take up backl ash m anually by attem pting to rotate the workpiece in the sam e di rection at each divisi on priorto Iocking the spindle ready formachining.In some cases a length of string wrapped around the chuck, workpiece,etc.and wei ghted willperform the task automaticaily foryou.

C hapter 4 T he M athem atics

Before one gets down to producing the case of a rotary table not fitted with a workpiece itwillbe necessary to establish dividing plate then the num ber of ibrations on the hand wheeldi alwillbe the set-up required.The process is m ore cal t h e i m por t ant f ac t or . com plex than m any w ill envisage, but For com mercialdividing heads 40 :1 fortunately the m athem atics itself is seems to be by farthe m ostcom mon,but relativel y simple. The two centralvalues when carrying in the case ofrotary tables there is more outthe required calculations are the worm/ variation,4O , 6O and 9O :1 aII being worm wheelratio and the numberofholes frequently available. Dividing plates by nature of their on the dividing plate ri ng being used.In the construction have more variation w ith smaller plates having around 6 rings of holes and larger ones 16 rings,m aybe m ore in some cases.The sm aller plates m ay have a m aximum num berofholes of 50 whilstthe Iargerplates may have up to 1OO holes.A smaller head is likely to be supplied with three plates making 18 rings in all,a largerhead willprobably have 2 plateswi th 16 holesmaking 32 ringsintotal. Anotherfactor is thatdividing heads from di fferentsourcesmay be provi ded wi th plates having di ffering numbers ofhofes in each ring,additionalplates also being available to fillsome ofthe gaps.This aI I

1.Shop made dividing head having the ability lo be used with differing worm/ worm wheelratios.

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adds up to considecable variation in what is available and increases the num berof permutations, m aking it im practicalto publish in thi s book tables covering every possibi li ty. Because ofthis and nothaving access to tables covering a wide range of si tuati ons,Ifirstly decided notincl ude any tables at all. However, l felt this was unfodunate and have as a resul tdeveloped a computerprogram to provide mewithany com bination possible. From this I have chosen to provide tables forplates having rings with 15,16,17,18,19,20,21,23, 27,29,31,33,37,39,41,43,47 and 49 holes.These appear to be very com rnin com binati ons.Values have been worked out using these and three worm/worm wheelratios 40:1, .60:1 and 90:1.Having provi ded these iiststhey wilbe ofparti cuîar benefit to those who make their own dividing heads.The lists can be found in Chapter11. lf purchasing a new com m ercia1 di viding head itwillofcourse be supplied with tables forthe divi ding plates included, butwillnotcoverany addi tionalplates you m ake or purchase Iater to fillin som e of the m issing divisions.A secondhand head may also be m issing its manual.ltis worth repeating here that only the m uch m ore expensive universaldi viding head willbe abl e to provide aIIdi visi ons,atIeastup to 380.A semiuniversalhead w illIikely be provided wi th di viding plates thatwillgive aIlnum bers up to 50 with progressively fewerthe higherthe number, typi cally 29 between 51 and 1O0 butonly 12 between 3O0 and 380.

Som e divisions considered To make the initi alexplanati ons as simple

#

as possible lhave chosen to base these on acombinationof60:1fortheworm/worm wheelratio and 60 holes in the di viding plate ring being used. Consi der creating a di alhaving 125 divi sions foruse with an eightthreads per

inch (TPI)Ieadscrew requiring an angle betweendivi sionsof2.88degrees(360/125 = 2.88),W i th the divi ding head having a 60:1 worm and worm wheelratio and fitted wi th a 60 hole di viding plate the rotation willbe 0.1 degree perhole movement.If we consider29 holes giving 2.9 degrees per division a sm allerror of plus 0.02 degrees willresult.As the error willbe accum ulati ve a finaldivision of only 0.4 degrees would result,obviously far too great.Had the error been very much

smal lerthenjustpossiblyitmayhavebeen possible to Ii ve with it. Taking now the requirementfora 127 t00th gear,as used in an im perialto metric conversion on the changewheels foran im peri alIathe. ln this case the required angle betweeneach division is 2.8346456. Usi ng 28 hol es on the divi sion plate would be the nearest but as a gear is being produced any error at a11 w ill be unacceptabl e. Otherdi visions willthough be m uch simpler.One hundred divisions,as required on a dialfor a 10 TPIIeadscrew,would require 3.6 degrees per division,easily being accom modated by 36 holes on the sixty hole division plate.

How m any holes? The above examples serve to show thati t isnecessarytochoose a di viding plate that willenable a requirementto be achieved exactly. Letus fudherconsi derthe 60:1 rati o

worm wi th a 60 hol e plate.To rotate the

R,H and D butwhatifattem pting to work

output of the dividing head one full outthe numberofholes (H)required on revoluti onthe worm willneed to rotate 60 the di viding plate fora given di vi sion (D). times.As the plate has 60 holes thi s will In this casethere are two unknowns (The resultin the setting device passing 60 x 60

whole number,W andthe num berofholes,

holes,thatis3600.Anywhole numberthat H)makingthe processtoappearlessthan divides exactly into thi s is an achi evable divisi on.Taking an easy example,say 40 divi sions,thi s willrequire the settingdevice to pass 3600/40 hol es perdivi sion,thatis 80 holes,being one fullturn plus 20 hol es. W hilst 360O is a relative1y Iarge number.the num berofpossible di visi ons is quite smalland are aI(whatone m i ght callsim ple numbers.These are 2,3,4,5, 6,8,1O,12,15,18,2O,24,3O,36,40,45, 50,6O,72,80,9O,100,120,150,180,200, 240,300,360,450,600,720,800,1200, 1800 and 3600. K now ing the dividing head ratio togetherwi th the number of holes on the divi ding plate itwillbe easy to determine i f the division required is achievable. However,you are Iikel y to have a setof

straightforward.However,the value forW can be any num berprovi ding itis whole. Consider the requirement for 57 divisions in whi ch the form ula willread. 60 x H W = 57 The values forR and D,60 and 57 in this eyam pl e,shoul d be simpli fied to their small estvalues,thatis by di viding both by 3,giving. 20 x H W = 19 From this i tcan be seen thatW wi/lbe whole providing H i s 19 orany m ul tiple of this,19,38,57,76,etc. Having established the dividing pl ate to be used i twillbe necessary to calcul ate the num ber of holes traversed for one di vision.Assuming a plate having 38 holes is chosen, the num ber of holes for a com pl ete revolution of the workpi ece will be, 60 x 38 = 2280 The number of holes traversed per division willtherefore be. 2280 = 40 57 Forty being achieved by one complete

di viding plates(orgears)thatcould resul t ina lotofcalculations,althoughsimple,until a suitabl e divi ding plate i s found,orm aybe notfound. On som e occasi ons,yourrequirement m ay not be covered by the published chads,oryou may nothave aIIthe plates quoted buthave others.In thi s case you w illhave to resort to calculation.The essentialrequirement is thatthe dividing

head rati o (R)ti m esthe numberofhol es onthe plate (H)di vided bythe numberof divisions required (0)mustbe a whote turn (38holes)plus2hol es. number(W ),thatis

w hilst60 is a com mon rati o,40 and

R xH

90arealsocommon,(40verycommonfor commercialdivi ding heads).

W =

D This is a simple cal culati on i fone has

.

.

.

Use fora dividing head in the hom e

<

.

$ ...5I.k.. di1 I +1fu..

workshop isIikelyto be Iimited and because ofthispurchasing anexpensive com mercial

item may be di ffi culttojusti fy.Because of this the workshop owner may choose to make one.Others may go down thi s route finding the manufacture ofa di vi ding head

an interesting projectin i tset f.One aspect of dividing heads made in the hom e workshop is that they m ay not have the restrictions inherent in those made com mercially. The dividing head seen in Photo 1 i s not Iimited to a single worm/worm wheel ratio as the design can cope with differing di ametergears.Ratios of30:1,35:1,etc. are thereforeobtainabl e.Tlle sim pl erhead, essentially an indexer,shown in Photo 2 can have additionalgears added as in Photo 3.Thi s opens up possibilities that are notavailable with a commercialunit. T he additionaIgears in P hoto 3 amountto a reducti on in angul armovement

2.Simple single geardividing head,also m ade in the workshop. 3,the formula given on page 34,willstill apply. However,ifa rati o of 9:5 is being betweenthe divi ding devi ce(ageari n thi s used thiswillgo into theform uia as 9/5 and case)and the divi dinghead spindleinthe would read. sam e Way as a worm and worm Wheel. 9x H However,the rati o with a worm and worm W = wheelwillalways be a sim pl e one 40:1, 5x D 60:1,etc.W ith two spurgears though the Considering a requirem ent for 63 resul tmay be com plex.Typically, gears of divi sions itwillread 60 and 20 teeth willhave a rati o of3:1 but gears of45 and 25 teeth willhave a ratio W = 9:5.W hetherthis is usable willdepend on 5 x 63 the numberof positions on the dividing Simpli fying this by di viding 9 and 63 device. by 9 gi ves The following shoul d gi ve an indi cati on 1x H as to whatis possibl e.One i m portantpoint W = to make is not to work outthe values in 5x 7 decimalterms asthis can so easilyshroud Five tim es seven being 35,W will the m eaning ofthe answer, do stick to therefore be whole i f H = 35 or70,both fracti onalform . standard change wheelsi zes.lnci dentally, W hen calculating the results for a 63 i s a value notachievabl e wi th my 40 :1 dividing head, typi cally as seen in Photo com mercialsem i-universaldividing head or

3.Two additionalgears fitted to the head seen J ' r?Photo 2 giving a greaterrange of possible divisions. gearand a 75/30 t00th gearchain that i s 15:6 The num berofholes passed for one com pl ete turn ofthe divi ding head output

is therefore 50x(75/30)= 125 hol es.The pointto note here is thateven though the rati oi s 15:6 itstilresul tsin a whol e num ber forthe numberofholes.Avai lable divisi ons are only 5,25 and 125,but 125 is useful

forcali brating a dialforan 8 TPI(threads perinch)leadscrew.Again this is not available with mycommercialdi viding head orrotary tabl e. Exam ple 3 Forthis example Iam suggesting a setup with a 40 t00th dividing gearand a 50/30 $00thgearchain,ratio 5:3,numberofhol es passedforone com pleteturnofthe divi ding m y 90:1 rotary table. 63 is frequently suggested as a good change wheelto use in pl ace of127 tcrtb wbeel,being smaller, when cutting metricthreadson an im perial Iathe.

headoutputi stherefore40x(5/3)= 66-2/3 holes,(noteweworkinfractions).Withan

answer of 66-2/3 hol es the set-up would appearto be worthless,as no numberwill divi de into this.lfhoweverin this case the outputisallowed to rotate afurthertwoturns then this results in 2O0 holes passed. This w illgi ve a range ofdivisi ons but Som e M ore Exam ples most obt ai nabl e by easi er gear Exam ple 1 Considera setup wi th a 30 t00th dividing com binations.However,200 willbe useful alin halfthou' s fora ten gearand a 60/20 t00th gearchain,thatis ifcalibrating a di 3:1.The formula i s therefore num ber of TPll eadscrew. hol es passed for1 rotati on ofthe di viding W ith the workpiece (dividing head

twillbefound headoutputis30x(60/20)=90holes.This output)rotatingthreetimes i willenable the 30 t00th wheelto give divisi ons of2,3,5,6,9,10,15,18,3O,45 and 9O.Nine,18,45 and90 beingadditi onal to those which woul d be possi ble with the 30 t00th geardirect. Exam ple 2 Considera setup with a 50 t00th dividing

thatthe di visions made onthe second and third rotation fal las required between those made on the firstrotati on.1was inclined to prove thi s m athemati cally butfeelthatthis maybe overcomplicating the explanati ons. Itwillthough workwherevera fraction ofa divi sionoccurswitha single outputrotati on.

Exam ple 4 Asa more extreme exam pl e,acom bi nati on ofa 40 t00th di viding gear and a pairof gears comprisi ng 45 and 35 giving a 9:7 rati o and 51-3/7 hol es.Seven com plete turns willtherefore give 51-3/7 x 7 which equals 360 holes passed.Itcan therefore be used when calibrating a rotary table i n degrees. These examples,which show onl ya m inute fraction of those possible,do,l hope,give an indication of the range of possibiliti esw i ththis setup,and asa result enables you to go down this avenue knowing thatitl s possible. Forthe sake Gfsimptlclty lhave i gnored the effectofusing a forked detentand for consistency with otherexpl anations,have referred to holes when using gears as the dividing device;the slot between two

plate errors.The extentwillno doubtcom e as a surprise to the uninitiated. Considera divi sion plate wi th 18 holes

theoreticall y spaced at20 degrees (360/ 18 = 20),buthaving an errorof1degree on the 12th hole,thatis spaced at19 and

21 degrees between its adjacentholes.If now attempting 9 divisions (4O degree spacing)and using a di viding head witha 6O:1ratio,the arm onthe dividing head will have to pass 120 hol esperdivision, thatis 60 x 18 /9.This i s 6 com plete rotations

plus 12 holes((6x 18)+ 12 = 120)forthe fi rstdivision. ln this exam ple,input rotation will

thereforbe,(6x360)+ (11x20)+ (1x19) degrees,thati s 2399 degrees forthe first

di visi on,andforthe nextdivi sion,(6x360) +(11x20)+ (1x21)degrees,thatis2401 degrees. A s the w orm /w orm w heeI

adjacentteethwillofcoursebeused.Ihave arrangement reduces the rotation by a aIso used for m y expIanations, changew heels having increm ents of 5 between si zes,25,30,35,etc,Ido realise thata few I atheswilluse otherincrements, typi cally by fours.W hiIstthe resul ts willbe di #erentthe m ethods wiltstillapply.

factorof60 the anglesatthe workpiece will

be 39.984 degrees (2399/60)and 40.016 degrees(2401/60).From thisitcanbeseen thatthe second division willbe accurately placed as itcom pensates forthe errorin

the first,(39.984 +40.016 = 8O)butthe+/ 1 degree error between the two on the di vision plate has been reduced to 0.016 degree,1/60th thaton the divi ding plate. The value of1/60th i s no coincidence but willal w ays be by the same factorasworm/ worm wheelratio. A feature ofthis is that even plates m ade in the hom e workshop by relatively inaccurate m ethodswillstillproduce results accurate enough for aII but the m ost demanding requirem ents.Even here you can use the dividing headto make asecond plate and use this forproducing the final product.Bythis meansyouwillbenefi tfrom improved accuracy twice over. -

Furtherdetails Chapter7 includes a design fora sim ple, three geardivi ding head and fudherdetails regarding the mathematics ofusing such a devi ce.A Iistofdivisions possi bl e is also included in Chapter11 based on a setof change wheels of20 to 75 by five's.

Division plate errors m inim ised W hiIstnota calculation thatthe reader willbe called to carry out,the foll owing illustrates how a worm and worm wheel

configurati on(ortwo spurgearsin a gear chain)reduces the effectofany division

37

v

,

A .,...I. w. kk wk!....

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(p -1)x 360

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v=Rsin---------------

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N calculating these values,even when there

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are m any m ore holes,willnot be that

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arduousifyOuhaveacalcul atorhaving

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andwhetheriti s positive ornegative.I would suggesttherefore thata basic

.

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Thiswillbetosetthedivi derstotheradius,

' J I j L /r) . '

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HolesonaPitchCircle

scribeacircl eandthenwiththescriberstill

e mathematicalmethod.

circle.Many applications willthough be

Diameter(PCD) Th

I

atthesamesettingsteproundthescribed

i

i tem in the home workshop.Even better would be to obtain a programmable

yz 1

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yj

sci entifi ccalcul atorshouldbeastandard

.

.

t a ri b gl o en sot mha etrn i co ar l mfa ul ny cti l o i sn tsv .au l un el i s ke upr t i o nt9 eo d 75 . d egreesacalculatorwilldealeasilywith the anglesabove 90deg.,giving thevalue

1Ii ij. .

,:5

4

1'5

(

--.

o I GIN kq= g

g

calculator,inthiscasejustentertheformula Y'=0 --ï4 - k5

once,thenentersequenti allythevariables F I VECICCLES0$AFI TCHCIFSLEDI AkE TE8 (lytgjj Ar rgyyyg rr yjrcjj cjycgg yj Aagrgy, (P only in thisexampl e)to arri ve ateach

'/ f 1 / /r I ! '. ' 1.h!

TheneedtoplaceholesonaPCD can muclahomo redifficultthanthatandanother occurformanydifferentreasonsbutmost met dwilbeneeded

requiredvalue.

.I j, . J . j 1 2; I ! II j I i11 .;

oftenonl yinvolvingasma11numberof Onesuchmethodi stoworkoutthe holes,typically3or4whenId r i l l i n g a b a c k p o s i t i o n o f e a c h h o l e i n t e rmslo fitsXandY n m odelling co-ordinatesand uses the m i ling machine plate to take a chuck. engineering 5 or6 holes to fix the cylinder leadscrew dials to position each hole.The coveronastationarysteam enginemaybe advantageofthismethodi sthati tachieves

i sthatitinvolvesb0thposi tiveandnegative negativevaluecreatingaminusminus c o o r d i n a t e s ma k i n g i t s o mewhatdifficult si tuati on,mathematicallythisequatestoa to equate the valuesto those to be read off plussituation

.

.

.

Theobviousfirstchoiceforpositi oning ahi ghdegreeofaccuracy,aisoanynumber

AdisadvantageofthemethodinsK.1

SK1 Insimpl eterms5-(-3)=5+ Some maysaythatholes bel ow the .

the leadscrew dials. This can easily be

overcomebychangingthereferencepoint

3 = 8.

lI' fIf j( I 'I

visi onsis achievable,exceptofcourse the holes willbe eithera divi ding head or ofdi

f romthecentreofthecircletoapointequal upperlineshouldbenegativeif norma! to the extreme upperand Ieftpositions as geom etric conventions are appli ed

1 II'I

rotarytable,However,someworkshopswill fora practicalmaximum.Before this can

in SK.2 making aIIco-ordinates positive. may be correctbutmycommentsrelateto

.

even i fthey are they m ay notadequatel y need to be calculated.SK.1 shows a very supporttheworkpiece, eitherdueto i tssi ze, simple exam ple requiring to positi on5 hol es

) fIfJ I f: lI 1II:

notbeequippedwithei theroftheseand beundedakentheposi tionsoftheboles

1 )1

.

: II

' I f J l j/ 1 1 . 1.I '

we ightors bape.Also,evenifasemi onagivendiametec universaldi viding head is available many

I ll i .' I11 .

b llnotbe achi evable. The Form ulae num ers over5o wi

s ' I l i I i k '?: 11 I 11

situationswillthereforearisewherean Theformulaforthe(X co-ordinatesis P-1)x360

alternative method uasto be found

.

.

probabl ythe mostlikely requi rem entwhere

II I

1 .I ;' I.

X = R Cos

a Iargernumberofholes are required will N be to make a di visi on plate,required,but whereR equalstheradius(thati sPCD/ notpresentl y available intheworkshop, 2)P equalsthe hol e number,e.g.1,2,3,4 Oneofthecommonestal ternati veswill or5 and N equalsthe numberofholes,5 be when the requirementisforsix hol es. inthiscase.SimilarlyfortheY co-ordinates

'

38 I1 I I .

- -

This

Therequiredco-ordi natesareeasi lyarrived increasingvaluesbeingreadonthe atforX by adding the radius ofthe PCD to each value the form ula forthis reads:

(P-1)x360

X = R Cos +

N

+

Ieadscrew dials.Adrillplacedoverthezero Iinewillmove towardsthe Iowerholeswhen

PCD- t hehandwheelisrotatedclockwiseand the reading onthe dialincreases .

2

However,

some mi lling machinesdo notconform to

ForYhowevertheval uearrivedatmustbe thi sstandardandthedatumpointwillhave

taken from the radius ofthe PCD,the tobeotherthaninthetopI eftasinSK2.I t formula therefore reads:

PCD Y=

willnotthough alterthe calcul ated values,

(P-1)x360

onlyi aow theyareapplied.

-R Sin-

2 N Hole generation Dotake notethatthevalueforR Sin((P-1) A variation is often used in the home x 36O/N)forsome hole positions willbea workshoptoenablelargeholestobecutin 39

I

'Qt. - -

.

.. '

. a. .:.. â1k.'

be drilled.However,do make the number divi sible by 4 as this wil make the holes identically positioned in each 90deg, segm ent and as a result reduces the amountofcalculationrequired appreciably, Having deci ded on the numberofholes

work outthe angle (A) between each adjacenthole,thatis360/N where N i sthe

1.Using X and Y . co-ordlnates to mark outposition ' Ofholes to remove the centre Ofa f) : , rotary table body. ' This to take the . table' s bearing. ?1 ' t

',

t);. y ' .. ' ' , 7

numberofholes.Chose an outerdiameter

(D)justsufficientforfinalfinishing.Thedrill diameter(d)can now be cal culated using the formuia in SK3.I fthe dri llworks outtoo sm alj,ortoo Iarge,then reduce orincrease

)q (5y?!

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) A AhGLE 8ETAEEN THE FOLES N NUkBEF 0F HOLES OUTEq CIAMETEF d EIqILL DIAMETEF FC() FCC ON AFIICH FICLES A9E 7' 0 8E ZiILLED :560 A = ---N

40

the num ber of holes estim ated and recalculate.The PCD on which the hol es are to be drilled willbe equalto D -d.The co-ordinates can now be calculated as in the firstexam pl e and then drilled, Iused this process when needing to produce a I arge hole i n a casting.Photo 1 shows the hole posi ti ons being initi ally marked out.After drilling it was very

gratifying when the centre justdropped away asthe Iasthole was drilled,Photo 2. This m ade the care taken wellworth the effortinvolved. W hen adopting the above approach i t is likely that in m any cases the angle betweeneach hol e willbea complexvalue, i twillthough be ofno consequence ifthe procedure is adopted fully.However,ifthe num berof holes is chosen to arrive ata

whole numberforthe angle,(say for36 holes,360/36= 10degrees),thena rotary

stadedontheprojectIhadanti cipatedthat thi s may prove qui te complex butin facti t is notthatdifficult,AIIthatis necessary is to roughly estimate the num berofholes to

.

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, ,)...

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...

N

a workpiece bydriling touching holes on a pitch circle as ill ustrated in SK3.W hiIstthis can be done using relati vel y inaccurate methods i talmostcertainly wi llresultinthe need for m ore m anual activity - filing, sawing,etc. In this case the PC D is not of im portance,whatis,isthe diameteron the outeredge ofthe drill ed hol es.The closer this is to the uItim ate hoIe size w iII considerably m inimise the finishing work necessary.W hat i s required is,thatwith the chosen hole size and onthe calculated PCD,thatthe outeredge is close to the hole size being m ade.This m ay seem relativel y straightforward butthe drillsi ze and PCD have to be chosen m ore carefull y and involves some calculation.Having

t ' . t y r j y j y y k ! . ' . @ t ? ;.ï. ax. l $.y t y .(, ) t'. jj .y .

.

----'

.

SK3

table could be used if available as the Iimited calibrations on a basic rotary table

willeasily cope with sucb angles. ln this case a1Ithatwillbe needed i s to calculate the hol e diameterand the PCD.

Using a com puter Having suggested using a programmabl e calculator,even betterwould be to use a computer spreadsheet program .This wili notonl y produce thecom pl ete Iistofvalues on screen butwilalso enable a printoutto be m ade for use in the workshop.Ifyou are an accomplished userofspreadsheets you willnotneed me to inform you ofthe advantages of using these for repeti tive calculations.Ifthough you have a com puter thatyouably useforotherapplicationsmay I encourage you to get to grips w ith spreadsheetappli cations. A spreadsheet is basically a form di vided into columns and rows and ateach pointwhere they cross a bl ock is created, called a cell,Columns being referencedA, B,C,D,etc,,and Rows num bered 1,2,3,

xI r' j ,

v

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2.Thecentre fell

Ini tiallycel lsA2and 82wilbeIeftblank

probablybe 8characterswide,5forwhole

away as the final

wjth cell02 having the formula forthe

pounds,1 forthe decimalpointand 2 for

h olewasdrilled,a very satisfactory

v olumeofacylinderadde d.However,the thepence.Thecolumntocarrythei tem formula Willnotappear i n cellC2 so the description willbe setup fortextand have

conclusion!

sheetwilli ni tially appearas above.The

' f 1 i I l j . ' j p l I /t I'

f unl a,nwoi l the ar l serwlih not e ho d p a n h aadld i n o So cr rm ee rl ma lln ya op np te he I oe we ne er ,e bo ut nh l y provl ids in gu cl ele lsx f ol ri inpt ua ttt ini gnv ui et so ant d

Ir 'j 1I '

whenthe ScreenCUrSO:iSgl aced inthat othersContai ni ngformul a,headingcellsare

1- , . 1 .

cell.The form ula,Volume = T:xR2xL will also included at the spreadsheetdesign

j 7 fp ' '

. ,r .k .. :. -

thoduWi ghllbte setupi ntermsofcellreferencesf s tage.Theoiameter,Lengthandvolume hereforebelx(A2/2)2xB2.Most,i headingsin the above are anexample,

:

an

'2 / '/J'

r'otall,Computerprogramsuse*asthe

'j' j# , j ,. , , j ,

4, etc

Typicall y therefore a cellin column

.

candrow swillbereferencedc5

, i tm aybe a ' si 'ngle cal culation from a very Iarge numberofentered item s,typi callythe

.

totalmoney spenton i tems purchased over a period oftime.On the otherhand itmay

a.(sry,;,.:z.sj njs js a very simple intje chapter, as this is the way form ulae

jonuutonethatwoul d beverywell iII have to be entered into the Wor a pptjjhoat setting up ifyou are a new comerto W spreadsj aeet. Spfeadsheets. You will soon find the

ayvaot ofbei abl elt ogt ent vajues ages fordiamet erng and or en h er andnew for M ultiple calculations

j ; I lji p

Each cellis established fora single purpose being chosen from fouravai lable

$ 1 I l g

d esai gn stagne ndtu sesheet df orsas uch urp oese sc hdi ap tees r,c l cul t i ntg al th eedXval and Yc oas headi ga for he ap wh ol or Or nat fa rom ja us t wo en ter ues, the

f II ' . I'i 1I1. 1

headi ngsforindividualcolumns. numberofholesandthediameteronwhi ch 2.To carrv t ex t en t er ed when t he t h es e a r e o l a c e d . ' I ti ssuch ' acaseasthisthatmakesthe

2 1l j y )yq Yesol* *****1 2VWUYE157$ a: t h.'

' 1/ ' 1. J ' tt f . I I ' 'I.1 1 1 1

a' oojicati onthiscouldbethedescriotionof useofsoreadsheetssousefulinthehome ''item purchase. ' 'p.Thespeedatwhichthistakes an worksho

-n lft , heitrogram cu ,rsori splacedonC2, slightlymorecomplexexamnleofworkina 'RORO'SeWrlereontnescreentheformula outthe touching holeso'na PCD aW s

j l fj u ses. makeamul ti tudeofcal cul ati onsfroma j ; 1.Tocarrytextenteredatthespreadsheet smallnumberofentered values.Inthis '

. j .I:. I I.

'' 1.

lI jl i jI . I.

') '1)

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soreadsheet is used. In a financial

a.To receive numericaldata asthe sheet placecanbe asurprise to a new userofa is being used. Again in a financial

personalcomputer. ,add new values forthe

item purchased. 4 To display numericalvalues calculated

requiredvalueswillappearalmostinstantl y.

application tl ni s coul d be tl ae value ofan

numberoflaoles and the PCD and the

t bOV jow l ue mn eow t oe an pt p ee rard i m i amet de i a rt o ef l y1 , 0anda a wn jd er w ehm er u el ti pe h l er ecar l c eum l aa ti oy n nsma orec i no vl om u l ve ns d joyjgtjjoygtjjosoraaswkjjreatj '' >., v

B

c

n ,

a ( y r o w s, sjat er j eet jt o caangl j ctja t eix ndY cn oor di nat es numer ous es saa good exampl e.However,unl essyouareexped i et ti ngout hef ormu l ai t llbeus netc essa rv tn ospr oceed st ys temat ical l ywi .Let ake the

fOrthat cell,' n * (A2/2)2 * B2'willbe

illustrated in SK3.

list.

calculated based on the outerradius and

background,the answers to very m any

headings, etc,setup atthe design stage,

di splayed,whilststi lldispl aying157.1inthe

First,the hole diametermustbe

zjje program willcalculate,inthe thenumberofhol eschosen ln addition to ,

il'!I

. by the spreadsheet using the values G ive ita try

decimalplaces butyou can formateach fourcellsmustbe provi ded asfollows:

:f .j: ?j

enteredasin3andusingformulaeentered The following i s anexampleofa simple

cojumntodisplayvalueswi ththenumber 1.Forthenumberofholeschosen.

11.

by the sheetdesi gneratthe spreadsheet s pr ds hder eets etupeto causer lcul ataedds thev olu of aea cy li n wher the bot hme i ts design stage

I,j

' IE II

' .

. There can be many entry cells and

diameterand length.

1''I' f rmulacal eg iviat ngedma nyues outpthat utvamak luese.l theis ,D,l- e'reR u.e- nf o to hese cul val t Y'oo-ze v '

a spreadsheetso useful.ln afinancialsheet 2

/

42

X. .. .=

of decim alplaces as appropriate to the 2 Forthe outerdiameter. Zp/li cati on.Allcolum nscan be individually 3. . The cel lin which the program displays configured in terms ofwidth,number of the cal culated value forthe angle between

decimalplaces,etc.Inafinanci alsheetthe a jThe acencel thloil eswhi .chthe program displays 4d n Colum n forthe value ofthe item purchased Willbe set up for num ericaldata and

.

the calculated value forthe hole diameter .

43

I

.

vj ae spreadsheet reall y proves i ts worth

III .. .

'

'haveabovepresentedtheformulain

ingle line format,ratherthan the easierto

multipli cationsignsotheformulawillread S understand multiline formatused earlier

.

i lF ' p 1 7 . 7' '@ ' jj I .

a much largerwidth,say 30 characters.

.

I I gj II -I

r

. II

J! ? '

Ili lf Ii .

' !'1. l

. ! I' I i ' I

1 fj:I jI

I! i' I ' l' j I 2 ! I,,j

Ii. I II. II

cetl's A2 and 82 should be set up for Changetheformulatoread=sin(cz/zland icalentry and c2 and o2 having the num er

formulaeforcalculatingtheanglebetween

A 1HOLES

B

c

D

OD

ANGLE

DRILLSlz.E

2 3

checkthis againstsome tri g tables oryour

sci entific calculator.The brackets ()are essentialelsetheprogram willgi veyouhal f

Dklplicate and Paste

and one ata time as the cellis selected.

Ifyou setup yourspreadsheetforup to

m

in the formula 1OO times and similarlyfor

to choosing the sine value. Again enter edit and change the

ei sewhere on the screen. tj ais stage thatas w ith aIIcom puter COSUIt,take note ofthe extra brackets.

theformulaedpnotappearli nthecellsbut Ormtllatoread=1/ (sin(C2/2))andcheck should add at tbattheVaiue 1divi ded bysin(C2/2)isthe

! I k,

! ! .i . l ') i . I ; : I ' ', II,

p '$

,

' rhe two screen prints helpto illustrate this, ore Iater.

1O0 holes,ormore,youmay feelthattyping

lfcarrcyi ngouttheabovewa stobea theotherthreecolumns,isataskbeyond one-ofcal ulationthenyouwoul dbebetter what is acceptable.Fortunately,the

Continue going through the following sequencechecki ng thevalue ateach stage,

wi thapieceofpaperandyourcal culator. If facisities ''duplicate''and d'paste''Iimitthe though you were expecting to make the work invol ved consi derabl y. vinchgeck type d cal culati on frequentl y then save the sheet the form ul a infora single IineHa and ed

k'p ' -tt, a ''=<'v =.->'L''.' ,''' :-, 'F?.u'--'--'-----'---- next=(1/(sin(c2/2)))+1andfinally,=B2/

toafil eandcallitupnexttimeyouneedi t, thatitworkeditcouldthenbeduplicated

there willbe subtl e differences px....l a.. pxcaoio rxrir aoïralo. i' aair-ntoa in tl ai q

II kk programs ,

.

isrequi red.The bracketssaydoC2/2pri or

II k lIIr:

I I jJ '' ) . E f ,J ! ' '

aough R, p and N willhave to be Work tl

coul d span rows 1 and 2 wi th the value Substitutedwiththe cellreferencesforthe being allocated to row 3 The colum ns do CelIS containing the values.

0 othavetobewideenoug htodisplaythe formulae these are displ ayed separately

holes and the drillsi ze.As alreadystated

, ,,

fholes'was preferred then the heading

o

th f evalueo fsineC2thatisquited ifferent rom the si ne value ofhalfofC2 bei ng what

i I

I ) gI f/ ' j

jfinthe caseofthe word 'holes' '''number COS((P-1)X 360/N).To make theformulae

C'Iapt er snou. Bu il d ing uwpopp, afoyr.mulaforaddingtoa ((j/(sin(c2y2)))+j)vjais may seem a slow ,

cellcanbe somewhatcompl exasthe Process but as the value appears

formula hastobe entered asa single Iine dthecorrectusageofbracketsobserved an to ensure thatparts ofthe formula are cal cusated in the correctorder.By farthe

immedi atelythe formula has been edited, itis no more thata couple ofminutes. Enteringtheformulaatonegoand Ieaving

outacrucialbracketcouldtakemuchI onger

a1Ithatwillbe needed woul d be to entef

into the remaining 99 cells. However,after

newval uesfornumberofholesand/orouter havingduplicated say4individuallythe diameter.lnanycase,asmenti onedabove, blockof5 coul d be duplicated and pasted evenforasingleappl icati onyoumay have in five at a time reducing the work to carry outthe calculati on a numberof significantl y.

timestoarri veatasuitabl eholesi ze

. Thisthoughis notthefuilstoryaswhilst As already mentioned, the r ealbenefi t the valuesforR (radius)and N (numberof

I i ' '.

bestway i s to build the formula in stages to solve,ofcourse ifyou are experienced

andobservethatthecorrectvalueisarrived FOuthenenteritaonego.

ofusingaspreadsheetbecomesapparent holes)areconstant,andwillbepickedfrom

I , II , 1 7 . 1 r. Ij 2 , ' r. II.1 'I

ateach stage. Onefudherpointregardingtheabove, startingfirstwi ththeeasyone c2.The FoUCSpreadsheetmayuse radians rather formulaforangleAisgi veninsK3and thandegreessotheaboveformulawill Whenplaced incellc2would read =36o/N needadjustingto suit Forthe readerwho

where multiple calculations are required thecell satthetopofthesheet(R isinC11 with the need for mul tiple X and Y co- and N is inA6 in the screen pri nt2),the

II. 1 ir '

II II'I

11. 1j :I I1'1:

If / J i I ;( I 1. I i1!1I i l i 'I !I

.

'

'

''

1

! I iII

pf.

'ji

I'I

.

w here N being the num berof holes.

L Q-

colum n A . Hole Number.

notIostaswhenthe formuia is pasted for

)progressesbyoneon each row.AIIis

.

Column B.X co-ordinate relati ve to the thefirsttime,theprogram willseekouteach circlecentre variableand aski fi ti sabsolute (the same

num ericalentry and could be 3 characters

Colum n D. X co-ordinate relative to

Theformul ainD2i sthoughmuchmore forA2 (upto 999 holes)and 7 characters complex being thatgiven for'd,in sK3. for 82 (up to 999.999 forthe outside

extreme co-ordinates. cOlum n E.v co-ordinate relative to

Build this up in stages starting attbe diameter).However,widthofcolumnswill often depend on the heading required at

extre meco-ordinates. easyand quickone andnc anbe done in 'rhe formulae forthese willbe those blocks ratherthan one Ii e at a time

rememberingthatAttheangle)isthevalue the top.Forexample column A may have

included in the paragraph headed 'The thoughsetuponlyafew Iinesandtestyour

spreadsheetprograms.

innerm ostexpression, that is A/2.Now

.

columnc.Y co-ordinate relativetothe valueeachtime)orRelative(changeseach Circlecentre.

time relative to anothercell).Having indi cated thatR and N are absolute and P rel ative i twillnotaskwhen the formul a is pasted infurlhercellssothe taskisstillan .

Do

in cellc2,therefore entertheform ula = c2/ 2 and observe tl aatthe resul t is hal f the

ve forthe to be 7 characterswide, thatis fi word 'holes' 'and a space either side to

Form ula''on page 38. Theywillneed to be Wri ttenon a single Iine with aIIthe brackets

angl e io c2.

separate itfrom its adjacentheading.

in the required pl aces,typically forthe X

This stillhas the column 'A''which

Headings can take up more than one row .

Co-ordinate relati ve to the circle centre = R

contains the hole posi tion num ber. 1f,for

44

program thoroughl y before pasting in the fullnum berofIines.

45

I -

v aluefortheholeposi tion''P''(A16,A17, etc

However, the value for N willhave been equaIs2x' n radians,Inthiscasesin(C2/2) entered in cellA2 therefore the formula in Would read .J:.c2/a6o)/2) c2 s hou lddi re ade d =and 360/A2. Nomul te t h a ty/i s si n((2and d f or vi * f or t i pl i n cel l s A2 B2 have been setup for use

Enter the edi t mode forthe celland

I

is notfamiliarwi th radians, 360 degrees

o rdinatesbeinganexcellentexample. yive columns willbe needed as follows:

x . .'..1. :

r

A 1 3 4 5 6

NUMBER OF HOLES

B C D CALCULATES CO-ORDINATES OF HOLES ON A PCD

USER ENTERED ITEYS

USE: ENTERED ITEMS Radius at hole centre 5 40.000

8 9 1G 11

NeMBER OF HOLES

1 2 3 4 5 0

0 1

RELATIVE T0 RELATIVE T0 CIRCLE CENTRE EXTREQE CO-ORDINATES Ce-ordlnate X C/-ardlnate Y X Y 40.000 0 80.000 40.000 12.361 38.042 52.361 1.958 -32.361 23.511 7.639 16.489 -32.361 -23.511 7.629 63.511 12.361 -38.042 52.361 78.042 0 0 0 0

B16/F@rmu1a: IF(A16jC6*C0S((A16-1)*(((2*PI)/360)*C11))jO)

could justtype in,when designing the spreadsheet,the num bers 1 to 1OO down colum n A.Thî s would work qui te wellbut when requesting co-ordinates for say 10 hoIes on a P C D it w ouId continue calculatingthrough the 100 Iines asthough you were going round the circle 10 times. W hen on the screen the lines beyond 11 up coul d be i gnored but ifas is Iikely a printoutis to be taken,extra sheets would be printed out.Iftherefore the sheetcould be made to stop at hole 10 an aIIround ti diersituation would result.

Ratherthanjusttyping1and upwards itcould be arranged forthe form ula to do this.Taking screen print2 as an exam pl e

cellA16 woul djusthavethefigure 1typed

PROGRAR CALCULATED ITEMS Hole diameter Radius at Angle hole Centre Between Holes 0.481 0.409 72.000

3 4

s BOLE NUQBER 6 17 18 19 0 1

RELATIVE T0 CIRCLE CENTRE 1 2 3 4 5 0

RELATIVE T0 EXTRERE CO-ORDINATES

Co-ordinate X Co-ordinate Y 0 .409 0 0 .127 0 .389 -0 .331 0 .241 -0 .331 0 .127 0

-0.241 -0.389 0

X 0.819 0 .536 0 .078

Y 0.409 0 .020 0 -169

0 .078 0.536 0

0 .650 0 .799 0

l6/Fermula: IF(A16,C11*C0S((A16-1)*(((2*PI)/360)*D11))j0)

CREEN PRINT 2

CREEN PRINT 1 exam ple,you are working to a m axim um num berofhol es of1OO in which case you

QUTER RADIUS 0.650

5

CALCULATED VALUES Angle Between holes 72.000

13 14 15 HOLE NU#BER 16 17 18 19 20 21

CALCULATES CO-ORBINATES OF HOLES IN AN ENCLOSEING OUTSIDE BIARETER

in.CellA17 woul d though have the simpte form ula = A16+1.W hen pasting in further cells and requesting these to be relative we would get,= A17+1,= A18+1,etc. Having now arri ved atthe hole number bythis means itwi llopen up the possibility ofextending the form ula so as to hal tthe progression atthe numberholes required. However,to do this willi nvolve including some l ogi c expressions in the formula and this w ill increase the complexity quite considerably and i fnew to spreadsheets it is bestI eftatthis stage.W i th this done the program willcalculate the values forthe numberofholes required butthen continue as i fgoing round a second time and more. To achieve a ti dy resultitwould then be a good i dea to copy the file to another and in thi s block delete aIIthe records

46

above the values required.Thiscould then be printed outforuse inthe workshop. The sheetwoul d stillcontain the form ulae up to thatvalue and could be used again with that num ber of holes for subsequent appl icati ons.

following shoul d give an insight into the

Logic

Trying to express this verbally the Iogic

Wi th this we are now arriving at a m uch more complex use of spreadsheets and one thatmostwilneversee a need to use. Iwould therefore suggestthatunless you are conversantwith Iogic expressions you leave yoursheetatthi s stage. Eventually, When becom ing fully conversant with Spreadsheets thi s far,you coul d then take time to Iookinto the l ogi c provisionsofyour program ,However,ratherthan I eaving the

Subjectwithoutanyfurtherexplanati onthe

subject.

Consider colum n A , carrying the rumberofeach hole, this could be setup w ith som e sim ple logic that halts the calculations once the hole num ber has reached the num ber of holes required.

would state, 'IF the value in the previous cellequals the num ber of holes required enter' dO''IF NOT entera value one higher than in the previ ous cell. W hiIstthis would work forthe firstIine after com pleting the num ber of holes Feguired itwould notwork forthe nextline as the previous value would notequalthe numberofholes required butwould equal nought.Itwoul d as a resul tadd one to this Starting the sequence again, because of

47

.

' >

...IkI,

thi s a m ul tiple IF hastherefore to be used. Thiswould state '' IF the value in the above cellequals .0' 'then enter'. 0,'1IF NOT1IF the value in the previ ouscellin the column equalsthe numberofholes required enter . '0' 'IF NOT entera value one higherthan in thatceII'. To ensure thatthe X and Y val ues also display a value ofzero the form ula should include an IF cl ause as illustrated atthe bottom ofthe screen prints i fthis were not done the co-ordinateswould be worked out on basis of zero in colum n '' A ' giving irreIevant vaIues. From these brief commentson the use oflogic i tcan beseen that the com plexity has been raised significantly.Please do nothowever1etthis putyou offattem pting to use spreadsheets in theirsimplerform .

repeating isthatyourspreadsheetm ayuse radians to describe angles rather than degrees.Therefore,as there are 2 Pi radi ans in a circle then say f0r70 degrees the numberofradians is 70*2pô/360.Thi s should be evidentin the formula forB16. The screen printsalso illustrateanother important feature ofspreadsheet desi gn. Considering screen print2 i tcan be seen thatIines 3 to 6 are used fordate inputed by the userwi th the remainderdi splaying values calculated by the spreadsheet. These though can be divided into two categories,iines 8 to 11 displaying values calculated the once only whilst Iines 13 upwards repeat basicaIIy the sam e calculation butforeach hole posi tion. Ofgreatersignifi cance is thatwhen a column has been all ocated a width ancl

formatted(numberofdecimalplaces,etc.) Screen Prints

thiswillapplyto aIIitems down the colum n. Iwi llfinish with a few brief com ments on Typically,had 'Outer radius' been placed ' in C6,hole the screen prints publi shed.Screen 1 i s in A6 and ''No.of Holes' for calculating holes on a PCD whilst numbers would have to display as 1.000, Screen 2 calculates values forholes in an 2.000,3.000,etc.and numberofholes as 5.000 -this can com plicate the desi gn of enclosing circl e. The onlymajordi fferencesare thatin som e sheets.Having m ade the point it 2 the ho1e diam eter (B11) has been m aybe that some m ore m odern and calculated bytheprogram as hasthe radius adaptable spreadsheetprograms do all ow onwhi chtheholesareto be placed (C11). characteristicsofa colum ntochange down It is not visible on the print butthe theirIength. cursorwas atcellB16 whenthe screenprint was taken and the form ul a forthatcellis Inthei rsimplestform spreadsheetsare displ ayed atthe bottom leftofthe screen. notthat com plex and yetstillvery useful M oving the cursor to other cells woul d and time saving.They also avoid any cause theirformula to be shown.Note the possibility oferrors due totapping incorrect IF com mand thatbasically states IFA 16 is vaiues into yourcalculator,easily done zero display zero in B16 IF NOT then when many hundreds of calculations are to be carried out.So whatare you wai ting calculate as perform ula. One other im portant point worth for,go on,give spreadsheets a try.

48

G hapter 6 S im ple S ho p-m ade D iv id ing D evices The nextfourchapters dealwith i tems that can be made in the home workshop. The reasonsforthis approach wi llbe num erous butare I ikely to be mainl y. 1.The workshop owner gets m uch sati sfaction in making workshop tools.

2.Becauseoflimitedusecannotjustifythe expense ofa com mercially made item . 3.ltem notavai lable comm ercially. Some ofthe i tems inthischapter, al ong with the Lining toolin Chapter 9 are

exam ples of item s not available com mercially.Itisassumed thatany reader

embarkingonprojectsinvolvingdi vidingwill havearrived ata reasonabie understanding ofbasic workshop practices. Because of this,details of m anufacturing techniques willbe reserved Iargely to areas ofgreater complexi ty,

Lathe m andrelgearm ounting The method ofmounting a gearonthe rear

1.The two parts thatm ake up the Iathe mandrel gearmounting assembly.

49

.. iLI.

r

'

-

ZLL.L ..'''. ,.

GgAR ...-..

yJ-ATFIE kAhE)@EL

.

---..--;-------------'

k

)

: -------------...-

--'-

1 : :' )uEEz Ez :: :)::::::::::i : i irqè-...;----,-ct.-.... ë - - . - ---

:ArEqIAksrEp:t-250::)7

. ----.--.----------... .

1.)r , j , .' 7 .7 ,

-

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2 Slitting the outer Thl ' sI J&O. Sthe Parl, directindexing disk seen behlnd the chuck to set the posltions

spuly MANOREL,ouTEn.

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JS T0 B2 FITT g

LATHE MANDREL GEAR ASSEM BLY

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SPLIT MANDREL,INNER .'.l' ('.'ë't', .'. s!. . . :t

2 j z z '-E:::-- zzjpx

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end of the lathe spindle for dividing purposes is show n in the published assem bly and parts drawings. No dimensions are gi ver astheywillvaryfrom Iathe to lathe. Manufacturing thi s is straightforward, placethe materialinthe chuckforthe outer

.-

t

.

ti' ..E :..:....... jhi7kféè-:;y .) ..tEE... ..,.. .

-4g -

FOLES k5 j 10 kk DEE?, 1 OFF kAJEFiAL 46 î 8 SIEEL 670:2C

DRIVING PLATE.

FOLES h 5.5 k: 08 9 kk / 5.5 02E? 8 1û kk C 30 kk 2 k6 RATEiI AL 40 k 8 STEEL 070k20

CO M PONENT PLATE. 50

I .1 ILk

.

.r? i:.k;:: 17LTt?.;;y

yyyyj yroy j

:j î . ' , . ' 4. r '.,.j i k, 6 , . g r y , y ! ., -. r:) .. ,:,,a$j.,

would seem a good starting point for consideration.The plungerin the detent assembly should be made a close sliding fitin i ts housing so asto m inimise backl ash when i n use.Despi te this,backlash should al w ays be taken up manuall y and in the sam e direction, even with a precision part and with sufficient projecting to dividing head.Again lhave not included machine the whole partatthi s stage.Use any details for the detentassem bly but leftand righthand kni fe tools to machine thoseinChapter8(parts41and42)shoul d the two parall elpodions withoutremoving prove a basis forany design. the partfrom the chuckensuring asa result Slitting the outer com ponent is thatthey are concentric. Setthe angie of illustrated in Photo 2 and as can be seen it the top sli de and make the internaltaper, is a simpl e operation forthe dividing head. Ieaving the top sli de atthis angle forthe However,itserves to illustrate a feature of Qtherpad. Nextmachine the parallelportion this form of head w hich has not been Ofthe inner pad making the thread also. mentioned so far.Just visible behind the Turn the pad in the chuck holding on the chuck is a plate containing a ring of24 portion justturned and machine thetaper holes.This permits directindexing ofany Photo 1showsthe two padson compl etion. division possible with a 24 hole plate,that Iam notincluding any details forthe is 2,3,4,6,8 and 12.The head is still design of the detent m ounting as the rotated via the worm/worm wheelfrom the method of mounting this wiltvary widely. m anualhandle butis Iocked in place using Bowever, usingthe changewheelquadrant a peg in the top hole posi tion.The peg i s

W

advanced using the sm allIever partly visible top rightofthe di viding head. Photo 3shows a typicalassembly,but do notbe confused,due to Iack ofspace the photograph hasbeen takenvia a mirror.

5.A chuck back plate being prepared for drilling holes fordividing purposes on the Iathe. make itdi fhcultto avoid the back plate's fixings.

Bullw heeldetent

Dividing plates

This is anothercase where the design will depend very m uch on the Iathe to which jt is being appli ed.However,once again the detentassem bl y in Chapter8 should prove a basis.Photo 4 showsthe m ethod applied to a Myford Series Seven lathe.

Having a sem i -uni versaldi viding head and a good range ofdi viding pl ates there will stillbe manydi visionsthatare notpossi ble, even atI owernum bers.Typicall y a need of 53 divi sions can only be achieved using a dividing plate having 53 divi sions itself.As this is a num bergreaterthan i s normally availabte on the sm aller plate sizes supplied,anotherplate woul d have to be FoI m ore information regarding prim e purchased, lf also it is for a one off num bers see Chapter 1O. application such an expense m ay be

Chuck back plate dividing

3Above.A typicaldetentm ounting,itwill ofcourse t/aly depending on the Iathe to which itis being fitted.Because ofIim ited space this photograph has been taken via a mirror. 4.Below.Bullwheeldividing attachment.

W hiIstnotapplicable to today'sworkshops to the same extentas in the past,divi ding using drilled holes around the chuck's backplate can stillhave its place.Drilling the holes maythoughseem a daunting task butasthe system isreally onlyappropriate forI owernum bers there is no reason why this should be. Using the method of a di vided paperstrip,asdescribed in Chapter 3,w illm ake the process an easy one. Photo 5 showsa chuckprepared fortwel ve divisi ons.The back plate is to be centre punched on the markings and then drilled withtbe required holes,3 mm diam eterand 3 mm deep would be aboutright.Care in positioning the m arked strip w iI1 be necessary to ensure thatthe holes do not conflictwith the back plate' s fixing as they may be close to the edge. Having also divisions of5 and 10 may be desirable inwhich case a ri ng of60 holes would appearto be required.This need not be the case as two rings ofholes,one of 10 and one of 12 could be made.ln this case,positi ons 1 and 7,from the 12 hol e ring,sharing the sam e places as hol es 1 and 6 from the 10 hole ring.Itmay though

difficultto justify.Making ones own may Using X and Y co-ordinates therefore be the way forward. An interesting feature of using a divi di ng plate vi a a worm/worm wheelwith

the samenumberofholesonthe plate (53 in thi s example)as there are divisions required, is that the num ber of holes traversed willalways be equalto the worm rati o.Assum ing a ratio of40:1 the num ber ofhole traversed forone turn atthe output wi llbe 40 x 53 and the num ber ofholes traversed forone di visi on willbe 40 y 53/ 53 = 40.This woul d be achi eved with one turn plus 13 holes. Having menti oned 53as a di visi onand Stating thatitcan only be achieved with a 53 hol e plate, Ishould pointoutthatthis is because i t is a prim e num ber. Unl ess a Uni versaldi viding head is available, rather than a Semiuni versalhead, prime numbers always require a plate having the sam e num ber of holes, excepting for Iower numberswherea m ul tiple maybe practical, typically for 19 di visi ons a 38 hol e plate.

Byfarthe m ostaccurate methodto produce a division plate would be to workoutthe X & Y co-ordinates and use the m illing machineto posi tion the holes.Forthe lower num bers thiswould cedainly be the way to proceed butforhi ghernum bersthe method could be a little tedi ous and great care would be needed in setting the table's position foreach division.As a precaution, Iightl y m ark outthe positions with a centre drilland inspectthe resultbefore rem ovi ng the plate from the table,Any obviousl y out ofposition hol e can be setup again using the avai lable co-ordinates,m arking again, this tim e with a Iargercentre drill,Do also mark the plates centre posi tion whi lston the table as concentrici ty ofthe m ounti ng position is im podant.

Using a Rotary table Using a rotary table would also be worth considering and where possible be as accurate asusing co-ordinates.The Iimiting

53

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6.A 125 dividing plate m ade using an actualsi ze CAD print. values from the star 'tcould be produced 7.826087, 15.652174, 23.478261 and 31.304348,etc.These could then be conveded to 1 turn + 1.826087,2 turns + 3.652174,3 turns + 5.478261 and 5 turns + 1.304348,etc.Finally they could be simplifi ed 1 turn + 1.83,2 turns + 3. 65,, 3 . turns + 5.48 and 5 turns + 1.30,etc.then used to setup each posi tion as accurately as is possibIe. This wouId avoid an accum ulating error and probably be acceptable w here a high degree of accuracy is notbeing aimed at.Ifthe plate were used vi a a worm /worm wheel, the im provem ent in accuracy this provides woul d be a distinctadvantage.

A Com puterM ethod If you, or a friend, have access to a com puter equipped wi th a CAD program factor would be if the angle between there is another method that may prove es on a di visi ons could easil y be establi shed using useful.Placing any num berofhol the tabl e' s calibration.Typically,a need for PCD using a CAD program is both easy 45 di visi onswoul d require anangl e of360/ andfast.Place a smallcircie forthe divi ding 45 perdi vision,being 8 degrees,W i th a pl atecentre plusjustone onthe PCD then 60:1 ratio worm/worm wheelthis would asking the program to repeatthis round the require,oneturn(6 degrees)plus2whole circle the numberoftim esrequired and the degrees using the table's calibration resultwillappearalm ostimm ediately. Print this out and fi x it to the plate making ita practicalproposition. I fthe need is for46 di vi sions,atfirst already prepared.Centre punch through sighta relati vely simple requirem ent,the each smallcircle,also thatin the centre, angl e between di visions would be 360/46, remove the printand exam ine resultlooking being 7. 826087 degrees.This would be forany punch marks which are obviously one turn plus 1.826087 degrees on the mi splaced. Make any adjustments tabIe's calibration for each division, necessary and use a centre drillto mark tion,again i nspect.The ring of obviousiy anim practicalproposi tion.Using each posi lled.Being very quick tbe rotary table bas therefore lim ited use holes can then be dri com pared to working to X/Y co-ordinates it form aking dividing plates. However,as a last resort, a Iist of is an idealm ethod for Iarge num bers of

54

7.Photo ofthe dividing plate printot/f. di visions.Photo 6 shows a pl ate having 125 di visi ons being used whi ch was made this way.Photo 7 shows the com puter printout.Making the circl es on the printout smali, say 1 m m diam eter,willhelp to ensure thatthe centre punch marks are made accuratel y. W hiIst fast, the system does have Iim i tations,m ainlyaccuracy.The com puter data is ofcourse extremely accurate but maynotbe maintained atthe printoutstage. Having tried the m ethod with a num berof printers very smallerrors in the printers' feed m echanism s result in the PCD sometim es being very slightl y oval.Not surprisingl y,a printoutusing a professi onal pen plotter produced a perfect resul t,at least as far as avaiIabIe m easuring methods would show.A cheap dotmatri x printer also did rem arkably weli,with ink

jets and Iaserprinters showing varying amounts oferror,even here no more than 1 m m on the PCD. Atworst,the errors in the diameter wouldproduce onsym i nute errorsin angular spacing and so would be perfectly adequate providing the radius errorofthe PC D could be accom m odated in the mechanics.lfusing a conventionaldetent its plungerwoul d notengage reli abl y due to the varying radius. Photo 6 shows how this was overcome using a heavy gauge, Spring wire detentthat would cope w i th Si deways errorswhil ststillholdingthe plate in terms ofrotation. Ifthe plate was to be used frequently on a norm aldivi ding head With plungertype detent, the plate could be used to makea second plate -the hol es Onthiswould then be on a constantradius.

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M ade on the Iathe's faceplate For a sim ple plate, using the m andrel m ounted gearas described atthe stad of this chapterwould be an obvi ous choice. Formore complex num bers,mounta disc onto the face plate and using a suitably di vided paperstrip round the outeredge of the facepl ate willenable the disc to be m arked for m aking a di viding plate. The positions can be established using an automaticcentre punch mounted on the top slide,or a sm alI drilling spindle. This process should m ake a dividing plate accurate enough for m ost applications,

55

&

8.Driving dog for between centres work,together

withdriver(on the ye#lstl pplied with the dividing head.

especially i fused via a worm/wormwheel, ratherthan direct.See Photo 4 Chapter3. Taking the need for 53 divi sions,as m enti oned earlierin the chapter.Ifusing a 15O mm diameterfaceplate and marked with 53 di visi onson its peripheryitwillresul t in divisions of 8.89 mm wide,so a fair degree ofaccuracy should be achievabl e.

A Betw een Centres driving dog N ot having been abIe to 1ocate a comm ercialdriving dog suitable for use when dividing betw een centres,I am including detai ls ofthe i tem 1have m ade

myself.As described in Chapter3 (see Photos 12 and 13)the dog needs to be firml y held bythe dri vingdevice onthe head itself.To cope w i th differing workpiece di ameters a flatpl ate is used forthe driven portion ofthe dog,it is atso drilled and tapped offcentre so thatitcan be rotated to extend the range even further, see assembiy drawing.Photo 8 shows the dri ver,suppli ed with the divi ding head,on the Ieft,and the dri ving dog on the right.

56

The publi shed drawi ng should gi ve aIlthe detail s necessary formanufacture.

Rotary tables M aking rotary tables inthe home workshop i s notreally in keeping w i th the title ofthi s chapter,'sim ple divi ding devi ces'',but is i ncluded as i tisthe mostappropriate place inview ofonl y a paragraph being required. A num ber of suppli ers to the home w orkshop provide designs and kits of material s formaking workshop equipment, rotarytables being one such item.Photo 9 showsa typical1O0m m tablethat has been m ade this way, Iarger sizes are also avail able. lf you Iike m aking workshop accessori esthen a rotarytabl e would make

9.A 100 mm home made rolary table.Casti ngs forthese are aval lable from a number ofsuppliers and y ' r?othersizes.

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an interesting project. None, to my knowledge include faciliti es for mounting divi sion plates butdesi gning thi s in shoul d notbe unduly di fficul t. The designfora basicdi viding head is

thesubjectofthenextchapter.Whi Istbasic, itwillbe sufficiently adaptabl e to meetthe needs ofmany workshop owners.

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C hapter 7 S ho p -m ad e B as ic D iv id in g H ead For many workshop ow ners the di viding head featured in this chapterwillbe more than adequate for their dividing requirements.ltdoes though rely on a set ofIathe changewheels being available as itusesthese to provide the divi sions.Using

a si ngle gear,Photo 1,wi llprovide many ofthe more usualdivi sions nam el y 2,3,4, 5,6,7,8,9,1O,11,12,13,14,15,2O,25, 3O,35,4O,45,50,55,6O,65,70,and 75 using the mostcom mon setofgears,20 to 75 by 5's.W ith a forked detent,Photo 2,

1.The single gearversion of the dividl hg head.

(indexer)

16, 18,22,24,26,28,8O,9O,100,110, 120,130,14O and 15O are also possibl e. The range is fudher extended by the addition ofa pairofgears as illustrated in Photo 3.This,essenti ally the same as adding a worm and wormwheelbetweer divi ding plate and the di viding head spindle. There is though one fundam ental difference, a worm/ w orm wheel configurationwillalwayshave a ratiotoone, typically 40 :1,60 :1,etc.A pairofspur gears may though have a more com pl ex ratio,forexam ple,gears of45 and 20 will have a ratio of9 :4. Considering the range ofgearsabove, butonly one ofeach,around 6OO different combinati ons willbe possibl e.Thi s based on,anyone ofthe gears being used as the dividing wheel.and as the driven.Also, replicating the si tuationwhen using aworm/ wormwheel,thatis inputruns fasterthan the output,the driverwheelmustalways 3.The three-gearversion.

59

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be sm allerthan the driven. W hi1st the arrangem ent w iII give divisions of up to 1050,doubl e this ifa forked detentis used there is consi derabl e duplication and very many gaps,Iarge at the higher numbers. On the basis that highernum bersare unlikel y to be required, the tabl e forthi s setup in Chapter11 gives aI1possible values up to 100 thatare not obtainable wi th a single gear.The tabl e, having been produaed w i th a com puter program I have developed,covers aII di visi ons mathematicall y possible.ln som e casesthe head m ay need adapti ng to m ake some combinationspossible.Typicall y,wi th the base plate fitted as seen in Photos 1 and 3 there is a Iimitto the si ze ofdi viding gearthatcan befitted.The base i s mounted usi ng 4 screws on a square formation that perm its it to be rotated 9O degrees rem oving the restricti on. In this posi tion howeveri tmay notbe easy to mounton a Myford Seri es Seven type bed.In anycase some readers maywish to use the table in connection wi tha divi di ng head oftheirown desi gn. An interesting and usefulfeature of these numbers is thatthey,with only two exceptions,are achieved stepping from geart00thto geart00th.There istherefore no need to move across more than one t00th avoi ding the com plication thatthis brings, Three possible hi gher numbers that mayfind a use are also incl uded' ,these are 125,2OO and 360.In the case of200 the workpi ece rotatestwi ce and for360,seven times,see Chapter4 foran explanation. From a divi di ng aspectthere really is no difficulty created by the workpiece rotating more than once and i f a gear is being m ade i tshouldnotpresenta di ffi cul ty.

60

Ifthougha diali s being m ade,probablywi th three Iine Iengths,ten's,five's and units, greatcare in the preplanningand theactual calibration willbe required to ensure the Iine Iengths are m ade in the correct posi tions.Probabl y the easiestwaywillbe to m ake aIIlines ini tially of equalI ength. then marking the position ofthe five' s and ten's wi th differentcoloured pencil s.The workpiece can thenbe rotated,againusing the di viding head,and the appropriate li nes lengthened.

4 BOn'ng the body to take the sleeve . bearlng.

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lti s notmy intention to go deepi y into the manufacturing processas to do thi sforthe shop m ade itemswould expand the size of the book beyond whatis acceptabl e.Itis also probabl e thatthe hom eworkshop user w ho has arrived at the need to use a dividing head willhave fairl y wellmastered the processes necessaryto make thisitem. Shoul d you though feela need forgreater help than this chapterprovides,you m ay consi derobtaining the foll owing additional books in the W orkshop Practice Series. Num ber 34,d'Lathework - A Complete Course'' and num ber 35 'M illing - A Com plete Course' '.Construction of the divi ding head in this chapteris covered in detailin num ber35.

The Body Part1 Photo 4 shows the 44m m diameterbores forthe sleeve bearing being made.Just visible in the photograph are three support pieces mounted on the machine table. Providing the initialbore is centralin the

width (5Omm)ofthe material,the suppod pi eceswilenable the partto be turnedover and bored w i thout the need forfurther measurement.A high degree ofaccuracy

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is not required as the bearings are eventuall y bored in situ. Fi tthe bearings using a two-partresin adhesive having made the bearing outer diameteraboutO.O5mm I essthanthe bore into whi ch itfi ts to allow space fora film of adhesive.Setupand bore tofinaldiam eter as shown in Photo 5,boring through both bearings atone pass to ensure alignment. Pay particular attention to clearance between boring head, workpiece and clamps, etc. Sli tting the body is straightforward,

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Photos 6 and 7 show ing the setups. However,depending on the diam eterofthe saw the setup in Photo 7 may notfully com plete the slit. In this case itw illbe necessaryto complete i twi th a pad saw,or a pi ece ofhack saw blade,pad wrapped in c10th to provide a safe handle.

Spindle Part12 First, make a plug gauge to accurately establish the diameterofthe bore in the bearing,This willgive you the dim ension forturning the spindle bearing surface that

5.Finalboring the bearings.

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willneed to be made using a I efthand knife tool,maki ng itim possibl e to use the spindl e itsel fto testits fit. The essential requirement for the spindle is concentricity of the four main surfaces the gear mounting,bearing, chuck mounting and the internalMorse taper.This can only be easil y achieved if these are made withoutrem oving the part from the chuck.To do this,machine the gearmounting,bearing surface and chuck mounting whil stsupported by the tailstock centre.Then,rflmove the centre,positi on fixed steady,re-engage centre,setsteady arms,rem ove centre,bore and m ake the taper.

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help. The drawingsdo notindicate a method forcoupli ng the two linked gears used on the three gearversion as i tisassum edthat the pads used on the lathe for a sim ilar requirementw i llbe used,or an identical set made forthe purpose.Also in thi s respect,the drawingsassume gearshavi ng a 5/8in bore and 3/8in w idth, som e dimensi ons mayneed changing i fthe gears to be used di #erfrom thi s.

Assem bly On assembly the detentmay need a Iittle

through 18O degreesand re-engage.There should be no perceptible rotation of the divi ding head spindl e.I fthere is,then fil ea very Iittle from the appropriate detent locating face and testagain,repeat as necessary,Carry outa similarsequence, this time with the innerfaces ofthe fork. O ther than that, assem bIy is strai ghtforward. In the next chapter a m uch m ore advanced dividing head i s detail ed,yet again being made from stock materials i s

wellwithi ntheabili tyofthemajorityofhome workshop owners. Being a m uch m ore

adjustment.Todeterminethis,fullyengage advanceddesign,making itshould provide the detentthencarefullyremove itand turn

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G hapter 8 A n A dvanced D esign D ividing Head The divi ding head featured here is m uch more advanced than that in the previous chapter, providing three m ethods of establishing the divi sions: 1. Directly off a gear mounted on the divi ding head spindle. 2.Directly of' fa di vision plate mounted on the di viding head spindle. 3.Interposing a worm and worm wheel between the dividing pl ate and spindle. T hese are based on the sam e assembly so the constructorcan choose to make any or alIof the arrangements using the one main frame,Photo 1.There would though be Iittie pointin constructing method one onl y as thatin the previous chapter is easier to m ake and m ore adaptable having the addi tionalgeartrain. W hen making the head employing method

carrier(Par141)enable the arrangement to cope w ith a w ide range of gear di ameters.ln Photo 3 a division plate has beenfittedand a detentassemblym ourted offthe arm using the two Sowerholes.The detentassembly has a slotted I owerarm

enablingittofunctionwith(di fferentdivisi on plate hole circle diameters). Photo 4 displays the reaireason for establishing the desi gn,the inclusion ofa worm and wormwheel.Unlike com merci al dividing heads, and probably m ost publi shed designs for hom e workshop constructi on,the design is notIimited to a single worm/worm wheelratio.Using 20 DP gears the assem bl y willcope wi th a gear up to 65 teeth,Iimited by the gear hitting the rearofthe divisi on plate.The holes in the end plate willthough provide sufhci ent

three,the method one system,(needing adjustmentforgears up to 75 teeth and Only two simple additi onali tems),is well could be accommodated by increasing the worth provi ding for its ease of use wi th Sim ple numbers,3,4, 5,6,etc. The plate on the I efthand side ofthe

65m m dimensi on for Part8 to 80mm and

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assembl y (Part3)can be swung into any comm on dividing head and rotary table posi tion and inPhoto 2 has beenfittedwith worm/worm wheelratios 4O,60 and 90:1, th 18 di fferent divisi on plate a sim pl e detentmechanism and moved into and fitted wi an easy to operate position.The rows of hole num bers w hich appear to be the tapped holes and the slot in the detent standard,The tables w illcover for the

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si zesandthe 18di vision pl ate num bersand attem pted to extricate from the Iists produced,those divisions thatcan onl y be achieved wi th thi s multi -ratio divi di ng head. majorityofthedi visionspossiblewiththese This is also included in Chapter11 though wi llalso be possible using ei thera 40 or60 lcannotguarantee thatlhave notm issed t00th gearand willbe covered by the Iisti ng an entry or two having had to do this ly. forthese.There are though a few thatcan manual only be achieved using another gear, As designed division plates up to typical ly wi th a 35 t00th gear,divisions of llom m can be accom modated and as the 77,91 and 133 are available butnotwith mounting arrangementcan be rotated into the 4O,60 and 90:1 rati o di viding heads. any posi tion there is actually no Iim i t in s as a result ofthe Ihave run my com puterprogram for diam eter.The Iim it i each one ofthe standard changewheel I engthofthe 'detentarm carri er''(54)and

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2.W ith only the addition ofa simple detentassem bly dividing using the I athe' s changewheels / ' s possible.

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can easil y be increased in length when 150m m diameterplates couldal so be used The uni tin the photos is being used wi th dividing plates m ade from sheet steel. These punched plates are m ade on computer controlled machines and are, even though veryeconom i call y priced,very accurate. They are 3m m thick but the design should cope with plates up to 6m m thick.Itwould though be tidierto increase the 14 x 1m m thread on Pad 8 to 15m m Iong and the 79m m dim ension on Part27 to 82mm .Some modi fication,perhaps a bush,m ay be,may be needed fordi ffering bore di am eters.lncidentally,being punched plates the holes go through w hist the commerciaithickplates have hol esthatare blind,surel y Iesseasyto keep them clean. The design isbased on being usedwith a3 l 4in centre heightMyford Seri es Seven Iathe to enable itto be used with the Iathe's taidstock for between centres work.For other centre heights the appropriate dim ensions can easiiy be changed. m ounting arrangem ents may also need changing. In practice,between centres workon the lathe is Iikely to be oflimited,if any,use in many workshopsin which case the usermaydecide to Ieave the design as published,i tsrealhom e being onthe milling machine.

free-cutting and i s also avail abl e in some square barSi zes.

Main assem bly vakethe endplates(2)todrawi ng,except for hole B that should only be 4Om m diam eter.Assemble togetherwi th items 5, 10,11and 12,and usi ng item l3mountonto the l athe as shown in Photo 5,The boring

bari slightlygrippedinthethree-jaw chuck, justsuficientl ytoallow li ghtcutstobetaken

butalso perm i tting the tail stockto feed the barthrough the end plate to m achine the bore.Do notuse the tailstock to feed the boring barthl 'ough both platesatone pass butdo one plate ata time.However,still leave tbeir bores a little undersi ze,say minus 0.5m m .

Having bored both ends adjustthe boring bar fora finalcut,doing firstone plate then Ioosening the chuck moving the barfurtherinto i tand then making a final #.The mostadaptable version,providing cuton the otherplate.Absolute accuracy as a resultthe widestrange ofpossible isnotcrucialas the bearings canbe turned divisions. '

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END PLATES 2.

to fitthe bore.The procedure wiflthough ensure both plates are bored to the same si ze,atcentre height and parallelto the uni t's base. Bearings -Parts 1 and 9 Place one piece ofcastiron in the three-

REAR BEARING 1. firstbearing.Machine the outerdiam eters and both endfacesto dim ension,doing this using Ieft and right had kni fe tools and withoutremovingthe partfrom tbe m andrel. This process ensuring thatend faces are square,and theouterdi am etersconcentric,

jaw chuck,facetheendandmakethebore withthebore.Repeatfortheotherbearing. a little undersize, say m inus O.5m m . Reverse the partin the chuck and face the otherend Ieaving i tabout1mm Iongerthan drawing. Repeatforthe second bearing. Leave the second par ' tinthe chuckand finall y bore to si ze. W i ththe toolstillsetat thatdiam eter, return the firstbearing and bore through atone pass ensuring as a resul tthatboth are the same si ze.This is essentialas both are to be machined on the same taperstub mandrel. M ake a taperstub mandreland fitthe

Slit,make recess and drilland tap M6. Positioning the hol e for tapping i s easil y done as illustrated in Photo 6 using the bush detailed in SKI. Spindle -Part8

Mounta I engthofmateri alinthethreejaw chuck supporting the outerend with the tailstock centre. Skim a shortIength ofthe outerdiameterforsupporting wi th a fixed steady.Remove the centre,posi ti on fixed steady, refit centre, set steady arm s, rem ove centre and make the 14 x 1mm

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finally the two slots as per stage 2 the process shoul d be relativel y easy.Photo 7 shows the partbeing setup formachining the firstsi de at3.5 degrees. Index fingers parts 29 and 31 Cut two pieces of 3m m brass,a Ii ttle oversi ze,and m ark outthe posi tion ofthe 18mm hol e.Setup on the Iathe facepl ate and bore the hole. Follow by boring to

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38m m diameterand 1.6mm deep.Repeat forsecond part,Mark outoutlines and cut roughly to size,then carefullyfile to profile. The 38mm diam etercounterbore giving the outline ofthe par 'tin thatarea. Division plate nutPart28 This is a strai ghtforward i tem to make but, when assembled,mustclam p the fingers So thatthey can be m oved,butonly wi th some resistance.lfeasy to move then they may move whilsttraversing the detentfrom One holeto another.Toachievethe requi red acti on,dish the frontfingerto give theeffect

ofa disc spring.Form this by clamping the fingerin the vice,backed by a pi ece ofsoft wood and wi th a piece of22m m di ameter steelpl aced centrall y overthe hole.Tighten

the vice very slightl y to form a dishjust suffi cientto provide adequate friction. The rem ai ni ng padsshould notpresent any problem .W hen assem bl ed youwilbe in possession of a quali ty di viding head, very robust,and able to provide m ore divisi ons thata commercialsem iuniversal dividing head.You willalso have the sati sfaction ofhaving made the item .

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draw n has ten teeth Iim iting repeat sequences to 2,5 or 10.Should another sequence be required a second ratchet wheelw ould have to be fitted,W hilst internal,changing the wheelis nota Iarge

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dividing and as such is outofplace in thi s s thougha Iini ng toolforengraving book. Iti dials and therefore used where dividing is very much a partofthe task being carriec! out.Also to my knowledge itis notan i tem that is availabl e com mercially and has therefore to be made in the workshop. The purpose ofthe tool,Photo 1,i s tc' enable Iines on a dialto be m ade to set Iengths and automatically in the order required. Typicallyforan imperialIeadscrew dialthe sequence would be,one long

cOntrolled by the stop disc (4)this is not adjustable butbeing such a simpl ei tem, otherscaneasil ybemadeasrequired.The sequence i s controlled bya ratchetwheel (12)and pawl(11).The ratchetwheelas

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0.025 mm making itvery close to 0.001' '.

Thiswould need a ratchethaving 8 teeth and giving, one Iong (0.1mm)threeshort (0.025 mm), one Iong (0.1mm)and three short(0025 mm). Tj ne uni twoul d normally be used on ,

the Iathes cross sli de with the top slide removed as seen in Photo 2. The positi on of the Iathe's saddle woul d be set and locked to give the line Iengths required and the cross sli de setto gi ve the depth ofcut.

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. Iengths by observing its Ieadscrew di al

readingsand having to countthe number ofIines made ofeach Iength.Photo 3 showsi tbeingusedinconjunctionwiththe basicassemblyofthe dividinghead inthe previ ous chaptermounted on the lathe's bed. Ideally the cutting tool should be nominallyatthe Iathescentre height, itmay therefore be necessary to change the 40 mm and 72 mm dim ensions ofthe angle

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the reason forthe flatonthe second side sli ghtsi dewaysmovementattheendofthe Oftheslide(1)ensuringthattheslidemates pawldoes notresul tin itcatching on the withslide body(2)on i tssides.Otherwise, slide and preventing itengaging with the

machine peadscrew dial. Thl s is using the ' dividing head from lheprevious chapterï

any clearance betweentheirtwo di ameters Would resultin the pads mating only in the bottom ofthe sem icircular portion ofthe Slide body. Anotherfeature forwhi ch the reason j S n0tobvious i s the 12 m m x 1 m m deep

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tailstock setoverifrequired. Next,cuta Iength ofm ateri alforthe slide body and machine itto Iength,the thicknessbeing I eftat16 mm atthis stage, machine the recess and slotC also holes A,B.

Carefuliy mark both ends ofthe body wi ththe centre posi tion ofthe 10m m radius and centre punch.M ountthe slide bodyor: the face pl ate using an angi e plate,or t) Keatsangle plate,asin Photo 4,and usinf ? a centre finderto accurately position tht e

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partforboring.Cuttwo smallpiecesofsteel and drillto fi tacross hol es B ateitherend. These allow each end to be bored 20 mm diameter as a continuous rather than an intermittentcut. Bore the sl ide recessinone end as shown in the photograph to be a Close fi ton the sli de. Remove,reverse and repeatthe operation on the other end. Flowever,ifyou have a suitable angle pl ate, ratherthan the Keats,and a Iong boring tooli twould be betterto machine bothends

without rem oving the part.This would guarantee alignment. Nextmillthe firstflaton the sli de -the one thatpasses along the whol eI ength of the pad.Iti s essentialthatthis i s the same depth along the Iength orelse the partwill getwi deral ong its Iength causing i tto be Ioose atone end ofi ts movement.W i th this in m ind,hold the partin a vice and wi th each end supported by preci sion parallels. Take a very Iightcutofabout0.05 mm deep

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al ong the Iength to check ifaIlis well.Any errorwillbe obvi ous in the varying width of the flatproduced. Drilland counterbore holes C,and use these to fix the partto the slide body for machining the second flat.Again support the ends on parallels and machi ne both slide and body to achi eve the 15 m m

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C hapter 10 P rim e N um bers

The reader may ask,w hat are prim e num bersand whatistheirrelevance to the

subjectofthisbook?

Taking prime num bers first.These are num bers whi ch are only di visible by 1 and i tself' ,typicall y,1,2,3,5,7,11,13, and on up to infi ni ty.There aretherefore very many and even a very advanced mathematician willbe unable to recognise beyond values ofa hundred orso.Forthis reasor tables of prim e num bers are an essential requirem ent w hen dealing with such num bers.

Prim e factors These are those prime numbersthatwhen m ul tiplied togethergive the num berbeing considered.Forexampie,prime factors of 1O5 are 3,5,and 7. 3x5x7 = 1O5 The method ofarriving atthese from a Iist ofprime num bers is as follows 1.Is the numbereven? No therefore start using the tables, 2.Look up 105 andthe smal lestfactoris 3. 3.Divide by 3 to give 35. 4.Look up 35 and the sm all estfactori s 5. 5.Divide by 5 to give 7.

6.Look up 7 which is found to be a primq num ber. 7.The prime factors are therefore 3,5 ancl Z'

Prim e factors ofan even num ber Forthisfi nd the prim e factors of132. 1.Being an even num ber the smallest factoris 2. 2.Di vide 132 by 2to give 66,again an even num ber. 3.Di vide 66 by 2 to give 33, 4.33 i s an odd num ber 5.Look up 33 and the smallestfactoris 3 6.Di vide 33 by 3 to give 11. 7.Look up 11which is found to be a prime num ber. 8.The prime factors are therefore 2, 2,7

41 willgive 47 also found to be a prime number.The primefactors therefore41x49 w 1927.Obviously a time consuming task wi thoutthe aid ofthe tabl es. Unfortunatelythough,space Iim i tations in this book restrictthe tabl e to 2000.To minimisethe taskoffinding prim e numbers above this a Iist ofprim e numbers up to 10 0O0 is also included.This willavoid looking forthe prime factors ofa num ber which is in itselfa prime num ber,typicall y, 4933.

having a worm /worm wheelratio of40:1 then for1turn atthe output40x49 holes on the plate willbe passed,thatis 1960.As explained in earlierchapters any number that divides exactly into this willbe a possible divi sion.To easily arrive atvalues possibl ei fwillbe necessaryto find itsprime num bers as explai ned above.Dividing by 2,2,2,5,7,7 giving 980,490,245,49,7, 1. The resul twillbe 2,2,2,5,7,7.which are ofcourse,num bers ofholes and m ust be divided into 1960 to determ ine the

Num bers greaterthan 2000

division possible.Ignoring the higher

Firsttake note thatno numberin the range 2000 to 10000 has a pri f' ne factolgreater than97, thatis24 possibili ties.Inthis range therefore take the following approach: First,ifeven,divi de by2,ifnottry 3,5, 7 jl etc. untilthe lowest prime factor is found. Having foundthis,divide the number by this value and in mostcases the result willthen be below 2000 and the table can then be used. lf not, repeat after first checking to see that it is not a prim e number, The procedure isunlikel y to be astime consuming as mayfirstbethoughtasevery third num beris di visibl e bythree,every fi fth num berby five,every seventh numberby Seven and so on.Asa resul tmostnumbers have a Iow firstprime factor.

numbers possible as they are unlikely to havea practi caluse, the more usefulvalues are typi cally. 1960/2x7=140 1960/2x2x5=98 1960/2x2x7=70 1960/5x7=56 1960/7x7=40 1960/2x5x7=28 1haveal so ignoredthe lowernum bers, such as 1960/2x2x5x7 = 14 as these are bound to be achievable by an easi erroute. As already m entioned the prim e factors relate to the number of holes traversed, therefore for 28 divisions the numberof hol estraversed willbe 2x5x7 = 7O.As the di vision plate being used has a ring of49 holes.thi s willbe achieved with 1 turn plus 21 holes.

and 11.

W hat use are Prim e num bers?

2x2x3x11 = 132 Both these examplesare sim pl e and prime

How thenwillthis be ofuse when divi ding? W ell,you may have acquired a second

factorscoul d easil ybeestabli shedwithout the aidtables.

hand dividing head withouti ts manual,or two initi alvalues,40 (2x2x2x5 )and 49 ( purchased an additionaiplate notonyour 7x7 ).However,lhave presented the

However,considerfinding the factors of1927.This,itwillbe found, isnotdivisible by 3,5,7, 11,13,nor 17, etc.Its I owest ding 1927 by prime factoris in fact41. Divi

manual's list,in these cases you willhave to resod to calculati ons. Considera divisi on plate having a ring Wi th 49 holes. Ifthi s is used wi th a head

90

Some readers willlam sure have noticed thatthe prim e factors 2,2,2,5,7 and 7 are in factthe prime factors ofthe explanation in this form as when working wi th thethree geardividing head inChapter 7 the ratio may notbe a sim pl e one,say 60:1,butwith gears of45 and 35 the ratio

91

i .p..11uad

willbe 9:7.In this situati on itwillbe easi er to work wi th the numberof hol es passed whi ch in some caseswillbewith more than one revolution of the w orkpiece, as expl ained in Chapter4 Exam ple 3. Having provided the Iist of prime num bers lhave decided to incl ude briefly som e details on another and m ore dem anding use for them . This is establishing gearsi zes forcomplex rati os, itmay ofcourse create a need to use your divi ding faciliti es formaking a specific size gear.

1000 315 Thatis multiplying both top and bottom by 1000. Factori sing both 1000 and 315,using the tables,we get 2x2x2x5x5x5

3x3x5x7 Cancelling outone 5 we get 2x2x2x5x5

3x3x7 I g n or i ng t h e f ac t t ha tgearscannotbe made Designing gearchains wi t h so f ew t eet h, we stillhave a problem A typi caluse for prime factortabl es is in as t her e ar e 5 dr i ven gears but only 3 thedesignofgearchains requi ring com pl ex .# ratios,such as when determining change drivers.Ifwe m ultiply 2x2 and 2x5we getdr i v er s and 3 dr i ven as f ol l ows wheelcom binations for cutting Metric 4x10x5 threads on a Iathe with an im perial Ieadscrew, and visa versa.Also when 3x3x7 cuttingworm wheel swhichsimilarlyrequire M ul t i pl yi ng each number by 10 willgive compsex ratios. practi calgearsizes and being in 10' s are Iikelyto be avail abl e with the Iathe.W e get Exam ple 40x100x5O Ifrequiring to cuta thread wi th a pitch oflmm or a I athe havi ng an8 TPII eadscrew 30x30x70 the l athe's mandrelwillhave to rotate 25,4 A 1OO t 00 t h gear is unlikel y to be available tim eswhilstthe leadscrew rotates 8 times. This therefore requires a rati o of25. 4 :8. so reducing this to 50 and the 70 to 35 will o. Expressi ng thi s as a fraction we get: retain the rati 40x50x50 8 1 =

25. 4 .314960629 Accepting that an exact rati o willnot be possible,this can be simplified to 1 .315 Asdecimalsare notappropriateto fractions thi s mustbe written as

92

30x30x35 The combinati on now requires two 30 acd two 50 t00th gears,but,increasing one of each bya factorof1.5they become 45 ard 75 $00th gears,again retaining the ratio. The resul ting gearchai n becomes: 40x75x50 30x45x35

' rhis gives a TPl 40x50x75 8 x = 25.3968254 30x35x45 pitch in m illimetres is very close at 25.4 = 1.OOO125m m 25.3968254 At25 pitches this is a totalerrorof plus 0.003125 m m or if you are imperially inclined O.O00123in in practice a negli gibl e amount, This can be sim plifi ed bydi viding 50 by 10

and 5,30 by 10 and 35 by 5 to give 4Ox1 x 75 8 A 3X 7 X 45 sim pli fied giving 40 x 75

8x

= 25.3968254 21 x 45 Thi s willbe easierto setup using onl y two drivers and two dri ven butrequires a 21 t00th gearthatwitlbe specialfor most changewheelsets.

5 .I

œ N X œ = X X & & X

ko (D. X r

X X LO a. C9 = O a. to X C)

= O & a. X

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n. (9 LD X X C<

v- X O &. >. =

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= X r œ X

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X = A X X

X O X X X

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C m1 X tD a. X

r< Q. X X v- Q.

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m

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94

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tD =

m co ko s. co (n X m ko g) c) % ok

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& œ&N œ

m

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m v D m r m N m c m

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m

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tn co (r)

c) = hu >.>. >. œ c) >. = = >.= v= co c)c) >. s.= = tn c) >.>.* o) % = v- = >.tn to >.>.co % = >.c9 v- vu7 fo >.co % c)tn m = >. tn œ) to ko r9 tn >.>.s.= cq % tn % e) v.cn tn tn = cu = c9 tn v= c9 = co >. m m x œ m c N > c & c c œ c v O = c O > & c & c & c ko = u7 Ctou ( un7 t%n

= >. = (D >.

vJ) = (n

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c) co ch (J: c) >. >. (9

(o v-

>. co to c)(n = >. (D co c)z) tn ot tn m (D to co to >.= c)=

Lo u7 >.xr = >. c)tn >. c9 to xr % (D >.(9 c9 5)vtw T) % co * = = = >.O tN to % (M to J) >.>.= = = % tn Cq e) >.= >.e) s. c % = v N X & = N O & O = & W = 0 = = 0 & œ & & o k koo œ tD >. u7 * to 0) 43

= = * b. = c)>.>. co = >. (D J) >.= m >.= >.(D tn e) (D = >. >.to to >.e) e) tn = to (D (o to cp tn = N tn = ko to to % co >. e) >.s.tn % % % tD >. (n tn cn = c) % tn ol(N to >. = = tn = tn tn G1cu ko = = >. >. c) œ œ = m % = O m m N = m m > œ N v O m = c c c c œ & (D = (D œ œ O c e%

= * O co ?< =

h= (D = =

= ry(7: 0) = tn F) = c) œ co m = = m >. (D = co >.>. >.= = c)= N (D >.(D >. = = tn O >. >.>.ço (D fo = tn v- >.v(D b.(o u7 (o tn v- >.= c) to (D ro % ko e)c)O tn c to to O O % v- tn c)c)>. to to = tN % >.>.= >.>. % O D œ = & O O X = œ = c œ = = > = X N W c & & & u (D3 ( (D DN œ * (D 0 œ)

>. = tn m

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N = v- >.tn (9 co (D (D = œ ot = m = c)(n œ)* co >.(0 = tn c) >.*-

% >.to >. to >.c)>.co (m % (D u7 N % m cp F) >.s. % * to (D co >.m = tn (n (n % % to e7 = cn s.= N cu tn tu % m (n = O c)o: m œ œ œ m N N N & N = & > W c N O X N m c c c = & t o( D >. >. >. >. * >.c) >. .

= 0) = * >. * O

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s.

c) c) c (m9

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co >.

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(p

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0) >. m c)m s.c u7 m cn N >. c) >. c) c (D u3 % œ

o) u7 e) >.o) N m * s. c >. = c) >. to >.œ >.>. %

>. c n ko u7 t a) m to >. >.c z) >. otv œ-

t nc cno ( >. co r w o >. (o (o co

O% c 9% % * >. >' = = cu t n c9 % %

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>. u7 O r = s. to u7 C9 (9

= fo = % =

= œ >. u7 t %n c)(n m (D to (9 >' v- =

p %- = u7 c>.( en ( %n = c9 m tn m % % c to tn (M N = = =

c) % o) ko c%o = = c œo N N m (n s. tu ko % >. to v- (9 = tn =

m fo s. to t cno c co v toc) o % v ko- = to >. s. t % N o = pv

% u7e)% ko % f ot o cu œ g)e)t nt o cq cu t o (n f o to % % u7 % co O >'= >'C >. X c)O >. O = O >.O c)c)c >.>. m tn tn c3 % % G1t n O c7c (N cu ko (n co ( n % cu m co cu u7 % cq < tn (J co

P C< <= O h w = m m

= Cq C X X X = &

>' * O N

= O = D

C tn9 * = = O m =

O (9 N. eq @) b. m >

X œ C >

N * = œ

= tY = v- 03 % >. ( 9 (9 = = m

= œ X œ >. (N O O >. >. 0 ) O m m m œ

= = >. m

O = Otn s. = o m

c) m = % m % s. c) co m m œ

c mp c cp oc cu t=n m co >. >. ( n s . c) m m a m m

m m X & D a

N c X c m œ

N m m = % œ

= œ O c mC

N m O c m m o c

m m = & c X

c m & & = X

c o O c œ X

N m = = c X

= m N & c X

O o m c & %

m m m c œ %

O m oN o%

*

; c) F) = >.*

v *. = >. c c) >. N N >. C m >. = O < to c >.>. O (9 = N >. œ œ) u7 ko œ ( c) to9 t %n X % m to to (M = = N >'

X m = = = œ

= m & & X œ

& c X m

N & N œ

= c X œ

O & O a

>.m o) V ; V >. 5) >. ;: (p c c) >. = 0) N tn = c) co m tn = 1Z c) D N c)(D ( D >. to cn m= c) >.cco) c) c

œ (9 X

N c m & X X

>. to tn to v-

o m N o % O

* m N & c O

96

c c = & N D

= m X & X X

N m mm m%

N o mN O%

= œ c c % %

m a O o c %

m m & o N %

o m m o œ %

t>o. p tox

c o O c c %

>.

ro

*' C) 05 s.

m X U< C) CD * f< = O N' J) * * (D

* O C) f< v'b'* b' >' = * * C1 (D

(X U* * C) *- O >' C X

0 )= tn * (M b.

m = =O * * >' (D

b .= O m X (D * * CD (D

> .N.vr< * N'

>. cj(7)c * h- c * m * m b.c(o D s. * s. (D c)s. o s.m (D (D % b.to cjm s. 0) s. to ID X (D * e)s. g) ro g)

to s.m b.?< (D e) lD c) (D ( kn o

= (M to C c) % v t<-

= O v- s.m u7 ?< * P- (D c % m (D s. U< m U- c) W)

v- v- = tn (D * tD ?< c) * c) (D >. kD s. (D

tD * O X * c)F) v- O vO m O LO O >. b.= @) c) = v' QN GlGl >m m m œ = N = O œ

m e) s.c) on % (D % O tO c s.x- s.co X V * O CD s. . s.m . X V O O to >.Ct= = = X CV O QN V v- b.= C) O (N œ v- = C9 X v X m X m N o O œ

co >.s. c)s. % LD u7 * O m m m tn ro * (D V * O uj s.c) .... W O O V O e)co s. s.m = O txl O W = U- = X * v- (N tNlGlvG o c & & & Om O v

p- e) s.v- c) X * & tD % x- x- ot g)m Y' O V O * s. cje) e) m * W V O O m (o ct o; oj O W O B1W X OtO U< C9 C) v' O = Gl m œ X m mm = m O o

c o o œ

v oo o

= o & c

m mc o

oj

O

vC) r< *

O =

mo o c

œ o œ o

m m o œ

c w % o

O o c o

No N o

Oo o o

X m c o

Xo = c

O o m c

D m X c

Xo % c

rw

m o c c

Om N c

Oo c c

m o o c

97 .u. u,

i

t ' ,.

One turn ofthe inputwillrotate the output those additi onalvalues possibl e,quoting

1*

o ntehd i vi sone ion,ofantd seiogbhvti ousl yvaisi chonievplaat ble oreeve achv u etheargeeaaIrIrart ioatre qui ehi dghto wi any hei een di e f achi ita .l They el ivel yr values.Twenty willsim il arly be arrived at withtwo ful lturns perdivision.Forthe same reason 20,30and 60 are om itted from table

a jes

numbers and mostly numbers oflittle use. O ne hundred and twenty-five is though on the listthat will be usefulf0r that dial

t woand30,45and90fr omtabl ethr ee, r equO i r e fdg f ore ran te8 rTs Pi I gI ea n i f d ics acn rc ew e.forthe Colum n Headings

workshop ownerwho constructs the

The column headings D,P,T and H relate to the following

divi ding head i sthatinaddi tion to the above extra di visions he orshe willget23 fudher

D. Division achi eved

divisions (51 being the lowest)using the

lnthe finalsection ofthi s booka numberof dividing head, Iists six set-ups for 24 tabl es are included to assistin arriving ata di vi sions.

p. Num berofholes on the divisi on pl ate to be used T. Numberofful lturns atthe input

60 :1 ratio thatare notavailable wi th the 40 :1 ratio,being by farthe most com mon com merciall y available di viding head.

suitable setup forthe projectin hand, Having purchased a new dividing head a Based on Division Plate sizes

H. The numberofaddi tionalholesto be trak 'ersed afterthe numberoffullLurns,if Three GearDividing Head

table ofpossible divi sions willbe included

any.

Thetabl escanofcourse onlycovera range

With it that will be fine in most ofdivi sion plate numbers and are based circumstances.However,a second hand head may have parted com pany from its manualand shoul d you makeone yourself, typicall y that in chapter 8,arriving at a suitabie setup woul d necessitate recourse

onthose com monlysuppliedwitha smaller head.These are 15, 16'17,18,19,2O,21, 23,27,29, 31,33,37,39,41,43,47 and 49.

to cal culati on.You mayhavenoti cedthatI Possible divisions notincluded

do notreferto 'the suitable setup''butto 'a sui table setup' 'this is because in many Cases there is more than one method of arriving at the sam e result,often many methods. The m anual supplied w ith a com mercialhead understandably quotes

To lim i tthe size ofthe tablesforpublication some values have been om itted. Values below 20 are notincluded astheycaneasil y be arrived at by observation and some y,values of sim pl e mathematics. O bviousl 15and above willuse divi sionplates having thatnum berofholes, say for 17 use the

justonesetupforeachvalue onthebasis pl ate wi th 17 holes,the holes traversed thatthe supplied division plates wilremain with the head.However,we aIIknow that this so often is notthe case and one plate may go m issing.Form any values this will notbe a probl em as the required divi sion willbe achievable using anotherplate.For thi s reason the published tables include every set-up that provides an achi evable di vision,typicall y,table one,for a 40 : 1

98

....n

being equalto the head' s ratio, say 40 for a 40:1 head. Typi cally forothervalues,for 7 divisions the 21 hole plate woul d be the obvious choice. Values above 200,36O excluded, have been om itted on the basis oftheirIim i ted application. O bserving table one, fora 40 :1head, i twillbe seen thatno values are given for either20 or40 di visi ons, This is because

The dividing head in chapter7 uses either

R.Ratio(gearsizelAppliestotable4only. asinglegearoraddi tionallyapairofgears Providingthe same functi onas a worm and

W orm /W orm W heelratios

wormwheel.TheratiowillthoughbeIower,

Tables 1,2 and 3 are for the commonly typicall y gears of60 and 20 having a rati o available ratios of40 :1,60 :1and 90 :1. of3 : 1,or com plex,gears of65 and 45 However,the head for hom e workshop having a rati o of13 :9.

construction in chapter8 can have 9

additionalratios,forexam ple,35 :1 using the 35 t00th changewheel.Publi shing 9 OtherIistsin fullisobviously impracticaland in any case m ostofthe possi bl e divisions With these willalso be achi evable w i th the 40 :1 and 60 :1 rati os Ii sted in tables one and tWO.

No tabl e isprovided forthe single gear as in this case the results should be obvious.Two exam plesbeing,a fodyt00th gearproviding 2,4,5,8,10 and 2O,a thirtyfive t00th gear5 and 7.Again the tabl e is Iim ited to 16 to 200 plus 36O but also Ieaving out divisions thatcan easily be

Forthatreason,tabl efourincludesonl y achieved wi th a single gear.

l ): i 1 ' ! r 1.DI VI SI ONSPOSSI BLEUSI NGA40: 1 DIVIDING HEA D D P T H 21 21 I 19 22 33 I 27 23 23 I 17 24 15 1 10 24 18 1 12

41 41 0 40 42 21 0 20 43 43 0 40 44 33 0 30 45 18 0 16

75 15 76 19 78 39 80 16 80 18

0 8 0 10 0 20 0 8 0 9

I32 33 I35 27 I36 17 I40 21 I40 49

0 0 0 0 0

24 21 24 11 24 33 24 39 25 15

I 14 l 18 I 22 1 26 I9

45 27 46 23 47 47 48 18 49 49

0 24 Q 20 0 40 0 15 0 40

80 20 :2 41 84 21 85 17 86 43

0 10 Q 2Q 0 10 0 8 0 20

I44 18 l45 29 I48 37 I50 15 I52 19

0 5 Q 8 0 10 0 4 05

25 20 26 39 27 27 28 21 28 49

I 12 I 21 I 13 I 9 I 21

50 15 50 20 52 39 54 27 55 33

0 12 0 16 0 30 0 2û 0 14

88 33 90 18 90 27 92 23 94 47

0 15 08 0 12 0 60 0 20

I55 31 I56 39 I60 16 l60 2û I64 4$

0 0 0 û Q

8 10 4 5 IQ

29 29 30 15 30 18 30 21 30 27

I 11 I 5 l 6 1 7 I9

56 21 56 49 58 29 60 15 60 18

0 15 0 35 0 20 0 10 0 12

95 19 0 8 98 49 0 20 I00 15 0 6 I00 20 0 8 104 39 0 15

I65 33 I68 21 I70 17 I72 43 I80 18

0 0 0 0 0

8 5 4 10 4

30 33 30 39 31 31 32 16 32 20 33 33 34 17 35 21 35 49 36 18 36 17 37 37 38 19 39 39

I 11 I 13 I 9 I 4 I5

60 21 60 27 60 33 60 39 62 31

0 14 0 18 0 22 0 26 0 20

I05 21 0 I08 27 0 II0 33 0 II5 23 0 I16 29 0

8 10 12 8 10

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l98 200 360 360

l35 I35 I35 I35 I35

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25 50 25 50 25 25 25 50 25 25 25 25 25 25 25 25 25 30 25 25 20 25

40 55 45 65 35 55 40 70 65 45 40 45 40 3 55 60 55 65 60 * 60 55 60 * 65

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HEADINGS DIV

DIVISION ACHIEVED

TURNS NUMBER OF WORKPIECE IURNS DG GEAR SG DIVIDING SVALL GEAR uc

COUPLua Tc OIVIOING GEA; UARG: UEAR vouNzEo os HEAD spINDLE

25 60 25 25 20 30 30 35

70 55 50 * 75 50 4 60 5

N OTES 1 USES A STEP UP RATIOI GEAR 0N

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