Minimizing Backlashin Spur Gears

Richard L. ThcenConsultant,

IMinneapo,lis. MN

NomenclatureB - Arc on both reference circles that corresponds to Bb 011

both base circles

Bb - Backlash on line of action

C - Distance between centers of reference circles

Cb - Ba ie center distance, {N + 1I)/2P or m(N + n)12

m -Module

N - umber of teeth on gear

n - Number of teeth on pinion

P - Diametral Pilch

p - Circular pitch on both reference circles, rrJP or 1m1

p12 - basic tooth thickness

R - Radin of gear reference circle, NI2P or mNI2

r - Radius of pinion reference circle, nl2P or mn12

Rb - Radius of gear base circle

rb - Radius of pinion base circle

S - Arc on both referenc circles that corresponds to Sb 011both base circles

Sb - Space online of action due to an increase in center distance

llT - Deviation From p/2 on gear reference circle

ill - Deviation from pl2 on pinion reference circle

IITb - Change in tooth thickness on gear base circle

tlrb - Change in tooth thickness on pinion base circle

~ Profile angle of generating rack

1/1, - Pressure angle

24, GEAR TECHNOLOGY

AbstractSimplified equations for backlash and roll

test center distance are derived, Unknownerrors in mea ured tooth, thickness are investi-gated, Ma tel: gear design is outlined, and analternative to the master gear method isde cribed, Defect in the test radius methodare enumerated. Procedures for calculatingbacklash and for preventing significant errorsin measurement are presented.

Introd uettonAn important part of designing for mini-

mum backlash is the prevention of significanterrors in measurement Circular tooth thick-ne s, for instance, . is not measured directly,but. is calculated from an equation that pre-sumes perfectteeth. A a result, a master gear- even if perfect - may not indicate thecenter di tance at which imperfect gear will.mesh tightly with each other. Moreover, themagnitude of the error in measured tooththickness varies with the number of teeth onthe master gear.

Mea ured tooth thickness also varies withdepth of contact and active face width, so therna ter gear should be representatlve of thegear that males with the wcrkgear, But mostmeasurements are made with general purpose,off-the- helf master gears.

Significant errors exit in many. if notmo t. roll test. measurements. Evidence forthl i the discrepancie in measurement thatarise when the arne gear is inspected by dif-ferent people (buyer and seller, inspection

andproductioll departments, differentinspectors using tile same te t equipment,etc ..). The typical respon e lias been to stan-dardize tile test equipment and in pectionprocedure - a maneuver which can reducethe discrepaneies In measurement, but notnecessarily the errors in measurement. Thepurpose of this article is to show how to' dealwith errors in measurement when de ignlngfor minimum backlash.

Basic 'GeometryA very simple way of treating backlash is

to start with no backlash (Fig. I) and thenexamine the effect of an inc rea e in centerdistance (Fig. 2),

In Fig. I the terms basic center di lance(Cb), profile angte (cI» and reference circle- which have been around for at lea t 30year (Refs. l-2) - are II ed in lieu of stan-dard center distance, tandard pressureangle and standard pitch circle. Consequent-ly, in Fig, 2 there is 110' need for the qualify.img adjective operating, which means thatoperating center distance, operating pressureangle and operating pitch circle can be short-ened to center distance (C), pressure angle(¢l) and pitch circle.

From Fig. 1 it is seen that

x + y = Rb tan cI>' + rb tan cI>',

and from Fig. 2 that

which. upon SUbstitution for x + y..becomes

where tan (})- ¢I' = inv ¢I and tan <p - III = inv,~, 0 that

Sb ""2(Rb + rb)(inv <p - inv 'cI»,. (])

Tills equation was first. derived in a some-what different way by Candee (Ref. 3), whoremarked that it is "the shortest and mostdirecteqaation," and "does 110\ require the

Fig. 1 - Reference circles and looth profiles in contact,

determination of tooth thickness on new pitchcircles" - an apparent reference to tne well-known derivation by Buckingham (Ref. 4).

From Fig. 2 and from the defininon of theinvolute (the curvegenerated by a point on astringes it is being unwound from a circle), itis seem that SII2 can be wrapped onto eitherbase circle. Con equently, with the gearfixed, tile pinion i free to rotate SII2rb radi-ans. both CW and CCW, or S,jrb total. Like-wise. with the pillion fixed, the gear is free torotate SII2Rb radians, both CW and CCW, orStlRb total.

It should be observed that space Sb couldbe eliminated by increasing the tooth thick-ness on either the gear (by setting IlTb :::::Sb)'ortbepinion (rub = Sb) or some combinationof both ( ATb + Atb"" Sb}. In general. with anallowance for backlash,

(2)

Ri:chalrd L Thoenis a consultant special-i~ingill medium GIld

fine.pitch gearing. He:is the author of severalarticles (lmi papers nil

measurement. invotm«mathematics. statisticalto/er(lI!cing and othergearing subjects,

MAY/JUNE 1994 25

where Sb is obtained from Eq. 1. And ithould be observed that if the teeth were

thinned, then both ATb and Atb would be neg-ative, which would make Bb greater than Sb'

Fig..2 ~ Effect of an increase in center distance.

Since tooth thickness on the base circlevaries with tooth number, it is customary wandwork with tooth thickness on the referencecircle. where for all tooth numbers, the tooththickness is, in most ca es, nearly equal. to thebasic tooth thickness (pI2). Further, it is con-venient to work with the deviations from p/2 •

namely. with AT and At •.mainly because theycan be exchanged for backlash (as will beseen III Eq. 4). These deviations from p/2 are,of course. related to the off et of the generat-ing rack; i.e .• At = 2AC tan ¢l where Ae is therack offset.

The relation of an are on the reference cir-cle to an arc on the ba~e circle is shown inFig. 2, where it is seen that arc S,)2 and 512subtend equal angle . Specifically, for thepinion,

Si/2 SI2. ... Sb-- =--.orSb=--·-rb r r

for the gear,

Sv'2 SI2 Sb-=-orSb=S·--

Rb R' R

where, from Fig. I, the Rb =R cos <1>, . 0 that

Sh ""S 'cos <P. ATb = tli cos <I> and Bb = B cos <1>.Thus. from Eqs .. I and 2 it is seen that

Sb = S cos <I> = (AT +- At ... B) cos <I> =2(Rb + 'b)(inv 4> - inv (I),),

or

Rb + rbB = 2 (mvl/> - il1vcJ» - ( T ...Ilt),cos ,q,

where. fromFigs. I.and2, the

so that

co 4> = Cb co 41"C

(3)

B = 2 ebOnv ¢I- inv (I) - (AT + At), (4)

(5)

Unknown Errors.It is important to remember that .1T and At

(in Eq, 4) are ubject to unknown errors inmeasured tooth thickness. becau e conven-tioaal measuring methods (span, pins, rna tergear) presume perfect teeth. For example,given a perfect 36T-12P-20° gear the mea-sured tooth thickne s can be found from

M = 10.8367 + Lltco 200,]2

where M i the pan measurement aero 4teeth and .10.8367 is the pan dimension for P

;;;;;I. and At = O. Thus. for a span measurementof 10.8367112, from

M = 1.0.8367 10.8367 -- 200--=-1-="2- = 12 + At cos - ,where, from Fig. 1,. the 'b = reos I), so that

Sb = S cos 4>. Similarly, for Alb and Bb in Eq.2, Alb = At co I) and Bb:= B cos <1>. Likewise, the At = O. andfor two uch gears me hed at a

26, GEAR TECHNOLOGY

center distance of C "" Cb"" (36 + 36)/(2 x12)= 3 inches, the backlash dong the line ofaction is zero; i.e, from Eqs. 3, 4, and 5, theB'b "" O. However. for the same span measure-ment on a gear that has been photographicallyreduced to P "" 12,05, from

10.8367 10.8367M ""---:-:-- - --:-:c-:-- + L1t cos 200

,

12 12.05

the Llt "" 0,00399, Cb"" 36H2,OS, and at C = 3,the Bb "" 0.0012, not Bb = 0 as for P "" 12.Similarly, for a gear that has been photo-graphically enlarged to P = 11.95, the L1t =- 0.00402, Cb = 36/11 ..95, and at C"" 3 the Bb=- 0.0009 (and interference).

For the aforesaid 36T-12P-20° gear, thedimension over 1.92 pins is 39.0886 for P "" Iand .L1I = O. Thus, for a pin measurement of39.0886/[2 on a P = 12.05 gear, the L1t =0.00502 (obtained by solving the pinequa-tions for L1t), and for two suchgears meshedat C "" 3, the Bb "" - 0.0008 (an interference,versus a backlash of 0.0012 for span). Simi-larly. for the same pin measurement on a P =11.95 gear, the L1t = - 0.00494 and Bb ""0.0008 (versus an interference of 0.0009 forspan) ..In short, for both P = 12.05 and 11.95.the backlash for pins was opposite in sign tothe backlash for span.

It is of interest to note that the error inbase pitch, relative to P "" 12, was 0.0010inches for both P = n.os and 11.95,. and thatan error of this magnitude is not uncommonin formed gearing (molded plastic, die cast,powder meta], stamped, cold-drawn). Further.when the combination of tolerances (for out-side radius, tip round. bearing clearance, cen-ter distance) is large relative to whole depth,as in the case of fine-pitch formed gearing, itusually is necessary to design for minimumbacklash. 0 as to avoid a contact ratio of lessthan unity (Ref. 5).

And it should be noted that no generaliza-tion can be drawn from these idealized exam-ples, since the error :in measured tooth thick-ness varies with the number of teeth spanned,the pin diameter and the tooth number.

When the two-flank roll test is not practi-cal, the .L1T and L1t (in Eq. 4) should beadjusted to account. fOT the allowable devia-

Fig..3 - A mesh in whlch some 'teeth fail to make contact.

tions in runout, tooth alignment, profile andspacing. An approximate adjustment can bededuced from similar designs, namely, fromthe discrepancy between calculated backlashand measured backlash - which alsoincludes the aforementioned unknown errorin measured tooth thickness. But if similardesigns are not available, then it is necessaryto estimate the individual adjustments (Ref.6). and to assume a value for the unknownerror in measured tooth thickness.

Characteristics of Master 'GearsWhen a master gea.r is used to measure

tooth thickness, the teeth will be thinner thanthe value indicated by the master gear. notthinner or thicker as when span or pins areused. To illustrate. consider two gears -one imperfect and one perfect - in meshwith a master gear at the same center dis-tance. It is clear that the teeth on the imper-fect gear will be thinner than the teeth on theperfect gear (Ref. 7). As in all functionalgaging (gears, splines, screw threads, bores),size is sacrificed for defects in form andposition, Consequently, since the teeth arethinner than that indicated by the mastergear, the backlash win be somewhat greaterthan that indicated by Eq. 4.

Moreover, the magnitude of the error inmeasured tooth thickness will vary with thenumber of teeth on the master gear. This canbe easily confirmed by meshing gears of vari-au tooth number with a ingle gear ofslightly different pitch and measuring boththe tooth-to-tooth composite variation (alsoknown as tooth-to-tooth composite tolerance,

MAYIJUNE 1994 21

28 GEAR TECHNOLOGY

iooth-to-tooth compo he error and originallyas klckaut [Refs, 8-9]) and center distance.

For example, when L92T-127 P-20oand180T- L20P-20° master gears are rolledtogether. the kickoutis only about 0.0002inches, despite the fact that the difference inba e pitch is 0.0014 inches. But when 120P-20° relatively high-grade gears (error in basepitch ~101?l of 0.0014 inches) of variou . toothnumbers are rolled with the 192T-l27P mas-ter gear, the kickout increases as the toothnumber decreases. to about 0.0019 inche fora 12T-120P pinion. This effect, which is well-known (Ref. 10), is the result of partial toothcontact (see Fig. 3), the degree of whichvaries with contaet ratio. Converely, themagnitude of the center distance error -namely, the amount by which the maximumcenter distance exceeds the sum of the refer-ence radii - decrea es as the tooth. numberdecreases, from about 0.006 inches for the180T-120P master gear to about 0.003 inchefor a 12T-120P pinion,

A single master gear, therefore. should notbe used to check any and all tooth numbers.The master gear method. has, in Mark Twain'swords, "a certain degree of merit .. " but Likechastity - it can be carried too far!"

Master Gear DesignThe ideal master gear would have the

same tooth number, same depth of contactand same pitch circle as the gear that matewith the gear to be inspected; i.e, with thework gear. And when the face width of themating gear .is less 'Ihanthe face width of thework gear, the idea] master gear would havethe arne face width a the active face widthof the mating gear.

Becau e depth of contact. depends uponthe outside radius of the mating gear, the tiprOUI1d or chamfer 011 the mating gear, and thecenter distance between mating gear and work:gear. all three of which vary, the master gearhould be designed to match the maximum

depth of contact. Specifically, the out ideradius of the master gear should be the sameas the maximum outside form radius of themating gear (maximum outside radius less theradial effect of minimum tip round or cham-fer}. And the tooth thickness of the mastergear should be such that the maximum roll

te t center distance between rna ter gear andwork gear is the same as the minimum centerdistance between the mating gear and thework: gear.. That is, for B = 0 in Eq. 4, the

(6)

where ..1T is for the rna tell' gear, inv fJ iobtained from Eq. 3 (wherein C i the mini-mum center distance between mating gear andwork gear), and At is for maximum tooththickne s of the work gear.

The tooth thickness of most work gearswill be less than maximum, so in most rolltests the depth of contact between master gearand work gear will be slightly greater than themaximum depth of contact. between matinggear and work gear ..

In most ca es an acceptable approximationto the ideal tooth number can be obtained bysimulating the roll test on a computer. In par-ticular. the idea] rna fer gear is meshed withthe work gear, and then akickout, equal to thekickout tolerance. is induced by alteringthediametral pitch of the work gear while main-taining the center distance by reducing thetooth thickness of the work gear. Next. thetooth number is altered, but only to the pointwhere there is no significant change in kick-om or tooth thickness,

It is of interest to note that the same com-puter simulation can be u eel to establish real-istic tolerances for rna ter gear . The proce-dure is the same. except that, instead of toothnumber. the diametral pitch of the mastergearis altered. The re ultingchange in diametralpitehi then converted into a change in basepitch. which. in tum. i converted into toler-ances for circular pitch and profile ..

And it is worth noting thai the arne corn-puler simulation can be used to determine thetight-mesh center distance for a gear pair withdifferent thermal expan ion . At elevatedtemperatures, for instance. the diametral pitchofa metal gear will be different from that of amating plastic gear, which means that 'theequations for tight-mesh center distance arenot valid.

Alternative to Master GearsFrom the tandpoint of the designer. tile

master gear method leaves milch to be

desired, Aside from the meed for a variety oftooth number and face widths (usually notavailable), there i the aforementionedunknown error in measured tooth thickness,Also, there is the uncertainty in kickoutbetween mating gears - a i evident fromthe fol]owingiUustrat.ion, Consider a 20Pma ter gear, two 19.95P gears, and two20,05P gears. an perfect and with the sametooth number. The kickout between the 20Pand the 19.95P win be about the same asbetween the 20P and the 20.0SP. But therewill be no kickout between the 19.95P gears,nor between the 20.05P gear. And the kick-out between the 19.9'5P and 20.05P wd[approach the slim of their kickouts against the20P. In short. the designer has no w,ay ofknowing how the kickouts win combine.

The unknown error in measured tooththickness and the uncertainly in kickout canbe avoided when the tooth thickness of oneor both members of a gear pair i readilyadjustable. For instance, when one memberi a bobbed pinion and the other a formedgear (molded plastic, die cast, powder metal.stamped), the hobbing machine operator cansimulate the assembly process by drawingparts at random from the geaJ lot and usingthem 10 check the pinions. The pinion speci-fieacion would read: ROLL TEST CENTERmST ANCE WITH ANY MATING GEARX.XXXXIX.XXXX, and K]CKOUT WITHANY MAHNG GEAR .XXXX. Am 0,

because all teeth are omewhat un ymmetri-cal and misaligned, the pinion drawingwould identify the mating Ilank and activeface width.

RoU Test Center DistanceEquations for roll test center distance are

derived from Eqs. 3 and 4. That. is, for B = 0in Eq. 4, the

m· LlT+m,inv 1/1 = inv '*' + IlT + Llt , - ................-

2 c,and from Eq. 3,.

cos tPC=Cb---,

cos 1/1

total composite en-or, and to ales er extent asrollout) must fall withinthe limits of ron te t

center distance, and since Eqs, 7 and 8 arevalid only for perfect teeth, it follows 'that thecorresponding limit of LIT and III are thelimits of perfect teeth. In other words, therollout must fall within the size tolerance.

It is pertinent to note that. the Germanequivalents of rollout and kickout are alsosingle words. namely, Walifehler (roll en-or)and Wdlzsprung (roll jump), respectively.

From Eqs, 7 and 8 it. is eenthat inv q,.must. be converted into cos 1/1. This conversioncan be easlly carried out on programmablepocket calc 1.1letors Lhat sell for a little as ·30(Ref.Ll ), Even so, the test radius method,which was devised to bypa s Eqs, 7 and 8, isstill in wide pread use.

Test Radius MethodOriginally, test radi.i were obtained by

simply adding LlII(2 tanlP) and IlTI(2 tan IP)to the reference radii of the work gear andmaster gear, respectively. But, since (Lit + Lin!(2 tan tP) is just the first term of an infiniteseries (Ref .. J 2) that method was subject to asignificant error.

Nowadays. the test radius of the workgear is defined as the radial distance from thecenter of the work: gear to the reference circleof the master gear, as obtained by etting AT:::::0. (in Eq, 7) and subtracting the referenceradius of the master gear from the center dis-tance (Eq. 8).

It is of course necessary to . et tAT :::::0 inEq, 7. since the test radius of the rna tel' gearis silfobtained in the original way •. i.e., byadding LlTI(2 tan ,qJ) to the reference radius.Asa result, the designer is forced to specifythe master gear tooth number (or range oftooth numbers), since the test radius of thework gear varies with number of teeth on themaster gear. (The raJlge usually narrows to a

(7) ingle number for large value of Ilt.) In addi-tion, the designer is forced to specify a toler-ance on master gear tootb thickne .• ince theerror in work gear tooth thickne s varies withmaster gear tooth thickness.

For example, given an 8T-20P-20° pinionwhose tooth thickness is 0.02830 inches

(8)

Because the total composite variation greater than the basic tooth thickness ('1.0

(also known as total compo: ite tolerance, eliminate undercut), and a 40T master gear.

M"VIJUNE 1i94 29

30 GEAR TECHNOLOGY

Accordingly, in Eq. 7 the .1t = 0.02830, LlT:=

0, II = 8, N = 40. P' = 20, Cb:= (n + N)/2P :=

1.2, and (JJ = 20° .. Then. 'from Sq. 8 the C =1.2353, so the pinion test radius is 1.2353 -N/2P ;;;:0.2353. (For a 30T master gear, thepinion te t radius would be 0.2346, not0.2351.) Next, given that the master gear1001.11 thickness is 0.00080 inches greater dumthe basic loath thickne (.1T = 0.00080). tilemaster gear test radius is NI2P + LlTI(2 tan f/)

= 1.00 l l , Consequently, the ron tester is setto 0.2353 + ].00 11 = 1.2364 inches.

But for thi center distance the piniontooth enlargement wiill not be .1J = 0.02830.Instead. for C = 1.2364 in Eq. 3 and LIT =0.00080 in Eq. 6, dIe & = 0.02845. which isin error by 0.000 15 inches. Because this erroris a biased error (not a random error). "goodgaging practice" dictates that it not be greaterthan about 5% of the tooth thicknes toler-ance. In other words. in this particular case'the test radius method would not be alid ifthe tooth thicknes tolerance were res thanabout 01.0030. inches.

Because the constraints 011 master geartooth thickness and tooth number can berather severe, the test radius method lends 10

raise the cost of manufacture, Another speci-fication that 'lends to raise the co t of manu-facture is rollout-e- a holdover from the dayswhen rollout did not fall within [he limit ofte t radius (Ref. 13). When 1I0t specified,rollout is free to increase as size variationdecreases, and whether the backlash iscau ed by rollout or size variation is usuallyof no con equence,

A rollout. specification is needed only inspecial case, uch a in high-speed gea.ringand in variou type of anti-backlash gearing.A rollout specification is not an effectivecheck for index error, since index error can benot only large for a small rollout (Refs.14-15), but al 0 small for a large rollout (aswhen the rollout opposes the accumulated. pacing error).

Ba.cklash CalculationsAssigning number to the center distance

(C in Eq, 3) and to the sum of tooth thickness-es (LIT + Lit im Eq. 4) is not an. easy matter.The task would be fairly simple if the axes ofrotation were parallel, but housing bores are

not coaxial, bearing clearances are not equal.journal are not coaxial, runouts of bearingraces are neither equal nor in phase. andmountings for gears on motors, encoders. etc ..are tilted. as are the fixed studs on whichgears are mounted. Furthermore, tooth reac-tions cause unequal radial shifts within thebearing clearance .

The center di lance and the increase in thesum of .tooth thi'cknesses, are obtained by pro-jecting both axes of rotation onto the axialand pitch planes, respectively. (The axialplane contains the nominal positions of bothhou ing axes .. The pitch plane Is perpendicu-lar tothe axial plane and parallel to the hou -ing axes.) In the axial plane the distancebetween the slewed axes al the point wherethe gears make contact is the center di lance.In the pitch plane the product of the anglebetween the axes (in radians) and the mailerof the two face widths is the effectiveincrease in the urn of tooth thicknesses.

If manufacturing distributions for all perti-Ilent dimension are available. then the back-Ia h distribution can be obtamed by simulat-ing the assembly process on a computer(Refs. 16-17). Parts are drawn at random,assembled on the computer, ami then thebacklash (D in Eq. 4) is computed for eachand every assembly. This method does notinvolve rnethernadeal statistics, Moreover, jl

can handle any type of frequency di trihution(bimodal, rectangular, skewed, decenrered,sorted. small quantities) ..

If manufacturing di tributicns are notavailable (as when i'l is not cost-effective [0

ascertain and control. both the shape and cen-trality of each distribution). then only themaximum backla h and minimum backla :11.

can be obtained. Specifically, maximumbacklash is that for maximum center distanceand minimum tooth thicknesses. Minimumbacklash is that for minimum center distanceand maximum tooth tbieknessesvplus theeffective increase in the urn of tooth thick-nesses due to mi alignment,

The minimum backla h can be slightlynegative (a potential •.but improbable interfer-ence). since the range of the backlash distrib-ution will be less than the difference betweenmaximum backlash and minimum backla h.

The magnitude of the potential interference isusually dependent upon the manufacturingprocess. In particular, parts made with hardtooling (molded plastic, die cast, powdermetal, stamped. cold-drawn) tend to havedecentered distributions, so the potentialinterference cannot be as great as for partswhose dimensions are readily adjustable dur-ing the course of manufacture.

In most designs the potential interferencecan range from 15 to 25 percent of the differ-ence between the maximum backlash andminimum backlash, depend:ing upon the shapeand centrality of the various distributions ..

Errors in RoU TestlngThe two-flank roll test has always been

plagued by significant errors in measurement.For example. about 30 years ago Michalecand Karsch conducted acorrelation study(Ref. 18), wherein a number of companiesinspected a large assortment of precisiongears for test radius, rollout and kickout, Thediscrepancies between the participants wereindeed s:ignificant. For instance. the averageof all the greatest discrepancies for test radius(each gear had a greatest discrepancy, name-ly. the difference between the highest andlowest reported measurements) was anincredible 0.0016 inches!

If a similar correlation study were to beconducted today. the di crepancies probablywould be similar. since there have been nomarked improvements in inspection practiceand test equipment, Techniques for prevent-ing significant errors in measurement havebeen known for some time (Refs. 19-20), butthey are seldom implemented.

To insure against significant errors illlmeasurement, the designer can. in self-defense. invoke a process specification (as isdone for heat treating, application of dry-filmlubricants, etc.) that spells out the roll testrequisites. namely. mounting method. axisalignment. permissible hy leresis and nonlin-earity of the movement, master gear dimen-sions and tolerances. and how to determinethe checking load, checking speed, load cor-rection and temperature correction. Withoutan effective process specification, the design-er can do little more than cross his/her fingersand hope for the best. •

References:

1. Dean, Paul M. "Center Distance." GearHandbook, Darle W. Dudley. ed .• McGraw-Hill, 1962. pp. 6 - 8.2. Candee, Allan H. Introduction to the Kine-malic Geometry of Gear Teeth, Chilton Co.,196[. pp ..24, 128.

3. Candee, Allan H. "Involute Gear Calcula-tions Simplified." American Machinist, Sept.13. 1945. pp. 122 - 124.4. Buckingham, Earle. Spur Gears, McGraw-Hill, 1928, pp. 55, 69 - 71.5. Thoen, Richard L. "Optimum Tooth Num-bers for Small Pinions," Machine Design. Oct25. 1984, pp, 107 - lO8.6. Dean, Paul M. "The Interrelationship ofTooth Thickness Measurements as Evaluatedby Various Measuring Techniques," GearTechnology, Sept.lOct., 1987, pp. 13 ff.7. Thoen, Richard L. "High-Grade Fine-PitchGearing," Machine Design. Jan. 19. 1961,Figs. 12,. 13.

8. Martin, Louis D. "Functional Checking ofGear Teeth." Machinery. April, 1950, p. 209.9. Martin. L. D. "New Standards for Gears,"Product Engineering, Dec. 22, 1958, p. 47.10. Martin, L. D. "Large Contact Ratios Mini-mize Effects of 'Gear Errors," Machinery, Feb.,1.967,.p. 97.I.1. Thoen, Richard L. "Solving the InvoluteFunction More Accurately," Machine Design,Aug. 25.1988, pp. 1.1.9-120.1.2. Thoen, Richard L. "Backlash in SpurGears," Machine Design, Jan. 23, 1958, p .. 147..B. Michalec, George W. Precision Gearing.John Wiley & SOilS, 1.966. Fig. 5-7(c).14. Thoen. "High-Grade Fine-Pitch Gearing ....Fig ..9.IS. Michalec. Fig. 4-13.1,6. Corlew, Gordon T & Oakland, Fred."Monte Carlo Simulation," Machine Design,May 6, 1976, pp. 91-92.17. Emick, Norbert Lloyd. Quality Controland Reliability, Industrial Press. 1977, pp.211-2.12.1.8. Michalec. pp. 599~OO.19. Thoen. Richard L. "Measurement Errors inGear Roll Testing," Machinery, May, 1960,pp. 145-150.20. Michalec. pp ..559-567.